Nonequilibrium relaxation transport of ultracold atoms

We analyze the equilibration process between two either fermionic or bosonic reservoirs containing ultracold atoms with a fixed total number of particles that are weakly connected via a few-level quantum system. We allow for both the temperatures and particle densities of the reservoirs to evolve in time. Subsequently, linearizing the resulting equations enables us to characterize the equilibration process and its time scales in terms of equilibrium reservoir properties and linear-response transport coefficients. Additionally, we investigate the use of such a device as particle transistor or particle capacitor and analyze its efficiency.

Figure 1

General two-terminal transport scheme with left and right reservoirs weakly coupled to a few-level quantum system. The reservoirs $\nu \in \{L,R\}$ are in thermal equilibrium and characterized by a chemical potential ${\mu }_{\nu }(T_{\nu },N_{\nu })$ that depends on the respective temperature $T_{\nu }$ and particle number $N_{\nu }$. The system dynamics is governed by the Hamiltonian ${\mathcal{H}}_S$ and the weak system-bath coupling is facilitated by energy-dependent tunneling rates ${\Gamma }_{\nu }\left(\omega \right)$ defined in Eq. (4).

Figure 2

Plot of the time evolution of the differences ${\Delta }_n=(N_L-N_R)/\mathcal{N}$, ${\Delta }_T=T_L-T_R$, and ${\Delta }_{\mu }={\mu }_L-{\mu }_R$ of the thermodynamic variables of the left and right reservoirs, for a quantum system with a single transition energy $\omega =0.9\overline{\omega }$. The solid lines correspond to the numeric simulation, and the dashed lines to the linear-response solution from Eqs. (27) and (28). The initial particle numbers are set to $N_L\left(0\right)=N_R\left(0\right)=0.5\mathcal{N}$ and the temperatures to $T_L\left(0\right)=0.45\overline{\omega }$ and $T_R\left(0\right)=0.25\overline{\omega }$. In panel (d), we show the evolution of the Fermi functions of the right (dotted-dashed line) and left (solid line) reservoirs.

Figure 3

(a) Time evolution of the particle-number difference ${\Delta }_n=(N_L-N_R)/\mathcal{N}$ for different values of the quantum system transition energy $\omega$ for the same initial parameters as in Fig. 2. (b) Plot of the difference ${\overline{n}}_L^{(-)}\left(\omega \right)-{\overline{n}}_R^{(-)}\left(\omega \right)$ of the initial Fermi functions versus the transition energy of the system. The dots correspond to the energies $\omega$ in panel (a). (c) Plot of the ratio between the steady-state particle-number bias and the initial temperature bias ${\Delta }_T\left(0\right)=0.2\overline{\omega }$ in dependence of the average temperature $T$ of the system, for a fixed transition energy $\omega =0.9\overline{\omega }$. The $y$ axis is measured in units $1/\overline{\omega }$. For high temperatures, the particle bias vanishes like $1/T$ for both fermions (solid line) and bosons (dashed line).

Figure 4

(a) Plot of the time evolution of the particle-number difference ${\Delta }_n=(N_L-N_R)/\mathcal{N}$ for a system with two different transition energies. (b) Plot of the linear-response results for the characteristic time scales $t_{<}$ from Eq. (32) and $t_{>}$ from Eq. (33) for a system with one (solid line) and two (dashed, dotted-dashed lines) transition energies in dependence of one of the energies. In the case of one channel, we compare it with the full numerics (dots). For the case of two transport channels, the lower transition energy is fixed to ${\omega }_1=0.9\overline{\omega }$ in both plots. The arrow indicates the fastest thermalization process corresponding to the minimum of $t_{>}$.

Figure 5

Plots of the numerically obtained efficiency for a single transport channel with $\omega /\overline{\omega }=2$. The initial parameters are set to $N_L\left(0\right)=N_R\left(0\right)=0.5\mathcal{N}$, $T_L\left(0\right)=0.45\overline{\omega }$, and $T_R\left(0\right)=0.25\overline{\omega }$. (a) Plot of the time evolution of the instantaneous efficiency from Eq. (37) (solid line), the cumulative efficiency from Eq. (39) (dotted-dashed line), and the instantaneous Carnot efficiency (dashed line). The insets show the steady-state total efficiency (dots) for different energies of the system and for different average temperatures $T=(T_L+T_R)/2$ of the reservoirs. Here, a cutoff appears (dashed line) beyond which no power can be extracted from the device. In panel (b) we show the corresponding total heat (dotted-dashed line) and work (dashed line) and in panel (c) the corresponding instantaneous heat flow from Eq. (36) (dotted-dashed line) and power from Eq. (38) (dashed line).

Figure 6

Plot of the time evolution of the differences ${\Delta }_n=(N_L-N_R)/\mathcal{N}$, ${\Delta }_T=T_L-T_R$, and ${\Delta }_{\mu }={\mu }_L-{\mu }_R$, of the thermodynamic variables of the left and right reservoirs, for a quantum system with two transition energies ${\omega }_1=0.9\overline{\omega }$ and ${\omega }_2=2.1\overline{\omega }$. The solid lines correspond to the numeric simulation, and the dashed lines to the linear-response solution. The initial particle numbers are set to $N_L\left(0\right)=N_R\left(0\right)=0.5\mathcal{N}$ and the temperatures to $T_L\left(0\right)=0.45\overline{\omega }$ and $T_R\left(0\right)=0.25\overline{\omega }$. In panel (d), we plot the Fermi functions of the right (dotted-dashed line) an left (solid line) reservoir for both transport channels. The upper branch corresponds to the transition energy ${\omega }_1$ and the lower branch to ${\omega }_2$.

Figure 7

(a) Time evolution of the temperature difference ${\Delta }_T$ in a bosonic transport setup, for different values of the system transition energy $\omega$ with the initial particle numbers $N_L\left(0\right)=0.6\mathcal{N}$ and $N_R\left(0\right)=0.4\mathcal{N}$ and the fixed average temperature $T=1.5\overline{\omega }$. (b) Plot of the initial temperature current ${\dot{T}}_L\left(0\right)$ versus the transition energy of the system. The dots correspond to the energies $\omega$ in panel (a). (c) Plot of the ratio between the initial particle-number bias and the steady-state temperature bias ${\Delta }_T\left(0\right)=0.2\overline{\omega }$ in dependence of the average temperature $T$ of the system, for a fixed transition energy. The $y$ axis is measured in units $1/\overline{\omega }$. For high temperatures, this ratio vanishes like $1/T$ for both fermions (solid line) and bosons (dashed line).