Dissipative dynamics of a qubit coupled to a nonlinear oscillator

We consider the dissipative dynamics of a qubit coupled to a nonlinear oscillator (NO) embedded in an Ohmic environment. By treating the nonlinearity up to first order and applying Van Vleck perturbation theory up to second order in the qubit-NO coupling, we derive an analytical expression for the eigenstates and eigenfunctions of the coupled qubit-NO system beyond the rotating wave approximation. In the regime of weak coupling to the thermal bath, analytical expressions for the time evolution of the qubit’s populations are derived: they describe a multiplicity of damped oscillations superposed to a complex relaxation part toward thermal equilibrium. The long-time dynamics is characterized by a single relaxation rate, which is maximal when the qubit is tuned to one of the resonances with the nonlinear oscillator.

Figure 1

Energy spectrum of the coupled qubit-nonlinear-oscillator system versus the linear oscillator frequency $\Omega$ (in units of the TLS tunneling splitting ${\Delta }_0$). Solid lines show the energy levels for the five lowest energy states $\left(|0⟩,|1⟩,|2⟩,|3⟩,|4⟩\right)$ with the TLS-NO coupling being switched on, $g=0.18{\Delta }_0$, and for finite nonlinearity, $\alpha =0.02\hslash {\Delta }_0$. The TLS is unbiased, $\epsilon =0$. The energy levels for the uncoupled case are given by the dotted lines. Due to the nonequidistant level spacing of the nonlinear oscillator the resonance condition (crossing of dotted lines), given in Eq. (16), is different for each doublet. This causes a shift of the exact crossings with respect to the linear case at zero coupling to lower frequencies. For finite coupling the spectrum exhibits avoided crossings around resonance, whereas it approaches the uncoupled case away from resonance.

Figure 2

(Color online) Top: dynamics of the population difference $P\left(t\right)$ for the unbiased, $\epsilon =0$, qubit-nonlinear-oscillator system at linear resonance $\left(\Omega ={\Delta }_0\right)$ [blue (dark gray) line]. We choose a nonlinearity $\alpha =0.02\hslash \Omega$, a TLS-NO coupling $g=0.18\Omega$, and inverse temperature $\beta =10{\left(\hslash \Omega \right)}^{-1}$. For comparison we plotted the corresponding linear case [orange (light gray) line]. Bottom: Fourier transform $F\left(\omega \right)$ of $P\left(t\right)$ for the unbiased system. The dominating frequencies are ${\omega }_{10}$ and ${\omega }_{20}$. To visualize the delta functions, finite widths have artificially been introduced.

Figure 3

(Color online) The relaxation rate ${\Gamma }_r$ given in Eq. (62) drawn against the oscillator frequency $\Omega$ [continuous blue (dark gray) line]. We used $ϵ=0.5{\Delta }_0$, corresponding to a frequency splitting ${\Delta }_b=1.118{\Delta }_0$, coupling $g=0.18{\Delta }_0$, and the nonlinearity $\alpha =0.02\hslash {\Delta }_0$. The damping constant is $\kappa =0.0154$ and the inverse temperature is $\beta =10{\left(\hslash {\Delta }_0\right)}^{-1}$. At resonance $\left(\Omega ={\Delta }_b-3\alpha /\hslash +g^2{\Delta }_0^2/{\Delta }_b^3\right)$ ${\Gamma }_r$ is maximal. For comparison also the second lowest eigenvalue is plotted [orange (light gray) dashed line]. The inset shows the two eigenvalues close to resonance.

Figure 4

(Color online) Comparison of the behavior of $P\left(t\right)$ and its Fourier transform $F\left(\omega \right)$ as obtained from the numerically exact solution [red (dark gray) line] and the three approximation schemes [orange (light gray) line] discussed in the text. Top: smallest eigenvalue approximation (SEA); middle: low temperature approximation (LTA); and bottom: partial secular approximation (PSA). The chosen parameters are $\alpha =0.02\hslash \Omega$, $g=0.18\Omega$, $\epsilon =0\Omega$, $\kappa =0.0154$, and $\beta =10{\left(\hslash \Omega \right)}^{-1}$. The dynamics is well reproduced within all approximations; however the agreement of the PSA with the exact numerics is the best. In the corresponding Fourier spectrum almost no deviations occur for all three approaches.

Figure 5

(Color online) Top: $P\left(t\right)$ for the linear [orange (light gray) line] and nonlinear [blue (dark gray) line] cases using the parameters $\alpha =0$ or $\alpha =0.02\hslash \Omega$, respectively, $g=0.18\Omega$, $\kappa =0.0154$, $\epsilon =0$, ${\Delta }_b=\Omega$, and $\beta =10{\left(\hslash \Omega \right)}^{-1}$. Bottom: corresponding Fourier transform $F\left(\omega \right)$.

Figure 6

(Color online) $P\left(t\right)$ and its Fourier transform $F\left(\omega \right)$ for the parameters $\alpha =0.02\hslash \Omega$, $g=0.18\Omega$, $\kappa =0.0154$, $\epsilon =0$, and ${\Delta }_b=\Omega$ as in Fig. 5, but $\beta =3{\left(\hslash \Omega \right)}^{-1}$. For comparison we plotted the linear case in orange (light gray).

Figure 7

(Color online) $P\left(t\right)$ and its Fourier transform $F\left(\omega \right)$ for the parameters $\alpha =0.02\hslash \Omega$, $g=0.18\Omega$, $\kappa =0.0154$, $\epsilon =0$, ${\Delta }_b=1.18\Omega$, corresponding to the oscillator transition from $|3⟩\to |2⟩$, and $\beta =3{\left(\hslash \Omega \right)}^{-1}$. For comparison we plotted the linear case in orange (light gray).