# Tuning the performance of a micrometer-sized Stirling engine through
reservoir engineering

Niloyendu Roy 111 Corresponding author. email<EMAIL_ADDRESS>Chemistry
and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific
Research, Jakkur, Bangalore - 560064, INDIA  Nathan Leroux  Unité Mixte de
Physique CNRS/Thales, 91767 Palaiseau, France  A K Sood  Department of
Physics, Indian Institute of Science, Bangalore- 560012, INDIA  International
Centre for Materials Science, Jawaharlal Nehru Centre for Advanced Scientific
Research, Jakkur, Bangalore - 560064, INDIA  Rajesh Ganapathy 222
Corresponding author. email<EMAIL_ADDRESS>International Centre for
Materials Science, Jawaharlal Nehru Centre for Advanced Scientific Research,
Jakkur, Bangalore - 560064, INDIA  School of Advanced Materials (SAMat),
Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore -
560064, INDIA

###### Abstract

Colloidal heat engines are paradigmatic models to understand the conversion of
heat into work in a noisy environment - a domain where biological and
synthetic nano/micro machines function. While the operation of these engines
across thermal baths is well-understood, how they function across baths with
noise statistics that is non-Gaussian and also lacks memory, the simplest
departure from equilibrium, remains unclear. Here we quantified the
performance of a colloidal Stirling engine operating between an engineered
memoryless non-Gaussian bath and a Gaussian one. In the quasistatic limit, the
non-Gaussian engine functioned like an equilibrium one as predicted by theory.
On increasing the operating speed, due to the nature of noise statistics, the
onset of irreversibility for the non-Gaussian engine preceded its thermal
counterpart and thus shifted the operating speed at which power is maximum.
The performance of nano/micro machines can be tuned by altering only the
nature of reservoir noise statistics.

Experimental advances in nano/micro manipulation has made feasible the
realization of mesoscale heat engines with only a single atom
rossnagel2016single  or colloidal particle blickle2012realization ;
quinto2014microscopic ; martinez2016brownian ; ciliberto2017experiments ;
martinez2017colloidal  as the working fluid. Even while the functioning of
these engines is strongly influenced by fluctuations in the local environment
with parameters like work and efficiency becoming stochastic quantities, when
operating between equilibrium heat baths, their cycle-averaged performance
mirrors their macroscopic counterparts and standard thermodynamic relations
apply sekimoto1998langevin ; sekimoto2010stochastic ; esposito2010efficiency ;
seifert2012stochastic ; verley2014unlikely ; rana2014single . Recently,
Krishnamurthy et al. krishnamurthy2016micrometre  experimentally realized an
active stochastic heat engine by replacing the isothermal branches of a
Stirling cycle with isoactive ones. Here, a colloidal particle in a time-
varying optical potential was periodically cycled across two bacterial
reservoirs characterized by different levels of activity. Unlike in
equilibrium thermal baths where the displacement distribution of the colloid,
$\rho(x)$, is a Gaussian, in active reservoirs, it was non-Gaussian and heavy-
tailed krishnamurthy2016micrometre ; wu2000particle . These rare large
displacement events resulted in large work output and the efficiency of this
active engine was found to surpass equilibrium engines; even those operating
between thermal baths with an infinite temperature difference. Since the
metabolic activity of the bacteria could not be altered rapidly, this engine
was operated only in the quasistatic limit, i.e. for a cycle duration $\tau$
larger than the relaxation time of the colloid. Subsequent theoretical
calculations for the $\tau\to\infty$ limit posited that a departure from
equilibrium efficiencies requires noise not just with non-Gaussian statistics
but also with memory, a feature typical of active baths due to the persistent
motion of the particles zakine2017stochastic . In fact, when the bath noise is
non-Gaussian and white, an effective temperature $T_{eff}$ defined through the
variance of $\rho(x)$ is thought to act like a bona fide temperature
zakine2017stochastic ; fodor2018non  and engines operating between such baths
are expected to perform like equilibrium ones in the quasistatic limit.
Whether this similarity persists when $\tau$ is reduced and irreversibility
begins to set in is not known and is worth exploring since real heat engines
never operate in the quasistatic limit as here their power $P\to 0$. On the
experimental front, memoryless non-Gaussian heat baths are yet to be realised
and predictions even in the quasistatic limit remain untested.

Here we engineered a memoryless non-Gaussian heat bath and then constructed
and quantified the functioning of a colloidal Stirling heat engine operating
between such a bath and a thermal one for different $\tau$. In the quasistatic
limit, the performance of this non-Gaussian engine mirrored a classical
Stirling engine operating between thermal/Gaussian baths in agreement with
theoretical predictions. Strikingly, due primarily to differences in the noise
statistics of the baths, the small $\tau$ behaviour of these engines was quite
different. On lowering $\tau$, not only did the distribution of work done per
cycle, $\rho(W_{\text{cyc}})$, by the non-Gaussian engine become increasingly
negatively skewed, unlike the standard Stirling case where it remained
Gaussian, the onset of irreversibility for these two engines was also
different. Importantly, we demonstrate that even sans memory, changing the
nature of noise statistics of the reservoirs between which an engine operates
allows tuning its performance characteristics, specifically, the $\tau$ at
which the power goes through a maximum.

## Results

### Reservoir engineering by flashing optical traps

Our experimental scheme for reservoir engineering is elaborated in Figure 1a.
A polystyrene colloidal particle of radius $R=2.5$ $\mu$m suspended in water
is held in a harmonic optical potential, $U={1\over 2}k_{1}\langle
x^{2}\rangle$, created by tightly focusing a laser beam (1064 nm ALS-IR-5-SF,
Azur Light Systems France) through a microscope objective (Leica Plan
Apochromat 100X, N.A. 1.4, oil) that is also used for imaging the particle
(see methods). Here, $k_{1}$ is the stiffness of this primary trap, $x$ is the
displacement of the colloid from the centre of the optical trap and
$\langle\rangle$ denotes an average. At equilibrium, the trap stiffness can be
determined through the equipartition relation ${1\over 2}k_{1}\langle
x^{2}\rangle={1\over 2}k_{B}T$ where $k_{B}$ is the Boltzmann constant and $T$
is the bath temperature, which in our experiments is fixed at 300 K. As a
first step, we attempted to engineer a reservoir that mimicked a thermal bath,
i.e. with Gaussian noise statistics, but with a desired $T_{eff}$. To this
end, we imposed an additional noise on the colloidal particle along one
spatial dimension, here the $x-$axis (Figure 1a), from a second optical trap
of fixed intensity but with a time-dependent centre that was flashed at a
distance $\delta a(t)$ away from the primary one (Figure 1b). This was made
possible by using a second laser (Excelsior 1064 nm, Spectra Physics USA)
coupled to the microscope through a spatial light modulator (Boulder Nonlinear
Systems USA) and the flashing frequency was held fixed at $34$ Hz (see
Methods). Earlier reservoir engineering studies wherein the colloidal particle
experienced only the potential from the flashing trap found that when $\delta
a$ was drawn from a Gaussian distribution, the particle indeed behaved like
one in a thermal bath but at a $T_{eff}>T$ and furthermore, when $\delta
a(t)<R$, the trap stiffness also remained unaltered. berut2014energy ;
chupeau2018thermal . Here we adhered to the same protocol and further ensured
that the peak of the $\delta a$ distribution coincided with the centre of the
primary trap. Thus, the effective trap stiffness in our experiments
$k=k_{1}+k_{2}$, where $k_{2}$ is the stiffness of the flashing trap. Like in
a thermal bath, $\rho(x)$ of the trapped colloidal particle was a Gaussian
(solid circles in Figure 1d) and its power spectral density (PSD) a
Lorentzian, allowing us to determine $k_{2}$ and hence $T_{eff}$
chupeau2018thermal  (Supplementary Figure 1 and Supplementary Note 1). For the
$\delta a(t)$ profile shown in Figure 1b, the particle experienced a
$T_{eff}=1331$ K.

Engineering a memoryless non-Gaussian reservoir involved only a small tweak to
the manner in which the external noise was imposed on the colloidal particle.
The instantaneous $\delta a$ was now drawn randomly from a distribution with
zero mean and skew, as before, but with a high kurtosis (see Methods and
Supplementary Figure 2). Such a distribution has a narrow central region with
heavy tails. The flashing optical trap is thus mostly coincident with the
primary trap, thereby confining the particle strongly, and is occasionally
positioned a large distance away from the centre leading to a large excursion
by the particle (Figure 1c and Supplementary Movie). The overall noise
experienced by the particle is $\delta$-correlated as the thermal and imposed
noise are individually $\delta$-correlated. Under the influence of such a
noise, the corresponding $\rho(x)$ of the colloidal particle was also non-
Gaussian (hollow squares in Figure 1d). The PSD of the particle could be fit
to a Lorentzian over the dynamic range accessible and since all other
experimental parameters are held fixed, the roll-off frequency of the PSD was
also same as the Gaussian case (Supplementary Figure 3 and Supplementary Note
1). For an appropriate choice of the variance and kurtosis of the $\delta a$
distribution, we could engineer the $T_{eff}$ of the non-Gaussian bath, again
defined through the variance of $\rho(x)$, to be nearly identical to that in a
Gaussian bath (Figure 1d).

### Performing a Stirling cycle between engineered reservoirs

Armed with the capability to engineer reservoirs, we built a colloidal
Stirling engine operating between a hot non-Gaussian and a cold Gaussian bath
held at temperatures $T_{eff}^{H}=1824$ K and $T_{eff}^{C}=1570$ K,
respectively. We also compared the performance of this non-Gaussian engine
with a standard Stirling engine operating between engineered Gaussian baths
with similar effective temperature difference (see Supplementary Note 2). The
Stirling cycle we executed with the trapped colloid (Figure 1e), like in
previous studies blickle2012realization ; krishnamurthy2016micrometre ;
schmiedl2007efficiency , comprised of an isothermal compression (path ① - ②)
and expansion step (path ③ - ④) linked by two isochoric transitions (paths ② -
③ and ④ - ①). In the isothermal compression (expansion) steps, $k$ was
increased (decreased) linearly from $k_{min}=$2.5 pN$\mu$m-1 to $k_{max}=$ 2.7
pN$\mu$m-1 by changing $k_{1}$ alone. The isochroric transitions were near
instantaneous and occurred on millisecond time scales. We exploited the
ability to rapidly alter $T_{eff}$ and also the nature of noise statistics
through the SLM to explore engine performance over a range of $\tau$ which
spanned from 2 s to 32 s (see Methods).

### Elucidating the origins of irreversibility in the non-Gaussian Stirling
engine

The framework of stochastic thermodynamics provides a prescription for
calculating thermodynamic quantities like the work, power, and efficiency of
mesoscopic machines sekimoto1998langevin ; sekimoto2010stochastic ;
seifert2012stochastic ; schmiedl2007efficiency . The work done per cycle,
$W_{cyc}$, by the particle due to a modulation in the stiffness of the trap is
just the change in potential energy and is given by
$W_{cyc}=\int_{t_{i}}^{t_{i}+\tau}\frac{\partial U}{\partial k}\circ
dk\equiv\frac{1}{2}\int_{t_{i}}^{t_{i}+\tau}x^{2}\circ dk$. Here, the $\circ$
signifies that the product is taken in the Stratonovich sense and $t_{i}$ is
the starting time of $i^{\text{th}}$ cycle. Owing to its stochastic nature,
$W_{cyc}$ of the engine fluctuates from cycle-to-cycle and we quantified the
nature of these fluctuations through the probability distribution function
$\rho(W_{cyc})$. Figure 2a and b show $\rho(W_{cyc})$ at different $\tau$ for
the thermal and non-Gaussian Stirling cycles, respectively. Focusing on the
large cycle duration ($\tau=32$ s) first, we observed that $\rho(W_{cyc})$ is
a Gaussian for the thermal and also for the non-Gaussian cycles (circles in
Figure 2a and b). The experimentally calculated average work done per cycle,
$\langle W_{cyc}\rangle$, is negative indicating that the engine extracts heat
from the bath to perform work on the surroundings. Further, $\tau=32$ s
corresponds to the quasistatic limit for both since the value of $\langle
W_{cyc}\rangle$ is in excellent agreement with the theoretically calculated
quasistatic Stirling work output,
$W_{\infty}=k_{B}(T_{eff}^{C}-T_{eff}^{H})\ln\sqrt{\frac{k_{max}}{k_{min}}}$
(short solid horizontal lines in Figure 2c).

On lowering $\tau$, $\rho(W_{cyc})$ for the thermal Stirling engine remained a
Gaussian (Figure 2a) and $\langle W_{cyc}(\tau)\rangle\approx\langle
W_{cyc}(\tau=32\text{ s})\rangle$ (hollow circles Figure 2c). As expected of
such a distribution, $\langle W_{cyc}\rangle$ was the same as the most-
probable work $W^{*}$ \- the value of $W_{cyc}$ where $\rho(W_{cyc})$ is a
maximum (solid circles Figure 2c). For the non-Gaussian engine on the other
hand, on reducing $\tau$, $\rho(W_{cyc})$ became increasingly negatively
skewed (Figure 2b) and $W^{*}(\tau)$ also became increasingly positive (solid
squares Figure 2c). $\langle W_{cyc}(\tau)\rangle$ however, was marginally
smaller than $\langle W_{cyc}(\tau=32\text{ s})\rangle$. (hollow squares
Figure 2c). We note that the work done by a thermal Stirling engine at a
finite $\tau$ is given by the empirical relation schmiedl2007efficiency ;
blickle2012realization

$W(\tau)=W_{\infty}+W_{diss}\equiv W_{\infty}+\frac{\Sigma}{\tau}$ (1)

where, $W_{diss}$ is the dissipative work which accounts for the particle's
inability to fully explore the available phase space when $k$ is rapidly
lowered during the hot isotherm and $\Sigma$ is a constant also called the
irreversibility parameter. Since $W_{diss}$ is a positive quantity as per
definition, at small enough $\tau$, the overall work done itself can be
positive indicating the stalling of the engine. Clearly there is no buildup of
irreversibility for the thermal engine as $\tau$ is lowered since $\langle
W_{cyc}(\tau)\rangle\equiv W^{*}(\tau)\approx W_{\infty}$, while for the non-
Gaussian one, there is, even if only in the most-probable sense ($\langle
W_{cyc}(\tau)\rangle\approx W_{\infty}<W^{*}(\tau)$), and the engine stalls
for $\tau\leq 10$ s. We also found excellent agreement between equation (1)
and our data allowing us to determine $\Sigma=0.11\text{ }k_{B}T_{eff}^{C}$
(red solid line in Figure 2c).

The observed behaviour of the non-Gaussian engine can be easily rationalized
by analyzing the relaxation of the particle in the hot isotherm at the level
of an individual cycle. For the particle to fully sample the statistical
properties of the non-Gaussian hot reservoir, it should also experience the
occasional large kicks that displaces it far from the center and not just the
ones that predominantly keep it confined close to it. As $\tau$ is lowered, in
most cycles, the probability that the particle encounters a large kick in the
isothermal expansion step also becomes increasingly small. Due to the
incomplete exploration of the available phase volume in these cycles, less
useful work is performed and $W^{*}(\tau)$ lifts off with decreasing $\tau$.
In a few cycles, where these large kicks are present, anomalously large work
is done by the engine and this results in $\rho(W_{cyc})$ being negatively
skewed. When an adequate number of cycles, which has to be increased when
$\tau$ is lowered, have been performed, all features of the noise are sampled
and the engine operates like one in the quasistatic limit in an average sense
with $\langle W_{cyc}(\tau)\rangle\to W_{\infty}$ (Figure 2c). This inference
can be strengthened by quantifying the equilibration of the particle over a
fixed, but limited, number of cycles for all $\tau$. In Figure 2d, we show
${k\langle x^{2}\rangle\over k_{B}T_{eff}^{H}}$ calculated over a small window
in the middle of the hot isotherm and averaged over $N=50$ cycles for the
thermal (squares) and the non-Gaussian engine (circles). Despite $N$ being
small, ${k\langle x^{2}\rangle\over k_{B}T_{eff}^{H}}$ is close to 1 at all
$\tau$ for the thermal engine implying that it is truly in the quasistatic
limit, while for the non-Gaussian engine this is the case only at large $\tau$
with a clear violation of quasistaticity setting in for $\tau\leq 10$ s.
Evidently, for a non-Gaussian engine $W^{*}(\tau)$, and not $\langle
W_{cyc}(\tau)\rangle$, is a more precise metric for performance.

### Tuning the performance of a Stirling engine through memoryless non-
Gaussian noise

We now examined how differences in the nature of noise-statistics affected the
power output of our engines. In the quasistatic limit $P(\tau)=-{\langle
W_{cyc}(\tau)\rangle\over\tau}\to 0$ since $\tau\to\infty$, while at high
cycle frequencies $W_{diss}$ is large and $P$ is once again small. At
intermediate $\tau$, however, these effects compete resulting in a maximum in
$P$ and this is a feature of both macroscopic and mesoscopic engines
blickle2012realization ; curzon1975efficiency . Figure 3a shows the most-
probable power, $P^{*}(\tau)={-W^{*}(\tau)\over\tau}$, for the Gaussian
Stirling engine (circles) and the non-Gaussian one (squares). Since for the
thermal engine, over the range of $\tau$ studied $\Sigma=0$, $P^{*}(\tau)$,
which is same as same as $P(\tau)$, only increases monotonically on lowering
$\tau$ and does not exhibit a maximum. Whereas for the non-Gaussian engine, on
reducing $\tau$, $P^{*}(\tau)$ first appears to increase slightly, crosses
zero for $\tau<10$ s and then becomes more negative indicating stalling of the
engine. We emphasize that for a Stirling cycle executed under conditions
identical to that in Figure 1e but where the non-Gaussian reservoir is
replaced by a Gaussian one with the same $T_{eff}^{H}$, the maximum in $P$ is
expected to be at a $\tau$ that is lower than even the Gaussian engine studied
here (see Supplementary Note 3). This clearly shows that, even sans memory,
altering the statistical properties of the noise bath alone allows for tuning
the performance characteristics of mesoscopic heat engines.

For a complete understanding of the operation of the non-Gaussian engine, we
calculated its efficiency at various $\tau$ and benchmarked it with the
thermal engine. Conventionally, the efficiency, $\varepsilon={W_{cyc}\over
Q}$, where $Q$ is the heat absorbed by the particle when it is in contact with
the hot reservoir. $Q$ is the sum of the isochoric heat during the transition
from state point ② to ③, $Q_{2\to 3}=-{1\over
2}k_{max}(T_{eff}^{H}-T_{eff}^{C})$ and the isothermal heat during transition
from ③ to ④, $Q_{3\to 4}=\int_{(3)}^{(4)}\frac{\partial U}{\partial
x}\dot{x}dt=W_{H}+Q_{boundary}$. Here,
$W_{H}=\frac{1}{2}\int_{(3)}^{(4)}x^{2}\circ dk$ is the work done in the hot
isotherm and $Q_{boundary}=-\frac{1}{2}[k(t)x^{2}(t)]_{(3)}^{(4)}$. For the
non-Gaussian engine, we naturally chose $W^{*}$ instead of $W_{cyc}$ and
defined the most-probable efficiency $\varepsilon^{*}={W^{*}\over{\langle
W_{H}\rangle+\langle Q_{boundary}\rangle+\langle Q_{isochoric}\rangle}}$ (See
Supplementary Note 4). For the thermal engine, the experimentally determined
$\varepsilon^{*}$ (black circles in Figure 3b) hovers around the theoretically
calculated saturation Stirling efficiency
$\varepsilon_{Sat}=\varepsilon_{c}[1+{\varepsilon_{c}\over\ln(k_{max}/k_{min})}]^{-1}$
(solid blue line). Here, $\varepsilon_{c}=1-{T_{eff}^{C}\over T_{eff}^{H}}$ is
the Carnot efficiency. Whereas for the non-Gaussian engine,
$\varepsilon^{*}(\tau)$ converges to $\varepsilon_{Sat}$ only at large $\tau$
(red squares in Figure 3a). When $\tau$ is reduced, $\varepsilon^{*}(\tau)$
drops and becomes negative for $\tau<10$ s indicating stalling of the engine.
Of particular importance in the operation of real heat engines is the
efficiency at maximum power $\varepsilon_{Max}$, which for the non-Gaussian
engine is at $\tau=10.3$ s with $\varepsilon_{Max}=0.025$. Most remarkably,
this value in excellent agreement with theoretically predicted Curzon-Ahlborn
efficiency, $\varepsilon_{CA}={\varepsilon_{Sat}\over
2-\alpha\varepsilon_{Sat}}=0.026$ curzon1975efficiency ;
schmiedl2007efficiency . In our experiments, $\alpha\sim 0$ is a constant
calculated from the irreversibility parameters corresponding to the work done
in the hot and cold isotherms (Supplementary Figure 4 and Supplementary Note
5). While it is known that $\varepsilon_{Max}\approx\varepsilon_{CA}$ for both
macro and mesoscopic thermal engines, ours is the first observation of this
being the case even for a non-Gaussian engine.

## Discussion

Collectively, our experiments show that a micrometer-sized Stirling engine
operating between a Gaussian and a non-Gaussian bath, without memory, indeed
performs like a conventional engine in the quasistatic limit as anticipated by
theory. On lowering the cycle times, the buildup of irreversibility in the
engine, due entirely to the non-Gaussian nature of noise, results in work
distributions that become increasingly negatively skewed unlike a thermal
engine where it remains Gaussian. Strikingly, this noise-induced enhancement
of irreversibility modulates the performance characteristics of the non-
Gaussian engine in a manner similar to predictions by Curzon and Ahlborn for
thermal engines where irreversibility sets in purely due to the rapid change
of the control parameter. Our experiments thus reveal a new strategy for
optimizing the performance of a mesoscale engine by tuning only the nature of
noise statistics. Importantly, the ease with which the noise can be engineered
and also applied locally, i.e. on the particle scale, in our approach presents
advantages over other reservoir engineering methods where this can prove to be
difficult, if not impossible martinez2013effective ; martinez2017colloidal .
This should now make feasible the experimental realization of new stochastic
machines like the non-Gaussian and the Buttiker–Landauer ratchet
luczka1997symmetric ; buttiker1987transport ; landauer1988motion .

## Methods

### Experimental set-up for Reservoir Engineering

In order to impart additional noise into the trapped colloid, a secondary
optical trap was flashed along a line passing through the time-averaged centre
of the particle at variable distances from the same. This was achieved by
coupling a second laser (Excelsior 1064 nm, Spectra Physics USA) to the
microscope which is reflected from a Spatial Light Modulator (Boulder
Nonlinear Systems USA). The Spatial Light Modulator (SLM) contains a
$512\times 512$ array of shiny electrodes covered with a transparent liquid
crystal layer so that an electric potential modulation across the electrodes
imposes an additional phase pattern on the incident beam. We interfaced the
SLM to a computer so that a series of desired phase patterns can be fed to the
SLM at a fixed frequency of $34Hz$. This enabled us to dynamically reconfigure
the position of the first order diffraction spot by applying a series of
linear diffraction grating patterns with varying periodicity which is
controlled through a computer. We blocked the zeroth order spot so that only
the first order spot is incident on the back of the microscope objective
resulting in a flickering optical trap in the vicinity of the tweezed
colloidal particle.

### Image acquisition and processing

Images of the trapped colloid was captured at $250$ Hz using a fast camera
(Photron 500K-M3) attached to the microscope. Position of particle's centre in
each frame was located at the subpixel level using the particle tracking codes
by R. Parthasarathy parthasarathy2012rapid . This allowed us to find the
particle's position within an accuracy of $5$ nm.

### Non-Gaussian Reservoir Engineering

For engineering the non-Gaussian reservoir, $\delta a$ were chosen from a
$\delta-correlated$ distribution with zero mean and skewness but an extremely
high kurtosis of 50. One such distribution with standard deviation of
$\sigma=0.28\mu m$ is represented in Supplementary Figure 2(b). To create this
distribution, we first generate two highly asymmetric distributions $\delta
a_{L}$ and $\delta a_{R}$ (Supplementary Figure 2(a)) with a standard
deviation of $0.28\mu m$, a kurtosis of $60$ and a skewness of $-6.5$ for
$\delta a_{L}$ and $+6.5$ for $\delta a_{R}$ through Pearson's protocol in
MATLAB. Next we add/subtract a suitable number to $\delta a_{L}$ and $\delta
a_{R}$ so that their peaks coincide at zero. Then we take union of $\delta
a_{L}$ and $\delta a_{R}$ and randomly permute all the elements to finally
obtain the set of $\delta a$. In order to realize a desired effective
temperature with such a noise, the standard deviation of $\delta a$ is
optimized. It should be noted that heavy tails rise due to extremely rare
events that can only be captured with a huge statistics. Since we are limited
by a flashing frequency of $34Hz$, it is not possible to completely sample the
statistics with in one isotherm even for the largest $\tau$. To address this
issue, the engine was cycled enough number of times (depending on $\tau$) so
that the collection of all the hot isotherms exhausts all the rare events.

### Instantaneous isochoric transitions

The isochoric transitions ②$\rightarrow$③ and ④$\rightarrow$① shown in Figure
1e of the main text is realised by changing the statistics and the variance of
$\delta a$-distribution. The transition ②$\rightarrow$③ is realised by
changing the $\delta a$ distribution from a Gaussian resulting in
$T_{eff}=1570K$ to a non-Gaussian producing $T_{eff}=1824.3K$ while the
transition ④$\rightarrow$① is realised by the reverse. Since the secondary
laser is diffracted by a computer controlled SLM, the distribution from which
$\delta a$s are chosen can be altered in $1/34$th of a second. Thus the
particle is decoupled and coupled from one engineered reservoir to the other
in less than $33ms$ which is almost negligible even in compared to the lowest
cycle time and hence instantaneous.

## Acknowledgements

N.R. thanks Dr. Sudeesh Krishnamurthy for fruitful discussions. N.R. thanks
Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR) for
financial support. AKS thanks Department of Science and Technology (DST),
Govt. of India for a Year of Science Fellowship. RG thanks JNCASR for
financial support.

## Author Contributions

N.R., N.L., A.K.S. and R.G. designed experiments. N.R., N.L. and R.G. devised
experimental procedures. N.R. performed experiments and carried out data
analysis. N.R. and R.G. wrote the paper with inputs from all N.L. and A.K.S.

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Figure 1: Experimental realization of a non-Gaussian Stirling heat engine. a
The big red spot represents the primary optical trap and the small red spots
represent the secondary flashing optical trap at different time instances
$t_{1}$, $t_{2}$ and $t_{3}$. b and c show the distance $\delta a(t)$ from the
primary trap at which the secondary trap was flashed as a function of $t$ for
engineering a Gaussian and a non-Gaussian reservoir, respectively. d shows the
probability distribution of particle displacements, $\rho(x)$, for the
engineered Gaussian/thermal (solid blue circles) and the non-Gaussian
reservoir (red hollow squares) for a nearly identical $T_{eff}$. e shows a
quintessential Stirling cycle between a hot non-Gaussian bath at
$T_{eff}^{H}=1824$ K and a cold Gaussian reservoir with $T_{eff}^{C}=1570$ K.
The trap stiffness $k$ is varied linearly in the expansion/compression steps.
Having a fixed primary trap and a second flashing optical trap, as opposed to
just the latter, prevented the trapped particle from escaping the trap and
allowed for long experiments. $\rho(x)$ of the particle measured at the four
state points labeled ① to ④ is also shown. The black lines are Gaussian fits.
Figure 2: Buildup of irreversibility in the non-Gaussian Stirling engine at
finite $\tau$. a and b show the probability distribution of work done per
cycle $\rho(W_{cyc})$ for the Gaussian and the non-Gaussian engine,
respectively, for $\tau=32$ s (blue circles), $10.6$ s (red triangles) and
$5.6$ s (black squares). Solid lines represent corresponding Gaussian fits to
the data. c Red hollow and solid squares show the average work done per cycle
$\langle W_{cyc}\rangle$ and the most-probable work $W*$, respectively, for
the non-Gaussian engine at various $\tau$. The red solid line is a fit to
Equation 1. Black hollow and solid circles show $\langle W_{cyc}\rangle$ and
$W^{*}$ respectively for the thermal/Gaussian engine. At large $\tau$, the
experimentally calculated work for these engines agrees with theoretically
calculated quasistatic work $W_{\infty}$ indicated by the Red and Black short
horizontal lines for the non-Gaussian and Gaussian engine respectively. d Red
squares (black circles) represent the ratio $k\langle
x^{2}\rangle/k_{B}T_{eff}^{H}$ calculated at the midpoint of the hot isotherm
of the non-Gaussian (Gaussian) engine at various $\tau$. The horizontal line
indicates the equilibrium condition, which is strongly violated inside the
shaded grey region, in case of the non-Gaussian engine. Figure 3: Quantifying
the performance of a non-Gaussian Stirling engine. In a, red squares (black
circles) show the most-probable power $P^{*}$ of the non-Gaussian (Gaussian)
engine at various $\tau$. $P^{*}$ increases slightly and then rapidly falls
for the non-Gaussian engine for $\tau\leq 10.6$ s. The red solid line is
calculated from the fit to Equation 1 and is overlaid on the experimental
data. b Red squares (black circles) represent the most-probable efficiency
$\varepsilon^{*}$ of the non-Gaussian (Gaussian) engine at various $\tau$. The
blue solid lines indicate the theoretically calculated saturation Stirling
saturation, $\varepsilon_{Sat}$. Efficiency $\varepsilon_{max}$ just before
the rapid drop in power ($\tau=10.6$) of the non-Gaussian engine agrees with
the Curzon-Ahlborn efficiency $\varepsilon_{CA}$. Note that the black vertical
line through the first data point (smallest $\tau$) is a portion of a large
error bar. The error bars at other $\tau$ values are smaller than the symbol
size.


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