### 63697 results for qubit oscillator frequency

Contributors: Bennett, Douglas A., Longobardi, Luigi, Patel, Vijay, Chen, Wei, Averin, Dmitri V., Lukens, James E.

Date: 2008-11-14

The effects of flux noise on the decay of the Rabi **oscillation**, of course, becomes much more pronounced for δ ≠ 0 . Figure rabidetunefig shows such Rabi **oscillations** at various detunings over a range of 0 ≤ δ 10 9 s -1 . At long microwave pulse times (not shown) the occupation of the excited state reaches the equilibrium values discussed in Sec secintraspec. Near resonance, the value of ρ 11 for long pulse lengths is determined mainly by the noise amplitude, while for large δ it is set by δ . The data cannot be fit to solutions of the Bloch equations using only the phenomenological decay constants. As in the spectroscopy, static Gaussian detuning noise must be included to fit the data over the whole range of detuning. The lines in Figure rabidetunefig correspond to calculations with an initial Rabi **frequency** of 0.59 n s -1 , Γ = 0.075 n s -1 and σ ν = 0.22 n s -1 . The detuning **frequency** is taken from φ x using the conversion from Fig. spec2dfig. For the range of δ in Fig. rabidetunefig Δ E 01 is in a region with relatively few spectral anomalies. However for δ < 0 , on the other side of resonance towards splittings in the spectroscopy, the data generally do not agree with theory suggesting that the spectroscopic splittings have a strong effect on the coherence, as expected....The components that create and combine the microwave pulses and readout pulse are shown at the top of Fig. fig:circuitdiag. The pulse generator is capable of producing measurement pulses with rise times as short as 200 ps. Two microwave mixers are used to modulate the envelope of the continuous microwave signal produced by the microwave source giving an on/off ratio of 10 3 . The output of the mixers is then amplified 20 dB and coupled through variable attenuators that set the amplitude of the microwave pulses applied to the **qubit**. Finally, these microwave pulses are combined with the video pulses using a hybrid coupler. The video pulses enter on the directly coupled port while the microwave pulse are coupled using the indirectly coupled port. The video pulses used for the data shown in the following sections are around typically 5 ns with a rise time of 0.5 ns. The directionality and **frequency** response of the coupler allows the two signals to be combined with a minimum amount of reflections and loss of power. This signal is then coupled to the **qubit** through a coaxial line that is filtered by a series of attenuators at 1.4 K (20 dB), 600 mK (10 dB) and at the **qubit** temperature (10 dB) followed by a lossy microstrip filter that cuts off around 1 GHz....The measurement sequence for coherent **oscillations** is very similar to that used to measure the lifetime of the excited state except that the duration of the (generally shorter) microwave pulse is varied and the microwave pulse is immediately followed by the readout flux pulse. This signal, on the high **frequency** line, is illustrated in the lower inset in Figure rabifig. Figure rabifig shows an example of the Rabi **oscillations** when δ = 0 and the microwave **frequency**, f x r f = 17.9 G H z . This bias point lies in a "clean" range of the spectrum ( 17.6 - 18 G H z ) as seen in Fig. spec2dfig, which should be the best region for observing coherent **oscillations** between the ground and excited states. Most of the time domain data, including the lifetime measurements of Fig. decayfig, have been taken in this **frequency** range. Each data point in Fig. spec2dfig corresponds to the average of several thousand measurements for a given pulse length....We report measurements of coherence times of an rf SQUID **qubit** using pulsed microwaves and rapid flux pulses. The modified rf SQUID, described by an double-well potential, has independent, in situ, controls for the tilt and barrier height of the potential. The decay of coherent **oscillations** is dominated by the lifetime of the excited state and low **frequency** flux noise and is consistent with independent measurement of these quantities obtained by microwave spectroscopy, resonant tunneling between fluxoid wells and decay of the excited state. The **oscillation's** waveform is compared to analytical results obtained for finite decay rates and detuning and averaged over low **frequency** flux noise....If the decay with lifetime T 1 were the dominant source of decoherence in the qubit, one would expect a resonance in the microwave occupation of the first excited state given, in the low power limit, by ρ 11 ∝ φ x r f 2 / δ 2 + Γ 1 2 / 4 where δ is the mean difference between the photon energy and the level spacing. For the potential parameters used for the resonant data shown in Fig. specfig the dependence of δ on Φ x ℓ f is δ = 1.70 × 10 9 s -1 / m Φ...(Color online) The occupation of the excited state as a function of detuning for microwave powers corresponding to attenuator settings of 39 (squares), 36 (circles), 33 dB (triangles). Lines are fits using Eq. av at microwave amplitudes corresponding to the measured Rabi **frequency** for each attenuator setting (0.017, 0.024, 0.034 n s -1 ) with Γ = 0.055 n s -1 convoluted with static Gaussian noise with σ ν = 0.235 n s -1 at the angular **frequencies** of the Rabi **oscillations** that correspond to these microwave powers...We report measurements of coherence times of an rf SQUID **qubit** using pulsed microwaves and rapid flux pulses. The modified rf SQUID, described by an double-well potential, has independent, in situ, controls for the tilt and barrier height of the potential. The decay of coherent oscillations is dominated by the lifetime of the excited state and low **frequency** flux noise and is consistent with independent measurement of these quantities obtained by microwave spectroscopy, resonant tunneling between fluxoid wells and decay of the excited state. The oscillation's waveform is compared to analytical results obtained for finite decay rates and detuning and averaged over low **frequency** flux noise....(Color online) (a) The occupation of the excited state on resonance versus microwave amplitude in units of the corresponding Rabi frequency. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , blue dotted line σ ν = 0 with Γ = 0.055 n s -1 , purple dashed line σ δ = 0 and Γ = 0.20 n s -1 . (b) The width of the spectroscopic peak from the Gaussian fits as a function of microwave amplitude in units of Rabi frequency. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , green dashed line σ ν = 0 with Γ = 0.055 n s -1 , blue dotted line σ ν = 0 and Γ = 0.585 n s -1 ....Decoherence \and Superconducting **Qubits** \and Flux **Qubit** \and SQUIDs...(Color online) The measured occupation of the excited state after a long microwave pulse expressed as color contours (blue being lowest and red being the highest) as function of both frequency and φ x . The solid lines are calculations of the energy level spitting between consecutive eigenstates states in the same well for β = 1.30 , L = 190 pH and C = 210 fF....(Color online) (a) The occupation of the excited state on resonance versus microwave amplitude in units of the corresponding Rabi **frequency**. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , blue dotted line σ ν = 0 with Γ = 0.055 n s -1 , purple dashed line σ δ = 0 and Γ = 0.20 n s -1 . (b) The width of the spectroscopic peak from the Gaussian fits as a function of microwave amplitude in units of Rabi **frequency**. The lines are calculations for the following parameters; red solid Γ = 0.055 n s -1 and σ ν = 0.235 n s -1 , green dashed line σ ν = 0 with Γ = 0.055 n s -1 , blue dotted line σ ν = 0 and Γ = 0.585 n s -1 ....The data for resonant Rabi **oscillations**, shown in Fig. rabifig. The solid line is a fit using Eq. fin including a 0.5 ns time delay for the rise time for the microwave pulse and averaged over quasi-static Gaussian noise in φ x equal to that obtained from the fits to the spectroscopy data in Fig. specfig. This gives a Rabi **frequency** f r a b i = 119 M H z and decay rate, η = 0.042 n s -1 , for the **oscillations**. Equation equ:rabidecay, with δ = 0 , together with our previously measured value for the decay rate of the first excited state Γ 1 (which gives the rate Γ in the equations above), and the observed Rabi decay rate η discussed below, imply Γ ν = 0.01 n s -1 . Even though Ω R a b i is not affected by flux noise to first order for δ = 0 , the amplitude of the flux noise in our **qubit** is large enough to cause a measurable effect. If the flux noise were neglected, it would be necessary to increase η to 0.058 n s -1 in order to account for the observed decay. In addition to increasing the decay rate, the low **frequency** noise also reduces the steady state occupation of the excited state , i.e. its value for long pulses, as calculated from Eq. av. The ordinate of Fig. specfig has been calibrated using this value and is consistent with the estimate obtained from the calculated tunneling rates from the two levels involved during the readout pulse....(Color online) (a) Schematic and (b) micrograph of an rf SQUID **qubit** and the readout magnetometer, (c) a cross section of the wafer around the junctions and (d) micrograph giving a detailed view of junctions....(Color online) (a) Schematic and (b) micrograph of an rf SQUID qubit and the readout magnetometer, (c) a cross section of the wafer around the junctions and (d) micrograph giving a detailed view of junctions....This solution, shows that when driven with microwaves, the population of the excited states **oscillates** in time, demonstrating the phenomenon of Rabi **oscillations**, which have been seen in a number of different superconducting **qubits** . Ω , the **frequency** of the **oscillations** for δ = 0 , is ideally proportional to the amplitude of the microwaves excitation φ x r f . This linear dependence is accurately seen in our **qubit** at low power levels (see Fig. rabifig upper inset) providing a convenient means to calibrate the amplitude of the microwaves incident on the **qubit** in terms of Ω ....The data for resonant Rabi oscillations, shown in Fig. rabifig. The solid line is a fit using Eq. fin including a 0.5 ns time delay for the rise time for the microwave pulse and averaged over quasi-static Gaussian noise in φ x equal to that obtained from the fits to the spectroscopy data in Fig. specfig. This gives a Rabi **frequency** f r a b i = 119 M H z and decay rate, η = 0.042 n s -1 , for the oscillations. Equation equ:rabidecay, with δ = 0 , together with our previously measured value for the decay rate of the first excited state Γ 1 (which gives the rate Γ in the equations above), and the observed Rabi decay rate η discussed below, imply Γ ν = 0.01 n s -1 . Even though Ω R a b i is not affected by flux noise to first order for δ = 0 , the amplitude of the flux noise in our qubit is large enough to cause a measurable effect. If the flux noise were neglected, it would be necessary to increase η to 0.058 n s -1 in order to account for the observed decay. In addition to increasing the decay rate, the low **frequency** noise also reduces the steady state occupation of the excited state , i.e. its value for long pulses, as calculated from Eq. av. The ordinate of Fig. specfig has been calibrated using this value and is consistent with the estimate obtained from the calculated tunneling rates from the two levels involved during the readout pulse....(Color online) Rabi **oscillations** for detunings going from top to bottom of 0.094, 0.211, 0.328, 0.562, 0.796 and 1.269 n s -1 with the corresponding fits using Γ = 0.075 n s -1 and σ ν = 0.22 n s -1...The upper inset to Figure rabifig shows the Rabi **frequency** extracted from Rabi oscillation measurements versus microwave amplitude in arbitrary units. As noted above, at low powers the Rabi **frequency** is linear in microwave amplitude as expected. At higher powers, however, it begins to saturate. There is evidence that at these higher power levels there is a small probability of excitation to the second excited state. However since the second excited state has orders of magnitude higher escape rate, this small probability can be significant....(Color online) The occupation of the excited state as a function of the length of the microwave pulse demonstrating Rabi **oscillations**. The line is a fit to Eq. fin for δ = 0 averaged over quasi-static noise with σ ν = 0.22 n s -1 . This gives f r a b i = 119 MHz and decay time T 2 ˜ = 24 n s -1 . The upper inset shows Rabi **frequency** as a function of amplitude of applied microwaves in arbitrary units. The line is a linear fit to the lower microwave amplitude data. The lower inset show the measurement pulse sequence....The measurement sequence for coherent oscillations is very similar to that used to measure the lifetime of the excited state except that the duration of the (generally shorter) microwave pulse is varied and the microwave pulse is immediately followed by the readout flux pulse. This signal, on the high **frequency** line, is illustrated in the lower inset in Figure rabifig. Figure rabifig shows an example of the Rabi oscillations when δ = 0 and the microwave **frequency**, f x r f = 17.9 G H z . This bias point lies in a "clean" range of the spectrum ( 17.6 - 18 G H z ) as seen in Fig. spec2dfig, which should be the best region for observing coherent oscillations between the ground and excited states. Most of the time domain data, including the lifetime measurements of Fig. decayfig, have been taken in this **frequency** range. Each data point in Fig. spec2dfig corresponds to the average of several thousand measurements for a given pulse length....(Color online) The occupation of the excited state as a function of detuning for microwave powers corresponding to attenuator settings of 39 (squares), 36 (circles), 33 dB (triangles). Lines are fits using Eq. av at microwave amplitudes corresponding to the measured Rabi frequency for each attenuator setting (0.017, 0.024, 0.034 n s -1 ) with Γ = 0.055 n s -1 convoluted with static Gaussian noise with σ ν = 0.235 n s -1 at the angular frequencies of the Rabi oscillations that correspond to these microwave powers...(Color online) The occupation of the excited state as a function of the length of the microwave pulse demonstrating Rabi oscillations. The line is a fit to Eq. fin for δ = 0 averaged over quasi-static noise with σ ν = 0.22 n s -1 . This gives f r a b i = 119 MHz and decay time T 2 ˜ = 24 n s -1 . The upper inset shows Rabi frequency as a function of amplitude of applied microwaves in arbitrary units. The line is a linear fit to the lower microwave amplitude data. The lower inset show the measurement pulse sequence....Decoherence in rf SQUID **Qubits** ... We report measurements of coherence times of an rf SQUID **qubit** using pulsed microwaves and rapid flux pulses. The modified rf SQUID, described by an double-well potential, has independent, in situ, controls for the tilt and barrier height of the potential. The decay of coherent **oscillations** is dominated by the lifetime of the excited state and low **frequency** flux noise and is consistent with independent measurement of these quantities obtained by microwave spectroscopy, resonant tunneling between fluxoid wells and decay of the excited state. The **oscillation's** waveform is compared to analytical results obtained for finite decay rates and detuning and averaged over low **frequency** flux noise.

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Contributors: Johansson, J., Saito, S., Meno, T., Nakano, H., Ueda, M., Semba, K., Takayanagi, H.

Date: 2005-10-17

Vacuum Rabi oscillations in a macroscopic superconducting **qubit** LC **oscillator** system...fig4 (a) Rabi oscillations when a 2 ns long pulse with frequency ν e x = 4.35 GHz and amplitude A M W ∝ 10 P o s c / 20 is inserted between** the **π pulse and** the **shift pulse. (b) Measured Rabi oscillations at different drive powers (symbols), and a fit (solid curve) to ∑ n = 0 3 a n cos n + 1 Ω R t exp - Γ t with a 0 , , a 3 and Ω R as** the **only fitting parameters ( Γ is fixed from a fit to the -100 dBm curve). (c) The weights of** the **different frequency components a 0 , , a 3 obtained from** the **fit as a function of** the **drive amplitude (The red arrows show** the **position of** the **curves in (b). The error-bars indicate** the **errors obtained from** the **fitting procedure....We have observed the coherent exchange of a single energy quantum between a flux **qubit** and a superconducting LC circuit acting as a quantum harmonic **oscillator**. The exchange of an energy quantum is known as the vacuum Rabi oscillations: the **qubit** is oscillating between the excited state and the ground state and the **oscillator** between the vacuum state and the first excited state. We have also obtained evidence of level quantization of the LC circuit by observing the change in the oscillation **frequency** when the LC circuit was not initially in the vacuum state....fig1 (a) SEM micrograph of the sample. The **qubit** and the detector SQUID enclosing it are the small square loops in the lower center picture. The square plates at the top of the picture are the top plates of the on-chip capacitors separated by an insulator from the large bottom plate. (b) A close up of the **qubit** and the SQUID. The **qubit** dimension is 10.2 × 10.4 μ m 2 . (c) Equivalent circuit of the sample. The Josephson junctions are indicated by crosses: three in the inner **qubit** loop and two in the SQUID. The LC mode is indicated by the dashed line. The inductance and capacitances are calculated from the geometry to be L = 140 pH and C = 10 pF, and the **qubit** LC **oscillator** mutual inductance to be M = 5.7 pH. The current and voltage lines are filtered through a series combination of copper powder filters and lossy coaxial cables at mixing chamber temperature and on–chip resistors ( R V = 3 k Ω and R I = 1 k Ω )....fig2 (a) Spectroscopic characterization of the **qubit**–**oscillator** system showing the LC **oscillator** at ν r = 4.35 GHz and the **qubit** dispersion around the gap of Δ = 2.1 GHz. (b) A close up of the region around 4.35 GHz (indicated by the red square in (a) showing an avoided crossing. The lines are guides for the eye. (c) Schematic of the pulse sequence used to obtain the spectroscopy in (b): the **qubit** operating point is fixed at 10.5 GHz and via the MW line a shift pulse of variable height moves the operating point to the vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to the shift pulse. Here the second MW pulse is phase shifted by 180 ∘ . The phase shifted second pulse damps the **oscillations** in the LC circuit , and is crucial in terms of resolving the relatively weak **qubit** signal in this region. After the MW pulses the **qubit** state is measured by applying a measurement pulse to the SQUID (green curve). The spectroscopy in Fig. 2(a) was obtained with the same scheme, but without the second phase shifted pulse....In our sample geometry, **the** **qubit** is spatially separated from **the** rest of **the** circuitry. **The** **qubit** is enclosed by a superconducting quantum interference device (SQUID) that is inductively coupled to **the** **qubit**. **The** SQUID functions as a detector for **the** **qubit** state: **the** switching current of **the** SQUID is sensitive to **the** flux produced by **the** current in **the** **qubit**. **The** **qubit** is also enclosed by a larger loop containing on–chip capacitors that provide a well–defined electromagnetic environment for **the** SQUID and filtering of **the** measurement leads. **The** lead inductance L and capacitance C in **the** outer loop constitute an **LC** **oscillator** [see Fig. fig1(b)] with **resonance** **frequency** ω r = 2 π ν r = 1 / L C . **The** **LC** **oscillator** is described by a simple **harmonic** **oscillator** Hamiltonian: H o s c = ℏ ω r a † a + 1 / 2 , where a † ( a ) is **the** plasmon creation (annihilation) operator. **The** **qubit** is coupled to **the** **LC** **oscillator** via **the** mutual inductance M , giving an interaction Hamiltonian H I = h λ σ z a † + a , where **the** coupling constant is h λ = M I p ℏ ω r / 2 L . **The** total system is thus described by a Jaynes-Cummings type of Hamiltonian H = h / 2 ϵ σ z + Δ σ x + ℏ ω r a † a + 1 / 2 + h λ σ z a † + a . We denote **the** state of **the** system by | Q , i , with **the** **qubit** either in **the** ground ( Q = g ) or excited ( Q = e ) state, and **the** **oscillator** in **the** Fock state ( i = 0 , 1 , 2 , ). **The** parameters of **the** system can readily be engineered during fabrication; **the** **qubit** gap is determined by α and **the** junction resistance, **the** **oscillator** **plasma** **frequency** is fully determined by L and C , and **the** coupling between **the** **qubit** and **the** **oscillator** can be tuned by M and L ....fig3 (a) Vacuum Rabi oscillations (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when** the **system is in resonance. (c) The **qubit** energy level diagram and pulse sequence for** the **vacuum Rabi measurements. The π pulse (4.6 ns long at -16 dBm) on the **qubit** brings** the **system from state 1 to 2 and** the **shift pulse changes** the **flux in the **qubit** by Φ s h i f t , which, in turn, changes** the **operating point from 2 to 3 where** the **system undergoes free evolution between | e , 0 and | g , 1 at** the **vacuum Rabi frequency Ω R until** the **shift pulse ends and** the **system returns to** the **initial operating point where** the **state is measured to be either in 2 or 4....Next we investigate the dynamics of the coupled system in the time domain. We performed a measurement cycle where we first excited the **qubit** and then brought the **qubit** and the **oscillator** into resonance where the exchange of a single energy quantum between the **qubit** and **oscillator** manifests itself as the vacuum Rabi **oscillation** | e , 0 ↔ | g , 1 (see Fig. 3). Figure. 3(c) is a schematic of the pulse sequence: We started by fixing the **qubit** operating point far from the resonance point [point 3 in Fig. fig3(c)] and prepared the **qubit** in the excited state by employing a π pulse. The π pulse was followed by a shift pulse, which brought the **qubit** into resonance with the **oscillator** for the duration of the shift pulse. After the shift pulse the **qubit** and the **oscillator** were brought back into off–resonance and the measurement pulse was applied to detect the state of the **qubit**. It is important to note that the rise time of the shift pulse, τ r i s e = 0.8 ns, is adiabatic with respect to both the **qubit** and the **oscillator**, τ r i s e > 1 / E 2 π / ω r , but non-adiabatic with respect to the coupling of the two systems, τ r i s e **oscillations** is thus different from that of normal Rabi **oscillations** where the system is driven by an external classical field and **oscillates** between two energy eigenstates. Also, in the normal Rabi **oscillations** the Rabi **frequency** is determined by the drive amplitude whereas the vacuum Rabi **oscillation** **frequency** is determined only by the system‘s intrinsic parameters. The observed Rabi **oscillations** are in excellent agreement with those calculated numerically [solid line in Fig. 3(a)]. The numerical calculation uses the total Hamiltonian and incorporates the effects of the decoherence of the **qubit** and the **oscillator**. The calculation was performed with the known **qubit** and LC **oscillator** parameters (obtained from spectroscopy and **qubit** experiments: **qubit** dephasing rate Γ φ = 0.1 GHz, **qubit** relaxation rate Γ e = 0.2 MHz, Δ = 2.1 GHz, ω r / 2 π = 4.35 GHz) and by treating the coupling constant and **oscillator** dephasing and relaxation rates as fitting parameters. From the fit we extracted the coupling constant λ = 0.2 GHz, **oscillator** dephasing rate Γ φ = 0.3 GHz, and relaxation rate Γ e = 0.02 GHz. The coupling constant extracted from the fit agrees well with that calculated with the mutual inductance λ = 0.216 GHz....fig3 (a) Vacuum Rabi **oscillations** (symbols) and a numerical fit (solid line). (b) The few lowest unperturbed and dressed energy levels when the system is in resonance. (c) The **qubit** energy level diagram and pulse sequence for the vacuum Rabi measurements. The π pulse (4.6 ns long at -16 dBm) on the **qubit** brings the system from state 1 to 2 and the shift pulse changes the flux in the **qubit** by Φ s h i f t , which, in turn, changes the operating point from 2 to 3 where the system undergoes free evolution between | e , 0 and | g , 1 at the vacuum Rabi **frequency** Ω R until the shift pulse ends and the system returns to the initial operating point where the state is measured to be either in 2 or 4....Next we investigate **the** dynamics of **the** coupled system in **the** time domain. We performed a measurement cycle where we first excited **the** **qubit** and then brought **the** **qubit** and **the** **oscillator** into **resonance** where **the** exchange of a single energy quantum between **the** **qubit** and **oscillator** manifests itself as **the** vacuum Rabi oscillation | e , 0 ↔ | g , 1 (see Fig. 3). Figure. 3(c) is a schematic of **the** pulse sequence: We started by fixing **the** **qubit** operating point far from **the** **resonance** point [point 3 in Fig. fig3(c)] and prepared **the** **qubit** in **the** excited state by employing a π pulse. **The** π pulse was followed by a shift pulse, which brought **the** **qubit** into **resonance** with **the** **oscillator** for **the** duration of **the** shift pulse. After **the** shift pulse **the** **qubit** and **the** **oscillator** were brought back into off–**resonance** and **the** measurement pulse was applied to detect **the** state of **the** **qubit**. It is important to note that **the** rise time of **the** shift pulse, τ r i s e = 0.8 ns, is adiabatic with respect to both **the** **qubit** and **the** **oscillator**, τ r i s e > 1 / E 2 π / ω r , but non-adiabatic with respect to **the** coupling of **the** two systems, τ r i s e i **frequency** is determined by **the** drive amplitude whereas **the** vacuum Rabi oscillation **frequency** is determined only by **the** system‘s intrinsic parameters. **The** observed Rabi oscillations are in excellent agreement with those calculated numerically [solid line in Fig. 3(a)]. **The** numerical calculation uses **the** total Hamiltonian and incorporates **the** effects of **the** decoherence of **the** **qubit** and **the** **oscillator**. **The** calculation was performed with **the** known **qubit** and **LC** **oscillator** parameters (obtained from spectroscopy and **qubit** experiments: **qubit** dephasing rate Γ φ = 0.1 GHz, **qubit** relaxation rate Γ e = 0.2 MHz, Δ = 2.1 GHz, ω r / 2 π = 4.35 GHz) and by treating **the** coupling constant and **oscillator** dephasing and relaxation rates as fitting parameters. From **the** fit we extracted **the** coupling constant λ = 0.2 GHz, **oscillator** dephasing rate Γ φ = 0.3 GHz, and relaxation rate Γ e = 0.02 GHz. **The** coupling constant extracted from **the** fit agrees well with that calculated with **the** mutual inductance λ = 0.216 GHz....In our sample geometry, the **qubit** is spatially separated from the rest of the circuitry. The **qubit** is enclosed by a superconducting quantum interference device (SQUID) that is inductively coupled to the **qubit**. The SQUID functions as a detector for the **qubit** state: the switching current of the SQUID is sensitive to the flux produced by the current in the **qubit**. The **qubit** is also enclosed by a larger loop containing on–chip capacitors that provide a well–defined electromagnetic environment for the SQUID and filtering of the measurement leads. The lead inductance L and capacitance C in the outer loop constitute an LC **oscillator** [see Fig. fig1(b)] with resonance **frequency** ω r = 2 π ν r = 1 / L C . The LC **oscillator** is described by a simple harmonic **oscillator** Hamiltonian: H o s c = ℏ ω r a † a + 1 / 2 , where a † ( a ) is the plasmon creation (annihilation) operator. The **qubit** is coupled to the LC **oscillator** via the mutual inductance M , giving an interaction Hamiltonian H I = h λ σ z a † + a , where the coupling constant is h λ = M I p ℏ ω r / 2 L . The total system is thus described by a Jaynes-Cummings type of Hamiltonian H = h / 2 ϵ σ z + Δ σ x + ℏ ω r a † a + 1 / 2 + h λ σ z a † + a . We denote the state of the system by | Q , i , with the **qubit** either in the ground ( Q = g ) or excited ( Q = e ) state, and the **oscillator** in the Fock state ( i = 0 , 1 , 2 , ). The parameters of the system can readily be engineered during fabrication; the **qubit** gap is determined by α and the junction resistance, the **oscillator** plasma **frequency** is fully determined by L and C , and the coupling between the **qubit** and the **oscillator** can be tuned by M and L ....fig4 (a) Rabi **oscillations** when a 2 ns long pulse with **frequency** ν e x = 4.35 GHz and amplitude A M W ∝ 10 P o s c / 20 is inserted between the π pulse and the shift pulse. (b) Measured Rabi **oscillations** at different drive powers (symbols), and a fit (solid curve) to ∑ n = 0 3 a n cos n + 1 Ω R t exp - Γ t with a 0 , , a 3 and Ω R as the only fitting parameters ( Γ is fixed from a fit to the -100 dBm curve). (c) The weights of the different **frequency** components a 0 , , a 3 obtained from the fit as a function of the drive amplitude (The red arrows show the position of the curves in (b). The error-bars indicate the errors obtained from the fitting procedure....We have observed the coherent exchange of a single energy quantum between a flux **qubit** and a superconducting LC circuit acting as a quantum harmonic **oscillator**. The exchange of an energy quantum is known as the vacuum Rabi **oscillations**: the **qubit** is **oscillating** between the excited state and the ground state and the **oscillator** between the vacuum state and the first excited state. We have also obtained evidence of level quantization of the LC circuit by observing the change in the **oscillation** **frequency** when the LC circuit was not initially in the vacuum state....fig2 (a) Spectroscopic characterization of the **qubit**–**oscillator** system showing** the **LC **oscillator** at ν r = 4.35 GHz and the **qubit** dispersion around** the **gap of Δ = 2.1 GHz. (b) A close up of** the **region around 4.35 GHz (indicated by** the **red square in (a) showing an avoided crossing. The lines are guides for** the **eye. (c) Schematic of** the **pulse sequence used to obtain** the **spectroscopy in (b): the **qubit** operating point is fixed at 10.5 GHz and via** the **MW line a shift pulse of variable height moves** the **operating point to** the **vicinity of 4.35 GHz. Two 50 ns MW pulses separated by 2 ns are added to** the **shift pulse. Here** the **second MW pulse is phase shifted by 180 ∘ . The phase shifted second pulse damps** the **oscillations in** the **LC circuit , and is crucial in terms of resolving** the **relatively **weak ****qubit** signal in this region. After** the **MW pulses the **qubit** state is measured by applying a measurement pulse to** the **SQUID (green curve). The spectroscopy in Fig. 2(a) was obtained with** the **same scheme, but without** the **second phase shifted pulse....fig1 (a) SEM micrograph of** the **sample. The **qubit** and** the **detector SQUID enclosing it are** the **small square loops in** the **lower center picture. The square plates at** the **top of** the **picture are** the **top plates of** the **on-chip capacitors separated by an insulator from** the **large bottom plate. (b) A close up of the **qubit** and** the **SQUID. The **qubit** dimension is 10.2 × 10.4 μ m 2 . (c) Equivalent circuit of** the **sample. The Josephson junctions are indicated by crosses: three in** the **inner **qubit** loop and two in** the **SQUID. The LC mode is indicated by** the **dashed line. The inductance and capacitances are calculated from** the **geometry to be L = 140 pH and C = 10 pF, and the **qubit** LC **oscillator** mutual inductance to be M = 5.7 pH. The current and voltage lines are filtered through a series combination of copper powder filters and lossy coaxial cables at mixing chamber temperature and on–chip resistors ( R V = 3 k Ω and R I = 1 k Ω ). ... We have observed the coherent exchange of a single energy quantum between a flux **qubit** and a superconducting LC circuit acting as a quantum harmonic **oscillator**. The exchange of an energy quantum is known as the vacuum Rabi **oscillations**: the **qubit** is **oscillating** between the excited state and the ground state and the **oscillator** between the vacuum state and the first excited state. We have also obtained evidence of level quantization of the LC circuit by observing the change in the **oscillation** **frequency** when the LC circuit was not initially in the vacuum state.

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Contributors: Kofman, A. G., Zhang, Q., Martinis, J. M., Korotkov, A. N.

Date: 2006-06-02

The circuit schematic of a flux-biased phase **qubit** and the corresponding potential profile (as a function of the phase difference δ across the Josephson junction). During the measurement the state | 1 escapes from the “left” well through the barrier, which is followed by **oscillations** in the “right” well. This dissipative evolution leads to the two-**qubit** crosstalk....3.17 **in ****the **absence of dissipation **in ****the **second qubit ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account **the **mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs. f4(a) and f7 shows that **in **both models **the **qubit energy remains small before a sharp increase **in **energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than **in ****the **oscillator model (**the **position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of **the **qubit may be to a much lower energy than for **the **oscillator; (c) After **the **sharp increase, **the **energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot **the **initial energy maximum; (d) The model now explicitly describes **the **qubit escape (switching) to **the **right well [Figs. f7(b) and f7(c)]; **in **contrast to **the **oscillator model, **the **escape may happen much later than initial energy increase; for example, **in **Fig. f7(b) **the **escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns....The **oscillating** term in Eq. ( 3.11a) describes the beating between the **oscillator** and driving force **frequencies**, with the difference **frequency** increasing in time, d t ~ 2 / d t = α t - t c , and amplitude of beating decreasing as 1 / t ~ (see dashed line in Fig. f4a). Notice that F 0 = 1 / 4 , F ∞ = 1 , and the maximum value is F 1.53 = 1.370 , so that E 0 is the long-time limit of the **oscillator** energy E 2 , while the maximum energy is 1.37 times larger:...The second **qubit** energy E 2 (in units of ℏ ω l 2 ) in the **oscillator** model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving **frequency** f d (see Fig. f3) is in resonance with ω l 2 / 2 π = 8.91 GHz....The second qubit energy E 2 (in units of ℏ ω l 2 ) in the oscillator model as a function of time t (in ns) for (a) C x = 5 fF and T 1 = 25 ns and (b) C x = 2.5 fF and 5 fF and T 1 = 500 ns, while N l 2 = 5 . Dashed line in (a) shows approximation using Eq. ( 3.10). The arrows show the moment t c when the driving frequency f d (see Fig. f3) is in resonance with ω l 2 / 2 π = 8.91 GHz....3.3 Figure f4 shows **the **time dependence of **the **energy E 2 t **in **units of ℏ ω l 2 for ω l 2 / 2 π = 8.91 GHz, which corresponds to N l 2 = 5 (parameters of **the **first qubit evolution have been discussed **in **Sec. III and correspond to Fig. f3). One can see that **the **energy E 2 remains very low until a sharp increase followed by gradually decreasing oscillations. This behavior can be easily explained by changing **in **time frequency f **d **of **the **driving force (Fig. f3) which passes through **the **resonance with **the **second qubit....The properties (a) and (b) can be understood by taking into account **the **fact that **the **oscillation frequency **in ****the **second qubit decreases with **the **energy increase (it should become formally zero at **the **top of **the **barrier), while **the **driving frequency increases with time (Fig. f3). Therefore, initially small out-of-resonance beatings when ω d increase of **the **qubit energy earlier then **the **condition ω d = ω l 2 is satisfied. The same mechanism is also responsible for lower qubit excitation, when compared to **the **harmonic oscillator model: **the **resonance cannot be as efficient as **in ****the **harmonic oscillator model since **the **qubit excitation quickly moves **the **qubit frequency out of **the **resonance. The property (c) is related to crossing of higher-order resonances, which occur when ω **d **t is commensurate with **the **oscillation frequency of **the **system, which itself depends on **the **energy E 2 t and hence on **the **time. Similar mechanism is responsible for **the **qubit switching at t ≫ t c ; **in **particular, **in **Fig. f7(b) **the **switching happens when **the **driving frequency f **d **becomes approximately twice larger than **the **second qubit frequency....We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability....Figure f3 shows that **the **oscillation frequency sharply increases initially and then slowly tends to **the **right-well plasma frequency ω r 1 / 2 π = 15.3 GHz (**the **dash-dotted horizontal line **in **Fig. f3). This is explained by **the **fact that **the **initial system energy is close to **the **barrier top, where **the **oscillation frequency is significantly lower (it tends to zero when **the **energy approaches **the **barrier top), while anharmonicity becomes relatively weak after **the **energy is no longer close to **the **barrier top....2.16 Figure f2 shows the **qubit** potential U δ for N l = 10 (corresponding to φ = 4.842 ), N l = 5 ( φ = 5.089 ), and N l = 1.355 ( φ = 5.308 ); the last value corresponds to the bias during the measurement pulse (see below). The **qubit** levels | 0 and | 1 are, respectively, the ground and the first excited levels in the left well....We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped oscillations of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability....The first-**qubit** **oscillation** **frequency** f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma **frequency** for the second **qubit**: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 ....Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [oscillator model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation frequency of the two-qubit imaginary-swap quantum gate....mcd05, a short flux pulse applied to the measured **qubit** decreases the barrier between the two wells (see Fig. f0), so that the upper **qubit** level becomes close to the barrier top. In the case when level | 1 is populated, there is a fast population transfer (tunneling) from the left well to the right well. Due to dissipation, the energy in the right well gradually decreases, until it reaches the bottom of the right well. In contrast, if the **qubit** is in state | 0 the tunneling essentially does not occur. The **qubit** state in one of the two potential minima (separated by almost Φ 0 ) is subsequently distinguished by a nearby SQUID, which completes the measurement process....T...Th...Now let us consider the effect of dissipation in the second **qubit**. ...The first-qubit oscillation frequency f d as a function of time t (normalized by the energy relaxation time T 1 ) for C x = 0 (solid line) and C x = 6 fF (dashed line), assuming N l 1 = 1.355 and parameters of Eq. ( 2.16). Dash-dotted horizontal line, ω r 1 / 2 π = 15.3 GHz, shows the long-time limit of f d t . Two dotted horizontal lines show the plasma frequency for the second qubit: ω l 2 / 2 π = 10.2 GHz for N l 2 = 10 and ω l 2 / 2 π = 8.91 GHz for N l 2 = 5 . The arrow shows the moment t c of exact resonance in the case N l 2 = 5 ....Dots: Rabi **frequencies** R k , k - 1 / 2 π for the left-well transitions at t = t c , for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Dashed line shows analytical dependence 1.1 k GHz....Solid lines: log-log contour plots for the values of the error (switching) probability P s = 0.01 , 0.1, and 0.3 on the plane of relaxation time T 1 (in ns) and coupling capacitance C x (in fF) in the quantum model for (a) N l 2 = 5 and (b) N l 2 = 10 . The corresponding results for C x , T T 1 in the classical models are shown by the dashed lines (actual potential model) and the dotted lines [**oscillator** model, Eq. ( bound1)]. The numerical data are represented by the points, connected by lines as guides for the eye. The scale at the right corresponds to the operation **frequency** of the two-**qubit** imaginary-swap quantum gate....Time dependence of the frequency detunings ω k , k - 1 - ω d t / 2 π of left-well transitions k - 1 ↔ k for N l = 10 , C x = 6 fF, and T 1 = 25 ns. Curves from top to bottom correspond to k = 1 , 2 11 ....3.17 in the absence of dissipation in the second **qubit** ( T 1 ' = ∞ ) for N l 2 = 5 and 10, while T 1 = 25 ns. (In this subsection we take into account the mass renormalization m → m ' ' explicitly, even though this does not lead to a noticeable change of results.) A comparison of Figs. f4(a) and f7 shows that in both models the **qubit** energy remains small before a sharp increase in energy. However, there are significant differences due to account of anharmonicity: (a) The sharp energy increase occurs earlier than in the **oscillator** model (the position of short-time energy maximum is shifted approximately from 3 ns to 2 ns); (b) The excitation of the **qubit** may be to a much lower energy than for the **oscillator**; (c) After the sharp increase, the energy occasionally undergoes noticeable upward (as well as downward) jumps, which may overshoot the initial energy maximum; (d) The model now explicitly describes the **qubit** escape (switching) to the right well [Figs. f7(b) and f7(c)]; in contrast to the **oscillator** model, the escape may happen much later than initial energy increase; for example, in Fig. f7(b) the escape happens at t ≃ 44 ns ≫ t c ≃ 2.1 ns....The second-qubit energy E 2 t in the classical model taking into account energy dissipation in the second qubit, for N l 2 = 5 , C x = 6 fF, and T 1 ' = T 1 = 25 ns. [Compare with Fig. f7(b).]...Theory of measurement crosstalk in superconducting phase **qubits** ... We analyze the crosstalk error mechanism in measurement of two capacitively coupled superconducting flux-biased phase **qubits**. The damped **oscillations** of the superconducting phase after the measurement of the first **qubit** may significantly excite the second **qubit**, leading to its measurement error. The first **qubit**, which is highly excited after the measurement, is described classically. The second **qubit** is treated both classically and quantum-mechanically. The results of the analysis are used to find the upper limit for the coupling capacitance (thus limiting the **frequency** of two-**qubit** operations) for a given tolerable value of the measurement error probability.

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Contributors: Xian-Ting Liang

Date: 2007-12-05

Decoherence and relaxation of a **qubit** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator**...The response functions of the Ohmic bath and effective bath, where Δ=5×109Hz, λκ=1050, ξ=0.01, Ω0=10Δ, T=0.01K, Γ=2.6×1011, the lower-**frequency** and high-**frequency** cut-off of the baths modes ω0=11Δ, and ωc=100Δ.
...Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator** (IHO). Here, we suppose the **oscillation** **frequencies** of the bath modes are higher than the IHO’s. When we choose suitable parameters the **qubits** in the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times of the **qubit** in the **qubit**-IHO-bath model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO....Using the numerical path integral method we investigate the decoherence and relaxation **of** **qubits** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator** (**IHO**). Here, we suppose the oscillation **frequencies** **of** the bath modes are higher than the **IHO**’s. When we choose suitable parameters the **qubits** **in** the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times **of** the **qubit** **in** the **qubit**-**IHO**-bath model can be modulated through changing the coupling coefficients **of** the **qubit**-**IHO** and **IHO**-bath and the oscillation **frequency** **of** the **IHO**....The response functions of the Ohmic bath and effective bath, where Δ=5×109Hz, λκ=1050, ξ=0.01, Ω0=10Δ, T=0.01K, Γ=2.6×1011, the lower-frequency and high-frequency cut-off of the baths modes ω0=11Δ, and ωc=100Δ.
... Using the numerical path integral method we investigate the decoherence and relaxation of **qubits** coupled to an Ohmic bath directly and via an intermediate harmonic **oscillator** (IHO). Here, we suppose the **oscillation** **frequencies** of the bath modes are higher than the IHO’s. When we choose suitable parameters the **qubits** in the two models may have almost same decoherence and relaxation times. However, the decoherence and relaxation times of the **qubit** in the **qubit**-IHO-bath model can be modulated through changing the coupling coefficients of the **qubit**-IHO and IHO-bath and the **oscillation** **frequency** of the IHO.

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Contributors: Fedorov, A., Feofanov, A. K., Macha, P., Forn-Díaz, P., Harmans, C. J. P. M., Mooij, J. E.

Date: 2010-04-09

fig:rabi_sym_2(color online). (a) MW **frequency** vs f ϵ . The dotted white line is obtained from Eq. ( total H1) with I p = 400 nA and Δ = ν o s c . The observed vacuum Rabi splitting is maximal due to fully transverse coupling of the **qubit** to the **oscillator** η = π / 2 . (b) Vacuum Rabi **oscillations** for different values of f ϵ . In the experiment f ϵ was controlled by the amplitude of the current pulse I ϵ while Δ was tuned to ν o s c by changing the external magnetic field B . The inset shows ν R extracted from data (red circles) and estimated from Eq. ( Rabi freq general) (blue line). The color indicates the switching probability of the SQUID minus 0.5....fig:spectrum1(color online). (a) Schematic representation of the control and measurement pulses to perform spectroscopy. (b) Diagram of Landau-Zener transitions transferring the excitation of the **oscillator** to the **qubit**. (c) MW **frequency** vs f ϵ (controlled by the amplitude of the current pulse I ϵ ). The color indicates the switching probability of the SQUID minus 0.5. The white dotted line is obtained from Eq. ( total H1) with Δ = 2.04 GHz, I p = 420 nA. The vacuum Rabi splitting of 180 MHz corresponds to the effective **qubit**-**oscillator** coupling strength reduced by sin η ....A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi oscillations. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible....Strong Coupling of a Quantum **Oscillator** to a Flux **Qubit** at its Symmetry Point...fig:rabi_sym_2(color online). (a) MW **frequency** vs f ϵ . The dotted white line is obtained from Eq. ( total H1) with I p = 400 nA and Δ = ν o s c . The observed vacuum Rabi splitting is maximal due** to** fully transverse coupling** of** the **qubit** to the **oscillator** η = π / 2 . (b) Vacuum Rabi oscillations for different values** of** f ϵ . In** the **experiment f ϵ was controlled by** the **amplitude** of**** the **current pulse I ϵ while Δ was tuned** to** ν o s c by changing** the **external magnetic field B . The inset shows ν R extracted from data (red circles) and estimated from Eq. ( Rabi freq general) (blue line). The color indicates** the **switching probability** of**** the **SQUID minus 0.5....fig:scheme(color online). (a) Circuit schematics: the tunable gap flux **qubit** (green) coupled** to** a lumped element superconducting LC **oscillator** (red) and controlled by** the **bias lines I ϵ , I ϵ , d c , I α (black). The SQUID (blue) measures** the **state** of** the **qubit**. The gradiometer loop (emphasized by a dashed line) is used** to** trap fluxoids. (b) Scanning Electron Micrograph (SEM) of** the **sample. (c) Energy diagram** of** the **qubit**-**oscillator** system. The minimum** of** energy splitting** of** the **qubit** Δ is reached at** the **symmetry point when one fluxoid is trapped in** the **gradiometer loop and** the **difference in magnetic fluxes f ϵ Φ 0 is 0 controlled by I ϵ and I ϵ , d c . By controlling the **flux **f α Φ 0 with I α and uniform field B one can tune Δ in resonance with **oscillator** **frequency** ν o s c ....A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible....fig:spectrum1(color online). (a) Schematic representation** of**** the **control and measurement pulses** to** perform spectroscopy. (b) Diagram** of** Landau-Zener transitions transferring** the **excitation** of** the **oscillator** to the **qubit**. (c) MW **frequency** vs f ϵ (controlled by** the **amplitude** of**** the **current pulse I ϵ ). The color indicates** the **switching probability** of**** the **SQUID minus 0.5. The white dotted line is obtained from Eq. ( total H1) with Δ = 2.04 GHz, I p = 420 nA. The vacuum Rabi splitting** of** 180 MHz corresponds** to** the **effective** **qubit**-**oscillator** coupling strength reduced by sin η ....fig:rabi_sym(color online). Vacuum Rabi oscillations (a) and MW **frequency** (b) vs magnetic f α . In** the **experiment the **qubit** was kept in its symmetry point ( ϵ = 0 ) by appropriately adjusting** the **amplitude** of**** the **current pulse I ϵ while Δ was changed by f α with use** of** external magnetic field B (a) or by applying** the **current pulse I α for fixed B (b). The color scale shows** the **switching probability** of**** the **SQUID minus 0.5. (c) Frequency** of**** the **vacuum Rabi oscillations extracted from data (a) and theoretical estimation (blue line) from Eq. ( Rabi **frequency**) as a function** of** f α . The minimum in ν R determines** the **bare **qubit**-**oscillator** coupling 2 g and corresponds** to**** the **resonance conditions Δ = ν o s c . (d) Single trace** of**** the **vacuum Rabi oscillations for Δ ≃ ν o s c ....fig:rabi_sym(color online). Vacuum Rabi **oscillations** (a) and MW **frequency** (b) vs magnetic f α . In the experiment the **qubit** was kept in its symmetry point ( ϵ = 0 ) by appropriately adjusting the amplitude of the current pulse I ϵ while Δ was changed by f α with use of external magnetic field B (a) or by applying the current pulse I α for fixed B (b). The color scale shows the switching probability of the SQUID minus 0.5. (c) **Frequency** of the vacuum Rabi **oscillations** extracted from data (a) and theoretical estimation (blue line) from Eq. ( Rabi **frequency**) as a function of f α . The minimum in ν R determines the bare **qubit**-**oscillator** coupling 2 g and corresponds to the resonance conditions Δ = ν o s c . (d) Single trace of the vacuum Rabi **oscillations** for Δ ≃ ν o s c ....fig:scheme(color online). (a) Circuit schematics: the tunable gap flux **qubit** (green) coupled to a lumped element superconducting LC **oscillator** (red) and controlled by the bias lines I ϵ , I ϵ , d c , I α (black). The SQUID (blue) measures the state of the **qubit**. The gradiometer loop (emphasized by a dashed line) is used to trap fluxoids. (b) Scanning Electron Micrograph (SEM) of the sample. (c) Energy diagram of the **qubit**-**oscillator** system. The minimum of energy splitting of the **qubit** Δ is reached at the symmetry point when one fluxoid is trapped in the gradiometer loop and the difference in magnetic fluxes f ϵ Φ 0 is 0 controlled by I ϵ and I ϵ , d c . By controlling the flux f α Φ 0 with I α and uniform field B one can tune Δ in resonance with **oscillator** **frequency** ν o s c . ... A flux **qubit** biased at its symmetry point shows a minimum in the energy splitting (the gap), providing protection against flux noise. We have fabricated a **qubit** whose gap can be tuned fast and have coupled this **qubit** strongly to an LC **oscillator**. We show full spectroscopy of the **qubit**-resonator system and generate vacuum Rabi **oscillations**. When the gap is made equal to the **oscillator** **frequency** $\nu_{osc}$ we find the strongest **qubit**-resonator coupling ($g/h\sim0.1\nu_{\rm osc}$). Here being at resonance coincides with the optimal coherence of the symmetry point. Significant further increase of the coupling is possible.

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Contributors: Altomare, Fabio, Cicak, Katarina, Sillanpää, Mika A., Allman, Michael S., Sirois, Adam J., Li, Dale, Park, Jae I., Strong, Joshua A., Teufel, John D., Whittaker, Jed D.

Date: 2010-02-02

(Color) Simulated energy for (a) the first **qubit**, (b) the CPW cavity, (c) the second **qubit**. (Red): ϕ 1 = 0.8949 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 . The first **qubit** decays exponentially up to t ≈ 123 ~ n s . At this time the **frequency** of the **oscillation** in right well matches the CPW cavity resonant **frequency** and the **qubit** transfers part of its energy to the CPW cavity. The second **qubit** is resonating at a different **frequency** and it is minimally exited by the incoming microwave voltage. This corresponds to the red x of Fig. 2(b) (Black): ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 . In this case the first **qubit** transfers part of its energy to the CPW cavity at t ≈ 103 ~ n s because it starts at a lower energy in the deep well. At this flux the second **qubit** is in resonance with the cavity and it is excited up to the sixth quantized level. This corresponds to the white x of Fig. 2(b) fig:singlelinecut...As can be seen from Eq. eq:eqofmotion, the treatment of the **qubits** and the CPW cavity is fully classical. The initial conditions for the solution of this system of differential equations are described below. The second **qubit** and the oscillator begin with zero kinetic energy ( ϕ 2 ̇ = ϕ r ̇ = 0 ) and have zero potential energy; zero energy is defined at the bottom of the left well. To understand the initial conditions for the first **qubit** **it **is useful to recall the physics of the measurement. When the MP is applied, the flux approaches the critical value (approximately 0.95 ϕ c ) over a short period of time, so that the first excited state tunnels out with unit probability. Once tunneled, this **qubit** can be assumed to have zero kinetic energy ( ϕ 1 ̇ = 0 ), to have a phase value just to the right side of the residual local maximum between the two wells (Fig. fig:QBpotential(b)), giving **it **an initial potential energy ∼ 0.2 Δ U ϕ e below the local maximum value. In addition, we assume that the decay rate in the right well is comparable to that in the left well, and the simulation is run for times ∼ 3 T 1 1 , after which the **qubit** phase has relaxed to rest. We have checked that small variations in these assumptions do not meaningfully affect the results of our simulations....Measurement crosstalk between two phase **qubits** coupled by a coplanar waveguide...For our experiment, we initially determine the optimal ’simultaneous’ timing between the two MPs that takes into account the different cabling and instrumental delays from the room-temperature equipment to the cold devices. Then, as a function of the flux applied to the two **qubits**, we measure the tunneling probability for the second (first) **qubit** after we purposely induce a tunneling event in the first (second) **qubit**. The results are shown in Fig. fig:experiment(a,c). The probability of finding the second (first) **qubit** in the excited state as a result of measurement crosstalk is significant only in a region around ϕ 2 / ϕ c 2 = Φ 2 / Φ c 2 ≈ Φ ¯ 2 / Φ c 2 ∼ 0.842 ( ϕ 1 / ϕ c 1 = Φ 1 / Φ c 1 ≈ Φ ¯ 1 / Φ c 1 ∼ 0.82 ) where the resonant **frequency** of the second (first) **qubit** is close to the CPW cavity **frequency**....AltomareX2009, the resonant **frequency** of both **qubits** exhibits an avoided crossing at the CPW cavity **frequency** ( ≈ 8.9 GHz). For the first **qubit** this happens at a flux Φ ¯ 1 = 0.82 Φ c 1 , and for the second at a flux Φ ¯ 2 = 0.842 Φ c 2 . For each **qubit**, Φ c i is the critical flux at which the left well of Fig. fig:QBpotential(b) disappears....(Color) Measurement crosstalk: (a) Experimental tunneling probability for **qubit** 2, after **qubit** 1 has already tunneled as function of the (dimensionless) flux applied to the **qubits**. The left ordinate displays the resonant **frequency** as measured from the **qubit** spectroscopy. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 2. (b) Simulation: ratio between the maximum energy acquired by the second **qubit** and the resonant **frequency** in the left well ( N l ) as a function of the flux applied to the **qubits**. The left ordinate displays the **oscillation** **frequency** as determined from the Fast Fourier Transform of the energy of **qubit** 2. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 1. Temporal traces corresponding to the two x’s are displayed in Fig. fig:singlelinecut. (c-d) Same as (a-b) after reversing the roles of the two **qubits**. fig:experiment...(Color) Measurement crosstalk: (a) Experimental tunneling probability for **qubit** 2, after **qubit** 1 has already tunneled as function of the (dimensionless) flux applied to the **qubits**. The left ordinate displays the resonant **frequency** as measured from the **qubit** spectroscopy. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 2. (b) Simulation: ratio between the maximum energy acquired by the second **qubit** and the resonant **frequency** in the left well ( N l ) as a function of the flux applied to the **qubits**. The left ordinate displays the oscillation **frequency** as determined from the Fast Fourier Transform of the energy of **qubit** 2. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 1. Temporal traces corresponding to the two x’s are displayed in Fig. fig:singlelinecut. (c-d) Same as (a-b) after reversing the roles of the two **qubits**. fig:experiment...(a) Equivalent electrical circuit for two flux-biased phase **qubits** coupled to a CPW cavity (modelled as a lumped element harmonic **oscillator**). C i is the total i - **qubit** (or CPW cavity) capacitance, L i the geometrical inductance, L j , i the Josephson inductance of the JJ, R i models the dissipation in the system. (b) U ϕ ϕ e is the potential energy of the phase **qubit** as function of superconducting phase difference ϕ across the JJ and the dimensionless external flux bias ϕ e = Φ 2 π / Φ 0 . Δ U ϕ e is the difference between the local potential maximum and the local potential minimum in the left well at the flux bias ϕ e . (c) During the MP, the potential barrier Δ U ϕ e between the two wells is lowered for a few nanoseconds allowing the | 1 state to tunnel into the right well where it will (classically) **oscillate** and lose energy due to the dissipation. fig:QBpotential...From these initial conditions the phase of the first **qubit**(classically) undergoes damped oscillations in the anharmonic right well. Because of the anharmonicity of the potential, when the amplitude of the oscillation is large, the **frequency** of the oscillations is lower than the unmeasured **qubit** **frequency**. As the system loses energy due to the damping, the oscillation **frequency** increases as seen by the CPW cavity. When the crosstalk voltage has a **frequency** close to the CPW cavity **frequency**, **it **can transfer energy to the CPW cavity. If the second **qubit**’s **frequency** matches that of the CPW cavity then the cavity’s excitation can be transferred to the second **qubit**. In Fig. fig:experiment(b) we plot**, for **the second **qubit**, the ratio ( N l ) between the maximum energy acquired and ℏ ω p , where ω **p **is the plasma **frequency** of the **qubit** in the left well, as a function of the fluxes in the two **qubits**. The crosstalk...At ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 (white x in Fig. fig:experiment(b)), the dynamics of the first **qubit** and the CPW cavity are essentially unchanged, except that the CPW cavity **frequency** is matched at a different time ( t = 103 ~ n s ) because the first **qubit** starts at a lower energy in the deep well (Fig. fig:singlelinecut (a-c)-Black). However, in this case, the second **qubit** is on resonance with the CPW cavity and is therefore excited to an energy N l ∼ 6 ....We analyze the measurement crosstalk between two flux-biased phase **qubits** coupled by a resonant coplanar waveguide cavity. After the first **qubit** is measured, the superconducting phase can undergo damped **oscillations** resulting in an a.c. voltage that produces a **frequency** chirped noise signal whose **frequency** crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second **qubit** when its **frequency** is far from the cavity's resonant **frequency**. We present a simple classical description of the **qubit** behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase **qubits** can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements....(Color) Simulated energy for (a) the **first** **qubit**, (b) the CPW cavity, (c) the **second** **qubit**. (Red): ϕ 1 = 0.8949 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 . The **first** **qubit** decays exponentially up to t ≈ 123 ~ n s . At this time the **frequency** of the oscillation in right well matches the CPW cavity resonant **frequency** and the **qubit** transfers part of its energy to the CPW cavity. The **second** **qubit** is resonating at a different **frequency** and it is minimally exited by the incoming microwave voltage. This corresponds to the red x of Fig. 2(b) (Black): ϕ 1 = 0.82 ϕ c 1 and ϕ 2 = 0.836 ϕ c 2 . In this case the **first** **qubit** transfers part of its energy to the CPW cavity at t ≈ 103 ~ n s because it starts at a lower energy in the deep well. At this flux the **second** **qubit** is in resonance with the cavity and it is excited up to the sixth quantized level. This corresponds to the white x of Fig. 2(b) fig:singlelinecut...We analyze the measurement crosstalk between two flux-biased phase **qubits** coupled by a resonant coplanar waveguide cavity. After the first **qubit** is measured, the superconducting phase can undergo damped oscillations resulting in an a.c. voltage that produces a **frequency** chirped noise signal whose **frequency** crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second **qubit** when its **frequency** is far from the cavity's resonant **frequency**. We present a simple classical description of the **qubit** behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase **qubits** can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements....From these initial conditions the phase of the first **qubit**(classically) undergoes damped oscillations in the anharmonic right well. Because of the anharmonicity of the potential, when the amplitude of the oscillation is large, the **frequency** of the oscillations is lower than the unmeasured **qubit** **frequency**. As the system loses energy due to the damping, the oscillation **frequency** increases as seen by the CPW cavity. When the crosstalk voltage has a **frequency** close to the CPW cavity **frequency**, **it **can transfer energy to the CPW cavity. If the second **qubit**’s **frequency** matches that of the CPW cavity then the cavity’s excitation can be transferred to the second **qubit**. In Fig. fig:experiment(b) we plot**, for **the second **qubit**, the ratio ( N l ) between the maximum energy acquired and ℏ ω p , where ω **p **is the plasma **frequency** of the **qubit** in the left well, as a function of the fluxes in the two **qubits**. The crosstalk, measured as the maximum energy transferred to the second **qubit**, is maximum at a flux ϕ 2 / ϕ c 2 ∼ 0.837 , where the second **qubit**’s **frequency** is ≈ 8.97 ~ G H z , determined by taking the Fast Fourier Transform of the oscillations in energy over time (see Fig. fig:singlelinecut (a-c)). Reversing the roles of the two **qubits**, we find that for the first **qubit** the crosstalk is maximum at a flux ∼ 0.825 ϕ c 1 , corresponding to an excitation **frequency** of ≈ 8.84 ~ G H z (Fig. fig:experiment(d)). These values were determined for **qubit** 2 (**qubit** 1) by performing a Gaussian fit of N l versus flux (or **frequency**) after averaging over the span of flux (or **frequency**) values for **qubit** 1 (**qubit** 2). Notice that the crosstalk transferred to **qubit** 2 (**qubit** 1) is flux independent of **qubit** 1 (**qubit** 2) and substantial only when the cavity **frequency** matches the **frequency** of **qubit** 2 (**qubit** 1). The results of the simulations are in good agreement with the experimental data. To gain additional insight into the dynamics of the system, we plot the time evolution of the energy for the **qubits** and the CPW cavity (Fig. fig:singlelinecut (a-c)) for two different sets of fluxes in the two **qubits**. At ϕ 1 = 0.895 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 (red x in Fig. fig:experiment(b)) the first **qubit** decays exponentially for a time **t **123 ~ **n **s (Fig. fig:singlelinecut (a-c)-Red). At t = 123 ~ **n **s there is a downward jump in the energy of the first **qubit** while the energy of the CPW cavity exhibits an upward jump. At this time, the **frequency** of oscillation in the right well matches the CPW cavity resonant **frequency**, so part of the **qubit** energy is transferred to the CPW cavity. However, since the second **qubit** is not on resonance with the CPW cavity, **it **does not get significantly excited by the microwave current passing through the capacitor C x ....(a) Equivalent electrical circuit for **two** flux-biased phase **qubits** coupled to a CPW cavity (modelled as a lumped element harmonic **oscillator**). C i is the total i - **qubit** (or CPW cavity) capacitance, L i the geometrical inductance, L j , i the Josephson inductance of the JJ, R i models the dissipation in the system. (b) U ϕ ϕ e is the potential energy of the phase **qubit** as function of superconducting phase difference ϕ across the JJ and the dimensionless external flux bias ϕ e = Φ 2 π / Φ 0 . Δ U ϕ e is the difference between the local potential maximum and the local potential minimum in the left well at the flux bias ϕ e . (c) During the MP, the potential barrier Δ U ϕ e between the **two** wells is lowered for a few nanoseconds allowing the | 1 state to tunnel into the right well where it will (classically) oscillate and lose energy due to the dissipation. fig:QBpotential...From these initial conditions the phase of the first **qubit**(classically) undergoes damped **oscillations** in the anharmonic right well. Because of the anharmonicity of the potential, when the amplitude of the **oscillation** is large, the **frequency** of the **oscillations** is lower than the unmeasured **qubit** **frequency**. As the system loses energy due to the damping, the **oscillation** **frequency** increases as seen by the CPW cavity. When the crosstalk voltage has a **frequency** close to the CPW cavity **frequency**, it can transfer energy to the CPW cavity. If the second **qubit**’s**frequency** matches that of the CPW cavity then the cavity’s excitation can be transferred to the second **qubit**. In Fig. fig:experiment(b) we plot, for the second **qubit**, the ratio ( N l ) between the maximum energy acquired and ℏ ω p , where ω p is the plasma **frequency** of the **qubit** in the left well, as a function of the fluxes in the two **qubits**. The crosstalk, measured as the maximum energy transferred to the second **qubit**, is maximum at a flux ϕ 2 / ϕ c 2 ∼ 0.837 , where the second **qubit**’s**frequency** is ≈ 8.97 ~ G H z , determined by taking the Fast Fourier Transform of the **oscillations** in energy over time (see Fig. fig:singlelinecut (a-c)). Reversing the roles of the two **qubits**, we find that for the first **qubit** the crosstalk is maximum at a flux ∼ 0.825 ϕ c 1 , corresponding to an excitation **frequency** of ≈ 8.84 ~ G H z (Fig. fig:experiment(d)). These values were determined for **qubit** 2 (**qubit** 1) by performing a Gaussian fit of N l versus flux (or **frequency**) after averaging over the span of flux (or **frequency**) values for **qubit** 1 (**qubit** 2). Notice that the crosstalk transferred to **qubit** 2 (**qubit** 1) is flux independent of **qubit** 1 (**qubit** 2) and substantial only when the cavity **frequency** matches the **frequency** of **qubit** 2 (**qubit** 1). The results of the simulations are in good agreement with the experimental data. To gain additional insight into the dynamics of the system, we plot the time evolution of the energy for the **qubits** and the CPW cavity (Fig. fig:singlelinecut (a-c)) for two different sets of fluxes in the two **qubits**. At ϕ 1 = 0.895 ϕ c 1 and ϕ 2 = 0.893 ϕ c 2 (red x in Fig. fig:experiment(b)) the first **qubit** decays exponentially for a time t 123 ~ n s (Fig. fig:singlelinecut (a-c)-Red). At t = 123 ~ n s there is a downward jump in the energy of the first **qubit** while the energy of the CPW cavity exhibits an upward jump. At this time, the **frequency** of **oscillation** in the right well matches the CPW cavity resonant **frequency**, so part of the **qubit** energy is transferred to the CPW cavity. However, since the second **qubit** is not on resonance with the CPW cavity, it does not get significantly excited by the microwave current passing through the capacitor C x ....(Color) Measurement crosstalk: (a) Experimental tunneling probability for **qubit** 2, after **qubit** 1 has already tunneled as function of the (dimensionless) flux applied to the **qubits**. The left ordinate displays the resonant **frequency** as measured from the **qubit** spectroscopy. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 2. (b) Simulation: ratio between the maximum energy acquired by the **second** **qubit** and the resonant **frequency** in the left well ( N l ) as a function of the flux applied to the **qubits**. The left ordinate displays the oscillation **frequency** as determined from the Fast Fourier Transform of the energy of **qubit** 2. The right ordinate displays the ratio between the applied flux and the critical flux for **qubit** 1. Temporal traces corresponding to the **two** x’s are displayed in Fig. fig:singlelinecut. (c-d) Same as (a-b) after reversing the roles of the **two** **qubits**. fig:experiment ... We analyze the measurement crosstalk between two flux-biased phase **qubits** coupled by a resonant coplanar waveguide cavity. After the first **qubit** is measured, the superconducting phase can undergo damped **oscillations** resulting in an a.c. voltage that produces a **frequency** chirped noise signal whose **frequency** crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second **qubit** when its **frequency** is far from the cavity's resonant **frequency**. We present a simple classical description of the **qubit** behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase **qubits** can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements.

Files:

Contributors: Serban, I., Dykman, M. I., Wilhelm, F. K.

Date: 2009-07-29

An important feature of **the** **qubit** relaxation **in**** the **presence of driving

**is**that

**stationary distribution over**

**the****the**

**qubit**states differs from

**thermal Boltzmann distribution. If**

**the**

**the****oscillator**-mediated decay is

**dominating**

**the****qubit**decay mechanism,

**the**

**qubit**distribution

**is**determined by

**ratio of**

**the****transition rates Γ e and Γ g . One can characterize it**

**the****by**effective temperature T e f f = ℏ ω q / k B ln Γ e / Γ g . If

**term**

**the****in**curly brackets

**in**

**numerator of Eq. ( eq:resonant_power_spectrum)**

**the****is**dominating, T e f f ≈ 2 T , but if

**field parameters are varied so that this term becomes comparatively smaller T e f f increases, diverges, and then becomes negative, approaching -2 T . Negative effective temperature corresponds**

**the****to**population inversion. The evolution of

**effective temperature with**

**the****intensity of**

**the****modulating field**

**the****is**illustrated

**in**Fig. fig:effect_temp....An important feature of the

**qubit**relaxation in the presence of driving is that the stationary distribution over the

**qubit**states differs from the thermal Boltzmann distribution. If the

**oscillator**-mediated decay is the dominating

**qubit**decay mechanism, the

**qubit**distribution is determined by the ratio of the transition rates Γ e and Γ g . One can characterize it by effective temperature T e f f = ℏ ω q / k B ln Γ e / Γ g . If the term in curly brackets in the numerator of Eq. ( eq:resonant_power_spectrum) is dominating, T e f f ≈ 2 T , but if the field parameters are varied so that this term becomes comparatively smaller T e f f increases, diverges, and then becomes negative, approaching -2 T . Negative effective temperature corresponds to population inversion. The evolution of the effective temperature with the intensity of the modulating field is illustrated in Fig. fig:effect_temp....Apart from

**proportionality**

**the****to**r a 2 ,

**the**attractor dependence of Γ 1

**is**also due

**to**

**different curvature of**

**the****effective potentials around**

**the****attractors. For weak**

**the****oscillator**damping κ ≪ ν a ,

**the**parameter ν a

**in**Eq. ( eq:resonant_power_spectrum) is

**the**

**frequency**of small-amplitude vibrations about attractor a . It sets

**spacing between**

**the****quasienergy levels,**

**the****the**eigenvalues of

**rotating frame Hamiltonian H S r close**

**the****to**

**attractor. The function Re N + - ω has sharp Lorentzian peaks at ω = ± ν a with halfwidth κ determined by**

**the**

**the****oscillator**decay rate. The dependence of ν a on

**control parameter β**

**the****is**illustrated

**in**Fig. fig:nu_a....Left panel: Squared scaled attractor radii r a 2 as functions of the dimensionless field intensity β for the dimensionless friction κ / | δ ω | = 0.3 . Right panel: The effective frequencies ν a / | δ ω | for the same κ / | δ ω . Curves 1 and 2 refer to small- and large amplitude attractors....Left panel: Squared scaled attractor radii r a 2 as functions of the dimensionless field intensity β for the dimensionless friction κ / | δ ω | = 0.3 . Right panel: The effective

**frequencies**ν a / | δ ω | for the same κ / | δ ω . Curves 1 and 2 refer to small- and large amplitude attractors....The scaled decay rate factors for the excited and ground states, curves 1 and 2, respectively, as functions of scaled difference between the

**qubit**

**frequency**and twice the modulation

**frequency**; Γ 0 = ℏ C Γ r a 2 / 6 γ S . Left and right panels refer to the small- and large-amplitude attractors, with the values of β being 0.14 and 0.12, respectively. Other parameters are κ / | δ ω = 0.3 , n ̄ = 0.5 ....The decay rate of

**excited state of**

**the****the**

**qubit**Γ e ∝ R e ~ N + - ω q - 2 ω

**F | w**sharply increases if

**the**

**qubit**

**frequency**ω q coincides with 2 ω

**F | w**± ν a , i.e., ω q - 2 ω

**F | w**resonates with

**inter-quasienergy level transition**

**the****frequency**. This new

**frequency**scale results from

**interplay of**

**the****system nonlinearity and**

**the****driving and**

**the****is**attractor-specific, as seen

**in**Fig. fig:nu_a. In

**experiment, for ω q close**

**the****to**2 ω 0 ,

**the**resonance can be achieved

**by**tuning

**driving**

**the****frequency**ω

**F | w**and/or driving amplitude

**F | w**0 . This quasienergy resonance destroys

**QND character of**

**the****measurement**

**the****by**inducing fast relaxation....The scaled decay rates Γ e , g as functions of detuning ω q - 2 ω

**F | w**are illustrated

**in**Fig. fig:decay_spectra. Even for comparatively strong damping,

**the**spectra display well-resolved quasi-energy resonances, particularly

**in**

**case of**

**the****large-amplitude attractor. As**

**the**

**the****oscillator**approaches bifurcation points

**where**

**corresponding attractor disappears,**

**the****the**frequencies ν a become small (cf. Fig. fig:nu_a) and

**peaks**

**the****in**

**the**

**frequency**dependence of Γ e , g move

**to**ω q = 2 ω

**F | w**and become very narrow, with width that scales as

**square root of**

**the****distance**

**the****to**

**bifurcation point. We note that**

**the****theory does not apply for very small ω q - 2 ω**

**the****F | w**|

**where**

**the**

**qubit**

**is**resonantly pumped;

**the**corresponding condition

**is**m ω 0 Δ q δ C r e s 2 r a 2 / ℏ ω q 2 ≪ T 1 T 2 -1 + ω q - 2 ω

**F | w**2 T 1 T 2 . For weak coupling

**to**

**the**

**qubit**, Γ e ≪ κ , it can be satisfied even at resonance....The scaled decay rates Γ e , g as functions of detuning ω q - 2 ω F are illustrated in Fig. fig:decay_spectra. Even for comparatively strong damping, the spectra display well-resolved quasi-energy resonances, particularly in the case of the large-amplitude attractor. As the

**oscillator**approaches bifurcation points where the corresponding attractor disappears, the

**frequencies**ν a become small (cf. Fig. fig:nu_a) and the peaks in the

**frequency**dependence of Γ e , g move to ω q = 2 ω F and become very narrow, with width that scales as the square root of the distance to the bifurcation point. We note that the theory does not apply for very small ω q - 2 ω F | where the

**qubit**is resonantly pumped; the corresponding condition is m ω 0 Δ q δ C r e s 2 r a 2 / ℏ ω q 2 ≪ T 1 T 2 -1 + ω q - 2 ω F 2 T 1 T 2 . For weak coupling to the

**qubit**, Γ e ≪ κ , it can be satisfied even at resonance....Relaxation of a

**qubit**measured by a driven Duffing

**oscillator**...The decay rate of the excited state of the

**qubit**Γ e ∝ R e ~ N + - ω q - 2 ω F sharply increases if the

**qubit**

**frequency**ω q coincides with 2 ω F ± ν a , i.e., ω q - 2 ω F resonates with the inter-quasienergy level transition

**frequency**. This new

**frequency**scale results from the interplay of the system nonlinearity and the driving and is attractor-specific, as seen in Fig. fig:nu_a. In the experiment, for ω q close to 2 ω 0 , the resonance can be achieved by tuning the driving

**frequency**ω F and/or driving amplitude F 0 . This quasienergy resonance destroys the QND character of the measurement by inducing fast relaxation....Apart from the proportionality to r a 2 , the attractor dependence of Γ 1 is also due to the different curvature of the effective potentials around the attractors. For weak

**oscillator**damping κ ≪ ν a , the parameter ν a in Eq. ( eq:resonant_power_spectrum) is the

**frequency**of small-amplitude vibrations about attractor a . It sets the spacing between the quasienergy levels, the eigenvalues of the rotating frame Hamiltonian H S r close to the attractor. The function Re N + - ω has sharp Lorentzian peaks at ω = ± ν a with halfwidth κ determined by the

**oscillator**decay rate. The dependence of ν a on the control parameter β is illustrated in Fig. fig:nu_a....We investigate the relaxation of a superconducting

**qubit**for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The

**qubit**relaxation rates are different in different states. They can display strong dependence on the

**qubit**

**frequency**and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven

**oscillator**changes the effective temperature of the

**qubit**....Picot08, and was increasing with the driving strength on the low-amplitude branch (branch 1 in the left panel of Fig. fig:nu_a), in qualitative agreement with the theory. It is not possible to make a direct quantitative comparison because of an uncertainty in the

**qubit**relaxation rates noted in Ref. ...The effective scaled

**qubit**temperature T e f f * = k B T e f f / ℏ ω q as function of the scaled field strength β in the region of bistability for the small- and large-amplitude attractors, left and right panels, respectively; ω q - 2 ω F / | δ ω | = - 0.2 and 0.1 in the left and right panels; other parameters are the same as in Fig. fig:decay_spectra. ... We investigate the relaxation of a superconducting

**qubit**for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The

**qubit**relaxation rates are different in different states. They can display strong dependence on the

**qubit**

**frequency**and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven

**oscillator**changes the effective temperature of the

**qubit**.

Files:

Contributors: Zorin, A. B., Chiarello, F.

Date: 2009-08-27

Figure f-shift shows this relative **frequency** shift versus parameter β L . One can see that for** the **rather conservative value of dimensionless coupling κ = 0.05 , the relative **frequency** shift can achieve** the **easily measured values of about 10 % . The efficiency of** the **dispersive readout can be improved in** the **non-linear regime with bifurcation . With our device this regime can be achieved in** the **resonance circuit including, for example, a Josephson junction (marked in** the **diagram in Fig. 1 by a dashed cross). Due to** the **high sensitivity of** the **amplitude (phase) bifurcation to** the **threshold determined by** the **effective resonance **frequency** of** the **circuit, one can expect a readout with high fidelity even at a rather weak coupling of the **qubit** and** the **resonator (compare with** the **readout of quantronium in Ref. ). Further improvement of** the **readout can be achieved in** the **QED-based circuit including this **qubit** ....(Color online) Position of the lowest six levels (solid lines) in the potential Eq. ( U-phi) for φ e = π as a function of parameter β L for typical values of L and C , yielding E J / E c ∼ E L / E c ≈ 5.1 × 10 4 . With an increase of β L , the spectrum crosses over from that of the harmonic oscillator type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the oscillator-type states in two separate wells. The spectrum in the central region β L ≈ 1 is strongly anharmonic. The dashed line shows the bottom energy of the potential U φ φ e = π , which in the case of β L > 1 is equal to - Δ U ≈ - 1.5 E L β L - 1 2 / β L (in other words, Δ U is the height of the energy barrier in the right inset) . The dotted (zero-level) line indicates the energy in the symmetry point φ = 0 , i.e. at the bottom of the single well ( β L ≤ 1 ) or at the top of the energy barrier ( β L > 1 ). The black dot shows the critical value β L c at which the ground state energy level touches the top of the barrier separating the two wells....(Color online) Position of the lowest six levels (solid lines) in the potential Eq. ( U-phi) for φ e = π as a function of parameter β L for typical values of L and C , yielding E J / E c ∼ E L / E c ≈ 5.1 × 10 4 . With an increase of β L , the spectrum crosses over from that of the harmonic **oscillator** type (left inset) to the set of the doublets (right inset), corresponding to the weak coupling of the **oscillator**-type states in two separate wells. The spectrum in the central region β L ≈ 1 is strongly anharmonic. The dashed line shows the bottom energy of the potential U φ φ e = π , which in the case of β L > 1 is equal to - Δ U ≈ - 1.5 E L β L - 1 2 / β L (in other words, Δ U is the height of the energy barrier in the right inset) . The dotted (zero-level) line indicates the energy in the symmetry point φ = 0 , i.e. at the bottom of the single well ( β L ≤ 1 ) or at the top of the energy barrier ( β L > 1 ). The black dot shows the critical value β L c at which the ground state energy level touches the top of the barrier separating the two wells....We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling....(a) Electric diagram of the **qubit** coupled to a resonant circuit and (b) possible equivalent compound (two-junction SQUID) circuit of the Josephson element included into the **qubit** loop. Capacitance C includes both the self-capacitance of the junction and the external capacitance. Due to inclusion in the resonant circuit of a Josephson junction JJ’, the resonator may operate in the nonlinear regime, enabling a bifurcation-based readout....(Color online) The resonance **frequency** shift in the circuit due to excitation of the **qubit** with the inductance value L = 50 pH and the set of capacitances C , decreasing from top to bottom. The dimensionless coupling coefficient κ = 0.05 ....Such a large, positive anharmonicity is a great advantage of** the **quartic potential **qubit** allowing manipulation within** the **two basis **qubit** states | 0 and | 1 not only when applying resonant microwave field, ν μ w ≈ ν 10 , but also when applying control microwave signals with large **frequency** detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic **qubit** **frequency** ν 10 = Δ E 0 / h and** the **anharmonicity factor δ computed from** the **Schrödinger equation for** the **original potential Eq. ( U-phi) in** the **range 0.9 ≤ β L ≤ 1.02 are shown in Fig. **frequency**-anharmonicity. One can see that** the **significant range in** the **tuning of the **qubit** **frequency** within** the **range of sufficiently large anharmonicity ( ∼ 20 - 50 % ) is attained at a rather fine (typically ± 1 - 2 % ) tuning of β L around** the **value β L = 1 . Such tuning of β L is possible in** the **circuit having** the **compound configuration shown in Fig. 1b. For values of β L > 1 , the symmetric energy potential has two minima and a barrier between them. The position of** the **ground state level depends on β L and** the **ratio of** the **characteristic energies E J / E c = β L E L / E c . The value of β L at which** the **ground state level touches** the **top of** the **barrier sets** the **upper limit β L c for** the **quartic **qubit** (marked in Fig. levels-beta by solid dot). At β L > β L c , the **qubit** energy dramatically decreases and the **qubit** states are nearly** the **symmetric and antisymmetric combinations of** the **states inside** the **two wells (see** the **right inset in Fig. 2). Although the **qubit** with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses , its readout can hardly be accomplished in a dispersive fashion....where Î is** the **operator of supercurrent circulating in the **qubit** loop. The dependence of** the **reverse inductance L J Φ e = Φ 0 / 2 n calculated numerically in** the **two lowest quantum states ( n = 0 and 1) for L = 50 pH and** the **same set of capacitances C as in Fig. **frequency**-anharmonicity is shown in Fig. inductance-L01. One can see that** the **ratio of** the **geometrical to Josephson inductances L / L J takes large and very different values that can be favorably used for** the **dispersive readout, ensuring a sufficiently large output signal. Note that for β L 1** the **inductance L J n = 1 changes** the **sign to positive....(Color online) (a) The **qubit** **frequency** as a function of parameter β L for fixed L = 50 pH and several values of capacitance C = 0.1 , 0.3, 1.0 and 3.0 pF (from top to bottom), corresponding to the values of the ratio E L / E c ≈ 1.7 × 10 4 , 5.1 × 10 4 , 1.7 × 10 5 and 5.1 × 10 5 . (b) Anharmonicity parameter δ as a function of parameter β L for the same as in (a) inductance L and capacitance values (from top to bottom)....(Color online) The resonance frequency shift in the circuit due to excitation of the **qubit** with the inductance value L = 50 pH and the set of capacitances C , decreasing from top to bottom. The dimensionless coupling coefficient κ = 0.05 ....where Î is the operator of supercurrent circulating in the **qubit** loop. The dependence of the reverse inductance L J Φ e = Φ 0 / 2 n calculated numerically in the two lowest quantum states ( n = 0 and 1) for L = 50 pH and the same set of capacitances C as in Fig. **frequency**-anharmonicity is shown in Fig. inductance-L01. One can see that the ratio of the geometrical to Josephson inductances L / L J takes large and very different values that can be favorably used for the dispersive readout, ensuring a sufficiently large output signal. Note that for β L 1 the inductance L J n = 1 changes the sign to positive....Figure f-shift shows this relative **frequency** shift versus parameter β L . One can see that for the rather conservative value of dimensionless coupling κ = 0.05 , the relative **frequency** shift can achieve the easily measured values of about 10 % . The efficiency of the dispersive readout can be improved in the non-linear regime with bifurcation . With our device this regime can be achieved in the resonance circuit including, for example, a Josephson junction (marked in the diagram in Fig. 1 by a dashed cross). Due to the high sensitivity of the amplitude (phase) bifurcation to the threshold determined by the effective resonance **frequency** of the circuit, one can expect a readout with high fidelity even at a rather weak coupling of the **qubit** and the resonator (compare with the readout of quantronium in Ref. ). Further improvement of the readout can be achieved in the QED-based circuit including this **qubit** ....Such a large, positive anharmonicity is a great advantage of the quartic potential **qubit** allowing manipulation within the two basis **qubit** states | 0 and | 1 not only when applying resonant microwave field, ν μ w ≈ ν 10 , but also when applying control microwave signals with large **frequency** detuning or using rather wide-spectrum rectangular-pulse control signals. The characteristic **qubit** **frequency** ν 10 = Δ E 0 / h and the anharmonicity factor δ computed from the Schrödinger equation for the original potential Eq. ( U-phi) in the range 0.9 ≤ β L ≤ 1.02 are shown in Fig. **frequency**-anharmonicity. One can see that the significant range in the tuning of the **qubit** **frequency** within the range of sufficiently large anharmonicity ( ∼ 20 - 50 % ) is attained at a rather fine (typically ± 1 - 2 % ) tuning of β L around the value β L = 1 . Such tuning of β L is possible in the circuit having the compound configuration shown in Fig. 1b. For values of β L > 1 , the symmetric energy potential has two minima and a barrier between them. The position of the ground state level depends on β L and the ratio of the characteristic energies E J / E c = β L E L / E c . The value of β L at which the ground state level touches the top of the barrier sets the upper limit β L c for the quartic **qubit** (marked in Fig. levels-beta by solid dot). At β L > β L c , the **qubit** energy dramatically decreases and the **qubit** states are nearly the symmetric and antisymmetric combinations of the states inside the two wells (see the right inset in Fig. 2). Although the **qubit** with such parameters has very large anharmonicity and can be nicely controlled by dc flux pulses , its readout can hardly be accomplished in a dispersive fashion....(Color online) (a) The **qubit** frequency as a function of parameter β L for fixed L = 50 pH and several values of capacitance C = 0.1 , 0.3, 1.0 and 3.0 pF (from top to bottom), corresponding to the values of the ratio E L / E c ≈ 1.7 × 10 4 , 5.1 × 10 4 , 1.7 × 10 5 and 5.1 × 10 5 . (b) Anharmonicity parameter δ as a function of parameter β L for the same as in (a) inductance L and capacitance values (from top to bottom)....(Color online) The values of the Josephson inductance of the quartic potential **qubit** in the ground (solid lines) and excited (dashed lines) states calculated for the geometric inductance value L = 50 pH and the set of capacitances C , increasing from top to bottom for both groups of curves....Superconducting phase **qubit** based on the Josephson **oscillator** with strong anharmonicity ... We propose a superconducting phase **qubit** on the basis of the radio-**frequency** SQUID with the screening parameter value $\beta_L = (2\pi/\Phi_0)LI_c \approx 1$, biased by a half flux quantum $\Phi_e=\Phi_0/2$. Significant anharmonicity ($> 30%$) can be achieved in this system due to the interplay of the cosine Josephson potential and the parabolic magnetic-energy potential that ultimately leads to the quartic polynomial shape of the well. The two lowest eigenstates in this global minimum perfectly suit for the **qubit** which is insensitive to the charge variable, biased in the optimal point and allows an efficient dispersive readout. Moreover, the transition **frequency** in this **qubit** can be tuned within an appreciable range allowing variable **qubit**-**qubit** coupling.

Files:

Contributors: Thrailkill, Z. E., Lambert, J. G., Ramos, R. C.

Date: 2009-09-20

The result of the simulation is shown in Fig. fig4 where the population in R2 is plotted versus the detuning Δ Q 2 , Q 1 , of Q2 from Q1, in the range of -2000 MHz to 2000 MHz. As per design, at zero detuning, the excitation is transferred after a time of 16 ns. This time can be chosen based on the desired time scale by adjusting the** qubit**-

**qubit**and

**-resonator coupling strengths. At large detuning, i.e., beyond ± 1.5 GHz, the coupling between Q1 and R2 becomes dispersive up to about 30 ns, showing that dispersive coupling can be weak enough to isolate the active**

**qubit****. Q2 can be detuned further to reduce the dispersive coupling if the desired time scale is longer, e.g., to perform operations on Q1. Thus, this simulation shows that a control**

**qubit****can be effectively used to turn coupling on and off between a**

**qubit****and an array of resonators....Multiple Resonators as a Multi-Channel Bus for Coupling Josephson Junction**

**qubit****Qubits**...figure3 (a) Placing a control

**qubit**(Q2) between the active

**qubit**(Q1) and the resonator array (R1, R2, R3) allows coupling to be turned on and off. (b) The system is initialized with an excitation in Q1, Q1 is in-resonance with R2, and the

**frequencies**of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). Each vertical cut represents the population in R2 over time at a particular detuning Δ Q 2 , Q 1 , of Q2 from Q1. At zero detuning, the excitation readily

**oscillates**in and out of the resonator R2. As Q2 is detuned further away, the coupling between Q1 and R2 becomes weaker, resulting in slower

**oscillations**of the excitation. The detuning of the control

**qubit**Q2 can be chosen based on the desired time scale, e.g. the time required to manipulate Q1....We consider an array of resonators used as a memory register. To accomplish this, one must be able to transfer an excitation from a

**to a specific resonator, without coupling to the other resonators in the array. This is implemented by designing resonators that are sufficiently detuned from each other. To determine the amount of detuning required to avoid crosstalk between resonators, we examine a single**

**qubit****coupled to an array of two resonators as shown in Fig. block1. The**

**qubit****(Q) and resonator 1 (R1) are fixed at the same frequency while resonator 2 (R2) is detuned. All**

**qubit****-resonator couplings are presumed to be identical with a strength of g i j = 110 M H z for all i , j , i.e., all coupling capacitances C c are equal. This value is typical in experiments such as in Ref. . We begin with Q in the excited state and let the system evolve over time while recording the population in R1, as we increase the detuning Δ R 2 , R 1 , of R2. The result of this simulation is shown in Fig. fig1....The second case where ω R 1 > ω R 2 > ω R 3 is shown in Fig. fig3. This is very similar to the first case, however, there is a slight overall increase in the state transfer rate. This is because the detuning is still relative to R2, but the detuning of R3 is now slightly less. As the**

**qubit****qubits**approach resonance with the resonators from below, the dispersive coupling strength becomes slightly larger because the

**frequency**of R3 is a little closer to the

**frequency**of the

**qubits**. The small differences in resonator

**frequencies**also cause slight non-uniformities in the high

**frequency**

**oscillations**or ripples that become more pronounced at lower detuning. This interference is not a factor if the

**qubits**are sufficiently detuned. Regardless of these artifacts, it is clear that an array of resonators that is used for memory storage can also be used to dispersively couple

**qubits**....In both cases, at large detuning of -2000 MHz, the excitation appears to smoothly

**oscillate**into

**qubit**2, as shown in Fig. fig2 and fig3. As the detuning decreases, two things happen: First, the time it takes for the excitation to move into

**qubit**2 gets shorter, indicating an increase in the effective coupling strength between the two

**qubits**. Second, small ripples, or

**oscillations**, begin to appear. These

**oscillations**are due to the detuning becoming small enough that the dispersive limit approximation starts to break down. This result shows us that using three identical resonators in parallel to dispersively couple

**qubits**is effectively similar to using only one resonator with three times the coupling strength. In general, an array of n resonators can be replaced by a single resonator with n times the coupling strength. The dispersive coupling can be increased by reducing the detuning, but at the cost of having significant

**oscillations**between the

**qubits**and resonators....figure2(a) Two qubits (Q1, Q2) dispersively coupled to an array of three resonators (R1, R2, R3) via identical coupling capacitors C c . (b),(c) For these two simulations, the system is initialized with a single excitation in Q1, and the two qubits maintain equal frequencies as they are simultaneously detuned from the resonators. Each vertical cut represents the population in R2 over time at a particular detuning Δ Q , R 2 , of the qubits from R2. In (b), the frequencies of all three resonators are equal ( ω R 1 = ω R 2 = ω R 3 ). At large detuning ( Δ Q , R 2 = -2000 MHz), the excitation smoothly oscillates between the two qubits without significant interference from the resonators. As the magnitude of the detuning decreases, the effective coupling between the two qubits strengthens, thus the oscillation of the excitation becomes more frequent. Also, the direct coupling of the qubits to the resonators strengthens, causing the small ripples. In (c), the frequencies of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). The excitation oscillates slightly faster than in (b) because the small offsets in frequency of R1 and R3 increases the coupling bandwidth, resulting in a small increase in coupling between the qubits over the same range of detuning. The offset of R1 and R3 from R2 also causes the ripples to be non-uniform at smaller detuning....figure2(a) Two

**qubits**(Q1, Q2) dispersively coupled to an array of three resonators (R1, R2, R3) via identical coupling capacitors C c . (b),(c) For these two simulations, the system is initialized with a single excitation in Q1, and the two

**qubits**maintain equal

**frequencies**as they are simultaneously detuned from the resonators. Each vertical cut represents the population in R2 over time at a particular detuning Δ Q , R 2 , of the

**qubits**from R2. In (b), the

**frequencies**of all three resonators are equal ( ω R 1 = ω R 2 = ω R 3 ). At large detuning ( Δ Q , R 2 = -2000 MHz), the excitation smoothly

**oscillates**between the two

**qubits**without significant interference from the resonators. As the magnitude of the detuning decreases, the effective coupling between the two

**qubits**strengthens, thus the

**oscillation**of the excitation becomes more frequent. Also, the direct coupling of the

**qubits**to the resonators strengthens, causing the small ripples. In (c), the

**frequencies**of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). The excitation

**oscillates**slightly faster than in (b) because the small offsets in

**frequency**of R1 and R3 increases the coupling bandwidth, resulting in a small increase in coupling between the

**qubits**over the same range of detuning. The offset of R1 and R3 from R2 also causes the ripples to be non-uniform at smaller detuning....The second case where ω R 1 > ω R 2 > ω R 3 is shown in Fig. fig3. This is very similar to the first case, however, there is a slight overall increase in the state transfer rate. This is because the detuning is still relative to R2, but the detuning of R3 is now slightly less. As the qubits approach resonance with the resonators from below, the dispersive coupling strength becomes slightly larger because the frequency of R3 is a little closer to the frequency of the qubits. The small differences in resonator frequencies also cause slight non-uniformities in the high frequency oscillations or ripples that become more pronounced at lower detuning. This interference is not a factor if the qubits are sufficiently detuned. Regardless of these artifacts, it is clear that an array of resonators that is used for memory storage can also be used to dispersively couple qubits....figure3 (a) Placing a control

**(Q2) between the active**

**qubit****(Q1) and the resonator array (R1, R2, R3) allows coupling to be turned on and off. (b) The system is initialized with an excitation in Q1, Q1 is in-resonance with R2, and the frequencies of R1 and R3 are set slightly above and below R2, respectively ( ω R 1 > ω R 2 > ω R 3 ). Each vertical cut represents the population in R2 over time at a particular detuning Δ Q 2 , Q 1 , of Q2 from Q1. At zero detuning, the excitation readily oscillates in and out of the resonator R2. As Q2 is detuned further away, the coupling between Q1 and R2 becomes weaker, resulting in slower oscillations of the excitation. The detuning of the control**

**qubit****Q2 can be chosen based on the desired time scale, e.g. the time required to manipulate Q1....We consider an array of resonators used as a memory register. To accomplish this, one must be able to transfer an excitation from a**

**qubit****qubit**to a specific resonator, without coupling to the other resonators in the array. This is implemented by designing resonators that are sufficiently detuned from each other. To determine the amount of detuning required to avoid crosstalk between resonators, we examine a single

**qubit**coupled to an array of two resonators as shown in Fig. block1. The

**qubit**(Q) and resonator 1 (R1) are fixed at the same

**frequency**while resonator 2 (R2) is detuned. All

**qubit**-resonator couplings are presumed to be identical with a strength of g i j = 110 M H z for all i , j , i.e., all coupling capacitances C c are equal. This value is typical in experiments such as in Ref. . We begin with Q in the excited state and let the system evolve over time while recording the population in R1, as we increase the detuning Δ R 2 , R 1 , of R2. The result of this simulation is shown in Fig. fig1....Next, we investigate the behavior of a system consisting of two qubits dispersively coupled to an array of three resonators, as shown in Fig. block2. We consider two cases: (1) All three resonators are designed with the same resonant frequency, and (2) the resonant frequencies of the three resonators are slightly detuned so that ω R 1 > ω R 2 > ω R 3 . We demonstrate how information in the form of an excitation is transferred dispersively from one

**to another through an array of resonators. In both cases, the system is initialized with a single excitation in Q1, and both qubits are held in resonance with each other as the magnitude of their detuning from R2 is decreased from, say, -2000 MHz to -400 MHz....The result of the simulation is shown in Fig. fig4 where the population in R2 is plotted versus the detuning Δ Q 2 , Q 1 , of Q2 from Q1, in the range of -2000 MHz to 2000 MHz. As per design, at zero detuning, the excitation is transferred after a time of 16 ns. This time can be chosen based on the desired time scale by adjusting the**

**qubit****qubit**-

**qubit**and

**qubit**-resonator coupling strengths. At large detuning, i.e., beyond ± 1.5 GHz, the coupling between Q1 and R2 becomes dispersive up to about 30 ns, showing that dispersive coupling can be weak enough to isolate the active

**qubit**. Q2 can be detuned further to reduce the dispersive coupling if the desired time scale is longer, e.g., to perform operations on Q1. Thus, this simulation shows that a control

**qubit**can be effectively used to turn coupling on and off between a

**qubit**and an array of resonators....figure1(a) Schematic of a

**(Q) coupled to an array of two resonators (R1, R2) via identical coupling capacitors C c . The**

**qubit****is characterized as having capacitance C J and critical current I c , with bias current I b . The resonators are characterized by inductance L i and capacitance C i , for i = 1 , 2 . (b) For this simulation, the system is initialized with a single excitation in Q, and Q is in-resonance with R1. Each vertical cut represents the population in R1 over time at a particular detuning Δ R 2 , R 1 , of R2 from R1. The**

**qubit****-resonator coupling strength is 110 MHz. At zero detuning, the excitation oscillates between the**

**qubit****and the two resonators; since the two resonators are identical here, each resonator is only half populated. As R2 is detuned, the excitation oscillates between Q and R1 with minimal population in R2. After more oscillations, R2 will accumulate some population, even at large detuning, which causes the appearance of ripples....In systems with a large array of resonators, there is a considerably large bandwidth used up, within which qubits cannot be operated without directly coupling to the resonators. This problem can be solved by introducing a “control”**

**qubit****qubit**placed between the resonator array and the active

**. This configuration is shown in Fig. block3 where Q1 is the active**

**qubit****and Q2 is the control**

**qubit****. In the simulation, the three resonators are identical to the ones in the previous discussion where ω R 1 > ω R 2 > ω R 3 . Q1 is held in resonance with R2 and Q1 is initialize in its excited state. We can enable or disable transmission of the excitation to R2 by tuning Q2....figure1(a) Schematic of a**

**qubit****qubit**(Q) coupled to an array of two resonators (R1, R2) via identical coupling capacitors C c . The

**qubit**is characterized as having capacitance C J and critical current I c , with bias current I b . The resonators are characterized by inductance L i and capacitance C i , for i = 1 , 2 . (b) For this simulation, the system is initialized with a single excitation in Q, and Q is in-resonance with R1. Each vertical cut represents the population in R1 over time at a particular detuning Δ R 2 , R 1 , of R2 from R1. The

**qubit**-resonator coupling strength is 110 MHz. At zero detuning, the excitation

**oscillates**between the

**qubit**and the two resonators; since the two resonators are identical here, each resonator is only half populated. As R2 is detuned, the excitation

**oscillates**between Q and R1 with minimal population in R2. After more

**oscillations**, R2 will accumulate some population, even at large detuning, which causes the appearance of ripples....Josephson junction-based

**qubits**have been shown to be promising components for a future quantum computer. A network of these superconducting

**qubits**will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple

**qubits**. We explore the use of multiple resonators with different resonant

**frequencies**to couple

**qubits**. We find that an array of resonators with different

**frequencies**can be individually addressed to store and retrieve information, while coupling

**qubits**dispersively. We show that a control

**qubit**can be used to effectively isolate an active

**qubit**from an array of resonators so that it can function within the same

**frequency**range used by the resonators....Next, we investigate the behavior of a system consisting of two

**qubits**dispersively coupled to an array of three resonators, as shown in Fig. block2. We consider two cases: (1) All three resonators are designed with the same resonant

**frequency**, and (2) the resonant

**frequencies**of the three resonators are slightly detuned so that ω R 1 > ω R 2 > ω R 3 . We demonstrate how information in the form of an excitation is transferred dispersively from one

**qubit**to another through an array of resonators. In both cases, the system is initialized with a single excitation in Q1, and both

**qubits**are held in resonance with each other as the magnitude of their detuning from R2 is decreased from, say, -2000 MHz to -400 MHz. ... Josephson junction-based

**qubits**have been shown to be promising components for a future quantum computer. A network of these superconducting

**qubits**will require quantum information to be stored in and transferred among them. Resonators made of superconducting metal strips are useful elements for this purpose because they have long coherence times and can dispersively couple

**qubits**. We explore the use of multiple resonators with different resonant

**frequencies**to couple

**qubits**. We find that an array of resonators with different

**frequencies**can be individually addressed to store and retrieve information, while coupling

**qubits**dispersively. We show that a control

**qubit**can be used to effectively isolate an active

**qubit**from an array of resonators so that it can function within the same

**frequency**range used by the resonators.

Files:

Contributors: Volkov, P. A., Fistul, M. V.

Date: 2013-05-31

We report a theoretical study of coherent collective quantum dynamic effects in an array of N **qubits** (two-level systems) incorporated into a low-dissipation resonant cavity. Individual **qubits** are characterized by energy level differences $\Delta_i$ and we take into account a spread of parameters $\Delta_i$. Non-interacting **qubits** display coherent quantum beatings with N different **frequencies**, i.e. $\omega_i=\Delta_i/\hbar$ . Virtual emission and absorption of cavity photons provides a long-range interaction between **qubits**. In the presence of such interaction we analyze quantum correlation functions of individual **qubits** $C_i(t)$ to obtain two collective quantum-mechanical coherent oscillations, characterized by **frequencies** $\omega_1=\bar{\Delta}/\hbar$ and $\omega_2=\tilde{\omega}_R$, where $\tilde{\omega}_R$ is the resonant **frequency** of the cavity renormalized by interaction. The amplitude of these oscillations can be strongly enhanced in the resonant case when $\omega_1 \simeq \omega_2$....Collective quantum coherent oscillations in a globally coupled array of **qubits**...The typical solution consisting of the single instanton "step" on i -th **qubit** (solid green (thick) line) and tails on other **qubits** (solid red (thin) lines)....We report a theoretical study of coherent collective quantum dynamic effects in an array of N **qubits** (two-level systems) incorporated into a low-dissipation resonant cavity. Individual **qubits** are characterized by energy level differences $\Delta_i$ and we take into account a spread of parameters $\Delta_i$. Non-interacting **qubits** display coherent quantum beatings with N different **frequencies**, i.e. $\omega_i=\Delta_i/\hbar$ . Virtual emission and absorption of cavity photons provides a long-range interaction between **qubits**. In the presence of such interaction we analyze quantum correlation functions of individual **qubits** $C_i(t)$ to obtain two collective quantum-mechanical coherent **oscillations**, characterized by **frequencies** $\omega_1=\bar{\Delta}/\hbar$ and $\omega_2=\tilde{\omega}_R$, where $\tilde{\omega}_R$ is the resonant **frequency** of the cavity renormalized by interaction. The amplitude of these **oscillations** can be strongly enhanced in the resonant case when $\omega_1 \simeq \omega_2$....The typical solution consisting of the single instanton "step" on i -th qubit (solid green (thick) line) and tails on other qubits (solid red (thin) lines). ... We report a theoretical study of coherent collective quantum dynamic effects in an array of N **qubits** (two-level systems) incorporated into a low-dissipation resonant cavity. Individual **qubits** are characterized by energy level differences $\Delta_i$ and we take into account a spread of parameters $\Delta_i$. Non-interacting **qubits** display coherent quantum beatings with N different **frequencies**, i.e. $\omega_i=\Delta_i/\hbar$ . Virtual emission and absorption of cavity photons provides a long-range interaction between **qubits**. In the presence of such interaction we analyze quantum correlation functions of individual **qubits** $C_i(t)$ to obtain two collective quantum-mechanical coherent **oscillations**, characterized by **frequencies** $\omega_1=\bar{\Delta}/\hbar$ and $\omega_2=\tilde{\omega}_R$, where $\tilde{\omega}_R$ is the resonant **frequency** of the cavity renormalized by interaction. The amplitude of these **oscillations** can be strongly enhanced in the resonant case when $\omega_1 \simeq \omega_2$.

Files: