### 54072 results for qubit oscillator frequency

Contributors: Jiang, Wei, Yu, Yang, Wei, Lianfu

Date: 2011-01-01

We theoretically study the measurement-induced dephasing caused by back action noise in quantum nondemolition measurements of a superconducting flux **qubit** which is coupled to a superconducting quantum interference device (SQUID). Our analytical results indicate that information on **qubit** flows from **qubit** to detector, while quantum fluctuations which may cause dephasing of the **qubit** also inject to **qubit**. Furthermore, the measurement probability is **frequency** dependent in a short time scale and has a close relationship with the measurement-induced dephasing. When the detuning between driven and bare resonator equals coupling strength, we will access the state of **qubit** more easily. In other words, we obtain the maximum measurement rate. Finally, we analyzed mixed effect caused by coupling between non-diagonal term and external variable. We found that the initial information of **qubit** is destroyed due to quantum tunneling between the **qubit** states....Recently, QND measurements have experimentally been achieved on superconducting flux **qubit** system by investigating the correlation between the results of two consecutive measurements. However, in general , the measuring apparatus is a mesoscopic system (e.g., dc-SQUID) which is coupled to environment. Therefore, noise can affect discrimination of signal which reflects the information of **qubit** via coupling between meter and **qubit** (see Fig. fig1(b)). There exist a larger number of works in mesoscopic system where one is forced to think about the quantum mechanics of detection process, and about the fundamental quantum limits which constrain the performance of the detector. In practice, noise plays an important role in quantum measurement: quantum noise from the meter acts back on the system, such as **qubit**, at the same time, the information about the variable conjugate to the measured variable is destroyed. This phenomenon is omnipresent, because information about system is carried away into the surrounding due to indirect environmental coupling via coupling to detector. For a weak measurement, the detection may be quantum limited that the signal-to-noise of measurement, defined as the ratio of the amplitude of the **oscillation** line in the output spectrum to background noise, is no more than 4....(color online) (a) Probability distribution considering zero point fluctuation for getting the measurement result for 0 and the initial state | 0 0 | , plotted as a function of detuning δ ω . (b) Probability distribution considering the back action noise instead of zero point fluctuation with the same parameter. The damping rate of resonator κ = 0.1 G H z , the couple strength between **qubit** and **oscillator** g = 0.3 G H z and the back action noise spectra density S I I = 2 / κ ....(color online) (a) ∼ (d) Probability of measuring 0 state for the initial state | 0 0 | with damping rate of **oscillator** κ = 0.1 ~ G H z , κ = 0.2 ~ G H z , κ = 0.3 G H z , κ = 0.4 G H z , respectively, and **qubit** coupling strength g = 0.3 G H z , t = 0.1 n s , noise spectra density S I I = 2 / κ . (e) ∼ (h) measurement-induced dephasing rate Γ m as a function of detuning δ ω between resonator and drive **frequency** with driving force f = 1 G H z . The parameters of (e) ∼ (h) are same with those in (a) ∼ (d), respectively....Then, if we know **oscillator** state is α i , the **qubit** state can be determined exactly. From Eq. ( 17) we find that the measurement-induced dephasing depends on the overlap of both two **oscillator** state α ± . In case that the **oscillator** states | α - and | α + are orthogonal, the measurement could be considered as a strong projective measurement, i.e., in the region e r f | A | t = 1 . For the maximum dephasing value, the **oscillator** states decay to a steady state significantly, which explains the feature of Fig. fig3 that both measurement-induced dephasing rate and measurement probability get a high value for matching the **qubit** **frequency**. In this situation, the **qubit** state is encoded in the amplitude of coherence rather than phase. It means that the easier we gain information about amplitude, the faster we lose information about phase. The measurement-induced dephasing rate can also be expressed as...(a) Schematic of flux **qubit** coupled to a SQUID which used for **qubit** readout as a detector. (b) Schematic of the process of information detraction from **qubit** and noise injection from one-port detector....In addition, as shown in Fig. fig3, with the value of damping rate κ increasing, both structures of probability distribution and measurement-induced dephasing rate become flat with synchronous tendency, which consistent with the expectation. Increasing of driving force f only change the overall scale of (e) ∼ (h) rather than the structure, which explains the trend that with the value of driving force increasing, the strong measurement region in (a) ∼ (h) becomes wider and the measurement probability increases significantly, especially for matching the **qubit** **frequency**....where n ± = | α ± s | 2 = f 2 / κ 2 / 4 + δ ω ± g 2 is average number of photons in the resonator. We find that for δ ω = ± g 2 + κ 2 / 4 we will get a large probability if measure the amplitude of the signal. In other words, when matching one of the **frequencies** of **qubit** (i.e., δ ω = ± g ), the information of **qubit** is encoded in the amplitude rather than phase which is the conjugate variable with amplitude. At the same time, the measurement-induced dephasing rate will get the maximum value. In Fig. fig3, we plot the probability distribution that measuring 0 state conditioned on initial state is prepared on | 0 0 | and measurement-induced dephasing as a function of detuning. When matching one of **qubit** **frequency**, both measurement-induced dephasing rate and measurement probability get a maximum value. In the region where e r f | A | t ≈ 1 , it corresponds to a strong projective measurement case. At the same time, the two coherent states of **oscillator** are well separated in phase space due to decay of coherence term at a measurement-induce dephaing rate Γ m . In general case, at time t = 0 , the system is in a product state ... We theoretically study the measurement-induced dephasing caused by back action noise in quantum nondemolition measurements of a superconducting flux **qubit** which is coupled to a superconducting quantum interference device (SQUID). Our analytical results indicate that information on **qubit** flows from **qubit** to detector, while quantum fluctuations which may cause dephasing of the **qubit** also inject to **qubit**. Furthermore, the measurement probability is **frequency** dependent in a short time scale and has a close relationship with the measurement-induced dephasing. When the detuning between driven and bare resonator equals coupling strength, we will access the state of **qubit** more easily. In other words, we obtain the maximum measurement rate. Finally, we analyzed mixed effect caused by coupling between non-diagonal term and external variable. We found that the initial information of **qubit** is destroyed due to quantum tunneling between the **qubit** states.

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Contributors: Alexander N. Korotkov

Date: 2005-03-01

We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**....Solid lines: synchronization degree D (and in-phase current quadrature 〈X〉) as functions of F for several values of the detection efficiency ηeff. Dashed and dotted lines illustrate the effects of the energy mismatch (ε≠0) and the **frequency** mismatch (Ω≠Ω0).
... We discuss an experimental proposal on quantum feedback control of a double-dot **qubit**, which seems to be within the reach of the present-day technology. Similar to the earlier proposal, the feedback loop is used to maintain the coherent **oscillations** in the **qubit** for an arbitrary long time; however, this is done in a significantly simpler way. The main idea is to use the quadrature components of the noisy detector current to monitor approximately the phase of **qubit** **oscillations**.

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Contributors: Chen, Yu, Neill, C., Roushan, P., Leung, N., Fang, M., Barends, R., Kelly, J., Campbell, B., Chen, Z., Chiaro, B.

Date: 2014-02-28

Note that the width of the wavefunction is set by the **oscillator** impedance Z o = 1 / ω C = ω L = L / C . Varying this impedance changes the widths of the charge and flux wavefunctions, as illustrated in Table tab:Zo. The impedance is also important since it is used to describe how strongly the **oscillator** couples to other modes. The flux and charge operators are conveniently expressed in terms of the raising and lowering operators...The next largest contribution to errors are from non-adiabatic transitions from the | 11 to | 02 state. We directly measure this transition using a Ramsey error filter technique ; the pulse sequence is shown inset in Fig. fig:budget(b). We initialize the system in the | 11 state and then apply two CZ gates separated by a variable delay time. Afer applying a π -pulse to each **qubit**, we measure the uncorrelated excited state probability for each **qubit**. The results are shown in Fig. fig:budget(b), where we see the expected **oscillations** that result from the interference between two CZ gates. The **frequency** of the **oscillation** is set by the detuning of the | 11 and | 02 states which was 130 MHz, corresponding to a period of 8 ns. The | 02 state leakage error is given as 1/4 of the **oscillation** amplitude (peak-to-peak). For our 30 ns CZ gate, we measured a non-adiabatic error of ∼ 0.25 % . This is suprisingly small considering such a short gate time, and can be exponentially surpressed with increasing gate length....We introduce a superconducting **qubit** architecture that combines high-coherence **qubits** and tunable **qubit**-**qubit** coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the **frequency** crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamic coupling by implementing a novel adiabatic controlled-Z gate, at a speed approaching that of single-**qubit** gates. Integrating coherence and scalable control, our "gmon" architecture is a promising path towards large-scale quantum computation and simulation....The most important part of constructing this tunable coupling architecture is to maintain the coherence inherent in the Xmon design. There are two primary sources of loss associated with the modifications that we have made: capacitive coupling to surface defects on the coupling structure and inductive coupling to the added bias line. The voltage divider created by L J and L g reduce capacitive losses by a factor of over 2000. The coupler bias line has a mutual inductance to the junction loop of 1 pH; this 1 pH coupling to a 50 Ohm line introduces a decoherence source with an associated T 1 of greater than 200 μ s at 80 MHz of coupling. We measure T 1 as a function of the **qubit** **frequency** and plot the results in Fig. fig:T1(a). These rsults are comparable to the performance of previous Xmon devices with similar capacitor geometry and growth conditions. We observe no indication that the T 1 is reduced as we vary the coupling strength, with data shown in Fig. fig:T1(b)....An important application of tunable coupling is to isolate individual **qubits** for local operations by turning off the coupling. We characterize the zero coupling of our architecture using a modified swap spectroscopy measurement. We bring the two **qubits** on resonance and vary the coupler flux bias. For each value of the coupling strength, we excite Q 1 , wait a variable delay time and measure its excited state probability. As the results in Fig. fig:off(a) show, over a wide range of biases, the two **qubits** can interact and swap an excitation. At a coupler bias of ∼ 0.32 Φ 0 , there is no excitation swapping between the two **qubits**, indicating that the coupling is turned off. Focusing on zero coupling, we examine the excited state probability P 1 of Q 1 over a extended delay time, with the results shown in Fig. fig:off(b). We see no indication of swapping between the two **qubits** after 6 μ s. This places an upper bound on residual coupling of 50 kHz, resulting in an on/off ratio > 1000 ....(a) Interleaved randomized benchmarking on a 20 ns two-**qubit** idle gate ( g = 0 ). We extract a fidelity of 99.56 % , which suggests a decoherence error of 0.66 % for the 30 ns CZ gate. (b) Inset: The pulse sequence for the Ramsey error filter technique. Main panel: The measured excited state probability P 1 + P 2 as a function of the delay between two CZ gates. We observe the expected sinusoidal **oscillation** with a peak-to-peak amplitude of 1 % . The non-adiabatic error from | 02 state leakage is 1/4 of the **oscillation** amplitude and is therefore 0.25 % ....(a) T 1 of Q 1 as a function of the **qubit** **frequency**, when g = 0 . These results are comparable to that of the Xmon with similar capacitor geometry and growth conditions. (b) T 1 of Q 1 as a function of the coupler bias, when the **qubit** **frequency** is set to 5.3 GHz. We find no dependence of the T 1 on the coupling strength....The results of this compensation protocol are shown in Fig. fig:calibration(a). For each value of the coupler flux bias, we sweep the microwave drive **frequency** and measure the excited state probability P 1 . The **frequency** is almost completely independent of the coupler bias, with a standard deviation of 110 kHz. We fit each vertical column of data for a peak and plot the results in blue in Fig. fig:calibration(b). We perform an identical measurement without calibration and overlay the results in green. We see that the **qubit** **frequency** shifts by over 60 MHz ( ∼ g / 2 π ) as we vary the coupler bias....(a) The **frequency** of Q 1 , as a function of the coupler flux bias while the second **qubit** is far detuned. For each value of the coupling strength, we compensate the **frequency** shift due to the change in inductance, sweep the microwave drive **frequency** and measure the **qubit** excited state probability P 1 . Each line is fit for a peak, with the results plotted in panel (b) in blue. The associated standard deviation is 110 kHz. The same experiment is performed without the calibration and overlayed in green....(a) Swap spectroscopy for Q 1 , as a function of the coupler flux bias, with the two **qubits** on resonance. For each value of the coupling strength, we excite Q 1 , wait a variable delay time and measure the excited state probability P 1 . We see no excitation swapping between the two **qubits** when coupler bias is ∼ 0.32 Φ 0 , indicating that the coupling is turned off. (b) We set the coupler bias to this value and examine the excited state probability P 1 of Q 1 over an extended delay time. We see no indication of swapping between the two **qubits** after 6 μ s (placing an upper bound on residual coupling of 50 kHz.) ... We introduce a superconducting **qubit** architecture that combines high-coherence **qubits** and tunable **qubit**-**qubit** coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the **frequency** crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamic coupling by implementing a novel adiabatic controlled-Z gate, at a speed approaching that of single-**qubit** gates. Integrating coherence and scalable control, our "gmon" architecture is a promising path towards large-scale quantum computation and simulation.

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Contributors: Shevchenko, S. N., Kiyko, A. S., Omelyanchouk, A. N., Krech, W.

Date: 2004-12-21

This means that at ω = Δ E there is resonance, P ¯ + = 1 2 , and P + t is an **oscillating** function with the **frequency** x 0 . This is illustrated in Fig. P(t)a. The width of the peak at ω = Δ E of the P ¯ + – ω curve at the half-maximum (i.e., at P + = 1 / 4 ) is approximately 2 x 0 (see the upper panels of Fig. P(w))....Dependence of the probability P ¯ + on the **frequency** ω for different x 0 at Γ φ = Γ r e l a x = 0 and at x o f f = 0 (solid line) and x o f f = 0.2 Δ (dashed line). (Only the first few resonant peaks are plotted; the others, which are very narrow, are not shown at the graphs.) Inset: enlargement of the low-**frequency** region....Finally, we illustrate the multiphoton resonant excitations of the interferometer-type charge **qubit**. Making use of the numerical solution of the master equation (Sec. II), we find the time-averaged probability P ¯ + plotted in Fig. P(w)_2. The position of the multiphoton resonant peaks is defined by the relation ω = Δ E / K , where Δ E = Δ E δ D C is supposed to be fixed. Alternatively, when the δ D C component of the phase is changed and the **frequency** ω is fixed, a similar graph can be plotted with resonances at δ D C = δ D C K defined by the relation Δ E δ D C K = K ω ....Dependence of the probability P ¯ + on the **frequency** ω for the phase-biased charge **qubit** at n g = 0.95 , E J 1 / E C = 12.4 , E J 2 / E C = 11 , Γ φ / E C = 5 ⋅ 10 -4 , Γ r e l a x / E C = 10 -4 , δ A C = 0.2 π , δ D C = π + 0.2 π ....Now making use of the numerical solution of Eqs. ( eq1– eq3) for the Hamiltonian ( Ham_Rabi), we study the dependence of the time-averaged probability P ¯ + on **frequency** ω and amplitude x 0 . For small amplitudes, x 0 ≪ Δ E , there are resonant peaks in the P ¯ + – ω dependence at ω ≃ Δ E / K , as described in Sec. III.A and illustrated in Fig. P(w). With increasing amplitude x 0 , the resonances shift to higher **frequencies**. For x o f f = 0 , the resonances appear at "odd" **frequencies** ( K = 1 , 3 , 5 , . . . ) only, as it was studied in Ref. 9. For x o f f ≠ 0 there are also resonances at "even" **frequencies** ( K = 2 , 4 , . . . ), which is demonstrated in Fig. P(w). We note that Fig. P(w) is plotted for the ideal case of the absence of decoherence and relaxation, Γ φ = Γ r e l a x = 0 , when the resonant value is P ¯ + = 1 / 2 . The effect of finite dephasing, Γ φ ≠ 0 , and relaxation, Γ r e l a x ≠ 0 , is to decrease the resonant values of P ¯ + and to widen the peaks for Γ φ > Γ r e l a x . Thus from the comparison of the theoretically calculated resonant peaks with the experimentally observed ones, the dephasing Γ φ and the relaxation rates Γ r e l a x can be obtained ....Time dependence of upper-level occupation probabilities. (a) Rabi **oscillations** in P + with the period T R = 2 π / x 0 , (b) LZ transition in P ↑ (see Sec. III.B), (c) and (d) P + probability evolution in the case of periodically swept parameters at x o f f = 0 and x o f f ≠ 0 . Here Γ φ = Γ r e l a x = 0 ; T = 2 π / ω ....We study the dynamic behaviour of a quantum two-level system with periodically varying parameters by solving the master equation for the density matrix. Two limiting cases are considered: multiphoton Rabi **oscillations** and Landau-Zener transitions. The approach is applied to the description of the dynamics of superconducting **qubits**. In particular, the case of the interferometer-type charge **qubit** with periodically varying parameters (gate voltage or magnetic flux) is investigated. The time-averaged energy level populations are calculated as funtions of the **qubit**'s control parameters. ... We study the dynamic behaviour of a quantum two-level system with periodically varying parameters by solving the master equation for the density matrix. Two limiting cases are considered: multiphoton Rabi **oscillations** and Landau-Zener transitions. The approach is applied to the description of the dynamics of superconducting **qubits**. In particular, the case of the interferometer-type charge **qubit** with periodically varying parameters (gate voltage or magnetic flux) is investigated. The time-averaged energy level populations are calculated as funtions of the **qubit**'s control parameters.

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Contributors: Oh, Sangchul, Kim, Jaewan

Date: 2006-05-03

The harmonic **oscillator** plays two competing roles: (i) making two **qubits** entangled by inducing the two-**qubit** coupling I e f f , (ii) having two **qubits** decoherent through entanglement with them, i.e. Γ R t . The role of the imaginary part Γ I t is different from that of the real part Γ R t . While Γ R t makes the two **qubit** decoherent, i.e. decaying the off-diagonal elements, Γ I t causes the two-**qubit** coupling to fluctuate as shown in Fig. Fig2(b). These could be uncovered by examining the reduced density matrix of the two **qubits**...(color online). (a) Maximum concurrence of the two **qubits** (b) entropy of two **qubits** at equilibrium as a function of α and ω 0 / ω c . (c) exp - Γ R ∞ as a function of temperature T and ω 0 / ω c for α = 0.25 ....As ω / λ becomes large, C goes to C i d e a l and S to 0 as depicted in Fig. Fig2. This is explained by means of the Born-Oppenheimer approximation that is based on the assumption of the weak coupling between the two **qubits** and the environment. The **frequency** ω of the harmonic **oscillator** is larger than that of the two **qubits**, θ . The harmonic **oscillator** **oscillates** very fast in comparison with the two **qubits**. So the two **qubits** feel the harmonic **oscillator** stays in the same state. However, the condition of ω ≫ λ is not a unique way to make the two **qubits** entangled maximally. Surprisingly, we find the maximum entanglement of the two **qubits**, C = 1 , at θ t = π / 4 under the condition that ω / λ = 4 n with n = 1 , 2 , as shown in Figs. Fig2(a) and Fig2(d). This is due to the fact that the **frequency** θ for entangling the two **qubits** is commensurate with the **frequency** ω of the harmonic **oscillator**. If the condition of ω / λ = 4 n , n = 1 , 2 , is not met, then the concurrence C does not reach 1 at θ t = π / 4 and 0 at θ t = π / 2 due to Γ t as shown in Fig. Fig2(b). The **oscillation** period of C (red solid line) does not coincide with that of the ideal case C i d e a l (thick black line). This implies that the two **qubit** coupling fluctuates due to Γ I t . For ω / λ **qubits** could not be entangled. Fig. Fig2(c) shows the case of ω = λ ....Fig. Fig2(d) shows two competing roles of the harmonic **oscillator**. Let us define the average concurrence C a v g for a period τ ≡ π / 2 θ by C a v g ≡ 1 τ ∫ 0 τ C t d t . For the first period 0 ≤ θ t **frequencies** ω and θ , S a v g and C a v g show the behavior of the stair case....The environment is characterized by the spectral density function J ω = ∑ j λ j 2 δ ω - ω j . As shown in the inset of Fig. Fig3-(a), let us consider an Ohmic environment with a gap ω 0 and an exponential cutoff function of the cutoff **frequency** ω c...The harmonic **oscillator** remains isolated always from the system, but it induces the indirect interaction between the two **qubits** and thus entanglement between them . For a pure two **qubits**, concurrence, an entanglement measure, reads C i d e a l = 2 | a d - e i 4 θ t b c | . Here the subscript, ideal, stands for the case of the **qubits** in a pure state. If a = b = c = d = 1 / 2 , we have C i d e a l = | sin 2 θ t | as shown in Fig. Fig2....Two circles refer to two **qubits** and an oval is an environment. The coupling between **qubits** and the common environment (solid arrow) induces the indirect interaction between two **qubits** (dashed arrow)....We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus. ... We study a system of two **qubits** interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two **qubits** and making them decoherent. For the environment of a single harmonic **oscillator**, if its **frequency** is commensurate with the induced two-**qubit** coupling strength, the two **qubits** could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two **qubits** at equilibrium and for it to be used as a quantum data bus.

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Contributors: Reed, M. D., Johnson, B. R., Houck, A. A., DiCarlo, L., Chow, J. M., Schuster, D. I., Frunzio, L., Schoelkopf, R. J.

Date: 2010-02-27

The efficacy of reset in this device is readily quantified using a modified Rabi **oscillation** sequence, described in Fig. fig:reset(a). Each experiment measures the degree to which the **qubit** is out of equilibrium after some reset time τ ; the protocol is insensitive to any equilibrium thermal population of the **qubit**. The non-equilibrium population is found to exhibit pure exponential decay over three orders of magnitude. The **qubit** can be reset to 99.9% in 120 ns or any other fidelity depending on τ . The sequence is also performed with the **qubit** remaining in the operating **frequency** during the delay to demonstrate the large dynamic range in T 1 available in this system. In the case of multi-**qubit** devices, it is possible that this reset process would affect other **qubits** coupled to the same bus, but this issue could be avoided by using separate coupling and reset cavities....Spontaneous emission through a coupled cavity can be a significant decay channel for **qubits** in circuit quantum electrodynamics. We present a circuit design that effectively eliminates spontaneous emission due to the Purcell effect while maintaining strong coupling to a low-Q cavity. Excellent agreement over a wide range in **frequency** is found between measured **qubit** relaxation times and the predictions of a circuit model. Using fast (nanosecond time-scale) flux biasing of the **qubit**, we demonstrate in situ control of **qubit** lifetime over a factor of 50. We realize **qubit** reset with 99.9% fidelity in 120 ns....We implement the Purcell filter with a transmission-line stub terminated in an open circuit placed outside the output capacitor C o u t [Fig. fig:circuit(a)]. The length of this stub is set such that it acts as a λ / 4 impedance transformer to short out the 50 ~ Ω environment at its resonance **frequency** ω f . We choose C o u t to be much larger than the input capacitor, C i n ≈ ~ C o u t / 15 , to ensure that the **qubit** would be overwhelmingly likely to decay through C o u t . The Purcell filter eliminates decay through this channel, leaving only the negligible decay rate through C i n . The combined total capacitance C t o t ≈ 80 ~ f F results in a small cavity Q . We use two identical stubs above and below the major axis of the chip [Fig. fig:circuit(b)] to keep the design symmetric in an effort to suppress any undesired on-chip modes. The cavity resonates at ω c / 2 π = 8.04 ~ G H z , the filter at ω f / 2 π = 6.33 ~ G H z , and a flux bias line (FBL) is used to address a single transmon **qubit** with a maximum **frequency** of 9.8 ~ G H z , a charging energy E C / 2 π of 350 ~ M H z , and a resonator coupling strength g / 2 π of 270 ~ M H z . Transmission through the cavity measured at 4.2 K was compared with our model to validate the microwave characteristics of the device [Fig. fig:circuit(c)]. There is a dip corresponding to inhibited decay through C o u t at ω f . The predicted and measured curves are also qualitatively similar, lending credence to the circuit model. This method provided a convenient validation before cooling the device to 25 mK in a helium dilution refrigerator....Fast **qubit** reset. (a) Schematic of a pulse sequence used to realize a **qubit** reset and characterize its performance. The fidelity of reset was quantified using a modified Rabi **oscillation** scheme. The **qubit** is first rotated around the x -axis by an angle θ at the operating **frequency** of 5.16 ~ G H z and then pulsed into near resonance with the cavity (solid line) or left at the operating **frequency** (dashed line) for a time τ . The state of the **qubit** is measured as a function of θ and τ after being pulsed back to 5.16 ~ G H z . (b) The Rabi-**oscillation** amplitude as a function of τ , normalized to the amplitude for τ =0. This ratio gives the deviation of the **qubit** state from equilibrium. Curves are fit to exponentials with decay constants of 16.9 ± 0.1 ns and 540 ± 20 ns respectively. Insets: Measured Rabi **oscillations** for τ =0 (lower left) and τ =80 ns (top right). The vertical scales differ by a factor of 100....where C q is the **qubit** capacitance [Fig. fig:circuit(a)] . Previous work has demonstrated that Eq. (1) accurately models the observed T 1 P u r c e l l when all modes of the cavity are taken into account in the calculation of Y . As the relationship holds for any admittance, this decay rate can be controlled by adjusting Y with conventional microwave engineering techniques. In particular, by manipulating Y to be purely reactive (imaginary-valued) at ω q , T 1 P u r c e l l diverges and the Purcell decay channel is turned off. This solution decouples the choice of cavity Q from the Purcell decay rate as desired, and, as we will see, has the advantage of using only conventional circuit elements placed in an experimentally convenient location....Design, realization, and diagnostic transmission data of the Purcell filter. (a) Circuit model of the Purcell-filtered cavity design. The Purcell filter, implemented with twin λ / 4 open-circuited transmission-line stubs, inhibits decay through C o u t near its resonance ω f . (b) Optical micrograph of the device with inset zoom on transmon **qubit**. Note the correspondence of the circuit elements directly above in (a). (c) Cavity transmission measured at 4.2 K and comparison to the circuit-model prediction. The Purcell filter shorts out the 50 Ω output environment at ω f , yielding a 30 dB drop in transmission (arrow). A circuit model involving only the parameters C i n , C o u t , ω c , and ω f shows excellent correspondence....**Qubit** T 1 as a function of **frequency** measured with two methods, and comparison to various models. The first method is a static measurement (circles): the **qubit** is excited and measured after a wait time τ . The second (triangles) is a dynamic measurement: the **qubit** **frequency** is tuned with a fast flux pulse to an interrogation **frequency**, excited, and allowed to decay for τ , and then returned to its operating **frequency** of 5.16 ~ G H z and measured. This method allows for accurate measurement even when T 1 is extremely short. Measurements using the two methods show near perfect overlap. The top dashed curve is the predicted T 1 P u r c e l l , while solid curve includes also non-radiative internal loss with best-fit Q N R = 2 π f T 1 N R ≈ 27 , 000 . The two lower curves correspond to an unfiltered device with the same C i n , C o u t , and ω c , with and without the internal loss. In this case, the Purcell filter gives a T 1 improvement by up to a factor of ∼ 50 ( 6.7 ~ G H z )....We measured the **qubit** T 1 as function of **frequency** and found it to be in excellent agreement with expectations. T 1 is well modeled by the sum of the Purcell rate predicted by our filtered circuit model and a non-radiative internal loss Q N R ≈ 27 , 000 (Fig. fig:lifetime). The source of this loss is a topic of current research, though some candidates are surface two level systems , dielectric loss of the tunnel barrier oxide or corundum substrate, and non-equilibrium quasiparticles . This model contains only the fit parameter Q N R combined with the independently measured values of g , E C , ω c , ω f , C i n , and C o u t . An improvement to T 1 due to the Purcell filter was found to be as much as a factor of 50 at 6.7 ~ G H z by comparison to an unfiltered circuit model with the same parameters. This would be much greater in the absence of Q N R . The device also exhibits a large dynamic range in T 1 : about a factor of 80 between the longest and shortest times measured. ... Spontaneous emission through a coupled cavity can be a significant decay channel for **qubits** in circuit quantum electrodynamics. We present a circuit design that effectively eliminates spontaneous emission due to the Purcell effect while maintaining strong coupling to a low-Q cavity. Excellent agreement over a wide range in **frequency** is found between measured **qubit** relaxation times and the predictions of a circuit model. Using fast (nanosecond time-scale) flux biasing of the **qubit**, we demonstrate in situ control of **qubit** lifetime over a factor of 50. We realize **qubit** reset with 99.9% fidelity in 120 ns.

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Contributors: Sun, Guozhu, Wen, Xueda, Mao, Bo, Zhou, Zhongyuan, Yu, Yang, Wu, Peiheng, Han, Siyuan

Date: 2010-04-26

fig:epsartTwo-photon transitions. Large Ω c hinders two-photon transitions, leading to a lower resonant peak in the measured spectrum as marked with arrows. The lower inset shows the energy levels in the **qubit**-TLS coupled system. The upper inset shows the spectrum density versus Ω c . The outside peaks are due to the stationary population from to 1 2 ( ). The middle peak is due to the two-photon transitions from to ....We present an analytical and comprehensive description of the quantum dynamics of a microwave resonantly driven superconducting phase **qubit** coupled to a microscopic two-level system (TLS), covering a wide range of the external microwave field strength. Our model predicts several interesting phenomena in such an ac driven four-level bipartite system including anomalous Rabi **oscillations**, high-contrast beatings of Rabi **oscillations**, and extraordinary two-photon transitions. Our experimental results in a coupled **qubit**-TLS system agree quantitatively very well with the predictions of the theoretical model....fig:epsart Spectroscopy and coherent **oscillations**. (a) Spectroscopy of the **qubit** versus the flux bias with a splitting at f = 16.572 GHz due to the coupling of the **qubit**-TLS. (b) Usual Rabi **oscillation** with the damping time T R ≈ 81.5 ns at f = 16.728 GHz (arrow in the left) where the effect of the TLS is negligible. (c) Coherent **oscillation** at the avoided crossing (arrow in the right) shows quantum beating due to the interference of Rabi **oscillations** in the coupled system. In (b) and (c), the red dots are the experimental results and the solid lines are the theoretical results....fig:epsartFrequencies Ω R in P φ . (a) and (b), Four **frequencies** with different weight (indicated by the color) in P 1 g and P 1 e versus Ω m , respectively. (c) Rabi **oscillations** with the microwave power, at the top of the fridge, increasing from -13 dBm to -1 dBm with a step of 1 dBm from bottom to top. Curves are shifted vertically for clarity. Quantum beating becomes more clear as the amplitude of microwave increases. (d) **Frequencies** (dots), obtained by the Fourier transformations of the corresponding Rabi **oscillations** in (c), versus the microwave amplitude. The color lines are the two **frequencies** in P 1 obtained from the theoretical analysis. (e) **Frequencies** in P 1 induced by the two-photon transitions versus Ω m with three different Ω c / 2 π : 15.0 MHz, 26.5 MHz, and 50.0 MHz. ... We present an analytical and comprehensive description of the quantum dynamics of a microwave resonantly driven superconducting phase **qubit** coupled to a microscopic two-level system (TLS), covering a wide range of the external microwave field strength. Our model predicts several interesting phenomena in such an ac driven four-level bipartite system including anomalous Rabi **oscillations**, high-contrast beatings of Rabi **oscillations**, and extraordinary two-photon transitions. Our experimental results in a coupled **qubit**-TLS system agree quantitatively very well with the predictions of the theoretical model.

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Contributors: Bishop, Lev S., Ginossar, Eran, Girvin, S. M.

Date: 2010-05-03

Transmitted heterodyne amplitude a as a function of drive detuning (normalized by the dispersive shift χ = g 2 / δ ) and drive amplitude (normalized by the amplitude to put n = 1 photon in the cavity in linear response, ξ 1 = κ / 2 ). Dark colors indicate larger amplitudes. (a) Experimental data , for a sample with a cavity at 9.07 G H z and 4 transmon **qubits** at 7.0 , 7.5 , 8.0 , 12.3 G H z . All **qubits** are initialized in their ground state, and the signal is integrated for the first 400 n s ≃ 4 / κ after switching on the drive. (b) Numerical results for the JC model of Eq. eq:master, with **qubit** fixed to the ground state and effective parameters δ / 2 π = - 1.0 G H z , g / 2 π = 0.2 G H z , κ / 2 π = 0.001 G H z . These parameters are only intended as representative numbers for circuit QED and were not optimized against the data of panel (a). Hilbert space is truncated at 10,000 excitations (some truncation artifacts are visible for the strongest drive), and results are shown for time t = 2.5 / κ . fig:latch000...We analyze the Jaynes-Cummings model of quantum optics, in the strong-dispersive regime. In the bad cavity limit and on timescales short compared to the atomic coherence time, the dynamics are those of a nonlinear **oscillator**. A steady-state non-perturbative semiclassical analysis exhibits a finite region of bistability delimited by a pair of critical points, unlike the usual dispersive bistability from a Kerr nonlinearity. This analysis explains our quantum trajectory simulations that show qualitative agreement with recent experiments from the field of circuit quantum electrodynamics....eq:chi it follows that for A ≫ 1 the response of the system will have an approximate symmetry of reflection with respect to the bare cavity **frequency** A Ω σ z = + 1 ≈ A - Ω σ z = - 1 . Therefore the response at the bare cavity **frequency** will be nearly independent of the state of the **qubit**. In order to translate the high gain available at the step into a **qubit** readout, it is necessary to break the symmetry of the response of the system between the **qubit** ground and excited states, such that the upper critical power ξ C 2 will be **qubit** state dependent. In the JC model the symmetry follows from the weak dependence of the decoupled Hamiltonian H ~ on the **qubit** state for high photon occupation. However, the experimentally-observed state dependence may be explained by a symmetry breaking caused by the higher levels of the weakly anharmonic transmon, or by the presence of more than one **qubit**, see Fig. fig:return. When designing a readout scheme that employs such a diminishing anharmonicity, the contrast of the readout is a product of both the symmetry breaking and the characteristic nonlinear response of the system near the critical point C 2 . Experiments were able to use this operating point to provide a scheme for **qubit** readout, which is attractive both because of the high fidelities achieved (approaching 90 % , significantly better than is typical for linear dispersive readout in circuit QED ) and because it does not require any auxiliary circuit elements in addition to the cavity and the **qubit**. During preparation of this manuscript we became aware of a theoretical modeling of the high-fidelity readout by Boissonneault et al. ....which we integrate numerically in a truncated Hilbert space using the method of quantum trajectories, after making the rotating wave approximation (RWA) with respect to the drive. The experiments we wish to describe are performed on a timescale short compared to the **qubit** decoherence times γ -1 , γ φ -1 and we therefore treat σ z as a constant of motion. The remaining degree of freedom constitutes a Jaynes–Cummings **oscillator**. Note that the **qubit** relaxation and dephasing terms that we have dropped involve the σ ± and σ z operators and would transform in a nontrivial way under the decoupling transformation T . The results of the numerical integration for σ z = - 1 are compared with recent experimental data in Fig. fig:latch000, where we show the average heterodyne amplitude a as a function of drive **frequency** and amplitude. Despite the presence of 4 **qubits** in the device, the fact that extensions beyond a two-level model would seem necessary since higher levels of the transmons are certainly occupied for such strong driving ...(Color online) Symmetry breaking. State-dependent transition **frequency** versus excitation number: (a) for the JC model, parameters as in Figs. fig:latch000 and fig:densclass; (b) for the model extended to 2 **qubits**, δ 1 = - 1.0 G H z , δ 2 = - 2.0 G H z , g 1 = g 2 = 0.25 G H z ; (c) for the model extended to one transmon **qubit**, tuned below the cavity, E C = 0.2 G H z , E J = 30 G H z , g = 0.29 G H z . In all panels, the transition **frequency** asymptotically returns to the bare cavity **frequency**. In (a) the **frequencies** within the σ z = ± 1 manifolds are (nearly) symmetric with respect to the bare cavity **frequency**. For (b), if the state of one (‘spectator’) **qubit** is held constant, then the **frequencies** are asymmetric with respect to flipping the other (‘active’) **qubit**. In (c), the symmetry is also broken due the existence of higher levels in the weakly anharmonic transmon. fig:return...eq:classic is plotted in Fig. fig:densclass for the same parameters as in Fig. fig:latch000. For weak driving the system response approaches the linear response of the dispersively shifted cavity. Above the lower critical amplitude ξ C 1 the **frequency** response bifurcates, and the JC **oscillator** enters a region of bistability. We denote by C 1 the point at which the bifurcation first appears. Dropping terms which are small according to the hierarchy of Eq. ...eq:classic, using the same parameters as Fig. fig:latch000b. (a) Amplitude response as a function of drive **frequency** and amplitude, for same parameters as in Fig. fig:latch000b. The region of bifurcation is indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr **oscillator** (Duffing **oscillator**), constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the **frequency** dependence of the classical solutions (solid blue). For comparison, the response for from the full quantum simulation of Fig. fig:latch000b is also plotted (dashed red) for the same parameters. (c) Cut through (a) for driving at the bare cavity **frequency**, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), on a linear scale. fig:densclass...(Color online). Solution to semiclassical equation eq:classic, using the same parameters as Fig. fig:latch000b. (a) Amplitude response as a function of drive **frequency** and amplitude, for same parameters as in Fig. fig:latch000b. The region of bifurcation is indicated by the shaded area, and has corners at the critical points C 1 , C 2 . The dashed lines indicate the boundaries of the bistable region for a Kerr **oscillator** (Duffing **oscillator**), constructed by making the power-series expansion of the Hamiltonian to second order in N / . The Kerr bistability region matches the JC region in the vicinity of C 1 but does not exhibit a second critical point. (b) Cut through (a) for a drive of 6.3 ξ 1 , showing the **frequency** dependence of the classical solutions (solid blue). For comparison, the response for from the full quantum simulation of Fig. fig:latch000b is also plotted (dashed red) for the same parameters. (c) Cut through (a) for driving at the bare cavity **frequency**, showing the large gain available close to C 2 (the ‘step’). Faint lines indicate linear response. (d) Same as (c), on a linear scale. fig:densclass ... We analyze the Jaynes-Cummings model of quantum optics, in the strong-dispersive regime. In the bad cavity limit and on timescales short compared to the atomic coherence time, the dynamics are those of a nonlinear **oscillator**. A steady-state non-perturbative semiclassical analysis exhibits a finite region of bistability delimited by a pair of critical points, unlike the usual dispersive bistability from a Kerr nonlinearity. This analysis explains our quantum trajectory simulations that show qualitative agreement with recent experiments from the field of circuit quantum electrodynamics.

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Contributors: He, Xiao-Ling, Liu, Yu-xi, You, J. Q., Nori, Franco

Date: 2007-03-16

(Color online) Schematic diagram of two charge **qubits** coupled by a (left) large Josephson junction (JJ) with coupling energy E J 0 and capacitance C J 0 . For the i th charge **qubit** (where i = 1 , 2 ), a superconducting island (denoted by a filled circle) is connected to two identical small JJs (each with coupling energy E J i and capacitance C J i ). Also, this island is biased by the voltage V i = V X i + V g i t via a gate capacitance C i , where V X i is a static (dc) gate voltage and V g i t is a time-dependent (ac) microwave gate voltage. Moreover, a static (dc) magnetic flux Φ e plus a microwave-field-induced magnetic flux Φ t (ac) are applied to the (yellow) region between the large JJ and the first charge **qubit**....Evolution κ q τ of the reduced total supercurrent expectation and reduced supercurrent fluctuation from the initial **qubit** state | g g + | e e / 2 in the presence of the quantum field initially in (a) coherent state | α with the phase ϕ = π / 2 ; (b) coherent state ϕ = 0 ; (c) superposition of coherent states | α + | - α / N + with ϕ = 0 ; (d) squeezed vacuum state | 0 , ζ . The irregularity of **oscillations** originates from the interference effect of the photon component of the above states....To implement quantum information processing, microwave fields are often used to manipulate superconuducting **qubits**. We study how the coupling between superconducting charge **qubits** can be controlled by variable-**frequency** magnetic fields. We also study the effects of the microwave fields on the readout of the charge-**qubit** states. The measurement of the charge-**qubit** states can be used to demonstrate the statistical properties of photons....From Eq. ( general fluctuation), we know that the macroscopic supercurrent expectation value Î can be described by κ q . Figure ( fig3) shows that the supercurrent of the charge **qubits** are different with the same initial **qubit** state | g g + | e e / 2 but with different initial states of the quantum field. From Eq. ( Iq), in the case of the coherent state | α with the phase ϕ = π / 2 , the total supercurrent Î displays a sinusoidal-like evolution, as shown in Fig. fig3(a). However, when ϕ = 0 , the total supercurrent is shown in Fig. fig3(b). If the quantized field is initially in a superposition of coherent states, the total supercurrent Î , as shown in Fig. fig3(c), demonstrates the collapse and partial-revival phenomena. In the case of the squeezed vacuum state, the total supercurrent approximately displays an ac current with a quasi-periodic evolution, which is demonstrated by Fig. fig3(d). All irregular **oscillations** of the supercurrent expectation or supercurrent fluctuation reflect the coherent interference that comes from the coherent superpositions of the different photon number states. The different initial photon states result in different output of the measurement of the charge-**qubit** states. Therefore, the measurement of the charge-**qubit** states can demonstrate the statistical properties of the photons, and charge **qubits** could be served as photon detectors....Now, we study how to apply our variable-**frequency**-controlled approach to the above charge-**qubit** circuits . We assume that besides the dc voltages V X i and the dc magnetic flux Φ e , an ac microwave voltage V g i t = V g i cos ω g i t with the **frequency** ω g i is applied to the superconduction island of the i th **qubit** via its gate capacitance, and an additional variable-**frequency** (ac) magnetic flux Φ t = Φ c sin ω t is also applied through the area between the large JJ and the first charge **qubit** (see Fig. fig1). To make our proposed charge-**qubit** more immune from the uncontrollable charge fluctuations, it is also assumed that two charge **qubits** work at their optimal points, i.e. the applied dc voltages V X i satisfy the condition ε i V X i = 0 . Considering these conditions, the Hamiltonian in Eq. ( eq:2) becomes ... To implement quantum information processing, microwave fields are often used to manipulate superconuducting **qubits**. We study how the coupling between superconducting charge **qubits** can be controlled by variable-**frequency** magnetic fields. We also study the effects of the microwave fields on the readout of the charge-**qubit** states. The measurement of the charge-**qubit** states can be used to demonstrate the statistical properties of photons.

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Contributors: Armour, A. D., Blencowe, M. P.

Date: 2008-04-14

The dispersive Hamiltonian shifts the mechanical **frequency** in a way that depends on the state of the TLS. This interaction can be used to probe the quantum coherence of the mechanical resonator. The idea is to perform a Ramsey interference experiment in which the **qubit** is prepared in a superposition of its eigenstates states using a control pulse, this superposition is then allowed to interact with the resonator for a time t before a second pulse is applied to the **qubit** and then a measurement of its state is performed. For an isolated TLS the probability of finding the system in one or other of its eigenstates at the end of the experiment will **oscillate** between zero and unity as a function of the time between the two control pulses. When the mechanical resonator is present the interaction with the superposition of TLS states leads to an overall superposition of states involving spatially separated mechanical states. For a sufficiently strong interaction, the separation of the resonator states coupled to the **qubit** states leads to a strong suppression of the **oscillations** in the final **qubit** state measurements. The coherence of the resonator can be inferred by inverting the state of the TLS midway between the two original control pulses. The scheme is illustrated schematically in figure fig:f0. In the absence of the resonator’s environment, such an inversion should lead to a reversal of its dynamics and hence the recovery of the **oscillations** in the final TLS state measurement . Very similar schemes have been demonstrated in optical systems ....Envelope of **oscillations** in P | + in an echo experiment with a π pulse applied at t = t 1 , measured at time t f = 2 t 1 as a function of t f . The full (dashed) curves are for κ = 0.1 ( κ = 0.2 ), with α 0 = 25 and n ¯ = m ¯ = 10 . The red curves are for Q = 10 3 and the blue curves are for Q = 10 4 . The black line is the result that would be obtained without any dissipation to the mechanical resonator....Envelope of **oscillations** in P | + in an echo experiment with a π pulse applied at t = t 1 = 0.2 ~ μ s, measured at time t f = 2 t 1 . The blue curves are for κ = 0.2 , with α 0 = 10 and n ¯ = m ¯ varied from 0 to 25. The red curves are for the same parameters but with n ¯ = m ¯ = 10 and α 0 varied from 0 to 25. In each case the dashed curve is for Q = 10 3 and the full curve is for Q = 10 4 . The black line is the result that would be obtained without any dissipation to the mechanical resonator....Envelope of **oscillations** in P | + in an echo experiment measured at time t f = 2 t 1 as a function of α 0 . The full (dashed) curves are for m ¯ = n ¯ ( m ¯ = 0 ) with κ = 0.2 , t 1 = 0.2 ~ μ s and Q = 10 4 . The red curves are for n ¯ = 20 and the blue curves are for n ¯ = 10 ....We propose a scheme in which the quantum coherence of a nanomechanical resonator can be probed using a superconducting **qubit**. We consider a mechanical resonator coupled capacitively to a Cooper-pair box and assume that the superconducting **qubit** is tuned to the degeneracy point so that its coherence time is maximised and the electro-mechanical coupling can be approximated by a dispersive Hamiltonian. When the **qubit** is prepared in a superposition of states this drives the mechanical resonator progressively into a superposition which in turn leads to apparent decoherence of the **qubit**. Applying a suitable control pulse to the **qubit** allows its population to be inverted resulting in a reversal of the resonator dynamics. However, the resonator's interactions with its environment mean that the dynamics is not completely reversible. We show that this irreversibility is largely due to the decoherence of the mechanical resonator and can be inferred from appropriate measurements on the **qubit** alone. Using estimates for the parameters involved based on a specific realization of the system we show that it should be possible to carry out this scheme with existing device technology....Schematic illustration of the evolution of the mechanical resonator in phase space during the echo sequence. Initially (a) the resonator is prepared in a coherent state and the **qubit** is prepared in a superposition of states. The two **qubit** states couple to the resonator leading to different effective **frequencies** ω ± ω 1 so that in the frame rotating at the resonator **frequency** the two mechanical states start to pull apart (b). A π pulse inverts the **qubit** state and hence interchanges the relative **frequencies** of the two resonator states (c). When the periods of evolution before and after the inversion of the **qubit** are the same the resonator will return to its initial state (d) in the absence of dissipation....Envelope of **oscillations** in P | + in an echo experiment with a π pulse applied at t = t 1 = 0.2 ~ μ s. The blue curves are for κ = 0.2 , n ¯ = m ¯ = 10 and α 0 = 25 . The red curves are for the same parameters but with m ¯ = 0 . In each case the full curve is for Q = 3000 and the dashed curve is for the case without any mechanical dissipation. The black curve is the result that would be obtained without any coupling to the mechanical resonator. ... We propose a scheme in which the quantum coherence of a nanomechanical resonator can be probed using a superconducting **qubit**. We consider a mechanical resonator coupled capacitively to a Cooper-pair box and assume that the superconducting **qubit** is tuned to the degeneracy point so that its coherence time is maximised and the electro-mechanical coupling can be approximated by a dispersive Hamiltonian. When the **qubit** is prepared in a superposition of states this drives the mechanical resonator progressively into a superposition which in turn leads to apparent decoherence of the **qubit**. Applying a suitable control pulse to the **qubit** allows its population to be inverted resulting in a reversal of the resonator dynamics. However, the resonator's interactions with its environment mean that the dynamics is not completely reversible. We show that this irreversibility is largely due to the decoherence of the mechanical resonator and can be inferred from appropriate measurements on the **qubit** alone. Using estimates for the parameters involved based on a specific realization of the system we show that it should be possible to carry out this scheme with existing device technology.

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