Nonadiabatic effects in periodically driven dissipative open quantum systems

We present a general method to calculate the periodic steady state of a driven-dissipative system coupled to a transmission line (and more generally, to a reservoir) under periodic modulation of its parameters. Using Floquet's theorem, we formulate the differential equation for the system's density operator which has to be solved for a single period of modulation. On this basis we also provide systematic expansions in both the adiabatic and high-frequency regime. Applying our method to three different systems—two- and three-level models as well as the driven nonlinear cavity—we propose periodic modulation protocols of parameters leading to a temporary suppression of effective dissipation rates, and study the arising nonadiabatic features in the response of these systems.

Figure 1

Open quantum system model: A local system is coupled to a transmission line supporting left- and right-propagating modes. Either the system's parameters or its coupling to the transmission line is periodically modulated. The input pulse into the transmission line is given by a coherent state $|{\mathrm{\Psi }}_0\rangle$ in the right-propagating mode ${\omega }_0$. The pulse's intensity is characterized by the photonic flux $f$.

Figure 2

Reflection from the qubit on resonance, $\delta =0$, for the time-modulated coupling $g\left(t\right)=g_{0}cos\mathrm{\Omega }t$ with slow $\mathrm{\Omega }=0.1\gamma$ (top) and fast $\mathrm{\Omega }=10\gamma$ (bottom) modulation frequencies, expressed in the units of $\gamma =\pi g_0^2$. (top) The adiabatic approximation (dashed line) obtained from Eq. (19) deviates from the numerical solution in the vicinities of time instants when the coupling is quenched. For all values of $f$, the reflection is suppressed at these points. At large $f$, the reflection is completely suppressed because of the qubit's saturation. (bottom) The high-frequency modulation suppresses the reflection for any input power $f$. The numerical result (solid lines) is well approximated by the high-frequency result (dashed line) obtained from Eqs. (25, 26, 27).

Figure 3

Inelastic power spectra of the qubit strongly driven $(f=200\gamma )$ on resonance ($\delta =0$) for the (top) sign-change protocol $g\left(t\right)=g_{0}cos\mathrm{\Omega }t$ and the (bottom) on-off protocol $g\left(t\right)=\frac{g_0}{2}(1+cos\mathrm{\Omega }t)$ for various modulation frequencies $\mathrm{\Omega }$. The Mollow triplet for the corresponding stationary case at coupling $g=g_0$ is shown for comparison in gray. Additional broadened peaks consistent with the Floquet spectrum arise due to the periodic modulation and may destructively interfere as seen, for example, in the missing main peak in the case of the sign-change protocol (top). Similar features in the power spectrum have been reported in Ref. [40] for a qubit subject to a pulsed excitation.

Figure 4

(top) Second-order coherence function $g_{LL}^{\left(2\right)}(\tau ,{\tau }_c)$ for the sign-change protocol $g\left(t\right)=g_{0}cos\left(\mathrm{\Omega }t\right)$ of the moderately driven $(f=10\gamma )$ qubit on resonance $(\delta =0)$ at fast modulation frequency $\mathrm{\Omega }=5\gamma$. The oscillations decay with the delay time $\tau$ at the rate $\gamma$ as can be seen, e.g., along the vertical cut at ${\tau }_c=T/4$ shown in the inset. Thus, this rapidly changing behavior takes place only for sufficiently fast modulations. (bottom) The strong oscillations in time ${\tau }_c$ between bunching and antibunching behavior reported in Ref. [37] become less pronounced with increasing input power $f$ as the qubit becomes saturated. The delay time $\tau$ is fixed at the value $3T/4$, which corresponds to the horizontal cut in the top figure.

Figure 5

Smallest dissipation rate ${\gamma }_{\min }$ of the $\mathrm{\Lambda }$ system as a function of the classical drive field amplitude $F$. Instead of directly modulating the coupling strength $g$, the three-level $\mathrm{\Lambda }$ system allows tuning of ${\gamma }_{\min }$ by means of $F$. The input power $f$ of the probe field has little effect on this behavior.

Figure 6

Transmission through the $\mathrm{\Lambda }$ system which is driven on resonance $({\delta }_1={\delta }_2=0)$ by both probe and drive pulses. The drive field has a periodically modulated amplitude $F\left(t\right)=10\gamma [1+cos\mathrm{\Omega }t]$ at $\mathrm{\Omega }=0.1\gamma$. The adiabatic approximation (dashed line) is only valid far away from the critical region defined by $\mathrm{\Omega }>{\gamma }_{\min }$ (cf. Fig. 5). In the time window where it breaks down, the system responds nonadiabatically. At large powers $f$ of the probe field, these effects are, however, washed out because of the system's saturation.

Figure 7

(top) Smallest dissipation rate ${\gamma }_{\min }$ of the undriven Kerr-nonlinearity model as a function of detuning $\delta$ for $U=-\gamma /2$ and $f=16\gamma$. Similar to the two- and three-level systems, the smallest dissipation rate is significantly suppressed within the critical region, although it remains finite. (bottom) One of the important signatures of the dissipative phase transition is a prominent increase in the stationary occupation number $\langle b^{†}b\rangle$ which peaks at the same parameter value for which the minimal dissipation rate is reached. In the following, the periodic modulation of $\delta$ is considered across the whole critical region with modulation frequency $\mathrm{\Omega }\ll \gamma$.

Figure 8

(left) Periodic steady-state occupation $\langle b^{†}b\rangle \left(t\right)$ of the Kerr-nonlinearity model under periodic modulation of detuning $\delta \left(t\right)=-15\gamma [1+cos\mathrm{\Omega }t]$ for various modulation frequencies $\mathrm{\Omega }$. (right) The same dependence in the parametric representation. For moderate modulation frequencies, the occupation is strongly enhanced compared with the true stationary state shown in the lower panel of Fig. 7. The adiabatic approximation based on Eq. (19) is given by the light brown curve (note that in the parametric representation it lies very close to the stationary-state result). The numerical results for rather small frequency $\mathrm{\Omega }=0.002\gamma$ (brown) still drastically deviate from the corresponding adiabatic approximation (light brown). As discussed by Casteels et al. [31], the dynamical hysteresis seen in the parametric plot is directly related to the breakdown of adiabaticity in the critical region, and the hysteresis area depends on the width of the parameter range where $\mathrm{\Omega }>{\gamma }_{\min }$.