# Modified Bose-Einstein condensation in an optical quantum gas

Mario Vretenar  Chris Toebes  Jan Klaers<EMAIL_ADDRESS>Adaptive Quantum
Optics (AQO), Complex Photonic Systems (COPS), MESA+ Institute for
Nanotechnology, University of Twente, PO Box 217, 7500 AE Enschede,
Netherlands

††preprint: APS/123-QED

Open quantum systems can be systematically controlled by making changes to
their environment. A well-known example is the spontaneous radiative decay of
an electronically excited emitter, such as an atom or a molecule, which is
significantly influenced by the feedback from the emitter’s environment, for
example, by the presence of reflecting surfaces. A prerequisite for a
deliberate control of an open quantum system is to reveal the physical
mechanisms that determine the state of the system. Here, we investigate the
Bose-Einstein condensation of a photonic Bose gas in an environment with
controlled dissipation and feedback realised by a potential landscape that
effectively acts as a Mach-Zehnder interferometer. Our measurements offer a
highly systematic picture of Bose-Einstein condensation under non-equilibrium
conditions. We show that the condensation process is an interplay between
minimising the energy of the condensate, minimising particle losses and
maximising constructive feedback from the environment. In this way our
experiments reveal physical mechanisms involved in the formation of a Bose-
Einstein condensate, which typically remain hidden when the system is close to
thermal equilibrium. Beyond a deeper understanding of Bose-Einstein
condensation, our results open new pathways in quantum simulation with optical
Bose-Einstein condensates.

When a Bose gas exceeds the critical phase space density, a condensation
process is triggered [1]. This process can be considered as a competition
between various possible condensate wavefunctions that experience different
growth rates under the given experimental conditions and in which ultimately
the state with the highest rate prevails. If these growth rates are given by
the respective Boltzmann factors, the system relaxes to thermal equilibrium
and the condensation takes place in the state that minimises the energy. It is
possible to modify these rates by moving to non-equilibrium conditions, for
example, by controlling the particle exchange between the condensate and its
environment. Intervening in the condensation process in this way is completely
analogous to the modification of the spontaneous emission rate for quantum
emitters by a deliberate control of dissipation and feedback from the
environment [2, 3]. If the conditions for reaching thermal equilibrium are not
met, the condensation will not necessarily occur in the lowest energy state.
Beyond that, the question arises: Is it possible to formulate what kind of
state a Bose condensate strives for under such non-equilibrium conditions? In
contrast to cold atomic gases [4, 5], which typically do not allow significant
particle exchange with their environment, optical quantum gases in photonic
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and polaritonic [17, 18, 19]
microcavity systems offer the possibility of answering such questions
experimentally.

A Bose-Einstein condensed photon gas represents a source of coherent light
that operates close to thermal equilibrium. Similar to a black-body radiator,
a thermalisation process can be created by repeated resonant interactions
between light and matter in a microcavity. In contrast to a black-body
radiator, however, the photon energy in such systems is significantly above
the thermal energy, which effectively creates a non-zero chemical potential
for the photons $\mu$ that is related to the excitation level in the optical
medium by $\text{e}^{\frac{\mu}{kT}}\propto\rho_{\uparrow}/\rho_{\downarrow}$
[8]. Here, $\rho_{\uparrow,\downarrow}$ denote the densities of electronically
excited and ground state molecules, which can be set by optically pumping the
medium. A complete thermalisation process in the system would make all net
energy and particle flows disappear. In real systems with imperfect photon
confinement, the thermalisation process is not always able to bring the system
into a global thermal equilibrium. This is especially true when highly
inhomogeneous optical pump geometries are used. Such pump geometries can lead
to non-equilibrium condensation phenomena in which the local chemical
potential of the photons (understood as the local excitation level of the
optical medium) varies across the system. This situation, however, still has
to be differentiated from laser-like operation in which particle losses
overcome photon reabsorption by the optical medium. The experiments described
in this work are carried out in a parameter range in which the probability
that a cavity photon is absorbed by the optical medium is significantly higher
than the probability for transmission through the cavity mirrors, but not so
high that the system relaxes into a global state of equilibrium with spatially
homogeneous chemical potential irrespective of the optical pumping geometry.

A particular class of non-equilibrium condensation phenomena are transport
processes, which are a current research topic in 2D optical quantum gases [20,
21, 22, 23]. In these systems, condensates propagating in the transverse plane
of the resonator can either be created by directly condensing into states with
higher kinetic energy, or by preparing condensates at rest and subsequently
setting the particles in motion. To achieve the latter, the condensates must
be generated in suitable potential landscapes  using spatially inhomogeneous
optical pump geometries. If the condensate is created at an elevated potential
energy level, for example, particles will gain kinetic energy as they fall
into regions of lower potential. This can be used to prepare a stream of
photons in a controlled state of motion. In our experiments, such photon
currents are directed into a potential landscape that acts as a Mach-Zehnder
interferometer. By partially or completely closing the outputs of the
interferometer, we can systematically vary the degree of dissipation and
feedback in the system, which allows us to identify the underlying physical
principles that determine the formation of Bose-Einstein condensates under
non-equilibrium conditions.

Results

Figure 1: Photon Bose-Einstein condensation in an open Mach-Zehnder
interferometer. a Microcavity formed by a planar mirror, a nanostructured
mirror, and an organic optical medium (top). The optical medium consists of a
water-based solution of a dye (rhodamine 6G) and a thermo-responsive polymer
(pNIPAM). The height profile of the nanostructured mirror (bottom), creates a
potential landscape for the photon gas that effectively acts as a Mach-Zehnder
interferometer. A laser beam at a wavelength of 532 nm is used to locally heat
the thermo-responsive polymer. This allows for reversible tuning of the local
refractive index, which can be used to create optical path length differences
in the internal arms of the interferometer. The system is off-resonantly
pumped with a tightly focused pulsed laser beam (pulse duration $\simeq
5\,\text{ns}$, wavelength 470 nm). The time delay between the heating pulse
and the optical pump pulse determines the path length difference that is
probed. The photon density within the cavity is determined by measuring the
transmitted cavity light with a camera. b Photon density for 4 time delays. c
Normalised switching function $I_{2}/(I_{1}+I_{2})$ (upper graph) and total
intensity $I_{1}+I_{2}$ (lower graph) of the open interferometer as a function
of the delay between heating and optical pumping. Data points are averages
over 20 measurements (optical pulses). Error bars indicate the standard error
of the mean.

Experimental system. In our experiment, we use a high-finesse optical
microcavity filled with a water-based solution of rhodamine 6G dye and the
thermo-responsive polymer pNIPAM, in which the photons are repeatedly absorbed
and re-emitted by the dye molecules, see Fig. 1. This process is governed by
the Kennard-Stepanov law linking the (broadband) absorption coefficient
$B_{12}(\omega)$ and the emission coefficient $B_{21}(\omega)$ by
$B_{12}(\omega)/B_{21}(\omega)=\exp[(\hbar(\omega-\omega_{\text{zpl}})/kT]$,
where $\omega_{\text{zpl}}$ corresponds to the zero-phonon line of the dye and
$T$ denotes the temperature. Low cavity losses allow the photons to be
absorbed and re-emitted multiple times before leaving the cavity, creating a
thermalisation process between the photon gas and the dye solution. Indeed,
Bose-Einstein distributed energies and the formation of Bose-Einstein
condensates at room temperature have been demonstrated several times in this
system [6, 7, 24, 14, 25]. The separation between the microcavity mirrors of
$D_{0}\simeq 10\,\upmu\text{m}$ is sufficiently small so that the longitudinal
mode number becomes a conserved quantity in the interaction between the
photons and the optical medium [6]. In this regime, the photon gas effectively
becomes two-dimensional and the photon energy can be approximated by

$E\simeq\frac{mc^{2}}{n_{0}^{2}}+\frac{(\hbar
k_{r})^{2}}{2m}-\frac{mc^{2}}{n_{0}^{2}}\left(\frac{\Delta
d}{D_{0}}+\frac{\Delta n}{n_{0}}\right),$ (1)

where $k_{r}$ is the transversal wavenumber describing the two-dimensional
motion in the cavity plane and $m$ denotes the effective photon mass. The
second and third term correspond to the kinetic and potential energy. The
potential energy is non-vanishing, if the distance between the mirrors
$D(x,y)=D_{0}+\Delta d(x,y)$ or the refractive index $n(x,y)=n_{0}+\Delta
n(x,y)$ is modified across the transverse plane of the resonator (we assume
$\Delta d\ll D_{0}$ and $\Delta n\ll n_{0}$).

Three different experimental techniques are combined to control the potential
landscape within the microresonator. First, a static nanostructuring of the
mirror surfaces of the cavity mirrors is performed via a direct laser writing
technique [26]. In this way, we can create smooth surface structures of up to
100 nm height with sub-nm precision in the growth direction. Second, a
reversible tuning of the potential landscape is performed by locally heating
the thermo-responsive polymer pNIPAM that is added to the optical medium with
a short laser pulse [24]. The absorbed laser energy increases the local
temperature of the optical medium by a few Kelvin such that it reaches the
lower critical solution temperature (LCST) of pNIPAM in water
($32^{\circ}\text{C}$). This leads to a significant increase of the index of
refraction which, with eq. (1), translates into a tunable potential for the
photons in the microcavity. Finally, the potential landscape can be further
adjusted by tilting one of the mirrors, effectively adding a potential
gradient to the photon gas.

In our experiments, we use non-resonant optical pumping to create regions of
high chemical potential in which photon Bose-Einstein condensates form [see
location ’high chemical potential’ in 1a]. Particles emitted from the
condensate into the microresonator plane are guided by two waveguide
potentials. The waveguides are at a lower potential energy, which means that
the condensate photons gain kinetic energy as they enter these regions. The
two waveguides first guide the photons into disjoint areas of the
microresonator plane before photons recombine in a region that is designed as
a 50:50 beamsplitter. In the following, the waveguides between the condensate
and the beamsplitter structure are referred to as the ’internal arms’ of the
interferometer, while the waveguides connected to the opposite end of the
beamsplitter are referred to as ’output arms’. Optical path length differences
between the internal arms of the interferometer can be generated with the help
of the thermo-responsive optical medium. For this purpose, we heat the upper
internal arm with a focused laser pulse, see location ”heating” in Fig. 1a, so
that the refractive index of the medium and thus the optical path length
changes significantly. After the initial heating, the refractive index relaxes
back to equilibrium. We observe that this decay is approximately exponential
in time. Interestingly, the decay time is found to depend on the energy of the
initial heating pulse. For sufficiently strong heating, it can become as large
as a few seconds.

Open Mach-Zehnder interferometer. Results for self-interference experiments
with photon BECs performed in this manner are shown in Fig. 1b. Here we show
the photon density in the microresonator plane for different time delays
between the initial heating pulse and the optical pumping. Depending on the
probed path length difference, a switching between constructive and
destructive interference in the outputs can be observed. This observation
confirms the coherence of the particle stream emitted by the photon Bose-
Einstein condensate [27]. Another indication is the interference pattern that
arises when the waveguide potentials at the end of the two outputs cease,
which is reminiscent of that of a double slit experiment. A quantitative
characterisation of the system is obtained by determining the intensities
$I_{1,2}$ in the lower and upper output arms of the interferometer [see
locations ’$I_{1}$’ and ’$I_{2}$’ in Fig. 1a]. The upper graph in Fig. 1c
shows the normalised intensity in the upper output arm $I_{2}/(I_{1}+I_{2})$
as function of the time delay. Every data point corresponds to an average over
20 optical excitations with a non-resonant pump pulse of $\simeq 5\,\text{ns}$
duration. These measurements demonstrate coherences close to 100% and a tuning
range of more than $8\pi$ for the phase difference in the internal arms. The
lower graph in Fig. 1c shows the total intensity integrated over both output
arms as a function of time delay. Similar results as shown in Fig. 1 have
previously been obtained in a related polaritonic system [28].

Semi-open Mach-Zehnder interferometer. In order to experimentally reveal the
underlying physical principles determining the condensation process under non-
equilibrium conditions, we will systematically vary the degree of dissipation
and feedback in the system. For this purpose, we close one or even both output
arms of the interferometer by creating additional potential walls at the ends
of the waveguide potentials. Particles that reach these walls are reflected
and propagate backwards through the interferometer. This backreflection
eventually reaches the location of the Bose-Einstein condensate, where it
interferes with the light field present there. Depending on the phase delays
that the particles accumulated on their way, the interference with the Bose-
Einstein condensate can be constructive or destructive. Generally speaking,
the condensation process is expected to be accelerated in the case of
constructive interference, as this increases the local field amplitude, which
triggers additional stimulated emission events. In contrast, destructive
interference is expected to slow down the growth of the condensate. This
feedback phenomenon is fully analogous to effects known from the modified
spontaneous emission of quantum emitters [2, 3]. The feedback provided by the
environment ultimately leads to a situation in which the condensation does not
necessarily occur in the lowest energy state as suggested by a gain-loss rule
of the Kennard-Stepanov type, but in a condensate state with higher kinetic
energy for which the feedback from the environment is more favourable. In the
Methods section we show that the initial evolution of the condensate
population upon optical pumping can be approximately described by

$\frac{\dot{n}}{n}=\Gamma_{\text{em}}-\Gamma_{\text{abs}}\,e^{\frac{\hbar(\dot{\theta}-\omega_{0})}{kT}}-\Gamma_{\text{env}}(1-\text{Re}[r(\dot{\theta})])\,\text{,}$
(2)

where the condensate wavefunction is described by a single complex number
$\psi=\sqrt{n}\exp(-i\theta)$ with time-dependent particle number $n=n(t)$ and
phase $\theta=\theta(t)$. The interaction with the optical medium is mediated
via stimulated absorption and emission with rates $\Gamma_{\text{abs}}$ and
$\Gamma_{\text{em}}$. Furthermore, $\Gamma_{\text{env}}$ denotes the rate of
photons that are emitted from the condensate to the environment
(interferometer), while the complex reflection coefficient $r=r(\dot{\theta})$
describes magnitude and phase delay of the coherent feedback created by the
closed arm(s) of the interferometer. In general, the reflection coefficient
depends on the frequency of the condensate $\dot{\theta}$ as the propagation
of particles through the interferometer is highly frequency-selective.

Figure 2: Semi-open Mach-Zehnder interferometer. a Height map of the
nanostructured mirror. b Normalized photon density for 4 specific time delays
between the heating pulse in the upper internal interferometer arm (’heating’)
and the optical pumping (’high chemical potential’). The observed standing
wave mode patterns indicate a superposition of an outgoing and a back
reflected wave. c Normalised switching function $I_{1}/(I_{1}+2I_{2})$ (upper
graph) and total intensity $I_{1}+I_{2}$ (lower graph) of the semi-open
interferometer. Data points are averages over 20 measurements (optical
pulses). Error bars indicate the standard error of the mean. Figure 3: Closed
Mach-Zehnder interferometer. a Height map of the nanostructured mirror (top).
For the results shown, the microcavity is operated in two configurations:
plane-parallel and tilted. By tilting one of the microcavity mirrors, a
potential gradient is introduced to the photon gas (bottom). This gradient
leads to different phase delays for photons that propagate in the different
output arms. b Normalised photon density at different time delays for both a
plane-parallel and a tilted microresonator. c Normalised switching function
$I_{2}/(I_{1}+I_{2})$ (upper graph) and total intensity $I_{1}+I_{2}$ (lower
graph) of the closed interferometer in plane-parallel configuration. d
Normalised switching function $I_{2}/(I_{1}+I_{2})$ (upper graph) and total
intensity $I_{1}+I_{2}$ (lower graph) of the closed interferometer in a tilted
configuration. Data points are averages over 20 measurements (optical pulses).
Error bars indicate the standard error of the mean.

Closing the lower output arm (Fig. 2a) reflects the particle flow back through
the interferometer and finally to the location of the Bose-Einstein
condensate. In Fig. 2b we show intensity patterns obtained under these
conditions for 4 different time delays after the initial heating pulse. A
notable difference to the measurements with the open interferometer is the
appearance of standing waves created by the superposition of forward and
backward propagating waves. The distance between the nodes in the wave
function can be taken as a measure of the velocity of the particles. The
varying node distances in the upper interferometer arm, see for example panel
i in Fig. 2b, shows that the particles are first accelerated and then
decelerated again when they run through the thermo-optically induced
potential. In this way they obtain a different phase delay than the particles
in the lower arm.

For the open interferometer the reflection amplitude $r$ vanishes and the
condensate growth rate $g=\dot{n}/n$ in eq. (2) reduces to the gain-loss rule
originating from the Kennard-Stepanov law. In this case, the condensation
occurs in the state that minimises the energy (vanishing kinetic energy)
regardless of phase differences in the internal arms of the interferometer.
This means that the switching function of the interferometer necessarily needs
to assume a sinusoidal shape, as was indeed shown in Fig. 1c. Conversely, any
deviation from a sinusoidal switching function indicates that the condensate
energy is not a constant during the scanning of the interferometer and,
consequently, that the condensation process is not solely determined by an
energy minimisation rule. In the upper graph of Fig. 2c we show the normalised
intensity in the lower (closed) output arm $\hat{I}_{1}=I_{1}/(I_{1}+2I_{2})$
as function of the time delay in the case of a semi-open interferometer. Since
we compare the intensity of a traveling wave with that of a standing wave we
have included an additional factor of 2 in the normalisation factor for
compensating the additional intensity contribution of the back-propagating
wave in the lower output arm. Interestingly, the switching function in Fig. 2c
shows a clear imbalance between the intensities in the upper and lower output
arms in favour of the intensity in the lower (closed) output. The latter
refers not only to the maximum signal levels in both arms, but also to the
time intervals for which $\hat{I}_{1}>0.5$. Even if one ignores the data for
time delays of $t>5\,\text{s}$, since the refractive index of the medium is
then largely relaxed, it is observed that the areas shaded in green
($\hat{I}_{1}>0.5$) occupy an area which is more than 4 times the size of the
areas shaded in orange ($\hat{I}_{1}<0.5$). This imbalance shows in particular
that the condensate energy is not constant while the interferometer is being
scanned. Indeed, for the semi-open interferometer the condensate growth rate
$g=\dot{n}/n$ in eq. (2) can be shown to favour condensate frequencies that
increase the level of constructive interference in the closed arm, see
Methods. As long as it does not become too detrimental in terms of energy (and
the correspondingly stronger absorption by the optical medium), the condensate
can adjust its kinetic energy in such a way that the particle loss via the
open output arm is reduced. The same picture emerges from the behaviour of the
total intensity $I_{1}+I_{2}$, which is given as a function of time delay in
the lower graph of Fig. 2c. The data indicates that the total intensity is
significantly higher when the conditions for constructive interference in the
closed arm are met (and vice versa). This simply means that the condensation
process experiences a higher growth rate when the particle loss is reduced.

Closed Mach-Zehnder interferometer. For an interferometer that is closed at
both output arms (see Fig. 3a), the particles emitted by the condensate return
to the location of the condensation after passing through the interferometer
twice (see the standing wave density profiles in Fig. 3b). These particles
interfere with the condensate, which alters the rate of the condensation
process. In the case where the optical path lengths in the two output arms are
identical, the system has no reason to prefer constructive or destructive
interference in either of the two output arms. Under these conditions, the
switching function can be expected to return to the sinusoidal shape as in the
case of the open interferometer, which can be confirmed experimentally, see
Fig. 3c. The observation of shoulders in the otherwise sinusoidal shape of the
switching function is related to transitions between wave functions with
different numbers of nodes.

In the case that the optical path lengths in the output arms differ, a
different scenario results. Differences in the optical path length of the
outputs can be created by tilting one of the cavity mirrors. This effectively
adds a potential gradient to the photon gas, which lifts the two output arms
to two different potentials, see Fig. 3a. Due to the resulting optical path
length difference, the feedback from the two output arms will not interfere in
the same way with the condensate. This means in particular that the feedback
cannot be maximally constructive if the photon density is distributed over
both interferometer outputs. Figure 3d shows experimental results for the case
of a tilted cavity. The data presented in Fig. 3d reveal an almost discrete
switching behaviour between maximum intensity in the upper and maximum
intensity in the lower output arm as the phase difference between the internal
arms of the interferometer is scanned. This demonstrates that the condensate
adjusts its frequency in such a way that a maximum degree of constructive
feedback is achieved for all optical path length differences.

Another interesting aspect related to the observed behaviour is the fact that
the switching function becomes quite steep as it jumps between completely
destructive and constructive interference. In these regions, the closed
interferometer is highly susceptible to changes in optical path length - which
is potentially interesting for sensing applications. Indeed, our measurement
can be regarded as a special form of self-mixing interferometry [29, 30]. The
observation that the switching functions for the plane-parallel and the tilted
resonator are different, furthermore suggests that the interferometer can also
be switched by changing the optical path length in the outputs. This is indeed
confirmed experimentally, see Fig. 4. The fact that the frequency of the
condensate (and thus the relative intensities in the outputs) can be
controlled by a refractive index change that is more than 100 micrometers away
from the location of the condensation clearly demonstrates the ’non-local’
character of the condensation process.

Figure 4: Closed Mach-Zehnder interferometer with optical path length tuning
in the upper output arm. a Height map of the nanostructured mirror. b
Normalised photon density for 4 specific time delays between the heating pulse
in the upper output arm (’heating’) and the optical pumping (’high chemical
potential’). c Normalised switching function $I_{2}/(I_{1}+I_{2})$ (upper
graph) and total intensity $I_{1}+I_{2}$ (lower graph) of the closed
interferometer with output arm tuning. Data points are averages over 20
measurements (optical pulses). Errors are smaller than the symbol size.

Conclusions

In conclusion, our work investigates the Bose-Einstein condensation of photons
in a Mach-Zehnder interferometer potential with controlled dissipation and
feedback. The switching behaviour of the interferometer is analysed in order
to reveal the physical mechanisms that control the formation of a Bose-
Einstein condensate. We show that by adjusting their frequency, Bose-Einstein
condensates naturally seek to minimise particle loss and destructive
interference in their environment. This ability remains hidden in thermal
equilibrium, but becomes visible when the condensation occurs under non-
equilibrium conditions. Beyond a deeper understanding of Bose-Einstein
condensation, our results open new pathways in novel computing schemes for the
solution of hard optimisation problems based on coherent networks of photonic
or polaritonic condensates [31, 32] and lasers [33, 34]. Understanding the
physical mechanisms that determine the state of a condensate under controlled
dissipation and feedback, as identified in our work, is essential to the
design of such systems.

Acknowledgements

This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 101001512).

Methods

Experimental Methods. Our experiment is based on a high finesse dye
microcavity as shown in Fig. 1. The mirror separation of $D_{0}\simeq
10\,\upmu\text{m}$ is larger than in earlier photon BEC experiments. In
general, short mirror spacings create experimental conditions in which the
free spectral range of the resonator becomes larger than the emission
bandwidth of the dye molecules. The longitudinal mode number then becomes a
conserved quantity in the interaction between the photons and the optical
medium. In our present experiments, we find that this is largely the case even
at mirror distances where the free spectral range is comparable but not
necessarily larger than the dye bandwidth. The advantage of working with a
larger mirror spacing results from the reduced pump power required to trigger
the condensation process and a lower susceptibility to imperfections in the
nanostructuring of the mirror.

In the regime, in which the longitudinal mode number is frozen out, the photon
gas becomes effectively two-dimensional. The photon energy in the cavity is
given by

$E=\frac{\hbar c}{n}\sqrt{k^{2}_{z}+k^{2}_{r}},$ (3)

in which $n=n(x,y)$ describes the index of refraction and the longitudinal and
transverse wavenumbers are denoted by $k_{z}$ and $k_{r}$, respectively. The
boundary conditions induced by the mirrors require $k_{z}=\pi q/D$, in which
$q$ is the longitudinal mode number and $D=D(x,y)$ denotes the mirror
separation across the transverse plane. We assume that both $D(x,y)$ and
$n(x,y)$ show only small variations across the cavity plane, i.e.,
$D(x,y)=D_{0}+\Delta d(x,y)$ with $\Delta d(x,y)\ll D_{0}$ and
$n(x,y)=n_{0}+\Delta n(x,y)$ with $\Delta n(x,y)\ll n_{0}$. In a paraxial
approximation ($k_{r}\ll k_{z}$), the photon energy in eq. (3) can be
rewritten as

$E\simeq\frac{mc^{2}}{n_{0}^{2}}+\frac{(\hbar
k_{r})^{2}}{2m}-\frac{mc^{2}}{n_{0}^{2}}\left(\frac{\Delta
d}{D_{0}}+\frac{\Delta n}{n_{0}}\right),$ (4)

where $m=\pi n_{0}q/cD_{0}$ is the effective photon mass. The first term
corresponds to the rest energy of the (two-dimensional) photon, which is
$mc^{2}/n_{0}^{2}\simeq 2.1$ eV in our experiment (yellow spectral regime).
The second term describes the (transverse) kinetic energy and the last term is
the potential energy of the photons. The potential energy becomes non-zero, if
either the mirror spacing or the refractive index changes locally. In our
experiment, variations of the mirror distance can be created by
nanostructuring one of the mirrors with a direct laser writing technique. With
this technique, a laser scans over the backside of the mirror, locally heating
an amorphous silicon layer located below the dielectric stack of the mirror.
This causes an irreversible expansion of the silicon layer, pushing the
dielectric stack outwards. In this way, we can create smooth surface
structures of up to 100 nm height with sub-nm precision in the growth
direction [26]. The height profiles of our mirrors are determined via Mirau
interferometry. For these measurements, we use a commercially available
interferometric microscope objective (20X Nikon CF IC Epi Plan DI).

The optical medium consists of a solution of rhodamine 6G in water
(concentration 10 mmol/L) with a 4% mass fraction of the thermo-responsive
polymer pNIPAM added. In our experiments, the dye molecules are excited by a
pulsed optical parametric oscillator (OPO) with a pulse duration of $\simeq$5
ns at a wavelength of 470 nm. This wavelength is outside the reflection band
of the cavity mirrors and therefore allows non-resonant excitation on the
optical axis. A second pulsed laser at a wavelength of 532 nm with pulse
duration 10-20 ms is used to heat the silicon layer on the mirror. This leads
to a temperature increase of a few Kelvin, which is enough to reach the lower
critical solution temperature (LCST) of pNIPAM in water. This causes the
polymer chains to collapse, which triggers a mass transport in the optical
medium that locally changes the refractive index ($\Delta n\simeq 0.1$). In
this way we can create variable potentials for the photon gas.

Theoretical Methods. The following section presents a theoretical model that
reproduces the results observed in our experiments. An illustration of the
model is given in Fig. 5.

Figure 5: Theoretical model. Our model describes the Bose-Einstein
condensation of photons in an environment with controlled dissipation and
feedback. A thermalization process is induced by absorption and emission
processes in an optical medium. This competes with the coherent feedback from
the environment, which is modelled as a Mach-Zehnder interferometer. The
outputs of the interferometer are assumed to be either open or closed, which
creates a varying degree of dissipation and feedback.

The wavefunction of the condensate is described by a single complex-valued
amplitude $\psi$. This amplitude is affected by both the interaction with the
optical medium, i.e. absorption and emission events, and the interaction with
the environment. For the optical medium, we assume the validity of the
Kennard-Stepanov law. We choose a particular implementation of the Kennard-
Stepanov law, in which the emission coefficient of the medium is assumed to be
frequency-independent and the absorption spectrum is assumed to increase
exponentially with the frequency of the light. This corresponds approximately
to the experimental conditions in our experiment. Taking this into account, we
expect that the time evolution of the condensate can be described by the
following dissipative Schrödinger equation:

$\frac{i\dot{\psi}}{\psi}=\omega_{0}+\omega_{r}+\frac{i}{2}\left(\Gamma_{\text{em}}-\Gamma_{\text{abs}}\,\text{e}^{\frac{\hbar(\dot{\theta}-\omega_{0})}{kT}}\right)-\frac{i}{2}\Gamma_{\text{env}}(1-r(\dot{\theta}))\,\text{.}$
(5)

Here, $\omega_{0}$ denotes the frequency of a condensate at rest, while
$\omega_{r}$ accounts for a possible non-vanishing kinetic energy of the
condensate. $\Gamma_{\text{em}}$ and $\Gamma_{\text{abs}}$ denote the emission
and absorption rates. The particle exchange with the environment is described
by the parameter $\Gamma_{\text{env}}$, which corresponds to the rate of
particles that are emitted by the condensate to the Mach-Zehnder
interferometer, and the reflection amplitude $r(\dot{\theta})$, which is a
complex-valued function that describes magnitude and phase delay of the
feedback created by the closed arm(s) of the interferometer. By describing the
interferometer feedback as a complex multiple of the instantaneous
wavefunction we implicitly have performed a slowly varying amplitude
approximation for the time evolution of the condensate, i.e., we assume that
the response time of the environment $\tau_{\text{env}}$ is so small that
$|\dot{\psi}|/|\psi|\ll\tau_{\text{env}}^{-1}$. Here, $\tau_{\text{env}}$
corresponds to the time the photons need to travel back and forth through the
Mach-Zehnder interferometer. Another approximation concerns the absorption
($\Gamma_{\text{abs}}$) and emission rates ($\Gamma_{\text{em}}$), which we
assume to be constant at the initial formation of the condensate. This
essentially means that eq. (5) only correctly describes the time evolution of
sufficiently small condensates. The growth rate for the condensate population
and the time evolution of the phase directly follow from eq. (5) by performing
a transformation $\psi=\sqrt{n}\text{e}^{-i\theta}$ with real-valued functions
$n=n(t)$ and $\theta=\theta(t)$:

$\displaystyle\frac{\dot{n}}{n}$
$\displaystyle=\Gamma_{\text{em}}-\Gamma_{\text{abs}}\,e^{\frac{\hbar(\dot{\theta}-\omega_{0})}{kT}}-\Gamma_{\text{env}}(1-\text{Re}[r(\dot{\theta})])\,\text{}$
(6) $\displaystyle\dot{\theta}$
$\displaystyle=\omega_{0}+\omega_{r}-\Gamma_{\text{env}}\,\text{Im}[r(\dot{\theta})]\,\text{.}$

In our experiments, the probabilistic character of the condensation process
manifests itself in the fact that the condensate state can vary from shot to
shot. In our model, we can introduce this kind of randomness by converting the
growth rate function $g=\dot{n}/n$ in eq. (6) into a probability distribution.
To do this, we define the probability for the occurrence of a condensate with
frequency $\dot{\theta}$ as $p(\dot{\theta})=\exp(g(\dot{\theta})\,t^{*})/Z$,
where $g(\dot{\theta})$ is the growth rate of the state, $t^{*}$ is an
effective time parameter, and $Z$ is a normalization parameter such that $\int
p(\dot{\theta})d\dot{\theta}=1$. Furthermore, an expectation value for an
observable $Q$ can be defined as $\langle Q\rangle=\langle
Q\rangle_{t^{*}}=\int p(\dot{\theta})Q(\dot{\theta})d\dot{\theta}$. Similar to
the temperature in thermal averages, the time parameter $t^{*}$ determines the
statistical composition of the expectation value. For small values of $t^{*}$,
many different states contribute almost equally to the average, while for
large values of $t^{*}$ the statistical average is dominated by the state that
maximizes the growth rate. In the following, we use the time parameter as a
global free parameter, which is chosen to obtain the best match between model
and experiment. It turns out, however, that the found value of this parameter
is close to the actual pulse duration.

Figure 6: Photon BEC in an open and semi-open Mach-Zehnder interferometer.
Theoretically expected relative photon density in the output arms and mode
frequency as a function of the optical path length difference. a Open MZI. b
Semi-open MZI. Parameters used to calculate these graphs are
$\hbar\omega_{0}=2.1\,\text{eV}$, $m=6.8\cdot 10^{-36}\,\text{kg}$,
$T=300\,\text{K}$, $D_{0}=10\,\mu\text{m}$, $\Delta d=4.5\,\text{nm}$,
$\Gamma_{\text{abs}}=100\,\text{GHz}$, $\Gamma_{\text{env}}=6.3\,\text{GHz}$,
and $t^{*}=4.75\,\text{ns}$. All other relevant parameters are given in the
figures. Note that some parameters such as $\Gamma_{\text{em}}$ have no
influence on the calculation of the expectation values shown here. Figure 7:
Photon BEC in a closed Mach-Zehnder interferometer. Theoretically expected
relative photon density in the output arms and mode frequency as a function of
the optical path length difference. a Closed MZI in plane-parallel
configuration. b Closed MZI in tilted configuration. c Closed MZI, for which
the optical path length tuning is performed in the upper output arm.
Parameters used to calculate these graphs are the same as in Fig. 6.

In order to reproduce the experimental results with the help of the so-defined
model, the frequency-dependent reflection amplitude $r=r(\dot{\theta})$ has to
be determined. For the open interferometer, this amplitude simply vanishes:
$r_{\text{open}}=0$. In general, $r$ is obtained by adding up the probability
amplitudes of all possible paths through the interferometer, which both start
and end at the condensate. As an intermediate step, we determine the
probability amplitudes at the two outputs of the interferometer, which,
assuming a perfect 50:50 beamsplitter, are given by
$a_{1}=(i\,\text{e}^{ik(\alpha+\gamma)}+\text{e}^{ik(\beta+\gamma)})/2$ and
$a_{2}=(\,\text{e}^{ik(\alpha+\delta)}+i\,\text{e}^{ik(\beta+\delta)})/2$.
Here, $\alpha$, $\beta$, $\gamma$, and $\delta$ denote the optical path
lengths of the 4 interferometer arms, see Fig. 5. The wavenumber
$k=k(\dot{\theta})$ of the photons propagating in the interferometer arms
follows from $k=(2m\,[\dot{\theta}-\omega_{0}(1-\Delta
d/D_{0})]/\hbar)^{1/2}$, where $\Delta d\simeq 4.5\,\text{nm}$ denotes the
height difference between the condensate location and the interferometer arms
on the nanostructured surface of our mirror (drop in potential energy). With
this, the frequency-dependent reflection amplitude in the semi-open
interferometer (output 1 closed, output 2 open) becomes
$r_{\text{semi}}=-a_{1}a_{1}$. For the closed interferometer, we obtain
$r_{\text{closed}}=-a_{1}a_{1}-a_{2}a_{2}$.

Figures 6 and 7 show results for the expected relative photon densities in the
output arms of the interferometer $\langle|a_{1,2}|^{2}\rangle$ (upper graphs)
and the mode frequency $\langle\dot{\theta}-\omega_{0}\rangle$ (lower graphs)
as a function of the optical path length difference. For the open
interferometer, our model reproduces the expected sinusoidal variation of the
output intensity. No variation of the condensate frequency is predicted. For
the semi-open interferometer, the imbalance in the switching function
predicted by our model is even more pronounced than in the experimental data
in Fig. 2. For large path length differences, this imbalance increases, which
suggests that, from a certain threshold value, the photon density is fully
concentrated in the closed interferometer output regardless of the optical
path length difference in the internal interferometer arms. In order to direct
the particle flow into the closed interferometer arm, the frequency of the
condensate can adapt to a certain extent, see lower graph in Fig. 6b. For the
closed interferometer, our theoretical model shows a transition between
sinusoidal and rectangular switching functions, depending on the path length
difference in the output arms. This is in good agreement with our experimental
results shown in Fig. 3. Qualitative agreement is furthermore obtained for the
closed interferometer with optical path length tuning in the upper output arm,
see Fig. 7c and Fig. 4.

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