Open quantum systems coupled to finite baths: A hierarchy of master equations

An open quantum system in contact with an infinite bath approaches equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium but it induces a dynamical evolution of the bath state. In this paper, we study the dynamics of open quantum systems in contact with finite baths. We obtain a hierarchy of master equations that improve their accuracy by including more dynamical information of the bath. For instance, as the least accurate but simplest description in the hierarchy, we obtain the conventional Born-Markov-secular master equation. Remarkably, our framework works even if the measurements of the bath energy are imperfect, which not only is more realistic but also unifies the theoretical description. Also, we discuss this formalism in detail for a particular noninteracting environment where the Boltzmann temperature and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our hierarchy of master equations to study the central spin model.

Figure 1

Sketch of the system (labeled by S) interacting with the finite bath (labeled by B). The bath energy levels are depicted with horizontal black lines, and their corresponding energy distribution $p\left(E_i\right)$ is plotted in the red solid line. The measurement apparatus (labeled by M) gives an output $E$ given an input $E_i$ according to the weighting function $W\left(E\right|E_i)$, which introduces a finite precision of order $\delta E$.

Figure 2

Sketch of the hierarchy and validity range of the different discussed master equations.

Figure 3

Sketch of the piecewise noninteracting spin bath. The central system (in blue, label S) is weakly coupled to several regions (in red, labels $R_1,R_2,R_3,...$) that conform a lattice $\mathfrak{L}$.

Figure 4

Sketch of the central spin model, where the central spin has an energy scale ${\omega }_{\text{S}}$ and the spins in the bath have different energy splittings ${\mathrm{\Omega }}_r$. The interaction strength between the system and the spins in the environment is of order $\lambda$.

Figure 5

Histogram of the energy levels of a noninteracting spin bath with random frequencies ${\mathrm{\Omega }}_r$ extracted from a single realization of the Gaussian distribution with mean ${\mathrm{\Omega }}_0$ and variance ${\sigma }_{\mathrm{\Omega }}=0.2{\mathrm{\Omega }}_0$ for $N=14$ spins. In orange, the comparison to the normal distribution $\mathcal{N}(e,{\sigma }_N)$ (solid red line).

Figure 6

Ratio $R\left(N\right)$ between the exact value of $f(E,E^{\prime };\mathrm{\Omega })$ in Eq. (46) and its approximated value as shown in Eq. (47) as a function of the number of spins $N$, whose frequencies are extracted from a single realization of the distribution $p_Z\left(\mathrm{\Omega }\right)=\mathcal{N}(\mathrm{\Omega }-{\mathrm{\Omega }}_0,{\sigma }_{\mathrm{\Omega }})$. The parameters are given in terms of the coarse-graining $\delta E$ and correspond to ${\mathrm{\Omega }}_0=\delta E, {\sigma }_{\mathrm{\Omega }}=0.2\delta E, \mathrm{\Omega }={\mathrm{\Omega }}_0, E=-\delta E$ and $E^{\prime }=-2\delta E$.

Figure 7

First column: Histogram of the Zeeman energies and fit to the underlying probability distribution $\mathcal{N}(\mathrm{\Omega }-{\mathrm{\Omega }}_0,{\sigma }_{\mathrm{\Omega }})$ (orange solid line). Second column: Real (orange solid line) and imaginary (blue dashed line) parts of the correlation function ${\langle B\left(\tau \right)\mathrm{\Pi }\left(E\right)B\rangle }_{E^{\prime }}$ with weighting function $W_{\text{I}}\left(E\right|E_i)$. Third column: Analytic expression of the dissipation rates in Eq. (48) (solid orange line) versus its numerical value computed using the numeric Fourier transform of the second column (blue dashed line). The particle numbers are $N=100$ (first row) and $N=1000$ (second row). In the three columns, each frequency ${\mathrm{\Omega }}_r$ is extracted from a single-shot realization of $p_Z\left(\mathrm{\Omega }\right)=\mathcal{N}(\mathrm{\Omega }-{\mathrm{\Omega }}_0,{\sigma }_{\mathrm{\Omega }})$ with ${\sigma }_{\mathrm{\Omega }}=0.2{\mathrm{\Omega }}_0$. For the second and third columns, we have chosen $\delta E={\mathrm{\Omega }}_0$, the energies $E=-8\delta E$ and $E^{\prime }=-9\delta E$, and fixed $c_r=c_0=1$. In the third column, we have set $\lambda =0.01\delta E$.

Figure 8

Left column: Comparison between the EMME dynamics (solid orange line), the BMS with the effective nonequilibrium temperature ${\beta }^{★}\left(t\right)$ (blue dashed line), and the BMS at fixed temperature ${\beta }_0$ (yellow dotted line) for (top row) the first excited state of a spin-1/2 particle and (bottom row) the third excited state of a spin-10 particle (right column) corresponding to evolution of the effective nonequilibrium temperature (solid orange line). Initially, the system-bath composite is found in the state $\rho \left(0\right)=|2s\rangle \langle 2s|\otimes {\pi }_{\text{B}}\left({\beta }_0\right)$ with ${\beta }_0=0.75\delta E$. We take the parameters $N=100$ spins, ${\omega }_{\text{S}}=\delta E, \lambda =0.01\delta E, {\mathrm{\Omega }}_0=\delta E, {\sigma }_{\mathrm{\Omega }}=0.2\delta E$, and a spin-independent coupling $c_r\to 1$.

Figure 9

Left column: Comparison between the stationary distributions of the system $p^{\text{st}}\left({\varepsilon }_k\right)$ and the bath $p^{\text{st}}\left(E\right)$ as predicted for the EMME dynamics (solid orange line), the BMS with the effective nonequilibrium temperature ${\beta }^{★}\left(t\right)$ (blue dashed line), and the BMS at fixed temperature ${\beta }_0$ (yellow dotted line). Top row: The first excited state of a spin-1/2 particle and (bottom row) the third excited state of a spin-10 particle. We take the initial state and parameters as in Fig. 8.

Figure 10

Comparison between the EMME dynamics (solid orange line), the BMS with the effective nonequilibrium temperature ${\beta }^{★}\left(t\right)$ (blue dashed line), and the BMS at fixed temperature ${\beta }_0$ (yellow dotted line) for a spin-10 particle. Top row (left): Comparative dynamics of the third excited state and (right) corresponding evolution of the effective nonequilibrium temperature (solid orange line). Bottom row: stationary value of the (left) system and (right) bath energy distributions. We take the initial bath state to be ${\omega }_{\text{B}}(E=-18\delta E)$ and the rest of the parameters as in Fig. 8.

Figure 11

Mutual information (53) for a state of the bath in a microcanonical state ${\omega }_{\text{B}}\left(E\right)$ with $E=-18\delta E$ and corresponding Boltzmann temperature $\beta \left(E\right)\approx 0.73\delta E^{-1}$ (orange solid and yellow dotted lines) or a canonical state at initial inverse temperature ${\beta }_0=0.75\delta E^{-1}$ (blue dashed and purple dot-dashed lines). The central particle has a spin of $\text{s}=1/2$ (solid orange and blue dashed lines) or $\text{s}=10$ (yellow dotted and purple dot-dashed lines). The horizontal gray dashed lines correspond to the maximal value of the mutual information $ln(2\text{s}+1)$ for $\text{s}=1/2$ and $\text{s}=10$. The rest of the parameters are set to $N=100, {\omega }_{\text{S}}=\delta E, {\mathrm{\Omega }}_0=\delta E, {\sigma }_{\mathrm{\Omega }}=0.2\delta E, c_r=1$, and $\lambda =0.01\delta E$.

Figure 12

Table showing the parallelism between the different levels of description, with corresponding second laws and Markovian approximations with their witness of validity. The notation $\mathrm{\Delta }\mathcal{S}\left(t\right)≔\mathcal{S}\left(t\right)-\mathcal{S}\left(0\right)$ represents the change in thermodynamic entropy at time $t$, with $\mathrm{\Delta }{\mathcal{S}}_{\text{S}}$ being the corresponding change of only the system entropy.

Figure 13

Function $\kappa (E;-{\omega }_{\text{S}})$ (solid orange line) and $\kappa (E;-{\omega }_{\text{S}})$ (blue dashed line) as a function of the energy $E$ for $N=100$ (first column) and $N=1000$ (second column) for a small central spin with $\text{s}=1/2$. The Zeeman frequencies of the bath are distributed according to $\mathcal{N}(\mathrm{\Omega }-{\mathrm{\Omega }}_0,{\sigma }_{\mathrm{\Omega }})$, and the central spin frequency is ${\omega }_{\text{S}}=\delta E$. In the third column, we show the block diagonal structure of the evolution matrix $\mathrm{\Lambda }({\varepsilon }_k,E;{\varepsilon }_q,E^{\prime })$ for $N=100$ in the basis $\{...,p(-{\omega }_{\text{S}}/2,E+\delta E),p({\omega }_{\text{S}}/2,E),...\}$. The numbers in the axis indicate the position in the basis, where the energies of the bath are ordered by increasing energy. The rest of the parameters are set to ${\mathrm{\Omega }}_0=\delta E, {\sigma }_{\mathrm{\Omega }}=0.2\delta E, \lambda =0.01\delta E$, and $cr=1$.