Non-stationary Statistics and Formation Jitter in Transient Photon Condensation
Benjamin T. Walker, João D. Rodrigues, Himadri S. Dhar, Rupert F. Oulton, Florian Mintert, Robert A. Nyman
NNon-stationary Statistics and Formation Jitter in Transient Photon Condensation
Benjamin T. Walker,
1, 2
Jo˜ao D. Rodrigues, ∗ Himadri S. Dhar, Rupert F. Oulton, Florian Mintert, and Robert A. Nyman Physics Department, Blackett Laboratory, Imperial College London, Prince Consort Road, SW7 2AZ, UK Centre for Doctoral Training in Controlled Quantum Dynamics, Imperial College London, Prince Consort Road, SW7 2AZ, UK (Dated: Friday 16 th August, 2019, 12:30am)While equilibrium phase transitions are well described by a free-energy landscape, there arefew tools to describe general features of their non-equilibrium counterparts. On the other hand,near-equilibrium free-energies are easily accessible but their full geometry is only explored in non-equilibrium conditions, e.g. after a quench. In the particular case of a non-stationary system,however, the concepts of an order parameter and free energy become ill-defined, and a comprehen-sive understanding of non-stationary (transient) phase transitions is still lacking. Here, we probetransient non-equilibrium dynamics of an optically pumped, dye-filled microcavity which exhibitsnear-equilibrium Bose-Einstein condensation under steady-state conditions. By rapidly exciting alarge number of dye molecules, we quench the system to a far-from-equilibrium state and, closeto a critical excitation energy, find delayed condensation, interpreted as a transient equivalent ofcritical slowing down. We introduce the two-time, non-stationary, second-order correlation function, g (2) ( t , t ), as a powerful experimental tool for probing the statistical properties of the transient re-laxation dynamics. In addition to number fluctuations near the critical excitation energy, we showthat transient phase transitions exhibit a different form of diverging fluctuations, namely timingjitter in the growth of the order parameter. This jitter is seeded by the randomness associatedwith spontaneous emission, with its effect being amplified near the critical point. The experimentalresults are accurately described by a microscopic model of light-matter interactions. The generalcharacter of our observations is then discussed based on the geometry of effective free-energy land-scapes. We thus identify universal features, such as the formation timing jitter, for a larger set ofsystems undergoing transient phase transitions. Our results carry immediate implications to diversesystems, including micro- and nano-lasers and growth of colloidal nanoparticles. I. Introduction
As far back as 1873 [1], Gibbs appreciated the powerof a geometric treatment of free-energies for understand-ing thermodynamic equilibria. Later, Jaynes generalisedthese ideas to non-equilibrium systems [2] and describedhow in cases where an entropy surface can be defined,its geometry fully determines how a “bubble” of prob-ability perturbed by fluctuations evolves in time. Sucha probability bubble encodes the statistical properties ofa non-equilibrium order parameter. A key prediction isthat, in regions where the entropy surface is concave, orequivalently, the free-energy is convex, the probabilitybubble becomes unstable, leading to a bifurcation. Thisinstability has tremendous implications for the dynamicsof order parameters in non-equilibrium systems, particu-larly in cases where a steady state has not or cannot bereached.The evolution of an ordered phase driven through con-figuration space by fluctuations (thermal, or quantum)is well described by an order parameter which evolvesthrough a free-energy landscape [3, 4], even though notall the free-energy surface is explored. Near a steadystate, for instance, fluctuations around thermal equilib-rium allow the free-energy landscape to be locally probed.An equivalent picture can be taken in non-equilibrium ∗ Correspondence to [email protected] systems that can be mapped onto equilibrium statisticalmechanics. A canonical example is the laser [5], a funda-mentally non-equilibrium system whose steady-state canbe described as the minimum of a properly defined ef-fective free-energy, corresponding to a detailed balancebetween driving and dissipation.While the previous arguments are relevant for systemsclose to a steady state, a sudden parameter change, of-ten called a quench, necessarily brings the system suf-ficiently far-from-equilibrium to question the validity ofthe such approaches. The meaning of a quench dependson context and, in particular, one can distinguish be-tween Hamiltonian and non-Hamiltonian cases. The for-mer consist of time-dependent variations in some sortof interaction term, involved, for instance, in the Mottinsulator-superfluid transition [6, 7] or the build-up ofanti-ferromagnetic correlations in Ising models [8]. Non-Hamiltonian quenches contain a more general class ofprocesses. In cold atoms, for instance, the Kibble-Zurekmechanism [9, 10] is observed by evaporatively coolingthe system at a finite rate, quenching the system througha BEC phase transition. We shall refer to a quench asa sudden change in one of the system parameters thatbrings it to a far-from-equilibrium state, without affect-ing its Hamiltonian.Here, we study the transient dynamics of photon con-densation that follows a quench in a dye-filled optical mi-crocavity. Besides measuring the ensemble-averaged pho-ton number dynamics, we introduce the non-stationary,two-time, second-order correlation function g (2) ( t , t ). a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug It provides access to the statistical properties of the pho-ton condensation transition, and is particularly relevantin non-stationary systems, when the full knowledge ofindividual realizations is inaccessible. While the usualstationary correlation function, g (2) ( τ ), accurately ac-counts for fluctuations in steady-state, g (2) ( t , t ) is theappropriate quantity to describe the evolution of tran-sient, non-equilibrium systems. The averaged conden-sate intensity as a function of time shows width broaden-ing, a manifestation of diverging jitter in the condensateformation time upon approaching the critical excitationenergy. This effect is directly witnessed by distinctiveoff-diagonal anti-correlations in g (2) ( t , t ) and originatesfrom quantum fluctuations associated with spontaneousemission. By properly defining an effective free-energy,we argue that such jitter is a universal feature of tran-sient phase transitions in systems obeying relatively gen-eral conditions on the convexity of their free-energy land-scape.This paper is organized as follows: In Sec. (II) wepresent a microscopic cavity model, the experimentalsetup, and discuss the experimental results. By averagingover all forms of correlations and fluctuations we demon-strate a transient equivalent of critical slowing down. Re-taining two-time correlations, we reveal the jitter in thecondensate formation time. From the theoretical model,we construct a quantum trajectories simulation and aquantum regression approach. The former, by keepingcorrelations to all orders, better describes the results. InSec. (III), we explicitly convert the microscopic modelinto an effective free-energy landscape and present anapproach that captures the qualitative features of thesystem dynamics, including formation jitter. The analy-sis in terms of the geometry of the free-energy landscape,being independent of the microscopic details of our par-ticular system, generalizes the phenomenology observedin here to a broader class of systems. In particular, wediscuss immediate implications in micro- and nano-lasersand in the growth of colloidal nanoparticles. II. Non-stationary StatisticsA. Microscopic Cavity Model
Despite the fundamentally multi-mode character of ouroptical cavity, the phenomenology described here is es-sentially that of a single-mode system. Cavity excitations(photons and excited molecules) can be lost by two pro-cesses: mirror transmission and molecular spontaneousemission into free-space, at rates κ and Γ ↓ , respectively.The essentials of the cavity dynamics are described bythe density operator ρ , for both photons and molecules, which obeys the master equation [11–13] dρdt = − i [ H , ρ ] + κ L [ˆ a ] ρ + N mol (cid:88) k =1 (cid:8) Γ ↓ L [ σ − k ] + Γ ↑ L [ σ + k ]+ A L [ˆ a σ + k ] + E L [ˆ a † σ − k ] (cid:9) ρ, (1)with the Hamiltonian for the bare cavity H = ˆ a † ˆ a , E and A the cavity photon emission and absorption rates,respectively, Γ ↑ the incoherent (external) pumping rateand N mol the total number of molecules inside the cav-ity. Due to the high collision rate between dye and sol-vent molecules, all the relevant cavity processes, includ-ing light-matter interactions, are incoherent.Mean-field rate equations are obtained by taking ex-pectation values and neglecting correlation terms inEq. (1). The number of cavity photons n = (cid:104) ˆ a † ˆ a (cid:105) andfraction of excited molecules f = (cid:80) k (cid:104) σ + k σ − k (cid:105) /N mol , with (cid:104)·(cid:105) denoting the (quantum-mechanical) ensemble average,are then determined by˙ n = ( E + A )( f − f c ) n + Ef, and (2)˙ f = − Γ ↓ f + AnN mol (1 − f ) − E ( n + 1) N mol f + Γ ↑ (1 − f ) . (3)Here, the critical excitation fraction is defined as f c = κ + AA + E (4)and, in the limit of high number of photons, corre-sponds to the transition point between a net increaseor decrease in the number of cavity photons with time.At f = f c , and in the absence of pumping and losses( κ = Γ ↓ = Γ ↑ = 0), an equilibrium between molecu-lar excitations and photons is established by a princi-ple of detailed balance [14]. The photon number in thisequilibrium state would show a phase transition as thetotal number of cavity excitations, N ex = n + f N mol ,which is the control parameter, is increased. The pho-ton number, or order parameter, ranges from a disor-dered phase ( n (cid:46)
1) dominated by spontaneous emissionto an ordered phase ( n (cid:29)
1) dominated by stimulatedemission. By exciting a large number of dye moleculesover a short period of time, the cavity can be quenchedthrough this phase transition to a far-from-equilibriumstate. The subsequent relaxation dynamics correspondto a non-stationary, transient counterpart of the equi-librium phase transition described above. However, thenon-linear coupling between photons and molecular exci-tations occurring during this transient relaxation processgives rise to non-trivial fluctuation and correlation prop-erties. Finally, given its lossy character, the light willtransition back to the phase dominated by spontaneousemission before all excitations are lost.A few notes are in order regarding the multi-mode na-ture of our cavity. Within the single-mode approxima-tion, the rate term Γ ↓ accounts for emission both into Dye
FIG. 1. Schematic representation of the experimental set-up. The dye-filled microcavity is composed of one planar (1)and one spherical (2) mirror. The pulsed pump (green) istransmitted through the cavity at an angle of approximately52 ◦ . The transverse ground-state mode (yellow) leaks throughone of the mirrors and is directed both into a spectrometer anda pair of single-photon detectors, B1 and B2, in an Hanbury-Brown-Twiss arrangement. A second pair of detectors, O1and O2, is used to time the beginning of the experiment. free-space and cavity modes that do not reach the regimeof stimulated emission (do not condense), which will bediscussed in more detail in Sec. (II D). Also, and despitenot being relevant for the results discussed here, effectsassociated with the multi-mode character as well as spa-tially resolved molecular reservoirs have been appreciatedin the context of gain clamping [15] and decondensationmechanisms [16]. B. Experimental Set-up
The experimental configuration is sketched in Fig. (1).The optical cavity is composed of one planar and onespherical mirror of 0.25 meters radius of curvature, whichtraps the photons. The cavity is filled with a 2 mM so-lution of Rhodamine-6G in ethylene glycol. All the es-sential dynamics occur at the 10th longitudinal mode,corresponding to a cavity length of approximately 2 µ m.A 40 ps laser pulse at 532 nm, typically ranging from 0.5to 2 nJ in energy, is used to rapidly excite the molecules,quenching the cavity to a far-from-equilibrium state. Inresponse, a much longer pulse ( (cid:38)
598 599 600 601 602 603
Wavelength (nm) − − S p e c t r u m ( a r b . un i t s ) P / P t h O u t p u t ( a r b . un i t s ) FIG. 2. Spectrum of the cavity output, above and below thecondensation threshold at the critical excitation energy P th .The spectral peak is located at the fundamental mode (groud-state) of the cavity, or cavity cutoff, at approximately 602 nm.The inset depicts the total cavity output (dots) and compar-ison with the single-mode mean-field model, with (full line)and without (dashed line) the contribution from the sponta-neous emission background. The arrow indicates the thresh-old point. experiment, with a measured uncertainty of about 10 ps.The cavity output light is directed onto two unsaturatedAPDs, B1 and B2, with an average of 0.1 detections perpulse, on each detector. The experiment is conducted ata repetition rate of 11 kHz. Such a low repetition rateensures a complete decay of all excitations and statisticalindependence between different realizations. We describethe experimental results in the form of three sets:– zero-time statistics: full time-averaged cavity out-put;– one-time statistics: time-resolved, but averagedover all forms of fluctuations and correlations inthe cavity output;– two-time statistics: unequal time, cross-correlatedsignal from detectors B1 and B2, providing accessto fluctuations in the cavity output. C. Zero-time Statistics
We begin by demonstrating the existence of a (con-densation) phase transition in the total amount of lightemitted by the cavity as the excitation energy, or pumpenergy, P is increased beyond a critical value P th , asshown in Fig. (2). The spectra also show a tendency to-wards thermalization, witnessed by robust condensationin the cavity ground-state and indicating a regime wherephoton reabsorption plays a significant role [14, 16]. Con-sequently, and despite the absence of a full thermal distri-bution, parallels may be drawn with Bose-Einstein con-densation of photons [17–20]. Despite only being strictly t (ps) C o un t s Experiment P / P t h t (ps) . . . . P / P t h C o un t s t (ps) N u m b e r o f p h o t o n s Theory P / P t h t (ps) . . . . P / P t h N u m b e r o f P h o t o n s FIG. 3. Output light intensity as a function of the time following the pump pulse. We observe a delay in the growth of thephoton condensate close to the critical excitation fraction P th , accompanied by a large pulse broadening. Experimental resultsare shown on top and simulations of the mean-field rate equations on the bottom. The cavity ground state is located at 602nm (cavity cutoff), the same conditions as in Fig. (2). The left panel depicts a subset of the data on the right. defined in thermal equilibrium as the macroscopic oc-cupation of the ground state, we are assuming here abroader concept of condensation, as discussed in such di-verse fields like physics, ecology, network theory or socialsciences [19, 21–23]. This can be thought as the processwhere a particular, or small set of modes, in a multi-modesystem becomes macroscopically occupied while the re-maining ones saturate or become depleted.By counting the rate of detection events in B1 and B2,we measure the total cavity output as a function of in-put pulse energy (input-output, or light-yield, curves), asshown in the inset of Fig. (2). The threshold, or critical,excitation energy can then be defined as the inflection inthe light-yield curve. The signal from the cavity is col-lected without any filtering and multimode fibres are usedto couple light into the detectors. Since non-condensedmodes are also coupled to the APDs, we model their con-tribution by defining the detected signal I det as the sumof the single condensing cavity mode and a backgroundof spontaneous emission, I det ∝ κn + α Γ ↓ f N mol . Here, α is an empirical parameter set by fitting the light-yieldcurve to the mean-field rate equations. Loosely speak-ing, α determines the fraction of spontaneous emissioninto non-modelled cavity modes and is expected to be asmall contribution, which will be verified in Sec. (II D). D. One-time Statistics
Here, we expand the time-averaged results of the pre-vious section to the time-dependent cavity output pulseshape, as shown in Fig. (3). By collecting unlabelled de-tections on both B1 and B2, we effectively average overany form of correlations and fluctuations. Pulses that . . . . P/P th F o r m a t i o n T i m e ( p s ) Mean Field ∝ ( P − P th ) − Experiment
FIG. 4. Condensate formation time, defined as the intervalbetween the pump pulse and the peak photon number, dis-playing a transient analogue of critical slowing down. Theformation time and corresponding error bars are obtained byfitting Gaussian profiles in a neighbouring region around thepeak photon number. The full-red and black-dashed lines arethe mean-field simulations and the critical (power-law) di-vergence of condensate formation time for the lossless case,respectively. The shaded area depicts the increasing pulsewidth upon approaching the critical excitation energy. form below the threshold excitation energy (
P < P th )display a simple exponential decay at a time scale ofabout τ ∼ t ( p s ) Experiment
P/P th = 2.05 t ( p s ) P/P th = 1.44 t (ps) t ( p s ) P/P th = 1.07 Quantum Trajectories t (ps) Regression Theorem . . . . . . . . . g ( ) ( t , t ) t (ps) FIG. 5. Two-time, non-stationary, second-order correlation function, g (2) ( t , t ), for various excitation energies. Experimentaldata (left column) can be compared with quantum trajectories simulations (middle column) and a semi-analytic quantum re-gression approach based on a master equation expansion up to second-order (right column). In both cases, good agreement withthe experiment requires a spontaneous emission background, as discussed in Sec. (II C). The dashed curves in the experimentaldata correspond to the diagonal and off-diagonal regions of g (2) ( t , t ) depicted in greater detail in Fig. (6). The cavity cutoffis set to λ = 595 nm. Despite the slightly lower cutoff wavelength than that of Sec. (II D), the qualitative features of theaverage cavity output are the same as before. tion, f , and the number of cavity photons, n . In equilib-rium, the molecular excitation fraction, f , cannot exceedits critical value, f c . Under non-equilibrium conditions,however, if at any instant f > f c (e.g. after a quench),the photon population will grow exponentially until f drops below f c . The specifics of this relaxation into equi-librium, dictated at a mean-field level by Eqs. (2) and (3),can show qualitatively different behaviour depending onthe parameters of the cavity. First, we consider the casewhere Γ ↓ is large compared to both κ and E . The evo-lution of the molecular excitation fraction is dominatedby the first term in Eq. (3) and the molecules act as aMarkovian bath of excitations, in the sense that theirdynamics are independent of n . The cavity populationcontinues to grow until loss through the Γ ↓ term causes f to decrease below f c . On the other hand, when Γ ↓ is comparable or small in relation to E and κ , and thesystem is quenched to a state with f > f c , emission intothe cavity plays a significant role in f decreasing below f c and the molecules act as a non-Markovian excitationbath, retaining memory about the lost photons. Thisgives rise to non-trivial correlations in the cavity output,as will be demonstrated in Sec. (II E).We fit Γ ↓ and κ to the results in Fig. (3) and findthe non-Markovian regime described earlier, with furtherevidence provided in Sec. (II E). In particular, κ =10 s − , corresponding to a cavity lifetime of 100 ps, and Γ ↓ =0.998Γ , with Γ = 1 /τ the molecular fluorescencedecay rate, meaning that only 0.2 percent of the totalmolecular emission couples to the cavity. Together withthe light-yield curves in Fig. (2), the contribution of spon-taneous emission coupling into the detectors is found tobe α = 0 .
13; small as expected. The emission and ab-sorption rates are not taken as fitting parameters butrather calculated from experimental absorption and emis-sion data for Rhodamine-6G [24]. The total number ofmolecules is calculated from the dye concentration andcavity volume to be N mol =1.9 × .As f approaches f c from above, the cavity dynamicsbecome slow, as dictated by Eq. (2). In particular, since˙ n ∝ f − f c , one might expect critical slowing down inthe condensate formation time, with a critical exponentof -1, assuming ˙ f ∼ f from remaining close to f c for long times and the en-tire relaxation process is necessarily transient. Despitethe mechanism of true critical slowing down being frus-trated, we still observe a slowing in the time taken forthe condensate to form as we approach the critical ex-citation energy from above, as shown in Fig. (4). Bycomparing to the critical divergence for the lossless case,where formation time is proportional to ( P − P th ) − , weobserve a broadening of the threshold region due to thelossy nature of the cavity. Besides this transient analogue . . . . . . . . g ( ) ( t , t ) Diagonal Correlations
P/P th =2.05 P/P th =1.44 P/P th =1.07 t (ps) C o un t s ( a r b . un i t s ) . . . . . . . g ( ) ( t − τ , t + τ ) Off-diagonal Correlations
P/P th =2.05 ( t =
260 ps)
P/P th =1.44 ( t =
460 ps)
P/P th =1.07 ( t =
740 ps) − − − −
100 0 100 200 300 400 τ (ps) . . . . . . . g ( ) ( t − τ , t + τ ) FIG. 6. Diagonal and off-diagonal correlations in the regions depicted by the dashed lines in Fig. (5). Top left: diagonalcorrelations, g (2) ( t, t ), shown on top, and the average condensate pulse, shown on the bottom left. The full and dashed linesdepict the experimental data and the quantum trajectories results, respectively. Top and bottom right: off-diagonal correlations, g (2) ( t − τ, t + τ ), with t the peak time of the average condensate pulse. Top right depicts the experimental results, with thequantum trajectories simulation on the bottom right. Here, the theory slightly deviate from the experimental results, whichmay be attributed to the error that propagates from determining the peak time t . of critical slowing down, a distinct feature emerges uponapproaching the critical excitation energy, the broaden-ing of the average output pulse. In the next section, weshow that this originates from a particular form of fluc-tuations that arise in such transient phase transitions:jitter in the condensate formation time. E. Two-time Statistics
Correlations and fluctuations of the cavity output cannow be investigated by retaining the labelling of de-tection timestamps in B1 and B2. We then constructthe two-time, non-stationary, second-order correlationfunction g (2) ( t , t ). Second-order correlations are typ-ically described by the single-time g (2) ( τ ) function, with τ = t − t , due to time-translation symmetry in steady-state conditions. In transient systems, however, the ab-sence of this symmetry means that the full two-time, t and t , dependence must be retained. We can then define g (2) ( t , t ) = (cid:104) a † ( t ) a † ( t ) a ( t ) a ( t ) (cid:105)(cid:104) a † ( t ) a ( t ) (cid:105)(cid:104) a † ( t ) a ( t ) (cid:105) ≈ P ( t , t ) P ( t ) P ( t ) , (5)where P ( t , t ) is the joint probability of photon detec-tion at times t and t in detectors B1 and B2, respec-tively. By marginalizing over the second detector, P ( t )and P ( t ) are obtained as the single-detector probabili-ties. The approximation in Eq. (5) is accurate as longas [ a † ( t ) , a ( t )] ≈ (cid:104) a † ( t ) a ( t ) (cid:105) (cid:29)
1. The former issatisfied when | t − t | is larger than the coherence time(much smaller than all relevant time scales involved in the cavity dynamics), and the latter is true for large photonnumbers, as verified in Fig. (3).The second-order correlation function is shown inFig. (5). The two main features to be retained here arethe diagonal positive correlation ( g (2) >
1) and the off-diagonal anti-correlation ( g (2) <
1) lobes. These featuresare mainly a manifestation of the same kind of fluctua-tions – jitter , or shot-to-shot timing fluctuations, in thecondensate formation – which become amplified near thecritical excitation energy. In the remainder of this sec-tion, we discuss this effect associated with transient phasetransitions.Let’s proceed by separately analysing diagonal andanti-diagonal correlations, as shown in Fig. (6). Forequal times, g (2) provides immediate information onnumber, or intensity, fluctuations, namely g (2) ( t, t ) (cid:39) (cid:104) ∆ n ( t ) (cid:105) / (cid:104) n ( t ) (cid:105) . As such, periods of larger fluc-tuations coincide with the inflection point of the aver-age pulse shape, consistent with a condensate forming atslightly different instants in each realization of the exper-iment. In a microscopic picture of the cavity dynamics,spontaneously emitted photons are required to seed thecondensate growth. The randomness associated with thequantum nature of spontaneous emission then leads tosuch shot-to-shot time fluctuations, or jitter in the con-densate formation. As we shall demonstrate in the nextsection, these periods of larger fluctuations correspondto a passage through the convex part of an effective free-energy landscape.The above interpretation is further supported by theoff-diagonal anti-correlation lobes, as seen in Figs. (5)and (6). Given the finite duration of the condensate t (ps) N u m b e r o f p h o t o n s P/P th =2.05 P/P th =1.07 FIG. 7. Quantum trajectories simulation. 50 Individual tra-jectories are shown in light colors while darker colors depicttheir (ensemble) average. The parameters match those ofFigs. (5) and (6). pulse, if a photon is detected at an early time, it is lesslikely that another photon will be detected at a latertime. In other words, the whole light pulse is eitherearly or late. Off-diagonal regions with g (2) < g (2) . However, the quantum trajectories methodallows us to easily appreciate the effect of formation jit-ter, depicted in Fig. (7). The good agreement between experiment and theory in Figs. (5) and (6) is evidencethat individual trajectories in the experiment have a sim-ilar form to those depicted in Fig. (7). Here, the forma-tion jitter becomes clear, with larger shot-to-shot fluc-tuations occurring close to the critical excitation energy,which is at the origin of the pulse broadening describedin Sec. (II D). The exact form of g (2) ( t , t ) depends onboth the individual pulse shapes and their uncertaintyin formation time. As it turns out, the earlier formingpulses (relatively far above threshold) are of shorter dura-tion than later forming pulses (close to the critical point).This effect competes with the larger fluctuations in for-mation time closer to threshold, such that a divergingbehaviour may not be extractable from the g (2) mapsalone. III. Effective Free-Energy
The results presented in Sec. (II) may now be re-interpreted in a more general thermodynamic framework.In that sense, a free energy may be constructed as a func-tion of an order parameter, the geometry of the formerdetermining the evolution of the latter. While free en-ergies are only strictly defined in thermodynamic equi-librium, there are effective analogues for non-equilibriumsystems [5, 35, 36]. In particular, the defining propertyof an effective free-energy F is that it determines theaverage dynamics of the order parameter ψ as [5] dψdt = − ∂ F ∂ψ . (6)If a particular microscopic model is known, in the formof rate equations, for instance, one can reverse engineerEq. (6) to define an effective free-energy landscape [5].Besides the formal equivalence between the two descrip-tions, the free-energy encodes all the relevant dynamicalfeatures in its geometrical properties while, at the same Number of photons n − − F r ee E n e r gy / E × f/f c = 0.9 f/f c = 1.4 f/f c = 1.6 f/f c = 1.9 FIG. 8. Effective free-energy landscape for the microcavityphoton number (order parameter), calculated for different val-ues of the initial molecular excitation fraction f . Besides be-ing the quantity directly varied in the experiment, f deter-mines the control parameter as N ex = n + fN mol Number of photons n − . − . − . . . . F r ee E n e r gy / E × time f/f c = 1.9 − − − − − P D F
261 (ps)348 (ps)391 (ps)435 (ps)
Number of photons n − . − . − . . . . F r ee E n e r gy / E × time f/f c = 2.0 − − − − − P D F
152 (ps)217 (ps)248 (ps)283 (ps)
FIG. 9. Probability distribution function (PDF) of photonnumber (colored lines), describing fluctuations of the orderparameter. The circles indicate the expected value of n at agiven time instant. The random walks (50000) are initializedat n = 0 and the descent through the effective free-energylandscape (dark lines), as defined in Eq. (7), is seeded byspontaneous emission. While the latter is essentially Gaus-sian noise, the free-energy convexity far from the equilibriumpoint induces non-Gaussian, heavy-tailed statistics in photonnumber. This skewed statistics partially explains the limita-tions of the quantum regression model described in Sec. (II). time, becoming independent of the microscopic details ofa particular system. This allows us to predict universalproperties for systems sharing similar free-energy geome-tries. Also, fluctuations can be built back into this simplemodel by including a stochastic term η ( t ) in Eq. (6) andturning it into a Langevin equation.Going back to Eqs. (2) and (3), and ignoring cavitylosses and pumping ( κ = Γ ↓ = Γ ↑ ≡ n ) = − (cid:90) n ˙ n (cid:48) dn (cid:48) = − EN ex N mol n − (cid:20) E ( N ex − N mol − A (cid:18) − N ex N mol (cid:19)(cid:21) n
2+ ( E + A ) N mol n . (7)Here, the order parameter is the number of cavity pho-tons ( ψ ≡ n ), while the control parameter is given by the total number of cavity excitations, as defined earlier.Fig. (8) depicts the free-energy for different (initial) ex-citation fractions, which allows for a direct analogy withthe experiment, where the pump energy determines theinitial excitation fraction, with the cavity being initial-ized with n = 0. Moreover, all the parameters used herematch those of Sec. (II).We consider the evolution of a probability distributionfunction (PDF) across the free-energy surface defined byEq. (7). This PDF encodes qualitative information aboutthe fluctuation properties of the order parameter as ittransiently evolves from the far-from-equilibrium statethat follows a quench. We consider a large set of randomwalks evolving over the free-energy landscape. Thesewalks (in photon number space) obey the Langevin equa-tion described before where η ( t ) accounts for spontaneousemission, such that (cid:104) η ( t ) η (0) (cid:105) = f E δ ( t ) [37, 38]. Fromthis ensemble of trajectories we estimate the order pa-rameter PDF, as shown in Fig. (9). An alternative, andformally equivalent, approach would be the constructiona Fokker-Planck equation with a drift term given by thederivative of the free energy, as in Eq. (6), and a diffu-sion term describing the fluctuations from spontaneousemission.Quite generally, fluctuations act to broaden the photonnumber PDF while a positive (concave) curvature in thefree-energy tends to localise it. At a second-order phasetransition, the curvature at the minimum of the free en-ergy (defining an equilibrium order parameter) vanishesand the PDF shows diverging fluctuations which per-sist for long times, giving rise to critical slowing down.In transient, non-equilibrium systems dramatic featuresalso occur, with regions of negative (convex) curvatureacting to amplify fluctuations. A PDF evolving throughthese regions while relaxing towards the free-energy min-imum experiences a short-lived but large increase of fluc-tuations, as shown in Fig. (9). The jitter described inSec. (II) is the immediate consequence of this. The max-imum of number fluctuations, marked by the peak in g (2) ( t, t ), and shown in Fig. (6), occurs when the averageorder parameter reaches the free-energy inflection point.From Eq. (6), this corresponds to the (temporal) inflec-tion point of the order parameter, in complete agreementwith the results depicted in the previous section. Thewidth of the photon number PDF then shrinks back asthe order parameter evolves to the concave part of F .While the free-energy landscape is more convex for largervalues of the control parameter, which increases numberfluctuations, these are longer-lived close to threshold. Atthe level of individual realizations, this means that thetiming jitter in the formation of the condensed phase islarger close to the critical excitation energy.In the presence of loss by cavity transmission, the free-energy and photon number PDF are coupled in a nontriv-ial way. For sufficiently large κ , as in the case of the ex-periment described in Sec. (II), the rate of change of thefree energy landscape depends on the photon number n ,which is itself described by a given PDF. The landscape isnow neither constant, nor a simple function of time, butrather coupled to the photon number history, such thatfor trajectories where the condensate forms early, it alsodecays early, leading to the anti-correlation lobes seenin g (2) ( t , t ). This is essentially the same result as de-picted by the quantum trajectories simulation in Fig. (7)but reinterpreted under the geometrical properties of theeffective free-energy landscape.The free-energy description assumes the total numberof cavity excitations N ex , the control parameter, to befixed, such that all molecular excitations are convertedinto cavity photons. In the experiment, this approxi-mation is only valid in the non-Markovian regime de-scribed earlier, where molecular de-excitation by stimu-lated emission dominates over direct emission into free-space. In the Markovian regime, where losses of molecu-lar excitation are dominated by the Γ ↓ term, both criticalslowing down and timing jitter in the condensate forma-tion vanish. As a final remark, Eq. (6) allows for a formalreconstruction of the free-energy landscape, F ( ψ ), fromthe observed average dynamics of the order parameter, ψ ( t ), although in practice the results are not very infor-mative. IV. Conclusions
In this work, we have described the transient non-equilibrium dynamics of light in a dye-filled optical cavityquenched through a condensation phase transition. Byrapidly exciting a large number of dye molecules, thesystem is brought to a far-from-equilibrium state. Byaveraging over all forms of fluctuations, we observed adelayed formation of the condensed phase, interpreted asa transient equivalent of critical slowing down. Whenquenched above the condensation threshold excitationenergy, the quantum fluctuations associated with spon-taneous emission seed the growth of the order parameteras the system relaxes into equilibrium. The relaxationdynamics is slower close to the critical point, a featureeasily interpreted under the geometrical properties of theeffective free-energy landscape, which becomes flat. Thesame mechanism is responsible for the usual critical slow-ing down in the relaxation rate of the ordered phase thatfollows a second-order phase transition. Also, despitethe absence of latent heat and the fact that we are deal-ing with second-order and not first-order phase transi-tions, analogies can be drawn with the precipitation insupercooled, or supersaturated, liquids. Even quenchedabove the critical point, a seed of spontaneously emittedphotons is needed to nucleate condensation, playing therole the seeding crystals in supercooled, or supersatu-rated, liquids. Also, once seeded, crystallization acrossthe entire liquid is faster for liquids quenched furtheracross their critical parameters, with temperature play-ing the same role as the excitation fraction that followsthe quench, in the optical cavity context.By measuring the statistical properties of this tran- sient condensation, we describe a novel form of divergingfluctuations around the critical point, jitter in the forma-tion of the ordered phase. These are witnessed by strongdiagonal correlations and off-diagonal anti-correlationsin the non-stationary, second-order correlation function, g (2) ( t , t ). More precisely, we demonstrated that whilethe diagonal of g (2) is a powerful probe of the geometricalproperties of the free-energy landscape, its off-diagonalelements reflect the relevant dissipation processes, withthe anti-correlation lobes a joint effect of jitter and cav-ity loss. Fluctuations, arising from spontaneous emission,are highly amplified as the order parameter goes throughthe convex part of the free-energy landscape towards itsequilibrium point. The condensation jitter is a directphysical manifestation of this increasing fluctuations, inaccordance with the ideas put forward in Ref. [2].The description in terms of the geometric propertiesof the effective free-energy landscape, being independentof the microscopical details of our particular system, al-lows us to generalize our observations. In particular,both the transient critical slowing down and the jitterin the formation of the order parameter are expectedto be universal features of the dynamics that followsa quench through a second-order phase transition. Inmicro- and nano-lasers, in particular, the full two-time,non-stationary analysis of the relaxation process has beengreatly overlooked and previous results [39–42] may nowbenefit from being re-examined. Despite some recent ef-forts in this direction [43–45], and to the best of ourknowledge, we present here for the first time a genericand comprehensive description of the relation betweentemporal and number fluctuations in the non-stationarydynamics of systems undergoing second-order phase tran-sitions. Implications of our results may also be of con-cern in the growth of colloidal nanoparticles. These resultfrom a nucleation process in a supersaturated chemicalsolution, and the growth mechanism can be describedby the minimization of a free-energy [46, 47]. Finally,the system studied in this work, as well as the relatedexamples stated above can be described by single-valueorder parameters. One may wonder on the generalizationof these effects in spatially extended systems, where theorder parameter is a function of both space and time. Acknowledgements
We acknowledge financial support from EPSRC (UK)through the grants EP/S000755/1, EP/J017027/1, theCentre for Doctoral Training in Controlled QuantumDynamics EP/L016524/1 and the European Commis-sion via the PhoQuS project (H2020-FETFLAG-2018-03) number 820392. We also thank Julian Schmitt forhelpful discussions. The data related to this paper maybe requested from the authors or via [email protected] .0 V. AppendixA. Second-order rate equations and the quantumregression theorem
From the non-equilibrium model introduced in Eq. (1),one can derive rate equations for the ensemble-averagedphoton number, (cid:104) n (cid:105) = (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) , and the number ofexcited molecules, (cid:104) m (cid:105) = (cid:80) k (cid:104) σ + k ( t ) σ − k ( t ) (cid:105) , as d (cid:104) n (cid:105) dt = − κ (cid:104) n (cid:105) + E {(cid:104) m (cid:105) + (cid:104) nm (cid:105)} − A {(cid:104) n (cid:105) N mol − (cid:104) nm (cid:105)} d (cid:104) m (cid:105) dt = − E {(cid:104) m (cid:105) + (cid:104) nm (cid:105)} + A {(cid:104) n (cid:105) N mol − (cid:104) nm (cid:105)}− Γ ↓ (cid:104) m (cid:105) + Γ ↑ ( N mol − (cid:104) m (cid:105) ) . (A1)The calculation of (cid:104) nm (cid:105) depends on the estimation of (cid:104) n m (cid:105) , (cid:104) nm (cid:105) , (cid:104) n (cid:105) , · · · , which requires solving a largenumber of ordinary differential equations. These can bereduced with an hierarchical set of approximations. Forinstance, in the semi-classical limit, the expectation val-ues for n and m are factorized, (cid:104) mn (cid:105) ≈ (cid:104) m (cid:105)(cid:104) n (cid:105) , reducingEqs. (A1) to˙ n = − κn + E f ( n + 1) − A n (1 − f )˙ f = − Γ ↓ f + A n (1 − f ) − E ( n + 1) f + Γ ↑ (1 − f ) . These are equivalent to Eqs. (2) and (3). Here, we define f = (cid:104) m (cid:105) /N mol as the molecular excitation fraction andset, for the ease of notation, n = (cid:104) n (cid:105) . Despite ignoringcorrelations all-together, this corresponds to a first levelapproximation to the non-equilibrium cavity dynamics.In order to account for correlations and fluctuations,one needs to go beyond the semi-classical approximation.In particular, the expectation values can be expanded ina hierarchical manner [26–29] given by σ xy = (cid:104) xy (cid:105) − (cid:104) x (cid:105)(cid:104) y (cid:105) (A2) σ xyz = (cid:104) xyz (cid:105) − (cid:88) σ xy (cid:104) z (cid:105) − (cid:104) x (cid:105)(cid:104) y (cid:105)(cid:104) z (cid:105) (A3) σ wxyz = (cid:104) wxyz (cid:105) − (cid:88) σ wxy (cid:104) z (cid:105) − (cid:88) σ wx (cid:104) y (cid:105)(cid:104) z (cid:105)− (cid:88) σ wx σ yz − (cid:104) w (cid:105)(cid:104) x (cid:105)(cid:104) y (cid:105)(cid:104) z (cid:105) . (A4)These represent the second, third, and fourth ordercumulants, with the summation referring to all possi-ble combination of variables. A minimal description ofcorrelations is constructed by truncating the hierarchyat second-order. In this way, and by defining σ x = (cid:104) x (cid:105) − (cid:104) x (cid:105) , with x = { n, m } , we explicitly write˙ n = − κn + E { ( n + 1) m + σ nm } − A { n ( N mol − m ) − σ nm } (A5)˙ m = − Γ ↓ m − E { ( n + 1) m + σ nm } + Γ ↑ ( N mol − m ) + A { n ( N mol − m ) − σ nm } (A6)˙ σ n = − κ ( n + 2 σ n ) + E { ( n + 1) m + 2 σ n m + σ nm (2 n + 1) }− A { n ( N mol − m ) + 2 σ n ( N mol − m ) − σ nm (2 n − } (A7)˙ σ m = − Γ ↓ ( m + 2 σ m ) − E {− ( n + 1) m + 2 σ m ( n + 1) + σ nm (2 m − } + A { n ( N mol − m ) − σ m n + σ nm ( − m + 2 N mol − } + Γ ↑ ( N mol − m − σ m ) (A8)˙ σ nm = − ( κ + Γ ↓ + Γ ↑ ) σ nm + E { ( n + 1)( − m + σ m ) − σ n m + σ nm ( m − n − } + A {− n ( N mol − m ) + σ m n + σ n ( N mol − m ) + σ nm ( m − n + 1 − N mol ) } . (A9)The second-order photon correlation function at zerotime delay, g (2) ( t ), follow immediately as g (2) ( t ) = (cid:104) ˆ a † ( t )ˆ a † ( t )ˆ a ( t )ˆ a ( t ) (cid:105)(cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) = (cid:104) n ( t ) (cid:105) − (cid:104) n ( t ) (cid:105)(cid:104) n ( t ) (cid:105) , = 1 + σ n ( t ) − n ( t ) n ( t ) . (A10)The two-time second-order correlation function canbe obtained by invoking the quantum regression theo-rem [30], which allows us to calculate any quantity of theform (cid:104) X ( t + τ ) Y ( t ) (cid:105) using two single-time evolutions. Letthe initial state of the system be χ (0), and the evolutionbe given by the map, χ ( t ) = V ( t, t (cid:48) ) χ ( t (cid:48) ). The two-timeexpectation value can then be written as (cid:104) X ( t + τ ) Y ( t ) (cid:105) = Tr[ X V ( t + τ, t ) { Y χ ( t ) } ] , = Tr[ X V ( t + τ, t ) { Y V ( t, χ (0) } ] . (A11) The two-time function is thus calculated by evolving χ (0)from 0 to t , followed by the conditional state Y χ ( t )from t to t + τ . For our cavity model, we begin byfirst evolving the density operator, ρ , from t = 0 to t = t , using the second-order rate Eqs. (A5) and(A9),obtaining g (2) ( t ). Second, the first-order rate Eqs. (2)and (3) are used to evolve the conditional state, ˜ ρ =ˆ a ( t ) ρ ˆ a † ( t ) / (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) from t = t to t = t . Follow-ing Eq. (A11), one then arrives at the two-time photoncorrelation function, g (2) ( t , t ) = (cid:104) ˆ a † ( t )ˆ a † ( t )ˆ a ( t )ˆ a ( t ) (cid:105)(cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) . (A12)1 B. Quantum trajectories approach
The second-order approach described above corre-sponds to a first-level approximation to the descriptionof correlations and fluctuations in the cavity dynamics.Moving to higher-order expansions increases the num-ber of ordinary differential equations needed to resolvethe dynamics, which soon becomes cumbersome and im-practical. An alternative approach to solve the masterEq. (1), is to use the quantum trajectories (or Monte-Carlo wavefunction) method [31–34]. Here, the Lindbladdynamics of the density operator ρ is replaced by a wave-function whose evolution is given by a non-Hermitian ef-fective Hamiltonian, interspersed with stochastic quan-tum jumps. Subsequently, evolution of ρ is approximatedby an ensemble-average of wavefunctions, or trajectories,say | ψ i (cid:105) . For a large number of trajectories, z , the aver-age of any observable is then given by (cid:104) ˆ X ( t ) (cid:105) = Tr[ ˆ Xρ ( t )] ≈ z z (cid:88) i =1 (cid:104) ψ i ( t ) | ˆ X | ψ i ( t ) (cid:105) . (B1)The effective non-Hermitian Hamiltonian for the non-equilibrium cavity model in Eq. (1) is given by H eff = H − i (cid:88) k J † k J k , (B2)where J k are the jump operators defining the stochas-tic dynamics. In the non-equilibrium cavity model, co- herences cannot be created by H eff . Hence, if a quan-tum trajectory starts in a particular number state, say | ψ i (0) (cid:105) = | n , m (cid:105) , the action of H eff alone does notchange the state in this number basis. The completedynamics of the trajectory is simply governed by thestochastic jumps J k , occuring at rates R k : √ κ ˆ a : | n, m (cid:105) → | n − , m (cid:105) ; R = κn (cid:112) Γ ↑ σ + : | n, m (cid:105) → | n, m + 1 (cid:105) ; R = Γ ↑ ( N − m ) (cid:112) Γ ↓ σ − : | n, m (cid:105) → | n, m − (cid:105) ; R = Γ ↓ n √ E ˆ a † σ − : | n, m (cid:105) → | n + 1 , m − (cid:105) ; R = E ( n + 1) m √ A ˆ aσ + : | n, m (cid:105) → | n − , m + 1 (cid:105) ; R = An ( N − m ) . 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