Dispersive regime of circuit QED: Photon-dependent qubit dephasing and relaxation rates

Maxime Boissonneault, J. M. Gambetta, and Alexandre Blais

Phys. Rev. A 79, 013819 – Published 23 January 2009

Abstract

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 nonlinear 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 nondemolition aspect of the readout. Using quantum trajectory theory, we show that this heat bath can induce quantum jumps in the qubit state. Nonlinear effects can also reduce the achievable signal-to-noise ratio of a homodyne measurement of the voltage.

Received 7 October 2008

DOI:https://doi.org/10.1103/PhysRevA.79.013819

Authors & Affiliations

Maxime Boissonneault¹, J. M. Gambetta², and Alexandre Blais¹

¹Département de Physique et Regroupement Québécois sur les Matériaux de Pointe, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
²Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

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Vol. 79, Iss. 1 — January 2009

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Images

Figure 1
(Color online) Schematic layout and lumped element version of the circuit QED implementation. A superconducting charge qubit (green) is fabricated inside a superconducting one-dimensional transmission line resonator (blue).Reuse & Permissions
Figure 2
(Color online) Trace distance between the simulation of the full master equation and the dispersive linear model presented in Ref. 21 versus time. (a) The initial state is $∣ 0 ⟩ (∣ g ⟩ + ∣ e ⟩) ∕ \sqrt{2}$ and $g ∕ 2 π = 50 MHz$ , $Δ ∕ 2 π = 2 GHz$ , $n_{crit} = 400$ , $κ ∕ 2 π = 2.5 MHz$ , $γ_{1} ∕ 2 π = 0.1 MHz$ , and $γ_{φ} ∕ 2 π = 0.3 MHz$ . A measurement drive of amplitude $ϵ_{m} ∕ 2 π = 10 MHz$ of shape $0.5 ϵ_{m} {\tanh [(t - t_{0}) ∕ σ] + 1}$ with $2 π t_{0} = 10.0 μ s$ and $2 π σ = 10.0 μ s$ is applied, corresponding to a mean of $⟨ a^{†} a ⟩ \approx 34 \approx 0.08 n_{crit}$ photons. (b) Maximum of the trace distance vs measurement amplitude. The vertical line indicates the measurement amplitude used in (a).Reuse & Permissions
Figure 3
Dispersive energy diagram. The full line represents the bare states while the straight dashed lines represent the qubit-resonator dressed states.Reuse & Permissions
Figure 4
(Color online) Comparison between the exact master equation (2.5) and the model (5.3). (a) A typical time evolution of $σ_{x}$ (full black line), $σ_{y}$ (full gray line), and $σ_{z}$ for the exact result (full black line), the nonlinear (dashed blue line), and the linear (dotted red line) models. The parameters and the initial state are the same as Fig. 2. (b) Maximum of the trace distance for the linear (red squares) and the non-linear (green triangles) models, for $γ_{φ} ∕ 2 π = 0.5, 0.3, 0.1 MHz$ (dotted, dashed, full lines).Reuse & Permissions
Figure 5
(Color online) (a) Steady state value of $⟨ σ_{z} ⟩$ and (b) effective decay rate $γ_{eff}$ as a function of measurement power for various values of $γ_{φ} ∕ 2 π = 0.0$ , 0.05, 0.2, and $0.5 MHz$ (black stars, gray squares, red circles, and blue triangles). Symbols are extracted from the numerical solution of Eq. (2.5). Lines correspond to Eqs (6.3, 6.1). The numerical simulations were done with the same parameters as Fig. 2, with the qubit initially in its excited state and the cavity in the corresponding steady state with a continuous measurement drive of amplitude $ϵ_{m}$ . The top $x$ axis is the approximate average photons number in the cavity.Reuse & Permissions
Figure 6
(Color online) Typical trajectories for $γ_{φ} ∕ 2 π = 0.5 MHz$ and measurement amplitudes of $ϵ_{m} ∕ 2 π = 0 MHz$ (bottom), $ϵ_{m} ∕ 2 π = 10 MHz$ (center), and $ϵ_{m} ∕ 2 π = 20 MHz$ (top). The other parameters are the same as in Fig. 2. The initial state has $⟨ σ_{z} ⟩ = 1$ with zero photons and a measurement drive starting shortly after $t = 0$ .Reuse & Permissions

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