Long-Term Behaviour in an Exactly Solvable Model of Pure Decoherence and the Problem of Markovian Embedding

Abstract

We consider a well-known, exactly solvable model of an open quantum system with pure decoherence. The aim of this paper is twofold. Firstly, decoherence is a property of open quantum systems important for both quantum technologies and the fundamental question of the quantum–classical transition. It is worth studying how the long-term rate of decoherence depends on the spectral density characterising the system–bath interaction in this exactly solvable model. Secondly, we address a more general problem of the Markovian embedding of non-Markovian open system dynamics. It is often assumed that a non-Markovian open quantum system can be embedded into a larger Markovian system. However, we show that such embedding is possible only for Ohmic spectral densities (for the case of a positive bath temperature) and is impossible for both sub- and super-Ohmic spectral densities. On the other hand, for Ohmic spectral densities, an asymptotic large-time Markovianity (in terms of the quantum regression formula) takes place.

1. Introduction

Decoherence is a fundamental property of quantum systems subject to external noise. It is important from both a fundamental and a practical point of view. Fundamentally, decoherence is often considered as the phenomenon responsible for the emergence of the description of macroscopic physical reality based on classical mechanics and classical (Kolmogorov) probability theory from the quantum description. Quantum coherence is a distinctive property of the quantum world and the decay of coherence is at least one of the factors of the quantum-to-classical transition. From the viewpoint of applications, decoherence is one of the obstacles to the construction of a quantum computer, which requires the preservation of coherence during the computation.

Mathematically, quantum systems subject to external noise are studied within the framework of the theory of open quantum systems, in which models of systems interacting with a thermal bath or another environment are considered. In this paper, we study a particular well-known, exactly solvable model of pure decoherence [1,2,3]. Despite the fact that it is well known and has attracted interest in a number of recent studies (especially with respect to the question of Markovianity or non-Markovianity of the dynamics) [4,5,6,7,8], precise asymptotic estimations of the decoherence rate depending on the spectral density (specifying the system–bath interaction) are absent. The purpose of this paper is to fill this gap.

Then, we discuss some consequences of these results, which can be important for present active discussions about non-Markovian open quantum dynamics. There is a hierarchy of mathematical definitions related to our intuitive understanding of “Markovianity” or the “absence of memory of the bath” in the context of open quantum systems [9]. According to one approach, Markovianity is associated with the semigroup property of the dynamics. A generator of this semigroup has the well-known Gorini–Kossakowski–Sudarshan–Lindblad form (GKSL; we consider only finite-dimensional systems). So, in this case, the dynamics of the reduced density operator of the system is simple and can be described by a GKSL quantum master equation. In the finite-dimensional case, this description is reduced to a system of linear ordinary differential equations with constant coefficients. Definitions of Markovianity related to certain properties of the dynamics of the reduced density operator of the system, like the decrease in the distinguishability of the evolving quantum states or CP-divisibility (“CP” means “completely positive”), cover wider ranges of models and include certain classes of time-dependent GKSL generators [5,9,10,11]. The definition of quantum Markovianity generalising the corresponding definition from the classical random process theory is related to the quantum regression formula [5,9,11,12,13]. In this sense, the definition of Markovianity for open quantum systems based on the quantum regression formula is the most natural one.

One of the popular ways of dealing with non-Markovian quantum dynamics is to embed an open quantum system into a larger system whose dynamics is Markovian in the sense that its dynamics is described by a quantum dynamical semigroup with a (time-independent) GKSL generator. Of course, the original system–bath unitary dynamics is Markovian. But the bath typically contains a continuum of oscillatory modes. In the described approach, we try to embed our system into a larger system, which is still finite-dimensional or, at least, has a discrete energy spectrum. In the latter case, it is possible to cut high energy levels and approximately reduce its (Markovian) dynamics to finite-dimensional (also Markovian) dynamics. The methods of pseudomodes [14,15,16,17,18] and reaction (collective) coordinates [19,20,21] are examples of this approach. The assumption of a possibility of a Markovian embedding is often used in data-driven prediction of the dynamics of open quantum systems [22,23]. The method of Markovian embedding is schematically depicted in Figure 1. However, we show that, for some bath spectral densities, such embedding is impossible. Namely, in the case of the bath with a positive temperature, only for the case of an Ohmic spectral density (i.e., very specific asymptotic behaviour in a vicinity of zero) is the Markovian embedding possible. But even in this case, there can be factors not described by Markovian embeddings. If the spectral density is either sub- or super-Ohmic (which are common in physics), such embedding is impossible.

Vice versa, for the Ohmic spectral density (again in the case of a positive temperature), we can observe the asymptotic Markovianity in the most natural sense related to the quantum regression formula. That is, even if the dynamics is non-Markovian, it becomes Markovian on large time scales, which simplifies its description. These results can be considered as a development of the results by Lonigro and Chruściński [7]. Namely, they show that the quantum regression formula for the considered pure decoherence model is satisfied exactly if the bath spectral density is flat for the whole real line of positive and negative frequencies of the bath. However, these conditions (especially negative frequencies) are unphysical. The authors, however, note that the flat spectral density can be a reasonable approximation in realistic scenarios. Here, we confirm this hypothesis: the asymptotic satisfiability of the quantum regression formula is explained by the fact that, for large time scales, only the behaviour of the spectral density in a vicinity of zero matters, so it can be considered to be approximately flat on the whole real line.

The following text is organised as follows. In Section 2, we describe the model. In Section 3, we prove the main theorem about the long-term rates of decoherence depending on the asymptotic behaviour of the spectral density in the vicinity of zero. Also, we discuss the relation of this theorem to known results about the impossibility of the exponential relaxation for some observables and its possibility for other observables, which has attracted interest for the current research [24,25,26,27,28]. In Section 4, we illustrate the theorem by two particular widely used spectral densities. In Section 5, we compare the obtained results with various rigorous results on the Davies GKSL generator of a quantum dynamical semigroup in the weak-coupling regime [29,30,31]. Section 6 is devoted to the impossibility of the Markovian embedding in the cases of sub- or super-Ohmic spectral densities. In Section 7, we prove asymptotic Markovianity (in the sense of the quantum regression formula) for Ohmic spectral densities. In all these sections, we consider the realistic case of a bath with positive temperature. However, the zero-temperature bath is often used as a reasonable approximation when the temperature is low. The results can be easily transferred to this case as well, which is discussed in Section 8.

2. Problem Statement

Let us consider a two-dimensional quantum system (a qubit) interacting with a finite number of N quantum harmonic oscillators. Mathematically speaking, we work in the Hilbert space ${\mathcal{H}}_{\mathrm{S}}\otimes {\mathcal{H}}_{\mathrm{B}}={\mathbb{C}}^2\otimes {\left({\ell }^2\right)}^{\otimes N}$. Consider the following Hamiltonian (self-adjoint operator) acting in this space:

$$H=H_{\mathrm{S}}+H_{\mathrm{B}}+H_{\mathrm{I}}=\frac{{\omega }_0}{2}{\sigma }_z+\sum _{k=1}^N{\omega }_k{a}_k^{†}{a}_k+\frac{{\sigma }_z}{2}\otimes \sum _{k=1}^N(g_k{a}_k^{†}+g_k^{*}{a}_k),$$

(1)

where

$${\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=|1\rangle \langle 1|-|0\rangle \langle 0|,|1\rangle =\left(\begin{array}{c}1\\ 0\end{array}\right),|0\rangle =\left(\begin{array}{c}0\\ 1\end{array}\right)$$

(2)

are one of the Pauli matrices and basis elements (the standard basis) of ${\mathbb{C}}^2$, respectively. Further, $a_k^{†}$ and $a_k$ are the bosonic creation and annihilation operators, respectively, ${\omega }_k>0$, ${\omega }_0$ is a real number, and $g_k$ are complex numbers. In Ref. [32], it is proved that formula (1) defines a self-adjoint operator. Formally, the proof is performed for the case of a continuum of oscillators in the bath, but it can be applied for a simpler case of a finite number of oscillators as well.

Consider the following initial density operator (quantum state):

$$\rho \left(0\right)={\rho }_{\mathrm{S}}\left(0\right)\otimes {\rho }_{\mathrm{B},\beta }\equiv {\rho }_{\mathrm{S}}\left(0\right)\otimes Z_{\mathrm{B}}^{-1}{e}^{-\beta H_{\mathrm{B}}},$$

(3)

where ${\rho }_{\mathrm{S}}\left(0\right)$ is the initial density operator of the system, ${\rho }_{\mathrm{B},\beta }$ is the thermal (Gibbs) state of the bath, $\beta$ is the inverse temperature, and $Z_{\mathrm{B}}=Tr{e}^{-\beta H_{\mathrm{B}}}$. The joint system–bath state at time t is given by

$$\rho \left(t\right)=e^{-itH}\rho \left(0\right)e^{itH}.$$

(4)

We are interested in the reduced density operator of the system:

$${\rho }_{\mathrm{S}}\left(t\right)={Tr}_{\mathrm{B}}\rho \left(t\right)={Tr}_{\mathrm{B}}\left[e^{-itH}\rho \left(0\right)e^{itH}\right],$$

(5)

where ${Tr}_{\mathrm{B}}$ denotes the partial trace over the space ${\mathcal{H}}_{\mathrm{B}}$. In the interaction representation:

$$\varrho \left(t\right)=e^{it{H}_{\mathrm{S}}}{\rho }_{\mathrm{S}}\left(t\right)e^{-it{H}_{\mathrm{S}}}=\left(\begin{array}{cc}{\varrho }_{11}\left(t\right)& {\varrho }_{10}\left(t\right)\\ {\varrho }_{01}\left(t\right)& {\varrho }_{00}\left(t\right)\end{array}\right),$$

(6)

where ${\varrho }_{jk}$ are the matrix elements of the operator $\varrho$ in the standard basis $\{|1\rangle ,|0\rangle \}$.

Since $[H,{\sigma }_z]=0$, ${\varrho }_{11}\left(t\right)={\varrho }_{11}\left(0\right)$ and ${\varrho }_{00}\left(t\right)={\varrho }_{00}\left(0\right)$. Also,

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right){Tr}_{\mathrm{B}}\left[e^{-it{H}_1}{\rho }_{\mathrm{B},\beta }{e}^{it{H}_0}\right],$$

where

$$H_j=\sum _{k=1}^N{\omega }_k{a}_k^{†}{a}_k+\frac{{(-1)}^{j-1}}{2}\sum _{k=1}^N(g_k{a}_k^{†}+g_k^{*}{a}_k),$$

(7)

$j=0,1$. One can show [1,2,3,7,11] that

$${Tr}_{\mathrm{B}}\left[e^{-it{H}_1}{\rho }_{\mathrm{B},\beta }{e}^{it{H}_0}\right]=e^{-\Gamma \left(t\right)},$$

(8)

where

$$\begin{array}{cc}\hfill \Gamma \left(t\right)& =\sum _{k=1}^N{\left|g_k\right|}^{2}coth\left(\frac{\beta {\omega }_k}{2}\right)\frac{1-cos{\omega }_{k}t}{{\omega }_k^2}\hfill \\ \hfill & ={\int }_0^{\infty }J\left(\omega \right)coth\left(\frac{\beta \omega }{2}\right)\frac{1-cos\omega t}{{\omega }^2}d\omega \hfill \\ & ={\int }_0^{\infty }{J}_{\mathrm{eff}}\left(\omega \right)\frac{1-cos\omega t}{{\omega }^2}d\omega .\hfill \end{array}$$

(9)

Here,

$$J\left(\omega \right)=\sum _{k=1}^N{\left|g_k\right|}^2\delta (\omega -{\omega }_k)$$

(10)

is the spectral density function of the bath and

$$J_{\mathrm{eff}}\left(\omega \right)=J\left(\omega \right)coth\left(\frac{\beta \omega }{2}\right)$$

(11)

is sometimes called the effective spectral density function.

The diagonal elements of the density matrix ${\varrho }_{11}\left(t\right)$ and ${\varrho }_{00}\left(t\right)$ are called the populations and the off-diagonal elements ${\varrho }_{10}\left(t\right)$ and ${\varrho }_{01}\left(t\right)={\varrho }_{10}{\left(t\right)}^{*}$ are called the coherences. The decrease in the off-diagonal elements is called decoherence. It is clear that $\Gamma \left(t\right)\ge \Gamma \left(0\right)=0$ and, hence, $|{\varrho }_{10}\left(t\right)|\le |{\varrho }_{10}\left(0\right)|$. If there is a finite number of oscillators N in the bath, then $\Gamma \left(t\right)$ and $\varrho \left(t\right)$ are periodic or almost periodic functions of t.

In the thermodynamic limit of the bath $N\to \infty$, the index k runs over a continuous set and we assume that $J\left(\omega \right)$ tends to an integrable function on the half-line $[0,\infty )$. Note that there is a mathematically rigorous way to start directly from the continuum of oscillators rather than to perform a thermodynamic limit of a finite number of oscillators [33,34]. However, for our purposes, this mathematically simpler approach with the thermodynamic limit will suffice.

We are interested in the behaviour of $\Gamma \left(t\right)$ given by the second (or third) line of Equation (9), where $J\left(\omega \right)$ is an integrable function on $[0,\infty )$, for large time scales $t\to \infty$. If $\Gamma \left(t\right)\to \infty$ and, thus, ${\varrho }_{10}\left(t\right)\to 0$ as $t\to \infty$, then we will say that the full decoherence occurs. If $\Gamma \left(t\right)$ is a bounded function of t, so $|{\varrho }_{10}\left(t\right)|\ge {\varrho }_{10}^{*}>0$, then we will say that the partial decoherence occurs. The partial decoherence for certain classes of spectral densities is a known theoretical prediction [2,3,8,35]. For completeness, we include these results into Theorem 1, but we are mainly interested in the case of full decoherence. Then, the question we are interested in is the following: what is the asymptotic rate of convergence of ${\varrho }_{10}\left(t\right)$ to zero for large t depending on the properties of $J\left(\omega \right)$? In book [1] and also in Ref. [2], only particular forms of $J\left(\omega \right)$ are considered. Here, we obtain a general answer.

The crucial point is the asymptotic behaviour of $J\left(\omega \right)$ for small $\omega$: pure decoherence is caused by interaction with the low frequencies of the bath. We consider only the case $J\left(\omega \right)\sim c{\omega }^{\gamma }$ as $\omega \to +0$ (i.e., ${lim}_{\omega \to +0}J\left(\omega \right){\omega }^{-\gamma }=c$), where $c,\gamma >0$. If $0<\gamma <1$, $\gamma =1$, or $\gamma >1$, the spectral density is called sub-Ohmic, Ohmic, or super-Ohmic, respectively.

3. Main Theorem

Theorem 1.

Let $J\left(\omega \right)$ be an integrable function on $[0,\infty )$.

1.

If $J\left(\omega \right)\sim c{\omega }^{2+\delta }$ as $\omega \to +0$, where $\delta >0$ and $c>0$, then

$$\underset{t\to \infty }{lim}\Gamma \left(t\right)={\int }_0^{\infty }\frac{J\left(\omega \right)}{{\omega }^2}coth\left(\frac{\beta \omega }{2}\right)d\omega \equiv {\Gamma }_{\infty },$$

(12)

hence, the decoherence is partial:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)e^{-{\Gamma }_{\infty }},$$

(13)

where $B\left(t\right)$ is a bounded function converging to 1 as $t\to \infty$.

2.

If $J\left(\omega \right)\sim c\omega$ as $\omega \to +0$, where $c>0$, there exists $J_{\mathrm{eff}}^{\prime }\left(0\right)={lim}_{\omega \to +0}{J}_{\mathrm{eff}}^{\prime }\left(\omega \right)$, and there exists a bounded second derivative $J_{\mathrm{eff}}^{″}\left(\omega \right)$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$, then

$$\Gamma \left(t\right)={\Gamma }_{0}t+\alpha lnt+C+o\left(1\right),t\to \infty ,$$

(14)

where C is a constant; hence, the decoherence is exponential:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)e^{-{\Gamma }_{0}t}/t^{\alpha }.$$

(15)

Here,

$$\begin{array}{cc}\hfill {\Gamma }_0& =\frac{\pi }{2}{J}_{\mathrm{eff}}\left(0\right)=\pi {\beta }^{-1}\underset{\omega \to +0}{lim}\frac{J\left(\omega \right)}{\omega }>0,\hfill \\ \hfill \alpha & =J_{\mathrm{eff}}^{\prime }\left(0\right)=2{\beta }^{-1}\underset{\omega \to +0}{lim}{\left(\frac{J\left(\omega \right)}{\omega }\right)}^{\prime },\hfill \end{array}$$

(16)

and $B\left(t\right)$ is a bounded function converging to $e^{-C}$ as $t\to \infty$.

3.

If $J\left(\omega \right)\sim c{\omega }^2$ as $\omega \to +0$, where $c>0$, there exists $J_{\mathrm{eff}}^{\prime }\left(0\right)={lim}_{\omega \to +0}{J}_{\mathrm{eff}}^{\prime }\left(\omega \right)$, and there exists a bounded second derivative $J_{\mathrm{eff}}^{″}\left(\omega \right)$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$, then the decoherence obeys a power law:

$$\Gamma \left(t\right)=\alpha lnt+C+o\left(1\right),t\to \infty ,$$

(17)

where $\alpha >0$ is given in Equation (16); hence,

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)/t^{\alpha },$$

(18)

where, again, $B\left(t\right)$ is a bounded function converging to $e^{-C}$ as $t\to \infty$.

4.

If $J\left(\omega \right)={\omega }^{1+\delta }G\left(\omega \right)$ as $\omega \to +0$, where $-1<\delta <1$ and $G\left(0\right)>0$, there exists a derivative of the function ${\omega }^{-\delta }{J}_{\mathrm{eff}}\left(\omega \right)$ at $\omega =0$ (which, as before, can be defined in terms of the limit $\omega \to +0$), and there exists a bounded second derivative ${\left[{\omega }^{-\delta }{J}_{\mathrm{eff}}\left(\omega \right)\right]}^{″}$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$, then

$$\Gamma \left(t\right)=A{t}^{1-\delta }+\tilde{C}+o\left(1\right),t\to \infty ,$$

(19)

if $0<\delta <1$, and

$$\Gamma \left(t\right)=A{t}^{1-\delta }+2{\beta }^{-1}{G}^{\prime }\left(0\right)O(t^{-\delta }lnt),t\to \infty ,$$

(20)

if $-1<\delta \le 0$, where

$$A=2{\beta }^{-1}G\left(0\right){\int }_0^{\infty }\frac{1-cos\upsilon }{{\upsilon }^{2-\delta }}d\upsilon$$

and $\tilde{C}$ is a constant. Thus, the decoherence is subexponential for $\delta >0$ and superexponential for $\delta <0$:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)\tilde{B}\left(t\right)e^{-A{t}^{1-\delta }}$$

(21)

if $0<\delta <1$, where $\tilde{B}\left(t\right)$ is a bounded function converging to $e^{-\tilde{C}}$ as $t\to \infty$, and

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)e^{-A{t}^{1-\delta }+2{\beta }^{-1}{G}^{\prime }\left(0\right)O(t^{-\delta }lnt)},t\to \infty$$

(22)

if $-1<\delta \le 0$.

The results of Theorem 1 can be summarised as follows. Let $J\left(\omega \right)\sim c{\omega }^{\gamma }$ (or, equivalently, $J_{\mathrm{eff}}\left(\omega \right)\sim c{\omega }^{\gamma -1}$) as $\omega \to +0$ for some $c>0$.

• If $0<\gamma <1$ (sub-Ohmic spectral density), then the decoherence is full and its rate is superexponential.
• If $\gamma =1$ (Ohmic spectral density), then the decoherence is full and its rate is exponential.
• If $1<\gamma <2$ (super-Ohmic spectral density), then the decoherence is full and its rate is subexponential, but faster than any degree of t.
• If $\gamma =2$ (super-Ohmic spectral density), then the decoherence is full and obeys a power law.
• If $\gamma >2$ (super-Ohmic spectral density), then the decoherence is partial.

Also, it can be noticed that all the decoherence constants in Theorem 1 are proportional to the temperature ${\beta }^{-1}$, which corresponds to the physical intuition that an environment with a higher temperature causes faster decoherence.

Remark 1.

The question of the possibility of exponential relaxation of various observables on large time scales has been studied in quantum mechanics for a long time. Though models of exponential relaxation are common in physics, it is known that, strictly speaking, exponential long-term decay of the so-called survival probability of a quasi-stationary quantum state is impossible whenever the Hamiltonian is bounded from below (i.e., has a ground state) [24,25] (see also Ref. [27]). Recently, this result was generalized to arbitrary positive observables [28]. For such observables, the exponential relaxation can only be an approximation.

However, Theorem 1 says that the exponential long-term relaxation of the coherence can take place for the considered model with Hamiltonian (1), which is bounded from below [32]. This does not contradict the aforementioned results, because the observables ${\sigma }_x$ and ${\sigma }_y$ (the first two Pauli matrices) related to the coherences are not positive, which is mentioned also in Ref. [28]. Moreover, in Ref. [26], a model of an exponential pure decoherence in a two-level system interacting with a quantum particle on a line (rather than with a continuum of oscillators) is presented. The present results say that the long-term exponential decoherence can be achieved in a certain class of systems (i.e., spin-boson with an Ohmic spectral density) rather than by a fine-tuned choice of the parameters. On the other hand, the Ohmic class of spectral densities, though being commonly used, can also be considered to be particular.

Proof of Theorem 1.

Let us consider all the cases.

Case 1. In this case, $J_{\mathrm{eff}}\left(\omega \right)\sim {\omega }^{1+\delta }$, $\delta >0$, and $J_{\mathrm{eff}}\left(\omega \right)/{\omega }^2\sim {\omega }^{-(1-\delta )}$. That is, $\frac{J_{\mathrm{eff}}\left(\omega \right)}{{\omega }^2}$ is an integrable function. Then, by the Riemann–Lebesgue lemma, ${\int }_0^{\infty }\frac{J_{\mathrm{eff}}\left(\omega \right)}{{\omega }^2}cos\omega td\omega$ tends to zero as $t\to \infty$, from which Equation (12) follows.

Note that if we impose additional conditions on the degree of smoothness of $J_{\mathrm{eff}}\left(\omega \right)$ to provide the possibility of a certain number of iterative integrations by parts, then the long-term rate of convergence of $\Gamma \left(t\right)$ to ${\Gamma }_{\infty }$ can be estimated.

Cases 2 and 3. In these cases, we can express

$$\begin{array}{cc}\hfill \Gamma \left(t\right)& ={\int }_0^{{\omega }_{\mathrm{c}}}\frac{J_{\mathrm{eff}}\left(\omega \right)-J_{\mathrm{eff}}\left(0\right)-J_{\mathrm{eff}}^{\prime }\left(0\right)\omega }{{\omega }^2}(1-cos\omega t)d\omega \hfill \\ \hfill & +J_{\mathrm{eff}}\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^2}d\omega \hfill \\ \hfill & +J_{\mathrm{eff}}^{\prime }\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{\omega }d\omega \hfill \\ \hfill & +{\int }_{{\omega }_{\mathrm{c}}}^{\infty }{J}_{\mathrm{eff}}\left(\omega \right)\frac{1-cos\omega t}{{\omega }^2}d\omega ,\hfill \end{array}$$

(23)

where ${\omega }_{\mathrm{c}}>0$ (“c” from “cutoff”) is arbitrary. Consider all the terms. The first factor in the integrand of the first term is a locally integrable function and, hence, by the Riemann–Lebesgue lemma, the first term tends to a constant. The same is true for the last term: $J_{\mathrm{eff}}\left(\omega \right)/{\omega }^2$ is integrable on Borel sets that do not include zero. Consider the second term:

$$\begin{array}{cc}\hfill {\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^2}d\omega & ={\int }_0^{\infty }\frac{1-cos\omega t}{{\omega }^2}d\omega -{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{1-cos\omega t}{{\omega }^2}d\omega \hfill \\ & =t{\int }_0^{\infty }\frac{1-cos\upsilon }{{\upsilon }^2}d\upsilon -\frac{1}{{\omega }_{\mathrm{c}}}+{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{cos\omega t}{{\omega }^2}d\omega \hfill \\ & =\frac{\pi }{2}t-\frac{1}{{\omega }_{\mathrm{c}}}+{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{cos\omega t}{{\omega }^2}d\omega .\hfill \end{array}$$

Here, in the second equality, we have performed the substitution $\omega t=\upsilon$. The last integral converges to zero as $t\to \infty$ by the Riemann–Lebesgue lemma.

In the second term of (23), we perform the same substitution $\omega t=\upsilon$ and then use a known formula for the integral of $(1-cos\upsilon )/\upsilon$ (see, e.g., Ref. [36]):

$${\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{\omega }d\omega ={\int }_0^{{\omega }_{\mathrm{c}}t}\frac{1-cos\upsilon }{\upsilon }d\upsilon =ln{\omega }_{\mathrm{c}}t+\gamma -\mathrm{Ci}\left(t\right),$$

(24)

where $\gamma$ is the Euler–Mascheroni constant and

$$\mathrm{Ci}\left(t\right)=-{\int }_t^{\infty }\frac{cosx}{x}dx$$

(25)

is the cosine integral, which is a bounded function converging to zero as $t\to \infty$.

The expressions for the constants ${\Gamma }_0$ and $\alpha$ follow from the previous calculations and the following observations:

$$\begin{array}{cc}\hfill J_{\mathrm{eff}}\left(0\right)& =\underset{\omega \to +0}{lim}J\left(\omega \right)coth\frac{\beta \omega }{2}=2{\beta }^{-1}\underset{\omega \to +0}{lim}\frac{J\left(\omega \right)}{\omega },\hfill \\ \hfill J_{\mathrm{eff}}^{\prime }\left(0\right)& =\underset{\omega \to +0}{lim}{\left(J\left(\omega \right)coth\frac{\beta \omega }{2}\right)}^{\prime }=2{\beta }^{-1}\underset{\omega \to +0}{lim}{\left(\frac{J\left(\omega \right)}{\omega }\right)}^{\prime }.\hfill \end{array}$$

We have proved cases 2 and 3 of the theorem.

Note that, if $J_{\mathrm{eff}}^{\prime }\left(0\right)=0$ (which is the case, e.g., for the Drude–Lorentz spectral density, see below), then we do not need to introduce a cutoff frequency ${\omega }_{\mathrm{c}}$ since both the first and the second integrals in Equation (23) are convergent even for an infinite upper limit of integration in this case.

Case 4. In this case, $J_{\mathrm{eff}}^{\prime }\left(0\right)$ is infinite unless $\delta =0$ (i.e., case 2). However, define $\tilde{G}\left(\omega \right)=\omega coth(\beta \omega /2)G\left(\omega \right)$, so that $J_{\mathrm{eff}}\left(\omega \right)={\omega }^{\delta }\tilde{G}\left(\omega \right)$. We can apply an expansion analogous to Equation (23) if we apply Taylor’s formula for $\tilde{G}$:

$$\begin{array}{cc}\hfill \Gamma \left(t\right)& ={\int }_0^{{\omega }_{\mathrm{c}}}\frac{\tilde{G}\left(\omega \right)-\tilde{G}\left(0\right)-{\tilde{G}}^{\prime }\left(0\right)\omega }{{\omega }^{2-\delta }}(1-cos\omega t)d\omega \hfill \\ \hfill & +\tilde{G}\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^{2-\delta }}d\omega \hfill \\ \hfill & +{\tilde{G}}^{\prime }\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^{1-\delta }}d\omega \hfill \\ \hfill & +{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\tilde{G}\left(\omega \right)\frac{1-cos\omega t}{{\omega }^{2-\delta }}d\omega .\hfill \end{array}$$

(26)

Applying the same arguments as for cases 2 and 3, we conclude that the first and the last terms tend to constants. The second term is analysed analogously to that of case 2:

$$\begin{array}{cc}\hfill {\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^{2-\delta }}d\omega & ={\int }_0^{\infty }\frac{1-cos\omega t}{{\omega }^{2-\delta }}d\omega -{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{1-cos\omega t}{{\omega }^{2-\delta }}d\omega \hfill \\ & =t^{1-\delta }{\int }_0^{\infty }\frac{1-cos\upsilon }{{\upsilon }^{2-\delta }}d\upsilon -\frac{1}{(1-\delta ){\omega }_{\mathrm{c}}^{1-\delta }}+{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{cos\omega t}{{\omega }^{2-\delta }}d\omega .\hfill \end{array}$$

The last integral converges to zero due to the Riemann–Lebesgue lemma.

Consider the integral in the third term in Equation (26). It can be calculated explicitly in terms of the generalised hypergeometric functions, but, for our purposes, the following estimations suffice. If $0<\delta <1$, then, using the substitution $\upsilon =\omega t$ again, we can perform the following:

$${\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^{1-\delta }}d\omega =\frac{{\omega }_{\mathrm{c}}^{\delta }}{\delta }-t^{-\delta }{\int }_0^{{\omega }_{\mathrm{c}}t}\frac{cos\upsilon }{{\upsilon }^{1-\delta }}d\upsilon .$$

The last integral converges to a constant (due to Dirichlet’s test), while $t^{-\delta }$ tends to zero as $t\to \infty$. If $-1<\delta \le 0$ (actually, the case $\delta =0$ corresponds to case 2, but we include it also here), then we have

$${\int }_0^{{\omega }_{\mathrm{c}}}\frac{1-cos\omega t}{{\omega }^{1-\delta }}d\omega =t^{-\delta }{\int }_0^{{\omega }_{\mathrm{c}}t}\frac{1-cos\upsilon }{{\upsilon }^{1-\delta }}d\upsilon \le t^{-\delta }{\int }_0^1\frac{1-cos\upsilon }{{\upsilon }^{1-\delta }}d\upsilon +t^{-\delta }{\int }_1^{{\omega }_{\mathrm{c}}t}\frac{1-cos\upsilon }{\upsilon }d\upsilon .$$

The first integral here is a constant. The second integral was analysed before: its principal term is $lnt$ as $t\to \infty$. Again, the expressions for the constants follow from these calculations and from the following:

$$\begin{array}{cc}\hfill \tilde{G}\left(0\right)& =\underset{\omega \to +0}{lim}G\left(\omega \right)\omega coth\frac{\beta \omega }{2}=2{\beta }^{-1}G\left(0\right),\hfill \\ \hfill {\tilde{G}}^{\prime }\left(0\right)& =\underset{\omega \to +0}{lim}{\left(G\left(\omega \right)\omega coth\frac{\beta \omega }{2}\right)}^{\prime }=2{\beta }^{-1}{G}^{\prime }\left(0\right).\hfill \end{array}$$

This finishes the proof of the last case of the theorem, case 4. □

4. Two Examples of Ohmic Spectral Densities

Let us consider two popular choices of Ohmic spectral densities. The first choice is the exponential cutoff:

$$J\left(\omega \right)=\omega e^{-\omega /\Omega },$$

where $\Omega$ is the characteristic frequency of the bath. Here ${\Gamma }_0=\pi {\beta }^{-1}$ and $\alpha =-2{(\beta \Omega )}^{-1}$; hence,

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)t^{2{(\beta \Omega )}^{-1}}{e}^{-\pi {\beta }^{-1}t}.$$

(27)

The second choice is the Drude–Lorentz spectral density:

$$J\left(\omega \right)=\frac{\omega {\Omega }^2}{{\omega }^2+{\Omega }^2}.$$

Here, ${\Gamma }_0=\pi {\beta }^{-1}$ and $\alpha =0$; hence,

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)e^{-\pi {\beta }^{-1}t}.$$

Thus, in the latter case, the decoherence is slightly faster due to different values of $J_{\mathrm{eff}}^{\prime }\left(0\right)$. If $\Omega$ is large, then the difference is negligible, but if $\Omega$ is small, then the difference can be significant.

Generally, large values of $|J_{\mathrm{eff}}^{\prime }\left(0\right)|$ (which corresponds to either a sharp peak or a sharp hollow at zero) can significantly modify the rate of decoherence.

Also, it is interesting that the long-term dynamics of coherences is determined by only two values: ${lim}_{\omega \to +0}J\left(\omega \right)/\omega$ and ${lim}_{\omega \to +0}{[J\left(\omega \right)/\omega ]}^{\prime }$, and does not depend on further details of the function $J\left(\omega \right)$.

5. Discussion of the Markovian (Weak-Coupling) Limit

In the theory of the weak system–bath coupling limit, one considers a Hamiltonian where the interaction Hamiltonian is multiplied by a small dimensionless parameter $\lambda$:

$$H=H_{\mathrm{S}}+H_{\mathrm{B}}+\lambda H_{\mathrm{I}},\lambda \to 0.$$

(28)

One can derive the Davies quantum dynamical semigroup (or, equivalently, the Davies Markovian quantum master equation) for the reduced density operator of the system [29,30,31]. It predicts that, for the considered model, only $J_{\mathrm{eff}}\left(0\right)$ matters: if this value is positive (an Ohmic spectral density), then the exponential full decoherence takes place and the rate of decoherence is proportional to ${\lambda }^2$. In the super-Ohmic case, $J_{\mathrm{eff}}\left(0\right)=0$ and neither full nor partial decoherence occurs.

Let us analyse this from the point of view of the presented analysis. We should replace $J\left(\omega \right)$ and, thus, $\Gamma \left(t\right)$ by ${\lambda }^{2}J\left(\omega \right)$ and ${\lambda }^2\Gamma \left(t\right)$, respectively. Then, if $J\left(\omega \right)\sim c{\omega }^{2+\delta }$ as $\omega \to +0$ for some $c>0$, the partial decoherence (12) and (13) is negligible in this limit. Indeed, $\Gamma \left(t\right)$ is a bounded function in this case and ${\lambda }^2\Gamma \left(t\right)$ tends to zero uniformly on $[0,\infty )$; hence, ${\varrho }_{10}\left(t\right)$ uniformly tends to ${\varrho }_{10}\left(0\right)$.

If the spectral density is Ohmic, then the Davies quantum dynamical semigroup correctly predicts the exponent in Equation (15). In particular, the decoherence rate is indeed proportional to ${\lambda }^2$. But the Markovian master equation does not capture the power term $t^{\alpha }$. From the point of view of the formal limit $\lambda \to 0$, the influence of the power-law terms disappears in the weak-coupling limit. This can be seen as follows: If $\tau ={\left({\lambda }^2{\Gamma }_0\right)}^{-1}$ is the characteristic time of decay of the exponential term, then ${\tau }^{{\lambda }^2\alpha }={\left({\lambda }^2{\Gamma }_0\right)}^{-{\lambda }^2\alpha }\to 1$ as $\lambda \to 0$. So, in this limit, the power-law terms significantly differ from 1 only on the time scales where the coherence is already suppressed by the exponential term. This formally justifies the Davies quantum dynamical semigroup. However, if we consider a concrete physical system, then $\lambda$ can be small but a constant. Then, a sharp peak or hollow at zero (large value of $|J_{\mathrm{eff}}^{\prime }\left(0\right)|$) can significantly modify the decoherence rate predicted by the Davies master equation. There is a particular case of a known fact that the Markovian approximation does not work in the case of rapid changes in spectral density near the Bohr frequencies (differences between the energy levels). Here, only the zero Bohr frequency is important.

In the case of the sub-Ohmic spectral density $J\left(\omega \right)\sim c{\omega }^{1+\delta }$ as $\omega \to +0$, where $-1<\delta \le 0$ and $c>0$, the decoherence is superexponential and the Davies quantum master equation cannot describe it, since $J_{\mathrm{eff}}\left(\omega \right)\to \infty$ as $\omega \to +0$ (which reflects a superexponential law of decoherence).

The “mild” super-Ohmic case $J\left(\omega \right)\sim c{\omega }^{1+\delta }$ as $\omega \to +0$ for $0<\delta \le 1$ and $c>0$ requires a bit more attention. In this case, the decoherence is slower than exponential, which also cannot be captured by the Markovian master equation: the full decoherence takes place, but the Markovian master equation predicts no decoherence.

Namely, let us denote ${\overline{\varrho }}_{10}\left(t\right)$ as the coherence predicted by the Davies master equation (in contrast to the exact value ${\varrho }_{10}\left(t\right)$). In our case, we have ${\overline{\varrho }}_{10}\left(t\right)={\varrho }_{10}\left(0\right)$ and ${\varrho }_{10}\left(t\right)\to 0$ as $t\to \infty$. For this particular simple model, the Davies theorem asserts that [29,30]

$$\underset{\lambda \to 0}{lim}\underset{0\le t\le T/{\lambda }^2}{sup}|{\overline{\varrho }}_{10}\left(t\right)-{\varrho }_{10}\left(t\right)|=0$$

(29)

for an arbitrary finite T. This does not contradict the aforementioned different long-term behaviour of ${\varrho }_{10}\left(t\right)$ and ${\overline{\varrho }}_{10}\left(t\right)$. Indeed, if one of the asymptotic formulas (Formula (17) or (19)) is valid, then ${\lambda }^2\Gamma \left(t\right)$ tends to 1 uniformly on $[0,T/{\lambda }^2]$ for any $T>0$. However, increasing T requires making $\lambda$ smaller and smaller to maintain a constant level of error.

The so-called resonance theory allows us to prove a stronger result than Equation (29) under certain additional conditions [31]. Namely, it is proved that the norm of the difference between the exact reduced density operator of the system and that given by the Davies quantum dynamical semigroup are bounded by $C{\lambda }^2$ for some C uniformly on the whole time half-line $t\in [0,\infty )$. Obviously, this is not true in our case. This is because the mentioned additional conditions are not satisfied in this model. Namely, one of the conditions (the so-called Fermi Golden Rule condition) says exactly that ${\lambda }^{-2}$ is the only characteristic time scale of dissipative dynamics, which is violated in the considered case. So, this comparison can serve as an example showing that the additional conditions in the mentioned theorem strengthening the Davies theorem are important (“physical”) and not merely “technical”.

The inclusion of a term proportional to ${\sigma }_x$ into the system Hamiltonian (i.e., transitions between the energy levels) restores the time scale ${\lambda }^{-2}$. In this case, the pure decoherence is accompanied by the exponential decoherence due to quantum transitions, which are described within the Markovian master equations. If the non-exponential pure decoherence is suppressed by the exponential decoherence due to transitions, then the error of the Markovian master equations is not large.

Note also that the quantum master equation for the case where the system–bath interaction is not weak, but an additional term proportional to ${\sigma }_x$ in the system Hamiltonian can be treated as a small perturbation, was proposed [37,38]. This is the so-called strong-decoherence regime: a strong pure decoherence is accompanied by slow transitions between the energy levels in the system.

6. Problem of Markovian Embedding

The case of non-exponential decoherence is interesting for one more problem. A popular way of dealing with non-Markovian (non-semigroup) dynamics of an open quantum system is to embed it into a larger system whose dynamics is Markovian (semigroup) (see Figure 1). Since the exact system–bath unitary dynamics is already, obviously, Markovian, an additional condition is usually assumed: the enlarged system should be finite-dimensional or, at least, have a discrete energy spectrum (e.g., a finite-dimensional system plus a finite number of harmonic oscillators). In the latter case, it is possible to cut high energy levels (depending on the temperature of the bath) and, again, to consider a finite-dimensional enlarged system. The usual physical interpretation of the extension of the system is that only part of the bath strongly interacts with the system, while the residual bath interacts with both the system and the separated part of the bath only weakly.

Namely, let $\mathrm{S}$ be a system with a corresponding finite-dimensional Hilbert space ${\mathcal{H}}_{\mathrm{S}}$ (in our case, ${\mathcal{H}}_{\mathrm{S}}={\mathbb{C}}^2$). Its open dynamics is given by a family of completely positive and trace-preserving maps ${\left\{{\Phi }_t\right\}}_{t\ge 0}$, so that ${\rho }_{\mathrm{S}}\left(t\right)={\Phi }_t{\rho }_{\mathrm{S}}\left(0\right)$, where ${\rho }_{\mathrm{S}}\left(t\right)$ is the density operator of the system. We comment on the Markovian embedding if ${\rho }_{\mathrm{S}}\left(t\right)={Tr}_{{\mathrm{S}}^{\prime }}{\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)$, where ${\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)$ is the density operator of the enlarged system with the (also finite-dimensional) Hilbert space ${\mathcal{H}}_{\mathrm{S}}\otimes {\mathcal{H}}_{{\mathrm{S}}^{\prime }}$ and ${\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)$ satisfies a master equation in the GKSL form:

$${\dot{\rho }}_{{\mathrm{SS}}^{\prime }}\left(t\right)=\mathcal{L}{\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)=-i[H_{{\mathrm{SS}}^{\prime }},{\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)]+\sum _{k=1}^K\left(L_k{\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)L_k^{†}-\frac{1}{2}\{L_k^{†}{L}_k,{\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)\}\right).$$

(30)

Here, $H_{{\mathrm{SS}}^{\prime }}$ is a self-adjoint operator (a Hamiltonian) in ${\mathcal{H}}_{\mathrm{S}}\otimes {\mathcal{H}}_{{\mathrm{S}}^{\prime }}$, $L_k$ are linear operators in ${\mathcal{H}}_{\mathrm{S}}\otimes {\mathcal{H}}_{{\mathrm{S}}^{\prime }}$, and $[·,·]$ and $\{·,·\}$ are a commutator and anti-commutator, respectively. In other words, ${\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)=e^{t\mathcal{L}}{\rho }_{{\mathrm{SS}}^{\prime }}\left(0\right)$, where $e^{t\mathcal{L}}$ is the quantum dynamical semigroup with the generator $\mathcal{L}$ [1,39].

However, in this case, according to the general theory of systems of ordinary differential equations and also general operator theory, ${\rho }_{{\mathrm{SS}}^{\prime }}\left(t\right)$ and, thus ${\rho }_{\mathrm{S}}\left(t\right)$, is a linear combination of terms of the form $t^{n_j}{e}^{-l_{j}t}$, where $-l_j$ are the eigenvalues of $\mathcal{L}$ and $n_j$ are non-negative integers. Moreover, from the positivity of $e^{t\mathcal{L}}$, we have $\mathrm{Re}{l}_j\ge 0$ and $n_j=0$ whenever $\mathrm{Re}{l}_j=0$.

So, such a Markovian embedding can describe only the exponential relaxation of ${\rho }_{\mathrm{S}}\left(t\right)$ to a stationary state (up to the factors $t^{n_j}$). In contrast, as we saw, the model of pure decoherence (1) allows the super- and subexponential and power-law decoherence, like $e^{-At\sqrt{t}}$, $e^{-A\sqrt{t}}$, and $t^{-\alpha }$, respectively. Such dynamics cannot be reproduced by a Markovian embedding. Even in the case of an Ohmic spectral density, the Markovian embedding cannot reproduce the power-law factor $t^{-\alpha }$ in Equation (15) (like in example (27)) if $\alpha$ is not a negative integer. Thus, in this particular model, the Markovian embedding is not excluded only for a very specific class of spectral densities.

It should be noted that the non-exponential decoherence is not exotic. For example, the decoherence law $e^{-A{t}^2}$ is observed in superconducting qubits as a consequence of flicker noise ($1/f$-noise), where the effective spectral density behaves as $J_{\mathrm{eff}}\left(\omega \right)\sim c/\omega$ as $\omega \to +0$ [40]. With such a spectral density, integral (9) diverges, but we can consider a regularisation, $J={\omega }^{\epsilon }G\left(\omega \right)$, where $\epsilon >0$ is small and $G\left(0\right)>0$, so that $J_{\mathrm{eff}}\sim 2{\beta }^{-1}G\left(0\right)/{\omega }^{1-\epsilon }$ as $\omega \to +0$. Then, according to Theorem 1 (namely, Equation (22)),

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)e^{-A{t}^{2-\epsilon }+2{\beta }^{-1}{G}^{\prime }\left(0\right)O(t^{1-\epsilon }lnt)},t\to \infty .$$

In Ref. [41], it is shown that the inclusion of a classical noise to a Markovian embedding can reproduce the non-exponential decoherence. In other words, the inclusion of classical noise (which can be non-Markovian, e.g., the aforementioned flicker noise) significantly extends the power of the method of Markovian embedding.

Two more comments should be made here:

• We still can hope to reproduce the dynamics of an open quantum system on a finite time interval by a Markovian embedding. But this is already not about a physically meaningful representation of the model (the aforementioned separation of the strongly interacting part of the bath), but rather about merely the mathematical approximation of the time dependence.
• A Markovian embedding can be used not only for the approximation of the dynamics, but also for the approximation of the equilibrium state [21], which is non-Gibbsian if the system–bath coupling is not negligibly weak [34]. Here, we do not consider this purpose of Markovian embeddings and write only about the problem of approximation of dynamics.

7. Asymptotic Markovianity

In the previous section, we have shown that the Markovian embedding (without additional means like classical noise) is impossible if the spectral density is not Ohmic and, thus, the decoherence is non-exponential. Let us consider now the case of the Ohmic spectral density and the exponential decoherence, i.e., case 2 of Theorem 1. Here, vice versa, the asymptotic Markovianity in the following sense takes place.

The exact solution $\varrho \left(t\right)$ (6) with the function $\Gamma \left(t\right)$ (9) obviously satisfies the following equation with a time-dependent GKSL generator:

$$\dot{\varrho }=\frac{\gamma \left(t\right)}{2}({\sigma }_z\varrho {\sigma }_z-\varrho ),$$

or, if we return to the Schrödinger picture,

$${\dot{\rho }}_{\mathrm{S}}=-i[H_{\mathrm{S}},{\rho }_{\mathrm{S}}]+\frac{\gamma \left(t\right)}{2}({\sigma }_z{\rho }_{\mathrm{S}}{\sigma }_z-{\rho }_{\mathrm{S}}),$$

where $\gamma \left(t\right)=\dot{\Gamma }\left(t\right)$. In the case $J\left(\omega \right)\sim c\omega$ as $\omega \to +0$, proceeding in the same manner as in Equation (23), we arrive at

$$\begin{array}{cc}\hfill \gamma \left(t\right)& =\dot{\Gamma }\left(t\right)={\int }_0^{\infty }{J}_{\mathrm{eff}}\left(\omega \right)\frac{sin\omega t}{\omega }d\omega \hfill \\ \hfill & ={\int }_0^{{\omega }_{\mathrm{c}}}\frac{J_{\mathrm{eff}}\left(\omega \right)-J_{\mathrm{eff}}\left(0\right)}{\omega }sin\omega td\omega +J_{\mathrm{eff}}\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{sin\omega t}{\omega }d\omega +{\int }_{{\omega }_{\mathrm{c}}}^{\infty }{J}_{\mathrm{eff}}\left(\omega \right)\frac{sin\omega t}{\omega }d\omega .\hfill \end{array}$$

Again, applying the Riemann–Lebesgue lemma, we obtain $\gamma \left(t\right)\to {\Gamma }_0=\mathrm{const}.$ (see Equation (16)) as $t\to \infty$ and, thus, for large time scales, we obtain a GKSL generator of a quantum dynamical semigroup. Of course, this can be seen also directly from Equations (14) and (15) since the linear term in $\Gamma \left(t\right)$ is principal for large time scales.

This asymptotic Markovianity is an interesting phenomenon. Usually, we expect the semigroup dynamics in the weak-coupling limit (or another limit). However, here, we have obtained that, for large time scales, the semigroup property is satisfied for any system–bath coupling strength. A peculiarity of the weak-coupling limit here is that the transient non-Markovian stage of the dynamics is infinitesimal and can be neglected in the principal order approximation, i.e., the second order in $\lambda$. We saw this in Section 5. Note that, in Ref. [42], it is shown that the effects of the transient dynamics should be included in the higher-order approximations.

However, as we mentioned in the introduction, the semigroup property is not the most general definition of quantum Markovianity. The definition generalising the corresponding definition from the classical theory of random processes is based on the quantum regression formula [5,9,11,12,13].

Let H, as before, be a system–bath Hamiltonian and the initial system–bath state be the product state ${\rho }_{\mathrm{S}}\otimes {\rho }_{\mathrm{B},\beta }$, where ${\rho }_{\mathrm{S}}$ is arbitrary and ${\rho }_{\mathrm{B},\beta }$ is the initial state of the bath (see Equation (3)). One says that the open quantum system satisfies the quantum regression formula if, for every ${\rho }_{\mathrm{S}}$, every n and every bounded operator $X_1,\dots ,X_n,Y_1,\dots ,Y_n$ in the Hilbert space of the system and all $t_1,\dots ,t_n\ge 0$, the following equality holds:

$${Tr}_{\mathrm{SB}}\left[{\tilde{\mathcal{E}}}_n{\mathcal{U}}_{t_n}\dots {\tilde{\mathcal{E}}}_1{\mathcal{U}}_{t_1}({\rho }_{\mathrm{S}}\otimes {\rho }_{\mathrm{B},\beta })\right]={Tr}_{\mathrm{S}}\left[{\mathcal{E}}_n{\Phi }_{t_n}\dots {\mathcal{E}}_1{\Phi }_{t_1}\left({\rho }_{\mathrm{S}}\right)\right],$$

(31)

where ${\mathcal{U}}_t=e^{-itH}(·)e^{itH}$, ${\Phi }_t={Tr}_{\mathrm{B}}\left[{\mathcal{U}}_t(·\otimes {\rho }_{\mathrm{B},\beta })\right]$, and

$${\tilde{\mathcal{E}}}_k=(X_k\otimes I_{\mathrm{B}})(·)(Y_k\otimes I_{\mathrm{B}}),{\mathcal{E}}_k=X_k(·)Y_k$$

with $I_{\mathrm{B}}$ being the identity operator in the bath Hilbert space. The quantum regression formula says that we can use the family of quantum dynamical maps ${\left\{{\Phi }_t\right\}}_{t\ge 0}$ not only for the description of the dynamics of the reduced density operator of the system and the prediction of the average values of observables, but also for multitime correlation functions. Obviously, the quantum regression formula cannot be satisfied exactly, except in trivial cases (e.g., $H_{\mathrm{I}}=0$). We can hope for the satisfaction of this formula only in certain limiting cases. In particular, the quantum regression formula is satisfied in the weak-coupling limit [43].

In Ref. [7] (Proposition 2.2), it is shown that, for the considered model of pure decoherence, the quantum regression formula (31) is satisfied if and only if the following formula holds for all n, $j_1,\dots ,j_n,l_1,\dots ,l_n\in \{0,1\}$, $t_1,\dots ,t_n\ge 0$:

$$Tr\left[e^{-i{t}_n{H}_{j_n}}\dots e^{-i{t}_1{H}_{j_1}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_1{H}_{l_1}}\dots e^{i{t}_n{H}_{l_n}}\right]=\prod _{k=1}^{n}Tr\left[e^{-i{t}_k{H}_{j_k}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_k{H}_{j_k}}\right],$$

(32)

where $H_j$ for $j=0,1$ are given in Equation (7).

Theorem 2.

Consider the thermodynamic limit $N\to \infty$ in Equation (10) such that $J\left(\omega \right)$ converges to a function (also denoted by $J\left(\omega \right)$) that is integrable on $[0,\infty )$. Let this limiting function satisfy the conditions of Case 2 of Theorem 1: $J\left(\omega \right)\sim c\omega$ as $\omega \to +0$ for some $c>0$, there exists $J_{\mathrm{eff}}^{\prime }\left(0\right)={lim}_{\omega \to +0}{J}_{\mathrm{eff}}^{\prime }\left(\omega \right)$, and there exists a bounded second derivative $J_{\mathrm{eff}}^{″}\left(\omega \right)$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$. Then, the quantum regression formula is asymptotically satisfied for large time scales in the following sense:

$$\underset{N\to \infty }{lim}Tr\left[e^{-i{t}_n{H}_{j_n}}\dots e^{-i{t}_1{H}_{j_1}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_1{H}_{l_1}}\dots e^{i{t}_n{H}_{l_n}}\right]=e^{-{\Gamma }_{0}T+O(ln(mint))},mint\to \infty ,$$

(33)

and

$$\underset{N\to \infty }{lim}\prod _{k=1}^{n}Tr\left[e^{-i{t}_k{H}_{j_k}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_k{H}_{l_k}}\right]=e^{-{\Gamma }_{0}T+O(ln(mint))},mint\to \infty .$$

(34)

Here,

$$T=\sum _{k:j_k\ne l_k}{t}_k,mint=\underset{k:j_k\ne l_k}{min}{t}_k,$$

and ${\Gamma }_0$ is given in Equation (16). Thus, both sides of Equation (32) are exponential functions whose arguments coincide in the principal terms as the minimal $t_k$ goes to infinity.

Note that the first limit in Equations (33) and (34) is thermodynamic, i.e., $N\to \infty$, and the second one is $mint\to \infty$.

Proof of Theorem 2.

Let us calculate (34) first. If $j_k=l_k$, then $Tr\left[e^{-i{t}_k{H}_{j_k}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_k{H}_{l_k}}\right]=1$. If $j_k\ne l_k$, then, according to Equations (8) and (14),

$$Tr\left[e^{-i{t}_k{H}_{j_k}}{\rho }_{\mathrm{B},\beta }{e}^{i{t}_k{H}_{l_k}}\right]=e^{-{\Gamma }_0{t}_k+O(ln{t}_k)},t_k\to \infty .$$

Taking the product over all k, we obtain Equation (34).

For the proof of Equation (33), we use the results of calculations from Ref. [7] (Proof of Proposition 3.2). According to them, the left-hand side of Equation (33) is equal to

$$exp\left\{-\frac{1}{2}{\int }_0^{\infty }\frac{J_{\mathrm{eff}}\left(\omega \right)}{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \right\},$$

where $T_k={\sum }_{k^{\prime }=1}^k{t}_{k^{\prime }}$. Now, we apply decomposition (23):

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }\frac{J_{\mathrm{eff}}\left(\omega \right)}{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \\ \hfill & ={\int }_0^{{\omega }_{\mathrm{c}}}\frac{J_{\mathrm{eff}}\left(\omega \right)-J_{\mathrm{eff}}\left(0\right)-J_{\mathrm{eff}}^{\prime }\left(0\right)\omega }{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \\ \hfill & +J_{\mathrm{eff}}\left(0\right){\int }_0^{\infty }\frac{1}{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \\ \hfill & -J_{\mathrm{eff}}\left(0\right){\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{1}{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \\ \hfill & +J_{\mathrm{eff}}^{\prime }\left(0\right){\int }_0^{{\omega }_{\mathrm{c}}}\frac{1}{\omega }{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \\ \hfill & +{\int }_{{\omega }_{\mathrm{c}}}^{\infty }\frac{J_{\mathrm{eff}}\left(\omega \right)}{{\omega }^2}{\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^{2}d\omega \hfill \end{array}$$

(35)

for some ${\omega }_{\mathrm{c}}>0$. Here, for all terms in the right-hand side, except the second one, we can apply the following estimation:

$$\begin{array}{cc}\hfill {\left|\sum _{k=1}^n(j_k-l_k)(e^{i\omega T_k}-e^{i\omega T_{k-1}})\right|}^2& \le \sum _{k=1}^n|j_k-l_k|·|e^{i\omega T_k}-e^{i\omega T_{k-1}}{|}^2\hfill \\ \hfill & \le 2\sum _{k=1}^n|j_k-l_k|·(1-cos\omega t_k).\hfill \end{array}$$

Thus, we can repeat the corresponding steps of the proof of case 2 of Theorem 1 and we can conclude that all these terms are $O(ln(mint\left)\right)$ as $mint\to \infty$. The integral in the second term on the right-hand side of Equation (35) was shown in Ref. [7] to be equal to $\pi T$, which concludes the proof of Equation (33) and, thus, the theorem. □

In Ref. [7], Lonigro and Chruściński show that the dynamics is exactly Markovian if we formally replace the lower limit of integration in Equation (9) for $\Gamma \left(t\right)$ by $-\infty$ and put $J_{\mathrm{eff}}\left(\omega \right)$ to be constant on the whole line $\omega \in (-\infty ,\infty )$. Of course, negative frequencies are unphysical. However, the authors write the following: “We point out that, while these choices of coupling may be considered unphysical, the corresponding results are indicative of what would be obtained in more realistic scenarios: we can expect an exponential dephasing in the regime in which the spin-boson interaction is ‘approximately flat’ in the energy regime of interest.” Here, we actually show that the spectral density can be considered to be “approximately flat” on large time scales, when only the behaviour of $J_{\mathrm{eff}}\left(\omega \right)$ in the vicinity of zero matters.

8. The Case of Zero Temperature

We have analysed a realistic case of a positive temperature. However, it is worth briefly mentioning the case of zero temperature as well since it is often used as an approximation for the case of low temperatures. In this case, we still have formula (8) for decoherence, but $\Gamma \left(t\right)$ is defined without the factor $coth(\beta \omega /2)$ in the integral [1,2,3,7]:

$$\Gamma \left(t\right)={\int }_0^{\infty }J\left(\omega \right)\frac{1-cos\omega t}{{\omega }^2}d\omega .$$

(36)

Theorem 3.

Let $J\left(\omega \right)$ be an integrable function on $[0,\infty )$.

1.

If $J\left(\omega \right)\sim c{\omega }^{1+\delta }$ as $\omega \to +0$, where $\delta >0$ and $c>0$, then

$$\underset{t\to \infty }{lim}\Gamma \left(t\right)={\int }_0^{\infty }\frac{J\left(\omega \right)}{{\omega }^2}coth\left(\frac{\beta \omega }{2}\right)d\omega \equiv {\Gamma }_{\infty },$$

(37)

hence, the decoherence is partial:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)e^{-{\Gamma }_{\infty }},$$

(38)

where $B\left(t\right)$ is a bounded function converging to 1 as $t\to \infty$.

2.

If $J\left(0\right)=c>0$, there exists $J^{\prime }\left(0\right)$, and there exists a bounded second derivative $J^{″}\left(\omega \right)$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$, then

$$\Gamma \left(t\right)={\Gamma }_{0}t+\alpha lnt+C+o\left(1\right),t\to \infty ,$$

(39)

where C is a constant; hence, the decoherence is exponential:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)e^{-{\Gamma }_{0}t}/t^{\alpha }.$$

(40)

Here,

$${\Gamma }_0=\frac{\pi }{2}J\left(0\right)>0,\alpha =J^{\prime }\left(0\right)$$

(41)

and $B\left(t\right)$ is a bounded function converging to $e^{-C}$ as $t\to \infty$.

3.

If $J\left(\omega \right)\sim c\omega$ as $\omega \to +0$, where $c>0$, then

$$\Gamma \left(t\right)=\alpha lnt+C+o\left(1\right),t\to \infty ,$$

(42)

where $\alpha >0$ is given in Equation (41); hence, the decoherence obeys a power law:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)B\left(t\right)/t^{\alpha },$$

(43)

where, again, $B\left(t\right)$ is a bounded function converging to $e^{-C}$ as $t\to \infty$.

4.

If $J\left(\omega \right)={\omega }^{\gamma }G\left(\omega \right)$ as $\omega \to +0$, where $-1<\gamma <1$ and $G\left(0\right)>0$, there exists $G^{\prime }\left(0\right)$, and there exists a bounded second derivative $G^{″}\left(\omega \right)$ on $(0,{\omega }_1)$ for some ${\omega }_1>0$, then

$$\Gamma \left(t\right)=A{t}^{1-\gamma }+\tilde{C}+o\left(1\right),t\to \infty ,$$

(44)

if $0<\gamma <1$, and

$$\Gamma \left(t\right)=A{t}^{1-\gamma }+G^{\prime }\left(0\right)O(t^{-\gamma }lnt),t\to \infty ,$$

(45)

if $-1<\gamma \le 0$, where

$$A=G\left(0\right){\int }_0^{\infty }\frac{1-cos\upsilon }{{\upsilon }^{2-\gamma }}d\upsilon$$

and $\tilde{C}$ is a constant. Thus, the decoherence is subexponential for $\delta >0$ and superexponential for $\gamma <0$:

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)\tilde{B}\left(t\right)e^{-A{t}^{1-\gamma }}$$

(46)

if $0<\gamma <1$, where $\tilde{B}\left(t\right)$ is a bounded function converging to $e^{-\tilde{C}}$ as $t\to \infty$, and

$${\varrho }_{10}\left(t\right)={\varrho }_{10}\left(0\right)e^{-A{t}^{1-\gamma }+G^{\prime }\left(0\right)O(t^{-\delta }lnt)},t\to \infty$$

(47)

if $-1<\delta \le 0$.

The results of Theorem 3 can be summarised as follows. Let $J\left(\omega \right)\sim c{\omega }^{\gamma }$ as $\omega \to +0$ for some $c>0$.

• If $-1<\gamma <0$ (sub-Ohmic spectral density), then the decoherence is full and its rate is superexponential.
• If $\gamma =0$ (sub-Ohmic spectral density), then the decoherence is full and its rate is exponential.
• If $0<\gamma <1$ (sub-Ohmic spectral density), then the decoherence is full and its rate is subexponential, but faster than any degree of t.
• If $\gamma =1$ (Ohmic spectral density), then the decoherence is full and obeys a power law.
• If $\gamma >1$ (super-Ohmic spectral density), then the decoherence is partial.

The proof of Theorem 3 is completely analogous to that of Theorem 1: we simply replace $J_{\mathrm{eff}}=J\left(\omega \right)coth(\beta \omega /2)$ by $J\left(\omega \right)$ everywhere. In other words, $J_{\mathrm{eff}}\left(\omega \right)=J\left(\omega \right)$ for $\beta =\infty$. Since $coth(\beta \omega /2)\sim {(\beta \omega /2)}^{-1}$ as $\omega \to +0$, the elimination of this factor leads to the reduction in $\gamma$ by 1 in all cases. For example, the case $J_{\mathrm{eff}}\left(0\right)=c>0$ corresponds to the exponential decoherence for both positive-temperature and zero-temperature cases. However, in the case of a positive temperature, this corresponds to $J\left(\omega \right)\sim (\beta /2)c\omega$ as $\omega \to +0$, while, in the case of zero temperature, this corresponds to $J\left(\omega \right)=c$.

Again, if the decoherence is not exponential ($\gamma \ne 0$), then a Markovian embedding is impossible, while, in the exponential case $\gamma =0$, the asymptotic Markovianity takes place.

Note that exact form of $\Gamma \left(t\right)$ for the particular super-Ohmic spectral density

$$J\left(\omega \right)=2\eta {\omega }_{\mathrm{c}}^{1-\gamma }{\omega }^{\gamma }{e}^{-\omega /{\omega }_{\mathrm{c}}}$$

(48)

for $\gamma >1$, where $\eta ,{\omega }_{\mathrm{c}}>0$, and the zero temperature was obtained in Ref. [8]. From this formula, the partial decoherence follows (which is discussed in the mentioned paper), in agreement with Theorem 3.

Let us consider now the case of a positive but very small temperature. How can the predictions of Theorems 1 and 3 agree? Consider, for example, the aforementioned spectral density (48) for $1<\gamma \le 2$. Theorem 3 predicts only a partial decoherence, while Theorem 1 predicts the full decoherence (though slower than exponential) for this case. Of course, the answer is in different time scales: all decoherence constants in Theorem 1 are proportional to the temperature ${\beta }^{-1}$, while those in Theorem 3 are not. So, at the beginning, the system partially decoheres and then, on a larger time scale proportional to $\beta$, the full decoherence occurs. Different time scales caused by vacuum and positive-temperature contributions are well known for this model [1,2,44].

9. Conclusions

The main result of this paper is Theorem 1 (for the case of the bath with positive temperature) and also Theorem 3 (for the case of zero temperature), where the long-time rate of decoherence in a known exactly solvable model of decoherence is related to the asymptotic behaviour of the bath spectral density at low frequencies. Though the considered model of pure decoherence is paradigmatic in the theory of open quantum systems, we are not aware of such a detailed analysis of the long-term behaviour of coherence in this model.

We have discussed the consequences of these results for the theory of the weak-coupling limit, for the possibility of Markovian embedding, and for asymptotic Markovianity. In particular, we see that the decoherence is not necessarily exponential, while Markovian embeddings can give only the exponential relaxation to a steady state. Hence, the Markovian embedding, which is widely used to model the non-Markovian dynamics of open quantum systems, is not always possible. On the other hand, if the spectral density is Ohmic (and the temperature is positive), the exponential decoherence dominates for large time scales. As a consequence, we have asymptotic Markovianity in the most natural (i.e., generalising the classical definition) sense: in the sense of the quantum regression formula.

Finally, it is worth mentioning works about the theoretical analysis of decoherence in the considered model with a special control technique called dynamical decoupling, which aims to suppress the decoherence [35,44]. It would be interesting to extend the results of the present paper about the long-term rates of decoherence depending on the asymptotic behaviour of the spectral density at low frequencies to the case of dynamical decoupling.