In this work we continue the inquiries two of us conducted in 2015 concerning the quantum thermodynamics and entanglement dynamics of two coupled oscillators, each with its private bath, for possible applications to quantum processes in systems involving two heat baths such as quantum transport and the design of quantum heat engines and other quantum devices. In [1] we show that for bilinear couplings between the oscillators, and between each oscillator and its own bath, described by a scalar field at temperatures $T_1>T_2$, the dynamics of the system can be solved exactly at arbitrary temperatures, even for strong coupling, thanks to the Gaussian structure. In particular, a nonequilibrium steady state (NESS) of this system can indeed exist at late times, its insensitivity to the initial conditions of the system testifies to the uniqueness of this NESS. In [2] we studied how entanglement between the two coupled oscillators depend on the temperatures of the two baths. We find that the valid entanglement criterion in general is not a function of the bath temperature difference, in contrast to thermal transport in the same setting. In fact lowering the temperature of one of the thermal baths does not necessarily help to safeguard the entanglement between the oscillators. Rather, quantum entanglement will disappear if any one of the thermal baths has a temperature higher than a critical temperature $T_c$. The baths in both of these studies are made up of scalar fields which are Ohmic (linear in frequency), and, with a full spectrum, there is no imposed cutoff in its frequency range. With no particular time scale involved they are memoryless. Scalar field baths are Markovian.

In this work we focus on the effects of nonMarkovian (nM) baths on the nonMarkovian dynamics of an open system in the same setup, with constant coupling between the two quantum oscillators, but not limiting our attention to very early or very late times. Rather, we aim at ranges where nonMarkovian dynamics are prominent and examine how the entanglement of the system evolves. In particular, we are interested in how the memory effects associated with the nonMarkovian dynamics affects the system’s entanglement throughout the evolution. Our next paper [3] will consider time-dependent coupling between the two oscillators and focus on the conditions which could allow entanglement to survive even at high temperatures, the so-called ‘hot entanglement’ first discovered by Galve et al. [4]. Since nonMarkovianity is the central theme of both papers we begin with a brief description of nonMarkovianity in open quantum systems.

Suppose we have a pair of coupled harmonic oscillators, each of which has its own private bath. Let ${\chi }_i\left(t\right)$ be the canonical position of the $i\mathrm{th}$ oscillator, whose private bath ${\varphi }_i$ has an initial temperature ${\beta }_i^{-1}$. We further assume that two oscillators have same physical frequency ω_p. The coupling strength between the oscillators is denoted by $\sigma$. In addition, the oscillators have the same mass m, coupling strength e with each individual’s bath, but each bath may not have the same cutoff scale in its spectral density.

The action describing such an interacting system is given by where ${\varphi }_i$ are two private baths with the spatial coordinate ${\mathit{x}}_i$, and the $i\mathrm{th}$ oscillator is located at the origin ${\mathit{z}}_i$. The private bath is initially prepared in a thermal states, but in the beginning, both baths are independent and uncorrelated. Then the equations of motion in the compact matrix form is [1,2,25] with $\mathrm{\Xi }\left(t\right)={\{{\chi }_1\left(t\right),{\chi }_2\left(t\right)\}}^T$ and $\mathrm{\Phi }\left(t\right)={\{{\varphi }_1\left(t\right),{\varphi }_2\left(t\right)\}}^T$, where T denotes the matrix transpose, and On the righthand side of (2), the operator $\mathrm{\Phi }$ describes a noise force due to quantum and thermal fluctuations of the bath. It satisfies the Gaussian statistics The expectation value is taken with respect to the initial thermal state of the bath. The Hadamard function ${\mathit{G}}_{H,0}^{\left(\mathrm{\Phi }\right)}(t-t^{\prime })$ is paired with the retarded Green’s function of the bath ${\mathit{G}}_{R,0}^{\mathrm{\Phi }}\left(\tau \right)$, to formulate the fluctuation-dissipation relation of the $i\mathrm{th}$ bath [25], so they are also respectively called the noise kernel and the dissipation kernel. We use the convention to define the Fourier transformation of the function $f\left(\tau \right)$.

We introduce the kernel function ${\mathrm{\Gamma }}^{\left(\mathrm{\Phi }\right)}\left(\tau \right)$ to isolate the contributions to renormalizations (When we carry out integration by parts on the integral expression in (2) with the identification (3), it yields a correction to ${\mathrm{\Omega }}_b^2$ in (2). When the cutoff scale is infinite as in the field theory, we conventionally call this procedure renormalization (of the frequency).) of or the corrections to the parameters due to nonMarkovianity of the private baths. Integration by parts leads to with $\gamma =e^2/\left(8\pi m\right)$, and Its formal solution can be formally found by the Laplace transformation of (8), and the Laplace transform of the solution is given by with Explicitly, ${\tilde{\mathit{D}}}_2$ is given by where Thus due to the nonvanishing inter-oscillator coupling $\sigma$, ${\tilde{\mathit{D}}}_2$ will intermingle dynamics of both oscillators. This is better seen from the example Explicitly the displacement of oscillator 1 is also imparted by the thermal noise from the private bath of oscillator 2, indirectly via oscillator 2 through the inter-oscillator coupling.

We may identify the normal modes by diagonalizing ${\tilde{\mathit{D}}}_2$; however the transformation matrix to the normal modes tends to be rather complicated in the asymmetric case, and in the time domain such a transformation is not local in time, so we will not pursue this approach, except for the symmetric setting.

We first start with the symmetric setting. In this case, both oscillator are assumed to have the same physical frequencies ${\omega }_{1,\mathrm{P}}={\omega }_{2,\mathrm{P}}={\omega }_{\mathrm{P}}$ and damping constants ${\gamma }_1={\gamma }_2=\gamma$. Likewise, both private baths will have the same power spectral densities and the cutoff scales ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2=\mathrm{\Lambda }$, so that their kernel functions are identical, ${\mathrm{\Gamma }}^{\left({\varphi }_1\right)}\left(\tau \right)={\mathrm{\Gamma }}^{\left({\varphi }_2\right)}\left(\tau \right)={\mathrm{\Gamma }}^{\left(\varphi \right)}\left(\tau \right)$. This setting is particularly convenient to investigate the effects of the bath’s nonMarkovianity, manifested in this case by the cutoff scale or the memory time.

In the symmetric setting, the normal modes are nothing but the center-of-mass and the relative superpositions of the original two modes, Then we have where ${\omega }_{\pm }^2={\omega }^2\pm \sigma$ represents the oscillating frequencies of the normal modes, and ${\varphi }_{\pm }$ is given by Here we remind that both private baths, associated with each oscillator, are initially uncorrelated and do not have direct coupling. Thus each mode acts as an Ohmic, nonMarkovian oscillator with different oscillating frequencies. Since for the equation of motion of the form (15) takes the generic form we will first use this equation to discuss the effects of the bath nonMarkovianity on the system’s dynamics.

The general solution of $\chi \left(t\right)$ is given by the inverse Laplace transform of with ${\tilde{d}}_1\left(z\right)=z{\tilde{d}}_2\left(z\right)$, and in which We will choose the double Lorentzian form for the bath spectral density. The two fundamental solutions $d_1\left(t\right)$ and $d_2\left(t\right)$ are two particularly useful homogeneous solutions to the equation of motion (17). They satisfy the initial conditions $d_1\left(0\right)=1$, ${\dot{d}}_1\left(0\right)=0$, $d_2\left(0\right)=0$ and ${\dot{d}}_2\left(0\right)=1$, and are used to construct the complete solution to (17).

For the double Lorentzian bath spectral density, the Laplace transform ${\tilde{d}}_2\left(z\right)$ has four poles: two negative real poles and two complex poles, whose real parts are real. Since for a given set of $\gamma$, ${\omega }_{\mathrm{P}}$, and $\mathrm{\Lambda }$, the two real poles are more negative than the real parts of the complex poles, so at late times, the damping behavior of $d_2\left(t\right)$ is controlled by the real part of the complex poles. Thus, we may construct an effective damping constant, which is approximately given by when $\gamma /\mathrm{\Lambda }\lesssim 1$, $\gamma /{\omega }_{\mathrm{P}}\lesssim 1$ in the long memory time limit. It implies that these two modes will have two different effective damping constants due to the difference in the normal mode frequencies, even though they have the same bath configuration.

Figure 1 shows how the effective damping constant depends the oscillating frequency and the cutoff scale. We can immediately identify two features: (1) for a given cutoff scale, the damping is weaker, that is, smaller effective damping constant, when the frequency is larger. It is clearly seen in the second row of Figure 1, and the difference is more significant for smaller cutoff scales. This feature is related to the second feature: (2) for a given oscillating frequency, the effective damping constant ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}$ decreases if we lower the cutoff scale $\mathrm{\Lambda }$, and such change is more dramatic when $\mathrm{\Lambda }$ is smaller than the oscillating frequency ${\omega }_{\mathrm{P}}$. The weakening of the effective damping constant can be understood from two aspects. Since the inverse cutoff scale can be interpreted as the memory time of the bath, which moderates the duration the system, with which the bath interacts, depends on its past history. From this viewpoint, when the memory time is longer than the typical period $2\pi /{\omega }_{\mathrm{P}}$ of the oscillatory motion, or when the cutoff scale $\mathrm{\Lambda }$ is smaller than the oscillating frequency ${\omega }_{\mathrm{p}}$, the system dynamics from the previous cycle is still in good coherence with the current one. Thus the motion is progressively superposed to compete against decaying due to dissipation. This effect is expected to be more prominent when the memory is much longer, because the contributions from more earlier cycles will coherently join in [23]. Alternatively, the weakening effect can also be understood from resonance absorption [26]. The equation of motion (17) more or less describes a driven, damped harmonic oscillator, although this may not be obvious when the nonlocal integral expression is present. However, we may start from the limit $\mathrm{\Lambda }\to \infty$, where the integral expression gives $2\gamma \dot{\chi }\left(t\right)$ and Equation (17) reduces to the standard form. It is known that such a system has peaked power spectrum, centered at around $\omega \sim {\omega }_{\mathrm{P}}$ with a width of the order $\mathcal{O}\left(\gamma \right)$. Now heuristically suppose we only allow bath modes whose frequencies are lower than the location of the resonance peak, ${\omega }_{\mathrm{P}}$. These modes will not be very efficient in exchanging energy between the oscillator and the bath, so neither can they effectively drive the oscillator, nor can drain the energy out of the oscillator. Therefore it leads to relatively weak damping to the oscillator’s motion.

By these arguments, we expect that for a fixed $\mathrm{\Lambda }$, in particular when $\mathrm{\Lambda }<{\omega }_{\mathrm{P}}$, a larger value of ${\omega }_{\mathrm{P}}$ means that the bath modes which participate in the interactions are further away from the resonance peak, thus rendering less effective energy exchange. Or, when the memory time becomes even longer than the system’s period, thus allowing more coherent cycles from the past history of the oscillator’s motion. These make the effective damping weaker.

Figure 2 shows the cutoff scale dependence of the effective damping constant by numerical method. We see that when $\mathrm{\Lambda }\lesssim 3{\omega }_{\mathrm{P}}$, the effective damping constant ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}$ monotonically decreases with $\mathrm{\Lambda }$, and is smaller than its Markovian counterpart $\gamma =0.3{\omega }_{\mathrm{P}}$. On the other hand, in this case when $\mathrm{\Lambda }$ is greater than $20{\omega }_{\mathrm{P}}$, the nonMarkovian effect is barely present. This raises an interesting observation. Here we choose $\gamma =0.3{\omega }_{\mathrm{P}}$, which typically falls in the strong oscillator-bath coupling regime. By introducing nonMarkovianity via engineering the bath’s spectrum, we seem to effectively render the reduced dynamics of the oscillator to behave like a weak coupling case. Or, by tuning the bath spectrum, we may be able to vary the effective coupling between the oscillator and the bath.

Based on the above arguments and the idea of effective damping constant, we may be led to ask an important strategic question: Can we find a weak-coupling Markovian effective system to mimic the strong-coupling nonMarkovian system? It can greatly facilitate the theoretical study of the nonMarkovian system, because the latter is computationally challenging and demanding. In contrast, the former is analytically and exactly solvable. The answer depends. We will come back to this in Section 4.

To compute the entanglement measure (Relevant material can be found in Appendix A) via negativity, we start from the covariance matrix elements of the normal modes, since each mode is “decoupled” (Strictly speaking this is not entirely correct because ${\varphi }_{\pm }$ are not independent). For the discussion of sustainability of entanglement, we suppose two oscillators are initially prepared in a two-mode squeezed vacuum state (see Appendix B for the discussion of the two-mode squeezed state). When the squeeze parameter $\eta$ is not zero, the initial state is already entangled, and the degree of entanglement grows with increasing $\eta$. With respect to the normal modes, the nonzero covariance matrix elements of the coupled oscillator in this initial state are These two conditions ensure the covariance matrix with respect to the normal mode remains blockwise diagonal at all times Thus we only need to compute a smaller set of covariance matrix elements, Here $d_i^{(\pm )}$ are the counterparts of the fundamental solutions, discussed in (18), of the normal modes.

Then we can restore the covariance matrix elements of the canonical variables of two modes by suitable superpositions of the covariance matrix elements of the normal modes. This step is essential because entanglement is partition dependent [27]. The entanglement measure, negativity, of a bipartite system depends on how we partition the bipartite system. Its values vary if we use the normal modes, instead of the canonical variables of the bipartite system, to construct the measure. This can be traced to the fact that the partial transposition does not belong to the symplectic transformation (see Appendix A for a brief discussion).

Before we proceed, we take a look at the time evolution of the selected covariance matrix element $\langle {\chi }_{+}^2\left(t\right)\rangle$ for different choices of the bath temperatures and the bath cutoff scales. In Figure 3, different rows correspond to different initial bath temperatures ${\beta }^{-1}$, while different columns are associated with different bath memory times ${\mathrm{\Lambda }}^{-1}$. We immediately see that $\langle {\chi }_{+}^2\left(t\right)\rangle$ relaxes more slowly for longer memory times, consistent with the time evolution of the solution shown in Figure 1.

We observe that each covariance matrix element can be decomposed into two components: (1) the intrinsic (active) component depends on the initial condition of the oscillator, and is damped by dissipation to the bath, and (2) the induced (passive) component is generated by the quantum and thermal fluctuations from the bath, independent of the oscillator’s initial condition. The latter is the only component that will survive at late times, so the late-time value of the covariance matrix elements will not depend on the initial conditions of the system. Both components obey different statistics and are not correlated for the linear system. The memory endowed by the bath has different effects in both components. In the intrinsic component, the nonMarkovian effect leads to a weaker damping rate, so we expect that the information on the initial conditions of the system will linger for an extended period of time. On the other hand, in the passive component, the coherent superposition of the oscillator’s motion over previous cycles, due to the memory effect, will compete with the accumulative intervention, resulting from the thermal fluctuations of the bath.

In each column, when the intrinsic component dies down, the value of the $\langle {\chi }_{+}^2\left(t\right)\rangle$ dispersion will not decay to zero; otherwise it will violate uncertainty principle. In fact it will more of less come a constant due to balance between the thermal fluctuations and damping, and this constant value grows with the bath temperature because at higher temperatures, the bath fluctuations are stronger and becomes more random, less correlated, but damping in this case is independent of the bath temperature.

A longer memory time implies a longer effective relaxation time scale, and thus a longer duration for the effects of thermal fluctuations to accumulate. This probably explains the larger values of the dispersions $\langle {\chi }_{+}^2\left(t\right)\rangle$ at late times when the cutoff scale $\mathrm{\Lambda }$ is smaller in each row of Figure 3. However, this difference is rather insignificant at higher bath temperature, signifying the ineffectiveness of nonMarkovian effects at higher temperatures.

We will transform the covariance matrix ${\mathbf{\sigma }}_{\pm }\left(t\right)$ back to that with respect to the modes ${\chi }_{1,2}$ by where ${\mathit{U}}_2$ is a global symplectic matrix, that is, ${\mathit{U}}_2\in Sp(4,\mathbb{R})$.

The symplectic eigenvalues of the partially transposed covariance matrix ${\mathbf{\sigma }}^{\mathrm{P}\mathrm{T}}$ is then given by with $\mathrm{\Delta }\left({\mathbf{\sigma }}^{\mathrm{P}\mathrm{T}}\right)$, $det{\mathbf{\sigma }}^{\mathrm{P}\mathrm{T}}$ spelled out below in terms of covariance matrix elements of the normal modes as Thus, the entanglement of the system at any given moment t is determined by the value ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}\left(t\right)<1/2$.

The time evolution of the symplectic eigenvalue ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}$ is shown in Figure 4. Since it contains products of various elements of the covariance matrix, we cannot simply decompose ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}$ into the intrinsic and the induced components as we did for each covariance matrix elements. However, we still expect that the initial entanglement between the oscillators will be gradually lost because contributions related to the oscillators’ initial conditions will be exponentially small, and that the late-time behavior of the symplectic eigenvalue ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}$ is predominantly controlled by the bath. Thus the existence of the entanglement at late times will be independent of the initial conditions of the oscillators, but determined by the configuration of the bath such as the memory time, temperature and the oscillator-bath coupling strength.

Since we choose an entangled initial state, we see that the memory effect is much more significant during the relaxation stage. The duration ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}$ for both oscillators to remain entangled is correlated with the length of memory time. In Figure 4 the coupled oscillators become separable when ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}\ge 1/2$. Comparing the plots in each row, we observe that the entanglement interval during the relaxation stage increases with the memory time ${\tau }_{\mathrm{M}\mathrm{E}\mathrm{M}}$. In other words, the entanglement is more robust against the thermal fluctuations and can be sustained at higher bath temperatures, owing to the nonMarkovian effect due to the bath’s memory. This point is particularly clearly seen when we compare the $\mathrm{\Lambda }=5{\omega }_{\mathrm{P}}$, $\beta =10{\omega }_{\mathrm{P}}^{-1}$ and $\mathrm{\Lambda }=1{\omega }_{\mathrm{P}}$, $\beta =1{\omega }_{\mathrm{P}}^{-1}$ cases. In the latter, the bath temperature has 10 times higher than the former, but the entanglement dies out at approximately the same time around $t=2{\omega }_{\mathrm{P}}^{-1}$, as shown in Figure 4. It is expected to be more dramatic for even smaller cutoff scale, as is inferred from the behavior of oscillator dynamics, shown in Figure 1. The result that ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\sim \mathcal{O}\left(1\right){\omega }_{\mathrm{P}}^{-1}$ may not seem significant as it appears. However, recall that the oscillators are strongly coupled to the bath because $\gamma =0.5{\omega }_{\mathrm{P}}$. In Figure 5, we will see that ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}$ for the corresponding Markovian bath (green curve) is merely ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\simeq 0.197{\omega }_{\mathrm{P}}^{-1}$. Furthermore, let us put it in a comparative perspective. For example, ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\simeq 5.86{\omega }_{\mathrm{P}}^{-1}$ for the $\mathrm{\Lambda }=1{\omega }_{\mathrm{P}}$, $\beta =1{\omega }_{\mathrm{P}}$ case. The effective damping constants for the two normal modes are approximately given by ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}^{(+)}\simeq 0.066{\omega }_{\mathrm{P}}$ and ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}^{(-)}\simeq 0.094{\omega }_{\mathrm{P}}$. Thus the effective relaxation time is roughly ${\tau }_{\mathrm{R}}\simeq 10\sim 15{\omega }_{\mathrm{P}}^{-1}$. The memory time in this case is ${\tau }_{\mathrm{M}\mathrm{E}\mathrm{M}}=1{\omega }_{\mathrm{P}}$. Thus the entanglement duration is about six-fold of the memory time and one third of relaxation time. Thus, this entanglement duration is quite appreciable in the relaxation process. It can be further improved if we enhance the initial squeezing.

In Figure 4, we also see that although the oscillators become disentangled after ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}$, determined by ${\omega }_{\mathrm{P}}$, $\gamma$, $\mathrm{\Lambda }$ and $\beta$, the entanglement can resurrect at a later time, when the memory time is sufficiently long and the bath temperature is low enough, as shown in the case $\mathrm{\Lambda }=1$, $\beta =10$. However, this late-time entanglement does not benefit from the initial squeezing because its effect has been dissipated away. The plots show that although the nonMarkovian effect may improve entanglement at late times, the benefit is marginal. To understand this better, let us focus on the behavior of the covariance matrix elements and the symplectic eigenvalue ${\eta }_{-}^{\mathrm{P}\mathrm{T}}$ after the oscillators are relaxed to the steady state.

The late-time result in this case is particularly simple, since ${\sigma }_{{\chi }_{+}{p}_{+}}(\infty )$ and ${\sigma }_{{\chi }_{-}{p}_{-}}(\infty )$ vanish. We are left with only four elements ${\sigma }_{{\chi }_{\pm }{\chi }_{\pm }}(\infty )$ and ${\sigma }_{p_{\pm }{p}_{\pm }}(\infty )$, and they are given by Figure 6 shows that the late-time value of ${\sigma }_{{\chi }_{+}{\chi }_{+}}\left(t\right)$ decreases with the larger cutoff scale $\mathrm{\Lambda }$ or the shorter memory time. It is consistent with earlier observations, and it can be clearly seen that the variation of ${\sigma }_{{\chi }_{+}{\chi }_{+}}\left(t\right)$ is less significant, implying that the nonMarkovian effect is much weaker at higher bath temperature. This is also demonstrated in Figure 7, where the curves corresponding to different cutoff scales essentially converge when the bath temperature $T_{\mathrm{B}}$ is of the order ${\omega }_{\mathrm{P}}$. Note that the late-time values of the covariance matrix elements are mainly governed by the bath. When the bath has a higher initial temperature ${\beta }^{-1}$, it imparts stronger thermal fluctuations to the system. Thus the values of ${\sigma }_{{\chi }_{+}{\chi }_{+}}(\infty )$ quickly increase with the bath temperature.

In Figure 8, we show the dependence of the symplectic eigenvalue ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}$ on the bath temperature and the bath cutoff scale at late times, after the dynamics of the oscillators are fully relaxed. Recall that a large damping constant in the strong oscillator-bath regime is chosen, so the entanglement between the oscillators is typically difficult to maintain at late times. We see a trend, though not very significant, that the longer memory time ${\tau }_{\mathrm{M}\mathrm{E}\mathrm{M}}={\mathrm{\Lambda }}^{-1}$ or the lower bath temperature $T_{\mathrm{B}}=1/\beta$ is prone to keep the oscillators’ entanglement at late times. In the left panel, with a longer memory time, the entanglement still exists at higher bath temperatures although this critical temperature does not change much with $\mathrm{\Lambda }$ and is still of the order $\mathcal{O}\left({\omega }_{\mathrm{P}}^{-1}\right)$. On the other hand, the right panel shows that with a lower bath temperature, the entanglement may still survive for shorter memory times. These examples illustrate the coherent superposition induced by the nonMarkovian memory effect is capable to counter, at least marginally, the debilitating effects accumulated over the whole course of evolution due to the thermal fluctuations.

In the previous section, we note that the dynamics of a coupled system strongly coupled to a nonMarkovian bath seems to behave in some aspects like that of a system weakly coupled to a Markovian bath, and raised the question whether we can use weakly coupled Markovian linear open systems to approximate strongly coupled nonMarkovian linear open systems? This issue is of significance because one may argue that even though many open system processes in the real world involve memories and are thus fundamentally nonMarkovian, we may in practice describe them by using a simpler effectively Markovian formulation. How valid are such prescriptions of convenience? We want to take a closer look at this issue with the help of a concrete example.

For the normal modes of the coupled oscillators described by (15), we suppose there correspond effective equations of motion of the normal modes for the weakly coupled Markovian oscillators For example, when ${\omega }_{\mathrm{P}}=1$, $\gamma =0.5$ and $\mathrm{\Lambda }=1$, the effective damping constants of the normal modes are ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}^{(+)}\simeq 0.066{\omega }_{\mathrm{P}}$ and ${\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}^{(-)}\simeq 0.094$. The presence of the delta function $\delta \left(t\right)$ in the Markovian case induces an instantaneous kick at the initial time. This gives a $\mathcal{O}\left({\gamma }_{\mathrm{E}\mathrm{F}\mathrm{F}}^{(\pm )}\right)$ distortion of $d_1^{(\pm )}\left(t\right)$ at time right after the initial time, compare to the case in the absence of the kick. Its effect then diminishes with time. On the other hand, the delta function term does not modify $d_2^{(\pm )}\left(t\right)$ because $d_2^{(\pm )}\left(0\right)=0$.

Figure 9 shows the time evolution of $d_1^{(+)}\left(t\right)$, the covariance matrix element ${\sigma }_{{\chi }_{+}{\chi }_{+}}\left(t\right)$, the uncertainty relation $I_{+}\left(t\right)$ for the normal mode +, and the symplectic eigenvalue ${\eta }_{-}^{\mathrm{P}\mathrm{T}}$. The blue curve corresponds to the strongly coupled nonMarkovian system (15), while the orange curve represents the effective, weakly coupled Markovian system (31). The fundamental solution in both cases look quite similar. They decay approximately with the same rate, but their phases vary with time. This minor difference starts showing its repercussion effects in the covariance matrix element ${\sigma }_{{\chi }_{+}{\chi }_{+}}\left(t\right)$, which accounts for the uncertainty of the ${\chi }_{+}$ mode, or the degree of coherence. Both curves notably disagree before relaxation. The situation deteriorates even more for $I_{+}$, the Robertson-Schrödinger relation of the mode +. Note that both modes are decoupled in the sense of (23). The plot shows that the nonMarkovian description of the same system seems to have a better control of coherence, compared with the effective Markovian description. Finally when we examine the symplectic eigenvalue ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}$, they give quite a distinct prediction. The original nonMarkovian description shows that the entanglement between the oscillators can be maintained up to ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\simeq 5.863{\omega }_{\mathrm{P}}^{-1}$, but the effective Markovian formalism predicts roughly $0.6{\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}$. Moreover, at late times, the original nonMarkovian description shows the presence of residual entanglement in the $\mathrm{\Lambda }=1{\omega }_{\mathrm{P}}$ and $\beta =10{\omega }_{\mathrm{P}}^{-1}$ case, which is in the strong oscillator-bath coupling, low temperature regime. Since the disparity is more than marginal, the Markovian approximation, instead, predicts that the state of the system is separable.

The results in Figure 9, structured as a hierarchy from the evolutionary phase of the canonical variables, to its dispersion, then to the uncertainty of the constituent of the system, and finally into the integrated correlation among the system, indicate that the approximated Markovian description fails to precisely grasp the phase information embedded in nonMarkovian dynamics of the reduced system of the coupled oscillators. The approximated description tends to give poor predictions of the quantities that involving phases, or coherence. Thus the effective, weak coupling, Markovian description cannot be a sufficiently accurate substitute of the original strong coupling, nonMarkovian, linear open systems, even though the former has extraordinary convenience in computations.

Next we turn to the final issue discussed in this paper: whether the memory effect can be transferred from one party to the other in a bipartite system?

Suppose in the system of two coupled oscillators, one of which (oscillator 1) has a finite memory time due to small cutoff scale of its private bath, but the other (oscillator 2) has a negligible memory time due to the large cutoff scale. From the previous discussions, we learn that when they stand alone, oscillator 1 has a much smaller effective damping constant, resulting from the memory effect, in comparison with oscillator 2, so they will relax with different paces. Now when we couple them together, how does the coupled system evolve? Is the evolution dominated by the memoryless system, or by the nonMarkovian system? Or, will the memory effects in oscillator 1 be transferred to oscillator 2, so that it is shared among them, causing the coupled system to evolve in a cooperative way? Further, how does this affect the entanglement dynamics.

Since both private baths have different cutoff scales, the reduced system of coupled oscillators has an asymmetric configuration. We will start from (8), (10) and (11) with for $i=1$, 2. Since working in the normal modes does not reduce computation hurdles, we will directly compute the covariance matrix elements of the canonical variables of the coupled system. For example, ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ will take the form As before, we suppose that the initial state of the coupled oscillator is a two-mode squeezed vacuum state.

We plot the time evolution of the covariance matrix element ${\sigma }_{{\chi }_i{\chi }_j}\left(t\right)$ in Figure 10. In the right plot, the blue curve corresponds to ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ when the private baths of two coupled oscillators have the same cutoff scale ${\mathrm{\Lambda }}_1=1{\omega }_{\mathrm{P}}$, while the green curve represents ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ when the private baths have the cutoff scale ${\mathrm{\Lambda }}_2=20{\omega }_{\mathrm{P}}$. They can be respectively compared to ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ and ${\sigma }_{{\chi }_2{\chi }_2}\left(t\right)$ in the left plot. For the memoryless case in the right plot (green curve), the covariance matrix element ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ decays very fast, and its time evolution is almost fully relaxed when $t\sim 4{\omega }_{\mathrm{P}}^{-1}$, which is about twice the relaxation time scale ${\gamma }^{-1}$. On the other hand, in the finite memory case, the blue curve falls off rather slowly. Its oscillatory behavior remains visible when $t=30{\omega }_{\mathrm{P}}^{-1}$. The left plot of Figure 10 describes the above asymmetric setting where the oscillator 1’s private bath has the cutoff scale ${\mathrm{\Lambda }}_1$, but the oscillator 2’s bath has ${\mathrm{\Lambda }}_2$. We immediately see that for such a hybrid system, ${\sigma }_{{\chi }_2{\chi }_2}\left(t\right)$ (green curve in the left plot) does not decay as fast as the green curve in the right plot. Nonetheless, ${\sigma }_{{\chi }_1{\chi }_1}\left(t\right)$ of oscillator 1 in the hybrid system falls off faster than its counterpart (blue curve) in the right plot. These results imply transfer of the memory effect. Oscillator 2, which is essentially memoryless, benefits from such a transfer because its covariance matrix element shows a prolonged relaxation time (to roughly $t\sim 5{\omega }_{\mathrm{P}}^{-1}$), but this transfer shortens the memory time of oscillator 1. Thus in the hybrid system, motion of both oscillators in fact takes a cooperative way, and is neither dominated by the memoryless component nor governed by the one with a finite memory. Will their entanglement dynamics shows a similar feature?

Figure 5 gives the time evolution of the symplectic eigenvalue ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}\left(t\right)$ of the partially transposed covariance matrix ${\sigma }^{\mathrm{P}\mathrm{T}}\left(t\right)$. The asymmetric setting is described by the orange curve, where two private baths have different cutoff scales ${\mathrm{\Lambda }}_1=1{\omega }_{\mathrm{P}}$, ${\mathrm{\Lambda }}_2=20{\omega }_{\mathrm{P}}$ such that one of the oscillator is Markovian and basically memoryless. The blue curve describes the symmetric setting when both private baths have the same cutoff scale ${\mathrm{\Lambda }}_1$. That is, both oscillators are nonMarkovian and have equal memory time. In contrast to the previous case, we have coupled (almost) Markovian oscillators (${\mathrm{\Lambda }}_2$) in the green curve. Since the damping constant $\gamma$ is chosen to be 0.5, we find ${\lambda }_{-}^{\mathrm{P}\mathrm{T}}\left(t\right)$ associated with the memoryless case settles down to an equilibrium value $0.929$ very quickly, compared to the other two cases. Its value shoots up past $1/2$ when ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}=t\simeq 0.197{\omega }_{\mathrm{P}}^{-1}$, even though the bath temperature $T_{\mathrm{B}}=0.1{\omega }_{\mathrm{P}}$, is rather low. In this case, once the state becomes separable, it remains disentangled. For the other extreme when both oscillators have the same, long memory time (blue curve), the initial entanglement is sustained up to ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\simeq 5.863{\omega }_{\mathrm{P}}^{-1}$, and at late times the curve falls below the $1/2$ level, whence the entanglement between the oscillators is revived. Finally, in the hybrid case, the duration the system remains entangled happens to fall between two cases of the symmetric setting. We find ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}\simeq 1.168{\omega }_{\mathrm{P}}^{-1}$, much shorter than the blue curve case but slightly better than the green curve case. At late times, we observe that the curve gradually dips down to 0.571, still above the $1/2$ level, so the system is not entangled at late time. This sloping-down behavior, also seen in the blue curve, may imply the memory effect in one of the private baths is still in effect, because the corresponding bath induces a longer effective relaxation time.

We observe that for the hybrid system during the relaxation regime, the duration the quantum entanglement is sustained over falls much shorter than the case when both oscillators have the same long memory time, and improves slightly compared to the Markovian, memoryless case. A similar phenomenon has been observed in [2]. There, two coupled oscillators in the two uncorrelated private bath setting are attached to their own Markovian private baths, which have different bath temperatures. In that case, we find that if the temperature in one of the private baths is raised beyond the critical temperature of the order ${\omega }_{\mathrm{P}}^{-1}$, then the entanglement between the oscillator is lost even though the other bath is kept at temperature much lower than the critical value. Similarly, here since one of the oscillator is memoryless, the coupled system formed purely by this oscillator is supposed to have a short entanglement time ${\tau }_{\mathrm{E}\mathrm{N}\mathrm{T}}$ (the green curve). Thus even it is coupled to a nonMarkovian oscillator, as in the hybrid case, the entanglement duration still fails to gain any improvement. That is, the entanglement duration of a bipartite system is predominantly controlled by the component that least favors the sustainability of system entanglement.

In this paper we present a detailed analysis of the effects of nonMarkovian dynamics on the entanglement dynamics between two coupled oscillators, each of which has its own private bath. In particular, we are interested in how the memory effects in the baths affect the full course of entanglement evolution–from the initial time to the moment the reduced system relaxes to the equilibrium state. We break down the analysis in parts to examine the imprints the nonMarkovian baths leave on the solutions of Langevin equation, time evolution of the covariance matrix, and on the negativity, serving as our entanglement measure. Based on the results we obtained we now can answer the questions posed in the Introduction in the following: