Exact solution of time-dependent Lindblad equations with closed algebras

Time-dependent Lindblad master equations have important applications in areas ranging from quantum thermodynamics to dissipative quantum computing. In this paper we outline a general method for writing down exact solutions of time-dependent Lindblad equations whose superoperators form closed algebras. We focus on the particular case of a single qubit and study the exact solution generated by both coherent and incoherent mechanisms. We also show that if the time dependence is periodic, the problem may be recast in terms of Floquet theory. As an application, we give an exact solution for a two-level quantum heat engine operating in a finite time.

Figure 1

Time evolution of the spin magnetization $\langle {\sigma }_3\rangle =tr\left({\rho }_t{\sigma }_3\right)$ for the two-level open quantum system in Eq. (26) with counteroscillating polarizers ${\mathrm{\Gamma }}_{+}=2.0+0.5sin\left(t\right),{\mathrm{\Gamma }}_{-}=3.0-0.5sin\left(t\right)$ and ${\mathrm{\Gamma }}_3=0,\mathrm{\Omega }=\sqrt{2}[1-cos\left(t\right)]$. The numerical exact time evolution (full line) is compared with the stroboscopic evolution (dots) given by Eq. (38). The system is prepared at $t=0$ in a state ${\rho }_0={\mathbb{I}}_2/2$.

Figure 2

(a) The incoherent driving is made through the polarizers in Eq. (39). In the plot we show the time evolution of ${\mathrm{\Gamma }}_{+}$ (thin line) and ${\mathrm{\Gamma }}_{-}$ (bold line) for three values of the timescale $t_s$, ranging from an adiabatic to a quench regime. The amplitude is set to the value $A=0.3$. (b) Time evolution of the population (43) of the excited (ground) state in thin line (bold line) of a single qubit subject to the polarizers in Eq. (39). We prepare our system at $t_0=-2$ in a high-temperature state ${\rho }_{t_0}={\mathbb{I}}_2/2$ with $A=0.3$ and $\mathrm{\Omega }=1.0$. We clearly see that the system changes its target state when the amplitudes of the two lasers cross each other. This effect becomes sharper for short timescale $t_s$. In particular, in the quench regime (dash-dotted line) the population presents a cusp point at the crossover related to the sudden switch of the laser direction.

Figure 3

Illustration of a two-level quantum heat engine. The Hamiltonian parameter $\mathrm{\Omega }\left(t\right)$ tunes the energy gap between the ground state $|\uparrow \rangle$ and the excited level $|\downarrow \rangle$. The thermal bath is modeled as in Eq. (45) with a couple of polarizers ${\mathrm{\Gamma }}_{+},{\mathrm{\Gamma }}_{-}$. The interactions of the system $\mathcal{S}$ with the thermal bath $\mathcal{R}$ are controlled by the interruptor $\gamma$.

Figure 4

A sketch of a Carnot cycle where the working parameter $\mathrm{\Omega }$ is varied in time as a linear piecewise function.

Figure 5

Carnot engine. Numerical results for the Floquet parameters in Eqs. (34) and (35) obtained by solving Eq. (33) with polarizers in Eq. (46) and energy level separation $\mathrm{\Omega }$ in Eq. (59), setting ${\mathrm{\Omega }}_a=1.8,{\mathrm{\Omega }}_b=1.3$ and bath temperatures $T^{\mathrm{hot}}=1.0,T^{\mathrm{cold}}=0.5$. The driving period is $\mathcal{T}=50$.

Figure 6

Operation of a finite-time Carnot engine. The plot shows the internal energy $E\left(t\right)$ as a function of the compression $1/\mathrm{\Omega }\left(t\right)$. In the figure we select $\mathrm{\Omega }$ as in Eq. (59), setting ${\mathrm{\Omega }}_a=1.8,{\mathrm{\Omega }}_b=1.3$ and the thermal bath in Eq. (46) with $T^{\mathrm{hot}}=1.0,T^{\mathrm{cold}}=0.5$. We see that increasing the value of the driving period $\mathcal{T}$ (from the innermost to the outermost), the cycle converges towards its quasistatic limit.

Figure 7

A numerical estimation of the deviations of the cycle areas ${\delta }_{\mathcal{A}}$ in Eq. (60) for the 2-level Carnot engine in Fig. 6. We see that the approach to the quasistatic limit is $\propto 1/\mathcal{T}$ with a proportionality constant $c=11.6$, extracted from our fitting data.

Figure 8

Otto engine. Numerical results for the Floquet parameters in Eqs. (34) and (35) obtained by solving Eq. (33) with polarizers in Eq. (46) and energy level separation $\mathrm{\Omega }$ in Eq. (61), setting ${\mathrm{\Omega }}_1=1.8,{\mathrm{\Omega }}_2=1.3$. The temperature is varied as in Eq. (62), requiring quasistatic reversibility conditions (63) and choosing $T_a=1.0,T_b=1.5$. The driving period is $\mathcal{T}=50$.

Figure 9

Operation of a finite-time Otto engine. The plot shows the internal energy $E\left(t\right)$ as a function of the compression $1/\mathrm{\Omega }\left(t\right)$. In the figure we select $\mathrm{\Omega }$ as in Eq. (61), setting ${\mathrm{\Omega }}_1=1.8,{\mathrm{\Omega }}_2=1.3$ and the thermal bath in Eq. (46) with temperature in Eq. (62). We require quasistatic reversibility conditions (63) and we chose $T_a=1.0,T_b=1.5$. We see that increasing the value of the driving period $\mathcal{T}$ (from the innermost to the outermost), the cycle converges towards its quasistatic limit.

Figure 10

A numerical estimation of the deviations of the cycle areas ${\delta }_{\mathcal{A}}$ in Eq. (60) for the 2-level Otto engine in Fig. 9. We see an approach ${\delta }_{\mathcal{A}}\propto 1/\mathcal{T}$ to the quasistatic limit with a proportionality constant $c=8.02$, extracted from our fitting data.

Figure 11

Operation of a finite-time Carnot engine obtained by a numerical solution of Eqs. (C8) with initial state ${\rho }_0={\mathbb{I}}_2/2$. The plots show the internal energy $E\left(t\right)$ as function of the compression $1/\mathrm{\Omega }\left(t\right)$. In the figures we select $\mathrm{\Omega }$ in Eq. (59) with ${\mathrm{\Omega }}_a=1.8,{\mathrm{\Omega }}_b=1.3$ and the thermal bath in Eq. (46) with $T^{\mathrm{hot}}=1.0,T^{\mathrm{cold}}=0.5$. (a) We show the solution in a time window $t\in [0,3\mathcal{T}]$ for different values of the driving period $\mathcal{T}$. We can see that increasing the value of $\mathcal{T}$ (from the innermost to the outermost), the cycle matches its quasistatic limit. (b) Inset showing the time evolution after $\mathcal{T}/\mathrm{\Gamma }$ where the dynamics becomes cyclic. The results are then in perfect agreement with those in Fig. 6.

Figure 12

Time evolution of the spin magnetization $\langle {\sigma }_3\rangle =tr\left({\rho }_t{\sigma }_3\right)$ for the single qubit in Eq. (26) with counteroscillating polarizers ${\mathrm{\Gamma }}_1=2.0+0.5sin\left(\omega t\right),{\mathrm{\Gamma }}_{-}=3.0-0.5sin\left(\omega t\right)$ and with ${\mathrm{\Gamma }}_3=0,\mathrm{\Omega }=\sqrt{2}[1-cos\left(\omega t\right)]$. The numerical exact time evolution (full line) is compared with the approximate time evolution (dashed line) obtained in high-frequency perturbation theory at the second order for three values of the driving frequency $\omega =2.0,5.0,10$. The dots show the stroboscopic evolution (circle: exact, in diamond: approximate). The system is prepared at time $t=0$ in a state ${\rho }_0={\mathbb{I}}_2/2$.