Single-piston quantum engine

The purpose of an engine is to convert energy into work while operating cyclically. Although essentially all commonplace implementations are rooted in classical physics for their operation, there is a growing interest in quantum engines. [1, 2] Unlike their classical analogs, where the working fluid is typically a liquid or a gas, quantum engines employ quantum components. Some examples of these quantum “working fluids” are two-level systems [3, 4, 5, 6], single [7, 8, 9, 10, 11, 12, 13, 14] or multiple [15, 16] harmonic oscillators, or photons [17]. Another feature that sets quantum and classical varieties apart is how they are powered. In particular, classical implementations rely on hot reservoirs as energy sources and require cold baths to expel waste heat. Quantum versions, on the other hand, can also obtain the required energy from measurements, [18, 19, 20, 21, 22, 23, 24, 25, 26, 27] which act as an effective hot bath.

When discussing quantum engines, it is generally assumed that there is a way to transfer the work released during the power stroke of the cycle to some “piston” without explicitly focusing on this engine component. This paper proposes an implementation of a quantum engine in which the piston plays an integral role. A recent study [28] demonstrated that it is possible to modify the coupling strength between quantum engine components by controlling their energies. In particular, heating two interacting harmonic oscillators increases the extent of their probability distributions, leading to a greater separation and weakening their interaction. This temperature-dependent interaction strength is at the heart of the proposed cycle.

A cycle employing this energy-controlled coupling is shown in Fig. 1. The system consists of a harmonic oscillator, acting as the working fluid, and a moving piston. The two components are coupled via an attractive interaction. The cycle starts with the oscillator in a low-energy state and the piston positioned close to the oscillator’s equilibrium point. As the first step, the oscillator’s energy is increased by adding heat to it. This heating reduces the interaction strength between the oscillator and the piston due to an increased extent of the oscillator wavefunction, allowing the piston to be retracted from the oscillator in the second step of the cycle with a reduced energy cost. With the piston retracted, the oscillator is cooled down using a cold bath. Finally, the piston is returned to its initial position. Due to the enhanced interaction because of the narrower oscillator wavefunction in the final step, more work is released during this phase than was required to retract it, resulting in a net-positive work output.

As mentioned above, the heat required for quantum engine operation can originate either from heat baths or measurements. Therefore, after introducing the general model for the engine in Sec. II, two prototypical approaches using a hot bath and measurements are demonstrated in Sec. III. Summary and conclusions of the main results and the outlook for future research are found in Sec. IV.

All computations are performed using julia. [29] The plots are made using Makie.jl package [30] using the color scheme designed for colorblind readers. [31] The scripts used for computing and plotting can be found at https://github.com/rodin-physics/quantum-oscillator-engine.

As shown in Fig. 1, the engine consists of a harmonic oscillator, acting as the working fluid, coupled to an externally controlled piston. When the working fluid is decoupled from the environment, the corresponding time-dependent Hamiltonian, expressed in terms of the oscillator energy $\hbar\Omega$, is given by

Here, the first term describes an independent oscillator using the second-quantization operators and $\Phi$ is the interaction between the two engine components, where $\hat{x}$ is the oscillator’s position operator and $y(\tau)$ is the piston coordinate. In addition to using $\hbar\Omega$ as the energy scale, it is convenient to express lengths in terms of the quantum oscillator length and time in terms of the oscillator periods so that $t=2\pi\tau/\Omega$.

For this study, the interaction is set to

There are two reasons for choosing this form of $\Phi$. First, the amplitude and the extent of the Gaussian are easily tunable, making this type of interaction very convenient for illustrating the relevant behavior. Second, $y$ and $\hat{x}$ are separable, simplifying the computational procedure. With this choice, the matrix elements of $\hat{H}(\tau)$ in the Fock space become

The form of Eq. (3) indicates that the interaction matrix must be computed only once for a particular choice of $\sigma$ and scaled based on $y(\tau)$ as the piston moves. Although $\langle j|e^{-\frac{\hat{x}^{2}}{2\sigma^{2}}}|k\rangle$ can be calculated analytically for the quantum oscillator wavefunctions, producing Gaussian hypergeometric functions, the integrals in the simulations are taken using Gaussian quadratures to avoid potential issues associated with the numerical implementation of these special functions. For simplicity, the piston is assumed to move at a constant speed so that $y(\tau)=y_{\mathrm{init}}+\tau(y_{\mathrm{final}}-y_{\mathrm{init}})/\tau_{p}$ for $0\leq\tau\leq\tau_{p}$, where $\tau_{p}$ is the duration of the retraction and advancing phases.

With the interaction form chosen, it is now possible to demonstrate the key component of the engine operation: the temperature-dependent coupling between the working fluid and the piston. For this illustration, the oscillator is assumed to be in a thermal state, described by the density operator $\hat{\rho}=e^{-\hat{H}/\omega_{T}}/\mathrm{tr}\left(e^{-\hat{H}/\omega_{T}}\right)$, where $\omega_{T}=k_{B}T/\hbar\Omega$ is the thermal frequency corresponding to temperature $T$ and $k_{B}$ is the Boltzmann constant. The interaction energy is then given by $\mathrm{tr}\left(\hat{\Phi}\hat{\rho}\right)$ with $\hat{\Phi}$ matrix elements given by the second term of Eq. (3). Setting $\Phi_{0}=-5$, the value that will be used in subsequent simulations, the energy is calculated as a function of $\omega_{T}$ and $y$ for several values of $\sigma$ with the results given in Fig. 2. As expected, the system’s energy is the lowest for a wide interaction term (large $\sigma$) when the piston is close to the oscillator and the temperature is low. Increasing either $\omega_{T}$ or $y$ reduces the magnitude of the interaction.

Figure 2 provides a convenient way of thinking about the cycle described by the schematic in Fig. 1. Let the state of the system be given by a point in the $\omega_{T}$-$y/\sigma$ plane, indicating that the working fluid is in a thermal state. Following Fig. 1, one starts at a small $y$ and $\omega_{T}$ and, by adding the heat to the system and assuming that the oscillator state remains in a thermal state, moves the point vertically. Next, with $\omega_{T}$ fixed, the piston is retracted and the point moves horizontally to the right. As the heat is removed from the system, the point moves down. Finally, the piston is advanced and the point returns to its starting position. An example of such a path is given in Fig. 2(a). If the piston is moved adiabatically, the state of the oscillator does not change and only the interaction energy is altered so that the amount of energy added to or removed from the system during these phases is proportional to the number of equicontours crossed by the path. Therefore, as long as more equicontours are crossed by the left-moving segment than the right-moving one, the engine outputs work.

It is useful to provide an estimate for the efficiency associated with this engine, given by $\eta=(W_{\mathrm{out}}-W_{\mathrm{in}})/Q_{\mathrm{added}}$, where $Q_{\mathrm{added}}$ corresponds to the heat added to the system during the first phase and $W$ is the work extracted from or added to the system during the second and the fourth phases. Let $\rho_{H/L}$ correspond to the oscillator’s density operator in its high/low energy state, $\Phi_{\mathrm{close}}$ and $\Phi_{\mathrm{far}}$ give the interaction when the piston is close to/far from the oscillator, and $H_{0}$ be the first term in Eq. (1). Assuming that the piston motion is adiabatic so that $\rho_{H/L}$ remains fixed, $\eta$ becomes

where the approximation in the final line holds when $\Phi_{\mathrm{far}}\rightarrow 0$.

Equation (4) indicates that, for a particular oscillator (fixed $H_{0}$), the efficiency is maximized by increasing $\mathrm{tr}\left[\left(\rho_{H}-\rho_{L}\right)\Phi_{\mathrm{close}}\right]$. Using the diagram in Fig. 2(a), this requirement means that one should maximize the number of equicontour lines that the upwards-moving segment crosses. Thus, from Fig. 2, for the parameters chosen, $\sigma=2$ is too wide as most of the energy delivered to the system comes from piston retraction rather than from heating. On the other hand, $\sigma=1/4$ is inferior to $\sigma=1/2$, where the equicontours are denser. This non-monotonic dependence of $\eta$ on $\sigma$ suggestes that one can improve the efficiency by exploring different interaction profiles not limited to a Gaussian form.

To estimate $\eta$, the piston movement was taken to be adiabatic. If it is not, the energy is added to the oscillator in a thermal state during the piston movement and has to be either removed as waste heat (after the piston retraction) or results in a reduced work output (during the piston advance). Of course, for the engine to deliver a finite power, the individual phases have to take a finite amount of time. Therefore, it is useful to explore how slow is “slow enough” when it comes to the piston motion using the following procedure.

Starting with the Hamiltionians corresponding to retracted ($y=10\sigma$) and advanced ($y=0$) piston, denoted by $\hat{H}_{R}$ and $\hat{H}_{A}$, respectively, one determines their ground states $|R\rangle$ and $|A\rangle$. If the piston were retracted (advanced) adiabatically, the energy of the system would end up as $\langle R|\hat{H}_{R}|R\rangle$ ($\langle A|\hat{H}_{A}|A\rangle$) because the system would remain in its ground state. Conversely, for an instantaneous change of piston position, the state would not have time to evolve and the final energy would be $\langle A|\hat{H}_{R}|A\rangle$ ($\langle R|\hat{H}_{A}|R\rangle$). Hence, the actual final energy is bounded by these two values.

To obtain the system energy for a finite $\tau_{p}$, one uses the fact that the state evolves following the time-dependent Schrödinger equation

where the factor of $2\pi$ originates from the definition of $\tau$. Solving Eq. (5) using the fifth order Runge-Kutta method starting with $|\Psi(0)\rangle=|A\rangle$ and $|\Psi(0)\rangle=|R\rangle$, the final energy can be computed as a function of $\tau_{p}$ for a constant-speed piston with the results given in Fig. 3. As expected, the finite-$\tau_{p}$ energies lie between the adiabatic and instantaneous results for both retraction and advance of the piston. Figure 3 suggests that $\tau_{p}\gtrapprox 5$ is sufficiently slow to avoid the non-adiabatic effects associated with the piston movement.

This work has introduced and simulated a realization of a single-piston quantum engine, where the role of the working fluid is played by a harmonic oscillator. By taking advantage of the fact that the interaction between the working fluid and the piston can be controlled by modifying the oscillator’s energy, it has been shown that by following a cycle comprised of heating/cooling of the oscillator and piston motion towards and away from the oscillator, the engine can output work. Two general protocols of engine fueling have been discussed: bath-powered and measurement-powered with both successfully demonstrating stable work output over multiple cycles, indicating a steady state of operation. For simplicity, the heat reservoirs used here comprised of single thermal harmonic oscillator modes. Even though these modes do not function as true thermodynamic baths, the fact that the engine produces work suggests that the engine operation scheme is robust and will perform better in the presence of real thermodynamic reservoirs.

There are several research directions that naturally flow from this study. For an experimental realization of this system, the harmonic oscillator component could be implemented using cold atoms in an optical trap. To extract the waste heat, one could employ laser cooling. Therefore, a theoretical study could explore the operation of the proposed engine where the role of the working fluid is played by a harmonic oscillator with a controllable damping. In a similar vein, powering the engine using resonant driving is an option. Naturally, when the piston is in the advanced position, the oscillator Fock states are no longer the eigenstates, simply driving at the natural oscillator frequency might not be optimal in every case. Finally, a two-piston setup could be explored, where some of the waste heat expelled by one of the “cylinders” goes to heating the other one.