**Physicists at the University of Bonn together with colleagues from the University of Antwerp and the University of Freiburg have established a fundamental relation connecting the temporal fluctuations and the response behaviour in a quantum gas made of photons, the particles of light. In their experiments, the researchers confirm the validity of the so-called quantum regression theorem, which so far could not be directly revealed for quantum gases. Their findings provide a powerful new tool to probe optical quantum gases in lattices with complex spatial structure or topological properties in the future. ****The study is published in the journal Nature Communications.**

In everyday life, many light sources, such as sunlight, LEDs, or lasers exist around us. However, a fundamentally different light source is the photon Bose-Einstein condensate (BEC). First created in 2010, the photon BEC realises a textbook example for a quantum gas and provides a testbed for exploring physical phenomena emerging in such quantum systems. The photon BEC is a macroscopic state of light where the particles behave in an orderly fashion and occupy the lowest energy state. Loosely speaking, the photon BEC can be thought of as an ordered ‘cold’ state of matter similar to ice, in which water molecules have arranged in a particular spatial order. For comparison, an uncondensed photon gas (like sunlight) resembles the liquid phase, where the water molecules are entirely disordered.

### Regression relation between fluctuations and response

The current study investigates the dynamics of a photon BEC that is coupled to a reservoir of dye molecules. In doing so, the researchers demonstrate the existence of a fundamental relation that links the photon BEC response after an external perturbation to its intrinsic microscopic variations, namely, the fluctuations. This relation, commonly known as the quantum regression theorem, is a fundamental prediction from statistical physics. It states that the temporal behaviour of the correlations of a specific observable in a system at two different times (for example, the particle number correlations between times *t*=0 and *t*=*t*’) is essentially the same as the temporal behaviour of the corresponding average observables at single times (for example, the photon number at time *t*’) following a perturbation. In this way, the study demonstrates a way to access the fluctuation dynamics of condensates, which are often hard to measure experimentally, by means of probing the response after a well-controlled perturbation.

As an illustration for the scheme to probe a system by its response, consider water and sugar syrup. The liquids cannot be distinguished by eye, but water has a high fluidity, while the syrup is rather viscous. One way to tell the liquids apart is by observing the motion of a test object (e.g. a small metal bead). In water, it will sink almost immediately, but in syrup it will do so much slower. In other words, by perturbing the liquids one can distinguish them and access their properties, in this case, the viscosity. The difference is also visible in the thermal fluctuations, namely the Brownian motion. If one adds a test particle (e.g. a small, coloured bead) one observes that the Brownian motion in water has a much larger amplitude and occurs faster than in syrup. This is no coincidence, as also in this system dissipation (the sinking object) and fluctuations (the Brownian motion) are linked by a (here classical) regression theorem.

### Kicking a gas of light

In their experiment, the team created and trapped the photon BEC inside a short optical cavity consisting of two mirrors filled with a liquid solution of dye molecules (Figure 1, left). To inject the photon gas, a laser was irradiated onto the molecules, causing the emission of fluorescence photons into the cavity. Through repeated absorption and emission of the photons by the molecules, the photon gas thermalised to a temperature set by the molecules (room temperature) and formed the BEC state. To perturb the system, an additional short laser pulse was irradiated onto the molecules and the subsequent response of the photon condensate population was monitored by recording photons that leaked out of the cavity. This measurement of the response was then compared to the independently measured number fluctuations of the photon condensate that result from the grand canonical particle exchange of the BEC with the dye reservoir. For weak perturbations, the researchers found that both the fluctuations and the response exhibit a corresponding behaviour as a function of time, confirming the validity of the regression theorem (Figure 1, right). Strikingly, even for large perturbations, where the condensate response became nonlinear, their theory model allowed them to confirm the regression theorem.

When perturbed, the condensate coupled to the dye reservoir behaved similar to a spring oscillator (Figure 2). In the classical world, depending on the spring’s stiffness, the oscillator will return to its initial position either with damped oscillations, or a simple decay without oscillations. Similarly, the BEC behaves with a “spring constant” given by the photon population, the molecule number, and the cavity loss. Despite its enormous, microscopic complexity, the molecule-photon system can be described by equations known long before the birth of quantum mechanics.

In the future, the findings pave the way for studies of the frequency-dependent fluctuation-dissipation theorem, so far not observed in BECs. Moreover, the scheme holds prospects to explore photon transport in complex lattices with topological or other exotic properties.

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