Examination of classical simulations for Heisenberg-Langevin equations for spin-1/2

Abstract

A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.

Figures (4)

• Figure 1: Three stochastic realizations of $S_z(t)$ under quantum zero-point noise. Each trajectory begins at $S_z(0) = +1$ and evolves stochastically. This figure demonstrates the need to average over many stochastic realizations to obtain a smooth curve, where the decay rate and long-time average can be analyzed. Here we chose the parameter set as $\Gamma=7.5$, $\omega_0=5$ and $\Gamma/\alpha=1$.
• Figure 2: This figure shows the ensemble-averaged decay of $S_z(t)$ under quantum noise averaged over $5,000$ realizations as well as the WW result with Lorentzian coupling. The parameters were chosen as $\Gamma/\alpha=1$ and $\omega_0=5$ in both cases, with two distinct choices for the remaining free parameter. These are $\Gamma_1=7.5$ and $\Gamma_2=20$. The solid lines result from the configuration $\Gamma_1$, whereas the dashed red and orange curves arise from the configuration $\Gamma_2$. Both parameter choices produce Markovian dynamics, where the second configuration is deeper in the Markovian limit with $\mu_{0,2}>\mu_{0,1}$. Comparing the dashed black line, which is the exponential decay resulting from the strictly Markovian WW calculation according to Eqs. (\ref{['WWres']}) and (\ref{['decayrate']}), one can see that both configurations excellently approximate this limit.
• Figure 3: Here, the ensemble-averaged decay of $S_z(t)$ is plotted over an average of $5,000$ realizations alongside the generalized WW result with Lorentzian coupling. The parameters were chosen so that $\Gamma/\alpha=1$, $\omega_0=5$ in both cases, and the remaining free parameter was varied with $\Gamma_3=0.01$, blue and turquoise, and $\Gamma_4=0.05$, red and orange. Damped oscillatory dynamics becomes clearly visible in both cases. The fourth configuration is less damped since $\mu_{0,3}<\mu_{0,4}$, where $\alpha/\Gamma=$ const..
• Figure 4: Here, the high temperature limit is plotted for three parameter choices at a temperature of $T=200$. The rapidly decaying turquoise curve originates from the classical HL simulation averaged over $25,000$ trajectories. The blue curve is the quantum mechanical expectation value computed along the lines of Eq. (\ref{['HTEX']}). The parameters chosen were $\Gamma_1=10$, $\Gamma_2=25$ and $\Gamma_3=50$, where all other parameters are dependently given by $\Gamma = 2\omega_0=10\alpha$. The parameters chosen yield dynamics moving deeper into the Markovian regime from left to right, since $\mu_{T,1}<\mu_{T,2}<\mu_{T,3}$.