Autonomous quantum thermodynamic machines

We investigate the dynamics of a quantum system consisting of a single spin coupled to an oscillator and sandwiched between two thermal baths at different temperatures. By means of an adequately designed Lindblad equation, it is shown that this device can function as a thermodynamic machine exhibiting Carnot-type cycles. For the present model, this means that when run as a heat engine, coherent motion of the oscillator is amplified. Contrary to the quantum computer, such a machine has a quantum as well as a classical limit. Away from the classical limit, it asymptotically approaches a stationary transport scenario.

Figure 1

System diagram of autonomous quantum thermodynamic machines.

Figure 2

Control functions ${\theta }^{\left(j\right)}\left(\tau \right)$ for switching on and off baths of temperature $T_j(j=U,L)$at different time slots (shown for the heat engine).

Figure 3

Modulus of the fidelities $F^{\left(j\right)}\left(\tau \right)$, Eq. (28), resulting from the respective control functions ${\theta }^{\left(j\right)}\left(\tau \right),j=U,L$, as shown in Fig. 2. Here we have used the cutoff version ${\widehat{\Pi }}^{\prime }$ according to Eq. (36) for the heat engine $M_1$ (see Sec. 5). The imaginary parts of $F^{\left(j\right)}\left(\tau \right)$ are typically two orders smaller than the respective real parts. The finite slope stems from the finite width of the oscillator state occupation number distribution.

Figure 4

Autonomous quantum heat engine $M_1$: Temperature $\overline{T_{\mathrm{G}}}$ and entropy $S_{\mathrm{G}}$ (four cycles superimposed; each of the four steps marked by boxed numbers). $\overline{T_{\mathrm{G}}}$ is given in units of $C_{\mathrm{C}}$ [Eq. (3)], $S_{\mathrm{G}}$ is dimensionless.

Figure 5

Autonomous quantum heat engine $M_1$: Average total system energy $⟨\widehat{H}⟩$ over four machine cycles. Process steps are numbered (cf. Fig. 4). Energy in units of $C_{\mathrm{C}}$.

Figure 6

Autonomous quantum heat engine $M_1$: Dimensionless amplitude of oscillator $C$ over four machine cycles. Note the coherence amplification for these parameters of the heat engine.

Figure 7

Same as Fig. 6, but over 30 machine cycles. Finally decoherence is stronger, and the initial coherence amplification vanishes.

Figure 8

Moment analysis of $\mid ⟨\widehat{x}⟩\mid$ for autonomous quantum heat engine $M_1$: normalization factor $N_0$ of the first off-diagonal (real part) of ${\widehat{\rho }}_{\mathrm{C}}$ over the cycle number (see text).

Figure 9

Moment analysis of $\mid ⟨\widehat{x}⟩\mid$ for autonomous quantum heat engine $M_1$: mean $\mu$ of the first off-diagonal (real part) of ${\widehat{\rho }}_{\mathrm{C}}$ over the cycle number (see text).

Figure 10

Moment analysis of $\mid ⟨\widehat{x}⟩\mid$ for autonomous quantum heat engine $M_1$: variance ${\sigma }^2$ of the first off-diagonal (real part) of ${\widehat{\rho }}_{\mathrm{C}}$ over the cycle number (see text).

Figure 11

Autonomous quantum heat pump $M_2$: Temperature $\overline{T_{\mathrm{G}}}$ in units of $C_{\mathrm{C}}$ and entropy $S_{\mathrm{G}}$ (four cycles superimposed; each of the four steps marked by boxed numbers).

Figure 12

Autonomous quantum heat pump $M_2$: Average total system energy $⟨\widehat{H}⟩$ in units of $C_{\mathrm{C}}$ over four machine cycles. Process steps are numbered (cf. Fig. 4).

Figure 13

Autonomous quantum heat pump $M_2$: Dimensionless amplitude of oscillator $C$ over four machine cycles. The amplitude decreases as coherently stored work is used to pump heat from the cold to the hot reservoir.

Figure 14

Autonomous quantum heat pump $M_2$: Initial adaptivedynamics in the plane of temperature $\overline{T_{\mathrm{G}}}$ and entropy $S_{\mathrm{G}}$ for eight cycles until a quasicyclic state for $G$ is established.

Figure 15

Autonomous quantum heat engine $M_3$: Temperature $\overline{T_{\mathrm{G}}}$ in units of $C_{\mathrm{C}}$ and entropy $S_{\mathrm{G}}$ (20 cycles superimposed). Spiraling in this plot is the effect of decoherence on the oscillator.

Figure 16

Autonomous quantum heat engine $M_3$: Average total system energy $⟨\widehat{H}⟩$ in units of $C_{\mathrm{C}}$ over 20 machine cycles. Note that even with severe decoherence there is machine function left (increase in not coherently stored energy).

Figure 17

Autonomous quantum heat engine $M_3$: Dimensionless displacement $⟨\widehat{x}⟩$ of oscillator $C$ over 20 machine cycles. The coherently stored energy decreases.

Figure 18

Autonomous quantum heat pump $M_2$: Diagonal elements in Fock representation of the reduced density operator ${\widehat{\rho }}_{\mathrm{C}}$ of the stationary state.

Figure 19

Fidelity $F\left(t\right)$ of Eq. (A7) for $\omega =1$ and $\alpha$ from the set {1,2,4,8,16}. The larger $\alpha$ is, the smaller is the decay time. $t$ is in units $C_{\mathrm{C}}^{-1}$.