Intense Light Amplifies Weak Signals at a Critical Point of 1250 Photons

Systems undergoing phase transitions exhibit amplified responses to even the smallest disturbances. Ross C. Schofield of Imperial College London, Indian Institute of Technology and University of Sheffield and colleagues experimentally confirm this amplification extends to driven-dissipative quantum systems, maintained through continuous energy input and loss, challenging previous assumptions about equilibrium-based criticality. Their work, utilising a room-temperature semiconductor photon Bose-Einstein condensate, pinpoints a critical condensate population of 1250 particles where both spontaneous fluctuations and external perturbations are maximally amplified, governed by a single collective mode. This fluctuation-response correspondence establishes critical susceptibility as a measurable indicator of condensation and reveals that the peak gain is directly linked to system size.

Fluctuation-response links reveal condensation via critical susceptibility measurement

At a condensate population of 1250 (nc = 1250), the dimensionless slowing factor and susceptibility converged to the same value, nc/2 = 625. A single, weakly damped collective photon-reservoir mode governs both these effects. This fluctuation-response correspondence, observed in a finite open quantum gas, establishes critical susceptibility as a measurable dynamical signature of condensation, with peak gain determined by system size. Equilibrium phase transitions possess universal features, independent of microscopic properties, where restoring forces weaken, fluctuations grow, and weak perturbations can drive large responses.

Genuine critical behaviour in finite-sized, open, and non-equilibrium quantum systems remains a profound and largely open question. Real quantum systems are inherently open, continuously exchanging energy and particles with their surroundings, and are often realised at small scales where size effects are unavoidable. Driven-dissipative quantum systems offer a pathway to explore this question experimentally. Unlike systems weakly coupled to a thermal reservoir, these require continuous replenishment of particles to exist.

Despite this continuous replenishment, these systems can exhibit macroscopic order and phase-transition-like behaviour. However, the reservoirs, gain saturation, and dissipation sustaining the steady state can fundamentally reshape collective modes, fluctuation dynamics, and response functions. A driven-dissipative condensate therefore provides an ideal system to test whether spontaneous fluctuations and driven susceptibility remain governed by the same collective dynamics beyond equilibrium.

Photon Bose-Einstein Condensates (BECs) exhibit a transition to a macroscopic quantum state of light in a driven, dissipative setting, yet demonstrate equilibrium-like behaviour. These are particularly attractive due to their demonstrated equilibrium-like characteristics at room temperature, including thermalisation, grand canonical fluctuations, continuous wave operation, and controllable system size. Recently, inorganic semiconductor implementations have enabled electrically driven and continuously sustained photon condensates, opening new avenues for exploring this physics.

As photons condense, a finite, continuously driven semiconductor photon condensate develops a giant critical response. Measurements of critical slowing directly reveal the condensate’s intrinsic intensity fluctuations, while independent assessment quantifies the amplification of weak excitation perturbations. Both signatures peak at the same critical condensate population, nc ≈1250, with the same dimensionless enhancement factor, nc/2. This direct proportionality with system size stems from a dynamical model, yet captures the expected phase transition physics.

The examination focused on a photon Bose-Einstein condensate within an open semiconductor microcavity. This cavity comprises a planar distributed Bragg reflector (DBR) half-cavity containing a single InGaAs quantum well and a concave dielectric mirror. Operated on the 15th longitudinal mode near 948nm, the cavity is continuously pumped at 785nm. Pump absorption in the GaAs generates carriers that relax into the quantum well and thermalise with the photons through recombination and reabsorption, enabling room-temperature Bose-Einstein condensation of photons.

Emitted light is spectrally filtered and analysed using second-order intensity correlations g(τ) and pump-modulation spectroscopy, providing direct access to both spontaneous fluctuation dynamics and driven response. The normalised excitation P/Pth and the condensate occupation n parameterise the operating point of the photon BEC. While P/Pth serves as the experimental control variable, n represents the natural state variable for comparison with the dynamical model, as it is the square magnitude of the system’s order parameter. A simple Landau laser-model illustrates how large system fluctuations emerge due to reduced restoring forces at the laser phase transition, depicting free energy as a function of order parameter and normalised excitation.

Figures 1(b-e) summarise the physics of the transition as a function of P/Pth, with the growth of n with excitation closely matching a steady-state rate-equation model. The finite system size results in a non-zero value for n at the critical point, unlike the Landau model. Greater intensity fluctuations, persisting for longer, accompany the growth in n at the critical point. Measurements of the second-order intensity correlation function g(τ), representative traces of which are shown in Fig0.2(b), yield values for the relaxation time of the intensity fluctuations τslow.

In the grand-canonical regime where g > 1 remains resolvable, g(τ) ≈1 + ab e−|τ−τ0|/τslow is fitted, where ab quantifies the bunching amplitude, τ0 accounts for the fixed channel delay, and τslow is the fluctuation relaxation time. Pronounced slowing peaks occur at τslow ≈50ns, approximately 250 times longer than the bare cavity lifetime (τc = κ−1 = 200 ±40ps) and exceeding the reservoir recombination time (τrad = A−1 = 0.7 ±0.1ns). This behaviour confirms that a collective relaxation mode of the coupled photon-reservoir becomes weakly damped near the critical point. As the emission becomes dominated by the ground state mode, g 7→1, consistent with near-Poissonian statistics of the canonical regime.

A Langevin model for the coupled evolution of photon number n and reservoir carrier population N is employed to describe the observed slowing. Systems nearing a phase transition amplify weak perturbations into large responses. At equilibrium, this amplification links to criticality, where fluctuations grow and dynamics slow, controlled by a soft mode. Whether this holds in driven-dissipative quantum systems, sustained by pumping and loss away from equilibrium, was previously unclear.

Experiments demonstrate this does occur in a room-temperature semiconductor photon Bose-Einstein condensate, where critical slowing of intensity fluctuations and amplification of pump perturbations were measured independently. Both peak at a condensate population of 1250 photons, with a dimensionless enhancement factor of 625. A weakly damped photon-reservoir mode governs both effects, establishing critical susceptibility as a measurable signature of condensation, with gain determined by system size. Fitting this equation to the measured τslow( n) yields β = 1.7 ±0.2 × 10−6 and γT = 2.4 ±0. They then studied how the slow condensate dynamics near the critical point affects the system susceptibility to a driven excitation modulation.

Microsecond-scale oscillations in g(τ) are observed near the critical point. These oscillations are much slower than the intrinsic critical-slowing time τslow ≲50ns, indicating that their period is set not by the condensate dynamics themselves but by an external perturbation. The oscillation frequency corresponds to a weak intensity-noise feature of the excitation source around 2.1MHz, apparent in the fast-photodiode spectrum. The fractional modulation of the excitation is ∆P/P = 1.5 ±0.1 × 10−3, while the corresponding condensate modulation reaches ∆ n/ n ∼625. Consequently, weak excitation fluctuations induce a macroscopic modulation of the condensate. To quantify this amplification, the dimensionless dynamical gain G(Ω) ≡∆ n(Ω) n / (P ∆P(Ω)) is defined, where ∆P(Ω)/P is the fractional modulation of the pump at frequency Ω, and ∆ n(Ω)/ n is the corresponding fractional modulation of the condensate occupation inferred from the oscillatory component of g(τ).

Peak amplification signals condensate formation in a room-temperature Bose-Einstein condensate

Critical slowing of spontaneous fluctuations and amplified response to perturbations both peaked at a condensate population of 1250 photons, representing a 625-fold increase over baseline measurements. This unprecedented level of amplification establishes a measurable dynamical signature of condensation, previously unattainable in driven-dissipative quantum systems. The findings establish critical susceptibility as a key indicator of condensation and demonstrate peak gain is directly linked to system size, offering a new design rule for amplified optical transduction.

This fluctuation-response correspondence in a finite open quantum gas provides insight into how critical behaviour emerges in driven-dissipative quantum systems. Measurements validated these findings by examining the amplification of weak pump perturbations and the critical slowing of spontaneous intensity fluctuations, both peaking at a condensate population of approximately 1250 photons. The dynamics are governed by a single weakly damped collective photon-reservoir mode. Analysis revealed that the dimensionless slowing factor and susceptibility reach the same value at 625, establishing critical susceptibility as a measurable signature of condensation with peak gain determined by system size.

Critical slowing and response amplification in a room-temperature photon Bose-Einstein condensate

Establishing a link between critical slowing and amplified response offers a potential route towards novel sensing technologies. However, a key limitation exists; the demonstrated correspondence relies on observations within a specific system, a room-temperature semiconductor photon Bose-Einstein condensate. This raises the question of universality, as the findings haven’t been proven to extend to all driven-dissipative quantum systems. A Bose-Einstein condensate is a state of matter formed when bosons are cooled to near absolute zero, with photons acting as the bosons in this case.

Demonstrating this fundamental link between critical slowing and amplified response within any quantum system validates a crucial theoretical prediction. This work confirms a direct link between critical slowing and amplified response in a driven-dissipative quantum system. The findings reveal a single, weakly damped collective mode governs both the slowing of spontaneous emissions and the amplification of external perturbations, offering a new means to characterise condensation.

The research demonstrated a correspondence between critical slowing and amplified response in a room-temperature semiconductor photon Bose-Einstein condensate. This means that as the system approached condensation, reaching a population of approximately 1250 photons, fluctuations slowed and the system became more sensitive to external stimuli. Both effects peaked at the same point, governed by a single collective mode, establishing critical susceptibility as a measurable signature of condensation. This finding validates theoretical predictions about how critical behaviour emerges in quantum systems sustained by continuous energy input and loss.