Geometric properties of adiabatic quantum thermal machines

We present a general unified approach for the study of quantum thermal machines, including both heat engines and refrigerators, operating under periodic adiabatic driving and in contact with thermal reservoirs kept at different temperatures. We show that many observables characterizing this operating mode and the performance of the machine are of geometric nature. Heat-work conversion mechanisms and dissipation of energy can be described, respectively, by the antisymmetric and symmetric components of a thermal geometric tensor defined in the space of time-dependent parameters generalized to include the temperature bias. The antisymmetric component can be identified as a Berry curvature, while the symmetric component defines the metric of the manifold. We show that the operation of adiabatic thermal machines, and consequently also their efficiency, are intimately related to these geometric aspects. We illustrate these ideas by discussing two specific cases: a slowly driven qubit asymmetrically coupled to two bosonic reservoirs kept at different temperatures, and a quantum dot driven by a rotating magnetic field and strongly coupled to electron reservoirs with different polarizations. Both examples are already amenable for experimental verification.

Figure 1

Geometrical thermal machine setup. A central, parametrically driven quantum system described by the Hamiltonian ${\mathcal{H}}_S$ is coupled to macroscopic reservoirs. A cycle of the machine is completely characterized by a closed path in the parameter space $\mathbf{X}$. After a complete cycle the averaged power $P$ is dissipated as heat in the reservoirs. The net transported energy $J_{\mathrm{tr}}^Q$ flows from one reservoir to the other.

Figure 2

Illustration of the q-bit coupled to two bosonic reservoirs by the Hamiltonian of Eq. (40) with ${\widehat{\tau }}_L={\widehat{\sigma }}_x$ and ${\widehat{\tau }}_R={\widehat{\sigma }}_z$, operating as a heat engine. (a) The q-bit is in one of the states $|x,\pm \rangle$ and couples to the reservoir $L$. (b) The q-bit is in one of the states $|z,\pm \rangle$ and couples only to the reservoir $R$. The driving changes the energy difference between the two levels.

Figure 3

Vectors ${\vec{A}}^A$ and ${\widetilde{\stackrel{⃗}{}}A}^S$. Black and red arrows represent the vector ${\vec{A}}_3^A\left(\vec{B}\right)\equiv \left({\mathrm{\Lambda }}_{3,1}^A\left(\vec{B}\right),{\mathrm{\Lambda }}_{3,2}^A\left(\vec{B}\right)\right)$ in the parameter space, while the green arrows represent the vector ${\widetilde{\stackrel{⃗}{}}A}^S$ defined in Eq. (32). The blue line is the closed path corresponding to the driving protocol in Eq. (44) with $B_{x,0}=B_{z,0}=0.2k_{\mathrm{B}}T, B_{x,1}=B_{z,1}=0.1k_{\mathrm{B}}T$. The other parameters are ${\mathrm{\Gamma }}_{\mathrm{L}}={\mathrm{\Gamma }}_R=0.2$ and ${\epsilon }_{\mathrm{C}}=100k_{\mathrm{B}}T$ and define the spectral properties of the bosonic bath as indicated in Eq. (50).

Figure 4

Adiabatically pumped heat $Q_{\mathrm{ac}}$ and total work $W$ vs the phase lag $\varphi$ in the weak pumping limit for $\mathrm{\Delta }T=0$. Same parameters as in Fig. 3.

Figure 5

(Top) Pumped heat and work vs the phase difference between adiabatically-driven system parameters for $k_{\mathrm{B}}T=0.01{\epsilon }_{\mathrm{C}}$. (Bottom) Normalized pumped heat currents flowing in the left and right lead for $\varphi =\pi /2$. We have used the following parameters: ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=1/5, B_{z,0}=0.06{\epsilon }_{\mathrm{C}}$ ($B_{z,0}=0.04{\epsilon }_{\mathrm{C}}$ for the dashed lines in the bottom panel), $B_{x,0}=0.03{\epsilon }_{\mathrm{C}}, B_{x,1}=B_{z,1}=0.07{\epsilon }_{\mathrm{C}}$, and $\mathrm{\Delta }T=0$.

Figure 6

Pumped heat and work vs reference temperature $T$. (Inset) Efficiency of a heat pump for $\mathrm{\Delta }T=0$ as a function of $k_{\mathrm{B}}T$. Same parameters as in Fig. 5 for the solid curves.

Figure 7

Coefficient of performance for refrigeration (black dashed curve for absolute value and red curve for normalized to the Carnot value) vs $\mathrm{\Delta }T$, for $\hslash \mathrm{\Omega }=k_{\mathrm{B}}T/100$ and vs $\mathrm{\Omega }$ (in the inset), for $\mathrm{\Delta }T=T/500$. We use the following parameters: ${\mathrm{\Gamma }}_{\mathrm{L}}={\mathrm{\Gamma }}_{\mathrm{R}}=0.2, B_{z,1}=10k_{\mathrm{B}}T, B_{x,0}=20k_{\mathrm{B}}T, B_{x,1}=30k_{\mathrm{B}}T, B_{z,0}=7k_{\mathrm{B}}T, {\epsilon }_{\mathrm{C}}=120k_{\mathrm{B}}T$, and $\varphi =\pi /2$.

Figure 8

Illustration of the quantum dot driven by a magnetic field and connected to electron reservoirs with different polarizations, represented by different orientations of the paraboloids. The hybridization strength is modified according to the magnetic field's pointing direction. In (a), the electron hopping between the quantum dot and the right ($z$-polarized) reservoir is favored, as is denoted by the thick arrow. In (b), the pointing direction of the magnetic field has changed to $x$ and now the quantum dot is stronger coupled to the left reservoir.

Figure 9

Pumped heat $Q_{\mathrm{tr},\mathrm{ac}}=Q_{\mathrm{tr}}$ (top) and work done by the ac sources $W$ (bottom) for $\mathrm{\Delta }T=0$ as functions of the phase difference in the protocol defined by $B_x\left(t\right)=B_{x,0}+B_{x,1}cos(\mathrm{\Omega }t+\varphi ), B_z\left(t\right)=B_{z,0}+B_{z,1}cos\left(\mathrm{\Omega }t\right)$ with $B_{x,0}=B_{z,0}=0.4k_{B}T$ and $B_{x,1}=B_{z,1}=B_1{k}_{B}T$. ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=0.4k_{B}T$ and $\hslash \mathrm{\Omega }=k_{B}T/800$. The plot with $B_1=0.1$ is multiplied by a factor 20 in order to be shown in the same scale.

Figure 10

Pumped heat $Q_{\mathrm{tr},\mathrm{ac}}=Q_{\mathrm{tr}}$ for the quantum dot with the same parameters as the q-bit operating with the protocol of Eq. (44) shown in Fig. 4 with $\varphi =\pi /2$.

Figure 11

Vector fields ${\vec{A}}_3^A\left(\vec{B}\right)$ for a driving protocol with $\vec{B}\left(t\right)=\left(B_x\left(t\right),0,B_z\left(t\right)\right)$, corresponding to a quantum dot coupled to reservoirs with different polarizations. $R$ ($L$) reservoirs is polarized along positive $z$ ($x$) direction. Left panel corresponds to ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$ and $V_g=0$, middle correspond to ${\mathrm{\Gamma }}_L=0.1{\mathrm{\Gamma }}_R$ and $V_g=0$, while right panel corresponds to ${\mathrm{\Gamma }}_L=0.1{\mathrm{\Gamma }}_R$ and $V_g=2{\mathrm{\Gamma }}_R$. The temperature of the reservoirs is $k_{B}T=0.5{\mathrm{\Gamma }}_R$.

Figure 12

Vector fields ${\vec{A}}_3^A\left(\vec{B}\right)$ (cyan) and ${\widetilde{\stackrel{⃗}{A}}}^S\left(\vec{B}\right)$ (red) over a closed path (solid blue curve) for the configuration shown in the lower panel of Fig. 11 (${\mathrm{\Gamma }}_R=0.1{\mathrm{\Gamma }}_L$). The driving protocol defining the path is $B_x\left(t\right)=B_{x,0}+B_{x,1}cos(\mathrm{\Omega }t+\varphi ), B_z\left(t\right)=B_{z,0}+B_{z,1}cos\left(\mathrm{\Omega }t\right)$ with $B_{x,0}=1.5{\mathrm{\Gamma }}_R,B_{z,0}={\mathrm{\Gamma }}_R, B_{x,1}=B_{z,1}={\mathrm{\Gamma }}_R$, and $\varphi =\pi /2$. The black arrows represent ${\vec{A}}_3\left(\vec{B}\right)$ outside the defined protocol.

Figure 13

Coefficient of performance for refrigeration (absolute in dashed black and normalized to the Carnot value in red) vs $\mathrm{\Delta }T$ for the protocol of Fig. 12 for $\hslash \mathrm{\Omega }={\mathrm{\Gamma }}_R/200$. (Inset) Normalized coefficient of performance for refrigeration as a function of $\hslash \mathrm{\Omega }$ for $\mathrm{\Delta }T=T/150$.

Figure 14

Vector fields ${\vec{A}}_1^S\left(\vec{B}\right)$ (left) and ${\vec{A}}_2^S\left(\vec{B}\right)$ (right) following Eqs. (68) and (75), for the parameters of the right panel of Fig. 11.

Figure 15

Components of the geometric magnetization $m_{1,2}^{\mathrm{geo}}$, defined in Eq. (80) as functions of the phase-lag $\varphi$ corresponding to paths of the form $B_x\left(t\right)=B_{x,0}+B_{x,1}cos(\mathrm{\Omega }t+\varphi ), B_z\left(t\right)=B_{z,0}+B_{z,1}cos\left(\mathrm{\Omega }t\right)$ with $B_{x,0}=1.5{\mathrm{\Gamma }}_R,B_{z,0}={\mathrm{\Gamma }}_R, B_{x,1}=B_{z,1}={\mathrm{\Gamma }}_R$, on the vector fields of Fig. 14.