Synchronization and Bistability of a Qubit Coupled to a Driven Dissipative Oscillator

O. V. Zhirov and D. L. Shepelyansky

Phys. Rev. Lett. 100, 014101 – Published 9 January 2008

Abstract

We study numerically the behavior of a qubit coupled to a quantum dissipative driven oscillator (resonator). Above a critical coupling strength the qubit rotations become synchronized with the oscillator phase. In the synchronized regime, at certain parameters, the qubit exhibits tunneling between two orientations with a macroscopic change of the number of photons in the resonator. The lifetimes in these metastable states can be enormously large. The synchronization leads to a drastic change of qubit radiation spectrum with the appearance of narrow lines corresponding to recently observed single artificial-atom lasing [O. Astafiev et al., Nature (London) 449, 588 (2007)].

Received 10 October 2007

DOI:https://doi.org/10.1103/PhysRevLett.100.014101

Authors & Affiliations

O. V. Zhirov¹ and D. L. Shepelyansky²

¹Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
²Laboratoire de Physique Théorique, UMR 5152 du CNRS, Université Toulouse III, 31062 Toulouse, France

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Vol. 100, Iss. 1 — 11 January 2008

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Images

Figure 1
Bistability of a qubit coupled to a driven oscillator with jumps between two metastable states. The top panel shows the average oscillator level number $⟨ n ⟩$ as a function of time $t$ at stroboscopic integer values $ω t / 2 π$ ; the middle panel shows the qubit polarization vector components $ξ_{x}$ [black (blue)] and $ξ_{z}$ [gray (green)] at the same moments of time; the bottom panel shows the degree of qubit polarization $ξ$ . Here the system parameters are $λ / ω_{0} = 0.02$ , $ω / ω_{0} = 1.01$ , $Ω / ω_{0} = 1.2$ , $f = ℏ λ \sqrt{n_{p}}$ , $n_{p} = 20$ , and $g = 0.04$ .Reuse & Permissions
Figure 2
Top panels: The Poincaré section taken at integer values of $ω t / 2 π$ for an oscillator with $x = ⟨ (\hat{a} + {\hat{a}}^{†}) / \sqrt{2} ⟩$ , $p = ⟨ (\hat{a} - {\hat{a}}^{†}) / \sqrt{2} i ⟩$ (left) and for qubit polarization with polarization angles ( $θ, ϕ$ ) defined in text (right). Middle panels: Same quantities shown at irrational moments of $ω t / 2 π$ . Bottom panels: Qubit phase $ϕ$ versus oscillator phase $φ$ ( $p / x = - \tan φ$ ) at time moments as in the middle panels for $g = 0.04$ (left) and $g = 0.004$ (right). Other parameters and the time interval are as in Fig. 1. The color of points is black (blue) for $ξ_{x} > 0$ and gray (red) for $ξ_{x} < 0$ .Reuse & Permissions
Figure 3
Right: Dependence of average qubit polarization components $ξ_{x}$ and $ξ_{z}$ (full curves and dashed curves) on $g$ , averaging is done over stroboscopic times (see Fig. 1) in the interval $100 \leq ω t / 2 π \leq 2 \times 10^{4}$ ; color is fixed as in Fig. 2. Left: Dependence of average level of oscillator in two metastable states on coupling $g$ , color is fixed by $ξ_{x}$ sign on the right panel [gray (red) for large $n_{+}$ and black (blue) for small $n_{-}$ ]; average is done over the quantum state and stroboscopic times as in the right panel; dashed curves show theory dependence (see text). Two QT are used with initial value $ξ_{x} = \pm 1$ . All parameters are as in Fig. 1 except $g$ .Reuse & Permissions
Figure 4
Dependence of average level $n_{\pm}$ of oscillator in two metastable states on the driving frequency $ω$ (average and color choice are the same as in the right panel of Fig. 3); coupling is $g = 0.04$ and $g = 0.08$ (dashed curves and full curves). Inset shows the variation of position of maximum at $ω = ω_{\pm}$ with coupling strength $g$ , $Δ ω_{\pm} = ω_{\pm} - ω_{0}$ . Other parameters are as in Fig. 1.Reuse & Permissions
Figure 5
Dependence of number of transitions $N_{f}$ between metastable states on rescaled qubit frequency $Ω / ω_{0}$ for parameters of Fig. 1; $N_{f}$ are computed along 2 QT of length $10^{5}$ driving periods. Inset shows lifetime dependence on $Ω / ω_{0}$ for two metastable states [ $τ_{+}$ for gray (red), $τ_{-}$ for black (blue), $τ_{\pm}$ are given in number of driving periods; color choice is as in Figs. 2, 3].Reuse & Permissions
Figure 6
Spectral density $S (ν)$ of qubit radiation $ξ_{z} (t)$ as function of driving power $n_{p}$ in the presence of phase noise in $ϕ$ with diffusion rate $η = 0.004 ω_{0}$ . Left: $Ω / ω_{0} = 1.2$ ; right: $Ω / ω_{0} = 1$ . Other parameters are as in Fig. 1. Grayness (color) shows $S (ν)$ in logarithmic scale [white (black) for maximal (zero)], $ν$ is given in units of $ω_{0}$ .Reuse & Permissions

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