Hadamard regularization of open quantum systems coupled to unstructured environments in the Schwinger-Keldysh formalism

Abstract

The theory of open quantum systems addresses how coupling to external degrees of freedom modifies observables and quantum coherence, a situation central to fundamental condensed-matter research and emerging quantum technologies. Schwinger-Keldysh field theory is a natural framework for both open- and nonequilibrium quantum systems in terms of functional integrals. However, its numerical solution is limited by a cubic scaling with the number of time steps. This is particularly prohibitive for scenarios with widely separated time scales, as is often the case for system and environmental scales. We consider a damped quantum harmonic oscillator as a toy model to study a separation-of-scales ansatz based on Hadamard regularization. A time-stepping algorithm for the Kadanoff-Baym equations on the slow system time-scale is presented that captures both low-temperature non-Markovianity and renormalization effects arising from the much faster environment scale.

Figures (7)

• Figure 1: (a): Comparison of the undamped and damped QHO amplitude variances for $\gamma/\omega_0 = 0.5$. The non-constant graphs, (red) for Drude regularization and (violet) for exponential regularization, show that the amplitude variance as a function of $\omega_c$ interpolates between the undamped oscillator (blue) and the ohmic limit \ref{['eq:field_variance_ohmic']} (green). For comparison, we also present the value for an undamped oscillator with the same resonance frequency, $\omega_\gamma$, as the damped one (yellow). (b): Scaling of the amplitude variance with damping strength $\gamma$ (green) compared to the undamped oscillator with the same resonance frequency (yellow) diverging at $\gamma = 2\omega_0$. (c)Comparison of the undamped and damped QHO momentum variances for $\gamma/\omega_0 = 0.5$ as a function of $\omega_c$. We compare the Drude regularization (red) with the exponential regularization (violet). Both converge to the isolated oscillator for $\omega_c \to 0$ and scale logarithmically for $\omega_c \to \infty$. (d): We show the momentum variance of the damped QHO at $\omega_c = 10\omega_0$ compared to the isolated QHO, both with the original frequency $\omega_0$ and with the same resonance frequency as the damped QHO, $\omega_\gamma$. We also compare the Drude regularization (red) with the exponential regularization (violet), both of which exhibit the same qualitative behavior: monotonically increasing as a function of $\gamma$.
• Figure 2: Illustration of the time-stepping algorithm in the time-plane for symmetric (left) and antisymmetric (right) correlators (in the style of schuler_time-dependent_2016). Each step from $t_n\to t_{n+1}$ consists of two sub-steps. First, evolve $G(t_n,t_k) \to G(t_{n+1},t_k)$ for all $k\leq n$ (violet arrows). The lower triangle contains redundant information, due to the respective symmetry, $G^{S/A}(t_k,t_{n+1}) = \pm G^{S/A}(t_{n+1},t_k)$ (dashed arrows). For the antisymmetric correlator, the diagonal vanishes exactly, so no further action is required (green dots). For the symmetric correlator, the diagonal elements capture the amplitude variance; on it, the KBE diverges logarithmically with $\omega_c$ (orange dots). It is obtained via the additional step $G^S(t_n,t_{n+1}) \to G^S(t_{n+1},t_{n+1})$ (red arrow).
• Figure 3: The terms of \ref{['eq:memory_term_interpretation']} which constitute the nontrivial part of the $f$ term in \ref{['eq:toy_model']} are shown as a function of $t_1$ for a fixed value of $t_2=100\, \Delta t$. The linear local approximations, implicit in the Verlet integration, are shown as dotted lines. X-markers highlight the points around $t_1=t_2$, where the linear approximation breaks down. Lastly, a black dashed line shows the Filon quadrature interpolation. The model parameters are $\omega_\gamma = 196$ meV, $\gamma = 200$ meV, $T = 26$ meV and $\omega_c = 100/\Delta t$, where $\Delta t = \frac{1}{30}\frac{2\pi}{\omega_0}$
• Figure 4: Thermalization benchmark of the numerical implementation across the temperature-damping parameter landscape. The relative error of the long-time asymptote $\lim_{t\to\infty}G^S(t,t)$ of the implementation with an environment at temperature $T$ compared to the exact thermal equilibrium variance $\expval{\varphi^2}_T$ obtained in the Matsubara formalism is shown. All axes use logarithmic scaling. The time step $\Delta t$ is $\frac{2\pi}{100}$ of the fastest system scale ($\omega_0$, $T$, or $\gamma$) and $\omega_c = 10^5 \omega_0$.
• Figure 5: Decay behaviour of the symmetric correlator in the ultra-cold regime ($T<\gamma$). The KBE solution (blue) is compared against the solution of the Lindblad equation (yellow). Whereas the Lindblad solution shows immediate exponential decay, the KBE solution enters a regime of quadratic decay (black dotted line) on the time-scale between $2/\gamma$ and the first Matsubara frequency, $1/2\pi T$, indicated by vertical lines. Here, $\gamma = 1$ and $T = 0.001$.