### 63617 results for qubit oscillator frequency

Contributors: Baur, M., Filipp, S., Bianchetti, R., Fink, J. M., Göppl, M., Steffen, L., Leek, P. J., Blais, A., Wallraff, A.

Date: 2008-12-23

depending linearly on the drive amplitude ε . Therefore, one would expect that the strong drive at the **qubit** transition **frequency** ω d ≈ ω g e should lead to a square-root dependence of the Autler-Townes and Mollow spectral lines on the drive power P d ∝ ε 2 . However, the Autler-Townes spectral lines show a clear power dependent shift, see Fig. fig:fig3, and the splitting of both pairs of lines scales weaker than linearly with ε ....We measure the Autler-Townes and the Mollow spectral lines according to the scheme shown in Fig. fig:fig1(b). First, we tune the qubit to the frequency ω g e / 2 π ≈ 4.811 G H z , where it is strongly detuned from the resonator by Δ / 2 π = 1.63 G H z . We then strongly drive the transition | g → | e with a first microwave tone of amplitude ε applied to the qubit at the fixed frequency ω d = 4.812 G H z . The drive field is described by the Hamiltonian H d = ℏ ε a † e - **i** ω d t + a e **i** ω d t where the drive amplitude ε is given in **units of **a frequency. The qubit spectrum is then probed by sweeping a weak second microwave signal over a wide range of frequencies ω p including ω g e and ω e f . Simultaneously, amplitude T and phase φ of a microwave signal applied to the resonator are measured . We have adjusted the measurement frequency to the qubit state-dependent resonance of the resonator under qubit driving for every value of ε . Figures fig:fig2(a) and (b) show the measurement response T and φ for selected values of ε . For drive amplitudes ε / 2 π > 65 M H z , two peaks emerge in amplitude from the single Lorentzian line at frequency ω e f corresponding to the Autler-Townes doublet, see Fig. fig:fig2(a). The signal corresponding to the sidebands of the Mollow triplet is visible at high drive amplitudes ε / 2 π > 730 M H z in phase, see Fig. fig:fig2(b). Black lines in Fig. fig:fig2 are fits of the data to Lorentzians from which the dressed qubit resonance frequencies are extracted....We measure the Autler-Townes and the Mollow spectral lines according to the scheme shown in Fig. fig:fig1(b). First, we tune the **qubit** to the **frequency** ω g e / 2 π ≈ 4.811 G H z , where it is strongly detuned from the resonator by Δ / 2 π = 1.63 G H z . We then strongly drive the transition | g → | e with a first microwave tone of amplitude ε applied to the **qubit** at the fixed **frequency** ω d = 4.812 G H z . The drive field is described by the Hamiltonian H d = ℏ ε a † e - i ω d t + a e i ω d t where the drive amplitude ε is given in units of a **frequency**. The **qubit** spectrum is then probed by sweeping a weak second microwave signal over a wide range of **frequencies** ω p including ω g e and ω e f . Simultaneously, amplitude T and phase φ of a microwave signal applied to the resonator are measured . We have adjusted the measurement **frequency** to the **qubit** state-dependent resonance of the resonator under **qubit** driving for every value of ε . Figures fig:fig2(a) and (b) show the measurement response T and φ for selected values of ε . For drive amplitudes ε / 2 π > 65 M H z , two peaks emerge in amplitude from the single Lorentzian line at **frequency** ω e f corresponding to the Autler-Townes doublet, see Fig. fig:fig2(a). The signal corresponding to the sidebands of the Mollow triplet is visible at high drive amplitudes ε / 2 π > 730 M H z in phase, see Fig. fig:fig2(b). Black lines in Fig. fig:fig2 are fits of the data to Lorentzians from which the dressed **qubit** resonance **frequencies** are extracted....(a) Extracted splitting **frequencies** of the Mollow triplet sidebands (red dots) and the Autler-Townes doublet (blue dots) as a function of the drive field amplitude. Dashed lines: Rabi **frequencies** obtained with Eq. ( eq:1). Black solid lines: Rabi **frequencies** calculated by numerically diagonalizing the Hamiltonian Eq. ( eq:2) taking into account 5 transmon levels. (b) Zoom in of the region in the orange rectangle in (a). Orange dots: Rabi **frequency** Ω g e vs. drive amplitude ε extracted from time resolved Rabi **oscillation** experiments, lines as in (a). (c) Rabi **oscillation** measurements between states | g and | e with Ω R / 2 π = 50 M H z and 85 M H z ....In the experiments presented here, we use a version of the Cooper pair box , called transmon qubit , as our multilevel quantum system. States of increasing energies are labelled | l with l = g , e , f , h , **i** , … The transition frequency ω g e between the ground | g and first excited state | e is approximated by ℏ ω g e ≈ 8 E C E J m a x | cos 2 π Φ / Φ 0 | - E C , where E C / h = 233 M H z is the charging energy and E J m a x / h = 32.8 G H z is the maximum Josephson energy. The transition frequency ω g e can be controlled by an external magnetic flux Φ applied to the SQUID loop formed by the two Josephson junctions of the qubit. The transition frequency from the first | e to the second excited state | f is given by ω e f = ω g e - α , where α ≈ 2 π E C / h is the qubit anharmonicity . The qubit is strongly coupled to a coplanar waveguide resonator with resonance frequency ω r / 2 π = 6.439 G H z and photon decay rate κ / 2 π ≈ 1.6 MHz. A schematic circuit diagram of the setup is shown in Fig fig:fig1(a)....fig:fig3 Measured Autler-Townes doublet (blue **dots**) and Mollow triplet sideband **frequencies** (red **dots**) vs. drive power P d at a fixed drive **frequency** ω d / 2 π = 4.812 G H z . Black solid lines are transition **frequencies** calculated by numerically diagonalizing the Hamiltonian ( eq:2) taking into account the lowest 5 transmon levels....(a) Extracted splitting **frequencies** of the Mollow triplet sidebands (red **dots**) and the Autler-Townes doublet (blue **dots**) as a function of the drive field amplitude. Dashed lines: **Rabi** **frequencies** obtained with Eq. ( eq:1). Black solid lines: **Rabi** **frequencies** calculated by numerically diagonalizing the Hamiltonian Eq. ( eq:2) taking into account 5 transmon levels. (b) Zoom in of the region in the orange rectangle in (a). Orange **dots**: **Rabi** **frequency** Ω g e vs. drive amplitude ε extracted from time resolved **Rabi** oscillation experiments, lines as in (a). (c) **Rabi** oscillation measurements between states | g and | e with Ω R / 2 π = 50 M H z and 85 M H z ....(a) Simplified circuit diagram of the measurement setup analogous to the one used in Ref. . In the center at the 20 mK stage, the qubit is coupled capacitively through C g to the resonator, represented by a parallel LC oscillator, and the resonator is coupled to the input and output transmission lines over capacitances C i n and C o u t . Three microwave signal generators are used to apply the measurement ν r f and drive and probe tones ν d r i v e / p r o b e to the input port of the resonator. The transmitted measurement signal is then amplified by an ultra-low noise amplifier at 1.5 K, down-converted with an IQ-mixer and a local oscillator (LO) to an intermediate **frequency** at 300K and digitized with an analog-to-digital converter (ADC). (b) Energy-level diagram of a bare three-level system with states | g , | e , | f ordered with increasing energy. Drive and probe transitions are indicated by black and red/blue arrows, respectively. (c) Energy-level diagram of the dipole coupled dressed states with the coherent drive tone. Possible transitions induced by the probe tone between the dressed states and the third qubit level ( ν - , f , ν + , f ) and between the dressed states ( ν - , + , ν + , - ) are indicated with blue and red arrows....We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives....To confirm the direct relationship between the measured dressed state splitting **frequency** and the Rabi **oscillation** **frequency** of the excited state population we have also performed time resolved measurements of the Rabi **frequency** up to 100 M H z , see Fig. fig:fig4(c). The extracted Rabi **frequencies** (orange data points) are in good agreement with the spectroscopically measured Rabi **frequencies** (blue squares) over the range of accessible ε , as shown in Fig. fig:fig4(b)...The frequencies of the Autler-Townes doublet (blue data points) and of the Mollow triplet sidebands (red data points) extracted from the Lorentzian fits in Fig. fig:fig2(a) and (b) are plotted in Fig. fig:fig3. The splitting of the spectral lines in pairs separated by Ω R and 2 Ω R , respectively, is observed for Rabi frequencies up to Ω R / 2 π ≈ 300 M H z corresponding to about 6% of the qubit transition frequency ω g e ....fig:fig3 Measured Autler-Townes doublet (blue dots) and Mollow triplet sideband **frequencies** (red dots) vs. drive power P d at a fixed drive **frequency** ω d / 2 π = 4.812 G H z . Black solid lines are transition **frequencies** calculated by numerically diagonalizing the Hamiltonian ( eq:2) taking into account the lowest 5 transmon levels....The **frequencies** of the Autler-Townes doublet (blue data points) and of the Mollow triplet sidebands (red data points) extracted from the Lorentzian fits in Fig. fig:fig2(a) and (b) are plotted in Fig. fig:fig3. The splitting of the spectral lines in pairs separated by Ω R and 2 Ω R , respectively, is observed for Rabi **frequencies** up to Ω R / 2 π ≈ 300 M H z corresponding to about 6% of the **qubit** transition **frequency** ω g e ....Measurement of Autler-Townes and Mollow transitions in a strongly driven superconducting **qubit**...(a) Simplified circuit diagram of the measurement setup analogous to the one used in Ref. . In the center at the 20 mK stage, the **qubit** is coupled capacitively through C g to the resonator, represented by a parallel LC **oscillator**, and the resonator is coupled to the input and output transmission lines over capacitances C i n and C o u t . Three microwave signal generators are used to apply the measurement ν r f and drive and probe tones ν d r i v e / p r o b e to the input port of the resonator. The transmitted measurement signal is then amplified by an ultra-low noise amplifier at 1.5 K, down-converted with an IQ-mixer and a local **oscillator** (LO) to an intermediate **frequency** at 300K and digitized with an analog-to-digital converter (ADC). (b) Energy-level diagram of a bare three-level system with states | g , | e , | f ordered with increasing energy. Drive and probe transitions are indicated by black and red/blue arrows, respectively. (c) Energy-level diagram of the dipole coupled dressed states with the coherent drive tone. Possible transitions induced by the probe tone between the dressed states and the third **qubit** level ( ν - , f , ν + , f ) and between the dressed states ( ν - , + , ν + , - ) are indicated with blue and red arrows....In the experiments presented here, we use a version of the Cooper pair box , called transmon **qubit** , as our multilevel quantum system. States of increasing energies are labelled | l with l = g , e , f , h , i , … The transition **frequency** ω g e between the ground | g and first excited state | e is approximated by ℏ ω g e ≈ 8 E C E J m a x | cos 2 π Φ / Φ 0 | - E C , where E C / h = 233 M H z is the charging energy and E J m a x / h = 32.8 G H z is the maximum Josephson energy. The transition **frequency** ω g e can be controlled by an external magnetic flux Φ applied to the SQUID loop formed by the two Josephson junctions of the **qubit**. The transition **frequency** from the first | e to the second excited state | f is given by ω e f = ω g e - α , where α ≈ 2 π E C / h is the **qubit** anharmonicity . The **qubit** is strongly coupled to a coplanar waveguide resonator with resonance **frequency** ω r / 2 π = 6.439 G H z and photon decay rate κ / 2 π ≈ 1.6 MHz. A schematic circuit diagram of the setup is shown in Fig fig:fig1(a). ... We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting **qubit**. The ground to first excited state transition of the **qubit** is strongly pumped while the resulting dressed **qubit** spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the **qubit** coupled off-resonantly to a microwave transmission line resonator. The observed **frequencies** of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited **qubit** states and dispersive level shifts due to off-resonant drives.

Files:

Contributors: Lupascu, A., Bertet, P., Driessen, E. F. C., Harmans, C. J. P. M., Mooij, J. E.

Date: 2008-10-03

fig1 (a) **Frequency** of the spectroscopy peaks p 1 (black squares), p 2 (black circles), and p 3 (triangles) versus Φ . The black lines are a fit for the peaks p 1 and p 2 with the expressions for E 01 q b + T L S and E 02 q b + T L S , yielding the following parameters: I p = 331 nA, Δ = 4.512 GHz, ν T L S = 4.706 GHz, and g = 0.104 GHz. The gray line is a plot of E 01 q b + T L S + E 02 q b + T L S / 2 with the above parameters. (b) Spectroscopy for different values of the microwave power P m w at Φ = 3 Φ 0 / 2 . The curves are vertically shifted for clarity....The observation of the two-photon transition brings important additional information about the coupled EQS. We observe (see Fig. fig1a) that the **frequency** of the peak p 3 is the average of the **frequencies** of peaks p 1 and p 2 . This is clearly shown by the gray line in Fig. fig1a, which is a plot of the average of the transition energies E 01 q b + T L S and E 02 q b + T L S , where E 01 q b + T L S and E 02 q b + T L S are given by the best fit to the **frequencies** of p 1 and p 2 . This particular position of the 2-photon peak is consistent with the hypothesis that the EQS is a TLS , but rules out that the EQS is a HO. This can be understood by considering the structure of levels for the coupled **qubit**-TLS and **qubit**-HO cases, as shown in Fig. fig3 a and b respectively. For the latter case the two-photon transition to the third excited state would have a **frequency** significantly lower than the average value of the one-photon transition **frequencies** to the first and second excited states. This conclusion holds for any type coupling between the **qubit** and the HO which is linear in the **oscillator** creation and annihilation operators. Making the distinction between coupled TLSs and HOs is important since HO modes coupled to the **qubit** can appear due to spurious resonances in the electromagnetic circuit used to control and read out the **qubit**....fig2 Measurements of Rabi **oscillations** at Φ = 3 Φ 0 / 2 , for a **qubit**-TLS configuration given by I p = 331 nA, Δ = 4.47 GHz, ν T L S = 4.39 GHz, and g = 0.099 GHz, for transitions p 1 (squares), p 2 (circles), and p 3 (triangles). (a) Rabi **oscillations** for microwave power P m w = 7 dBm. (b) Rabi **frequency** F Rabi vs P m w . The lines are power law fits for the one- (black) and two-photon (gray) transitions. Only values of F Rabi smaller than 40 MHz are considered for the fit....Spectroscopy is performed by repeating, typically 10 6 times, the following steps : the **qubit** is first prepared in the ground state by energy relaxation. Transitions to excited states are then induced with microwaves at power P m w and **frequency** f m w , applied for a time T m w ; for spectroscopy measurements we take T m w > > T 1 , T 2 . As a final step the driving of the resonant circuit used for readout is switched on and the amplitude V a c is measured. The information on the **qubit** state is provided by the average value of V a c , . In Fig. fig1a the position of the observed spectroscopy peaks for low power is shown as black squares and circles. Away from the symmetry point of the **qubit** ( Φ = 3 Φ 0 / 2 ) the spectrum is similar to the usual flux **qubit** spectrum: we observe a single peak at **frequency** f m w ≈ 2 I p / h Φ - 3 Φ 0 / 2 2 + Δ 2 , corresponding to the transition between the ground and excited states of the **qubit**. However, around Φ = 3 Φ 0 / 2 , we observe two peaks (labeled p 1 and p 2 ) with a Φ dependence characteristic of an anticrossing. This reveals the presence of an EQS with a **frequency** close to the **qubit** parameter Δ . At larger microwave power a third peak is observed (labeled p 3 ) between the peaks p 1 and p 2 . In Fig. fig1b we plot the average as a function of the microwave **frequency** at Φ = 3 Φ 0 / 2 , for increasing microwave power....The observation of** the **two-photon transition brings important additional information about

**coupled EQS. We observe (see Fig. fig1a) that**

**the****the**

**frequency**of

**peak p 3 is**

**the****average of**

**the****frequencies**

**the****of**peaks p 1

**and**p 2 . This is clearly shown by

**gray line in Fig. fig1a, which is**

**the****a**plot of

**average of**

**the****transition energies E 01 q b + T L S**

**the****and**E 02 q b + T L S , where E 01 q b + T L S

**and**E 02 q b + T L S are given by

**best fit to**

**the****frequencies**

**the****of**p 1

**and**p 2 . This particular position of

**2-photon peak is consistent with**

**the****hypothesis that**

**the****EQS is**

**the****a**TLS , but rules out that

**EQS is**

**the****a**HO. This can be understood by considering

**structure**

**the****of**levels for

**coupled**

**the****qubit**-TLS

**and**

**qubit**-HO cases, as shown in Fig. fig3

**a**

**and**b respectively. For

**latter case**

**the****two-photon transition to**

**the****third excited state would have**

**the****a**

**frequency**significantly lower than

**average value of**

**the****one-photon transition frequencies to**

**the****first**

**the****and**second excited states. This conclusion holds for any type coupling between

**the**

**qubit**and

**HO which is linear in**

**the****oscillator creation**

**the****and**annihilation operators. Making

**distinction between coupled TLSs**

**the****and**HOs is important since HO

**modes**coupled to

**the**

**qubit**can appear due to spurious resonances in

**electromagnetic circuit used to control**

**the****and**read out

**the**

**qubit**....fig1 (a) Frequency of the spectroscopy peaks p 1 (black squares), p 2 (black circles), and p 3 (triangles) versus Φ . The black lines are a fit for the peaks p 1 and p 2 with the expressions for E 01 q b + T L S and E 02 q b + T L S , yielding the following parameters: I p = 331 nA, Δ = 4.512 GHz, ν T L S = 4.706 GHz, and g = 0.104 GHz. The gray line is a plot of E 01 q b + T L S + E 02 q b + T L S / 2 with the above parameters. (b) Spectroscopy for different values of the microwave power P m w at Φ = 3 Φ 0 / 2 . The curves are vertically shifted for clarity....Spectroscopy is performed by repeating, typically 10 6 times,

**the**following steps :

**the**

**qubit**is first prepared in

**ground state by energy relaxation. Transitions to excited states are then induced with microwaves**

**the****at**power P m w

**and**

**frequency**f m w , applied for

**a**time T m w ; for spectroscopy measurements we take T m w > > T 1 , T 2 . As

**a**final step

**driving of**

**the****resonant circuit used for readout is switched on**

**the****and**

**amplitude V**

**the****a**c is measured. The information on

**the**

**qubit**state is provided by

**average value**

**the****of**V

**a**c , . In Fig. fig1a

**position of**

**the****observed spectroscopy peaks for low power is shown as black squares**

**the****and**circles. Away from

**symmetry point**

**the****of**

**the**

**qubit**( Φ = 3 Φ 0 / 2 )

**the**spectrum is similar to

**usual**

**the****flux**

**qubit**spectrum: we observe

**a**single peak

**at**

**frequency**f m w ≈ 2 I p / h Φ - 3 Φ 0 / 2 2 + Δ 2 , corresponding to

**transition between**

**the****ground**

**the****and**excited states

**of**

**the**

**qubit**. However, around Φ = 3 Φ 0 / 2 , we observe two peaks (labeled p 1

**and**p 2 ) with

**a**Φ dependence characteristic

**of**an anticrossing. This reveals

**presence**

**the****of**an EQS with

**a**

**frequency**close to

**the**

**qubit**parameter Δ . At larger microwave power

**a**third peak is observed (labeled p 3 ) between

**peaks p 1**

**the****and**p 2 . In Fig. fig1b we plot

**average as**

**the****a**function of

**microwave**

**the****frequency**

**at**Φ = 3 Φ 0 / 2 , for increasing microwave power....We now discuss

**origin of**

**the****peak p 3 in**

**the****spectroscopy signal shown in Fig. fig1. Further understanding on this peak is provided by**

**the****analysis**

**the****of**Rabi oscillations, observed

**at**strong microwave driving. These are shown in Fig. fig2a for three different frequencies, corresponding respectively to

**peaks p 1 , p 2 , and p 3 . It is interesting to note that**

**the****measurement of**

**the****Rabi oscillations shows that**

**the****EQS has coherence times comparable to those**

**the****of**

**the**

**qubit**. The microwave amplitude dependence of

**Rabi**

**the****frequency**is shown in Fig. fig2b for

**three different transitions. For low microwave power, we observe**

**the****a**power law behavior with exponent 1.0 for peaks p 1

**and**p 2 , and 1.8 for peak p 3 . This confirms that p 1

**and**p 2 are one-photon transitions. We attribute p 3 to

**a**two-photon transition to

**third excited state of**

**the****coupled system. The value of**

**the****exponent of**

**the****amplitude dependence, 1.8 , is smaller than**

**the****ideal value**

**the****of**2 . This is consistent with numerical simulations of

**driven dynamics. We attribute this difference to**

**the****partial excitation of**

**the****first two excited states of**

**the****coupled system....One- and two-photon spectroscopy of a flux**

**the****qubit**coupled to a microscopic defect...where σ x , y , z T L S are TLS operators and g is the coupling strength. This Hamiltonian is easily diagonalized, yielding the eigenenergies E n q b + T L S ( n = 0 to 3 ) and the transition energies E m n q b + T L S = E n q b + T L S - E m q b + T L S . The continuous lines in In Fig. fig1b are a combined fit of the Φ -dependent transition energies E 01 q b + T L S and E 02 q b + T L S with the

**frequency**of the peaks p 1 and p 2 . This fit yields the parameters I p , Δ , ν T L S , and g . The agreement of the model with the data is very good. We note that the good agreement does not justify the specific model for the interaction in Eq. eq_Hamiltonian_interaction, as discussed in more detail below, but it justifies the model of resonant interaction and moreover it provides the value of the coupling g ....We observed the dynamics of a superconducting flux

**qubit**coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the

**qubit**. The excitation of a two-photon transition to the third excited state of the

**qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the

**qubit**. The transition

**frequency**and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin....fig2 Measurements of Rabi oscillations at Φ = 3 Φ 0 / 2 , for a

**qubit**-TLS configuration given by I p = 331 nA, Δ = 4.47 GHz, ν T L S = 4.39 GHz, and g = 0.099 GHz, for transitions p 1 (squares), p 2 (circles), and p 3 (triangles). (a) Rabi oscillations for microwave power P m w = 7 dBm. (b) Rabi

**frequency**F Rabi vs P m w . The lines are power law fits for the one- (black) and two-photon (gray) transitions. Only values of F Rabi smaller than 40 MHz are considered for the fit....fig3 Energy level structure for the

**qubit**(qb) coupled to (a) a two level system (TLS) or (b) an harmonic

**oscillator**(HO). The dotted lines indicates energy levels for the uncoupled system. Black/gray arrows indicate one-/two-photon transitions starting in the ground state and are labeled by p n , with n the final state of the coupled system....During the experiments we observed important changes of the energy-level structure of the combined

**qubit**-TLS system. Our

**qubit**sample was used in two experiments, A and B. Between these two experiments our cryostat was warmed up to room temperature. In experiment A a few different configurations of the energy-level structure were observed. In Fig. fig4a we present the spectroscopy data at small power (only lines p 1 and p 2 ) for two such configurations. The measured spectroscopy is in both cases well described by the

**qubit**-TLS model (given by Eqs. eq_Hamiltonian_qubit, eq_Hamiltonian_TLS, and eq_Hamiltonian_interaction), but with different

**frequency**and coupling of the TLS: ν T L S = 4.706 GHz and g = 0.104 GHz for the first configuration and ν T L S = 4.493 GHz and g = 0.099 GHz for the second configuration. During the first experiment we observed a few changes between such configurations. The change between two configurations was fast on the time scale of a few tens of minutes, which is the time necessary to acquire the data in order to characterize the spectroscopic structure. Each configuration was in turn stable over times of the order of days. We observed for each of these configurations the two-photon transition and Rabi

**oscillations**on all the three transitions. In experiment B we observed a similar spectrum (see Fig. fig4b), with a configuration given by I p = 350 nA, Δ = 4.565 GHz, ν T L S = 5.039 GHz, and g = 0.036 GHz. The

**frequency**and the coupling of the TLS are significantly different. In contrast to experiment A, the spectrum was stable over all the duration of the experiment (two months). These observations are consistent with other experiments and support the idea that the coupled TLS is of microscopic origin....fig3 Energy level structure for the

**qubit**(qb) coupled to (a) a two level system (TLS) or (b) an harmonic oscillator (HO). The dotted lines indicates energy levels for the uncoupled system. Black/gray arrows indicate one-/two-photon transitions starting in the ground state and are labeled by p n , with n the final state of the coupled system....During

**experiments we observed important changes of**

**the****energy-level structure of**

**the****combined**

**the****qubit**-TLS system

**. Our**

**qubit**sample was used in two experiments, A

**and**B. Between these two experiments our cryostat was warmed up to room temperature. In experiment A

**a**few different configurations of

**energy-level structure were observed. In Fig. fig4a we present**

**the****spectroscopy data**

**the****at**small power (only lines p 1

**and**p 2 ) for two such configurations. The measured spectroscopy is in both cases well described by

**the**

**qubit**-TLS model (given by Eqs. eq_Hamiltonian_qubit, eq_Hamiltonian_TLS, and eq_Hamiltonian_interaction), but with different

**frequency**and coupling of

**TLS: ν T L S = 4.706 GHz**

**the****and**g = 0.104 GHz for

**first configuration**

**the****and**ν T L S = 4.493 GHz

**and**g = 0.099 GHz for

**second configuration. During**

**the****first experiment we observed**

**the****a**few changes between such configurations. The change between two configurations was fast on

**time scale**

**the****of**

**a**few tens

**of**minutes, which is

**time necessary to acquire**

**the****data in order to characterize**

**the****spectroscopic structure. Each configuration was in turn stable over times of**

**the****order**

**the****of**days. We observed for each

**of**these configurations

**two-photon transition**

**the****and**Rabi oscillations on all

**three transitions. In experiment B we observed**

**the****a**similar spectrum (see Fig. fig4b), with

**a**configuration given by I p = 350 nA, Δ = 4.565 GHz, ν T L S = 5.039 GHz, and g = 0.036 GHz. The

**frequency**and

**coupling of**

**the****TLS are significantly different. In contrast to experiment A,**

**the****the**spectrum was stable over all

**duration of**

**the****experiment (two months). These observations are consistent with other experiments and support**

**the****idea that**

**the****coupled TLS is**

**the****of**microscopic origin....We now discuss the origin of the peak p 3 in the spectroscopy signal shown in Fig. fig1. Further understanding on this peak is provided by the analysis of Rabi

**oscillations**, observed at strong microwave driving. These are shown in Fig. fig2a for three different

**frequencies**, corresponding respectively to the peaks p 1 , p 2 , and p 3 . It is interesting to note that the measurement of the Rabi

**oscillations**shows that the EQS has coherence times comparable to those of the

**qubit**. The microwave amplitude dependence of the Rabi

**frequency**is shown in Fig. fig2b for the three different transitions. For low microwave power, we observe a power law behavior with exponent 1.0 for peaks p 1 and p 2 , and 1.8 for peak p 3 . This confirms that p 1 and p 2 are one-photon transitions. We attribute p 3 to a two-photon transition to the third excited state of the coupled system. The value of the exponent of the amplitude dependence, 1.8 , is smaller than the ideal value of 2 . This is consistent with numerical simulations of the driven dynamics. We attribute this difference to the partial excitation of the first two excited states of the coupled system. ... We observed the dynamics of a superconducting flux

**qubit**coupled to an extrinsic quantum system (EQS). The presence of the EQS is revealed by an anticrossing in the spectroscopy of the

**qubit**. The excitation of a two-photon transition to the third excited state of the

**qubit**-EQS system allows us to extract detailed information about the energy level structure and the coupling of the EQS. We deduce that the EQS is a two-level system, with a transverse coupling to the

**qubit**. The transition

**frequency**and the coupling of the EQS changed during experiments, which supports the idea that the EQS is a two-level system of microscopic origin.

Files:

Contributors: Hoffman, Anthony J., Srinivasan, Srikanth J., Gambetta, Jay M., Houck, Andrew A.

Date: 2011-08-12

We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi oscillations are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures....figure1 Energy level diagram for the TCQ showing the hybridized energy levels. The transitions that have a high probability of occurring are indicated by arrows. Considering that the system is primarily in the | 00 or | 10 ˜ states, single-photon transitions leading out of these two states have the maximum transition probabilities and they are indicated by arrows. The red, solid arrows indicate transitions with low coupling strengths and the blue, dashed arrows indicate transitions with high coupling strengths. The levels shown here are for the bare energy levels of the device; there are no effects of coupling to a cavity. In this work, the | 00 and | 10 ˜ states are used as the logical states of the **qubit**....figure2 (a) Observed cavity transmission versus the two control voltages with a fixed spectroscopy tone at 7.5 G H z . Both the dressed **qubit** **frequency** and coupling strength are functions of the control voltages. The contour shows where the **qubit** is resonant with the 7.5 G H z tone and is therefore driven between the ground and excited states. (b) Measured dressed **frequency** response of the **qubit** while moving along the 7.5 G H z contour in Fig. 1. The dressed **qubit** **frequency** remains constant at 7.5 G H z . The amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. The point where the signal disappears corresponds to coupling strengths where the **qubit** cannot be driven by the spectroscopy tone. The dotted and dashed lines indicate g 10 - 00 control values where the measurements were performed for Fig. 3....figure3 Rabi oscillations for three different **qubit**-cavity coupling strengths and a fixed dressed **qubit** frequency of 7.5 G H z . Panels (a), (b), and (c) correspond to the dashed, dot-dash, and dotted lines in Fig. 2, respectively. In (a), a spectroscopy power of -32 dBm is used. To keep the number of oscillations approximately the same for the lower **qubit**-cavity coupling strength in (b), the spectroscopy power is increased to -22 dBm. In panel (c), 27 dBm more power than that in (a) is applied and no oscillations are observed. Given the measurement noise, we put a bound of 1 / 10 of a Rabi oscillation....figure4 (a) Observed Rabi **oscillations** when the **qubit** starts in the g 10 - 00 = 0 state and is simulataneously moved to a large g 10 - 00 state and driven by a 7.5 G H z spectroscopy pulse of varying amplitude. The fast flux pulse is 60 ns in duration and is followed by an identical pulse of the opposite sign so that the total pulse integral is zero; these zero integral pulses help reduce slow transients. (b) Pulsed measurements showing the probability of the **qubit** being in the excited state as a function of delay following a π -pulse. The **qubit** starts in the g 10 - 00 = 0 state and is excited with a π -pulse in the manner described in (a); a pulsing scheme is included as an inset to the figure. The measured T 1 is 1.6 μ s. (c) Hahn echo measurements with the **qubit** starting in the g 10 - 00 = 0 state. Each of the pulses in the Hahn sequence is synchronized with a pair of fast flux pulses. A pulsing scheme is included as an inset to the figure. The measured T 2 time is 1.9 μ s....Using these fast flux bias pulses, we first measure T 1 by applying a π -pulse that is synchronized with the fast flux pulse, and measure the **qubit** excitation probability after a delay. We measure T 2 using a Hahn echo experiment. The **qubit** is returned to the g 10 - 00 state after each pulse in the Hahn echo sequence. The results of these measurements and the pulse schemes are shown in Fig figure4b, c. The measured T 1 and T 2 times are 1.6 and 1.9 μ s , respectively. The times are only slightly shorter than the 1.9 and 2.8 μ s times recorded at high g 10 - 00 without any fast flux pulses....Time domain measurements provide a more quantitative assessment of any residual coupling at the g 10 - 00 = 0 point. The rate of Rabi driving is proportional to the coupling strength g 10 - 00 and the applied drive amplitude as per the equation Ω R a b i = g 10 - 00 n , where n is the number of drive photons . In Fig figure3, we demonstrate Rabi **oscillations** at three different points on the constant **frequency** contour; these three points are marked on Fig. figure2b. Figure figure3a, b show Rabi **oscillations** at high and medium coupling respectively. In the two panels, the **oscillation** rate is kept nearly constant by increasing the applied rf spectroscopy power by 10 dB to compensate for the reduction in **qubit**-cavity coupling. Fig. figure3c shows the measurement at the g 10 - 00 = 0 point, with 27 dB more rf power than at the high coupling point. No excitation is visible. Given the measurement noise, we should easily be able to detect a tenth of a Rabi **oscillation**; that we see no excitation puts a lower bound on the change in the Rabi rate of a factor of 80. Together with the much higher excitation power, we estimate that the coupling is at least 1500 times smaller at the g 10 - 00 = 0 point compared with the high coupling point. If several **qubits** were in a single cavity, this tuning provides protection against single **qubit** gate errors in one **qubit** while a second **qubit** is driven....figure2 (a) Observed cavity transmission versus the two control voltages with a fixed spectroscopy tone at 7.5 G H z . Both the dressed **qubit** frequency and coupling strength are functions of the control voltages. The contour shows where the **qubit** is resonant with the 7.5 G H z tone and is therefore driven between the ground and excited states. (b) Measured dressed frequency response of the **qubit** while moving along the 7.5 G H z contour in Fig. 1. The dressed **qubit** frequency remains constant at 7.5 G H z . The amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. The point where the signal disappears corresponds to coupling strengths where the **qubit** cannot be driven by the spectroscopy tone. The dotted and dashed lines indicate g 10 - 00 control values where the measurements were performed for Fig. 3....Along this contour of constant **qubit** **frequency**, the **qubit**-cavity coupling strength, g 10 - 00 , changes due to the quantum interference of the two transmon-like halves of the TCQ. In Fig. figure2b, we measure the **frequency** response of the **qubit** while moving along the parameterized contour and can clearly see that the dressed **qubit** **frequency** remains 7.500 G H z . Moreover, in this constant power measurement, the amplitude of the response is related to the coupling strength between the **qubit** and the superconducting resonator. When the coupling is small, little response is seen because the **qubit** cannot be driven. The disappearance of a signal corresponds to the situation where the **qubit**-cavity coupling is tuned through zero....Time domain measurements provide a more quantitative assessment of any residual coupling at the g 10 - 00 = 0 point. The rate of Rabi driving is proportional to the coupling strength g 10 - 00 and the applied drive amplitude as per the equation Ω R a b i = g 10 - 00 n , where n is the number of drive photons . In Fig figure3, we demonstrate Rabi oscillations at three different points on the constant **frequency** contour; these three points are marked on Fig. figure2b. Figure figure3a, b show Rabi oscillations at high and medium coupling respectively. In the two panels, the oscillation rate is kept nearly constant by increasing the applied rf spectroscopy power by 10 dB to compensate for the reduction in **qubit**-cavity coupling. Fig. figure3c shows the measurement at the g 10 - 00 = 0 point, with 27 dB more rf power than at the high coupling point. No excitation is visible. Given the measurement noise, we should easily be able to detect a tenth of a Rabi oscillation; that we see no excitation puts a lower bound on the change in the Rabi rate of a factor of 80. Together with the much higher excitation power, we estimate that the coupling is at least 1500 times smaller at the g 10 - 00 = 0 point compared with the high coupling point. If several **qubits** were in a single cavity, this tuning provides protection against single **qubit** gate errors in one **qubit** while a second **qubit** is driven....In this work, we are mainly concerned with changing only the coupling strength of the **qubit** to the cavity while keeping the **qubit** **frequency** fixed. Since the flux controls allow for a wide range of coupling strengths and dressed **qubit** transition frequencies, it is necessary to find the control subspace that corresponds to constant dressed **qubit** **frequency**. This subspace accounts for any dispersive shifts due to changes in **qubit**-cavity coupling. To accomplish this, we use standard dispersive readout techniques of cQED: monitoring the amplitude and phase of cavity transmission while applying a second spectroscopy tone. Here, though, we keep the spectroscopy tone at a constant **frequency** of 7.500 G H z while sweeping the two control fluxes. When the dressed **qubit** **frequency**, which is a function of the two control fluxes, is resonant with the 7.500 G H z spectroscopy tone, a change in the cavity transmission is measured . Over a wide range of control voltages, it is then possible to extract a contour that corresponds to where the dressed **qubit** **frequency** is 7.500 G H z ; such a contour is shown in Fig. figure2a....In this work, we are mainly concerned with changing only the coupling strength of the **qubit** to the cavity while keeping the **qubit** **frequency** fixed. Since the flux controls allow for a wide range of coupling strengths and dressed **qubit** transition **frequencies**, it is necessary to find the control subspace that corresponds to constant dressed **qubit** **frequency**. This subspace accounts for any dispersive shifts due to changes in **qubit**-cavity coupling. To accomplish this, we use standard dispersive readout techniques of cQED: monitoring the amplitude and phase of cavity transmission while applying a second spectroscopy tone. Here, though, we keep the spectroscopy tone at a constant **frequency** of 7.500 G H z while sweeping the two control fluxes. When the dressed **qubit** **frequency**, which is a function of the two control fluxes, is resonant with the 7.500 G H z spectroscopy tone, a change in the cavity transmission is measured . Over a wide range of control voltages, it is then possible to extract a contour that corresponds to where the dressed **qubit** **frequency** is 7.500 G H z ; such a contour is shown in Fig. figure2a....figure3 Rabi **oscillations** for three different **qubit**-cavity coupling strengths and a fixed dressed **qubit** **frequency** of 7.5 G H z . Panels (a), (b), and (c) correspond to the dashed, dot-dash, and dotted lines in Fig. 2, respectively. In (a), a spectroscopy power of -32 dBm is used. To keep the number of **oscillations** approximately the same for the lower **qubit**-cavity coupling strength in (b), the spectroscopy power is increased to -22 dBm. In panel (c), 27 dBm more power than that in (a) is applied and no **oscillations** are observed. Given the measurement noise, we put a bound of 1 / 10 of a Rabi **oscillation**....Coherent Control of a Superconducting **Qubit** with Dynamically Tunable **Qubit**-cavity Coupling...figure4 (a) Observed Rabi oscillations when the **qubit** starts in the g 10 - 00 = 0 state and is simulataneously moved to a large g 10 - 00 state and driven by a 7.5 G H z spectroscopy pulse of varying amplitude. The fast flux pulse is 60 ns in duration and is followed by an identical pulse of the opposite sign so that the total pulse integral is zero; these zero integral pulses help reduce slow transients. (b) Pulsed measurements showing the probability of the **qubit** being in the excited state as a function of delay following a π -pulse. The **qubit** starts in the g 10 - 00 = 0 state and is excited with a π -pulse in the manner described in (a); a pulsing scheme is included as an inset to the figure. The measured T 1 is 1.6 μ s. (c) Hahn echo measurements with the **qubit** starting in the g 10 - 00 = 0 state. Each of the pulses in the Hahn sequence is synchronized with a pair of fast flux pulses. A pulsing scheme is included as an inset to the figure. The measured T 2 time is 1.9 μ s....We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures. ... We demonstrate coherent control and measurement of a superconducting **qubit** coupled to a superconducting coplanar waveguide resonator with a dynamically tunable **qubit**-cavity coupling strength. Rabi **oscillations** are measured for several coupling strengths showing that the **qubit** transition can be turned off by a factor of more than 1500. We show how the **qubit** can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this **qubit** can be incorporated into quantum computing architectures.

Files:

Contributors: Ku, Li-Chung, Yu, Clare C.

Date: 2004-08-31

Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi oscillations that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**....We have used our algorithm to study **the** effect of TLS energy decay via phonon emission on Rabi oscillations. The results are shown in Fig. fig:10ns. The dotted lines show **the** beating that occurs when **the** **qubit** and TLS are in resonance with no TLS decay. The solid lines shows **the** rapid damping of **the** Rabi oscillations and **the** dephasing that occurs when **the** TLS can decay via phonon emission. In Fig. fig:10nsa there is no direct coupling between **the** microwaves and **the** TLS, whereas in Fig. fig:10nsb, **the** microwaves are directly coupled **both** to **the** **qubit** and to **the** TLS. We see that in **the** latter case beating is damped out more quickly. The Rabi decay time τ R a b i can be defined as **the** time for **the** envelope of **the** Rabi oscillations to decay by 1/e of their original amplitude. Fig. fig:10ns shows that τ R a b i is longer than τ p h simply because **the** excited state of **the** TLS is not always occupied and available for decay. Fig. fig:QB4L shows that states | 0 , e and | 1 , e are essentially unoccupied at early times and that **the** TLS energy decay can take place only when **the** excited state of **the** TLS is sufficiently populated. Therefore in Fig. fig:10nsa **the** solid and dotted lines are in phase for **the** first few Rabi cycles....The **qubit**-TLS system starts in its ground state at t = 0 . A microwave π or 3 π pulse (from 0 to 5 ns) puts the **qubit**-TLS system in the **qubit** excited state | 1 that is a superposition of** the **two entangled states | ψ ' 1 ≡ | 0 e + | 1 g / 2 and | ψ ' 2 ≡ | 0 e - | 1 g / 2 . After** the **microwaves are turned off, the occupation probability starts oscillating coherently. Values of g T L S indicated in** the **figure are normalized by ℏ ω 10 . The rest of** the **parameters are** the **same as in Fig. 1. (a) No energy decay of** the **excited TLS, i.e., τ p h = ∞ . Coherent oscillations with various values of g T L S . (b) Oscillations following a π -pulse with τ p h = 40 ns and various values of g T L S . (c) Oscillations following a π -pulse and a 3 π -pulse with τ p h = 40 ns and g T L S = 0.004 . The dip in** the **dot-dash line is one and a half Rabi cycles....where the **qubit** energy levels ( | 0 and | 1 ) are the basis states and the noise is produced by a single TLS. Our calculations are oriented to the experimental conditions and the results are shown in Fig. fig:RTN. In Fig. fig:RTNa-c the characteristic fluctuation rate t T L S -1 = 0.6 GHz. Panel fig:RTNa shows that the **qubit** essentially stays coherent when the level fluctuations are small ( δ ω 10 / ω 10 = 0.001 ). Panel fig:RTNa shows that when the level fluctuations increase to 0.006, the Rabi **oscillations** decay within 100 ns. The Rabi relaxation time also depends on the Rabi **frequency** as panel fig:RTNc shows. The faster the Rabi **oscillations**, the longer they last. This is because the low-**frequency** noise is essentially constant over several rapid Rabi **oscillations** . Alternatively, one can explain it by the noise power spectrum S I f . Since the noise from a single TLS is a random process characterized by a single characteristic time scale t T L S , it has a Lorentzian power spectrum...Solid lines show** the **Rabi oscillation decay due to **qubit** level fluctuations caused by a single fluctuating two level system trapped inside** the **insulating tunnel barrier. The TLS produces random telegraph noise in I o that modulates the **qubit** energy level splitting ω 10 . (a) The level fluctuation δ ω 10 / ω 10 = 0.001 . The characteristic fluctuation rate t T L S -1 = 0.6 GHz. The Rabi frequency f R = 0.1 GHz. The dotted lines show** the **usual Rabi oscillations without any noise source. (b) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.1 GHz. The dotted lines show** the **usual Rabi oscillations without any noise source. (c) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.5 GHz. (d) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.06 GHz, and f R = 0.5 GHz. Note that** the **scales of** the **horizontal axes in (a)-(c) are** the **same. They are different from that in (d)....Solid line represents Rabi oscillations in** the **presence of both TLS decoherence mechanisms: resonant interaction between** the **TLS and the **qubit**, and low frequency **qubit** energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and** the **energy decay time for** the **TLS is τ p h = 10 ns, the same as in Fig. 2b. The size of the **qubit** level fluctuations is** the **same as in Fig. fig:RTNb. The dotted line shows** the **unperturbed Rabi oscillations....Solid line represents Rabi **oscillations** in the presence of both TLS decoherence mechanisms: resonant interaction between the TLS and the **qubit**, and low **frequency** **qubit** energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and the energy decay time for the TLS is τ p h = 10 ns, the same as in Fig. 2b. The size of the **qubit** level fluctuations is the same as in Fig. fig:RTNb. The dotted line shows the unperturbed Rabi **oscillations**....Rabi **oscillations** of a resonantly coupled **qubit**-TLS system with ε T L S = ℏ ω 10 . There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) P 1 is the occupation probability in the **qubit** state | 1 ; (b) P 0 g is the occupation probability in the state | 0 , g ; (c) P 0 e is the occupation probability of the state | 0 , e ; (d) P 1 g is the occupation probability of the state | 1 , g ; and (e) P 1 e is the occupation probability of the state | 1 , e . Notice the beating with **frequency** 2 η . Throughout the paper, ω 10 / 2 π = 10 GHz. Parameters are chosen mainly according to the experiment in Ref. : η / ℏ ω 10 = 0.0005 , g q b / ℏ ω 10 = 0.01 , and g T L S = 0 . The dotted line in panel (a) shows the usual Rabi **oscillations** without resonant interaction, i.e. η = 0 ....Solid lines show the Rabi **oscillation** decay due to **qubit** level fluctuations caused by a single fluctuating two level system trapped inside the insulating tunnel barrier. The TLS produces random telegraph noise in I o that modulates the **qubit** energy level splitting ω 10 . (a) The level fluctuation δ ω 10 / ω 10 = 0.001 . The characteristic fluctuation rate t T L S -1 = 0.6 GHz. The Rabi **frequency** f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (b) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.1 GHz. The dotted lines show the usual Rabi **oscillations** without any noise source. (c) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.6 GHz, and f R = 0.5 GHz. (d) δ ω 10 / ω 10 = 0.006 , t T L S -1 = 0.06 GHz, and f R = 0.5 GHz. Note that the scales of the horizontal axes in (a)-(c) are the same. They are different from that in (d)....Solid line represents Rabi oscillations in **the** presence of **both** TLS decoherence mechanisms: resonant interaction between **the** TLS and **the** **qubit**, and low frequency **qubit** energy level fluctuations caused by a single fluctuating TLS. The TLS couples to microwaves ( g T L S / ℏ ω 10 = 0.008 ) and **the** energy decay time for **the** TLS is τ p h = 10 ns, **the** same as in Fig. 2b. The size of **the** **qubit** level fluctuations is **the** same as in Fig. fig:RTNb. The dotted line shows **the** unperturbed Rabi oscillations....We first consider the case of strong driving with g T L S = 0 and with the TLS in resonance with the **qubit**, i.e. ε T L S = ℏ ω 10 . If there is no coupling between the **qubit** and the TLS, then the four states of the system are the ground state | 0 , g , the highest energy state | 1 , e , and two degenerate states in the middle | 1 , g and | 0 , e . If the **qubit** and the TLS are coupled with coupling strength η , the degeneracy is split by an energy 2 η . Figure fig:QB4L shows the coherent **oscillations** of the resonant **qubit**-TLS system. We define a projection operator P ̂ 1 ≡ | 1 , g 1 , g | + | 1 , e 1 , e | so that P ̂ 1 corresponds to the occupation probability of the **qubit** to be in state | 1 as in the phase-**qubit** experiment. Instead of being sinusoidal like typical Rabi **oscillations** (the dotted curve), the occupation probability P 1 exhibits beating (Fig. 1a) because the two entangled states that are linear combinations of | 1 , g and | 0 , e have a small energy splitting 2 η , and this small splitting is the beat **frequency**. Without any source of decoherence, the resonant beating will not decay. Thus far the beating phenomenon has not yet been experimentally verified. The lack of experiment...where **the** **qubit** energy levels ( | 0 and | 1 ) are **the** basis states and **the** noise is produced by a single TLS. Our calculations are oriented to **the** experimental conditions and **the** results are shown in Fig. fig:RTN. In Fig. fig:RTNa-c **the** characteristic fluctuation rate t T L S -1 = 0.6 GHz. Panel fig:RTNa shows that **the** **qubit** essentially stays coherent when **the** level fluctuations are small ( δ ω 10 / ω 10 = 0.001 ). Panel fig:RTNa shows that when **the** level fluctuations increase to 0.006, **the** Rabi oscillations decay within 100 ns. The Rabi relaxation time also depends on **the** Rabi frequency as panel fig:RTNc shows. The faster **the** Rabi oscillations, **the** longer they last. This is because **the** low-frequency noise is essentially constant over several rapid Rabi oscillations . Alternatively, one can explain it by **the** noise power spectrum S I f . Since **the** noise from a single TLS is a random process characterized by a single characteristic time scale t T L S , it has a Lorentzian power spectrum...The **qubit**-TLS system starts in its ground state at t = 0 . A microwave π or 3 π pulse (from 0 to 5 ns) puts the **qubit**-TLS system in the **qubit** excited state | 1 that is a superposition of the two entangled states | ψ ' 1 ≡ | 0 e + | 1 g / 2 and | ψ ' 2 ≡ | 0 e - | 1 g / 2 . After the microwaves are turned off, the occupation probability starts **oscillating** coherently. Values of g T L S indicated in the figure are normalized by ℏ ω 10 . The rest of the parameters are the same as in Fig. 1. (a) No energy decay of the excited TLS, i.e., τ p h = ∞ . Coherent **oscillations** with various values of g T L S . (b) **Oscillations** following a π -pulse with τ p h = 40 ns and various values of g T L S . (c) **Oscillations** following a π -pulse and a 3 π -pulse with τ p h = 40 ns and g T L S = 0.004 . The dip in the dot-dash line is one and a half Rabi cycles....Experimentally, **the** two TLS decoherence mechanisms (resonant interaction and low-frequency level fluctuations) can **both** be active at **the** same time. We have calculated **the** Rabi oscillations in **the** presence of **both** of these decoherence sources by using **the** **qubit**-TLS Hamiltonian in eq. ( eq:ham) with a fluctuating ω 10 t that is generated in **the** same way and with **the** same amplitude as in Figure fig:RTNb. We show **the** result in Fig. fig:RTN_decay. By comparing Fig. fig:RTN_decay with Fig. 2b, we note that adding level fluctuations reduces **the** Rabi amplitude and renormalizes **the** Rabi frequency. The result in Fig. fig:RTN_decay is closer to what is seen experimentally ....(a)-(b): Rabi **oscillations** in the presence of 1/f noise in the **qubit** energy level splitting. Dotted curves show the Rabi **oscillations** without the influence of noise. Panel (c) shows the two noise power spectra S f ≡ | δ ω 10 f / ω 10 | 2 of the fluctuations in ω 10 that were used to produce the solid curves in panels (a) and (b). Rabi **frequency** f R = 0.1 GHz....Decoherence of a Josephson **qubit** due to coupling to two level systems...(a)-(b): Rabi oscillations in** the **presence of 1/f noise in the **qubit** energy level splitting. Dotted curves show** the **Rabi oscillations without** the **influence of noise. Panel (c) shows** the **two noise power spectra S f ≡ | δ ω 10 f / ω 10 | 2 of** the **fluctuations in ω 10 that were used to produce** the **solid curves in panels (a) and (b). Rabi frequency f R = 0.1 GHz....Experimentally, the two TLS decoherence mechanisms (resonant interaction and low-**frequency** level fluctuations) can both be active at the same time. We have calculated the Rabi **oscillations** in the presence of both of these decoherence sources by using the **qubit**-TLS Hamiltonian in eq. ( eq:ham) with a fluctuating ω 10 t that is generated in the same way and with the same amplitude as in Figure fig:RTNb. We show the result in Fig. fig:RTN_decay. By comparing Fig. fig:RTN_decay with Fig. 2b, we note that adding level fluctuations reduces the Rabi amplitude and renormalizes the Rabi **frequency**. The result in Fig. fig:RTN_decay is closer to what is seen experimentally ....We do not expect Rabi **oscillations** to be sensitive to noise at **frequencies** much greater than the **frequency** of the Rabi **oscillations** because the higher the **frequency** f , the smaller the noise power and because the Rabi **oscillations** will tend to average over the noise. Rabi dynamics are sensitive to the noise at **frequencies** comparable to the Rabi **frequency**. In addition, the characteristic fluctuation rate plays an important role in the rate of relaxation of the Rabi **oscillations**. It has been shown that t T L S -1 can be thermally activated for TLS in a metal-insulator-metal tunnel junction. If the thermally activated behavior applies here, the decoherence time τ R a b i should decrease as temperature increases. In Fig. fig:RTNd, the characteristic fluctuation rate has been lowered to 0.06 GHz (which is much lower than ω 10 / 2 π ≈ 10 GHz). The noise still causes **qubit** decoherence but affects the **qubit** less than in Fig. fig:RTNc. Fig. fig:RTN shows that the noise primarily affects the Rabi amplitude rather than the phase....Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**....Rabi oscillations of a resonantly coupled **qubit**-TLS system with ε T L S = ℏ ω 10 . There is no mechanism for energy decay. Occupation probabilities of various states are plotted as functions of time. (a) P 1 is** the **occupation probability in the **qubit** state | 1 ; (b) P 0 g is** the **occupation probability in** the **state | 0 , g ; (c) P 0 e is** the **occupation probability of** the **state | 0 , e ; (d) P 1 g is** the **occupation probability of** the **state | 1 , g ; and (e) P 1 e is** the **occupation probability of** the **state | 1 , e . Notice** the **beating with frequency 2 η . Throughout** the **paper, ω 10 / 2 π = 10 GHz. Parameters are chosen mainly according to** the **experiment in Ref. : η / ℏ ω 10 = 0.0005 , g q b / ℏ ω 10 = 0.01 , and g T L S = 0 . The dotted line in panel (a) shows** the **usual Rabi oscillations without resonant interaction, i.e. η = 0 . ... Noise and decoherence are major obstacles to the implementation of Josephson junction **qubits** in quantum computing. Recent experiments suggest that two level systems (TLS) in the oxide tunnel barrier are a source of decoherence. We explore two decoherence mechanisms in which these two level systems lead to the decay of Rabi **oscillations** that result when Josephson junction **qubits** are subjected to strong microwave driving. (A) We consider a Josephson **qubit** coupled resonantly to a two level system, i.e., the **qubit** and TLS have equal energy splittings. As a result of this resonant interaction, the occupation probability of the excited state of the **qubit** exhibits beating. Decoherence of the **qubit** results when the two level system decays from its excited state by emitting a phonon. (B) Fluctuations of the two level systems in the oxide barrier produce fluctuations and 1/f noise in the Josephson junction critical current I_o. This in turn leads to fluctuations in the **qubit** energy splitting that degrades the **qubit** coherence. We compare our results with experiments on Josephson junction phase **qubits**.

Files:

Contributors: Allman, M. S., Altomare, F., Whittaker, J. D., Cicak, K., Li, D., Sirois, A., Strong, J., Teufel, J. D., Simmonds, R. W.

Date: 2010-01-06

For our design parameters g r e s i d u a l ∼ 10 k H z , much too weak to account for** the **residual effect seen in** the **data. We believe** the **residual coupling effect is due to weakly coupled, spurious two-level system fluctuators (TLSs) interacting with the **qubit** at this **frequency** . We have used a scan of vacuum Rabi data that confirms these types of weak oscillations throughout** the **entire spectroscopic range, even at frequencies far detuned from** the **resonator. This indicates interactions with weakly coupled TLSs not seen in traditional spectroscopy measurements. Figure T1 compares** the **vacuum Rabi data taken at Φ x / Φ 0 = - 0.421 and** the **exponential and non-exponential T 1 data taken at **qubit** frequencies far detuned from** the **resonator and where no TLS splittings were visible in** the **spectroscopic data....The coupler is first calibrated by sweeping its external flux bias, Φ x , and measuring** the **effect on** the **tunneling probability of the | g state of the **qubit**. By tracking** the **required applied **qubit** flux Φ q , to maintain a constant total **qubit** flux φ q = Φ q + M q c I c / Φ 0 such that the | g state tunneling probability is approximately 10 % , we can determine** the **circulating current in** the **coupler as a function of Φ x . Figure IGFCombined(a) shows** the **measured coupler circulating current as a function of applied coupler bias flux....where ω r 0 = 1 / L r C r . The measured resonator **frequency** is shown in Figure IGFCombined(b)....Measurements of the dependence of I c , ω r , and g c on applied coupler flux, Φ x / Φ 0 . The vertical dashed lines bracket the applied flux ranges for the waterfall data shown in Figure CombinedWFall. (a) The measured circulating coupler current as a function of applied coupler flux along with the theoretical fit giving β c = 0.51 . (b) Measured resonator frequency as a function of applied coupler flux, along with theoretical fit using β c extracted from (a). The fit yields ω r 0 / 2 π = 7.710 G H z . (c) Measured coupling strength as a function of applied coupler flux along with the theoretical fit using parameters extracted from the theory fits in (a) and (b)....Spectroscopic and time-domain data over the range Φ x / Φ 0 = - 0.462 to Φ x / Φ 0 = - 0.366 bounded by the vertical dashed lines in Figure IGFCombined. (a) Waterfall plot of the spectroscopic measurements of the | ± states showing the splitting transition from g c -0.462 / π ≃ 50 M H z through g c -0.421 / π = 0 to g c -0.366 / π ≃ 40 M H z . The inset to the left is a 3D plot of the **qubit** spectroscopy showing the avoided crossing transition through zero for applied coupler flux values close to Φ x = - 0.421 . (b) The corresponding vacuum Rabi measurements demonstrating coherent modulation in the coupling strength g c Φ x ....Measurements of the dependence of I c , ω r , and g c on applied coupler flux, Φ x / Φ 0 . The vertical dashed lines bracket the applied flux ranges for the waterfall data shown in Figure CombinedWFall. (a) The measured circulating coupler current as a function of applied coupler flux along with the theoretical fit giving β c = 0.51 . (b) Measured resonator **frequency** as a function of applied coupler flux, along with theoretical fit using β c extracted from (a). The fit yields ω r 0 / 2 π = 7.710 G H z . (c) Measured coupling strength as a function of applied coupler flux along with the theoretical fit using parameters extracted from the theory fits in (a) and (b)....A higher resolution trace of the occupation probability of the | e 0 state when Φ x / Φ 0 = - 0.421 along with exponential T 1 and non-exponential T 1 measurements taken at a **qubit** **frequencies** largely detuned from the resonator. The non-exponential T 1 trace showed no evidence of a TLS interaction in the corresponding spectroscopy....RFSQUID-Mediated Coherent Tunable Coupling Between a Superconducting Phase **Qubit** and a Lumped Element Resonator...The next step in** the **experiment is to demonstrate** the **effect of** the **coupler on** the **quantum mechanical interactions between the **qubit** and cavity. We first look for a cavity interaction using well-established spectroscopic techniques . By use of figure IGFCombined(a) the coupler is set to** the **desired coupling strength and then **qubit** spectroscopic measurements are performed. When the **qubit** transition **frequency** nears** the **resonant **frequency** of** the **resonator, an avoided crossing occurs, splitting** the **resonant peak into two peaks. When the **qubit** **frequency** exactly matches** the **resonator’s **frequency** ( Δ = 0 ) the size of** the **spectroscopic splitting is minimized to g Φ x / π . This whole cycle is repeated for different flux biases applied to** the **coupler. We observe** the **size of** the **zero-detuning splitting modulate from a maximum of g m a x / π ≈ 100 M H z down to no splitting (Figure CombinedWFall (a)). The spectroscopic measurements are a good indicator that** the **coupler is working, but we do not consider them to be proof of coherent coupling between the **qubit** and resonator, because** the **length of** the **microwave pulse is longer ( ≃ 500 n s ) than** the **lifetime of the **qubit**....We demonstrate coherent tunable coupling between a superconducting phase **qubit** and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase **qubit** and resonator . We verify the modulation of coupling strength from 0 to $100 MHz$ by observing modulation in the size of the splitting in the phase **qubit**'s spectroscopy, as well as coherently by observing modulation in the vacuum Rabi **oscillation** **frequency** when on resonance. The measured spectroscopic splittings and vacuum Rabi **oscillations** agree well with theoretical predictions....We demonstrate coherent tunable coupling between a superconducting phase **qubit** and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase **qubit** and resonator . We verify the modulation of coupling strength from 0 to $100 MHz$ by observing modulation in the size of the splitting in the phase **qubit**'s spectroscopy, as well as coherently by observing modulation in the vacuum Rabi oscillation **frequency** when on resonance. The measured spectroscopic splittings and vacuum Rabi oscillations agree well with theoretical predictions....The next step in the experiment is to demonstrate the effect of the coupler on the quantum mechanical interactions between the **qubit** and cavity. We first look for a cavity interaction using well-established spectroscopic techniques . By use of figure IGFCombined(a) the coupler is set to the desired coupling strength and then **qubit** spectroscopic measurements are performed. When the **qubit** transition **frequency** nears the resonant **frequency** of the resonator, an avoided crossing occurs, splitting the resonant peak into two peaks. When the **qubit** **frequency** exactly matches the resonator’s **frequency** ( Δ = 0 ) the size of the spectroscopic splitting is minimized to g Φ x / π . This whole cycle is repeated for different flux biases applied to the coupler. We observe the size of the zero-detuning splitting modulate from a maximum of g m a x / π ≈ 100 M H z down to no splitting (Figure CombinedWFall (a)). The spectroscopic measurements are a good indicator that the coupler is working, but we do not consider them to be proof of coherent coupling between the **qubit** and resonator, because the length of the microwave pulse is longer ( ≃ 500 n s ) than the lifetime of the **qubit**....The coupler is first calibrated by sweeping its external flux bias, Φ x , and measuring the effect on the tunneling probability of the | g state of the **qubit**. By tracking the required applied **qubit** flux Φ q , to maintain a constant total **qubit** flux φ q = Φ q + M q c I c / Φ 0 such that the | g state tunneling probability is approximately 10 % , we can determine the circulating current in the coupler as a function of Φ x . Figure IGFCombined(a) shows the measured coupler circulating current as a function of applied coupler bias flux....For our design parameters g r e s i d u a l ∼ 10 k H z , much too weak to account for the residual effect seen in the data. We believe the residual coupling effect is due to weakly coupled, spurious two-level system fluctuators (TLSs) interacting with the **qubit** at this **frequency** . We have used a scan of vacuum Rabi data that confirms these types of weak **oscillations** throughout the entire spectroscopic range, even at **frequencies** far detuned from the resonator. This indicates interactions with weakly coupled TLSs not seen in traditional spectroscopy measurements. Figure T1 compares the vacuum Rabi data taken at Φ x / Φ 0 = - 0.421 and the exponential and non-exponential T 1 data taken at **qubit** **frequencies** far detuned from the resonator and where no TLS splittings were visible in the spectroscopic data....Experimentally, we excite the | e 0 state by applying a short τ p ≃ 5 - 10 n s pulse with the **qubit** on resonance with the resonator. The pulse is fast enough that the resonator remains in its ground state during state preparation. We then measure the state of the **qubit** as a function of time. Figures IGFCombined(b,c) and CombinedWFall summarize the spectroscopic and time domain measurements. For g Φ x / π > 10 M H z , the vacuum Rabi data are used to determine the coupling strength by applying a Fast Fourier Transform (FFT) to the measured probability data. For g Φ x / π **oscillation** in the data (Figure T1)....(a) Circuit diagram for the phase **qubit**, coupler and resonator. The **qubit** parameters are I q 0 ≃ 0.6 μ A , C q s ≃ 0.6 p F , L q ≃ 1000 p H , β q ≃ 1.8 , and M q c ≃ 60 p H . The coupler parameters are I c 0 ≃ 0.9 μ A , L c ≃ 200 p H , C j c ≃ 0.3 p F and β c ≃ 0.5 . The resonator parameters are L r ≃ 1000 p H , C r ≃ 0.4 p F , and M c r ≃ 60 p H . (b) Optical micrograph of the circuit....A higher resolution trace of the occupation probability of the | e 0 state when Φ x / Φ 0 = - 0.421 along with exponential T 1 and non-exponential T 1 measurements taken at a **qubit** frequencies largely detuned from the resonator. The non-exponential T 1 trace showed no evidence of a TLS interaction in the corresponding spectroscopy. ... We demonstrate coherent tunable coupling between a superconducting phase **qubit** and a lumped element resonator. The coupling strength is mediated by a flux-biased RF SQUID operated in the non-hysteretic regime. By tuning the applied flux bias to the RF SQUID we change the effective mutual inductance, and thus the coupling energy, between the phase **qubit** and resonator . We verify the modulation of coupling strength from 0 to $100 MHz$ by observing modulation in the size of the splitting in the phase **qubit**'s spectroscopy, as well as coherently by observing modulation in the vacuum Rabi **oscillation** **frequency** when on resonance. The measured spectroscopic splittings and vacuum Rabi **oscillations** agree well with theoretical predictions.

Files:

Contributors: Plourde, B. L. T., Robertson, T. L., Reichardt, P. A., Hime, T., Linzen, S., Wu, C. -E., Clarke, John

Date: 2005-01-27

(a) SQUID switching probability vs. amplitude of bias current pulse near **qubit** 2 transition. The two curves represent the states corresponding to Φ Q 2 = 0.48 Φ 0 (red) and Φ Q 2 = 0.52 Φ 0 (blue); Φ S is held constant. Each curve contains 100 points averaged 8 , 000 times. (b) I s 50 % vs. Φ S . Each period of **oscillation** contains ∼ 5 , 000 flux values, and each switching current is averaged 8 , 000 times. (c) Dependence of I s 50 % on Φ Q 1 for constant Φ S . (d) **Qubit** flux map. fig:flux-map...We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip....(a) Chip layout. Dark gray represents Al traces, light gray AuCu traces. Pads near upper edge of chip provide two independent flux lines; wirebonded Al jumpers couple left and right halves. Pads near lower edge of chip supply current pulses to the readout SQUID and sense any resulting voltage. (b) Photograph of center region of completed device. Segments of flux lines are visible to left and right of SQUID, which surrounds the two **qubits**. fig:layout...(a) SQUID switching probability vs. amplitude of bias current pulse near **qubit** 2 transition. The two curves represent the states corresponding to Φ Q 2 = 0.48 Φ 0 (red) and Φ Q 2 = 0.52 Φ 0 (blue); Φ S is held constant. Each curve contains 100 points averaged 8 , 000 times. (b) I s 50 % vs. Φ S . Each period of oscillation contains ∼ 5 , 000 flux values, and each switching current is averaged 8 , 000 times. (c) Dependence of I s 50 % on Φ Q 1 for constant Φ S . (d) **Qubit** flux map. fig:flux-map...Flux **Qubits** and Readout Device with Two Independent Flux Lines...Spectroscopy of **qubit** 2. Enhancement and suppression of I s 50 % is shown as a function of Φ Q 2 and f m relative to measurements in the absence of microwaves. Dashed lines indicate fit to hyperbolic dispersion for 1- and 2-photon **qubit** excitations. The 2-photon fit is one-half the **frequency** of the 1-photon fit. Inset containing ∼ 23 , 000 points is at higher resolution. fig:spectroscopy...Coherent manipulation of **qubit** state. (a) Rabi oscillations, scaled to measured SQUID fidelity, as a function of width of 10.0 GHz microwave pulses. (b) Rabi **frequency** vs. 10.0 GHz pulse amplitude; line is least squares fit to the data. (c) Ramsey fringes for **qubit** splitting of 9.95 GHz, microwave **frequency** of 10.095 GHz. (d) Ramsey fringe **frequency** vs. microwave **frequency**. Lines with slopes ± 1 are fits to data. fig:rabi...We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi oscillations and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip....Coherent manipulation of **qubit** state. (a) Rabi **oscillations**, scaled to measured SQUID fidelity, as a function of width of 10.0 GHz microwave pulses. (b) Rabi **frequency** vs. 10.0 GHz pulse amplitude; line is least squares fit to the data. (c) Ramsey fringes for **qubit** splitting of 9.95 GHz, microwave **frequency** of 10.095 GHz. (d) Ramsey fringe **frequency** vs. microwave **frequency**. Lines with slopes ± 1 are fits to data. fig:rabi ... We report measurements on two superconducting flux **qubits** coupled to a readout Superconducting QUantum Interference Device (SQUID). Two on-chip flux bias lines allow independent flux control of any two of the three elements, as illustrated by a two-dimensional **qubit** flux map. The application of microwaves yields a **frequency**-flux dispersion curve for 1- and 2-photon driving of the single-**qubit** excited state, and coherent manipulation of the single-**qubit** state results in Rabi **oscillations** and Ramsey fringes. This architecture should be scalable to many **qubits** and SQUIDs on a single chip.

Files:

Contributors: Schmidt, Thomas L., Borkje, Kjetil, Bruder, Christoph, Trauzettel, Bjoern

Date: 2010-02-25

which contain Lorentz and Fano shaped resonances at the characteristic **frequencies** of the system, ω = 0 , Ω , 2 Δ . The complete expression for the noise reads S ω = ∑ X S X ω X where X denotes all combinations of **qubit** and **oscillator** operators contained in the EVM ( chi). As mentioned above, it turns out that all except three of the prefactors S X ω are nonvanishing and are distinguishable combinations of the functions α 1 , 2 ω and β 1 , 2 , 3 ω . Results for S X ω can be found in Appendix app:noise. A plot of the relevant cross-correlations’ prefactors is shown in Fig. NoisePlot. Since the shapes of these functions are rather distinct, the expectation values constituting the EVM can be recovered from the total measurable noise S ω ....In this article, we propose a system which allows the detection of entanglement between an **oscillator** and a **qubit** using an electronic measurement in an atomic point contact (APC). The electronic system is based on a tunneling contact, a readout device which is known to be quantum-limited. We find that the measurement of the current and the symmetrized current noise in this system allows the evaluation of a criterion for entanglement based on the density matrix of the **oscillator**-**qubit** system. This allows for the detection of entanglement in arbitrary pure or mixed states. All elements of the proposed setup have been realized separately in different experiments. Moreover, it has been shown that the current and the noise of an APC can be measured with a high accuracy. Therefore, it should be possible to combine both elements into one functional device as schematically shown in Fig. FigScheme and to measure its current and noise properties....where δ 1 = arg 1 + c e - i Φ / Φ 0 . Hence, this setup provides a way to obtain a tunneling Hamiltonian with tunable δ 1 . This has been important for** the **calculation of** the **noise. The setup can easily be extended to achieve** the **second tunable phase δ 2 which we used in** the **calculation of** the **current. Here, we need a third junction with** an** amplitude t c = t c 0 , which is decoupled from both

**oscillator**and

**qubit**, and two magnetic fluxes Φ 1 , 2 . A schematic is shown in Fig. fig:phasesb....In general, the amplitudes γ j = | γ j | e i δ j ( j = 0 , 1 , 2 ) can be complex. Since the global phase is irrelevant, we set δ 0 = 0 . Finite phases δ 1 , 2 can be realized experimentally by closing the electric circuit using an additional tunnel junction as shown in Fig. FigScheme. Threading the loop with a magnetic flux causes Aharonov-Bohm phases which can be absorbed in the tunneling amplitudes and generally lead to finite phases δ 1 and δ 2 . This is discussed in more detail in Appendix app:phases. The benefits of a controllable δ 1 have been investigated for a system consisting of an APC and an

**oscillator**: while for δ 1 = 0 , the current noise only depends on the

**oscillator**position x 2 , a finite δ 1 leads to terms proportional to p 2 and thus contains information about the

**oscillator**momentum. Similarly, the presence of tunable phases δ 1 , 2 increases the number of measurable

**oscillator**and

**qubit**properties....(Color online) Schematic density plot of the prefactors S X ω X = x σ x x σ y … of the frequency-dependent noise S ω = ∑ X S X ω X as a function of δ 1 and ω ....(Color online) Schematic density plot of the prefactors S X ω X = x σ x x σ y … of the

**frequency**-dependent noise S ω = ∑ X S X ω X as a function of δ 1 and ω ....Detection of

**qubit**-

**oscillator**entanglement in nanoelectromechanical systems...The state of the

**qubit**-

**oscillator**system modulates the tunneling amplitude γ of the APC. If the

**oscillator**acts as one of the electron reservoirs of the APC as shown in Fig. FigScheme, the tunneling gap depends on the

**oscillator**displacement x . For small x one obtains γ ∝ γ 0 + γ 1 x . The same dependence can also be realized for capacitive coupling. The

**qubit**can be realized as a Cooper pair box in which case a depletion of the electron reservoirs of the APC depending on the state of the

**qubit**leads to an additional term γ 2 σ z in the tunneling amplitude. Irrespective of the concrete realization, to lowest order the combined effect of the

**oscillator**and the

**qubit**leads to...(Color online) Possible experimental setup consisting of a qubit and an oscillator coupled to an atomic point contact (APC). Electrons tunnel at the APC ( t a ) and a fixed tunnel junction ( t b ), which are both biased with a voltage V . The area enclosed by the junctions (red dashed line) is threaded with a magnetic flux to create an Aharonov-Bohm phase. The qubit is realized as a Cooper pair box (CPB, yellow). Its state can be tuned using the gate voltage V g and it couples capacitively to both junctions. The oscillation of the nanomechanical resonator (NR, green) modulates the tunneling amplitude t a . As discussed in more detail in Appendix app:phases, this setup can be used to realize the tunneling amplitude ( gamma)....(Color online) Possible experimental setup consisting of a

**qubit**and an

**oscillator**coupled to an atomic point contact (APC). Electrons tunnel at the APC ( t a ) and a fixed tunnel junction ( t b ), which are both biased with a voltage V . The area enclosed by the junctions (red dashed line) is threaded with a magnetic flux to create an Aharonov-Bohm phase. The

**qubit**is realized as a Cooper pair box (CPB, yellow). Its state can be tuned using the gate voltage V g and it couples capacitively to both junctions. The

**oscillation**of the nanomechanical resonator (NR, green) modulates the tunneling amplitude t a . As discussed in more detail in Appendix app:phases, this setup can be used to realize the tunneling amplitude ( gamma)....The state of the

**qubit**-

**oscillator**system modulates

**the**tunneling amplitude γ of

**the**APC. If

**the**

**oscillator**acts as one of

**the**electron reservoirs of

**the**APC as shown in Fig. FigScheme, the tunneling gap depends

**on**

**the**

**oscillator**displacement x . For small x one obtains γ ∝ γ 0 + γ 1 x . The same dependence can also be realized for capacitive coupling. The

**qubit**can be realized as a Cooper pair box in which case a depletion of

**the**electron reservoirs of

**the**APC depending

**on**

**the**state of the

**qubit**leads to

**additional term γ 2 σ z in**

**an****the**tunneling amplitude. Irrespective of

**the**concrete realization, to lowest order

**the**combined effect of

**the**

**oscillator**and the

**qubit**leads to...In general, the amplitudes γ j = | γ j | e i δ j ( j = 0 , 1 , 2 ) can be complex. Since

**the**global phase is irrelevant, we set δ 0 = 0 . Finite phases δ 1 , 2 can be realized experimentally by closing

**the**electric circuit using

**additional tunnel junction as shown in Fig. FigScheme. Threading**

**an****the**loop with a magnetic flux causes Aharonov-Bohm phases which can be absorbed in

**the**tunneling amplitudes and generally lead to finite phases δ 1 and δ 2 . This is discussed in more detail in Appendix app:phases. The benefits of a controllable δ 1 have been investigated for a system consisting of

**APC and**

**an**

**an****oscillator**: while for δ 1 = 0 , the current noise only depends

**on**

**the**

**oscillator**position x 2 , a finite δ 1 leads to terms proportional to p 2 and thus contains information about

**the**

**oscillator**momentum. Similarly, the presence of tunable phases δ 1 , 2 increases

**the**number of measurable

**oscillator**and

**qubit**properties....In this article, we propose a system which allows

**the**detection of entanglement between

**an****oscillator**and a

**qubit**using

**electronic measurement in**

**an****atomic point contact (APC). The electronic system is based**

**an****on**a tunneling contact, a readout device which is known to be quantum-limited. We find that

**the**measurement of

**the**current and

**the**symmetrized current noise in this system allows

**the**evaluation of a criterion for entanglement based

**on**

**the**density matrix of

**the**

**oscillator**-

**qubit**system. This allows for

**the**detection of entanglement in arbitrary pure or mixed states. All elements of

**the**proposed setup have been realized separately in different experiments. Moreover, it has been shown that

**the**current and

**the**noise of

**APC can be measured with a high accuracy. Therefore, it should be possible to combine both elements into one functional device as schematically shown in Fig. FigScheme and to measure its current and noise properties....which contain Lorentz and Fano shaped resonances at**

**an****the**characteristic frequencies of

**the**system, ω = 0 , Ω , 2 Δ . The complete expression for

**the**noise reads S ω = ∑ X S X ω X where X denotes all combinations of

**qubit**and

**oscillator**operators contained in

**the**EVM ( chi). As mentioned above, it turns

**out**that all except three of

**the**prefactors S X ω are nonvanishing and are distinguishable combinations of

**the**functions α 1 , 2 ω and β 1 , 2 , 3 ω . Results for S X ω can be found in Appendix app:noise. A plot of

**the**relevant cross-correlations’ prefactors is shown in Fig. NoisePlot. Since

**the**shapes of these functions are rather distinct, the expectation values constituting

**the**EVM can be recovered from

**the**total measurable noise S ω ....where δ 1 = arg 1 + c e - i Φ / Φ 0 . Hence, this setup provides a way to obtain a tunneling Hamiltonian with tunable δ 1 . This has been important for the calculation of the noise. The setup can easily be extended to achieve the second tunable phase δ 2 which we used in the calculation of the current. Here, we need a third junction with an amplitude t c = t c 0 , which is decoupled from both

**oscillator**and

**qubit**, and two magnetic fluxes Φ 1 , 2 . A schematic is shown in Fig. fig:phasesb....Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting

**qubits**close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a

**qubit**in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an

**oscillator**and a

**qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state. ... Experiments over the past years have demonstrated that it is possible to bring nanomechanical resonators and superconducting

**qubits**close to the quantum regime and to measure their properties with an accuracy close to the Heisenberg uncertainty limit. Therefore, it is just a question of time before we will routinely see true quantum effects in nanomechanical systems. One of the hallmarks of quantum mechanics is the existence of entangled states. We propose a realistic scenario making it possible to detect entanglement of a mechanical resonator and a

**qubit**in a nanoelectromechanical setup. The detection scheme involves only standard current and noise measurements of an atomic point contact coupled to an

**oscillator**and a

**qubit**. This setup could allow for the first observation of entanglement between a continuous and a discrete quantum system in the solid state.

Files:

Contributors: Johansson, G., Tornberg, L., Shumeiko, V. S., Wendin, G.

Date: 2006-02-24

Resonant circuits for read-out: a) A lumped element LC-**oscillator** coupled to a driving source and a radio-**frequency** detector through a transmission line. b) The radio-**frequency** single-electron transistor measuring the charge of a charge **qubit** (SCB). The current through the SET determines the dissipation in the resonant circuit. The dissipation is determined by measuring the amplitude of the reflected signal. c) Setup for measuring the quantum capacitance of the charge **qubit**. The **qubit** capacitance influences the resonance **frequency** of the **oscillator**. The capacitance is measured by determining the phase-shift of the reflected signal....up to a constant phase depending on the length of the transmission line. Here Q is the resonator’s quality factor, which for the circuitry in Fig. ResonantCircuitsFig a) is determined by the characteristic impedance on the transmission line Z 0 through Q = ω 0 L C 2 / C c 2 Z 0 . Since there is no dissipation in the **oscillator** we have | Γ ω | = 1 . Driving the **oscillator** at the bare resonance **frequency** ω d = ω 0 the phase-difference between the ground and excited state of the **qubit** will be...The quantum capacitance of the Cooper-pair box is related to the parametric capacitance of small Josephson junctions which is a dual to the Josephson inductance. The origin of the quantum capacitance of a single-Cooper-pair box (SCB) can be understood as follows. Assume that we put a constant voltage V m on the measurement capacitance of the SCB, i.e. we put a voltage source between the open circles in Fig. ResonantCircuitsFigc. The amount of charge on the measurement capacitance q m g / e V m V g will be a nonlinear function of the voltage V m as well as the gate voltage V g and whether the qubit is in the ground or excited state. We may define an effective (differential) capacitance...At the charge degeneracy point the effective capacitance of the SCB in the ground and excited state differs by 2 C Q m a x . Imbedding the SCB in a resonant circuit as shown in Fig. ResonantCircuitsFig a) and c) we can detect the corresponding change in the oscillators resonance frequency ω 0 g / e = 1 / L C ± C Q m a x = ω 0 1 ∓ C Q m a x / 2 C , where ω 0 = 1 / L C is the bare resonance frequency. The voltage reflection amplitude Γ ω = V o u t ω / V d ω seen from the driving side of the transmission line can for a high quality oscillator be written...Single-contact flux **qubit** inductively coupled to a linear **oscillator**....At the charge degeneracy point the effective capacitance of the SCB in the ground and excited state differs by 2 C Q m a x . Imbedding the SCB in a resonant circuit as shown in Fig. ResonantCircuitsFig a) and c) we can detect the corresponding change in the **oscillators** resonance **frequency** ω 0 g / e = 1 / L C ± C Q m a x = ω 0 1 ∓ C Q m a x / 2 C , where ω 0 = 1 / L C is the bare resonance **frequency**. The voltage reflection amplitude Γ ω = V o u t ω / V d ω seen from the driving side of the transmission line can for a high quality **oscillator** be written...Readout methods and devices for Josephson-junction-based solid-state **qubits**...Circuit diagrams and 2-level energy spectrum of two basic JJ-**qubit** designs: the SCB charge **qubit** with LC-**oscillator** readout (left), and persistent-current flux **qubit** with SQUID **oscillator** readout (right). For the charge **qubit**, the control variable ϵ on the horizontal axis of the energy spectrum (middle) represents the external gate voltage (induced charge), and the splitting is given by the Josephson tunneling energy mixing the charge states. For the flux **qubit**, the variable ϵ represents the external magnetic flux. In both cases, the energy of the **qubit** can be "tuned" and the working point controlled. Away from the origin (asymptotically) the levels represent pure charge states (zero or one Cooper pair on the SCB island) or pure flux states (left or right rotating currents in the SQUID ring)....LagrangianSubsection The circuit for performing read-out through the quantum capacitance is presented in figure fig:circuit. A Josephson charge **qubit** is capacitatively coupled to a harmonic **oscillator**, which is coupled to a transmission line. Through this line, all measurement on the **qubit** is performed. We model the line as a semi-infinite line of LC-circuits in series. The working point of the Josephson junction can be chosen using the bias V g . In writing down the Lagrangian we are free to chose any quantities as our coordinates as long as they give a full description of our circuit. Since we are treating a system including a Josephson junction, the phases Φ i t = ∫ t d t ' V i t ' across the circuit elements are natural coordinates, as discussed by Devoret in ref. ...Double-well potential and energy levels of the flux **qubit** ( f q = π )....Double-well potential and energy levels of the **flux **qubit ( f q = π )....up to a constant phase depending on the length of the transmission line. Here Q is the resonator’s quality factor, which for the circuitry in Fig. ResonantCircuitsFig a) is determined by the characteristic impedance on the transmission line Z 0 through Q = ω 0 L C 2 / C c 2 Z 0 . Since there is no dissipation in the oscillator we have | Γ ω | = 1 . Driving the oscillator at the bare resonance frequency ω d = ω 0 the phase-difference between the ground and excited state of the qubit will be...Resonant circuits for read-out: a) A lumped element LC-oscillator coupled to a driving source and a radio-frequency detector through a transmission line. b) **The radio**-frequency single-electron transistor measuring the charge of a charge qubit (SCB). The current through the SET determines the dissipation in the resonant circuit. The dissipation is determined by measuring the amplitude of the reflected signal. c) Setup for measuring the quantum capacitance of the charge qubit. The qubit capacitance influences the **resonance** frequency of the oscillator. The capacitance is measured by determining the phase-shift of the reflected signal....The quantum capacitance of the Cooper-pair box is related to the parametric capacitance of small Josephson junctions which is a dual to the Josephson inductance. The origin of the quantum capacitance of a single-Cooper-pair box (SCB) can be understood as follows. Assume that we put a constant voltage V m on the measurement capacitance of the SCB, i.e. we put a voltage source between the open circles in Fig. ResonantCircuitsFigc. The amount of charge on the measurement capacitance q m g / e V m V g will be a nonlinear function of the voltage V m as well as the gate voltage V g and whether the **qubit** is in the ground or excited state. We may define an effective (differential) capacitance...Single-contact **flux **qubit inductively coupled to a linear oscillator....We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time....LagrangianSubsection The circuit for performing read-out through the quantum capacitance is presented in figure fig:circuit. A Josephson charge qubit is capacitatively coupled to a harmonic oscillator, which is coupled to a transmission line. Through this line, all measurement on the qubit is performed. We model the line as a semi-infinite line of LC-circuits in series. The working point of the Josephson junction can be chosen using the bias V g . In writing down the Lagrangian we are free to chose any quantities as our coordinates as long as they give a full description of our circuit. Since we are treating a system including a Josephson junction, the phases Φ i t = ∫ t d t ' V i t ' across the circuit elements are natural coordinates, as discussed by Devoret in ref. ...Circuit diagrams and 2-level energy spectrum of two basic JJ-qubit designs: the SCB charge qubit with LC-oscillator readout (left), and persistent-current **flux **qubit with SQUID oscillator readout (right). For the charge qubit, the control variable ϵ on the horizontal axis of the energy spectrum (middle) represents the external gate voltage (induced charge), and the splitting is given by the Josephson tunneling energy mixing the charge states. For the **flux **qubit, the variable ϵ represents the external magnetic flux. In both cases, the energy of the qubit can be "tuned" and the working point controlled. Away from the origin (asymptotically) the levels represent pure charge states (zero or one Cooper pair on the SCB island) or pure **flux **states (left or right rotating currents in the SQUID ring). ... We discuss the current situation concerning measurement and readout of Josephson-junction based **qubits**. In particular we focus attention of dispersive low-dissipation techniques involving reflection of radiation from an **oscillator** circuit coupled to a **qubit**, allowing single-shot determination of the state of the **qubit**. In particular we develop a formalism describing a charge **qubit** read out by measuring its effective (quantum) capacitance. To exemplify, we also give explicit formulas for the readout time.

Files:

Contributors: Majer, J., Chow, J. M., Gambetta, J. M., Koch, Jens, Johnson, B. R., Schreier, J. A., Frunzio, L., Schuster, D. I., Houck, A. A., Wallraff, A.

Date: 2007-09-13

Superconducting circuits are promising candidates for constructing quantum bits (**qubits**) in a quantum computer; single-**qubit** operations are now routine, and several examples of two **qubit** interactions and gates having been demonstrated. These experiments show that two nearby **qubits** can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant **qubits** is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the **qubits** to a quantum bus, which distributes quantum information among the **qubits**. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting **qubits** on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the **qubits** to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the **qubits**. The cavity is also used to perform multiplexed control and measurement of the **qubit** states. This approach can be expanded to more than two **qubits**, and is an attractive architecture for quantum information processing on a chip....Cavity transmission and spectroscopy of single and coupled **qubits**. a The transmission through **the** cavity as a function of applied magnetic field is shown in **the** **frequency** range between 5 GHz and 5.4 GHz. When either of **the** **qubits** is in resonance with **the** cavity, **the** cavity transmission shows an avoided crossing due to **the** vacuum Rabi splitting. The maximal vacuum Rabi splitting for **the** **two** **qubits** is **the** same within **the** measurement uncertainty and is ∼ 105 MHz. Above 5.5 GHz, spectroscopic measurements of **the** **two** **qubit** transitions are displayed. A second microwave signal is used to excite **the** **qubit** and **the** dispersive shift of **the** cavity **frequency** is measured. The dashed lines show **the** resonance **frequencies** of **the** **two** **qubits**, which are a function of **the** applied flux according to ω 1 , 2 = ω 1 , 2 m a x cos π Φ / Φ 0 . The maximum transition **frequency** for **the** first **qubit** is ω 1 m a x / 2 π = 7.8 G H z and for **the** second **qubit** is ω 2 m a x / 2 π = 6.45 G H z . For strong drive powers, additional resonances between higher **qubit** levels are visible. b Spectroscopy of **the** **two**-**qubit** crossing. The **qubit** levels show a clear avoided crossing with a minimal distance of 2 J / 2 π = 26 M H z . At **the** crossing **the** eigenstates of **the** system are symmetric and anti-symmetric superpositions of **the** **two** **qubit** states. The spectroscopic drive is anti-symmetric and therefore unable to drive any transitions to **the** symmetric state, resulting in a dark state. c Predicted spectroscopy at **the** **qubit**-**qubit** crossing using a Markovian master equation that takes into account higher modes of **the** cavity. The parameters for this calculation are obtained from **the** vacuum Rabi splitting and **the** single **qubit** spectroscopy. TransmissionSpec...To realize **the** cavity bus, we place **two** superconducting **qubits** 5 mm apart at opposite ends ** of **a superconducting transmission line resonator (Fig. SchemePicturea, SchemePictureb

**).**The

**qubits**are transmons, a modified version

**of****the**Cooper-pair box. In this type

**of**

**qubit**,

**the**Josephson energy is larger than

**the**charging energy ( E J ≫ E C )

**and**

**the**transition

**frequency**between

**the**ground state

**and**

**the**first excited state is given by ω ≈ 8 E J E C / ℏ . The Josephson junctions are arranged in a split-pair geometry, so that

**the**Josephson energy, E J = E J m a x cos π Φ / Φ 0 depends

**on**

**the**magnetic flux Φ applied through

**the**split-pair loop. Hence,

**the**transition

**frequency**

**of****the**

**qubits**, ω 1 , 2 = ω 1 , 2 m a x cos π Φ / Φ 0 can be tuned in-situ

**with**

**the**applied flux. The size

**of****the**

**two**loops is different

**and**incommensurate, so that control

**of****the**

**two**transition

**frequencies**is attainable

**with**a certain degree

**independence. To probe**

**of****the**state

**of****the**system, homodyne detection

**of****the**transmitted signal is performed

**and**

**both**quadrature voltages are recorded, which allows reconstruction

**of****the**phase

**and**amplitude

**of****the**transmitted signal....Controllable effective coupling

**and**coherent state transfer via off-resonant Stark shift. a Spectroscopy

**of****qubits**versus applied Stark tone power. Taking into account an attenuation

**67 dB before**

**of****the**cavity

**and**

**the**filtering effect

**of****the**cavity, 0.77 mW corresponds to an average

**one photon in**

**of****the**resonator. The

**qubit**transition

**frequencies**(starting at ω 1 / 2 π = 6.469 G H z

**and**ω 2 / 2 π = 6.546 G H z ) are brought into resonance

**with**a Stark pulse applied at 6.675 G H z . An avoided crossing is observed

**with**one

**of****the**

**qubit**transition levels becoming dark as in Figure TransmissionSpec. b Protocol for

**the**coherent state transfer using

**the**Stark shift. The pulse sequence consists

**a Gaussian-shaped π pulse (red)**

**of****on**one

**of****the**

**qubits**at its transition

**frequency**ω 1 , 2 followed by a Stark pulse (brown)

**varying duration Δ t**

**of****and**amplitude A detuned from

**the**

**qubits**,

**and**finally a square measurement pulse (blue) at

**the**cavity

**frequency**. The time between

**the**π pulse

**and**

**the**measurement is kept fixed at 130 ns. c Coherent state transfer between

**the**

**qubits**according to

**the**protocol above. The plot shows

**the**measured homodyne voltage (average

**3,000,000 traces)**

**of****with**

**the**π pulse applied to

**qubit**1 (green dots)

**and**to

**qubit**2 (red dots) as a function

**of****the**Stark pulse length Δ t . For reference,

**the**black dots show

**the**signal without any π pulse applied to either

**qubit**. The overall increase

**of****the**signal is caused by

**the**residual Rabi driving due to

**the**off-resonant Stark tone, which is also reproduced by

**the**theory. Improved designs featuring different coupling strengths for

**the**individual

**qubits**could easily avoid this effect. The thin solid lines show

**the**signal in

**the**absence

**a Stark pulse. Adding**

**of****the**background trace (black dots) to these, we construct

**the**curves consisting

**open circles, which correctly reproduce**

**of****the**upper

**and**lower limits

**of****the**oscillating signals due to coherent state transfer. d The oscillation

**frequency**(red)

**of****the**time domain state transfer measurement (c)

**and**

**the**splitting

**frequency**(blue)

**of****the**continuous wave spectroscopy (a) versus power

**of****the**Stark tone. The agreement shows that

**the**oscillations are indeed due to

**the**coupling between

**the**

**qubits**. StarkSwap...Multiplexed control and read-out of uncoupled

**qubits**. a Predicted cavity transmission for

**the**four uncoupled

**qubit**states. In

**the**dispersive limit ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 ),

**the**

**frequency**is shifted by χ 1 σ 1 z + χ 2 σ 2 z . Operating

**the**

**qubits**at transition

**frequencies**ω 1 / 2 π = 6.617 G H z and ω 2 / 2 π = 6.529 G H z , we find χ 1 / 2 π = - 5.9 M H z and χ 2 / 2 π = - 7.4 M H z . Measurement is achieved by placing a probe at a

**frequency**where

**the**four cavity transmissions are distinguishable. The

**two**-

**qubit**state can then be reconstructed from

**the**homodyne measurement of

**the**cavity. Rabi oscillations of b

**qubit**1 and c

**qubit**2. A drive pulse of increasing duration is applied at

**the**

**qubit**transition

**frequency**and

**the**response of

**the**cavity transmission is measured after

**the**pulse is turned off. Oscillations of quadrature voltages are measured for each of

**the**

**qubits**and mapped onto

**the**polarization σ 1 , 2 z . The solid line shows results from a master equation simulation, which takes into account

**the**full dynamics of

**the**

**two**

**qubits**and

**the**cavity. The absence of beating in both traces is a signature of

**the**suppression of

**the**

**qubit**-

**qubit**coupling at this detuning. d The homodyne response (average of 1,000,000 traces) of

**the**cavity after a π pulse on

**qubit**1 (green),

**qubit**2 (red), and both

**qubits**(blue). The black trace shows

**the**level when no pulses are applied. The contrasts(i.e.

**the**amplitude of

**the**pulse relative to its ideal maximum value) for these pulses are 60% (green), 61% (green) and 65% (blue). The solid line shows

**the**simulated value including

**the**

**qubit**relaxation and

**the**turn-on time of

**the**cavity. The agreement between

**the**theoretical prediction and

**the**data indicates

**the**measured contrast is

**the**maximum observable. From

**the**theoretical calculation one can estimate

**the**selectivity (see text for details) for each π -pulse to be

**87**% (

**qubit**1) and

**94**% (

**qubit**2). We note that this figure of merit is not at all intrinsic and that it could be improved by increasing

**the**detuning between

**the**

**two**

**qubits**for instance, or using shaped excitation pulses. MultiplexedControl...Sample and scheme used to couple two

**qubits**to an on-chip microwave cavity. Circuit a and optical micrograph b of the chip with two transmon

**qubits**coupled by a microwave cavity. The cavity is formed by a coplanar waveguide (light blue) interrupted by two coupling capacitors (purple). The resonant

**frequency**of the cavity is ω r / 2 π = 5.19 G H z and its width is κ / 2 π = 33 M H z , determined be the coupling capacitors. The cavity is operated as a half-wave resonator ( L = λ / 2 = 12.3 m m ) and the electric field in the cavity is indicated by the gray line. The two transmon

**qubits**(optimized Cooper-pair boxes) are located at opposite ends of the cavity where the electric field has an antinode. Each transmon

**qubit**consists of two superconducting islands connected by a pair of Josephson junctions and an extra shunting capacitor (interdigitated finger structure in the green inset). The left

**qubit**(

**qubit**1) has a charging energy of E C 1 / h = 424 M H z and maximum Josephson energy of E J 1 m a x / h = 14.9 G H z . The right

**qubit**(

**qubit**2) has a charging energy of E C 2 / h = 442 M H z and maximum Josephson energy of E J 2 m a x / h = 18.9 G H z . The loop area between the Josephson junctions for the two transmon

**qubits**differs by a factor of approximately 5 / 8 , allowing a differential flux bias. The microwave signals enter the chip from the left, and the response of the cavity is amplified and measured on the right. c Scheme of the dispersive

**qubit**-

**qubit**coupling. When the

**qubits**are detuned from the cavity ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 ) the

**qubits**both dispersively shift the cavity. The excited state in the left

**qubit**↑ ↓ 0 interacts with the excited state in the right

**qubit**↓ ↑ 0 via the exchange of a virtual photon ↓ ↓ 1 in the cavity. SchemePicture...Cavity transmission and spectroscopy of single and coupled

**qubits**. a The transmission through the cavity as a function of applied magnetic field is shown in the

**frequency**range between 5 GHz and 5.4 GHz. When either of the

**qubits**is in resonance with the cavity, the cavity transmission shows an avoided crossing due to the vacuum Rabi splitting. The maximal vacuum Rabi splitting for the two

**qubits**is the same within the measurement uncertainty and is ∼ 105 MHz. Above 5.5 GHz, spectroscopic measurements of the two

**qubit**transitions are displayed. A second microwave signal is used to excite the

**qubit**and the dispersive shift of the cavity

**frequency**is measured. The dashed lines show the resonance

**frequencies**of the two

**qubits**, which are a function of the applied flux according to ω 1 , 2 = ω 1 , 2 m a x cos π Φ / Φ 0 . The maximum transition

**frequency**for the first

**qubit**is ω 1 m a x / 2 π = 7.8 G H z and for the second

**qubit**is ω 2 m a x / 2 π = 6.45 G H z . For strong drive powers, additional resonances between higher

**qubit**levels are visible. b Spectroscopy of the two-

**qubit**crossing. The

**qubit**levels show a clear avoided crossing with a minimal distance of 2 J / 2 π = 26 M H z . At the crossing the eigenstates of the system are symmetric and anti-symmetric superpositions of the two

**qubit**states. The spectroscopic drive is anti-symmetric and therefore unable to drive any transitions to the symmetric state, resulting in a dark state. c Predicted spectroscopy at the

**qubit**-

**qubit**crossing using a Markovian master equation that takes into account higher modes of the cavity. The parameters for this calculation are obtained from the vacuum Rabi splitting and the single

**qubit**spectroscopy. TransmissionSpec...Controllable effective coupling and coherent state transfer via off-resonant Stark shift. a Spectroscopy of

**qubits**versus applied Stark tone power. Taking into account an attenuation of 67 dB before

**the**cavity and

**the**filtering effect of

**the**cavity, 0.77 mW corresponds to an average of one photon in

**the**resonator. The

**qubit**transition

**frequencies**(starting at ω 1 / 2 π = 6.469 G H z and ω 2 / 2 π = 6.546 G H z ) are brought into resonance with a Stark pulse applied at 6.675 G H z . An avoided crossing is observed with one of

**the**

**qubit**transition levels becoming dark as in Figure TransmissionSpec. b Protocol for

**the**coherent state transfer using

**the**Stark shift. The pulse sequence consists of a Gaussian-shaped π pulse (red) on one of

**the**

**qubits**at its transition

**frequency**ω 1 , 2 followed by a Stark pulse (brown) of varying duration Δ t and amplitude A detuned from

**the**

**qubits**, and finally a square measurement pulse (blue) at

**the**cavity

**frequency**. The time between

**the**π pulse and

**the**measurement is kept fixed at 130 ns. c Coherent state transfer between

**the**

**qubits**according to

**the**protocol above. The plot shows

**the**measured homodyne voltage (average of 3,000,000 traces) with

**the**π pulse applied to

**qubit**1 (green dots) and to

**qubit**2 (red dots) as a function of

**the**Stark pulse length Δ t . For reference,

**the**black dots show

**the**signal without any π pulse applied to either

**qubit**. The overall increase of

**the**signal is caused by

**the**residual Rabi driving due to

**the**off-resonant Stark tone, which is also reproduced by

**the**theory. Improved designs featuring different coupling strengths for

**the**individual

**qubits**could easily avoid this effect. The thin solid lines show

**the**signal in

**the**absence of a Stark pulse. Adding

**the**background trace (black dots) to these, we construct

**the**curves consisting of open circles, which correctly reproduce

**the**upper and lower limits of

**the**oscillating signals due to coherent state transfer. d The oscillation

**frequency**(red) of

**the**time domain state transfer measurement (c) and

**the**splitting

**frequency**(blue) of

**the**continuous wave spectroscopy (a) versus power of

**the**Stark tone. The agreement shows that

**the**oscillations are indeed due to

**the**coupling between

**the**

**qubits**. StarkSwap...In addition to acting as a quantum bus, the cavity can also be used for multiplexed read-out and control of the two

**qubits**. Here, “multiplexed" refers to acquisition of information or control of more than one

**qubit**via a single channel. To address the

**qubits**independently, the flux is tuned such that the

**qubit**

**frequencies**are 88 MHz apart ( ω 1 = 6.617 G H z , ω 2 = 6.529 G H z ), making the

**qubit**-

**qubit**coupling negligible. Rabi experiments showing individual control are performed by applying an rf-pulse at the resonant

**frequency**of either

**qubit**, followed by a measurement pulse at the resonator

**frequency**. The response (see Fig. MultiplexedControlb and MultiplexedControlc) is consistent with that of a single

**qubit**

**oscillation**and shows no beating, indicating that the coupling does not affect single-

**qubit**operations and read-out. With similar measurements the relaxation times ( T 1 ) of the two

**qubits**are determined to be 78 ns and 120 ns, and with Ramsey measurements the coherence times ( T 2 ) are found to be 120 ns and 160 ns. The ability to simultaneously read-out the states of both

**qubits**using a single line is shown by measuring the cavity phase shift, proportional to χ 1 σ 1 z + χ 2 σ 2 z (see Eq. Hamiltonian), after applying a π -pulse to one or both of the

**qubits**. Figure MultiplexedControld shows the response of the cavity after a π -pulse has been applied on the first

**qubit**(green points), on the second

**qubit**(red points) or on both

**qubits**(blue points). For comparison the response of the cavity without any pulse applied (black points) is shown. Since the cavity

**frequency**shifts for the two

**qubits**are different ( χ 1 ≠ χ 2 ), so we are able to distinguish the four states ↓ ↓ , ↓ ↑ , ↑ ↓ , ↑ ↑ of the

**qubits**with a single read-out line. One can show that this measurement, with sufficient signal to noise and combined with single-

**qubit**rotations, should in principle allow for a full reconstruction of the density matrix (state tomography), although not demonstrated in the present experiment....Controllable effective coupling and coherent state transfer via off-resonant Stark shift. a Spectroscopy of

**qubits**versus applied Stark tone power. Taking into account an attenuation of 67 dB before the cavity and the filtering effect of the cavity, 0.77 mW corresponds to an average of one photon in the resonator. The

**qubit**transition

**frequencies**(starting at ω 1 / 2 π = 6.469 G H z and ω 2 / 2 π = 6.546 G H z ) are brought into resonance with a Stark pulse applied at 6.675 G H z . An avoided crossing is observed with one of the

**qubit**transition levels becoming dark as in Figure TransmissionSpec. b Protocol for the coherent state transfer using the Stark shift. The pulse sequence consists of a Gaussian-shaped π pulse (red) on one of the

**qubits**at its transition

**frequency**ω 1 , 2 followed by a Stark pulse (brown) of varying duration Δ t and amplitude A detuned from the

**qubits**, and finally a square measurement pulse (blue) at the cavity

**frequency**. The time between the π pulse and the measurement is kept fixed at 130 ns. c Coherent state transfer between the

**qubits**according to the protocol above. The plot shows the measured homodyne voltage (average of 3,000,000 traces) with the π pulse applied to

**qubit**1 (green dots) and to

**qubit**2 (red dots) as a function of the Stark pulse length Δ t . For reference, the black dots show the signal without any π pulse applied to either

**qubit**. The overall increase of the signal is caused by the residual Rabi driving due to the off-resonant Stark tone, which is also reproduced by the theory. Improved designs featuring different coupling strengths for the individual

**qubits**could easily avoid this effect. The thin solid lines show the signal in the absence of a Stark pulse. Adding the background trace (black dots) to these, we construct the curves consisting of open circles, which correctly reproduce the upper and lower limits of the

**oscillating**signals due to coherent state transfer. d The

**oscillation**

**frequency**(red) of the time domain state transfer measurement (c) and the splitting

**frequency**(blue) of the continuous wave spectroscopy (a) versus power of the Stark tone. The agreement shows that the

**oscillations**are indeed due to the coupling between the

**qubits**. StarkSwap...Sample and scheme used to couple

**two**

**qubits**to an on-chip microwave cavity. Circuit a and optical micrograph b of

**the**chip with

**two**transmon

**qubits**coupled by a microwave cavity. The cavity is formed by a coplanar waveguide (light blue) interrupted by

**two**coupling capacitors (purple). The resonant

**frequency**of

**the**cavity is ω r / 2 π = 5.19 G H z and its width is κ / 2 π = 33 M H z , determined be

**the**coupling capacitors. The cavity is operated as a half-wave resonator ( L = λ / 2 = 12.3 m m ) and

**the**electric field in

**the**cavity is indicated by

**the**gray line. The

**two**transmon

**qubits**(optimized Cooper-pair boxes) are located at opposite ends of

**the**cavity where

**the**electric field has an antinode. Each transmon

**qubit**consists of

**two**superconducting islands connected by a pair of Josephson junctions and an extra shunting capacitor (interdigitated finger structure in

**the**green inset). The left

**qubit**(

**qubit**1) has a charging energy of E C 1 / h = 424 M H z and maximum Josephson energy of E J 1 m a x / h = 14.9 G H z . The right

**qubit**(

**qubit**2) has a charging energy of E C 2 / h = 442 M H z and maximum Josephson energy of E J 2 m a x / h = 18.9 G H z . The loop area between

**the**Josephson junctions for

**the**

**two**transmon

**qubits**differs by a factor of approximately 5 / 8 , allowing a differential flux bias. The microwave signals enter

**the**chip from

**the**left, and

**the**response of

**the**cavity is amplified and measured on

**the**right. c Scheme of

**the**dispersive

**qubit**-

**qubit**coupling. When

**the**

**qubits**are detuned from

**the**cavity ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 )

**the**

**qubits**both dispersively shift

**the**cavity. The excited state in

**the**left

**qubit**↑ ↓ 0 interacts with

**the**excited state in

**the**right

**qubit**↓ ↑ 0 via

**the**exchange of a virtual photon ↓ ↓ 1 in

**the**cavity. SchemePicture...Coupling Superconducting

**Qubits**via a Cavity Bus...Multiplexed control and read-out of uncoupled

**qubits**. a Predicted cavity transmission for the four uncoupled

**qubit**states. In the dispersive limit ( Δ 1 , 2 = ω 1 , 2 - ω r ≫ g 1 , 2 ), the

**frequency**is shifted by χ 1 σ 1 z + χ 2 σ 2 z . Operating the

**qubits**at transition

**frequencies**ω 1 / 2 π = 6.617 G H z and ω 2 / 2 π = 6.529 G H z , we find χ 1 / 2 π = - 5.9 M H z and χ 2 / 2 π = - 7.4 M H z . Measurement is achieved by placing a probe at a

**frequency**where the four cavity transmissions are distinguishable. The two-

**qubit**state can then be reconstructed from the homodyne measurement of the cavity. Rabi

**oscillations**of b

**qubit**1 and c

**qubit**2. A drive pulse of increasing duration is applied at the

**qubit**transition

**frequency**and the response of the cavity transmission is measured after the pulse is turned off.

**Oscillations**of quadrature voltages are measured for each of the

**qubits**and mapped onto the polarization σ 1 , 2 z . The solid line shows results from a master equation simulation, which takes into account the full dynamics of the two

**qubits**and the cavity. The absence of beating in both traces is a signature of the suppression of the

**qubit**-

**qubit**coupling at this detuning. d The homodyne response (average of 1,000,000 traces) of the cavity after a π pulse on

**qubit**1 (green),

**qubit**2 (red), and both

**qubits**(blue). The black trace shows the level when no pulses are applied. The contrasts(i.e. the amplitude of the pulse relative to its ideal maximum value) for these pulses are 60% (green), 61% (green) and 65% (blue). The solid line shows the simulated value including the

**qubit**relaxation and the turn-on time of the cavity. The agreement between the theoretical prediction and the data indicates the measured contrast is the maximum observable. From the theoretical calculation one can estimate the selectivity (see text for details) for each π -pulse to be 87% (

**qubit**1) and 94% (

**qubit**2). We note that this figure of merit is not at all intrinsic and that it could be improved by increasing the detuning between the two

**qubits**for instance, or using shaped excitation pulses. MultiplexedControl...We first observe

**the**coherent interaction between

**the**

**two**

**qubits**via

**the**cavity by performing spectroscopy

**their transition**

**of****frequencies**(see Fig. TransmissionSpec

**).**This is done by monitoring

**the**change in cavity transmission when

**the**

**qubits**are probed by a second microwave signal. By applying a magnetic flux

**the**

**qubits**can be tuned through resonance

**with**each other (see Fig. TransmissionSpecb), revealing an avoided crossing. The magnitude

**of****the**splitting agrees well

**with**

**the**theoretical value 2 J = 2 g 1 g 2 / Δ = 2 π ⋅ 26 M H z when one takes into account that g 1 , 2 vary

**with**

**frequency**for a transmon

**qubit**. The splitting is well resolved,

**with**a magnitude J much greater than

**the**

**qubit**line widths, indicating a coherent coupling

**and**that

**the**

**qubits**are in

**the**strong dispersive limit. Note that although

**the**coupling strength J is smaller than

**the**cavity decay rate κ / 2 π ∼ 33 M H z ,

**the**avoided crossing is nearly unaffected by

**the**cavity loss. This is possible in such a large- κ cavity, required for fast measurements, because only virtual photons are exchanged; if real photons were used,

**the**cavity induced relaxation

**of****the**

**qubits**(Purcell effect) would make coherent state transfer unfeasible....We can perform coherent state transfer in the time domain by rapidly turning the effective

**qubit**-

**qubit**coupling on and off. Rather than the slow flux tuning discussed above, we now make use of a strongly detuned rf-drive, which results in an off-resonant Stark shift of the

**qubit**

**frequencies**on the nanosecond time scale. Figure StarkSwapa shows the spectroscopy of the two

**qubits**when this off-resonant Stark drive is applied with increasing power. The

**qubit**

**frequencies**are pushed into resonance and a similar avoided crossing is observed as in Fig. TransmissionSpecb. With the Stark drive’s ability to quickly tune the

**qubits**into resonance, it is possible to observe coherent

**oscillations**between the

**qubits**, using the following protocol (see Fig. StarkSwapb): Initially the

**qubits**are 80 MHz detuned from each other, where their effective coupling is small, and they are allowed to relax to the ground state ↓ ↓ . Next, a π -pulse is applied to one of the

**qubits**to either create the state ↑ ↓ or ↓ ↑ . Then a Stark pulse of power P A C is applied bringing the

**qubits**into resonance for a variable time Δ t . Since ↑ ↓ and ↓ ↑ are not eigenstates of the coupled system,

**oscillations**between these two states occur, as shown in Fig. StarkSwapc. Fig. StarkSwapd shows the

**frequency**of these

**oscillations**for different powers P A C of the Stark pulse, which agrees with the

**frequency**domain measurement of the

**frequency**splitting observed in Fig. StarkSwapa. These data are strong evidence that the

**oscillations**are due to the coupling between the

**qubits**and that the state of the

**qubits**is transferred from one to the other. A quarter period of these

**oscillations**should correspond to a i S W A P , which would be a universal gate. Future experiments will seek to demonstrate the performance and accuracy of this state transfer....In the first measurement we observe strong coupling of each of the

**qubits**separately to the cavity. By varying the flux, each of the two

**qubits**can be tuned into resonance with the cavity (see Fig. TransmissionSpeca). Whenever a

**qubit**and the cavity are degenerate, the transmission is split into two well-resolved peaks in

**frequency**, an effect called vacuum Rabi splitting, demonstrating that each

**qubit**is in the strong coupling limit with the cavity. Each of the peaks corresponds to a superposition of

**qubit**excitation and a cavity photon in which the energy is shared between the two systems. From the

**frequency**difference at the maximal splitting, the coupling parameters g 1 , 2 ≈ 105 M H z can be determined for each

**qubit**. The transition

**frequency**of each of the two

**qubits**(see Fig. TransmissionSpeca) can also be measured far from the cavity

**frequency**as described below....In addition to acting as a quantum bus,

**the**cavity can also be used for multiplexed read-out

**and**control

**of****the**

**two**

**qubits**. Here, “multiplexed" refers to acquisition

**information or control**

**of****more than one**

**of****qubit**via a

**single**channel. To address

**the**

**qubits**independently,

**the**flux is tuned such that

**the**

**qubit**

**frequencies**are 88 MHz apart ( ω 1 = 6.617 G H z , ω 2 = 6.529 G H z ), making

**the**

**qubit**-

**qubit**coupling negligible. Rabi experiments showing individual control are performed by applying an rf-pulse at

**the**resonant

**frequency**

**either**

**of****qubit**, followed by a measurement pulse at

**the**resonator

**frequency**. The response (see Fig. MultiplexedControlb

**and**MultiplexedControlc) is consistent

**with**that

**a**

**of****single**

**qubit**oscillation

**and**shows no beating, indicating that

**the**coupling does not affect

**single**-

**qubit**operations

**and**read-out. With similar measurements

**the**relaxation times ( T 1 )

**of****the**

**two**

**qubits**are determined to be 78 ns

**and**120 ns,

**and**

**with**Ramsey measurements

**the**coherence times ( T 2 ) are found to be 120 ns

**and**160 ns. The ability to simultaneously read-out

**the**states

**of****both**

**qubits**using a

**single**line is shown by measuring

**the**cavity phase shift, proportional to χ 1 σ 1 z + χ 2 σ 2 z (see Eq. Hamiltonian), after applying a π -pulse to one or

**both**

**of****the**

**qubits**. Figure MultiplexedControld shows

**the**response

**of****the**cavity after a π -pulse has been applied

**on**

**the**first

**qubit**(green points),

**on**

**the**second

**qubit**(red points) or

**on**

**both**

**qubits**(blue points

**).**For comparison

**the**response

**of****the**cavity without any pulse applied (black points) is shown. Since

**the**cavity

**frequency**shifts for

**the**

**two**

**qubits**are different ( χ 1 ≠ χ 2 ), so we are able to distinguish

**the**four states ↓ ↓ , ↓ ↑ , ↑ ↓ , ↑ ↑

**of****the**

**qubits**

**with**a

**single**read-out line. One can show that this measurement,

**with**sufficient signal to noise

**and**combined

**with**

**single**-

**qubit**rotations, should in principle allow for a full reconstruction

**of****the**density matrix (state tomography), although not demonstrated in

**the**present experiment....In this regime, no energy is exchanged with the cavity. However, the

**qubits**and cavity are still dispersively coupled, resulting in a

**qubit**-state-dependent shift ± χ 1 , 2 of the cavity

**frequency**(see Fig. MultiplexedControla) or equivalently an AC Stark shift of the

**qubit**

**frequencies**. The

**frequency**shift χ 1 , 2 can be calculated from the detuning Δ 1 , 2 and the measured coupling strength g 1 , 2 . The last term describes the interaction between the

**qubits**, which is a transverse exchange interaction of strength J = g 1 g 2 1 / Δ 1 + 1 / Δ 2 / 2 (See Fig. SchemePicturec). The

**qubit**-

**qubit**interaction is a result of virtual exchange of photons with the cavity. When the

**qubits**are degenerate with each other, an excitation in one

**qubit**can be transferred to the other

**qubit**by virtually becoming a photon in the cavity (see Fig. MultiplexedControlb). However, when the

**qubits**are non-degenerate | ω 1 - ω 2 | ≫ J this process does not conserve energy, and therefore the interaction is effectively turned off. Thus, instead of modifying the actual coupling constant, we control the effective coupling strength by tuning the

**qubit**transition

**frequencies**. This is possible since the

**qubit**-

**qubit**coupling is transverse, which also distinguishes our experiment from the situation in liquid-state NMR quantum computation, where an effective switching-off can only be achieved by repeatedly applying decoupling pulses....In

**the**first measurement we observe strong coupling

**each**

**of**

**of****the**

**qubits**separately to

**the**cavity. By varying

**the**flux, each

**of****the**

**two**

**qubits**can be tuned into resonance

**with**

**the**cavity (see Fig. TransmissionSpeca

**).**Whenever a

**qubit**

**and**

**the**cavity are degenerate,

**the**transmission is split into

**two**well-resolved peaks in

**frequency**, an effect called vacuum Rabi splitting, demonstrating that each

**qubit**is in

**the**strong coupling limit

**with**

**the**cavity. Each

**of****the**peaks corresponds to a superposition

**of**

**qubit**excitation

**and**a cavity photon in which

**the**energy is shared between

**the**

**two**systems. From

**the**

**frequency**difference at

**the**maximal splitting,

**the**coupling parameters g 1 , 2 ≈ 105 M H z can be determined for each

**qubit**. The transition

**frequency**

**each**

**of**

**of****the**

**two**

**qubits**(see Fig. TransmissionSpeca) can also be measured far from

**the**cavity

**frequency**as described below. ... Superconducting circuits are promising candidates for constructing quantum bits (

**qubits**) in a quantum computer; single-

**qubit**operations are now routine, and several examples of two

**qubit**interactions and gates having been demonstrated. These experiments show that two nearby

**qubits**can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant

**qubits**is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the

**qubits**to a quantum bus, which distributes quantum information among the

**qubits**. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting

**qubits**on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the

**qubits**to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the

**qubits**. The cavity is also used to perform multiplexed control and measurement of the

**qubit**states. This approach can be expanded to more than two

**qubits**, and is an attractive architecture for quantum information processing on a chip.

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Contributors: Zhang, Jing, Liu, Yu-xi, Zhang, Wei-Min, Wu, Lian-Ao, Wu, Re-Bing, Tarn, Tzyh-Jong

Date: 2011-01-17

(color online) Decoherence suppression by the auxiliary chaotic setup. (a) the evolution of the coherence C x y = S ̂ x 2 + S ̂ y 2 of the state of the qubit, where the red asterisk curve and the black triangle curve represent the ideal trajectory without any decoherence and the trajectory under natural decoherence and without corrections; and the green curve with plus signs and the blue solid curve denote the trajectories with I 0 / ω q = 5 and 30 . With these parameters, the dynamics of the Duffing oscillator exhibits periodic and chaotic behaviors. τ = 2 π / ω q is a normalized time scale. (b) and (c) are the energy spectra of δ q t with I 0 / ω q = 5 (the periodic case) and 30 (the chaotic case). The energy spectrum S δ q ω is in unit of decibel (dB). (d) the normalized decoherence rates Γ / ω q versus the normalized driving strength I 0 / ω q ....(color online) Decoherence suppression by the auxiliary chaotic setup. (a) the evolution of the coherence C x y = S ̂ x 2 + S ̂ y 2 of the state of the **qubit**, where the red asterisk curve and the black triangle curve represent the ideal trajectory without any decoherence and the trajectory under natural decoherence and without corrections; and the green curve with plus signs and the blue solid curve denote the trajectories with I 0 / ω q = 5 and 30 . With these parameters, the dynamics of the Duffing **oscillator** exhibits periodic and chaotic behaviors. τ = 2 π / ω q is a normalized time scale. (b) and (c) are the energy spectra of δ q t with I 0 / ω q = 5 (the periodic case) and 30 (the chaotic case). The energy spectrum S δ q ω is in unit of decibel (dB). (d) the normalized decoherence rates Γ / ω q versus the normalized driving strength I 0 / ω q ....We propose a strategy to suppress decoherence of a solid-state **qubit** coupled to non-Markovian noises by attaching the **qubit** to a chaotic setup with the broad power distribution in particular in the high-**frequency** domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-**frequency** components of our control are generated by the chaotic setup driven by a low-**frequency** field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide **frequency** domain, including low-**frequency** $1/f$, high-**frequency** Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the **qubits** to a Duffing **oscillator** as the chaotic setup. Significantly, the decoherence time of the **qubit** is prolonged approximately $100$ times in magnitude. ... We propose a strategy to suppress decoherence of a solid-state **qubit** coupled to non-Markovian noises by attaching the **qubit** to a chaotic setup with the broad power distribution in particular in the high-**frequency** domain. Different from the existing decoherence control methods such as the usual dynamics decoupling control, high-**frequency** components of our control are generated by the chaotic setup driven by a low-**frequency** field, and the generation of complex optimized control pulses is not necessary. We apply the scheme to superconducting quantum circuits and find that various noises in a wide **frequency** domain, including low-**frequency** $1/f$, high-**frequency** Ohmic, sub-Ohmic, and super-Ohmic noises, can be efficiently suppressed by coupling the **qubits** to a Duffing **oscillator** as the chaotic setup. Significantly, the decoherence time of the **qubit** is prolonged approximately $100$ times in magnitude.

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