Thermodynamic engine with a quantum degenerate working fluid

Classical thermodynamic engines have been critical to human technology since the industrial revolution. In the past decade, the capabilities of quantum thermodynamic engines have been explored theoretically  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20], and recent years have seen experimental demonstrations of both quantum and nanoscopic classical engines using single ions [21, 22], nuclear spins [23], cold atoms [24, 25, 26], nitrogen-vacancy centers [27], and quantum gases  [28, 29]. A natural question is whether quantum phenomena can enhance the performance of a thermodynamic engine [30, 31, 32]. Perhaps the simplest experimental approach to this question — the direct comparison of an engine using a classical working fluid to an equivalent one using a quantum degenerate working fluid — has remained unexplored.

In this work, we experimentally realize and characterize an isentropic thermodynamic engine with a quantum degenerate working fluid. The engine cycle interleaves compression and decompression of an optical trap with Feshbach enhancement and suppression of interparticle scattering to pump energy from a magnetic field to an optical field via a trapped ensemble of ultracold neutral lithium. We observe that quantum degeneracy significantly enhances the output power. We measure the dependence of efficiency and power on cycle time, and investigate the effects on engine performance of compression ratio and interaction strength ratio. Results agree quantitatively with both approximation-free finite temperature interacting numerical simulations and mean-field analytics.

The experiments begin by preparing a Bose-Einstein condensate (BEC) of 300,000 to 1 million ⁷Li atoms in a far-detuned crossed optical dipole trap with a mean trap frequency $~\bar{\omega}=2\pi\times 133$ Hz, at a temperature of 170 nK, corresponding to a condensate fraction of 0.95. After evaporative cooling to degeneracy, the $s$-wave interparticle scattering length is Feshbach-tuned to 100$a_{0}$, where $a_{0}$ is the Bohr radius. This sets the initial condition (labeled $A$ in figures). Interleaved variation of the trap intensity and Feshbach field then execute the thermodynamic cycle illustrated in Fig. 1a. Between steps $A$ and $B$ (stroke $AB$), the trap power is increased with a functional form such that $\bar{\omega}$ increases from $\bar{\omega}_{A}$ to $\bar{\omega}_{B}$ at a constant rate. This is the compression stroke of the engine. In stroke $BC$, the trap frequency is held constant as the interaction strength is ramped from $a_{s}^{B}=100a_{0}$ to a larger value $a_{s}^{C}$ at a constant rate. Subsequently, the trap frequency and then the interactions are ramped linearly back to their initial values. Such a cycle pumps energy between the magnetic and optical control fields, because the work performed by the strongly interacting gas during decompression is not equivalent to the work done to compress the more weakly interacting gas. Performing the strokes of the cycle in the order shown in Fig. 1a results in a net transfer of energy from magnetic to optical fields. Appendix B details an intuitively useful analogy between this isentropic cycle and the Otto heat-engine cycle.

The second-quantized Hamiltonian describing the working fluid includes kinetic, interaction, and potential terms:

where $g(t)=4\pi\hbar^{2}m^{-1}a_{s}(t)$ is the interaction coupling constant, $a_{s}(t)$ is the scattering length, $m$ is the mass, and $\omega_{\bm{\mathrm{k}}}(t)$ is the trap frequency of mode k at time $t$. To measure release energy (defined below) at each step, we first abruptly switch off the trap, quenching to zero the last term of the Hamiltonian $\hat{H}_{\mathrm{pot}}$. Following 12 ms of free expansion we measure the column-integrated density distribution by absorption imaging and reconstruct the 3D distribution via Abel inversion [33]. After expansion, not only is the initial momentum distribution converted to a position distribution, but also essentially all the initial interaction energy is converted to kinetic energy [34], so the distribution provides a measure of the condensate’s release energy $E_{\mathrm{rel}}=E_{\mathrm{kin}}+E_{\mathrm{int}}$. We report release energies per atom, while plotted powers represent the total engine power.

Engine performance can be characterized by efficiency and power. We define work done on the condensate as positive. As in refs. [7, 12], we define the efficiency

and the power

Here $W_{ij}^{k}$ is the work done on the BEC by the field $k$ (laser or magnetic) in stroke $ij$ of the cycle and $T_{\mathrm{cycle}}$ is the total cycle time.

While we measure the release energy rather than the total energy, thse quantities can be simply related via the Gross-Pitaevskii description of an interacting gas. In the Thomas-Fermi regime, the total energy is given by [35]

where $\mu$ is the chemical potential, $E_{\mathrm{pot}}=(2/7)\mu$, $E_{\mathrm{int}}=(3/7)\mu$ and $E_{\mathrm{kin}}\approx 0$. The ratio between the total energy and the measured energy is then

Therefore, a power measurement based on release energy will be reduced from the true power by a factor of 2.5, while measured efficiency will give the true value. As shown later, we have verified the validity of these assumptions using approximation-free numerics.

Fig. 1b demonstrates the stark contrast between the behavior of degenerate and non-degenerate gases subjected to similar thermodynamic cycles. The thermal gas is prepared via inhibited evaporation at a small scattering length of 57$a_{0}$, resulting in a density of $\small{\sim}6\times 10^{11}\,\mathrm{atoms/cm^{3}}$ at a temperature of 890 nK. The density of the condensate is $\small{\sim}2\times 10^{13}\,\mathrm{atoms/cm^{3}}$, about 33 times larger, at a temperature of 170 nK. Much of this enhancement in density is a direct result of bosonic quantum statistics. While the thermal gas and condensate are prepared at different trap frequencies, the compression ratio $\nu=\bar{\omega}_{B}/\bar{\omega}_{A}\approx 2$ is the same for both. As interaction strength increases, the low density of the thermal gas results in a negligible change in release energy, while the Bose-enhanced density of the quantum degenerate sample results in a significant change. The measured efficiency of the engine with a thermal working fluid is consistent with zero, while the measured efficiency of the engine with a quantum degenerate working fluid is $0.45\pm 0.1$, near the maximum theoretical value of 0.55 for this compression ratio.

Reversibility can be tested by comparing the results of forward (${A}$-${B}$-${C}$-${D}$-${A}$) and reverse (${A}$-${D}$-${C}$-${B}$-${A}$) cycles. Fig. 2a shows the experimental results of such a comparison, demonstrating a high degree of reversibility and confirming that the reverse cycle results in a net transfer of energy from optical to magnetic fields, opposite to the forward cycle. Fig. 2b shows that the same cycle can be performed many times. The repeated return of the condensate to its initial release energy indicates that it can mediate energy transfer between magnetic and optical fields without significant net absorption of energy.

To estimate the degree of adiabaticity, one can apply the Landau-Zener formalism [36] to approximate the probability of low-lying collective excitations [37]. Considering only the ground state and lowest-lying collective excitation, the probability of diabatic passage between them is $P_{D}=\exp(-1/\Theta)$, where the adiabaticity parameter $\Theta=\dot{\omega}_{E}/(2\pi\omega_{E}^{2})$ depends on the energy gap $\hbar\omega_{E}$ to the nearest excited level. Taking our trap to be approximately axially symmetric, and using the results of [37] with a known ramp speed $\dot{\bar{\omega}}=2\pi\times 1$ Hz/ms, we estimate a maximum adiabaticity parameter of $\Theta\simeq 0.001$ for the cycles shown in Figs. 1 and 2, with cycle times of 530 ms.

Varying the engine cycle time affects both efficiency and power. Fig. 3a compares measured efficiency to the ideal Thomas-Fermi efficiency, which is independent of cycle time. Measurements for a range of cycle times cluster near this ideal. However, at long cycle times three-body loss, one-body loss, and heating can degrade efficiency, while at the shortest cycle times a combination of technical limitations (for example inductive limits on magnet current ramp rate) and decreasing adiabaticity affect engine performance.

Measuring engine power, we observe the expected inverse dependence over a range of cycle times, as shown in Fig. 3b. Power increases for faster cycles, deviating somewhat from the adiabatic prediction of Eq. 7 as the cycle time is reduced. The breakdown of engine performance at very short cycle times is also visible. These results indicate an optimal range of working speeds; as with any engine, there is a balance to be struck between power and efficiency [38]. Related theoretical work has explored the possibility of bypassing this trade-off using shortcuts to adiabaticity [7, 6, 39].

To investigate the validity of our theoretical approximations, we compare experimentally measured release energies to the results of fit-parameter free, finite temperature equilibrium simulations reproducing the experimental particle number, scattering length, and confinement. The engine’s demonstrated reversibility and adiabaticity justify the use of multiple equilibrium simulations and an assumption of isentropic evolution. At steps ${A}$ through ${D}$, we model the system as an interacting confined Bose gas using a path integral over complex-conjugate coherent states fields $\phi$ and $\phi^{*}$ with the action given in continuous imaginary time notation as [40]

where the notation $\tau+$ indicates the field should be evaluated at an advanced position on the $\tau$ contour, with $\tau\in[0,\beta]$ and $\beta=1/k_{B}T$. $U_{\mathrm{ext}}(\bm{r})=\frac{1}{2}m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})$ is the confinement potential, with $\omega_{i}$ the angular trap frequency in the $i^{\mathrm{th}}$ direction. Interactions are modeled as pairwise contact repulsions. The chemical potential $\mu$ is constrained such that total particle number $N$ is constant in each simulation [41]. We sample configurations of this field theory using the complex Langevin (CL) technique, a stochastic method of evaluating integrals that is robust for actions with a sign problem [42, 43] such as that in Eq. 6. Observables are calculated by time averaging field operators, obtained from thermodynamic derivatives of the partition function. This method does not require simplifying approximations, even at finite temperature, so it fully accounts for quantum and thermal fluctuations. The use of fields rather than particle coordinates allows full-scale replication of the experimental system on readily-available GPU hardware. Further details of the numerical methods appear in Appendix A.

Fig. 2a shows close agreement between numerically calculated and experimentally measured release energies. This correspondence provides additional retroactive justification for the isentropic assumption, and also demonstrates that approximate analytic expressions describing only the condensate with $N_{c}$ particles provide relatively accurate estimates of the energy. The analytic formulas for release energy and total energy are [44]

and

with $\alpha=\mu_{0}/(k_{B}T_{c}^{0})$, $\mu_{0}$ the zero-temperature chemical potential, $t=T/T_{c}^{0}$ the reduced temperature, $T_{c}^{0}$ the critical temperature of a harmonically confined Bose gas, $\zeta(x)$ the Riemann zeta function, and $k_{B}$ the Boltzmann constant. These results are accurate to within about 3% of approximation-free numerical simulations at the measured condensate fraction. The numerical results shown in Fig. 2b do indicate that $E_{\mathrm{kin}}$ accounts for 10% to 15% of the total energy in steps $A_{1}$ through $A_{5}$, violating to some extent the Thomas-Fermi approximation. Evaluating the ratios $P_{\mathrm{tot}}/P_{\mathrm{rel}}$ and $\eta_{\mathrm{tot}}/\eta_{\mathrm{rel}}$ using energies obtained from simulations shows that the former is 1% to 3% larger than predicted while the latter is 0.1% to 0.6% larger.

A natural parameter to tune in order to maximize efficiency is the compression ratio $\nu=\bar{\omega}^{B}/\bar{\omega}^{A}$. Fig. 4a shows measured release energy evolution over one cycle for different values of $\nu$. At higher compression ratios we observe distinctly higher release energies for steps ${B}$ and ${C}$ but no significant changes to the values at steps ${A}$ and ${D}$, in agreement with expectations from Eqs. 7 and 8. Fig. 4b demonstrates that increasing the compression ratio increases the efficiency $\eta$, which asymptotically approaches unity. In the Thomas-Fermi approximation this can be understood by analyzing the change in energy per particle $E_{\mathrm{tot}}\propto a_{s}^{2/5}\bar{\omega}^{\;6/5}$ [35]. Defining $\kappa=a_{s}^{C}/a_{s}^{A}$ as the interaction ratio between steps ${C}$ and ${A}$, the efficiency can be expressed as

As $\nu\rightarrow\infty,\eta\rightarrow 1$, and $\eta$ is independent of the interaction strength. This can be compared in loose analogy to the Otto cycle efficiency $\eta_{\mathrm{Otto}}=1-\nu^{1-\gamma}$ with $\nu$ the compression ratio and $\gamma$ the specific heat ratio.

Similarly, we can isolate the effects of interaction strength ratio $\kappa$ by holding the compression ratio constant. Following the same procedure used to derive Eq. 9, we find $P\propto(\kappa^{2/5}-1)(\nu^{6/5}-1)$: the power is determined solely by the interaction ratio $\kappa$ for a fixed compression ratio $\nu$. Fig. 5a shows release energy evolution over one cycle for various interaction strength ratios corresponding to step ${C}$ interaction strengths of 120, 160 and 200$a_{0}$, at a constant compression ratio $\nu=1.94$ and a particle number approximately 60% larger than in Fig. 2. Fig. 5b shows that the output power indeed increases with $\kappa$, with a departure from theoretical predictions at larger values of $\kappa$ a possible hint of beyond-mean-field behavior. These results emphasize the importance of interaction effects in the engine: Feshbach tuning is the key parameter controlling energy transfer between magnetic and optical fields. This power enhancement is completely decoupled from the boost to efficiency achieved through stronger compression, and from the power enhancement due to decreased cycle time.

In conclusion, we have realized an isentropic thermodynamic engine with a quantum degenerate working fluid and demonstrated that it outperforms a classical counterpart. Experimental measurements of engine performance for various values of control parameters and degrees of adiabaticity are in good agreement with both low-temperature analytics and approximation-free numerical simulations. This work opens up a variety of interesting directions for future exploration. These include optimizing performance with shortcuts to adiabaticity [6, 12, 7, 39], realizing a quantum Otto refrigerator [45, 46, 47, 48, 49], applying similar techniques to quantum heat engines involving trapped reservoirs of hot and cold atoms, investigating the role of criticality [8], and experimentally exploring the effects of entanglement on quantum thermodynamic engines [50, 51, 52].