Transient temperature dynamics of reservoirs connected through an open quantum system

The dynamics of open quantum systems connected with several reservoirs attract great attention due to their importance in quantum optics, biology, quantum thermodynamics, transport phenomena, etc. In many problems, the Born approximation is applicable, which implies that the influence of the open quantum system on the reservoirs can be neglected. However, in the case of long-time dynamics or mesoscopic reservoirs, the reverse influence can be crucial. In this paper, we investigate the transient dynamics of several bosonic reservoirs connected through an open quantum system. We use an adiabatic approach to study the temporal dynamics of temperatures of the reservoirs during relaxation to thermodynamic equilibrium. We show that there are various types of temperature dynamics that strongly depend on the values of dissipative rates and initial temperatures. We demonstrate that temperatures of the reservoirs, including the hottest and coldest ones, can exhibit nonmonotonic behavior. Moreover, there are moments of time during which the reservoir with an initially intermediate temperature becomes the hottest or coldest reservoir. The obtained results pave the way for managing energy flows in mesoscale and nanoscale systems.

Figure 1

The system considered in Sec. 3: $n$ bosonic reservoirs of quasiparticles at temperature $T_1,...,T_n$ interacting through an open system represented by a set of oscillators. Oscillators' frequencies are indexed with ${\omega }_{\kappa }$ [see Eq. (7)]. The interaction between $j\mathrm{th}$ reservoir and oscillator with frequency ${\omega }_{\kappa }$ is described via function ${\gamma }_j\left({\omega }_{\kappa }\right)$ [see Eq. (13)].

Figure 2

The dependence of $t_{\mathrm{eq}}(\omega ,T)$ on the ratio $\omega /2T$. Parameters of the reservoirs are set to $A_1=A_2=1, V_1=1, V_2=1, {\alpha }_1={\alpha }_2=3$ [product $AV$ is not dimensionless, however, here and after we consider $A$ and $V$ to be measured in such units, that temperature in Eq. (18) is measured in units of energy].

Figure 3

The time dependence of temperature of three reservoirs connected through a set of seven oscillators with frequencies ${\omega }_1=1, {\omega }_2=1.1{\omega }_1, {\omega }_3=1.5{\omega }_1, {\omega }_4=0.9{\omega }_1, {\omega }_5=2{\omega }_1, {\omega }_6=2.2{\omega }_1, {\omega }_7=3{\omega }_1$. The parameters: ${\alpha }_j=\alpha =3, {\gamma }_1/{\omega }_1={10}^{-4}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_2/{\omega }_1=2\times {10}^{-2}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_3/{\omega }_1=2\times {10}^{-2}{(\omega /{\omega }_1)}^{\alpha }, A_j=1, V_j=1$. Initial temperatures of the reservoirs are $T_1/{\omega }_1=0.105, T_2/{\omega }_1=0.085, T_3/{\omega }_1=0.107$. The horizontal dot-dashed gray line depicts $T_{\mathrm{eq}}$ from Eq. (23).

Figure 4

The time dependence of temperatures of the five reservoirs connected through a set of seven oscillators with fre- quencies ${\omega }_1=1, {\omega }_2=1.1{\omega }_1, {\omega }_3=1.5{\omega }_1, {\omega }_4=0.9{\omega }_1, {\omega }_5=2{\omega }_1, {\omega }_6=2.2{\omega }_1, {\omega }_7=3{\omega }_1$. The parameters: ${\alpha }_j=\alpha =3, {\gamma }_1/{\omega }_1={10}^{-2}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_2/{\omega }_1={10}^{-3}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_3/{\omega }_1={10}^{-6}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_4/{\omega }_1={10}^{-7}{(\omega /{\omega }_1)}^{\alpha }, {\gamma }_5/{\omega }_1=3\times {10}^{-9}{(\omega /{\omega }_1)}^{\alpha }, A_j=1, V_j=1$. Initial temperatures of the reservoirs are $T_1/{\omega }_1=0.1, T_2/{\omega }_1=0.07, T_3/{\omega }_1=0.1155, T_4/{\omega }_1=0.055, T_5/{\omega }_1=0.127$. The horizontal dot-dashed gray line depicts $T_{\mathrm{eq}}$ from Eq. (23).