Trapped-ion toolbox to simulate quantum Otto heat engines

Abstract

We present a scheme that utilizes an ion confined within a bi-dimensional trap to simulate a quantum Otto heat engine whose working substance is a two-level system. In this scheme, the electronic component of the ion (the two-level system) can interact with effective heat reservoirs of different types. We specifically focus on effective thermal reservoirs (those with positive temperatures), effective heat reservoirs with apparent negative temperatures, and effective squeezed thermal reservoirs. We show how to generate these effective reservoirs and provide numerical results to illustrate the applicability of the presented scheme. Finally, considering the same types of effective heat reservoirs, we briefly discuss the simulation of a quantum Otto heat engine where a quantum harmonic oscillator serves as the working substance.

Figures (2)

• Figure 1: Scheme displaying the quantum Otto cycle described in the text. The boxes represent the cycle strokes. Starting with the cooling stroke, $\Delta_{e}^{c}$ corresponds to the modified electronic frequency of the ion, $\omega_{m}$ represents the motional frequency of the ion along the $x$- and $y$-directions, $\kappa$ denotes the motional decay rate, and $H_{int}^{c}$ stands for the interaction Hamiltonian between the electronic and motional components of the ion. As the text explains, $H_{int}^{c}$ determines the effective cold heat reservoir for the electronic component of the ion (ECI). In the expansion stroke, the two-level system undergoes a unitary evolution governed by $U_{e}^{exp}\left(\tau\right)$, which changes the modified electronic frequency from $\Delta_{e}^{c}$ to $\Delta_{e}^{h}$ ($\Delta_{e}^{c}<\Delta_{e}^{h}$) after time $\tau$. During the heating stroke, the interaction Hamiltonian is now $H_{int}^{h}$, which leads to the effective hot heat reservoir for the ECI. Finally, in the compression stroke, the unitary operator $U_{e}^{comp}\left(\tau\right)$ reverses the change, bringing the modified electronic frequency back from from $\Delta_{e}^{h}$ to $\Delta_{e}^{c}$.
• Figure 2: Engine efficiency $\eta$ as a function of the transition probability $\xi$. Panel (a) displays curves associated with the effective hot thermal reservoir, panel (b) shows curves related to the effective hot reservoir with an apparent negative temperature, and panel (c) depicts curves linked with the effective hot squeezed thermal reservoir. In panels (a)-(c), we assume a cold thermal reservoir with $n_{R}^{c}=0.6$, defined by the Bose-Einstein distribution. In panel (a), we choose $n_{R}^{h}=1.2$, determined by the Bose-Einstein distribution; in panel (b), $n_{R}^{h}=0.8$, given by the Fermi-Dirac distribution; and in panel (c), $n_{R}^{h}=0.4$, provided by the Bose-Einstein distribution, and $r=1.5$. In addition, we set $\Delta_{e}^{c}=2\pi\times\left(0.2\ \text{MHz}\right)$, $\Delta_{e}^{h}=2\pi\times\left(0.4\ \text{MHz}\right)$, $\lambda=10^{-2}$, $\Omega=2\pi\times\left(10\ \text{MHz}\right)$, $\kappa=2\pi\times\left(1\ \text{MHz}\right)$, and $\gamma=2\pi\times\left(10^{-3}\ \text{MHz}\right)$. The red dots correspond to the case in which we apply the adiabatic elimination in the heating and cooling strokes (effective dynamics), the black solid lines correspond to the situation in which we do not perform the adiabatic elimination (full dynamics), the gray dotted lines show the Otto efficiency $\eta_{O}$, and the gray dot-dashed lines display the Carnot efficiency $\eta_{C}$.