Extreme-temperature single-particle heat engine

TL;DR

The paper demonstrates an extreme-temperature, underdamped single-particle heat engine using a charged microparticle levitated in a Paul trap and driven by a synthetic, spatially varying heat bath created with noisy electric fields. A Langevin-based model incorporating multiplicative, position-dependent noise yields a Fokker-Planck equation with $D(z)=D_0+D_1(z-z_0)+D_2(z-z_0)^2$, which accurately captures the observed large heat/work fluctuations and the nontrivial diffusion behavior. The analysis reveals wide heat-distribution tails, stochastic efficiencies exceeding 100%, and a breakdown of equipartition due to the $D_2$ term, highlighting the importance of multiplicative noise in microscale thermodynamics and biological-relevant transport. The experimental platform enables exploration of position-dependent diffusion and non-equilibrium energetics at mega-Kelvin temperatures, with potential implications for nanoscale engines and biological processes.

Abstract

There are many exotic thermodynamic processes that are hard to study in nature. Here, we synthesize a structured environment to explore the extremes of thermodynamics. We present an engine running at extreme temperatures of above ten Mega-Kelvin. Our underdamped engine is realised by electrically levitating and controlling a charged microparticle in vacuum. Giant fluctuations are observed in the engine's heat exchange with the environment, while its efficiency shows stochastic events where more work is performed by the engine than heat consumed. Moreover, the non-uniformity of the synthetic environment leads to the particle experiencing position dependent diffusion, a critical phenomenon in microscale biological processes. We theoretically account for the effects of multiplicative noise and find excellent agreement with the observed behavior.

Figures (7)

• Figure 1: Schematic of the single particle engine.a) A charged silica microparticle (image of the motion in blue/black, recorded by an event based camera Ren2022) is levitated within a linear Paul trap under vacuum conditions. Coaxial endcap control electrodes provide harmonic confinement along the $z$-axis, with a frequency that can be varied by changing the voltages $U_0$. An additional fluctuating voltage $U_T$ with white-noise statistics applied to one control electrode generates a spatially varying synthetic heat bath (red lines, amplitude qualitatively indicates noise strength in space), with temperatures in excess of $10^7$ K. b) Probability distribution function of particle position in the $y-z$ plane for a particle in hot (red, $T_h~=~2.7\times10^6$ K) and cold (blue, $T_c~=~1.2\times10^5$ K) baths.
• Figure 2: Engine cycle with position dependent diffusion.a) A measured position trajectory (black trace) of a levitated microparticle undergoing a Stirling engine cycle (green). In the isochoric heating step the temperature is changed from $T_\mathrm{c}$ (blue shaded region) to $T_\mathrm{h}$ (red shaded region). The trap frequency is subsequently changed linearly from $f_1 = 341.4 \pm 0.1$ Hz to $f_2 = 316.6 \pm 0.1$ Hz in a 10 s ramp, realizing isothermal expansion. b) Velocity variance $\sigma_v$ (coloured traces) over 1000 cycles with a time-bin of 1 ms. Hot bath temperatures indicate average temperatures $T_\mathrm{h} = (T^\mathrm{h}_1 + T^\mathrm{h}_2)/2$, where the effective temperatures at the two trap frequencies are defined as $T^\mathrm{h}_{1,2} = \frac{m}{k_\mathrm{B}} \, \sigma_v(f_{1,2})$. Note that here $\sigma_v$ denotes the variance, $\langle v^2 \rangle - \langle v \rangle^2$, not the standard deviation $\sqrt{\langle v^2 \rangle - \langle v \rangle^2}$. Solid black lines are obtained by numerically solving equation \ref{['eq:fokkerplanck']}, see Supplemental Material SM. The white dashed line indicates the predicted variance for standard Brownian motion with temperature $T^\mathrm{h}_{1}$ for the largest noise magnitude. The observed deviation of the levitated particle's velocity variance from standard Brownian motion is a clear indication of position dependent diffusion.
• Figure 3: Stochastic engine heat distributions.a) Measured distribution of the heat $Q$ the levitated particle exchanges with the cold (blue) and hot (red) environment, respectively, as a function of temperature ratio $T_{\mathrm{h}}/T_{\mathrm{c}}$. The highly stochastic nature of the heat exchange is evident in the wide range of heat values. Distributions at the highest temperature ratio have been truncated for clarity. Negative (positive) heat values signify energy is transferred from (to) the particle's motion to (from) the bath. The mean of these experimental distributions are marked with a black circle, with the theoretical prediction (Eq. (\ref{['eq:heat_avg']})) indicated by a dashed line. Values of heat reach $1.5 \times 10^{-15}$ J, equivalent to $\sim900\,k_{\mathrm{B}}T_{\mathrm{c}}$, much larger than previous single-particle heat engines ($<1\,k_{\mathrm{B}}T_{\mathrm{c}}$Martinez2016Li2024) due to the extreme temperatures involved. b) A pair of heat distributions at $T_{\mathrm{h}}/T_{\mathrm{c}} = 3.3$. The stochastic nature is particularly pronounced, as evidenced by the negative values of the particle's heat absorption from the hot bath (red), i.e. the particle sometimes dumps heat into the hot bath, and vice versa cools the cold bath (blue, positive values). The experimental data is compared to predictions based on standard Brownian motion (dotted lines) and our model with position-dependent diffusion (solid lines, Eq. \ref{['eq:heat_avg']}), the latter showing better agreement with the data.
• Figure 4: Engine efficiency and power over $T_{\mathrm{h}}/T_{\mathrm{c}}$.a) The measured efficiency $\eta$ (green circles) agrees well with the theoretical prediction (shaded green line includes parameter uncertainties). Inset shows the histogram of the single-cycle efficiencies, $\eta_s = W/Q_h$, i.e. the work/heat-ratio calculated for each engine cycle realisation, at temperature ratio 3.3. The histogram highlights the highly stochastic nature of the single-cycle efficiency. Logarithmically spaced bins are used to clearly represent the full range of data, with positive and negative values separated by the grey dashed line. (Note, that $\eta = \frac{\langle W \rangle}{\langle Q_h \rangle} \neq \langle \eta_s \rangle = \langle \frac{W}{Q_h} \rangle$, due to the non-linear nature of the efficiency). Efficiency distributions with a similar bimodal shape have been discussed in Refs. Martinez2016Polettini2015. b) Measured average power $P$ (purple circles) compared to theoretical curve (purple shaded region). In contrast to the saturating efficiency $\eta$, the power $P$ continues to increase with increasing temperature ratio $T_{\mathrm{h}}/T_{\mathrm{c}}$.
• Figure 5: Image of custom built Paul trap. The custom built linear Paul trap sits within metal shields to protect it from stray fields. The coaxially aligned endcap control electrodes have additional shielding to minimize cross-talk between the outer trapping and the inner control electrodes.