### 63448 results for qubit oscillator frequency

Contributors: Wallquist, M., Shumeiko, V. S., Wendin, G.

Date: 2006-08-09

...Pulse sequence producing (trivial) diagonal gate: during time T 1 , **qubit** 1 swaps its state onto the **oscillator**, then the **oscillator** interacts with **qubit** 2 before swapping its state back onto **qubit** 1; free evolution during time T 3 is added to annihilate two-photon state in the cavity....Protocol for creating a Bell-pair: the cavity **frequency** is sequentially swept through resonances with both **qubits**; at the first resonance the **oscillator** is entangled with **qubit** 1, at the next resonance the **oscillator** swaps its state onto **qubit** 2 and ends up in the ground state. A Bell measurement is performed by applying Rabi pulses to non-interacting **qubits**, and projecting on the **qubit** eigenbasis, | g | e , by measuring quantum capacitance....Sketch of the device: charge **qubits** (single Cooper pair boxes, SCB) coupled capacitively ( C c ) to a stripline cavity integrated **with **a dc-SQUID formed by two large Josephson junctions (JJ); cavity eigenfrequency is controlled by magnetic flux Φ through the SQUID....Equivalent circuit for the device in Fig. Sketch: chain of L C -**oscillators** represents the stripline cavity, φ 1 and φ N are superconducting phase values at the ends of the cavity, φ j and φ l are local phase values where the **qubits** are attached; attached dc-SQUID has effective flux-dependent Josephson energy, E J s f , and capacitance C s , control line for tuning the SQUID is shown at the right; SCB **qubits** are coupled to the cavity via small capacitances, C c 1 and C c 2 ....Sketch of the device: charge **qubits** (single Cooper pair boxes, SCB) coupled capacitively ( C c ) to a stripline cavity integrated with a dc-SQUID formed by two large Josephson junctions (JJ); cavity eigenfrequency is controlled by magnetic flux Φ through the SQUID....**The** experimental setup with **the** **qubit** **coupling** to **a** distributed **oscillator** - stripline **cavity** possesses potential for scalability - several **qubits** can be coupled to** the **

**cavity**. In this paper we investigate

**possibility to use this setup for implementation**

**the****of**tunable

**qubit**-

**qubit**

**coupling**and simple gate operations. Tunable

**qubit**-

**cavity**

**coupling**is achieved by varying

**the****cavity**

**frequency**by controlling magnetic flux through

**a**dc-SQUID

**attached**to

**the****cavity**(see Fig. Sketch). An advantage

**of**this method is

**possibility to keep**

**the**

**the****qubits**at

**optimal points with respect to decoherence during**

**the****whole**

**the****two**-

**qubit**operation.

**The**

**qubits**coupled to

**the****cavity**must have

**different**frequencies, and

**the****cavity**in

**idle regime must be tuned away from resonance with all**

**the****of**

**the****qubits**. Selective addressing

**of**

**a**

**particular**

**qubit**is achieved by relatively slow passage through

**resonance**

**the****of**

**a**

**selected**

**qubit**, while other resonances are rapidly passed.

**The**speed

**of**

**active resonant passage should be comparable to**

**the****the**

**qubit**-

**cavity**

**coupling**

**frequency**while

**rapid passages should be fast on this scale, but slow on**

**the****scale**

**the****of**

**the****cavity**eigenfrequency in order to avoid

**cavity**excitation. This strategy requires narrow width

**of**

**the**

**qubit**-

**cavity**resonances compared to

**differences in**

**the****the**

**qubit**frequencies, determined by

**available interval**

**the****of**

**the****cavity**

**frequency**divided by

**number**

**the****of**

**attached**

**qubits**. This consideration simultaneously imposes

**a**limit on

**maximum number**

**the****of**

**employed**

**qubits**. Denoting

**difference in**

**the****the**

**qubit**energies, Δ E J ,

**the**

**coupling**energy, κ ,

**the**maximum variation

**of**

**the****cavity**

**frequency**, Δ ω k , and

**number**

**the****of**

**qubits**, N , we summarize

**above arguments with relations, κ ≪ Δ E J , N ∼ ℏ Δ ω k / Δ E J . In**

**the****off-resonance state,**

**the****the**

**qubit**-

**qubit**

**coupling**strength is smaller than

**on-resonance**

**the****coupling**by

**ratio, κ / ℏ ω k - E J ≪ 1 ....In this section we modify the Bell state construction to implementing a control-phase (CPHASE) two-**

**the****qubit**gate. This gate has the diagonal form: | α β 0 → exp i φ α β | α β 0 ( φ 00 = φ 01 = φ 10 = 0 , φ 11 = π ), and it is equivalent to the CNOT gate (up to local rotations). To generate such a diagonal gate, we adopt the following strategy: first tune the

**oscillator**through resonance with both

**qubits**performing π -pulse swaps in every step, and then reverse the sequence, as shown in figure fig_prot_naive. With an even number of swaps at every level, clearly the resulting gate will be diagonal....Equivalent circuit for the device in Fig. Sketch: chain of

**L**C -oscillators represents the stripline cavity, φ 1 and φ N are superconducting phase values at the ends of the cavity, φ j and φ l are local phase values where the

**qubits**are attached; attached dc-SQUID has effective flux-dependent Josephson energy, E J s f , and capacitance C s , control line for tuning the SQUID is shown at the right; SCB

**qubits**are coupled to the cavity via small capacitances, C c 1 and C c 2 ....Selective coupling of superconducting

**qubits**via tunable stripline cavity...Jonn,NewJP:

**the**duration

**of**

**gate operation in**

**the****latter case is h / 8 in**

**the****units**

**the****of**inverse

**coupling**energy, while it is 2.7 h for

**protocol presented in Fig. fig_prot_SK. This illustrates**

**the****advantage**

**the****of**longitudinal, z z

**coupling**(in

**the**

**qubit**eigenbasis), which is achieved for

**charge**

**the****qubits**biased at

**charge degeneracy point by current-current**

**the****coupling**. More common for charge

**qubits**is

**capacitive**

**the****coupling**, however there

**situation is**

**the****different**: this

**coupling**has x x symmetry at

**charge degeneracy point, and because**

**the****of**inevitable difference in

**the**

**qubit**frequencies,

**the**gate operation takes much longer time, prolonged by

**ratio between**

**the**

**the****qubits**

**frequency**asymmetry and

**the**

**coupling**

**frequency**. Recent suggestions to employ dynamic control methods to effectively bring

**the****qubits**into resonance can speed up

**gate operation. For these protocols,**

**the****the**gate duration is ∼ h in units

**of**direct

**coupling**energy, which is longer than in

**case**

**the****of**z z

**coupling**, but somewhat shorter than in our case. However,

**the**protocol considered in this paper might be made faster by using pulse shaping....Jonn,NewJP: the duration of the gate operation in the latter case is h / 8 in the units of inverse coupling energy, while it is 2.7 h for the protocol presented in Fig. fig_prot_SK. This illustrates the advantage of longitudinal, z z coupling (in the

**qubit**eigenbasis), which is achieved for the charge

**qubits**biased at the charge degeneracy point by current-current coupling. More common for charge

**qubits**is the capacitive coupling, however there the situation is different: this coupling has x x symmetry at the charge degeneracy point, and because of inevitable difference in the

**qubit**

**frequencies**, the gate operation takes much longer time, prolonged by the ratio between the

**qubits**

**frequency**asymmetry and the coupling

**frequency**. Recent suggestions to employ dynamic control methods to effectively bring the

**qubits**into resonance can speed up the gate operation. For these protocols, the gate duration is ∼ h in units of direct coupling energy, which is longer than in the case of z z coupling, but somewhat shorter than in our case. However, the protocol considered in this paper might be made faster by using pulse shaping....For a given eigenmode, the integrated stripline + SQUID system behaves as a lumped

**oscillator**with variable

**frequency**. Our goal in this section will be to derive an effective classical Lagrangian for this

**oscillator**. To this end we consider in Fig. 2qubit_circuit an equivalent circuit for the device depicted in Fig. Sketch. A discrete chain of identical L C -

**oscillators**, with phases φ i across the chain capacitors (i=1,…,N), represents the stripline cavity; the dc SQUID is directly attached at the right end of the chain, while the superconducting Cooper pair boxes (SCB) are attached via small coupling capacitors, C c 1 and C c 2 to the chain nodes with local phases, φ j and φ l (for simplicity we consider only two attached SCBs). The classical Lagrangian for this circuit,...Gate circuit for constructing a CNOT gate using the control-phase gate: a z-axis rotation is applied to

**qubit**1, and Hadamard gates H are applied to the second

**qubit**....Pulse sequence producing (trivial) diagonal gate: during time T

**1 ,**qubit 1 swaps its state

**onto**the oscillator, then the oscillator interacts

**with**qubit 2 before swapping its state back

**onto**qubit 1; free evolution during time T 3 is added to annihilate two-photon state in the cavity....For

**a**given eigenmode,

**the**integrated stripline + SQUID system behaves as

**a**lumped

**oscillator**with variable

**frequency**. Our goal in this section will be to derive an effective classical Lagrangian for this

**oscillator**. To this end we consider in Fig. 2qubit_circuit an equivalent circuit for

**device depicted in Fig. Sketch. A discrete chain**

**the****of**identical L C -oscillators, with phases φ i across

**chain capacitors (i=1,…,N), represents**

**the****stripline**

**the****cavity**;

**the**dc SQUID is directly

**attached**at

**right end**

**the****of**

**chain, while**

**the****superconducting Cooper pair boxes (SCB) are**

**the****attached**via small

**coupling**capacitors, C

**c**1 and C

**c**2 to

**chain nodes with local phases, φ j and φ l (for simplicity we consider only**

**the****two**

**attached**SCBs).

**The**classical Lagrangian for this circuit,...

**The**reso...The experimental setup with the

**qubit**coupling to a distributed

**oscillator**- stripline cavity possesses potential for scalability - several

**qubits**can be coupled to the cavity. In this paper we investigate the possibility to use this setup for implementation of tunable

**qubit**-

**qubit**coupling and simple gate operations. Tunable

**qubit**-cavity coupling is achieved by varying the cavity

**frequency**by controlling magnetic flux through a dc-SQUID attached to the cavity (see Fig. Sketch). An advantage of this method is the possibility to keep the

**qubits**at the optimal points with respect to decoherence during the whole two-

**qubit**operation. The

**qubits**coupled to the cavity must have different

**frequencies**, and the cavity in the idle regime must be tuned away from resonance with all of the

**qubits**. Selective addressing of a particular

**qubit**is achieved by relatively slow passage through the resonance of a selected

**qubit**, while other resonances are rapidly passed. The speed of the active resonant passage should be comparable to the

**qubit**-cavity coupling

**frequency**while the rapid passages should be fast on this scale, but slow on the scale of the cavity eigenfrequency in order to avoid cavity excitation. This strategy requires narrow width of the

**qubit**-cavity resonances compared to the differences in the

**qubit**

**frequencies**, determined by the available interval of the cavity

**frequency**divided by the number of attached

**qubits**. This consideration simultaneously imposes a limit on the maximum number of employed

**qubits**. Denoting the difference in the

**qubit**energies, Δ E J , the coupling energy, κ , the maximum variation of the cavity

**frequency**, Δ ω k , and the number of

**qubits**, N , we summarize the above arguments with relations, κ ≪ Δ E J , N ∼ ℏ Δ ω k / Δ E J . In the off-resonance state, the

**qubit**-

**qubit**coupling strength is smaller than the on-resonance coupling by the ratio, κ / ℏ ω k - E J ≪ 1 ....We theoretically investigate selective coupling of superconducting charge

**qubits**mediated by a superconducting stripline cavity with a tunable resonance

**frequency**. The

**frequency**control is provided by a flux biased dc-SQUID attached to the cavity. Selective entanglement of the

**qubit**states is achieved by sweeping the cavity

**frequency**through the

**qubit**-cavity resonances. The circuit is scalable, and allows to keep the

**qubits**at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-

**qubit**-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate....Gate circuit for constructing a CNOT gate using the control-phase gate: a z-axis rotation is applied to qubit 1, and Hadamard gates H are applied to the second qubit....Protocol for creating a Bell-pair: the cavity frequency is sequentially swept through resonances

**with**both

**qubits**; at the first resonance the oscillator is entangled

**with**qubit 1, at the next resonance the oscillator swaps its state

**onto**qubit 2 and ends up in the ground state. A Bell measurement is performed by applying Rabi pulses to non-interacting

**qubits**, and projecting on the qubit eigenbasis, | g | e , by measuring quantum capacitance. ... We theoretically investigate selective coupling of superconducting charge

**qubits**mediated by a superconducting stripline cavity with a tunable resonance

**frequency**. The

**frequency**control is provided by a flux biased dc-SQUID attached to the cavity. Selective entanglement of the

**qubit**states is achieved by sweeping the cavity

**frequency**through the

**qubit**-cavity resonances. The circuit is scalable, and allows to keep the

**qubits**at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-

**qubit**-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate.

Files:

Contributors: Cooper, K. B., Steffen, Matthias, McDermott, R., Simmonds, R. W., Oh, Seongshik, Hite, D. A., Pappas, D. P., Martinis, John M.

Date: 2004-05-31

(a) Detail of the **qubit** spectroscopy near Δ U / ℏ ω p = 3.55 , showing splittings of strengths S ≈ 44 MHz and 24 MHz. (b) Tunneling probability versus measurement delay time τ D after application of π -pulse. Solid (dashed) line is taken at a well depth of solid (dashed) arrow in (a), corresponding to a resonant (off-resonant) bias. Inset illustrates how the **qubit** probability amplitude first moves to state | 1 g and then **oscillates** between | 1 g and | 0 e . (c) and (d) Tunneling probability (gray scale) versus well depth and τ D for experimental data (c) and numerical simulation (d). The peak **oscillation** periods are observed to correspond to the spectroscopic splittings....Spectroscopy of ω 10 obtained using the current-pulse measurement method, as a function of well depth Δ U / ℏ ω p . For each value of Δ U / ℏ ω p , the grayscale intensity is the normalized tunneling probability, with an original peak height of 0.1 - 0.3 . Insets: A given splitting in the spectroscopy of magnitude S comes from a critical-current fluctuator coupled to the **qubit** with strength h S / 2 . On resonance, the **qubit**-fluctuator eigenstates are linear combinations of the states | 1 g and | 0 e , where | g and | e are the two states of the fluctuator....(a) Room temperature measurement of the fast current pulse. (b) Tunneling probability versus δ I m a x with the **qubit** in state | 0 (solid circles) and in an equal mixture of states | 1 and | 0 (open circles). Fit to data is shown by the solid line. The plateau, being less than 0.5, corresponds to a maximum measurement fidelity of 0.63....Observation of quantum oscillations between a Josephson phase **qubit** and a microscopic resonator using fast readout...(a) Schematic of the **qubit** circuitry. For the **qubit** used in Fig. 2, the Josephson critical-current and junction capacitance are I 0 ≈ 10 μ A and C ≈ 2 pF; in Figs. 3 and 4, each of these values is about 5 times smaller. (b) Potential energy landscape and quantized energy levels for I φ = I d c prior to the state measurement. (c) At the peak of δ I t , the **qubit** well is much shallower and state | 1 rapidly tunnels to the right hand well....We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain....(a) Detail of the **qubit** spectroscopy near Δ U / ℏ ω p = 3.55 , showing splittings of strengths S ≈ 44 MHz and 24 MHz. (b) Tunneling probability versus measurement delay time τ D after application of π -pulse. Solid (dashed) line is taken at a well depth of solid (dashed) arrow in (a), corresponding to a resonant (off-resonant) bias. Inset illustrates how the **qubit** probability amplitude first moves to state | 1 g and then oscillates between | 1 g and | 0 e . (c) and (d) Tunneling probability (gray scale) versus well depth and τ D for experimental data (c) and numerical simulation (d). The peak oscillation periods are observed to correspond to the spectroscopic splittings....We have detected coherent quantum oscillations between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the oscillations is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain. ... We have detected coherent quantum **oscillations** between Josephson phase **qubits** and microscopic critical-current fluctuators by implementing a new state readout technique that is an order of magnitude faster than previous methods. The period of the **oscillations** is consistent with the spectroscopic splittings observed in the **qubit**'s resonant **frequency**. The results point to a possible mechanism for decoherence and reduced measurement fidelity in superconducting **qubits** and demonstrate the means to measure two-**qubit** interactions in the time domain.

Files:

Contributors: Chirolli, Luca, Burkard, Guido

Date: 2009-06-04

where x i t = 2 σ R e α i t** and **G x = exp - x 2 / σ 2 / σ π is a Gaussian of width σ schematically depicted **in **Fig. Fig2. Introducing a rate κ that describes** the **Markovian damping of** the **harmonic oscillator by a zero-temperature bath of harmonic oscillators, the coherent state qubit-dependent amplitude α i t is found to be...The QND character of the **qubit** measurement is studied by repeating the measurement. A perfect QND setup guarantees identical outcomes for the two repeated measurement with certainty. In order to fully characterize the properties of the measurement, we can initialize the **qubit** in the state | 0 , then rotate the **qubit** by applying a pulse of duration τ 1 before the first measurement and a second pulse of duration τ 2 between the first and the second measurement. The conditional probability to detect the **qubit** in the states s and s ' is expected to be independent of the first pulse, and to show sinusoidal **oscillation** with amplitude 1 in τ 2 . Deviations from this expectation witness a deviation from a perfect QND measurement. The sequence of **qubit** pulses and **oscillator** driving is depicted in Fig. Fig1b). The conditional probability P 0 | 0 to detect the **qubit** in the state "0" twice in sequence is plotted versus τ 1 and τ 2 in Fig. Fig1c) for Δ = 0 , and in Fig. Fig1d) for Δ / ϵ = 0.1 . We anticipate here that a dependence on τ 1 is visible when the **qubit** undergoes a flip in the first rotation. Such a dependence is due to the imperfections of the mapping between the **qubit** state and the **oscillator** state, and is present also in the case Δ = 0 . The effect of the non-QND term Δ σ X results in an overall reduction of P 0 | 0 ....(Color online) Conditional probability to obtain a) s ' = s = 1 , b) s ' = - s = 1 , c) s ' = - s = - 1 , and d) s ' = s = - 1 for the case Δ t = Δ / ϵ = 0.1 and T 1 = 10 ~ n s , when rotating the **qubit** around the y axis before the first measurement for a time τ 1 and between the first and the second measurement for a time τ 2 , starting with the **qubit** in the state | 0 0 | . Correction in Δ t are up to second order. The harmonic **oscillator** is driven at resonance with the bare harmonic **frequency** and a strong driving together with a strong damping of the **oscillator** are assumed, with f / 2 π = 20 ~ G H z and κ / 2 π = 1.5 ~ G H z . Fig6...We now investigate whether it is possible to identify the contribution of different mechanisms that generate deviations from a perfect QND measurement. In Fig. Fig7 we study separately the effect of **qubit** relaxation and **qubit** tunneling on the conditional probability P 0 | 0 . In Fig. Fig7 a) we set Δ = 0 and T 1 = ∞ . The main feature appearing is a sudden change of the conditional probability P → 1 - P when the **qubit** is flipped in the first rotation. This is due to imperfection in the mapping between the **qubit** state and the state of the harmonic **oscillator**, already at the level of a single measurement. The inclusion of a phenomenological **qubit** relaxation time T 1 = 2 ~ n s , intentionally chosen very short, yields a strong damping of the **oscillation** along τ 2 and washes out the response change when the **qubit** is flipped during the first rotation. This is shown in Fig. Fig7 b). The manifestation of the non-QND term comes as a global reduction of the visibility of the **oscillations**, as clearly shown in Fig. Fig7 c)....For driving at resonance with the bare harmonic **oscillator** **frequency** ω h o , the state of the **qubit** is encoded in the phase of the signal, with φ 1 = - φ 0 , and the amplitude of the signal is actually reduced, as also shown in Fig. Fig3 for Δ ω = 0 . When matching one of the two **frequencies** ω i the **qubit** state is encoded in the amplitude of the signal, as also clearly shown in Fig. Fig3 for Δ ω = ± g . Driving away from resonance can give rise to significant deviation from 0 and 1 to the outcome probability, therefore resulting in an imprecise mapping between **qubit** state and measurement outcomes and a weak **qubit** measurement....(Color online) Schematic description of the single measurement procedure. In the bottom panel the coherent states | α 0 and | α 1 , associated with the **qubit** states | 0 and | 1 , are represented for illustrative purposes by a contour line in the phase space at HWHM of their Wigner distributions, defined as W α α * = 2 / π 2 exp 2 | α | 2 ∫ d β - β | ρ | β exp β α * - β * α . The corresponding Gaussian probability distributions of width σ centered about the **qubit**-dependent "position" x s are shown in the top panel. Fig2...The dynamics governed by U R 0 t produces a coherent state of** the **oscillato...We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**....In Fig. Fig5 we plot the second order correction to the probability to obtain "1" having prepared the **qubit** in the initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only the deviation from the unperturbed probability because we want to highlight the contribution to spin-flip purely due to tunneling in the **qubit** Hamiltonian. In fact most of the contribution to spin-flip arises from the unperturbed probability, as it is clear from Fig. Fig3. Around the two **qubit**-shifted **frequencies**, the probability has a two-peak structure whose characteristics come entirely from the behavior of the phase ψ around the resonances Δ ω ≈ ± g . We note that the tunneling term can be responsible for a probability correction up to ∼ 4 % around the **qubit**-shifted **frequency**....In Fig. Fig5 we plot** the **second order correction to** the **probability to obtain "1" having prepared** the **qubit in** the **initial state ρ 0 = | 0 0 | , corresponding to F 2 t , for Δ t = Δ / ϵ = 0.1 . We choose to plot only** the **deviation from** the **unperturbed probability because we want to highlight** the **contribution to spin-flip purely due to tunneling in** the **qubit Hamiltonian. In fact most of** the **contribution to spin-flip arises from** the **unperturbed probability, as it is clear from Fig. Fig3. Around** the **two qubit-shifted frequencies, the probability has a two-peak structure whose characteristics come entirely from** the **behavior of** the **phase ψ around** the **resonances Δ ω ≈ ± g . We note that** the **tunneling term can be responsible for a probability correction up to ∼ 4 % around** the **qubit-shifted frequency....(Color online) Comparison of the deviations from QND behavior originating from different mechanisms. Conditional probability P 0 | 0 versus **qubit** driving time τ 1 and τ 2 starting with the **qubit** in the state | 0 0 | , for a) Δ = 0 and T 1 = ∞ , b) Δ = 0 and T 1 = 2 ~ n s , and c) Δ = 0.1 ~ ϵ and T 1 = ∞ . The **oscillator** driving amplitude is f / 2 π = 20 ~ G H z and a damping rate κ / 2 π = 1.5 ~ G H z is assumed. Fig7...The combined effect of** the **quantum fluctuations of** the **oscillator together with** the **tunneling between** the **qubit states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by** the **non-QND tunneling term. Such a conclusion pertains to a model **in **which** the **qubit QND measurement is studied in** the **regime of strong projective qubit measurement** and **qubit relaxation is taken into account only phenomenologically. We compared** the **conditional probabilities plotted **in **Fig. Fig6** and **Fig. Fig7 directly to Fig. 4 **in **Ref. [...Quantum non-demolition measurements of a **qubit** coupled to a harmonic **oscillator**...In Fig. Fig1 c) we plot** the **conditional probability P 0 | 0 for** the **case Δ = 0 , when strongly driving** the **harmonic oscillator at resonance with** the **bare harmonic frequency, Δ ω = 0** . The **initial qubit state is chosen to be | 0 0 | . No dependence **on τ** 1 appears and** the **outcomes s** and **s ' play a symmetric role. This is indeed what we expect from a perfect QND measurement. In Fig. Fig6 we plot** the **four combinations of conditional probability P s ' | s up to second order corrections **in **Δ t = Δ / ϵ = 0.1** and **with a phenomenological qubit relaxation time T 1 = 10 ~ n s . We choose Δ ω = 0 , that is at resonance with** the **bare harmonic frequency. The initial qubit state is | 0 0 | . Three features appear: i) a global reduction of** the **visibility of** the **oscillations, ii) a strong dependence **on τ** 1 when** the **qubit is completely flipped in** the **first rotation** and **iii) an asymmetry under change of** the **outcome of** the **first measurement, with an enhanced reduction of** the **visibility when** the **first measurement produces a result that is opposite with respect to** the **initial qubit preparation | 0 0 | . Furthermore, we find a weak dependence of** the **visibility **on τ** 1 ....The combined effect of the quantum fluctuations of the **oscillator** together with the tunneling between the **qubit** states is therefore responsible for deviation from a perfect QND behavior, although a major role is played, as expected, by the non-QND tunneling term. Such a conclusion pertains to a model in which the **qubit** QND measurement is studied in the regime of strong projective **qubit** measurement and **qubit** relaxation is taken into account only phenomenologically. We compared the conditional probabilities plotted in Fig. Fig6 and Fig. Fig7 directly to Fig. 4 in Ref. [...(Color online) a) Schematics of the 4-Josephson junction superconducting flux **qubit** surrounded by a SQUID. b) Measurement scheme: b1) two short pulses at **frequency** ϵ 2 + Δ 2 , before and between two measurements prepare the **qubit** in a generic state. Here, ϵ and Δ represent the energy difference and the tunneling amplitude between the two **qubit** states. b2) Two pulses of amplitude f and duration τ 1 = τ 2 = 0.1 ~ n s drive the harmonic **oscillator** to a **qubit**-dependent state. c) Perfect QND: conditional probability P 0 | 0 for Δ = 0 to detect the **qubit** in the state "0" vs driving time τ 1 and τ 2 , at Rabi **frequency** of 1 ~ G H z . The **oscillator** driving amplitude is chosen to be f / 2 π = 50 ~ G H z and the damping rate κ / 2 π = 1 ~ G H z . d) Conditional probability P 0 | 0 for Δ / ϵ = 0.1 , f / 2 π = 20 ~ G H z , κ / 2 π = 1.5 ~ G H z . A phenomenological **qubit** relaxation time T 1 = 10 ~ n s is assumed. Fig1 ... We theoretically describe the weak measurement of a two-level system (**qubit**) and quantify the degree to which such a **qubit** measurement has a quantum non-demolition (QND) character. The **qubit** is coupled to a harmonic **oscillator** which undergoes a projective measurement. Information on the **qubit** state is extracted from the **oscillator** measurement outcomes, and the QND character of the measurement is inferred by the result of subsequent measurements of the **oscillator**. We use the positive operator valued measure (POVM) formalism to describe the **qubit** measurement. Two mechanisms lead to deviations from a perfect QND measurement: (i) the quantum fluctuations of the **oscillator**, and (ii) quantum tunneling between the **qubit** states $|0>$ and $|1>$ during measurements. Our theory can be applied to QND measurements performed on superconducting **qubits** coupled to a circuit **oscillator**.

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Contributors: Saiko, A. P., Fedaruk, R.

Date: 2010-12-10

Energy-level diagram of a **qubit** and transitions created by a bichromatic field at double resonance ( ω 0 = ω , ω 1 = ω r f )....Multiplication of **Qubits** in a Doubly Resonant Bichromatic Field...Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi oscillation in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account....Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account. ... Multiplication of spin **qubits** arises at double resonance in a bichromatic field when the **frequency** of the radio-**frequency** (rf) field is close to that of the Rabi **oscillation** in the microwave field, provided its **frequency** equals the Larmor **frequency** of the initial **qubit**. We show that the operational multiphoton transitions of dressed **qubits** can be selected by the choice of both the rotating frame and the rf phase. In order to enhance the precision of dressed **qubit** operations in the strong-field regime, the counter-rotating component of the rf field is taken into account.

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Contributors: Bertet, P., Chiorescu, I., Harmans, C. J. P. M, Mooij, J. E.

Date: 2005-07-13

(a) **qubit** biased by Φ x and SQUID biased by current I b . (b) Simplified electrical scheme : the SQUID-**qubit** system is seen as an inductance L J connected to the shunct capacitor C s h through inductance L s h . Φ a is the flux across the two inductances L J and L s h in series....**Qubit** **frequency** ν q as a function of the bias ϵ for Δ = 5.5 G H z (minimum **frequency** in the figure). The dashed line indicates the phase-noise insensitive bias point ϵ = 0 where d ν q / d ϵ = 0...The hamiltonian eq:qubit_hamiltonian yields a **qubit** transition **frequency** ν q = Δ 2 + ϵ 2 . The corresponding dependence is plotted in figure fig:nuq for realistic parameters. An interesting property is that when the **qubit** is biased at ϵ = 0 (dashed line in figure fig:nuq), it is insensitive to first order to noise in** the **bias variable ϵ ....The** flux**-**qubit** is a superconducting loop containing three Josephson junctions threaded by an external** flux** Φ x ≡ f Φ 0 / 2 π . It is coupled to a DC-SQUID detector shunted by an external capacitor C s h whose role is to limit phase fluctuations across** the **SQUID as well as to filter high-**frequency** noise from** the **dissipative impedance. The SQUID is threaded by a** flux** Φ S q ≡ f ' Φ 0 / 2 π . The circuit diagram is shown in figure fig1a. There, the** flux**-**qubit** is** the **loop in red containing** the **three junctions of phases φ i and capacitances C i ( i = 1 , 2 , 3 ). It also includes an inductance L 1 which models** the **branch inductance and eventually** the **inductance of a fourth larger junction . The two inductances K 1 and K 2 model** the **kinetic inductance of** the **line shared by** the **SQUID and the **qubit**. The SQUID is** the **larger loop in blue. The junction phases are called φ 4 and φ 5 and their capacitances C 4 and C 5 . The critical current of** the **circuit junctions is written I C i ( i = 1 to 5 ). The SQUID loop also contains two inductances K 3 and L 2 which model its self-inductance. The SQUID is connected to** the **capacitor C s h through superconducting lines of parasitic inductance L s . The phase across** the **stray inductance and** the **SQUID is denoted φ A . The whole circuit is biased by a current source I b in parallel with a dissipative admittance Y ω . Since our goal is primarily to determine the **qubit**-plasma mode coupling hamiltonian, we will neglect** the **admittance Y ω ....Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed....The flux-**qubit** is a superconducting loop containing three Josephson junctions threaded by an external flux Φ x ≡ f Φ 0 / 2 π . It is coupled to a DC-SQUID detector shunted by an external capacitor C s h whose role is to limit phase fluctuations across the SQUID as well as to filter high-**frequency** noise from the dissipative impedance. The SQUID is threaded by a flux Φ S q ≡ f ' Φ 0 / 2 π . The circuit diagram is shown in figure fig1a. There, the flux-**qubit** is the loop in red containing the three junctions of phases φ i and capacitances C i ( i = 1 , 2 , 3 ). It also includes an inductance L 1 which models the branch inductance and eventually the inductance of a fourth larger junction . The two inductances K 1 and K 2 model the kinetic inductance of the line shared by the SQUID and the **qubit**. The SQUID is the larger loop in blue. The junction phases are called φ 4 and φ 5 and their capacitances C 4 and C 5 . The critical current of the circuit junctions is written I C i ( i = 1 to 5 ). The SQUID loop also contains two inductances K 3 and L 2 which model its self-inductance. The SQUID is connected to the capacitor C s h through superconducting lines of parasitic inductance L s . The phase across the stray inductance and the SQUID is denoted φ A . The whole circuit is biased by a current source I b in parallel with a dissipative admittance Y ω . Since our goal is primarily to determine the **qubit**-plasma mode coupling hamiltonian, we will neglect the admittance Y ω ....(a)...We will now discuss quantitatively** the **behaviour of g 1 and g 2 for actual sample parameters : I C = 3.4 μ A , M = 6.5 p H , I p = 240 n A , Δ = 5.5 G H z , ν p = 3.1 G H z , L J = 300 p H , f ' / 2 = 1.45 π . We will restrict ourselves to a range of bias conditions relevant for our conditions, supposing that I b varies between ± 300 n A and that f ' / 2 varies by d f ' = ± 4 ⋅ 10 -3 π around 1.45 π . We chose such an interval for f ' because it corresponds to changing the **qubit** bias point ϵ by ± 2 G H z around 0 . The constants g 1 and g 2 are plotted in figure fig:couplings as a function of I b for two different values of f ' ( g 1 is shown as a full line, g 2 as a dashed line, and** the **two different values of f ' are symbolized by gray for d f ' = - 2 π 4 ⋅ 10 -3 and black for d f ' = 0 ). It can be seen that** the **coupling constants only weakly depend on** the **value of** the **flux in this range, so that we will neglect this dependence in** the **following and consider that g 1 and g 2 only depend on** the **bias current I b . Moreover we see from figure fig:couplings that** the **approximations made in equation eq:g1g2approx are justified in this range of parameters since g 1 is closely linear in I b and g 2 nearly constant. We also note that g 1 = 0 for I b = 0 . This fact can be generalized to** the **case where** the **SQUID-**qubit** coupling is not symmetric and** the **junctions critical current are dissimilar : in certain conditions these asymmetries can be compensated for by applying a bias current I b * for which g 1 I b * = 0 . At** the **current I b * , the **qubit** is effectively decoupled from** the **measuring circuit fluctuations to first order....We stress that these biasing conditions are non-trivial in** the **sense that they do not satisfy an obvious symmetry in** the **circuit. This point is emphasized in figure fig:deltanu0 where we plotted as a dashed line** the **bias conditions ϵ = 0 for which the **qubit** is insensitive to phase noise (due to** flux** or bias current noise) ; and as a dotted line** the **decoupling current conditions I b = I b * for which the **qubit** is effectively decoupled from its measuring circuit. The ϵ m I b line shares only one point with these two curves : the point I b * ϵ which is optimal with respect to** flux**, bias current, and photon noise. For** the **rest, the three lines are obviously distinct. This makes it possible to experimentally discriminate between** the **various noise sources limiting the **qubit** coherence by studying** the **dependence of τ φ on bias parameters....Contrary to the shift produced by the linear coupling term, the sign of this **frequency** shift now depends on ϵ . Since g 2 is negative (see figure fig:couplings), δ ν 0 2 actually has the same sign as ϵ . We also note that the quadratic term has no effect on the **qubit** when ϵ = 0 , since at that point the average flux generated by both **qubit** states | 0 and | 1 averages out to zero so that the SQUID Josephson inductance is unchanged....Qubit frequency ν q as a function of the bias ϵ for Δ = 5.5 G H z (minimum frequency in the figure). The dashed line indicates the phase-noise insensitive bias point ϵ = 0 where d ν q / d ϵ = 0...We will now discuss quantitatively the behaviour of g 1 and g 2 for actual sample parameters : I C = 3.4 μ A , M = 6.5 p H , I p = 240 n A , Δ = 5.5 G H z , ν p = 3.1 G H z , L J = 300 p H , f ' / 2 = 1.45 π . We will restrict ourselves to a range of bias conditions relevant for our conditions, supposing that I b varies between ± 300 n A and that f ' / 2 varies by d f ' = ± 4 ⋅ 10 -3 π around 1.45 π . We chose such an interval for f ' because it corresponds to changing the **qubit** bias point ϵ by ± 2 G H z around 0 . The constants g 1 and g 2 are plotted in figure fig:couplings as a function of I b for two different values of f ' ( g 1 is shown as a full line, g 2 as a dashed line, and the two different values of f ' are symbolized by gray for d f ' = - 2 π 4 ⋅ 10 -3 and black for d f ' = 0 ). It can be seen that the coupling constants only weakly depend on the value of the flux in this range, so that we will neglect this dependence in the following and consider that g 1 and g 2 only depend on the bias current I b . Moreover we see from figure fig:couplings that the approximations made in equation eq:g1g2approx are justified in this range of parameters since g 1 is closely linear in I b and g 2 nearly constant. We also note that g 1 = 0 for I b = 0 . This fact can be generalized to the case where the SQUID-**qubit** coupling is not symmetric and the junctions critical current are dissimilar : in certain conditions these asymmetries can be compensated for by applying a bias current I b * for which g 1 I b * = 0 . At the current I b * , the **qubit** is effectively decoupled from the measuring circuit fluctuations to first order....**Frequency** shift per photon δ ν 0 as a function of I b and ϵ . The white regions correspond to -15 M H z and the black to + 35 M H z . The solid line ϵ m I b indicates the bias conditions for which δ ν 0 = 0 . The dashed line indicates the phase noise insensitive point ϵ = 0 ; the dotted line indicates the decoupling current I b = I b * ....Coupling constants g 1 (solid line) and g 2 (dashed line) as a function of the bias current, for two values of the reduced **SQUID** flux bias f ' differing by d f ' = 4 ⋅ 10 -3 (gray and black lines)....The hamiltonian eq:qubit_hamiltonian yields a **qubit** transition **frequency** ν q = Δ 2 + ϵ 2 . The corresponding dependence is plotted in figure fig:nuq for realistic parameters. An interesting property is that when the **qubit** is biased at ϵ = 0 (dashed line in figure fig:nuq), it is insensitive to first order to noise in the bias variable ϵ ....We stress that these biasing conditions are non-trivial in the sense that they do not satisfy an obvious symmetry in the circuit. This point is emphasized in figure fig:deltanu0 where we plotted as a dashed line the bias conditions ϵ = 0 for which the **qubit** is insensitive to phase noise (due to flux or bias current noise) ; and as a dotted line the decoupling current conditions I b = I b * for which the **qubit** is effectively decoupled from its measuring circuit. The ϵ m I b line shares only one point with these two curves : the point I b * ϵ which is optimal with respect to flux, bias current, and photon noise. For the rest, the three lines are obviously distinct. This makes it possible to experimentally discriminate between the various noise sources limiting the **qubit** coherence by studying the dependence of τ φ on bias parameters....Dephasing of a flux-**qubit** coupled to a harmonic **oscillator**...Contrary to** the **shift produced by** the **linear coupling term, the sign of this **frequency** shift now depends on ϵ . Since g 2 is negative (see figure fig:couplings), δ ν 0 2 actually has** the **same sign as ϵ . We also note that** the **quadratic term has no effect on the **qubit** when ϵ = 0 , since at that point** the **average** flux** generated by both **qubit** states | 0 and | 1 averages out to zero so that** the **SQUID Josephson inductance is unchanged....(a) qubit biased by Φ x and **SQUID** biased by current I b . (b) Simplified electrical scheme : the **SQUID**-qubit system is seen as an inductance L J connected to the shunct capacitor C s h through inductance L s h . Φ a is the flux across the two inductances L J and L s h in series. ... Decoherence in superconducting **qubits** is known to arise because of a variety of environmental degrees of freedom. In this article, we focus on the influence of thermal fluctuations in a weakly damped circuit resonance coupled to the **qubit**. Because of the coupling, the **qubit** **frequency** is shifted by an amount $n \delta \nu_0$ if the resonator contains $n$ energy quanta. Thermal fluctuations induce temporal variations $n(t)$ and thus dephasing. We give an approximate formula for the **qubit** dephasing time as a function of $\delta \nu_0$. We discuss the specific case of a flux-**qubit** coupled to the plasma mode of its DC-SQUID detector. We first derive a plasma mode-**qubit** interaction hamiltonian which, in addition to the usual Jaynes-Cummings term, has a coupling term quadratic in the **oscillator** variables coming from the flux-dependence of the SQUID Josephson inductance. Our model predicts that $\delta \nu_0$ cancels in certain non-trivial bias conditions for which dephasing due to thermal fluctuations should be suppressed.

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Contributors: Murch, K. W., Ginossar, E., Weber, S. J., Vijay, R., Girvin, S. M., Siddiqi, I.

Date: 2012-08-22

fig3 Experimental device: a transmon **qubit** coupled to a nonlinear superconducting cavity. (a) Circuit diagram of the device; the anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from the 50 Ω environment by coupling capacitors C c and coupled to a transmon **qubit** characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of the the device with the resonator and **qubit** junctions (lower and upper insets)....When a **frequency** chirped excitation is applied to a classical high-Q nonlinear **oscillator**, its motion becomes dynamically synchronized to the drive and large **oscillation** amplitude is observed, provided the drive strength exceeds the critical threshold for autoresonance. We demonstrate that when such an **oscillator** is strongly coupled to a quantized superconducting **qubit**, both the effective nonlinearity and the threshold become a non-trivial function of the **qubit**-**oscillator** detuning. Moreover, the autoresonant threshold is sensitive to the quantum state of the **qubit** and may be used to realize a high fidelity, latching readout whose speed is not limited by the **oscillator** Q....fig2 Energy levels and the effective nonlinearity λ of the strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus **qubit**-cavity detuning for the **qubit** prepared in the ground state in the low excitation regime. Theory curves for the model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear classical **oscillators** (red) are shown. The arrows indicate the locations of avoided crossings of the level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of the resonator when driven with tone at ω d that occupied the resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of the **qubit**-**oscillator** model with N l = 7 show the avoided crossings in the 4 and 5 excitation manifold. (c) Quantum trajectory simulation of the system exhibits a general trends of increasing effective nonlinearity λ with diminishing **qubit**-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in the 4 and 5 excitation manifold. For these simulations, the **qubit** energy levels were modeled as a Duffing nonlinearity....fig2 Energy levels and **the** effective nonlinearity λ of **the** strongly coupled system. (a) The measured coefficient of nonlinear response for a strongly coupled system versus **qubit**-cavity detuning for **the** **qubit** prepared in **the** ground state in **the** low excitation regime. Theory curves for **the** model Eq. ( eq:1) with N l = 2 (blue), N l = 3 (gray), and for a model of coupled nonlinear **classical** oscillators (red) are shown. The arrows indicate **the** locations of avoided crossings of **the** level pairs | 1 , 3 ↔ | 0 , 4 and | 1 , 4 ↔ | 0 , 5 . (Inset) The transmission of **the** resonator when driven with tone at ω d that occupied **the** resonator with = 0.4 (black) and = 10 (gray) off-resonant photons. (b) Energy levels of **the** **qubit**-**oscillator** model with N l = 7 show **the** avoided crossings in **the** 4 and 5 excitation manifold. (c) Quantum trajectory simulation of **the** system exhibits a general trends of increasing effective nonlinearity λ with **diminishing** **qubit**-cavity detuning ( Δ ) with abrupt reductions associated with avoided crossings in **the** 4 and 5 excitation manifold. For these simulations, **the** **qubit** energy levels were modeled as a Duffing nonlinearity....fig3 Experimental device: a transmon **qubit** coupled to a nonlinear superconducting cavity. (a) Circuit diagram of **the** device; **the** anharmonic resonator is formed from a meander inductor embedded with a Josephson junction and an interdigitated capacitor. The resonator is isolated from **the** 50 Ω environment by coupling capacitors C c and coupled to a transmon **qubit** characterized by Josephson and charging energy scales E J and E c respectively. The coupling rate is g . (b) Scanning electron micrograph of **the** **the** device with **the** resonator and **qubit** junctions (lower and upper insets)....fig2 Measured autoresonance and threshold sensitivity on the level structure and initial **qubit** state. (a) Color plot shows S | 1 versus **qubit** detuning. The dashed line indicates the AR threshold, V | 1 . AR measurements were not taken for small values of the detuning as indicated by the hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate the location of avoided crossings in the 4 and 5 excitation manifold....When a **frequency** chirped excitation is applied to a classical high-Q nonlinear **oscillator**, its motion becomes dynamically synchronized to the drive and large oscillation amplitude is observed, provided the drive strength exceeds the critical threshold for autoresonance. We demonstrate that when such an **oscillator** is strongly coupled to a quantized superconducting **qubit**, both the effective nonlinearity and the threshold become a non-trivial function of the **qubit**-**oscillator** detuning. Moreover, the autoresonant threshold is sensitive to the quantum state of the **qubit** and may be used to realize a high fidelity, latching readout whose speed is not limited by the **oscillator** Q....fig2 Measured autoresonance and threshold sensitivity on **the** level structure and initial **qubit** state. (a) Color plot shows S | 1 versus **qubit** detuning. The dashed line indicates **the** AR threshold, V | 1 . AR measurements were not taken for small values of **the** detuning as indicated by **the** hatched region. (b) Color plot of S | 0 with V | 0 indicated as a solid black line. The AR threshold, V | 1 , is also plotted for comparison as a dashed black line. The two arrows indicate **the** location of avoided crossings in **the** 4 and 5 excitation manifold....fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) **the** AR threshold. (Inset) Pulse sequence: **the** **qubit** manipulation pulse was applied immediately before **the** start of **the** chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz....fig3duffing (a) The transmitted magnitude for chirp sequences with drive voltages that were above (black), near (dashed), and below (gray) the AR threshold. (Inset) Pulse sequence: the **qubit** manipulation pulse was applied immediately before the start of the chirp sequence. (b) The average transmitted magnitude near 400 ns versus drive voltage shows S | 0 (black) and S | 1 (red) for Δ / 2 π = 0.59 GHz. ... When a **frequency** chirped excitation is applied to a classical high-Q nonlinear **oscillator**, its motion becomes dynamically synchronized to the drive and large **oscillation** amplitude is observed, provided the drive strength exceeds the critical threshold for autoresonance. We demonstrate that when such an **oscillator** is strongly coupled to a quantized superconducting **qubit**, both the effective nonlinearity and the threshold become a non-trivial function of the **qubit**-**oscillator** detuning. Moreover, the autoresonant threshold is sensitive to the quantum state of the **qubit** and may be used to realize a high fidelity, latching readout whose speed is not limited by the **oscillator** Q.

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Contributors: Dial, O. E., Shulman, M. D., Harvey, S. P., Bluhm, H., Umansky, V., Yacoby, A.

Date: 2012-08-09

Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity....Two level quantum systems (**qubits**) are emerging as promising candidates both for quantum information processing and for sensitive metrology . When prepared in a superposition of two states and allowed to evolve, the state of the system precesses with a **frequency** proportional to the splitting between the states. However, on a timescale of the coherence time, T 2 , the **qubit** loses its quantum information due to interactions with its noisy environment. This causes **qubit** **oscillations** to decay and limits the fidelity of quantum control and the precision of **qubit**-based measurements. In this work we study singlet-triplet ( S - T 0 ) **qubits**, a particular realization of spin **qubits** , which store quantum information in the joint spin state of two electrons. We form the **qubit** in two gate-defined lateral quantum dots (QD) in a GaAs/AlGaAs heterostructure (Fig. pulsesa). The QDs are depleted until there is exactly one electron left in each, so that the system occupies the so-called 1 1 charge configuration. Here n L n R describes a double QD with n L electrons in the left dot and n R electrons in the right dot. This two-electron system has four possible spin states: S , T + , T 0 , and T - . The S , T 0 subspace is used as the logical subspace for this **qubit** because it is insensitive to homogeneous magnetic field fluctuations and is manipulable using only pulsed DC electric fields . The relevant low-lying energy levels of this **qubit** are shown in Fig. pulsesc. Two distinct rotations are possible in these devices: rotations around the x -axis of the Bloch sphere driven by difference in magnetic field between the QDs, Δ B z (provided in this experiment by feedback-stabilized hyperfine interactions), and rotations around the z -axis driven by the exchange interaction, J (Fig. pulsesb) . A S can be prepared quickly with high fidelity by exchanging an electron with the QD leads, and the projection of the state of the **qubit** along the z -axis can be measured using RF reflectometery with an adjacent sensing QD (green arrow in Fig. pulsesa)....Using a modified pulse sequence that changes** the **clock **frequency** of our waveform generators to achieve picosecond timing resolution (Fig. S1)), we measure exchange oscillations in 0 2 as a function of ϵ and time (Fig. t2stare) and we extract both J (Fig. t2starc) and T 2 * (Fig. t2stard) as a function of ϵ . Indeed, the predicted behavior is observed: for moderate ϵ we see fast oscillations that decay after a few ns, and for** the **largest ϵ we see even faster oscillations that decay slowly. Here, too, we observe that T 2 * ∝ d J d ϵ -1 (Fig. t2stard), which indicates that FID oscillations in 0 2 are also primarily dephased by low **frequency** voltage noise. We note, however, that we extract a different constant of proportionality between T 2 * and d J / d ϵ -1 for 1 1 and 0 2 . This is expected given that** the **charge distributions associated with the **qubit** states are very different in these two regimes and thus have different sensitivities to applied electric fields. We note that in** the **regions of largest d J / d ϵ (near ϵ = 0 ), T 2 * is shorter than** the **rise time of our signal generator and we systematically underestimate J and overestimate T 2 * (Fig. S1)....Electrometry Using Coherent Exchange Oscillations in a Singlet-Triplet-**Qubit**...Spin-echo measurements reveal high **frequency** bath dynamics. a, The pulse sequence used to measure exchange echo rotations. b, A typical echo signal. The overall shape of the envelope reflects T 2 * , while the amplitude of the envelope as a function of τ (not pictured) reflects T 2 e c h o . c, T 2 e c h o and Q ≡ J T 2 e c h o / 2 π as a function of J . A comparison of the two noise models: power law and a mixture of white and 1 / f noise. Noise with a power law spectrum fits over a wide range of **frequencies** (constant β ), but the relative contributions of white and 1 / f noise change as a function of ϵ . d, A typical echo decay is non-exponential but is well fit by exp - τ / T 2 e c h o β + 1 . e, T 2 e c h o varies with d J / d ϵ in a fashion consistent with dephasing due to power law voltage fluctuations. echo...The device used in these measurements is a gate-defined S - T 0 **qubit** with an integrated RF sensing dot. a The detuning ϵ is the voltage applied to the dedicated high-**frequency** control leads pictured. b, The Bloch sphere that describes the logical subspace of this device features two rotation axes ( J and Δ B Z ) both controlled with DC voltage pulses. c, An energy diagram of the relevant low-lying states as a function of ϵ . States outside of the logical subspace of the **qubit** are grayed out. d, J ϵ and d J / d ϵ in three regions; the 1 1 region where J and d J / d ϵ are both small and S - T 0 **qubits** are typically operated, the transitional region where J and d J / d ϵ are both large where the **qubit** is loaded and measured, and the 0 2 region where J is large but d J / d ϵ is small and large quality oscillations are possible. pulses...Ramsey oscilllations reveal low **frequency** enivronmental dynamics. a, The pulse sequence used to measure exchange oscillations uses a stabilized nuclear gradient to prepare and readout the **qubit** and gives good contrast over a wide range of J . b, Exchange oscillations measured over a variety of detunings ϵ and timescales consistently show larger T 2 * as d J / d ϵ shrinks until dephasing due to nuclear fluctuations sets in at very negative ϵ . c, Extracted values of J and d J / d ϵ as a function of ϵ . d, The decay curve of FID exchange oscillations shows Gaussian decay. e, Extracted values of T 2 * and d J / d ϵ as a function of ϵ . T 2 * is proportional to d J / d ϵ -1 , indicating that voltage noise is the cause of dephasing of charge oscillations. f, Charge oscillations measured in 0 2 . This figure portrays the three basic regions we can operate our device in: a region of low **frequency** oscillations and small d J / d ϵ , a region of large **frequency** oscillations and large d J / d ϵ , and a region where oscillations are fast but d J / d ϵ is comparatively small. t2star...Previous work on S - T 0 **qubits** focused almost entirely on x ( Δ B Z ) rotations, which are dephased by fluctuations in** the **nuclear bath . In this work, we focus on** the **exchange interaction, which creates a splitting, J , between** the **S and T 0 states once** the **1 1 and 0 2 S states of** the **double QD are brought near resonance (Fig. pulsesc). The value of J depends on** the **energy detuning, ϵ , between** the **QDs. The exchange interaction drives single **qubit** rotations in S - T 0 and exchange-only **qubits** and is** the **foundation for two-**qubit** operations in single spin, S - T 0 , and exchange-only **qubits** (). Exchange oscillations are dephased by fluctuations in J (Fig. pulsesc) driven, for example, by ϵ (voltage) fluctuations between** the **dots with a tunable sensitivity proportional to d J / d ϵ (Fig. pulsesd) . We will show that this controllable sensitivity is a useful experimental tool for probing** the **noise bath dynamics. Previous studies have shown** the **decay of exchange oscillations within a few π rotations , but a detailed study of** the **nature of** the **noise bath giving rise to this decay is still lacking. In this work, using nuclear feedback to control x -rotations, we systematically explore** the **low **frequency** noise portion of** the **voltage noise bath and its temperature dependence, as well as introduce a new Hahn-echo based measurement of** the **high **frequency** components of** the **voltage noise and its temperature dependence....The **oscillations** in these FID experiments decay due to voltage noise from DC up to a **frequency** of approximately 1 / t . As the relaxation time, T 1 is in excess of 100 μ s in this regime, T 1 decay is not an important source of decoherence (Fig. S4). The shape of the decay envelope and the scaling of coherence time with d J / d ϵ (which effectively changes the magnitude of the noise) reveal information about the underlying noise spectrum. White (Markovian) noise, for example, results in an exponential decay of e - t / T 2 * where T 2 * ∝ d J / d ϵ -2 is the inhomogeneously broadened coherence time . However, we find that the decay is Gaussian (Fig. t2stard) and that T 2 * (black line in Fig. t2stare) is proportional to d J / d ϵ -1 (red solid line in Fig. t2stare) across two orders of magnitude of T 2 * . Both of these findings can be explained by quasistatic noise, which is low **frequency** compared to 1 / T 2 * . In such a case, one expects an amplitude decay of the form exp - t / T 2 * 2 , where T 2 * = 1 2 π d J / d ϵ ϵ R M S and ϵ R M S is the root-mean-squared fluctuation in ϵ (Eq. S3). From the ratio of T 2 * to d J / d ϵ -1 , we calculate ϵ R M S = 8 μ V in our device. At very negative ϵ , J becomes smaller than Δ B z , and nuclear noise limits T 2 * to approximately 90ns, which is consistent with previous work . We confirm that this effect explains deviations of T 2 * from d J / d ϵ -1 by using a model that includes the independently measured T 2 , n u c l e a r * and Δ B z (Eq. S1) and observe that it agrees well with measured T 2 * at large negative ϵ (dashed red line in Fig. t2stare)....Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** oscillations to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange oscillations. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity....The device used in these measurements is a gate-defined S - T 0 **qubit** with an integrated RF sensing dot. a The detuning ϵ is the voltage applied to the dedicated high-**frequency** control leads pictured. b, The Bloch sphere that describes the logical subspace of this device features two rotation axes ( J and Δ B Z ) both controlled with DC voltage pulses. c, An energy diagram of the relevant low-lying states as a function of ϵ . States outside of the logical subspace of the **qubit** are grayed out. d, J ϵ and d J / d ϵ in three regions; the 1 1 region where J and d J / d ϵ are both small and S - T 0 **qubits** are typically operated, the transitional region where J and d J / d ϵ are both large where the **qubit** is loaded and measured, and the 0 2 region where J is large but d J / d ϵ is small and large quality **oscillations** are possible. pulses...Ramsey oscilllations reveal low **frequency** enivronmental dynamics. a, The pulse sequence used to measure exchange **oscillations** uses a stabilized nuclear gradient to prepare and readout the **qubit** and gives good contrast over a wide range of J . b, Exchange **oscillations** measured over a variety of detunings ϵ and timescales consistently show larger T 2 * as d J / d ϵ shrinks until dephasing due to nuclear fluctuations sets in at very negative ϵ . c, Extracted values of J and d J / d ϵ as a function of ϵ . d, The decay curve of FID exchange **oscillations** shows Gaussian decay. e, Extracted values of T 2 * and d J / d ϵ as a function of ϵ . T 2 * is proportional to d J / d ϵ -1 , indicating that voltage noise is the cause of dephasing of charge **oscillations**. f, Charge **oscillations** measured in 0 2 . This figure portrays the three basic regions we can operate our device in: a region of low **frequency** **oscillations** and small d J / d ϵ , a region of large **frequency** **oscillations** and large d J / d ϵ , and a region where **oscillations** are fast but d J / d ϵ is comparatively small. t2star...Since we observe J to be approximately an exponential function of ϵ , ( d J / d ϵ ∼ J ), we expect and observe the quality (number of coherent **oscillations**) of these FID **oscillations**, Q ≡ J T 2 * / 2 π ∼ J d J / d ϵ -1 , to be approximately constant regardless of ϵ . However, when ϵ is made very positive and J is large, an avoided crossing occurs between the 1 1 T 0 and the 0 2 T 0 state, making the 0 2 S and 0 2 T 0 states electrostatically virtually identical. Here, as ϵ is increased, J increases but d J / d ϵ decreases(Fig. pulsesd), allowing us to probe high quality exchange rotations and test our charge noise model in a regime that has never before been explored....The simplest probe of J and its fluctuations is a free induction decay (FID) experiment, in which the **qubit** is allowed to freely precess for a time t under** the **influence of** the **exchange splitting. For FID measurements, we use a π / 2 pulse around** the **x -axis to prepare and readout** the **state of the **qubit** along** the **y -axis (Fig. t2stara, Fig. S1). Fig. t2starb shows **qubit** oscillations as a function of t for many different values of ϵ . By measuring** the **period of these oscillations with t we extract J ϵ , and we calculate d J / d ϵ by fitting J ϵ to a smooth function and differentiating (Fig. t2starc). For negative ϵ (small J ), we empirically find across many devices and tunings that J is well described by J ϵ ≃ J 0 + J 1 e x p - ϵ / ϵ 0 ....Two level quantum systems (**qubits**) are emerging as promising candidates both for quantum information processing and for sensitive metrology . When prepared in a superposition of two states and allowed to evolve, the state of** the **system precesses with a **frequency** proportional to** the **splitting between** the **states. However, on a timescale of** the **coherence time, T 2 , the **qubit** loses its quantum information due to interactions with its noisy environment. This causes **qubit** oscillations to decay and limits** the **fidelity of quantum control and** the **precision of **qubit**-based measurements. In this work we study singlet-triplet ( S - T 0 ) **qubits**, a particular realization of spin **qubits** , which store quantum information in** the **joint spin state of two electrons. We form the **qubit** in two gate-defined lateral quantum dots (QD) in a GaAs/AlGaAs heterostructure (Fig. pulsesa). The QDs are depleted until there is exactly one electron left in each, so that** the **system occupies** the **so-called 1 1 charge configuration. Here n L n R describes a double QD with n L electrons in** the **left dot and n R electrons in** the **right dot. This two-electron system has four possible spin states: S , T + , T 0 , and T - . The S , T 0 subspace is used as** the **logical subspace for this **qubit** because it is insensitive to homogeneous magnetic field fluctuations and is manipulable using only pulsed DC electric fields . The relevant low-lying energy levels of this **qubit** are shown in Fig. pulsesc. Two distinct rotations are possible in these devices: rotations around** the **x -axis of** the **Bloch sphere driven by difference in magnetic field between** the **QDs, Δ B z (provided in this experiment by feedback-stabilized hyperfine interactions), and rotations around** the **z -axis driven by** the **exchange interaction, J (Fig. pulsesb) . A S can be prepared quickly with high fidelity by exchanging an electron with** the **QD leads, and** the **projection of** the **state of the **qubit** along** the **z -axis can be measured using RF reflectometery with an adjacent sensing QD (green arrow in Fig. pulsesa)....Using a modified pulse sequence that changes the clock **frequency** of our waveform generators to achieve picosecond timing resolution (Fig. S1)), we measure exchange **oscillations** in 0 2 as a function of ϵ and time (Fig. t2stare) and we extract both J (Fig. t2starc) and T 2 * (Fig. t2stard) as a function of ϵ . Indeed, the predicted behavior is observed: for moderate ϵ we see fast **oscillations** that decay after a few ns, and for the largest ϵ we see even faster **oscillations** that decay slowly. Here, too, we observe that T 2 * ∝ d J d ϵ -1 (Fig. t2stard), which indicates that FID **oscillations** in 0 2 are also primarily dephased by low **frequency** voltage noise. We note, however, that we extract a different constant of proportionality between T 2 * and d J / d ϵ -1 for 1 1 and 0 2 . This is expected given that the charge distributions associated with the **qubit** states are very different in these two regimes and thus have different sensitivities to applied electric fields. We note that in the regions of largest d J / d ϵ (near ϵ = 0 ), T 2 * is shorter than the rise time of our signal generator and we systematically underestimate J and overestimate T 2 * (Fig. S1)....In our echo measurements, we select a fixed ϵ inside 1 1 for** the **free evolutions, and we sweep** the **length of** the **evolution following** the **π -pulse time by small increments δ t to reveal an echo envelope (Fig. echoa-b). The maximum amplitude of this observed envelope reveals** the **extent to which** the **state has dephased during** the **echo process, while** the **Gaussian shape and width of** the **envelope arise from an effective single-**qubit** rotation for a time δ t , and thus reflect** the **same T 2 * and low **frequency** noise measured in FID experiments. We note that this exchange echo is distinct from** the **echo measurements previously performed in singlet-triplet **qubits** in that we use Δ B z rotations to echo away voltage noise, rather than J rotations to echo away noise in** the **nuclear bath....The use of Hahn echo dramatically improves coherence times, with T 2 e c h o (the τ at which the observed echo amplitude has decayed by 1 / e ) as large as 9 μ s , corresponding to qualities ( Q ≡ T 2 e c h o J / 2 π ) larger than 600 (Fig. echoc). If at high **frequencies** (50kHz-1MHz) the voltage noise were white (Markovian), we would observe exponential decay of the echo amplitude with τ . However, we find that the decay of the echo signal is non-exponential (Fig. echod), indicating that even in this relatively high-**frequency** band being probed by this measurement, the noise bath is not white. ... Two level systems that can be reliably controlled and measured hold promise in both metrology and as **qubits** for quantum information science (QIS). When prepared in a superposition of two states and allowed to evolve freely, the state of the system precesses with a **frequency** proportional to the splitting between the states. In QIS,this precession forms the basis for universal control of the **qubit**,and in metrology the **frequency** of the precession provides a sensitive measurement of the splitting. However, on a timescale of the coherence time, $T_2$, the **qubit** loses its quantum information due to interactions with its noisy environment, causing **qubit** **oscillations** to decay and setting a limit on the fidelity of quantum control and the precision of **qubit**-based measurements. Understanding how the **qubit** couples to its environment and the dynamics of the noise in the environment are therefore key to effective QIS experiments and metrology. Here we show measurements of the level splitting and dephasing due to voltage noise of a GaAs singlet-triplet **qubit** during exchange **oscillations**. Using free evolution and Hahn echo experiments we probe the low **frequency** and high **frequency** environmental fluctuations, respectively. The measured fluctuations at high **frequencies** are small, allowing the **qubit** to be used as a charge sensor with a sensitivity of $2 \times 10^{-8} e/\sqrt{\mathrm{Hz}}$, two orders of magnitude better than the quantum limit for an RF single electron transistor (RF-SET). We find that the dephasing is due to non-Markovian voltage fluctuations in both regimes and exhibits an unexpected temperature dependence. Based on these measurements we provide recommendations for improving $T_2$ in future experiments, allowing for higher fidelity operations and improved charge sensitivity.

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Contributors: Strauch, F. W., Dutta, S. K., Paik, Hanhee, Palomaki, T. A., Mitra, K., Cooper, B. K., Lewis, R. M., Anderson, J. R., Dragt, A. J., Lobb, C. J.

Date: 2007-03-02

The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the **oscillation** **frequency** Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 ....Experimental microwave spectroscopy of a Josephson phase **qubit**, scanned in **frequency** (vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 ....Rabi **frequency** Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....The ac Stark shift Δ ω 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. The inset shows the oscillation **frequency** Ω ̄ R , 01 as a function of the level spacing ω 01 for I a c = 5.87 nA and the fit using ( rabif) to obtain Ω R , 01 and Δ ω 01 ....Strong-field effects in the Rabi oscillations of the superconducting phase **qubit**...Rabi **frequency** Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi **oscillations** of the escape rate for I a c = 16.5 nA....Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment....Experimental microwave spectroscopy of a Josephson phase qubit, scanned in **frequency** (vertical) and bias current (horizontal). Dark points indicate experimental microwave enhancement of the tunneling escape rate, while white dashed lines are quantum mechanical calculations of (from right to left) ω 01 , ω 02 / 2 , ω 12 , ω 13 / 2 , and ω 23 ....Rabi **frequency** Ω R , 02 of the two-photon 0 2 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed line perturbative results. Inset shows Rabi oscillations of the escape rate for I a c = 16.5 nA....Rabi **frequency** Ω R , 01 of the one-photon 0 1 transition as function of microwave current I a c . The dots are experimental data, the solid line predictions from the three-level model, and the dashed lines are the lowest-order results ( rabi1) (top) and second-order ( rabi2) (bottom) perturbative results. The inset shows Rabi oscillations of the escape rate for I a c = 16.5 nA....Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment. ... Rabi **oscillations** have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (**qubits**) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase **qubit** (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi **frequency**, and two-photon Rabi **frequency** are compared to measurements made on a dc SQUID phase **qubit** with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.

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Contributors: Averin, D. V.

Date: 2002-02-05

The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics....Spin representation of the QND measurement of the quantum coherent oscillations of a **qubit**. The oscillations are represented as a spin rotation in the z - y plane with frequency Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with frequency Ω ≃ Δ ....Schematic of the Josephson-junction **qubit** structure that enables measurements of the two non-commuting observables of the **qubit**, σ z and σ y , as required in the QND Hamiltonian ( 2). For discussion see text....Spin representation of the QND measurement of the quantum coherent **oscillations** of a **qubit**. The **oscillations** are represented as a spin rotation in the z - y plane with **frequency** Δ . QND measurement is realized if the measurement frame (dashed lines) rotates with **frequency** Ω ≃ Δ ....The concept of quantum nondemolition (QND) measurement is extended to coherent oscillations in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these oscillations avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics....Quantum nondemolition measurements of a **qubit** ... The concept of quantum nondemolition (QND) measurement is extended to coherent **oscillations** in an individual two-state system. Such a measurement enables direct observation of intrinsic spectrum of these **oscillations** avoiding the detector-induced dephasing that affects the standard (non-QND) measurements. The suggested scheme can be realized in Josephson-junction **qubits** which combine flux and charge dynamics.

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Contributors: Dutta, S. K., Strauch, Frederick W., Lewis, R. M., Mitra, Kaushik, Paik, Hanhee, Palomaki, T. A., Tiesinga, Eite, Anderson, J. R., Dragt, Alex J., Lobb, C. J.

Date: 2008-06-28

F031806Dfosc Rabi oscillation frequency Ω R , 0 1 at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig. F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930 μ A and C 1 = 4.50 p F with ω r f / 2 π = 6.2 G H z ....F031806Dfosc Rabi **oscillation** **frequency** Ω R , 0 1 at fixed bias as a function of microwave current I r f . Extracted values from data (including the plots in Fig. F031806DGtot) are shown as circles, while the rotating wave solution is shown for two- (dashed line) and five- (solid) level simulations, calculated using I 01 = 17.930 μ A and C 1 = 4.50 p F with ω r f / 2 π = 6.2 G H z ....FDeviceThe dc SQUID phase **qubit**. (a) The **qubit** junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the phase difference γ 1 across the **qubit** junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n differs in energy from m by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c) The photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-**qubit** experiments; the second SQUID was kept unbiased throughout the course of this work....F010206H1 (Color online) Multiphoton, multilevel Rabi oscillations plotted in the time and frequency domains. (a**) The** escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the qubit; Γ ranges from 0 (white) to 3 × 10 8 1 / s (black). (b**) The** normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi frequencies obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828 μ A and C 1 = 4.52 p F , and microwave current I r f = 24.4 n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d)....In** **order** **to** **follow** **experimentally** **the** **shift** **of** **the** **resonance** **condition, it** **was** **necessary** **to** **measure** **Rabi** **oscillations** **for** **different** **detunings** **of** **the** **microwave** **drive. We** **chose** **to** **do** **this** **by** **keeping** **the** **drive** **frequency** **ω** **r** **f fixed** **and** **changing** **the** **level** **spacing** **ω** **0** **1 (through** **I** **b ), because** **the** **power** **transmitted** **by** **the** **microwave** **lines** **had** **a** **nontrivial** **frequency** **dependence. Figure F010206H1(a) shows** **a** **grayscale** **plot** **of** **Rabi** **oscillations** **measured** **from** **such** **an** **experiment, where** **black** **represents** **a** **high** **escape** **rate. Each** **horizontal** **line** **is** **the** **escape** **rate** **versus** **time** **due** **to** **a** **microwave** **current** **of** **6.5** **GHz** **and -11** **dBm, which** **was** **turned** **on** **at** **the** **value** **of** **the** **current** **bias** **I** **b** **indicated** **on** **the** **vertical** **axis. While** **the** **measurements** **were** **performed** **at** **110** **mK, this** **is** **not** **expected** **to** **have** **a** **significant** **impact** **on** **the** **Rabi** **oscillations, as** **the** **temperature** **was** **well** **below** **ℏ** **ω** **0** **1 / k** **B ≈ 325 m** **K ....We present Rabi oscillation measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi oscillations were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed oscillation **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states....F031806DGtot Rabi **oscillations** in the escape rate Γ were induced at I b = 17.746 μ A by switching on a microwave current at t = 0 with a **frequency** of 6.2 GHz (resonant with the 0 → 1 transition) and source powers P S between -12 and -32 dBm, as labeled. The measurements were taken at 20 mK. The solid lines are from a five-level density-matrix simulation with I 01 = 17.930 μ A , C 1 = 4.50 p F , T 1 = 17 n s , and T φ = 16 n s ....Figure F010206H1(b) shows that the minimum **oscillation** **frequency** Ω R , 0 1 m i n / 2 π = 540 M H z of the first (experimental) band occurs at I b = 17.624 μ A , for which ω 0 1 / 2 π = 6.4 G H z . This again indicates an ac Stark shift of this transition, which we denote by Δ ω 0 1 ≡ ω r f - ω 0 1 ≈ 2 π × 100 M H z . In addition, the higher levels have suppressed the **oscillation** **frequency** below the bare Rabi **frequency** of Ω 0 1 / 2 π = 620 M H z [calculated with Eq. ( eqf)]....Multilevel effects in the Rabi oscillations of a Josephson phase **qubit**...F032206MNstats (Color online**) The** (a) on-resonance Rabi oscillation frequencies Ω R , 0 1 m i n and Ω R , 0 2 m i n and (b) resonance frequency shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110 m K with a microwave drive of frequency ω r f / 2 π = 6.5 G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10 d B m . Values extracted from data for the 0 → 1 ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736 μ A and C 1 = 4.49 p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system....We fit the escape rates in Fig. F031806DGtot (and additional data for other powers not shown) to a decaying sinusoid with an offset. The extracted **frequencies** are shown with circles in Fig. F031806Dfosc. To compare to theory, Ω R , 0 1 , calculated using the rotating wave solution for a system with five levels, is shown with a solid line. The implied assumption that the **oscillation** **frequencies** of Γ and ρ 11 are equal, even at high power in a multilevel system, will be addressed in Sec. SSummary. In plotting the data, we have introduced a single fitting parameter 117 n A / m W that converts the power P S at the microwave source to the current amplitude I r f at the **qubit**. Good agreement is found over the full range of power....FEnergyGamma Qubit energy-level spectroscopy and tunneling escape rates. (a) Open circles show the resonance frequency of the transition between the ground and first excited states of the qubit, measured at 20 mK. The scatter in the values is indicative of the uncertainty in the measurement. Also plotted are theoretical values of ω 0 1 (solid), ω 1 2 (dashed), and ω 2 3 (dotted) for I 01 = 17.930 μ A and C 1 = 4.50 p F . (b) At high bias, the measured background escape rate (open circles) agrees with the predicted ground-state escape rate Γ 0 (solid line) for the junction parameters given above. Calculated Γ 1 , Γ 2 , and Γ 3 are plotted as dashed, dotted, and dashed-dotted lines. In both plots, the bottom axes show the total bias current I b , while the top axes indicate the normalized barrier height N s , calculated using the extracted junction parameters....We next consider the time dependence of the escape rate for the data plotted in Fig. F032106FN10. Here, a 6.2 GHz microwave pulse nominally 30 ns long was applied on resonance with the 0 → 1 transition of the **qubit** junction. The measured escape rate shows Rabi **oscillations** followed by a decay back to the ground state once the microwave drive has turned off. This decay appears to be governed by three time constants. Nontrivial decays have previously been reported in phase **qubits** and we have found them in several of our devices....We** **fit** **the** **escape** **rates** **in** **Fig. F031806DGtot (and** **additional** **data** **for** **other** **powers** **not** **shown) to** **a** **decaying** **sinusoid** **with** **an** **offset. The** **extracted** **frequencies** **are** **shown** **with** **circles** **in** **Fig. F031806Dfosc. To** **compare** **to** **theory, Ω** **R , 0** **1 , calculated** **using** **the** **rotating** **wave** **solution** **for** **a** **system** **with** **five** **levels, is** **shown** **with** **a** **solid** **line. The** **implied** **assumption** **that** **the** **oscillation** **frequencies** **of** **Γ** **and** **ρ** **11** **are** **equal, even** **at** **high** **power** **in** **a** **multilevel** **system, will** **be** **addressed** **in** **Sec. SSummary. In** **plotting** **the** **data, we** **have** **introduced** **a** **single** **fitting** **parameter** **117 n** **A / m** **W** **that** **converts** **the** **power** **P** **S at** **the** **microwave** **source** **to** **the** **current** **amplitude** **I** **r** **f at** **the** **qubit. Good** **agreement** **is** **found** **over** **the** **full** **range** **of** **power....As I r f increases in Fig. F031806Dfosc, the **oscillation** **frequency** is smaller than the expected linear relationship for a two-level system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase **qubit**. There are two distinct phenomena that affect 0 → 1 Rabi **oscillations** in such a device. To describe...FDeviceThe dc SQUID **phase** qubit. (a**) The** qubit junction J 1 (with critical current I 01 and capacitance C 1 ) is isolated from the current bias leads by an auxiliary junction J 2 (with I 02 and C 2 ) and geometrical inductances L 1 and L 2 . The device is controlled with a current bias I b and a flux current I f which generates flux Φ a through mutual inductance M . Transitions can be induced by a microwave current I r f , which is coupled to J 1 via C r f . (b) When biased appropriately, the dynamics of the **phase** difference γ 1 across the qubit junction are analogous to those of a ball in a one-dimensional tilted washboard potential U . The metastable state n differs in energy from m by ℏ ω n m and tunnels to the voltage state with a rate Γ n . (c**) The** photograph shows a Nb/AlO x /Nb device. Not seen is an identical SQUID coupled to this device intended for two-qubit experiments; the second SQUID was kept unbiased throughout the course of this work....F032206MNstats (Color online) The (a) on-resonance Rabi **oscillation** **frequencies** Ω R , 0 1 m i n and Ω R , 0 2 m i n and (b) resonance **frequency** shifts Δ ω 0 1 = ω r f - ω 0 1 and Δ ω 0 2 = 2 ω r f - ω 0 2 are plotted as a function of the microwave current, for data taken at 110 m K with a microwave drive of **frequency** ω r f / 2 π = 6.5 G H z and powers P S = - 23 , - 20 , - 17 , - 15 , - 10 d B m . Values extracted from data for the 0 → 1 ( 0 → 2 ) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with I 01 = 17.736 μ A and C 1 = 4.49 p F are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a two-level system....We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states....The** **resonant** **oscillation** **frequencies** **Ω** **R , 0** **1** **m** **i** **n and** **Ω** **R , 0** **2** **m** **i** **n in** **Fig. F032206MNstats(a) are** **well** **described** **by** **Eq. ( eqd) over** **the** **full** **range** **of** **I** **r** **f . The** **deviation** **between** **the** **0 → 1 oscillation** **frequency** **and** **the** **values** **expected** **in** **a** **two-level** **system (dotted** **line) increases** **with** **I** **r** **f . Similar** **measurements** **taken** **at** **a** **higher** **I** **b** **show** **a** **smaller** **frequency** **suppression** **over** **a** **similar** **range** **of** **I** **r** **f , as** **expected** **for** **a** **system** **with** **stronger** **anharmonicity....F010206H1 (Color online) Multiphoton, multilevel Rabi **oscillations** plotted in the time and **frequency** domains. (a) The escape rate Γ (measured at 110 mK) is plotted as a function of the time after which a 6.5 GHz, -11 dBm microwave drive was turned on and the current bias I b of the **qubit**; Γ ranges from 0 (white) to 3 × 10 8 1 / s (black). (b) The normalized power spectral density of the time-domain data from t = 1 to 45 ns is shown with a grayscale plot. The dashed line segments indicate the Rabi **frequencies** obtained from the rotating wave model for transitions involving (from top to bottom) 1, 2, 3, and 4 photons, evaluated with junction parameters I 01 = 17.828 μ A and C 1 = 4.52 p F , and microwave current I r f = 24.4 n A . Corresponding grayscale plots calculated with a seven-level density-matrix simulation are shown in (c) and (d)....Figure F010206H1(b) shows** **that** **the** **minimum** **oscillation** **frequency** **Ω** **R , 0** **1** **m** **i** **n / 2** **π = 540 M** **H** **z** **of** **the** **first (experimental) band** **occurs** **at** **I** **b = 17.624 μ** **A , for** **which** **ω** **0** **1 / 2** **π = 6.4 G** **H** **z . This** **again** **indicates** **an** **ac** **Stark** **shift** **of** **this** **transition, which** **we** **denote** **by** **Δ** **ω** **0** **1 ≡ ω** **r** **f - ω** **0** **1 ≈ 2** **π × 100 M** **H** **z . In** **addition, the** **higher** **levels** **have** **suppressed** **the** **oscillation** **frequency** **below** **the** **bare** **Rabi** **frequency** **of** **Ω** **0** **1 / 2** **π = 620 M** **H** **z [calculated** **with** **Eq. ( eqf)]....For this data set, the level spacing ω 0 1 / 2 π is equal to the microwave **frequency** ω r f / 2 π = 6.5 G H z at I b = 17.614 μ A . The band with the highest current in Fig. F010206H1(b) is centered about I b = 17.624 μ A , suggesting that 0 → 1 Rabi **oscillations** are the dominant process near this bias. For slightly higher or lower I b , the **oscillation** **frequency** increases as Ω R , 0 1 ≈ Ω 01 ′ 2 + ω r f - ω 0 1 2 , in agreement with simple two-level Rabi theory, leading to the curved band in the grayscale plot....Figure FDevice(a) shows** **the** **circuit** **schematic** **for** **our** **dc** **SQUID** **phase** **qubit. The** **qubit** **junction** **J** **1 (with** **critical** **current** **I** **01** **and** **capacitance** **C** **1 ) is** **shown** **on** **the** **left. It** **is** **isolated** **from** **the** **current** **bias** **source** **I** **b** **by** **geometrical** **inductances** **L** **1** **and** **L** **2** **and** **the** **second** **junction** **J** **2 (with** **I** **02** **and** **C** **2 ). In** **order** **to** **independently** **control** **the** **currents** **in** **the** **two** **arms** **of** **the** **resulting** **dc** **SQUID, a** **current** **source** **I** **f** **applies** **a** **flux** **Φ** **a** **to** **the** **SQUID** **loop** **through** **mutual** **inductance** **M . Good** **isolation** **of** **the** **qubit** **junction** **is** **obtained** **when** **L** **1 / M ≫ 1** **and** **L** **1 / L** **2 + L** **J** **2 ≫ 1 , where** **L** **J** **2** **is** **the** **Josephson** **inductance** **of** **the** **isolation** **junction. ... We present Rabi **oscillation** measurements of a Nb/AlOx/Nb dc superconducting quantum interference device (SQUID) phase **qubit** with a 100 um^2 area junction acquired over a range of microwave drive power and **frequency** detuning. Given the slightly anharmonic level structure of the device, several excited states play an important role in the **qubit** dynamics, particularly at high power. To investigate the effects of these levels, multiphoton Rabi **oscillations** were monitored by measuring the tunneling escape rate of the device to the voltage state, which is particularly sensitive to excited state population. We compare the observed **oscillation** **frequencies** with a simplified model constructed from the full phase **qubit** Hamiltonian and also compare time-dependent escape rate measurements with a more complete density-matrix simulation. Good quantitative agreement is found between the data and simulations, allowing us to identify a shift in resonance (analogous to the ac Stark effect), a suppression of the Rabi **frequency**, and leakage to the higher excited states.

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