Time-dependent Markovian quantum master equation

We construct a quantum Markovian master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the master equation to be extended beyond the adiabatic limit. The nonadiabatic master equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic master equation, the NAME leads to coupled equations of motion for the population and coherence. We employ the NAME to solve the example of an open driven time-dependent harmonic oscillator. For the harmonic oscillator the NAME predicts the emergence of coherence associated with the dissipation term. As a result of the nonadiabatic driving the thermalization rate is suppressed. The solution is compared with both numerical calculations and the adiabatic master equation.

Figure 1

System dynamics as generated from the NAME for different coupling strengths $g$ ($g=0$ represents isolated dynamics). The left panel shows the expectation value of the energy as a function of time, and the right panel shows the position momentum correlation $\langle \widehat{C}\rangle$ as a function of time. The chosen parameters are $\mu =-0.1,\omega \left(0\right)=40$, and $T=20$, where the initial conditions are $\beta \left(0\right)=-1$ and $\gamma \left(0\right)=0.5$. This corresponds to an initial state described by $\langle {\widehat{H}}_S\left(0\right)\rangle \approx 55,\langle \widehat{L}\left(0\right)\rangle \approx -20.5$, and $\langle \widehat{C}\left(0\right)\rangle \approx 3.7$.

Figure 2

The system dynamics generated from the NAME (pink, dark gray) is compared to the isolated quantum system (blue, dashed light gray) and the instantaneous attractor (fixed point) of the adiabatic solution (green, dashed gray) for a parametric harmonic oscillator. The dynamics are represented by the system variables $\langle {\widehat{H}}_S\left(t\right)\rangle ,\langle \widehat{L}\left(t\right)\rangle$, and $\langle \widehat{C}\left(t\right)\rangle$. Here the chosen parameters are $\mu =-0.1,\omega \left(0\right)=40,T=20$, and $g=1$, where the initial conditions include no coherence $\langle {\widehat{H}}_S\rangle =60,\langle \widehat{L}\left(0\right)\rangle =\langle \widehat{C}\left(0\right)\rangle =0$.

Figure 3

The dynamics of the coherence, $\mathcal{C}\equiv \frac{\sqrt{{\langle \widehat{L}\rangle }^2+{\langle \widehat{C}\rangle }^2}}{\hslash \omega \left(t\right)}$, is presented for different system-bath coupling strengths. Increasing the system-bath coupling induces an increase in coherence associated with the nonadiabatic driving. Model parameters and initial conditions are identical to Fig. 2.

Figure 4

(a) The energy value of the instantaneous attractor as a function of time for different values of the constant adiabatic parameter $\mu$. (b) The difference between the initial and the temporary value of the coherence of the instantaneous attractor, $\mathrm{\Delta }\langle {\widehat{C}}_{ia}\rangle =\langle {\widehat{C}}_{ia}\left(\tau \right)\rangle -\langle {\widehat{C}}_{ia}\left(0\right)\rangle$. (c) The time-dependent rate coefficient (atomic units are used). The initial frequency is $\omega \left(0\right)=40$ and temperature of the bath $T=20$, with initial values $\langle \widehat{H}\left(0\right)\rangle \approx \{26.3,26.2,26.2\}$ for $\mu =\{{10}^{-1},{10}^{-2},{10}^{-3}\}$ and similarly for $\langle \widehat{C}\left(0\right)\rangle =1.31\times \{1,{10}^{-1},{10}^{-2}\}$.

Figure 5

The energy as a function of time for different solutions. The frequency decreases for a negative $\mu$, leading to a decrease in the energy. The initial state is of a Gibbs form: ${\widehat{\rho }}_S=exp\left(\beta \left(0\right){\widehat{b}}^{†}\widehat{b}\right)$. The model parameters are shown in a table in Appendix pp8.