Three-dimensional harmonic oscillator as a quantum Otto engine

Open Access

Three-dimensional harmonic oscillator as a quantum Otto engine

A. Rodin

Phys. Rev. Research 6, 013180 – Published 20 February 2024

Abstract

A quantum Otto engine based on a three-dimensional harmonic oscillator is proposed. One of the modes of this oscillator functions as the working fluid, while the other two play the role of baths. The coupling between the working fluid and the baths is controlled using an external central potential. All four strokes of the engine are simulated numerically, exploring the nonadiabatic effects in the compression and expansion phases, as well as the energy transfer during the working fluid's contact with the baths. The efficiency and power of several realizations of the proposed engine are also computed with the former agreeing well with the theoretical predictions for the quantum Otto cycle.

Received 7 December 2023
Accepted 31 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013180

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum harmonic oscillator

Quantum heat engines & refrigerators

Quantum Information, Science & Technology

Authors & Affiliations

A. Rodin

Yale-NUS College, 16 College Avenue West, 138527, Singapore; Centre for Advanced 2D Materials, National University of Singapore, 117546, Singapore; and Department of Materials Science and Engineering, National University of Singapore, 117575, Singapore

Article Text

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Issue

Vol. 6, Iss. 1 — February - April 2024

Subject Areas

Quantum Physics

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Images

Figure 1
Engine schematic. The proposed engine consists of a three-dimensional harmonic oscillator with different force constants in the three dimensions. The force constant of one of the modes corresponding to the working fluid, can be modified during the engine operation. The other two modes are kept in thermal states by coupling them to external reservoirs. The working fluid mode can be coupled to the other two using symmetry-breaking central potentials $Φ_{c} (x_{c}, x_{g})$ and $Φ_{h} (x_{h}, x_{g})$ , where $x_{i}$ 's are the displacements of the oscillator in the direction of the modes.
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Figure 2
Engine cycle. An illustration of the engine cycle, indicating the portions of the Hamiltonian that need to be considered for each phase. All four phases need to include the component corresponding to the gas, given by the expression inside the rectangle. During the heating and cooling phases, the terms positioned next to the corresponding lines need to be included.
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Figure 3
Nonadiabatic effects. A system initialized at a 41-level thermal state at temperature $ω_{T} = k_{B} T / ℏ Ω$ with $α = 0$ (top row) or $α = ω^{2} - 1$ (bottom row). Next, the state is either compressed (until it arrives at $α = ω^{2} - 1$ ) or expanded (until $α = 0$ ) uniformly over time $τ_{α}$ , which can be chosen independently of of $α$ . The final energy of the system is then divided by what would be the energy of the system if the modification were adiabatic. The color of the heatmap gives the magnitude of this ratio with smaller numbers corresponding to greater adiabaticity. The nonadiabatic effects are most prevalent when the rate of the confinement change is high (corresponding to large $ω$ and small $τ_{α}$ ). High-temperature compression demonstrates the effects of the finite Fock space, as discussed in the main text.
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Figure 4
Interaction with a series of baths. (a) Trace distance between the gas and bath modes as a function of time. The density operators include 41 Fock states. “Expanded” (“compressed”) corresponds to $α = 0$ ( $α = 8$ ). “Heat” means that the oscillator starts at $ω_{T} = 1$ and the bath is at $ω_{T} = 5$ ; for “cool” the temperatures are switched. The interaction between the modes is Gaussian with $σ = 1$ and offset $x_{0} = 1$ . The interaction strength starts with 1, later being reduced to $1 / 5$ and, finally, to $1 / 20$ . The vertical lines mark the times when the bath is switched our for a new one. For dashed lines, $Φ_{0}$ is unchanged; solid lines show where the interaction strength is reduced. (b) Trace distance between the final states from panel (a) and thermal states for a range of $ω_{T}$ . Vertical lines indicate the temperatures of the cold and hot baths. These lines coincide with the minima of the distance curves, indicating that the final gas state approaches the correct thermal state.
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Figure 5
Engine operation. [(a)–(d)] Energies of the gas in an Otto engine at each phase of the cycle as a function of time. For the fast engine (top row), the time of each stroke is 1. For the slow engine (bottom row), the time of expansion/compression is 4, while the bath contact time is 10. The system is evolved following Eq. (10) with $ω_{T}^{hot} = 5$ and $ω_{T}^{cold} = 1 / 10$ . The interaction between the baths and the gas is the same as in Fig. 4 with $Φ_{0} = 1$ . For the compression and expansion phases, $\hat{U}$ is calculated for a 41-state basis. (e) Efficiency for each of the engines as a function of the cycle number, computed by dividing the total work output by heat input. (f) The power of the engines, obtained by dividing the total work output by the duration of a cycle.
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Figure 6
Numerical benchmarks. A system is initialized in a 101-dimensional thermal mixed state with $ω_{T} = 5$ and $α = 0$ at $τ = 0$ . It is then evolved numerically until $τ = 5$ using the fifth-order Runge-Kutta method. For one set of simulations, $α$ is set to a finite value at $τ = 0^{+}$ and kept constant (labeled $α_{fixed}$ in the legend). For the second set, $α$ increases linearly until it reaches its maximum value at $τ = 5$ (labeled as $α_{\max}$ ). The final state is calculated for $3 \leq ɛ / δ τ \leq 16$ and the trace distance is obtained between the smallest time step and all others for each configuration. The trace distance is then taken as the error.
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