Thermodynamic Power of the Squeezed Thermal Reservoir

Abstract

The last part of this thesis is related to the performance of quantum thermal machines. Quantum thermal machines have been introduced in Chap. 3 as a topic which has attracted increasing attention within the new field of quantum thermodynamics. They generically consist of small quantum devices performing some useful thermodynamic task, such as refrigeration, heat pumping, or work extraction, while powered by out-of-equilibrium thermodynamic or mechanical forces. Their importance comes from the fact that they can be used to investigate fundamental questions related to the laws of thermodynamics as well as being useful in practical applications.

Appendices

Appendix

A.1 Reservoir Entropy Changes

In the main text we claim that the effective entropy flow, $\dot{\Phi }$, appearing in the generalized second law inequality, Eq. (10.9) in Sect. 10.1, equals the entropy decrease in the reservoir due to the interaction with the bosonic mode. We demonstrate here this relation from the collisional model introduced in Sect. 2.3.2, where the system bosonic mode interacts sequentially with a ‘fresh’ reservoir mode k in the same squeezed thermal state at inverse temperature $\beta$, and squeezing parameter $\xi = r e^{i\theta }$ with $r\geqslant 0$ and $\theta \in [0, 2\pi ]$:

$$\begin{aligned} \rho _R^{(k)}&= \hat{\mathcal {S}}_k(\xi ) \frac{e^{-\beta \hat{H}_R(\Omega _k)}}{Z_R} \hat{\mathcal {S}}_k^\dagger (\xi ) \nonumber \\&= \sum _{\nu } \left( \frac{e^{- \beta \hslash \Omega _k \nu }}{Z_R} \right) \hat{\mathcal {S}}_k(\xi ) |\nu _k\rangle \langle \nu _k | \hat{\mathcal {S}}_k^\dagger (\xi ) \end{aligned}$$

(A.1)

where $\hat{\mathcal {S}}_k(\xi ) \equiv \exp {\frac{r}{2}(b_k^2 e^{- i \theta } - b_k^{\dagger 2} e^{i \theta })}$, stands for the squeezing operator on the reservoir mode k, and in the last equality we decomposed the Gibbs state in its Fock basis $\{ |\nu _k\rangle \}$. It’s easy to see from the above equation that the eigenvalues and eigenvectors of $\hat{\rho }_R$ are given by:

$$\begin{aligned} \epsilon _\nu ^{(k)} = \frac{e^{- \beta \hslash \Omega _k \nu }}{Z_R},\quad |\epsilon _\nu ^{(k)}\rangle = \hat{\mathcal {S}}_k(\xi ) |\nu _k\rangle , \end{aligned}$$

(A.2)

i.e. the state $\rho _R^{(k)}$ can be viewed as a classical mixture of squeezed Fock states $|\epsilon _v^{(k)}\rangle$ with Boltzmann weights $\epsilon _\nu ^{(k)}$.

We can estimate the reservoir entropy change during the evolution by constructing, analogously to what have been done for the system bosonic mode, a coarse-grained time derivative by partial tracing Eq. (2.63) over the system degrees of freedom:

$$\begin{aligned} \dot{\rho }_{R}^{(k)} \simeq \frac{1}{\delta t}[\rho _R^{(k)}(t + \delta t) -\rho _R^{(k)}] = \mathcal {R} ~[\rho _R^{(k)}(t + \tau ) - \rho _R^{(k)}] \end{aligned}$$

(A.3)

for the interaction of duration $\tau \ll g_k^{-1}$ between system and a particular mode k in the reservoir.

Using Eqs. (2.64) and (2.65) we obtain:

$$\begin{aligned} \dot{\rho }_{R}^{(k)}&= - i [\Delta \hat{H}_R(\Omega _k) , \rho _R^{(k)}] + [\epsilon _k^*\langle \hat{a}\rangle _t \hat{b}_k^\dagger - \epsilon _k \langle \hat{a}^\dagger \rangle _t \hat{b}_k , \rho _R^{(k)}] \nonumber \\&+ c_k \langle \hat{a} \hat{a}^\dagger \rangle _{t} \left( \hat{b}_k \rho _R^{(k)} \hat{b}_k^\dagger - \frac{1}{2}\{ \hat{b}_k^\dagger \hat{b}_k, \rho _R^{(k)}\} \right) \nonumber \\&+ c_k \langle \hat{a}^\dagger \hat{a} \rangle _{t} \left( \hat{b}_k^\dagger \rho _R^{(k)} \hat{b}_k - \frac{1}{2}\{ \hat{b}_k \hat{b}_k^\dagger , \rho _R^{(k)}\} \right) \nonumber \\&- c_k e^{-i \Delta _k (2t + \tau )} \langle \hat{a}^2 \rangle _{t} \left( \hat{b}_k^\dagger \rho _R^{(k)} \hat{b}_k^\dagger - \frac{1}{2}\{\hat{b}_k^{\dagger 2} , \rho _R^{(k)} \} \right) \nonumber \\&- c_k e^{i \Delta _k (2t + \tau )} \langle \hat{a}^{\dagger 2} \rangle _{t} \left( \hat{b}_k \rho _R^{(k)} \hat{b}_k - \frac{1}{2}\{\hat{b}_k^2 , \rho _R^{(k)} \} \right) , \end{aligned}$$

(A.4)

where $\langle \hat{O} \rangle _t = \mathrm {Tr}_{\mathcal {S}}[\hat{O} \rho _t]$ are the system expectation values at time t, and $\Delta _k = \omega - \Omega _k$. In the above equation we defined

$$\begin{aligned} \epsilon _k&\equiv \mathcal {R}~ \tau ~ g_k ~{\mathrm {sinc}}(\Delta _k \tau /2) ~e^{i\Delta _k (t + \tau /2)}, \nonumber \\ c_k&\equiv \mathcal {R}~ \tau ^2 g_k^2 ~{\mathrm {sinc}}^2(\Delta _k \tau /2), \end{aligned}$$

(A.5)

together with the mode dependent frequency-shift in the reservoir

$$\begin{aligned} \Delta \hat{H}_R(\Omega _k)&\equiv \mathcal {R}~ \frac{g_k^2 \tau }{\Delta _k} \big [ \hat{b}^\dagger \hat{b} \left( {\mathrm {sinc}}(\Delta _k \tau /2)\cos (\Delta _k \tau /2) - 1 \right) + 1 \\&- {\mathrm {sinc}}(\Delta _k \tau /2)\left( 2\langle \hat{a}^\dagger \hat{a} \rangle _t (\cos (\Delta _k \tau /2) - 1) + e^{i\Delta _k \tau /2}\right) \big ], \end{aligned}$$

which is analogous to the system frequency shift, and will be neglected as well. Notice that Eq. (A.4) give us the average evolution of the reservoir modes k when it interacts once at a time with the system at random times (as specified by the rate $\mathcal {R}$). However, we don’t know the frequency of the reservoir mode interacting with the system in each collision, so we must assume that the system interacts with all modes in the reservoir with certain probability, given by the density of states in the reservoir $\vartheta (\Omega _k)$. Therefore the average reservoir entropy change due to the entropy change in all reservoir modes during the evolution should read

$$\begin{aligned} \dot{S}_R = \sum _k \vartheta (\Omega _k) \dot{S}_R^{(k)} = - \sum _k \vartheta (\Omega _k) \mathrm {Tr}_R[\dot{\rho }_R^{(k)} \ln \rho _R^{(k)}]. \end{aligned}$$

(A.6)

In the following we introduce the explicit form of $\rho _R^{(k)}$ as given in Eq. (A.1) into the above expression for the average reservoir entropy change, and exploit Eq. (A.4). We obtain:

$$\begin{aligned} \dot{S}_R&= \beta \sum _k \vartheta (\Omega _k)~\mathrm {Tr}_R[\dot{\rho }_R^{(k)} ~ \hat{\mathcal {S}}_k(\xi ) \hat{H}_R(\Omega _k) \hat{\mathcal {S}}^\dagger _{k}(\xi )] = \nonumber \\&= - \beta ~\mathrm {Tr}_{\mathcal {S}}[\dot{\rho }_t \hat{\mathcal {S}}(\xi ) \hat{H}_{\mathcal {S}} \hat{\mathcal {S}}^\dagger (\xi )] = - \dot{\Phi } \end{aligned}$$

(A.7)

where the second line follows after a little of operator algebra, by expanding $\hat{\mathcal {S}}_k(\xi ) \hat{H}_R(\Omega _k) \hat{\mathcal {S}}_{k}^\dagger (\xi )$ and using Eqs. (A.4) and (10.3). As a hint, first notice that the first order term in Eq. (A.4) does not contribute to the entropy. Secondly notice that once the trace over the reservoir degrees of freedom have been performed, one can take the continuum limit over the reservoir spectra by introducing the spectral density, $J(\Omega )$, to recover the system master equation decay factors in Eq. (2.73) after integrating over frequencies.

Henceforth the entropy flow entering the system during the evolution, as given by $\dot{\Phi }(t) = - \mathrm {Tr}[\dot{\rho }_t \ln \pi ]$, Eq. (10.9) in Sect. 10.1, is the average entropy lost in the the reservoir in the sequence of collisions. This implies that the non-adiabatic entropy production [41,42,43, 55], $\Delta _{\mathrm {i}} S_{\mathrm {na}}$ in Eq. (10.8), corresponds indeed the total entropy produced in the process. In terms of the rates:

$$\begin{aligned} \dot{S}_{\mathrm {na}} \equiv - \frac{d}{dt} D(\rho _t || \pi ) = \dot{S} + \dot{S}_R \geqslant 0 \end{aligned}$$

(A.8)

where $D(\rho ||\sigma ) = \mathrm {Tr}[\rho (\ln \rho - \ln \sigma )]$ is the quantum relative entropy. As a consequence the adiabatic (or house-keeping) contribution due to non-equilibrium external constraints [41, 42] is always zero in the present case. An important consequence of the above finding is that no entropy is produced in order to maintain the non-equilibrium steady state $\pi$, Eq. (10.7), provided we have access to an arbitrarily big ensemble of reservoir modes in the state $\rho _R$.

A.2 Equations of Motion

From the Master Equation (10.3) in Sect. 10.1, one can derive the following equations of motion for the expectation values of the Lindblad operators expectation values and its combinations:

$$\begin{aligned}&\frac{d}{dt} \langle \hat{R} \rangle _t = - \frac{\gamma _0}{2} \langle \hat{R} \rangle _t \end{aligned}$$

(A.9)

$$\begin{aligned}&\frac{d}{dt} \langle \hat{R}^2 \rangle _t = - \gamma _0 \langle \hat{R}^2 \rangle _t, \end{aligned}$$

(A.10)

$$\begin{aligned}&\frac{d}{dt} \langle \hat{R}^\dagger \hat{R} \rangle _t = - \gamma _0 \left( \langle \hat{R}^\dagger \hat{R} \rangle _t - n_{\mathrm {th}}(\omega ) \right) . \end{aligned}$$

(A.11)

They can then be employed to explicitly asses the dynamics of the different contributions appearing in the effective entropy flow, $\dot{\Phi }$ in Eq. (10.9). Indeed by rewriting

$$\begin{aligned} \hat{a}&= \hat{R} \cosh (r)~ - \hat{R}^\dagger \sinh (r)e^{i \theta }, \end{aligned}$$

(A.12)

$$\begin{aligned} \hat{a}^\dagger&= \hat{R}^\dagger \cosh (r) - \hat{R} \sinh (r) e^{-i \theta }, \end{aligned}$$

(A.13)

and substituting into the expressions $\dot{Q}(t) = \dot{U}_{\mathcal {S}}(t) = \mathrm {Tr}[\hat{H}_{\mathcal {S}} \dot{\rho }_t]$ for the heat flux entering from the reservoir, and $\dot{\mathcal {A}}(t) = \hslash \omega \mathrm {Tr}[\hat{A}_\theta \dot{\rho }_t]$ with $\hat{A}_\theta = -\frac{1}{2} (\hat{a}^{\dagger 2} e^{i \theta } + \hat{a}^{2} e^{-i \theta })$, for the extra non-thermal contribution, we obtain the following equations

$$\begin{aligned} \dot{Q}(t)= & {} - \gamma _0 \left( Q(t) + \langle \hat{H}_{\mathcal {S}} \rangle _{\rho _0} - \langle \hat{H}_{\mathcal {S}} \rangle _{\pi } \right) , \nonumber \\ \dot{\mathcal {A}}(t)= & {} - \gamma _0 \left( \mathcal {A}(t) - \hslash \omega \langle \hat{A}_\theta \rangle _{\pi } \right) . \end{aligned}$$

(A.14)

In the above equations we introduced the steady state expectation values $\langle \hat{H}_{\mathcal {S}} \rangle _{\pi } = \hslash \omega N_\omega$ and $\langle \hat{A}_\theta \rangle _{\pi } = |M_\omega |$, being $\pi$ given in Eq. (10.7), with the reservoir expectation values, $N_\omega = \langle \hat{b}_k^\dagger \hat{b}_k \rangle _{\rho _R}$ and $M_\omega = \langle \hat{b}_k^2 \rangle _{\rho _R}$ as defined in (2.74) for a mode with resonant frequency $\Omega _k = \omega$ in the state $\rho _R$. We notice that both flows behave monotonically, yielding to an exponential decay as discussed in Sect. 10.1.