Quantum spin-heat engine with trapped ions

TL;DR

The paper addresses the limitation of conventional heat engines that require two energy reservoirs by proposing an ion-trap realization of the Vaccaro–Barnett spin-heat engine (SHE) that operates between a hot energy reservoir and a spin reservoir. It develops an effective two-level model under far-detuned, resolved-sideband conditions, where vibrational heat is converted into optical work via a two-photon Raman process, and the spin state is reset through a spin bath, exchanging angular momentum as spinlabor $\mathcal{L}$ and spintherm $\mathcal{Q}$. It also derives optimization criteria for maximizing work, establishes entropy-based bounds on extractable work, and demonstrates that the spin and energy channels bear equal unitless work in certain limits, all within a full cycle that includes re-thermalization. The work illustrates how quantum-coherent exchange among multiple conserved quantities can enable beyond-Carnot performance, laying groundwork for experimental exploration of entropy-driven engines and generalized thermodynamics in trapped-ion systems.

Abstract

We propose an ion-trap implementation of the Vaccaro, Barnett and Wright et al. spin-heat engine (SHE); a hypothetical engine that operates between energy and spin thermal reservoirs rather than two energy reservoirs. The SHE operates in two steps: first, in the work extraction stage, heat from a thermal energy reservoir is converted into optical work via a two photon Raman transition resonant with close-to energy degenerate spin states; second, the internal spin states are brought back to their initial state via non-energetic information erasure using a spin reservoir. The latter incurs no energy cost, but rather the reset occurs at the cost of angular momentum from a spin bath that acts as the thermal spin reservoir. The SHE represents an important first step toward demonstrating heat engines that operate beyond the conventional paradigm of requiring two thermal reservoirs, paving the way to harness quantum coherence in arbitrary conserved quantities via similar machines.

Figures (8)

• Figure 1: Conceptual diagrams for (A) a conventional Carnot heat engine and (B) the VB spin-heat engine. In the Carnot cycle, energy is transferred from a hot to a cold reservoir producing useful work along the way. In the VB-SHE, work is extracted from a single hot thermal reservoir. The cycle is completed by resetting the system’s entropy in a process where the initial spin supplied by an external source (the spinlabor $\mathcal{L}_s$) is dissipated into a spin reservoir in the form of spintherm, $\mathcal{Q}_s$. The latter is the spin equivalent of the heat dissipated in the cold reservoir in the Carnot cycle.
• Figure 2: A schematic of an operation cycle of the optical SHE. The working fluid comprises a trapped ion with three electronic levels and a quantized vibrational mode. The latter is represented by the equally spaced levels of a harmonic oscillator, while the former contains two energy degenerate levels $\ket{\uparrow}$ and $\ket{\downarrow}$ and the upper state $\ket{u}$. Panel (a) represents the first stage of the engine cycle, where the working fluid is initialized in the spin up state (full green circle) and the vibrational state is in a thermal state due to the contact with the reservoir at temperature $T_H$. Panel (b) represents the work extraction stage, where energy from the thermal reservoir (coming from the vibrational degrees of freedom of the ion) is transferred in the form of coherent light (optical work). The resetting stage is represented in (c) where the system is brought into contact with a spin reservoir: the entropy in the internal spin states is then reset to zero at the expense of increasing the entropy in the spin reservoir.
• Figure 3: Top panels: Evolution of the average number of vibrational quanta as a function of time for $\eta=0.05$ (left) and 0.4 (right). The arrows represent the final time $t_f$ for the stage, chosen to be the one at which the work extracted is maximized ($\bar{n}$ is minimum). Bottom panels: Comparison between the probability distributions $P(n)$ at the beginning (orange) and at the end (green) of the stage
• Figure 4: Work extracted in a cycle (in units of $\hbar\nu$) for $\kappa = 1$. The work ($W_{t_f}$) is calculated from the time evolution, as in Fig.\ref{['fig: dynamics of nbar']}), and plotted as a function of the Lamb-Dicke parameter for different values of the initial temperature. These curves show that there is an optimal $\eta$ ($\eta_\text{opt}$) that maximizes the work extracted.
• Figure 5: Optimal Lamb-Dicke parameter ($\eta_\text{opt}$, as defined in Fig. \ref{['fig: Work extracted for different eta']}) as a function of the initial average motional excitation $\bar{n}_0$. The curve shows that the higher the initial vibrational temperature, the smaller the value of $\eta_\text{opt}$.