Work extraction and energy storage in the Dicke model

We study work extraction from the Dicke model achieved using simple unitary cyclic transformations keeping into account both a nonoptimal unitary protocol and the energetic cost of creating the initial state. By analyzing the role of entanglement, we find that highly entangled states can be inefficient for energy storage when considering the energetic cost of creating the state. Such a surprising result holds notwithstanding the fact that the criticality of the model at hand can sensibly improve the extraction of work. While showing the advantages of using a many-body system for work extraction, our results demonstrate that entanglement is not necessarily advantageous for energy storage purposes, when nonoptimal processes are considered. Our work shows the importance of better understanding the complex interconnections between nonequilibrium thermodynamics of quantum systems and correlations among their subparts.

Figure 1

(a) Four different cyclic time protocols, where the dotted black line is an instantaneous quench, the solid blue line is a linear quench, the dotted red line is a quadratic quench, and the solid green line is a quartic quench. (b) Work-energy ratio, for the protocols reported in (a), as a function of the renormalized coupling parameter. The initial state is an eigenenergy Fock state ${\left|m=10\right.〉}_d{\left|n=0\right.〉}_c$ in the polariton basis.

Figure 2

Diagrammatic representation of the cycles. Initially the system is prepared in a locally thermal state, at inverse temperatures ${\beta }_a$ and ${\beta }_b$. The cyclic unitary transformation $\widehat{U}\left(t\right)$ in the parameter space $(\lambda ,\omega )$ is highlighted on the right: quench $A\left(C\right)\to B\left(D\right)$, evolution in $B\left(D\right)$, quench $B\left(D\right)\to A\left(C\right)$, with a final extraction of work.

Figure 3

Panel (a): Work-energy ratio for the A-B cycle in Fig. 2, with $\mathrm{\Delta }\omega /2\pi =\omega /2\pi =15\text{MHz}$ and ${\omega }_0/2\pi =8.3\text{kHz}$. (b) Work-energy ratio for the C-D cycle in Fig. 2, with $\mathrm{\Delta }\omega =0.1\omega ,\mathrm{\Delta }\lambda =-0.1{\lambda }_{\text{cr}}$. The free evolution time at point $B$ is ${\tau }_B=0.003$ s. Dashed purple: ${\widehat{\rho }}_1^{{\beta }_a{\beta }_b}$, solid red: ${\widehat{\rho }}_2^{{\beta }_a{\beta }_b}$, dashed black: ${\widehat{\rho }}_3^{{\beta }_a{\beta }_b}$, solid blue: ${\widehat{\rho }}_4^{{\beta }_a{\beta }_b}$, where ${\beta }_J=1/k_B{T}_J$, with $T_b=0.01K,T_a^1={10}^{-1}\text{K},T_a^2={10}^{-1.5}\text{K},T_a^3={10}^{-2.5}\text{K},T_a^4={10}^{-3}\text{K}$. The green curve is the work-ergotropy ratio.

Figure 4

Entanglement in the polariton partition $d-c$ for four locally passive entangled states, with different values of $\beta =1/K_{B}T$. Dotted black: $T={10}^{-4}$, solid blue: $T=2\times {10}^{-4}$, dotted red: $T=3\times {10}^{-4}$, solid green: $T=4\times {10}^{-4}$.

Figure 5

(a) Work-ergotropy ratio for the locally passive entangled state in Eq. (17) for the A-B cycle in Fig. 2, with $\mathrm{\Delta }\omega /2\pi =\omega /2\pi =15\text{MHz}$, and ${\omega }_0/2\pi =8.3\text{kHz}$. (b): Total work for the same cycles as in panel (a).