Collective performance of a finite-time quantum Otto cycle

We study the finite-time effects in a quantum Otto cycle where a collective spin system is used as the working fluid. Starting from a simple one-qubit system we analyze the transition to the limit cycle in the case of a finite-time thermalization. If the system consists of a large sample of independent qubits interacting coherently with the heat bath, then the super-radiant equilibration is observed. We show that this phenomenon can boost the power of the engine. Mutual interaction of qubits in the working fluid is modeled by the Lipkin-Meshkov-Glick Hamiltonian. We demonstrate that in this case the quantum phase transitions for the ground and excited states may have a strong negative effect on the performance of the machine. Conversely, by analyzing the work output we can distinguish between the operational regimes with and without a phase transition.

Figure 1

Schematic Otto cycle. $S$ is entropy and $\lambda$ a nonthermal control parameter of the WF.

Figure 2

Single qubit in Otto cycle with parameters ${\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$. (a) Cycle with a full thermalization. The numbers in circles correspond to Fig. 1. The dotted curves are analytic. (b) A visual demonstration of the Carnot bound. For description see the main text. (c) System approaching the limit cycle in the case of a finite-time thermalization. Duration of the contact with the heat baths is fixed as $t_{\mathrm{th}}=1{\omega }^{-1}$. The arrows show how the limit cycle is reached by “winding” around the full thermalization cycle (plotted with dotted curves).

Figure 3

Collective spin system (14) in Otto cycle with parameters $j=20,{\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$. (a) Cycle with a complete thermalization. The numbers in circles correspond to Fig. 1. The dotted curves represent truly thermal states for ${\lambda }_i$ and ${\lambda }_f$. (b) Distance between the actual state $\rho$ and the thermal reference state ${\rho }^{*}$ during the thermalization stroke $2\to 3$ measured by fidelity $\mathcal{F}$. (c) System approaching the limit cycle in the case of a finite-time thermalization. Duration of the contact with the heat baths is fixed as $t_{\mathrm{th}}=0.1{\omega }^{-1}$. The dotted cycle represents the full thermalization case.

Figure 4

Power of the collective-spin heat engine as a function of the duration of the thermal strokes $t_{\mathrm{th}}$. The curve for $j\to \infty$ is analytic according to Eq. (17). Parameters are $T_c=1\omega ,T_h=8\omega ,{\lambda }_i=1,{\lambda }_f=3,\gamma =0.1\omega$. Inset: Power as a function of the size of the system $j$ computed for fixed $t_{\mathrm{th}}=1{\omega }^{-1}$ (denoted with a thin vertical line in the main part) and two different values of $T_h$ as indicated (other parameters are the same as in the main part of the figure). The dashed curve represents the quadratic fit for $j\le 5$. The dotted line indicates the saturation value of power $\overline{P}\approx 12.26{\omega }^2$ [given by Eq. (17)] for the $T_h=40\omega$ bath. The maximal power for the $T_h=80\omega$ bath is $\overline{P}\approx 25.59{\omega }^2$.

Figure 5

Thermalization time $t_T$ as a function of $j$ according to Eq. (15). Fidelity $\mathcal{F}\left(\rho \right|\left|{\rho }_{T_f}\right)$ was used as a measure of the distance between the actual state $\rho$ and the final thermal state ${\rho }_{T_f}$, see Eq. (5). The case shown corresponds to cooling of the thermal state from $T_i=4\omega$ to $T_f=1\omega$ with $\lambda =1$ fixed, $\gamma =0.1\omega$. The tolerance to establish $t_T$ was chosen as $1-\mathcal{F}\le {10}^{-5}$. The green curve is a $1/j$ fit.

Figure 6

Analytic solution to the thermalization from a thermal state of $T_i=8\omega$ to $T_f=4\omega$ using Eq. (23) for three different values of $j$. We define $\mathrm{\Delta }m\left(t\right)=m\left(t\right)-m_0$. Value $\lambda =1$ is fixed, $\gamma =0.1\omega$. The dotted lines are the numerical solutions.

Figure 7

(a) Energy spectrum of the LMG model during the unitary stroke as a function of time. The blue full lines correspond to even parity while the red dashed lines to odd parity. Parameters of the model are $j=20,{\lambda }_i=1,{\lambda }_f=3,\overline{\mathrm{\Gamma }}=15$. For $t=0$ and $t=t_u$ the energy spectrum is equidistant. (b) Detail of the spectrum from panel (a), the QPT and a chain of ESQPTs are marked. (c) A sketch of the dependence $\mathrm{\Gamma }\left(\lambda \right)$. If the protocol is critical, then for certain values of $\lambda$ the system enters the symmetry-broken phase.

Figure 8

Quantum nonadiabatic evolution during the unitary strokes of the LMG model. The thermalization strokes are considered as perfect. The black dotted lines correspond to the fully thermalized cycle with no interaction as in Fig. 3. The panels differ according to the finite value of the time $t_u$ of the unitary strokes. (a) $t_u=6{\omega }^{-1}$, (b) $t_u=8{\omega }^{-1}$, (c) $t_u=10{\omega }^{-1}$, (d) $t_u=15{\omega }^{-1}$, (e) $t_u=20{\omega }^{-1}$, and (f) $t_u=100{\omega }^{-1}$. The parameter value $\overline{\mathrm{\Gamma }}=3$ guarantees that during the unitary stroke the system crosses the QPT. Other parameters are $j=20,{\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$.

Figure 9

Quantum nonadiabatic evolution during the unitary stroke $1\to 2$ of the LMG model for different values of $j$. Duration of the stroke is $t_u=8{\omega }^{-1}$ as in Fig. 8. Other parameters are $\overline{\mathrm{\Gamma }}=3,{\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$. The dip indicating the QPT becomes sharper with growing $j$.

Figure 10

Extracted work in a cycle $W_c^{\prime }$ as a function of the duration of the unitary stroke $t_u$. (a) $\overline{\mathrm{\Gamma }}=0.75$ (the QPT is not crossed). (b) $\overline{\mathrm{\Gamma }}=3$, same as in Fig. 7. The QPT is crossed. The red dashed line marks zero work output level. The inset shows the result for longer timescale $t_u\in [0,40]{\omega }^{-1}$. Other parameters are $j=20,{\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$.

Figure 11

Extracted work in a cycle $W_c^{\prime }$ as a function of parameter $\overline{\mathrm{\Gamma }}$ from Eq. (32) for several values of $j$. The duration of the unitary stroke is $j$ dependent and set to $t_u=j{\omega }^{-1}$. The arrow indicates the critical value ${\overline{\mathrm{\Gamma }}}_c=1.87$ when the QPT is crossed. Other parameters are ${\lambda }_i=1,{\lambda }_f=3,T_c=1\omega ,T_h=8\omega ,\gamma =0.1\omega$.