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Markovian quantum master equations are exponentially accurate in the weak coupling regime

Johannes Agerskov, Frederik Nathan

Abstract

We consider the evolution of open quantum systems coupled to one or more Gaussian environments. We demonstrate that such systems can be described by a Markovian quantum master equation (MQME) up to a correction that decreases exponentially with the inverse system-bath coupling strength. We provide an explicit expression for this MQME, along with rigorous bounds on its residual correction, and numerically benchmark it for an exactly solvable model. The MQME is obtained via a generalized Born-Markov approximation that can be iterated to arbitrary orders in the system-bath coupling; our error bound converges asymptotically to zero with the iteration order. Our results thus demonstrate that the non-Markovian component in the evolution of an open quantum system, while possibly inevitable, can be exponentially suppressed at weak coupling.

Markovian quantum master equations are exponentially accurate in the weak coupling regime

Abstract

We consider the evolution of open quantum systems coupled to one or more Gaussian environments. We demonstrate that such systems can be described by a Markovian quantum master equation (MQME) up to a correction that decreases exponentially with the inverse system-bath coupling strength. We provide an explicit expression for this MQME, along with rigorous bounds on its residual correction, and numerically benchmark it for an exactly solvable model. The MQME is obtained via a generalized Born-Markov approximation that can be iterated to arbitrary orders in the system-bath coupling; our error bound converges asymptotically to zero with the iteration order. Our results thus demonstrate that the non-Markovian component in the evolution of an open quantum system, while possibly inevitable, can be exponentially suppressed at weak coupling.
Paper Structure (6 sections, 22 theorems, 111 equations, 2 figures)

This paper contains 6 sections, 22 theorems, 111 equations, 2 figures.

Table of Contents

  1. S.I: Gaussian baths
  2. S.III: proof of Proposition \ref{['prop:higher_order_born']}: the generalized Born approximation
  3. S.V: Proof of Lemma \ref{['ref:bk_lemma']}: bound on moments of the memory kernel
  4. S.VII: proof of Proposition \ref{['proposition:TightestBound']}: the generalized Markov approximation
  5. S.IX: proof of Lemma \ref{['lemma:simple_bound']}: simplified bound on $\lvert \xi_{mn}(t) \rangle\space\rangle$.
  6. S.XI: Proof of Theorem \ref{['ThmExpDecay']}

Key Result

Proposition 1

Let $\hat{\rho}(t)$ be the solution of Eq. EqSchrodinger. Then with $\left\lVert {{\xi^{\rm B}_n}(t)} \right\rVert_{\operatorname{tr}}\leq \varepsilon_n$, uniformly in $t$, where Here $\mathcal{W}^n_m$ denotes the set of weak compositions of $n$ into $m$ parts, i.e, the set of tuples of non-negative, not necessarily distinct, integers $(q_1,\ldots q_m)$ for which $\sum_{i} q_i = n$.

Figures (2)

  • Figure 1: Exponential suppression of error for Markovian quantum master equations. We show that any open quantum system coupled to Gaussian baths can be described by a Markovian quantum master equation with dissipator $\Delta_{mn}(t)$ up to a bounded residual error $\xi_{mn}$, Eqs. (\ref{['eq:bm_dis']},\ref{['EqBornMarkovApproxm']}). In (a) we show our bounds on $\left\lVert \xi_{nn} \right\rVert_{\rm tr}/\Gamma$ in terms of $\Gamma \tau$, with $\Gamma$ and $\tau$ scales for bath coupling strength and correlation time defined in Eqs. (\ref{['eq:gamma_mu_def']}-\ref{['eq:taudef']}). Curve labels indicate $n$, while P2 and T1 refer to bounds from Proposition \ref{['proposition:TightestBound']} and Theorem \ref{['ThmExpDecay']}, respectively. The bounds for $n=n_{* }$, as defined in Eq. \ref{['eq:mexpdef']}, decrease exponentially with $1/\sqrt{\Gamma \tau}$. (b) Evolution of $z$-spin error $\delta \langle \sigma_z(t)\rangle$ resulting from $\Delta_{11}$ ($\cdot$) and $\Delta_{22}$ ($\cdot$) for an exactly solvable spin-boson model; see below Eq. \ref{['eq:benchmark_model']} for details. Corresponding points in (a) indicate $\delta \langle \sigma_z(t)\rangle/t$ at $t=10/\gamma$.
  • Figure S1: Numerical data proving Eq. \ref{['eqa:thm1v2']} holds for $\sqrt{\Gamma \tau}\geq 0.042$ ($1/\Gamma \tau\leq 567$). Green: bound on $\left\lVert \xi_{nn} \right\rVert_{\operatorname{tr}}$ from Eq. \ref{['eq: bound relaxed22']}. Red: right-hand side of Eq. \ref{['eqa:thm1v2']}. For convenience, we also depict the other curves shown in Fig. \ref{['fig:1']} of the main text: Blue and orange depict bounds on $\left\lVert \xi_{nn} \right\rVert_{\operatorname{tr}}$ from Proposition \ref{['proposition:TightestBound appendix']}, for $n=1,2$, respectively. Black curve depicts bound on $\left\lVert \xi_{nn} \right\rVert_{\operatorname{tr}}$ from Proposition \ref{['proposition:TightestBound appendix']} for a part of the interval. Here we use that $\mu_i \leq i ! \tau$ for $i=1,\ldots n_{* }$.

Theorems & Definitions (49)

  • Definition 1
  • Definition 2: Correlation time
  • Definition 3: Memory kernel
  • Proposition 1
  • Lemma 1
  • Definition 4: Dissipator
  • Proposition 2: Tightest bound
  • Lemma 2: Simple bound
  • Definition 5
  • Theorem 1: Exponential accuracy of MQMEs
  • ...and 39 more