Stochastic simulation of dissipative quantum oscillators

Abstract

Generic open quantum systems are notoriously difficult to simulate unless one looks at specific regimes. In contrast, classical dissipative systems can often be effectively described by stochastic processes, which are generally less computationally expensive. Here, we use the paradigmatic case of a dissipative quantum oscillator to give a pedagogic introduction into the modelling of open quantum systems using quasiclassical methods, i.e. classical stochastic methods that use a 'quantum' noise spectrum to capture the influence of the environment on the system. Such quasiclassical methods have the potential to offer insights into the impact of the quantum nature of the environment on the dynamics of the system of interest whilst still being computationally tractable.

Figures (4)

• Figure 1: Stochastic force. Panel a) shows a single realization of the stochastic force as a function of time, $F(t)$, which has the Gaussian probability distribution given in b). Panel d) shows the power spectral density $P_F(\omega, T) = \pi \hbar J(\omega) \coth{( \frac{\hbar \omega}{2 k_\mathrm{B}T} )}$ of the stochastic noise. Choosing a Lorentzian form of the spectral density $J(\omega)$ gives rise to colored noise, leading to correlations, which are shown by the skewed distribution of the two-time correlation function in panel c).
• Figure 2: Panels a) - c): agreement of the dynamical covariances $\sigma_{xx}$(t), $\sigma_{xp}(t)$ and $\sigma_{pp}(t)$, see Eqs. \ref{['eq:covs']} and \ref{['eq:I']}, from the OQu (dark blue lines) and StoQu (dark blue dots) approaches. Also plotted are the dynamical covariances from the OCl approach (light blue lines) and the StoCl approach (light blue dots). Panel d) shows the violation of the uncertainty relation by the classical dynamics, indicated by the drop of the light-blue line into the grey region. In panels $\mathbf{a)}$ - $\mathbf{c)}$, the variances have been normalised by the zero-point magnitudes $x_{\rm{zpm}} = \sqrt{\hbar / (2 m \Omega)}$ and $p_{\rm{zpm}} = \sqrt{(\hbar m \Omega)/2}$, and the Lorentzian parameters are $\lambda = 0.3 \, \Omega^2 m^{1/2}$, $\omega_0 = 0.5 \, \Omega$ and $\Gamma = 0.1 \, \Omega$, and $T = 0.1 \hbar \Omega/k_\mathrm{B}$. The initial variances are $\sigma_{xx}(0) = \sigma_{pp}(0) = 0.5$ and $\sigma_{xp}(0) = 0$.
• Figure 3: Agreement of the steady-state covariance $\sigma_{xx}(\infty)$, see Eq. \ref{['eq:cov_xx_ss']}, from the OQu (lines) and StoQu (markers) approaches for environmental coupling strengths of $\lambda = 0.3 \, \Omega^2 m^{1/2}$ (solid line, circle markers) and $\lambda = 2 \, \Omega^2 m^{1/2}$ (dashed line, square markers). Also plotted is $\sigma_{xx}$ calculated with the Gibbs state \ref{['eq:gibbs']} (grey dotted line). One can see that increasing $\lambda$ consistently leads to a deviation of the steady state from the Gibbs state. Here, the variances have been normalised by the zero-point magnitude $x_{\rm{zpm}} = \sqrt{\hbar / (2 m \Omega)}$, and the other Lorentzian parameters are $\omega_0 = 0.5 \, \Omega$ and $\Gamma = 0.1 \, \Omega$, and $T = 0.1 \hbar \Omega/k_\mathrm{B}$. The initial variances are $\sigma_{xx}(0) = \sigma_{pp}(0) = 0.5$ and $\sigma_{xp}(0) = 0$.
• Figure 4: Agreement of the steady-state heat currents across a two oscillator network calculated using the open quantum systems approach (OQu, lines, \ref{['eq:qu_currents']}) and the quasiclassical stochastic approach (StoQu, dots, \ref{['eq:sto_currents']}) techniques. The oscillators are coupled together with strength $\kappa = 0.1 \hbar \Omega/m^2$ and the temperature gradient is $T_{\mathrm{H}} = 10 T_{\mathrm{C}}, T_{\mathrm{C}} = T$. As the system is in its steady state, the two heat currents have the same magnitude but opposite sign. The parameters that characterize the Lorentzian spectral density $J_\mathrm{Lor}(\omega)$ for each oscillator are $\lambda = 0.3 \, \Omega^2 m^{1/2}$, $\omega_0 = 0.5 \, \Omega$ and $\Gamma = 0.8 \, \Omega$. For each oscillator, the initial variances are $\sigma_{xx}(0) = \sigma_{pp}(0) = 0.5$ and $\sigma_{xp}(0) = 0$.