We present a theoretical study of a superconducting charge qubit dispersively coupled to a transmission line resonator. Starting from a master equation description of this coupled system and using a polaron transformation, we obtain an exact effective master equation for the qubit. We then use quantum trajectory theory to investigate the measurement of the qubit by continuous homodyne measurement of the resonator out-field. Using the same porlaron transformation, a stochastic master equation for the conditional state of the qubit is obtained. From this result, various definitions of the measurement time are studied. Furthermore, we find that in the limit of strong homodyne measurement, typical quantum trajectories for the qubit exhibit a crossover from diffusive to jumplike behavior. Finally, in the presence of Rabi drive on the qubit, the qubit dynamics is shown to exhibit quantum Zeno behavior.

Superconducting electrical circuits can be used to study the physics of cavity quantum electrodynamics (QED) in new regimes, therefore realizing circuit QED. For quantum information processing and quantum optics, an interesting regime of circuit QED is the dispersive regime, where the detuning between the qubit transition frequency and the resonator frequency is much larger than the interaction strength. In this paper, we investigate how non-linear corrections to the dispersive regime affect the measurement process. We find that in the presence of pure qubit dephasing, photon population of the resonator used for the measurement of the qubit act as an effective heat bath, inducing incoherent relaxation and excitation of the qubit. Measurement thus induces both dephasing and mixing of the qubit, something that can reduce the quantum non-demolition aspect of the readout. Using quantum trajectory theory, we show that this heat bath can induce quantum jumps in the qubit state and reduce the achievable signal-to-noise ratio of a homodyne measurement of the voltage.

In dispersive readout schemes, qubit-induced nonlinearity typically limits the measurement fidelity by reducing the signal-to-noise ratio (SNR) when the measurement power is increased. Contrary to seeing the nonlinearity as a problem, here we propose to use it to our advantage in a regime where it can increase the SNR. We show analytically that such a regime exists if the qubit has a many-level structure. We also show how this physics can account for the high-fidelity avalanchelike measurement recently reported by Reed et al. [arXiv:1004.4323v1].PACS numbers: 03.65. Yz, 42.50.Pq, 42.50.Lc, 74.50.+r Quantum measurements are crucial to all quantum information protocols. In solid-state systems, readout can be performed by connecting the qubits to noisy measurement electronics, such as single-electron transistors [1]. Dispersive readouts based on coupling qubits to high-Q resonators are however much less disruptive as all of the energy is dissipated away from the qubit [2]. This type of measurement leads to quantum nondemolition (QND) readout. Unfortunately, so far the typical signal-to-noise ratio (SNR) has been relatively low, requiring sometimes up to 10 6 repetitions of the experiment to average out the noise added by the amplifiers [3,4]. Increasing the qubit-resonator coupling is one approach to improve the SNR as it increases the amount of information about the qubit state carried by the measurement photons. This is however at the cost of an increased Purcell decay rate [5], which decreases the useful integration time and in turn the SNR. Another approach has been to exploit bifurcation in a nonlinear resonator [6]. This has already led to the experimental demonstration [7] of single-shot readout of a transmon-type superconducting qubit [8]. Very recently, Reed et al. have also shown that single-shot measurement of a transmon qubit can also be realized in a linear resonator but working at very large measurement power [9].Motivated by these results, we study how the nonlinearity induced by the qubit in a linear resonator can lead to improvement of the measurement. We first explore the low driving power regime before considering the high-power regime studied in Ref. [9]. Using a simple model, we find qualitative agreement with these experimental results.For simplicity, we first focus on a two-level system (TLS), of states labeled {|0 , |1 }, dispersively coupled to a microwave resonator. This system is well described by the Jaynes-Cummings Hamiltonian, expressed here in the dispersive basis (h = 1) [10]where Π ij = |i j| for the TLS, and N = a † a. In this expression, ω r is the resonator frequency and ω a (ω a ) the bare (Lamb-shifted) TLS transition frequency. The qubitresonator coupling is characterized by χ = g 2 (1 − λ 2 )/∆ the dispersive coupling strength and ζ = −gλ 3 is a Kerrtype nonlinearity, with g the bare qubit-resonator coupling strength, ∆ = ω a − ω r the qubit-resonator detuning and λ = −g/∆. The second line of Eq. (1) is valid to fourth order in λ and at photon numbersn ≪ n crit , where n c...