# Laser-induced heating for the experimental study of critical Casimir forces
with optical trapping.

I. A. Martínez<EMAIL_ADDRESS>Ecole Normale Supérieure de Lyon,
Laboratoire de Physique , CNRS. UMR5672, 46, Allée d’Italie, 69364 Lyon,
France  LCP Group, ELIS Department, Ghent University, Technologiepark 126,
9052 Gent, Belgium  A. Petrosyan  Ecole Normale Supérieure de Lyon,
Laboratoire de Physique , CNRS. UMR5672, 46, Allée d’Italie, 69364 Lyon,
France  S. Ciliberto  Ecole Normale Supérieure de Lyon, Laboratoire de
Physique , CNRS. UMR5672, 46, Allée d’Italie, 69364 Lyon, France

###### Abstract

Critical Casimir interactions represent a perfect example of bath-induced
forces at mesoscales. These forces may have a relevant role in the living
systems as well as a role in the design of nanomachines fueled by
environmental fluctuations. Since the thermal fluctuations are enhanced in the
vicinity of a demixing point of a second-order phase transition, we can
modulate the magnitude and range of these Casimir-like forces by slight
changes in the temperature. Here, we consider two optical trapped colloidal
beads inside a binary mixture. The Casimir interaction is controlled by
warming the mixture by laser-induced heating, whose local application ensures
high reproducibility. Once this two-particle system is warmed, the critical
behavior of different observables allows the system to become its self-
thermometer. We use this experimental scheme for analyzing the energetics of a
critical colloidal system under a non-equilibrium-driven protocol. We quantify
how the injected work can be dissipated to the environment as heat or stored
as free energy. Indeed, our system allows us to use the fluctuation theorems
framework for analyzing the performance of this critically driven toy model.
Our work paves the way for future experimental studies on the non-equilibrium
features of bath-induced forces and the design of critically driven
nanosystems.

##  I Introduction

Temperature is a physical quantity that defines the amount of energy a system
stores. In particular, Brownian motion observed in the mesoscopic systems is
an intrinsic feature of the bath temperature. However, controlling temperature
in the micrometrical scale in micromanipulation setups is far from standard.
The usual approach consists in working with macroscopic thermal baths which
modify the temperature of the microscopic system, although light-induced local
heating has been applied either directly [1] or by using highly absorbing
spots [2]. Nevertheless, the protocols are limited to stationary regimes [3]
or to study the thermalization after energy quenches [4].

A non-trivial example of thermal fluctuations with extreme sensitivity to
temperature changes is liquid mixture close to its critical demixing point. An
upcoming second order phase transition enhances the thermal fluctuations by
modifying the correlation length $\xi$ and the relaxation time $\tau$ of the
fluctuations-field $\phi$ of the fluid. Once the liquid approaches its
critical temperature $T_{c}$, both parameters of the fluctuations, $\xi$ and
$\tau$, diverge from their intrinsic values $\xi_{0}$ and $\tau_{0}$ following
a universal scaling, $\xi\approx\xi_{0}(\Delta T/T_{c})^{-\nu}$and
$\tau\approx\tau_{0}(\Delta T/T_{c})^{-\nu z}$ where $\nu$ and $z$ are the
static and dynamic exponents respectively and $\Delta T=T_{c}-T$ is the
distance to the criticality (from now critical distance). Critical binary
liquid mixtures have been experimentally tested to produce Casimir-like forces
between microscopic objects (critical Casimir forces, CCF) [5], to transfer
energy in multi-particle systems[6], to react to the bacterial swimming[7] or
to generate self-assembly [8], and they have been proposed to induce non-
Gaussian fluctuations [9], to react to the chemical affinity of the tracers by
changing their viscosity[10] or to play a fundamental role in the
intracellular dynamics [11].

Critical Casimir force (CCF) is a paradigmatic case of bath-induced force. In
the vicinity of a critical demixing point, the confinement of the thermal
fluctuations produces a force between the confining walls. The sense of force
depends on the symmetry of the boundary conditions, attractive in the case of
symmetric boundaries, and repulsive in the case of antisymmetric ones. In the
case of two identically coated spheres of radius $R$ with central positions
$x_{1}$ and $x_{2}$ acting as confining walls, and under the Derjaguin
approximation, Casimir-like potential can be written as:

$U_{\rm cas}(d,\xi)=-\frac{AR\pi kT}{\xi}\exp\left(-\frac{d}{\xi}\right)$ (1)

where $d=x_{2}-x_{1}-2R$ is the distance between the surfaces, $kT$ is the
thermal energy, and $A\approx 1.3$ is a numerical constant from the numerical
approximation [12]. Notice how the temperature can be extracted from $U_{\rm
cas}$ since the correlation length $\xi$ is a function of the distance to the
criticality.

This article presents a method for accurately adjusting the interaction force
between colloids using a laser source. We demonstrate that by gradually
modulating the intensity of the laser on the colloidal solution, we can alter
the force acting on two beads trapped in optical tweezers and change the
energy fluxes. This method holds the potential for understanding the impact of
bath-induced forces, which may be crucial in membrane interactions, and for
constructing nano-devices that utilize forces tunable by slight temperature
changes. Additionally, it opens avenues for quantitative investigations into
the non-equilibrium characteristics of critical baths.

##  II Experimental system.

Figure 1: Sketch of the system. a) The heating laser raises the local
temperature of an amount ($\delta T$) over the ground temperature
($T_{g}\approx 28{\rm{}^{o}C}$). The width of the heating beam (40$\mu$m) is
larger than the region of interest (ROI) around the particles (size $\simeq
15\mu$m). The power of the heating laser is slowly changed to access different
values of $\delta T$. The temperature field is assumed to be homogeneous and
with no thermophoretic flows within the ROI. b) Two particles are held in two
independent optical traps (1 and 2). The two particles (2.5 microns radius
each) interact via the critical Casimir force produced by confining the
concentration field fluctuations between the particle surfaces. c) The two
particles are held in two independent tweezers at a distance of $\Delta x_{\rm
trap}=5570{\rm nm}$ between them. We aim at extracting independent information
from the system by using two parameters of each video frame. The positions,
$x_{1}$ and $x_{2}$, of the particles are tracked while we analyze the
dispersion $\delta I^{2}=\sum_{i,j}(I_{i,j}-I)^{2}$ of the intensity at the
four corners ($I_{ij}$, 30px x 30 px each) being $I$ the mean intensity).

Our critical bath consists of a micelle-solvent mixture, ${\rm C_{12}E_{5}}$
in water, at its critical concentration whose demixing point is at
$T_{c}\approx 30.1$°C, see Methods. The sample has an intrinsic correlation
length of 1.3 nm which is about 5 times larger than previously studied liquids
mixtures such as lutidine-water. This feature enhances the critical features,
which become non-negligible even at relatively long critical distances. The
fluid mixture is contained in a transparent cell of thickness $40\mu m$.

Inside the mixture, two beads (P1 and P2) are held by two optical tweezers (T1
and T2) at a distance of $\Delta x_{T}$ between their equilibrium positions
(see Fig.1). The two beads are equal (Silica, 5$\mu$m diameter), so the
boundary conditions are symmetric and the critical Casimir force is
attractive. Besides the trapping laser beam ($\lambda=1064nm$) an extra beam
of wavelength 1550 nm is sent into the cell (see Fig.1) to modulate the
temperature around the particles by the light absorbed by the mixture. This
beam has a negligible effect on the trapping strength (see Supplementary Video
1) because it has a focal depth of 400 microns and a diameter of $40\mu$m,
much larger than the region of interest (ROI) around the two particles whose
size is about $15\mu m$ Its power $P_{h}$ can be changed from $0$ to $200$mW.
Being the cell kept at $T_{g}\simeq 28.00^{\rm o}$C, the critical distance
$\Delta T$ in the heated volume is about $2.0$ K. The cell is shined by white
light to image the two beads on a CCD camera (see Fig. 1c). The positions
$x_{1},x_{2}$ of the two-particle centers are recorded at a sampling frequency
of $400$Hz at different values of $P_{h}$.

Furthermore, to obtain an independent measure of the critical distance without
including extra devices, we analyze the variance $\delta I^{2}$ of the
illumination light fluctuations in the corners of the image in the spirit of
dynamic light scattering Fig. 1c). Since the micelle-rich phase has a
different refractive index than the micelle-poor phase, once the transition is
crossed we expect a huge change in $\delta I^{2}$ as a function of $\Delta T$,
that allows us to distinguish between the two phases. Furthermore, since the
refractive index is expected to depend on the critical distance, we aim at
quantifying the behavior of $\delta I^{2}(\Delta T)$.

##  III Results

### CCF as nanothermoter.

In this section, we demonstrate the reproducibility of CCF with light-induced
heating and how it can be used as the system’s self-thermometer. In the
physical system described before, the trap distance $\Delta x_{T}=5570{\rm
nm}$ is kept fixed while we slightly change the power of the heating laser
$P_{h}(t)$. The following protocol is time-symmetrical and it has a triangular
shape with a constant increase of the irradiated power of $v_{\rm
heat}=90\mu{\rm Ws}^{-1}=\max(P_{h})/\tau_{\rm heat}$, where
$\max({P_{h}})=180{\rm mW}$ is the maximum laser power achieved and
$\tau_{h}=2000{\rm s}$ is half the duration of the heating cycle, see the
black solid line in Fig. 2a) as a guide to the eye. The trajectory of the two
particles is followed at a frequency of 400 Hz which is more than enough since
the trap stiffness is small ($\kappa=0.5{\rm pN\mu m^{-1}}$) hence the beads
relaxation time is about $80$ milliseconds.This time is much shorter than
$\tau_{h}$ showing that the heating protocol is quasi-stationary, i.e. the
system’s temperature is always close to equilibrium. In Fig. 2a), we show the
time-evolution of the observables extracted from the analysis of $\delta
I^{2}$ (left axis, blue line) and the beads trajectory (mean distance between
the walls $d$ over a 20 seconds block, right axis, red lines). Notice that the
total time of the experiment is above one day with the same set of particles.
Since the time evolutions of $d$ and $\delta I^{2}$ are correlated with $P$,
we performed an average conditioned to the $P$ values on all of the cycles.
The results $\langle d\rangle$ and $\langle\delta I^{2}\rangle$ of this
conditional average are plotted in Fig. 2b) as a function of $P$. The variance
$\langle\delta I^{2}\rangle$ has remarkable reproducibility and stability as a
function of $P_{h}$ and shows that the scattering increases when the critical
point is approached.

Figure 2: Critical interactions as local nanothermometers. a) Time-evolution
of $\delta I^{2}$ (blue, left axis) and $d=x_{2}-x_{1}-2R$ (red, right axis).
The black solid line is a guide to the eye without a y-scale showing the power
of the heating laser $P(t)$. Notice the reproducibility over tens of
repetitions over more than 24 hours of the experiment. b) $\langle\delta
I^{2}\rangle$ (blues circles and line) and $\langle d\rangle$ (red points) are
plotted as a function of the input heating power $P_{\rm heat}$. For
increasing power, the mean distance decreases showing the appearance of an
attractive force, whereas $\langle\delta I^{2}\rangle$ increases approaching
the critical point. c) The attraction potentials measured at two different
heating powers $P_{h}=0$ mW (blue) and $P_{h}=100$ mW (red) are plotted as a
function of $d$. The correlation length is extracted from the critical Casimir
potential using Derjaguin approximation (blue empty squares), d) The measured
$\xi$ is plotted as a function of $P$ (blue dots). Since
$\xi/\xi_{0}=\left(\Delta T/T_{c}\right)^{-\nu}$, we can infer a critical
distance $\Delta T$ for each value of the input power (red points, right axis)
obtaining a heating rate of $C_{h}=(4.87\pm 0.20){\rm mKmW^{-1}}$.

In Fig. 2b), we see that the mean distance $\langle d\rangle$ between the two
beads decreases for growing $P_{h}$ demonstrating the appearance of an
attractive force. The appearance of this force can be understood by measuring
the probability distribution $\rho(d)$ hence the potential
$U(d)\propto\ln(\rho(d))$, which is plotted in Fig. 2c) for two values of
$P_{h}$. At $P_{h}=0$ mW, the equilibrium position defined by the Casimir
force and the electrostatic repulsion is small compared to that of the optical
trap. Instead at $P_{h}=100$ mW, i.e. $\Delta T\to 0$ the combination of the
different interactions produces an energy potential landscape with two
comparable equilibrium positions. For the sake of simplicity during the
reading, we will call the optical trap and the Casimir wells ’OW’ and ’CW’
respectively. Since the profile of the critical force depends on $\Delta T$,
the occupation of each well also depends on $\Delta T$ following the detailed
balance between the wells. This explains the behavior of $\langle d\rangle$ as
a function of $P_{h}$ in Fig. 2b). The mean value between surfaces $\langle
d\rangle$ gets progressively closer from $\langle d\rangle=500$ nm up to
saturation at 110 nm. However, this is not a continuous approach between the
particles, but a change in the proportion of the permanence time in OW and CW.
Indeed, if we get even closer, the total potential would evolve to a single
equilibrium position landscape since the range of the potential scales with
$\xi$.

As described in previous sections, the correlation length is obtained from
$U(d,P_{h})$, plotted in Fig. 2c). We fit the critical Casimir force
contribution $U_{CC}(d)$ and hence obtain the correlation length $\xi$ of the
fluid assuming Derjaguin approximation. Assuming an Ising 3D class of
universality, $\xi(\Delta T)=\xi_{0}(\Delta T/T_{c})^{-\nu}$ with $\nu\simeq
0.63$ we can assign a $\Delta T$ to each value of the heating power $P_{h}$.
The measured $\xi$ (blue points) and $\Delta T$ (red dots) are plotted in Fig.
2d) as a function of $P_{h}$. The blue dashed line is the estimated
$\xi(\Delta T)$ with $\nu=0.63$. This allows us to find a relationship between
the temperature $T_{ROI}$ of the ROI and the heating power, specifically
$T_{\rm ROI}=C_{h}P_{h}+T_{g}$ with a heating rate of $C_{h}\approx(4.87\pm
0.2){\rm mK\ mW^{-1}}$.

Using this relationship between $\Delta T$ and $P_{h}$, we observe that
$\langle\delta I^{2}\rangle$ has a power law behavior (blue dashed line in
Fig. 2b) as a function of $\Delta T$ but with an exponent much larger of
$\nu$. Specifically comparing $\langle\delta I^{2}\rangle$ with $\xi$ we find
$\langle\delta I^{2}\rangle\propto\xi^{\alpha}\propto\Delta T^{-\nu\alpha}$,
where $\alpha=4.7\pm 0.6$.

Figure 3: a) Energetic sketch of the system. Two particles are held by
independent optical tweezers within a critical bath with correlation length
$\xi$ at a given critical distance ($\Delta T$). Trap 1 remains static while
trap 2 is moved back and forward at a constant velocity between two
equilibrium positions. b) Color plot showing the temporal evolution of the
energetic seascape $U(d)$ felt by the relative distance degree of freedom
$d=x_{2}-x_{1}-2R$. The two-colloidal particle system has different options
through the protocol. The single trajectories can start synchronized at the
optical trapping well (OW) or the critical well (CW) to remain in the same
state,$\rm{OW}\to\rm{OW}$ (yellow triangles) or $\rm{CW}\to\rm{CW}$ (red solid
line), or change it, $\rm{OW}\to\rm{CW}$ (purple squares) or
$\rm{CW}\to\rm{OW}$ (magenta circles). For the sake of simplicity in the
analysis of the energetics, we will focus on the analysis of
$\rm{OW}\to\rm{OW}$ and $\rm{CW}\to\rm{CW}$, defining them as OFF and ON
respectively. c-d) Energetics of the system. Work (blue circles), the heat
released by particle 1 (red triangles) and heat released by particle 2 (black
squared), and the change of internal energy (green diamonds) are shown for the
forward and backward process ($X_{f}$ filled symbols, $-X_{b}$ empty symbols).
c) Mean value $\langle X\rangle$ and d) variance $\sigma_{X}^{2}$ as functions
of the input energy $P_{\rm h}$.

### Switching the energy transfer between colloids by light-induced heating

In this section, we apply this laser-induced heating method in a dynamical
colloidal system. We designed a toy machine based on a protocol in which
thermodynamic magnitudes (work, heat,…) can be easily defined for analyzing
their statistics under the stochastic thermodynamics framework. We inject
energy as work in P1 for studying how the energy is dissipated via released
heat in both particles, see the sketch in Fig. 3a). For that purpose, we
displace the position of trap 2 following a time-symmetric protocol $\Gamma$.
The position of the movable trap is changed from $\Delta x_{1}^{T}=5500{\rm
nm}$ to position $\Delta x_{2}^{T}=5840{\rm nm}$ at a constant velocity
$v=342{\rm nm/s}$, see black solid line in Fig. 3b). We call the action of
pushing T2 away from T1 the forward (F) protocol while the action of getting
the traps closer is the backward one (B). Both processes, of one second each,
are connected by two equilibrium positions of the same duration to warrant the
equilibration of the colloids within their corresponding traps giving a
protocol total time of 4 seconds. The stiffness of the traps is kept small to
allow a broader exploration of the phase space ($\kappa=0.5{\rm pN\mu
m^{-1}}$). The dynamics can be drastically modified by small temperature
changes which determines the ratio between the critical (CW) and optical (OW)
wells. Indeed, in Fig. 3b) it is shown how the system, the colloidal
particles, can evolve in different ways along the potential seascape
represented by the color map in the background image. At the beginning of each
cycle, the system can mainly lie either in OW (yellow and purple time-series
in Fig. 3b)) or CW (magenta and red). If the traps are close, the thermal
transitions between the states are allowed and the system may erase its
previous state. Once T2 is pushed away, the system must make a choice, see the
different trajectories on Fig. 3b). As the probability of choosing each option
depends on the ratio between the well’s depth, the performance of the toy
machine can be modulated by changing the distance to the criticality hence the
heating laser power. In this analysis, our focus will be on two events with
higher statistical significance, specifically those that do not alter the
well. We refer to the events where particles remain within the optical trap
well (OW) as OFF events, while those with the particle residing in the
critical well (CW) are labeled as ON events. The trajectories of the ON and
OFF events are depicted by red and yellow curves, respectively, in Fig. 3b).

In Fig. 3c) and d) we show how the ensemble average of the system’s energetics
changes as a function of $P_{h}$, that is, the temperature. From the
trajectories, we obtain the values of the stochastic work ($W$) and both
released heats ($Q_{1}$ and $Q_{2}$) within the framework of stochastic
thermodynamics, see Methods. We calculate the ensemble average of each
quantity $\langle X\rangle$ and its probability density function $\rho(X)$.
The mean value of the heat released by each particle $\langle Q_{i}\rangle$
and the mean injected work $\langle W\rangle$ are shown in Fig. 3c) with their
standard deviations, Fig. 3d). Indeed, $\langle W\rangle$ increases with
temperature during F and B protocol, but there is a discrepancy between them.
The same features seem to appear in other magnitudes like the heat released by
P2 ($\langle Q_{2}\rangle$, black ) but not by P1 (red, $\langle
Q_{2}\rangle$). Finally, the mean change of internal energy ($\langle
U\rangle=\langle W\rangle+\langle Q_{1}\rangle+\langle Q_{2}\rangle$) remains
constant during both protocols as expected although changes with the critical
distance. The variance of the same quantities is shown in Fig. 3d), where we
observe different behaviors between F and B. Indeed, the existence of
different options during the process produces a bimodal distribution in the
energetics, see Fig. 4). If we compare the distribution of the injected work
$\rho(W)$ for different $\Delta T$, we observe how the critical interactions
start to dominate at small $\Delta T$, the CW dominates OW. In Fig. 4 we also
compare the distributions of the forward $\rho_{F}(W)$ and backward
$\rho_{B}(-W)$ protocols in the Crook theorem spirit,
$\rho_{F}(W)/\rho_{B}(-W)=\exp{(W-\Delta F)/kT}$. The global energetics of the
system is the combination of pure dissipative events (OFF) and events that
change the free energy of the system (ON).

Figure 4: Work injected into the system during the forward $\rho_{F}(W)$, full
squares and colored area, and backward process $\rho_{B}(-W)$, empty circles.
a) Statistics of W far from the criticality, $P=0{\rm mW}$ with $\xi\approx$
65 nm. b) Close to the criticality, $P=150{\rm mW}$ with $\xi\approx$ 140 nm.
The bimodality of all the distributions can be interpreted as the contribution
of the two possible equilibrium states. Moreover, the distributions can be
decoupled and analyzed in the Crooks-theorem spirit with a crossing of the
forward and backward distributions close to zero for the OFF events and a
crossing different from zero for the ON events.

The same analysis can be performed for each quantity as it is presented in
Fig. 5, which shows how the ON events dissipate much less energy than the OFF
events. Indeed, the OFF events are dragging-like processes in a more complex
environment (the non-negligible hydrodynamics due to the surfaces’ proximity
derives in a non-homogeneous viscosity, the Casimir interactions,…). The ON
events allow a higher proximity between the surfaces, and pulling away the
traps between them increases the effect of the critical interaction since the
increasing distance between the particle and the trap equilibrium position
suggests a higher importance of the CCF. It is this increase of the critical
Casimir force once the traps are pulled away that increases the free energy of
the system while the small change in the relative distance between the
particles is the reason for the small energy dissipation during ON events.

Figure 5:  Probability density function of the energetics ($W$ blue circles,
$Q_{1}$ red triangles, and $Q_{2}$ black squares) discriminating between a) ON
events and b) OFF events (colored areas with full symbols and non-shadowed
areas with empty symbols respectively) during the forward process at $P=150$
mW. The decoupling of the energetics between the two different possibilities
shows a higher dissipation during OFF events inferred from the non-zero mean
distribution of both released heats (black and red open symbols with non-
colored curve) compared with the zero mean heat distributions of the ON events
(black and red full symbols with colored area). The difference between the
work distributions of the ON and OFF events comes from the storage of free
energy in the case of ON (both particles within the critical Casimir well).

##  IV Discussion

Since the temperature of the fluid is increased locally, there is no direct
measurement of the temperature via thermometry. Indeed, obtaining a
reproducible nano-thermometric technique from our typical experiments is one
of the objectives of the article. Here, we have based our temperature
measurement, or more precisely the critical distance $\Delta T$ from $T_{c}$,
on i) the trajectory of the particles via CCF, and 2) the fluctuations of the
pixels’ luminosity. Assuming that the critical temperature is well defined at
$T_{c}=30.1{\rm{}^{o}C}$, we can infer the temperature of the fluid as the
distance to this reference $T=\Delta T+T_{c}$. Indeed, this is one of the
typical fingerprints of studying CCF using optical traps. The possible local
heating of the laser trap and the illumination for video tracking may produce
a mismatch between the temperature of the thermal bath and the temperature of
the physical system. The influence of the critical distance on the ratio of
the two potentials makes the statistics of the jumps as well as the population
rates very sensitive to the temperature changes in its environment. However,
this method could be used by any physical system to detect temperature changes
autonomously, for example, confining a vesicle filled with a critical fluid
and sensoring its size or shape.

Using light as the heating source allows us to compare the changes of the
different observables, the experiment is clear, very reproducible, and with no
dependence on a statistic of jumps, so very accurate. However, the optical
configuration of our setup makes difficult an analytical expression for the
account of the change of the dispersion with the critical distance via
critical exponents, $\langle\delta I^{2}\rangle\propto(\Delta T)^{\mu}$, where
$\mu=(3.0\pm 0.5)$ seems to be a combination of different critical exponents.

Fluctuation theorems are powerful theoretical tools that generalize the second
law of thermodynamics allowing us to connect out-of-equilibrium measures with
equilibrium magnitudes like free energy or entropy changes. In particular,
Crooks’ theorem is very visual for discriminating between pure-dissipative and
non-pure dissipative processes. Being
$kT\log{[\rho_{F}(W)/\rho_{B}(-W)]}=W-\Delta F$, on the one hand, if $\Delta
F=0$ (all the injected work is dissipated as released heat) both distribution
cross at zero. On the other hand, if there is a storing or release of energy
in the system, i.e. $\Delta F\neq 0$, the pdfs will cross at $W=\Delta F$.
Since in our system, the initial and final states are multiple-defined, as
shown in previous sections, we build the pdf corresponding to each event
(ON/OFF) and we can extract information. Namely, Fig. 4 shows multimodal
distributions for $\rho_{F}(W)$ and $\rho_{B}(-W)$. However, we can easily
notice that the distributions are not symmetric. This is due to the different
contributions of the ON and OFF events. While the ON events activate the
critical well, increasing $\Delta F$ along the forward protocol or decreasing
it along the backward, the OFF events only dissipate the injected work as
released heat. This is the reason why the forward and backward distribution of
the ON events are almost identical (a very small amount of energy is
dissipated) while the OFF distributions crosses each other at $W=0kT$, see
Fig. 5. However, we expect that an increase in the resolution for a given
$\Delta T$ will reveal differences between $F$ and $B$ also at the OFF events.
The energy dissipation is easier to visualize in Fig. 5b), where we show the
heat distribution for both particles as well as the work injected by selecting
only the desired realizations. We observe how in the case of ON events the
heat distribution is centered almost at zero, for both particles, while the
distribution does not peak at zero in the case of OFF events. Indeed, the
values of the mean work and mean heat shown in Fig. 3c) mainly evolve due to
the changes in the probability of choosing between the different events. We
want to point out how the distributions of work forward and backward overlap
each other in the case of the ON events. We interpret it as the fact that the
critical Casimir force is always in equilibrium for those displacements due to
its fast relaxation time.

##  V Conclusions

In this article, we have used light-induced heating to control the bath-
induced forces between two optically trapped colloidal particles. The heating
laser beam is focused by the upper objective irradiating the sample with a
cylindrical shape that allows us to control the temperature in a range of 1K
below the demixing point ($\approx 30.1{\rm{}^{o}C}$) with a precision of
${\pm 10{\rm mK}}$. The temperature is smoothly changed during hours with a
symmetric protocol for ensuring quasistatic warming. We do not observe any
hysteresis in the sample. Since the temperature calibration is done via the
correlation length of the fluid obtained by non-linear fits to the total
energy landscape, the precision of the method is highly affected by the
duration of the measurement. To have an extra observable with the same optical
setup, we study how the luminosity of the pixels has a clear dependence on the
critical distance. However, since the scattering geometry is highly non-
trivial and out of the interest of this article, the results remain open to
link them with a quantitative value of the correlation length.

The temperature modulation via laser-induced heating is applied in a two-
particle toy model whose flux of energy is changed as a function of the local
temperature. The temperature control is stable enough to study the stochastic
energetics of the two-particle system from a statistical point of view and to
use the fluctuation theorems in the interpretation of the dissipation sources.
From the use of the Crooks’ theorem, we interpret that the critical Casimir
force is always in equilibrium since the distributions of the work in the
forward and backward process for the ON events overlaps each other. However,
the light-induced heating technique paves the way for time-driven protocols in
the local temperature for future experiments in the out-of-equilibrium
performance of critical Casimir forces such as its equilibration or its
influence in the rheology of the sample.

We are just scratching the possibilities of fluctuation-induced forces in the
performance of nanomachines or their use in nanothermometry. Indeed, although
the range would be limited, critical interactions may be used in specific
experiments whose temperature dependence is large in a narrow temperature
range. In line with the final sentence of [13], ”the consciousness of the
environment as a part of the whole system is important not only in the ecology
but also at the micron or nanoscale physics”, we expect these results pave the
way to future experiments where the thermal bath is not just a passive actor
but a tunable agent during a physical process.

##  VI Acknowledgements

IAM acknowledges funding from the Spanish Government through grants Contract
(FIS2017-83709-R) and Juan de la Cierva program as well as the CNRS visiting
researcher program and the MSCA-IF NEQLIQ - 101030465. AP and SC acknowledge
funding from ANR18-CE30-0013. This work was partially financed by ERC-
OUTEFLUCOP.

##  Appendix A Methods

Experimental methods. Our experiments are done in a low critical temperature
micelle-solvent solution, ${\rm C_{12}\rm E_{5}}$ in Milli-Q water at 1.2$\%$
mass concentration. This mixture has a correlation length of $\xi_{0}\approx
1.4$nm and a critical temperature $T_{C}\approx(30.5\pm 0.1)^{\rm o}$C [14,
15]. A few microspheres (Fluka silica, $R=(2.50\pm 0.35)\mu$m) per milliliter
are added to the final mixture in a low concentration to allow long-term
measurements without intrusive particles. The mixture is injected into a
custom-made cell $40\mu$m thick and mechanically sealed to avoid
contamination. This chamber is made by sandwiching a parafilm mask between a
microscope slide and a sapphire optical window. The chamber thickness is
reduced at minima to avoid thermophoretic or convective effects. Within the
fluid cell, the two optical traps are created by a near-infrared laser beam
(LaserQuantum, $\lambda=$1064nm ) which is focused thanks to a high NA
immersion oil objective (Leica $\times$63, NA=1.4). The dynamic optical trap
is based on the control of the laser beam by an acousto-optical deflector (AA
optoelectronics) which allows us to create two different traps using the time-
sharing regime at 10 kHz. The two optical traps are kept $15\mu$m from the
cell bottom slide. The bead’s images are acquired by a high-speed camera
(Mikrotron MC1310) and their positions are tracked in real-time by suitable
software. The tracking resolution is $\pm$5 nm with a conversion rate of
$S=105.4{\rm nm/px}$. The acquisition frequency is fixed at 400 frames per
second for all experiments. The optical traps are calibrated using standard
methods such as power spectrum density. From the time series, we obtain the
total energy potential by the Boltzmann relation
$\rho(r)\propto\exp\left(-U(r)/kT\right)$. The total potential can be split
into its different components: electrostatical $U_{el}$, Casimir $U_{CCF}$ and
trapping $U_{OT}$. The Critical Casimir contribution, $U_{CCF}$ is fitted
assuming the Derjaguin approximation (for $d/R\ll 1$). The correlation length
is obtained from the non-linear fit of $U_{\rm cas}$ at different values of
the irradiating power $P_{h}$ and hence $\Delta T=T_{c}-T$.

The ground temperature ($T_{g}$) is controlled by a double feedback system one
on the objective and one inside the cell. Temperature is registered with two
independent sensors (Pt 1000$\Omega$) and sent to a programmable temperature
controller (PID Stanford research instruments). The objective and the cell are
heated with heater mats (Minco 40 $\Omega$ and 80 $\Omega$ respectively). The
whole system is thermally isolated from the environment to reduce the effect
of environmental perturbations both on the position of the particles and on
the temperature. Once the bulk is thermalized, $T_{g}=28.00{\rm{}^{o}C}$ the
sample is irradiated using an infrared heating laser (Thorlabs,
$\lambda=1550$nm) focused by the upper objective (Leica, NA 0.53). The heating
laser beam has a width of 20$\mu m$ and a depth of field of 400 $\mu m$ that
is much larger than the chamber thickness (40 $\mu m$). Therefore, a
cylindrical shape can be assumed in the description of the irradiating heating
beam.

Stochastic thermodynamics. Work ($W$) and heat ($Q$) as the exchange of energy
of the colloidal system with the external agent (change in the movable trap
position $x_{T}$) and the thermal bath respectively. Work and heat are defined
as $\delta W_{i}=\kappa(x_{i}-x^{T}_{i})\circ{\rm d}x^{T}_{i}$ and $\delta
Q_{i}=-\kappa(x_{i}-x^{T}_{i})\circ{\rm d}x_{i}$, where $i=1,2$ are the two
particles and $\circ$ stands for Stratonovich integration. From this
definition, the work is always injected in particle 2, since $x^{T}_{1}$ is
static. The events are discriminated between ON and OFF by comparing the mean
distance during a single protocol with a threshold between the colloidal
surfaces: if the mean relative distance is always below $d<400$ nm along a
single protocol, we consider it as an ON event. Ensemble average $\langle
X\rangle$ of any quantity $X$ over $N$ process of $M$ points each is defined
as $\langle X(t_{j})\rangle=\sum_{i=1}^{N}\sum_{k=1}^{j}\delta X_{i}(t_{k})$.

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