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arXiv:0907.5181 · Physical Review A

Relaxation of a qubit measured by a driven Duffing oscillator

I. Serban, M. I. Dykman, Frank K. Wilhelm

Published Feb 5, 2010• 24 citations
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Abstract

We investigate the relaxation of a superconducting qubit for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The qubit relaxation rates are different in different states. They can display strong dependence on the qubit frequency and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven oscillator changes the effective temperature of the qubit.

DOI
10.1103/physreva.81.022305

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We investigate the relaxation of a superconducting qubit for the case when its detector, the Josephson bifurcation amplifier, remains latched in one of its two (meta)stable states of forced vibrations. The qubit relaxation rates are different in different states. They can display strong dependence on the qubit frequency and resonant enhancement, which is due to quasienergy resonances. Coupling to the driven oscillator changes the effective temperature of the qubit.
We analyze the relaxation of a superconducting flux qubit during measurement. The qubit state is measured with a nonlinear oscillator driven across the threshold of bifurcation, acting as a switching dispersive detector. This readout scheme is of quantum nondemolition type. Two successive readouts are used to analyze the evolution of the qubit and the detector during the measurement. We introduce a simple transition rate model for characterizing the qubit relaxation and the detector switching process. Corrected for qubit relaxation the readout fidelity is at least 95%. Qubit relaxation strongly depends on the driving strength and the state of the oscillator.
Recent experiments on quantum state engineering with superconducting circuits realized concepts originally introduced in the field of quantum optics. Motivated by one such experiment we investigate a Josephson qubit coupled to a slow LC oscillator with frequency much lower than the qubit’s energy splitting. The qubit is ac‐driven to perform Rabi oscillations, and the Rabi frequency is tuned to resonance with the oscillator. The properties of this driven circuit QED system depend strongly on relaxation and decoherence effects in the qubit. We investigate both one‐photon and two‐photon qubit‐oscillator coupling, the latter being dominant at the symmetry point of the qubit. When the qubit driving frequency is blue detuned, we find that the system exhibits lasing behavior; for red detuning the qubit cools the oscillator. Similar behavior is expected in an accessible range of parameters for a Josephson qubit coupled to a nano‐mechanical oscillator. In a different parameter regime, furthering the analogies between superconducting and quantum optical systems, we investigate Sisyphus damping, which is the key element of the Sisyphus cooling protocol, as well as its exact opposite, Sisyphus amplification.
We analyzed the Josephson bifurcation amplifier (JBA) readout process of a superconducting qubit quantum mechanically by calculating the dynamics of the density operator of a driven nonlinear oscillator and a qubit coupled system during the measurement process. In purely quantum cases, bifurcation is impossible. Introducing decoherence enables us to reproduce the bifurcation with a finite hysteresis. When a qubit is initially in a superposition state, we have observed the qubit-probe (JBA) entangled state, and it is divided into two separable states at the moment the JBA transition begins. This corresponds to "projection." To readout the measurement result, however, we must wait until the two JBA states are macroscopically well separated. The waiting time is determined by the strength of the decoherence in the JBA.
We analyze the dynamics of a qubit-resonator system coupled with a thermal bath and external electromagnetic fields. Using the evolution equations for the set of Heisenberg operators that describe the whole system, we derive an expression for the resonator field, that includes the resonator-drive, the resonator-bath, and resonator-qubit interactions. The renormalization of the resonator frequency, caused by the qubit-resonator interaction, is accounted for. Using the solutions for the resonator field, we derive the equation that describes the qubit dynamics. The dependence of the qubit evolution during the measurement time on the fidelity of a single-shot measurement is studied. The relation between the fidelity and measurement time is shown explicitly. We proposed a novel adiabatic method for the phase qubit measurement. The method utilizes a low-frequency, quasi-classical resonator inductively coupled to the qubit. The resonator modulates the qubit energy, and the back reaction of the qubit causes a shift in the phase of the resonator. The resonator phase shift can be used to determine the qubit state. We have simulated this measurement taking into the account the energy levels outside the phase qubit manifold. We have shown that, for qubit frequencies in the range of 8-12GHZ, a resonator frequency of 500 MHz and a measurement time of 100 ns, the phase difference between the two qubit states is greater than 0.2 rad. This phase difference exceeds the measurement uncertainty, and can be detected using a classical phase-meter. A fidelity of 0.9999 can be achieved for a relaxation time of 0.5 ms. We also model and simulate a microstrip-SQUID amplifier of frequency about 500 MHz, which could be used to amplify the resonator oscillations in the phase qubit adiabatic measurement. The voltage gain and the amplifier noise temperature are calculated. We simulate the preparation of a generalized Bell state and compute the relaxation times required for achieving high fidelities. We consider two capacitively coupled phase qubits similar to that used in the experimental work of R. Bialczak et al. (Nature Physics, 6, 409, 2010). The Bell state was created using a NOT gate and a universal entangling gate. We solved the equation of motion for the density matrix with the Bloch relaxation times, T{sub 1} and T{sub 2}. For every qubit we took into consideration all the states in the shallow well including the states outside the qubit manifold. Also we took into consideration unwanted interaction between the qubits during the application of the NOT gate. The NOT gate was implemented using a Gaussian pulse assuming the frequency difference between the two qubits is 200 MHz. An entangling gate was implemented reducing this frequency difference to zero. We have found the optimal parameters for the Gaussian pulse and the optimal duration for the entangling gate. We have shown that the maximum fidelity of 0.99 can be achieved for the minimum relaxation time, T{sub 1} = 1.2 {micro}s, if T{sub 2} = 2T{sub 1}. By slightly increasing the required value of T{sub 1}, one can sharply reduce the required value for T{sub 2}.
The Josephson bifurcation amplifier (JBA) readout process of a superconducting qubit is analyzed quantum mechanically. We examined the dynamics of the density operator of a detector (a driven nonlinear oscillator) and a qubit coupled system during the measurement process. We have observed the qubit-detector (JBA) entangled state and it is divided into two separable states at the moment the JBA transition begins. Moreover, we discuss the process from the viewpoint of time variations in information quantities, such as mutual information between the qubit and detector. We show that the measurement process corresponds to an information conversion from quantum to classical one.
It is widely known that spin-locking noise spectroscopy is a powerful technique for the characterization of low-frequency noise mechanisms in superconducting qubits. Here we show that the relaxation rate of the driven spin-locking state of a qubit can be significantly affected by the presence of an off-resonant high-frequency two-level-system defect. Thus, both low- and high-frequency defects should be taken into account in the interpretation of spin-locking measurements and other types of driven-state noise spectroscopy.