From polariton condensates to highly photonic quantum degenerate states of bosonic matter

Abstract

Bose–Einstein condensation (BEC) is a thermodynamic phase transition of an interacting Bose gas. Its key signatures are remarkable quantum effects like superfluidity and a phonon-like Bogoliubov excitation spectrum, which have been verified for atomic BECs. In the solid state, BEC of exciton–polaritons has been reported. Polaritons are strongly coupled light-matter quasiparticles in semiconductor microcavities and composite bosons. However, they are subject to dephasing and decay and need external pumping to reach a steady state. Accordingly the polariton BEC is a nonequilibrium process of a degenerate polariton gas in self-equilibrium, but out of equilibrium with the baths it is coupled to and therefore deviates from the thermodynamic phase transition seen in atomic BECs. Here we show that key signatures of BEC can even be observed without fulfilling the self-equilibrium condition in a highly photonic quantum degenerate nonequilibrium system.

Microcavity polaritons are composite bosons, which are partly photons and partly excitons as quantified by the Hopfield coefficients |C²| and |X²| giving the relative photonic and excitonic content (1), respectively, and are expected to condense at high temperatures because of their light mass (2). Moreover, the photonic and excitonic contents of polaritons can be precisely adjusted by changing the detuning Δ = E_c - E_x between the bare cavity mode and the bare exciton mode. Unlike Bose–Einstein condensates (BECs) in atomic gases, solid-state systems are subject to strong dephasing and decay on timescales on the order of the particle lifetimes. As a consequence, external pumping is required to achieve a steady state. Despite this nonequilibrium character, degenerate polariton systems show several textbook features of BECs (3), including spontaneous build up of coherence (4) and polarization (5), quantized vortices (6, 7), spatial condensation (8), and superfluidity (9, 10). Usually this behavior is attributed to the system undergoing an equilibrium phase transition toward a condensed state: Although the polariton gas is not necessarily in equilibrium with the lattice, it is in self-equilibrium, if the relaxation kinetics of excited carriers is fast enough compared to the leakage of the photonic component out of the cavity, which is usually the case for positive detunings Δ≥0. In this case |X²| is larger than 50%, an effective temperature can be defined and the polariton gas can be considered as a thermodynamic equilibrium state, which is out of equilibrium with the baths it is coupled to. This degenerate polariton gas is distinguishable from a simple photon laser (11). It is often pointed out that this intrinsic nonequilibrium situation and the two-dimensional order parameter of polariton BECs give rise to interesting phenomena like half-vortices (12) and a diffusive Goldstone mode (13, 14), which do not occur in equilibrium condensates. Accordingly, the next interesting questions are whether the same or even unique phenomena occur, if the system is driven even further from equilibrium into a regime that cannot be described by an effective temperature and whether it is indeed necessary to consider a thermodynamic equilibrium phase transition and thermalized polariton gases to have condensation effects. It should be pointed out that the polariton system as a whole is almost never in thermal equilibrium. Usually it will consist of the low-energy part of the polariton distribution which may be described by an effective temperature if this part reaches a local self-equilibrium, the high-energy exciton-like part thermalized at the lattice temperature, and the intermediate bottleneck regime which is almost always out of equilibrium. It has been stated that such a local self-equilibrium condensed state is directly related to BEC, whereas one which does not obtain a local self-equilibrium can hardly be described in the framework of BEC (15). Although this statement is true in terms of thermodynamic properties, it is not immediately clear which of the signatures seen in local self-equilibrium polariton condensates will still prevail far from even local self-equilibrium. Recent theoretical analysis (16) predicts condensation effects even when the local self-equilibrium condition is not fulfilled and also not only for the equilibrium Bose–Einstein distribution, but also for a wide range of nonequilibrium distribution functions. However, although there are theoretical studies classifying the condensate ground state for different detunings and mean polariton separations (17), from an experimental point of view, detailed studies of this regime without local self-equilibrium are still missing.

We realize a system far from equilibrium using polaritons with high-photonic content, characterized by a negative detuning Δ < 0 between the bare cavity and exciton modes, and excited by short laser pulses with duration of 1.5 ps. The fast decay times because of the high-photonic content of the polaritons ensure that the polariton distribution is neither in thermal equilibrium with the lattice nor in local self-equilibrium and a stationary situation is never reached. The thermalization times for our sample agree well with values previously reported (18) where the polariton thermalization time becomes shorter than their lifetime above the degeneracy threshold for positive detunings, but is always longer than their lifetime for negative detunings. Although there have been experiments where this regime was called out of equilibrium condensation (19) or metastable condensate (15), so far no systematic studies characterizing the condensate-like properties in this regime where it is not possible to define an effective temperature have been performed. We will refer to highly photonic (HI-P) polariton states when discussing this nonequilibrium regime in order to stress that it is different from common polariton condensates which are also nonequilibrium states, but can be described by an effective temperature. In the following we test HI-P states for several common signatures of condensation like build up of a macroscopic ground-state occupation, suppressed quantum fluctuations, and linearization of the excitation spectrum.

Basic characteristics of condensation and the excitation spectrum are manifested in the polariton dispersion. Above threshold a blueshift of the k_|| = 0 lower polariton (LP) is expected and the LP dispersion is supposed to change from a parabolic shape in the uncondensed case toward a different dispersion in the low-momentum condensed regime |k_∥ξ| < 1, where is the healing length of the condensate (20). In standard homogeneous equilibrium Bogoliubov theory, the expected dispersion is a phonon-like linear one at small k given by

Here g and g_r are coupling constants describing the interaction between two condensate polaritons and between condensate polaritons and reservoir excitons, respectively, and n_c and n_r are their densities. Using resonant excitation, the reservoir exciton contributions are expected to be negligible. Highly photonic condensed states are neither in thermal equilibrium, nor spatially homogeneous. Their spatial extent is given by the finite size of the pump spot. Therefore one might expect their dispersion to show strong deviations from the ideal homogeneous equilibrium Bogoliubov dispersion. In particular, the finite pump spot size causes quantization and the large amount of pump and decay causes the excitation spectrum to become diffusive at low-momentum (13, 14),

where is the effective decay rate of the Bogoliubov mode, γ is the bare cavity decay rate, , where thr means threshold, gives the relative pump strength compared to the threshold value, and β quantifies the dependence of the condensate amplification rate because of stimulated scattering of polaritons on the reservoir population. In the following we assume the effect of β to be negligible above threshold, so that the variation of Γ between zero at threshold and γ at large pump rates is governed by α alone. This diffusive spectrum is expected to manifest itself as a flat region in the low-momentum region of the real part of the excitation spectrum. The range of this flat region is expected to increase with α, starting from an unmodified Bogoliubov dispersion exactly at the threshold, where α is exactly zero. In Fig. 1 we compare the LP dispersion of the emission copolarized with the excitation for detunings of 0, -2, -4, and -7 meV to the theoretical predictions below and above threshold. Below threshold (Fig. 1 A, E, I, and M) the standard quadratic LP dispersion is observed in all cases. Increasing the excitation power near the threshold value (Fig. 1 B, F, J, and N), a blueshift of the k_|| = 0 emission energy becomes apparent, whereas the cross-circularly polarized emission component never shows threshold-like behavior. In addition to the blueshift of the copolarized emission, the dispersion starts to differ from the standard quadratic dispersion. Blue and white lines in Fig. 1 give the quadratic LP and calculated equilibrium Bogoliubov dispersions for the condensate blueshift. We use the experimentally determined values of the blueshift for the interaction energies in calculating ω_Bog. The prediction for the diffusive modes ω_± is not shown because at the threshold the real part of the positive branch of the diffusive mode does not differ from the equilibrium Bogoliubov dispersion. It is apparent that at threshold the equilibrium Bogoliubov dispersion cannot reproduce the measured dispersion accurately for Δ = 0, -2, or -4 meV. Here the measured dispersions lie approximately in the middle between both theoretical dispersions, and the dispersions are not even symmetric with respect to k_|| = 0. The latter feature becomes more apparent with increasing negative detuning and is a signature of the nonequilibrium state favoring the presence of polaritons having a wavevector with the same sign as the pump incidence wavevector, although the first can be attributed to the inhomogeneity of the system: Although the LPs with k_|| = 0 are stationary, LPs with k_|| ≠ 0 will move across the excitation spot and experience a different interaction energy given by the spatial pump pulse profile. However, it is striking that the theoretical prediction again matches the experimental data well for the Δ = -7 meV dispersion recorded at threshold. Further above threshold (Fig. 1 C, G, K, and O) the k-space region of highest intensity moves significantly closer toward k_|| = 0. The green lines give the calculated diffusive Goldstone mode dispersion. At negative detunings there still is no complete reflection symmetry between k_|| and -k_||, but the positive wavevector half of the dispersion now shows reasonable agreement with the equilibrium Bogoliubov dispersion and the diffusive Goldstone mode dispersion up to k_|| ≈ 0.75 μm^-1 without usage of any fitting parameters. This modified dispersion indeed is a sign of collective behavior as expected in a condensed and thermalized polariton gas. However, it is not immediately clear whether the equilibrium Bogoliubov dispersion or the diffusive Goldstone mode dispersion corresponds better to the experimental results because their main difference lies in the low-momentum regime where the excitation spectrum and the condensate luminescence are superimposed. Note that the luminescence from higher energy gets gradually larger with increased photonic content of the LP which is a clear indication of the occupation achieving gradually a more nonequilibrium character. Further increase of the excitation power (Fig. 1 D, H, L, and P) results in a more pronounced occupation of the condensate ground state. Although the reflection symmetry between positive and negative k_|| is still not perfect, both parts of the dispersion can now be described by the same equilibrium Bogoliubov dispersion with accuracy. We interpret this behavior as a sign of effective redistribution by polariton scattering processes. Although the excitation spectrum is now strongly masked by the condensate luminescence, the occupation at larger energies still varies with the detuning. Accordingly we are approaching a more thermalized, but still nonequilibrium, regime for negative detunings. Anyway, it should be noted that the shape of the excitation spectrum shows no significant dependence on the occupation and whether it is close to equilibrium or not. Again, it is difficult to differentiate whether the equilibrium Bogoliubov or diffusive Goldstone mode dispersion matches the experiment better. Obviously there is some flat region at low momenta, especially at detunings of -2 or -4 meV, but because the emission from the condensate masks the emission from the excitation spectrum even stronger than at lower pump rates, this observation is not necessarily conclusive evidence for the presence of the flat dispersion in momentum space.

Fig. 1.

Polariton dispersions on a logarithmic scale at several excitation densities along the threshold for detunings of Δ = 0 meV (Top), -2 meV (Second Row), -4 meV (Third Row), and -7 meV (Bottom). Black, blue, white, and green lines represent the calculated LP, blue-shifted LP, homogeneous equilibrium Bogoliubov, and diffusive Goldstone mode dispersions, respectively.

Because of their large photonic content, one might imagine that the coherence properties of HI-P states are not different from common photon lasers. This assumption can be tested by probing the strength of quantum fluctuations via normal-ordered second- and third-order equal-time correlations of the photon number n defined as

where the double stops denote normal ordering of the underlying photon field operators. Photon lasers are expected to show statistically independent emission as described by Poissonian statistics [corresponding to g⁽²⁾(0) = g⁽³⁾(0) = 1] above threshold, on the other hand increased fluctuations because of nonresonant scattering processes between polariton pairs (21–23) are expected for operation well above the stimulation threshold for polaritonic condensates, resulting in g⁽²⁾(0) > 1. Below threshold both are supposed to emit thermal light with g⁽²⁾(0) = 2 and g⁽³⁾(0) = 6. To test these predictions we resonantly excite LPs with large transverse momentum k_|| = 5.8 μm^-1 using circularly and linearly polarized laser pulses. Subsequently those polaritons relax toward the ground state by acoustic phonon emission or polariton–polariton scattering. If the occupation of the ground state becomes large enough, bosonic final-state stimulation is expected to significantly increase the scattering toward this state. We make use of a recently developed streak camera technique to determine the second- and third-order correlation functions (24, 25). Although there have been experimental investigations of the second-order coherence properties of polariton condensates for GaAs (26, 27) and CdTe (28, 29) structures before, the results are contradictory. We resolve this question by performing systematic studies of the higher order coherence properties in terms of g⁽²⁾(0) and g⁽³⁾(0) for different detunings and excitation schemes and show that they are indeed strongly dependent on both.

As can be seen in Fig. 2, g⁽²⁾(0) does not reach the expected thermal value of two for any detuning as we do not single out one fundamental mode like in earlier publications (25), but detect both realizations of the spin-degenerate ground state simultaneously. Polaritons with different spin states will not interfere with each other and therefore the values of g⁽²⁾(0) and g⁽³⁾(0) expected in the thermal regime of two superposed modes are 1.5 and 3, respectively. Note further that if plotted on a comparable scale g⁽²⁾(0) and g⁽³⁾(0) give results which are in quantitative agreement. Under linear excitation all detunings between Δ = +2 and Δ = -10 meV show a threshold, which is evidenced by a decrease in g⁽²⁾(0) and g⁽³⁾(0) and agrees well with the position of the threshold (shown as green dashed lines in Fig. 2) evidenced in measurements of the input–output curve. We define the threshold as the excitation power at which the emission from the condensed state becomes stronger than the emission from the LP. In the case of strong negative detuning of Δ = -10 meV no differences from a common photon lasing transition are observed under linearly polarized pumping. Additional dispersion measurements evidence that above threshold the photons are indeed emitted from the bare cavity mode in this case. Under circularly polarized pumping the threshold is not even reached for a detuning of -10 meV. This result is in accordance with previous results showing that polariton relaxation is less efficient under circularly polarized pumping, which in turn leads to a higher threshold excitation power P_thr (30). For all other detunings significant deviations from a simple photon laser behavior emerge. Even at high-excitation powers the ground-state emission has lower energy compared to the bare cavity mode with an energy difference of at least 4 meV. Although this behavior is not necessarily a proof that the strong coupling regime is still intact (31), it is at least very likely. The second- and third-order correlation functions give further evidence that the system is not a simple photon laser in the range of detunings between +2 and -7 meV, corresponding to photonic contents in the range from |C²| ≈ 44–70%. Although at first a decay toward one is seen for linearly polarized excitation, especially for a detuning of -7 meV, an increase is evidenced for further increased excitation densities. Depending on the detuning g⁽²⁾(0) can reach values even higher than the thermal value of 1.5. For further increased excitation power a smooth decrease back toward one is observed. For circularly polarized excitation the general behavior of the correlation functions is similar to the linearly polarized case as a decrease and a reoccurrence of the degenerate mode quantum fluctuations can be identified for detunings between +2 meV and -7 meV. However, in this case the correlation functions can also show increased fluctuations slightly above threshold as can be nicely seen for a detuning of -4 meV. This increase is caused by the build up of polarization. The thermal regime value of 1.5 is just valid for unpolarized two-mode emission. As the excitation power reaches the threshold, the emission will also start to polarize and the two modes will not contribute equally to the correlation functions anymore (Materials and Methods). This onset of polarized emission will lead to an increase in g⁽²⁾(0) whereas the build up of coherence will lead to a decrease. Although these results are in fair qualitative agreement with mean-field and reservoir calculations of the second-order correlation function of a polariton BEC (21, 22), they are not sufficient evidence for deviations from a photon laser as the nonmonotonous behavior of the correlation function can also be a result of the interplay of coherence and polarization.

Fig. 2.

Measured g⁽²⁾(0) (black dots) and g⁽³⁾(0) (red triangles) determined by simultaneous two- and three-photon detections of the whole fundamental mode emission for a wide range of excitation powers and detunings of -10 meV (|C²| ≈ 77%), -7 meV (|C²| ≈ 70%), -4 meV (|C²| ≈ 62%), -2 meV (|C²| ≈ 56%), 0 meV (|C²| ≈ 50%), and +2 meV (|C²| ≈ 44%) under linearly polarized (Upper) and circularly polarized (Lower) excitation. Red (blue) lines denote the coherent (thermal) limit. Green dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion. Below threshold the necessary integration time for the correlation measurements increases drastically. Only data points for excitation powers where the results are not limited by the long-term stability of the setup are considered.

To make sure the onset of polarized emission is not the main influence on the observed behavior of the correlation function we studied g⁽²⁾(0) under circularly polarized excitation also for the cocircularly polarized emission only. As shown in Fig. 3, here the expected values of g⁽²⁾(0) = 2 are approximately reached in the limit of low-excitation power for all detunings except +2 meV. At this detuning the emitted intensity below threshold is too small to perform sensible measurements using our setup. Above threshold the shape of g⁽²⁾(0) shows qualitative agreement with theoretical results (21). On resonance it is apparent that full coherence is not reached within the available excitation power range. Instead g⁽²⁾(0) decreases monotonically toward a value between 1.3 and 1.4. There is a trend toward further decrease at high-excitation power, however, the dependence on pump power is very weak. Going to more negative detunings, the dip already seen without polarization-sensitive detection still occurs. Apparently, the excitation power corresponding to the occurrence of the dip takes on smaller values compared to P_thr for larger negative detuning. Moreover, the dip in g⁽²⁾(0) is most pronounced for the most negative detuning (-7 meV). In fact, for a detuning of -7 meV almost complete coherence is reached at an excitation power of approximately 1.1 P_thr and a steep rise to g⁽²⁾(0) > 1.6 is evidenced at 1.5 P_thr. Calculations of the second-order correlation function are generally done using one of two different approaches (21): Mean-field calculations predict a decrease of g⁽²⁾(0) toward approximately 1.2 at the threshold, followed by a short rise for increasing excitation power until a constant value of about 1.3 is reached. This prediction is in agreement with our results for no or small negative detuning. A two-reservoir model predicts a sharp decrease of g⁽²⁾(0) at the threshold, a strong recurrence of the photon bunching up to values of g⁽²⁾(0) = 1.6 for excitation powers moderately larger than the threshold value and a slow drop for even higher excitation powers. This model better reproduces our results for large negative detuning. We conclude that because of the increasing relaxation bottleneck and decreased scattering rate expected for large negative detuning the two-reservoir model appears to be a valid description in the HI-P regime and the dip seen for several detunings is a sign of inefficient scattering between polaritons from the degenerate ground state and those in excited states. Because the photon statistics of emission depends strongly on the detuning we suppose that one might also find variations depending on the Rabi splitting and the excitonic and photonic decay constants. Therefore, microcavities operated in the HI-P regime may open up the possibility to introduce a high-intensity light source with tunable photon statistics.

Fig. 3.

Measured g⁽²⁾(0) of the cocircularly polarized fundamental mode for a wide range of detunings and excitation powers under circularly polarized excitation. Red (blue) lines denote the coherent (thermal) limit. Green dashed lines give the position of the degeneracy threshold determined by measurements of the dispersion.

In conclusion we have driven a microcavity polariton system into a completely out of equilibrium degenerate HI-P state by increasing its photonic content up to |C²| ≈ 70% and examined whether the system bears more similarities to an inverted system; i.e., a laser or a thermalized polariton BEC. Surprisingly, although the matter component is small and the degeneracy transition we have demonstrated cannot be considered a thermodynamic equilibrium phase transition by any means, the system shows many features one would expect from a BEC. In particular we have observed a modified dispersion and compared it to standard equilibrium Bogoliubov theory predicting a linearized spectrum and nonequilibrium Bogoliubov theory predicting a diffusive Goldstone mode and demonstrated the detuning dependence of the emitted photon statistics. We believe our results can open up the road for tunable photon statistics light sources and deepen the understanding of nonequilibrium phase transitions.

Materials and Methods

Experimental Details.

In our experiments the sample was kept at 8 K in a helium flow cryostat. The microcavity structure is basically identical to that used in earlier measurements (20), but was fabricated in another growth run. It consists comprehensively of 12 GaAs/AlAs quantum wells embedded in a planar microcavity with 16 (21) AlGaAs/AlAs mirror pairs in the top (bottom) distributed Bragg reflector. Reflectivity measurements gave results of 1.6158 eV for the bare exciton energy and 13.8 meV for the Rabi splitting. Using a Ti:Sapphire laser with a pulse duration of 1.5 ps and a repetition rate of 75.39 MHz the pump was focused to a spot approximately 30 μm in diameter on the sample at an angle of 45° from normal incidence. The pump was resonant with the LP branch at an in-plane wavenumber of k_|| = 5.8 μm^-1 for resonant polariton injection or tuned to the first minimum of the cavity reflectivity curve at a wavelength of 744 nm for nonresonant and incoherent pumping. We collected the emitted signal using a microscope objective with a numerical aperture of 0.26. The signal was then focused on a streak camera for correlation measurements, or the Fourier plane was imaged on a monochromator for dispersion measurements.

Details of Correlation Measurements.

The basics of our photon correlation method were already explained in the references given in the text. Accessing extremely short pulses, like the emission from HI-P states in the degenerate regime, introduces additional difficulties. In this regime the timing jitter becomes comparable to the emission pulse duration. Therefore the momentary intensity detected at a certain position of the CCD will be determined by the photon statistics as well as by the momentary influence of timing jitter, which will cause deviations from the real value of g⁽²⁾. The deviation is systematic, so it is possible to correct for this jitter effect. From a theoretical point of view, the effect of jitter can be modeled as a Gaussian distribution having a certain FWHM describing the peak position of the emitted pulse on the CCD. This timing jitter will cause a broadening of the duration of the averaged intensity compared to the real emission pulses. Accordingly g⁽²⁾(0) will be overestimated whereas g⁽²⁾(τ) will be underestimated for large enough τ. The exact measured g⁽²⁾ depends on the ratio of the timing jitter FWHM J to the pulse FWHM P. This situation can be simulated theoretically and shows that the measured and depend on the real g⁽²⁾(0) and g⁽³⁾(0) corresponding to the following relation,

Accordingly it is possible to extract the real values from the measured ones if the emission pulse widths and the jitter widths are known. The jitter width can be determined by measuring for a light source with well-defined P and g⁽²⁾. We used a laser pulse with P = 1.5 ps as determined by using an autocorrelator and g⁽²⁾ = 1. The resulting is discussed in detail in ref. 32 and is well-reproduced by a simulation using J = 1.8 ps.

P can be determined directly from the integrated streak camera data, which give the jitter-broadened emission pulse widths. Because all essential parameters are known, it is possible to calculate g⁽²⁾(0) from .

In the case of a superposition of two noninterfering modes A and B, the resulting measured is given by

where R_A and R_B are the relative intensity ratios of mode A and B to the total intensity. Therefore it is possible to calculate from if the relative intensity ratios and are known:

Comparing single-mode g⁽²⁾ measurements to the values obtained by two-mode measurements shows that for circularly polarized excitation the two modes are indeed independent and the cross-circularly polarized mode stays thermal.