Summary We describe the signatures of exciton-polariton condensation without a periodically modulated potential, focusing on the spatial coherence properties and condensation in momentum space. The characteristics of the exciton-polariton condensate form the basis of the study of the exciton-polariton condensate array. We also discuss the diffraction patterns of zerostate and π-state considering the finite transmittance through the metallic strips and spatial coherence. A rate-equation model is used to describe the dynamics and evolution of the zerostate and π-state as a function of the pumping rate. Characterization of exciton-polariton condensates In two-dimensional (D) systems with continuous symmetry, the phase fluctuations of the order parameters prevent formation of condensates with off-diagonal long-range order 1-3 at finite temperatures. However, a finite-size bosonic D fluid can still undergo Bose-Einstein condensation (BEC) 30. In addition, quasi-condensates with a topological long-range order 31 or long range coherence 3 as well as superfluidity enabled by interaction 33 (eg. Berezinskii- Kosterlitz-Thouless transition 31,34,35 ) can exist in D systems. Above the critical density or below the critical temperature of BEC or BKT transition, the phase can be correlated from region to region as long as the order exists locally. Here we describe experimental observations of the condensation in momentum space and long range spatial coherence of LP condensates in the absence of a periodic potential modulation. Under an elliptical excitation spot with an axial-to-radial aspect ratio of :1 (~60 μm 30 μm), the exciton-polariton momentum distribution is isotropic as shown in Fig. S1. With increasing pumping rates, strong nonlinear LP emission develops from a small area, and the momentum distribution evolves into an anisotropic distribution with a reversed axial-to-radial aspect ratio. Furthermore, the product of the spreads of the LP distribution in momentum and coordinate spaces is only ~ to 4 times the Heisenberg uncertainty limit 36. The anisotropic and narrow momentum distribution above a threshold pumping rate is supportive of a long-range spatial coherence across the condensate. www.nature.com/nature 1
The effects of condensation in the momentum space are further revealed by the energy versus in-plane momentum LP distribution shown in Fig. S1c. Above the condensation threshold, macro-occupation of a low energy state and sharp reduction of the LP energy linewidth at k = 0 are observed (see also ref. 4,5 ). The spatial coherence characteristic of the exciton-polariton condensate is quantitatively determined by Young s double-slit interference experiment 36 with double-slits inserted at the conjugate image plane (see the main text, Fig. a). The slits are positioned at Y =± d / with respect to the axis of symmetry (optical axis) and the center of the condensate to ensure equal intensity between two slits, where d is the separation of the two probing slits. The plane of observation is the focal plane of the spectrometer whose entrance slit coincides with the Fourier transform plane (momentum space) of the double slits. Under such an arrangement, the visibility V of the fringes is equal to the degree of coherence 37,38 as shown in Fig. S. The determined spatial coherence is expected to reflect the correlation function, which is proportional to the off- diagonal element of the single-particle density matrix ρ( d) ψˆ ( d/ ) ψˆ( d/) =. Below the pumping threshold the correlation function can be approximated by exp ( π r / λ T ) 11, where r is the distance between two probed locations and the spatial coherence length is characterized by * the thermal de Broglie wavelength λ h/ πm k T. At a temperature T = 10 K, λ.5 μm T for a typical polariton effective * m = 10 4 me. For a homogeneous Bose gas, the correlation function can be expressed as ( ) ( ) B ( D η ) ( ) + ψˆ r ψˆ 0 r exp r/ ξ, where D is the dimensionality, η a critical exponent, ξ c the correlation length 39. The visibility decreases approximately exponentially with increasing slit separations at large separations, where the power low decay is negligible for D =. The measured correlation length increase from ~ μm up to ~18 μm (limited by the pumping spot size) at just above the condensation threshold. At just above the threshold, the visibility decreases from more than 90% to an invariant ~30% until the slits reach the edges of the condensate (Fig. S). At a high pump rate, the correlation length decreases to ~5 μm, likely due to dephasing of the condensate induced by polariton-polariton interaction. The role of polariton-polariton interaction and types of phase transitions (BEC versus BKT transition) in D systems are beyond the scope of this letter. The measured coherence c T www.nature.com/nature
doi: 10.1038/nature06334 length far exceeding the thermal de Broglie wavelength confirms that spatial coherence can develop across the whole condensate. Figure S1 Near-field and far-field imaging and spectroscopy of an exciton-polariton condensate in the absence of metallic strips. The pumping rate is Pth/3, 1.5 Pth, and 6 Pth (Pth 30mW) from left to right panels. T 8 K and the LP emission wavelength λ 780 nm. The condensate is studied at a location with a cavity detuning EC E X 6 mev, near the region with metallic strips where data shown in Fig. 3c and Fig. 4 are measured. a, Pseudo-color (log scale) near-field images of the LP emissions. b, Far-field images showing the LP distribution in momentum space. An anisotropic distribution with a reversed axial-to-radial aspect ratio in coordinate space develops above threshold. c, Energy versus in-plane momentum LP distribution, where a slice of the far-field ( emission with in-plane momentum within k X π / λ sin 0 ± 0.4 D ) is dispersed by the spectrometer and imaged by a CCD. Above the threshold, the LPs exhibit narrow distribution in energy and momentum. www.nature.com/nature 3
doi: 10.1038/nature06334 Figure S Spatial coherence measured by Young s double-slit interference measurement. Young s interference experiment is conducted in a symmetrical arrangement with slits positioned at an equal distance from the optical axis as shown in Fig. a. Two slits with an effective width of a = 0.8 μm on the sample are positioned symmetrically at ± d / from the optical axis, where the center of the condensate image resides. The quasimonochromatic condition is satisfied (i.e. optical path difference is much less than the longitudinal coherence length, d sin (θ ) c / Δν, where Δν is the spectral resolution). Across the plane of observation, the intensities produced by two slits are equal, I (1) ( v ) = I ( ) ( v ) sin (π av ) / π v, where v is the coordinate at the plane of observation. The intensity of the interference pattern thus can be expressed as I ( v ) = I (1) + I ( ) + I (1) I ( ) V cos ( π d v λf ), where V is the visibility and corresponds to the degree of spatial coherence. a, Selected spectrally resolved interference fringes measured at a pumping power of ~45 mw, which is ~1.5 times above the transition pump threshold, Pth = 30 mw. Here the cavity detuning Ec E X 6 mev similar to the case shown in Fig. 3c. The separation between the two slits is gradually increased from near coincidence ( μm) to the dimension of the condensate (~0 μm). The appearance of the fringes is a manifestation of spatial coherence between the LP emissions from the two slits. b, Cross sections of the interference fringes (red dots) and the fittings based on I ( v ) (blue lines). The dips or spikes near the center are attributed to the finite transmission through the area outside of the double slits. This background LP emission is included in the fitting functions. c, The visibility as a function of the normalized pumping rate for various slit separations. The visibility increases dramatically above the threshold pumping rate. d, www.nature.com/nature 4
The visibility as a function of the slit separation for various pumping rates. The visibility drops exponentially with increasing slit separation. At just above the threshold, the visibility remains finite until the slits separation reaches the dimension of the condensate (solid blue line in the inset). The coherence length decreases with increasing pumping rate despite of the increase in the dimension of the condensate (magenta triangles and lines). The finite coherence length (~ μm) is attributed to the finite thermal de Broglie wavelength of the LPs as well as the spatial resolution of the optical system. Diffraction patterns of the zero-state and π-state The field of LP emissions is the product of the probability amplitude of the LP wave functions and the spatial transmission function of the metallic strips, (, ) ψ (, ) (, ) U X Y = X Y t X Y. The field in momentum space (the back focal plane of the 0 nk, m objective) is related to U0 ( X, Y) at the object plane by a Fourier transform 1 π U f ( u, v) U0 ( X, Y) exp j ( Xu Yv) dxdy jλ f +, neglecting the finite resolution of the obj λ fobj optical system and the finite spatial coherence. The intensity distribution in the diffraction pattern is I ( uv, ) U ( uv, ) f. When the effect of spatial coherence of the condensate is included, the intensity distribution at momentum space is given by the convolution of the Fourier transform of mutual intensity function and the autocorrelation function of the complex function U0 ( X, Y ) [Schell's theorem] 37,38. The relative transmittance t 0 50% through the metal as compared to that through the bare cavity is determined independently using a uniform metallic film of similar thickness and composition. The Bloch wave functions in the presence of a weak periodic potential are approximately constant for zero-state and sinusoidal for π-state as shown in Fig. S3a. The relative intensity ratio of the first-order diffraction peaks (i.e. peaks at ~ ± 16 for zero-state and ~ ± 4 for π-state ) to the zero-order peaks are 1 π ( t0 ) ( 1+ t ) 0 4% and 1 t0 π / 1 + /+ 1 t ( π ) 0 7% for zero-state and π-state, respectively. These relative intensity ratios are in approximate agreement with the experimental observation. If the www.nature.com/nature 5
transmission is uniform ( t 0 = 1), the diffraction pattern reflects the Bloch wave functions of the states and the intensity of first order diffraction peaks will be weaker. The width of the diffraction peaks is limited by the finite size of the condensate as well as the spatial coherence. The correlation function of LP condensate decays approximately exponentially, thus the mutual intensity can be expressed as μ ( X) ( X ξ ) 0 Δ exp Δ / c. The diffraction peaks in momentum space are thus expected to be Lorentzians with a width Δk ξ (FWHM, full-width-at-half-maximum). The width of the diffraction peaks from the / c Δ / 0.3 μm-1 ( Δθ ), corresponding to a typical coherence length π-state is k ( k G ) 0 ξ 6 7 μm (in agreement with the Fourier transform limit). The typical width of the c diffraction peaks from the zero-state is k ( k ) Δ 0 0.9 1.0 μm -1 ( Δθ 7 ). The product Δ k ξ c is thus about three to four times of the Fourier transform limit for the zero-state. The wave function of the zero-state is only weakly disturbed in the presence of the periodic potential. Therefore, the zero-state is expected to exhibit similar coherence properties as the unperturbed condensate. The product Δ k ξ across the polariton array is increased by ~1.5 to times as compared to that of the condensates in the absence of the metallic strips. c Figure S3 Diffraction of LP emissions. a, The Bloch wave functions of the zero-state and π -state in a weak periodic potential. The relative transmittance of LP emissions through the metallic strips is t 0 50%. b, Diffraction peaks and relative intensities for the zero-state and π -state. www.nature.com/nature 6
Analysis of dynamics and competition of the zero-state and π-state Formation of band structure A spatial modulation in the cavity resonance, induced by a periodic array of metal strips, introduces a spatial modulation of the lower polariton energy in the strong coupling regime like our experimental system. Our experimental results show a polariton plane wave perturbed by the periodic boundary condition (Fig. 1 and Fig. 3). The energy versus in-plane momentum dispersion ( E vs. k ) characteristics and corresponding near-field pattern (π-state consisting of p-state Bloch functions in coordinate space) support that the elementary excitation in our system is a polariton with heavier mass than a bare cavity photon (Fig 3a). Additional experimental results that preclude the interpretation in terms of simple optical coupling model include: (i) the type of the Bloch function in the π-state, (ii) the energy blue shift of the condensed states, and (iii) evolution of zero-state and π-state as a function of pumping rate (see subsection Dynamics and mode competition ). Above threshold, the bosonic final state stimulation decreases abruptly the spectral linewidth and increases the population of the two metastable states, which manifests the mode competition of discrete π-state and zero-state (see below for mode competition analysis). The blue shift of these condensed states suggests that a large number of polaritons occupy the same spatial mode and exert the on-site repulsive interaction. The energy shift is quantitatively understood in terms of the polariton-polariton interaction due to phase space filling and fermionic exchange interaction. The metastable π-state condensate consisting of p-type-like wavefunctions in momentum and coordinate space as shown in both near-field and far-field images (Fig. ) is an essential manifestation of the band structure in an array of polariton condensates (Fig. 3). For instance, in a semiconductor array laser with periodic gain/loss spatial modulation, the π-state mode has a lower laser threshold than the zero-state mode because the π-state mode has a zero electric field amplitude at loss regions but the zero-state mode has a finite electric field amplitude and so suffers from a higher loss. However, in this case the lasing π-state mode has a maximum at a center of the trap (gap between metal strips) where the gain is maximal. In this case, the Bloch wave function is s-like. As shown in Fig. 3, the condensed π-state has not only π-phase difference between adjacent traps but also a minimum intensity at the center of the gap. This indicates that the Bloch wavefunction is p-like (see Fig. 3d). This experimental result is www.nature.com/nature 7
understood by the fact that the π-state with p-type Bloch wavefunction (upper state at the anticrossing point, Point A in Fig. 3b) is metastable while the π-state with s-type Bloch wavefunction (lower state at the anti-crossing point, Point B in Fig. 3b) is unstable. In conclusion, the present experimental results cannot be explained by the simple optical coupling/diffraction model. Optical coupling is ruled out by the simultaneous π-phase shift observed in both near-field and far-field emissions as illustrated in Fig. and Fig. 3. In the context of s-wave-like zero-state and p-wave-like π-state in an array of coupled multiple condensates, we thus make the analogy to the π-state observed in a superfluid 3 He weak link (where the underling p-wave coupling plays a role) and coupled Josephson junction arrays. Dynamics and mode competition The π-state consisting of local p-type wavefunctions is a metastable state with a higher energy than the zero-state consisting of s-type wavefunctions. Both states decay to the crystal ground state by emitting photons with a finite lifetime. After high-energy exciton-like lower polaritons (LPs) with large in-plane wavenumber k are optically injected into the system, they cool down to the metastable π-state first by continuously emitting phonons or via polaritonpolariton scattering. If the decay rate of the π-state is slower than the loading rate into the π-state, the population of the π-state exceeds one (quantum degeneracy condition) and induces the bosonic final state stimulated scattering into this mode. This is similar to a so-called bottle-neck condensation effect 40,41. Here the LPs with finite k dynamically condense due to the slow relaxation rate into the k 0 region, where the density of states is reduced. When the pump rate is further increased, the zero-state eventually acquires the population greater than one. The onset of stimulated scattering into the zero-state from the metastable π- state favors the condensation of the zero-state. The reverse stimulated process from the zero-state to the π-state requires the absorption of phonons, thus is energetically unfavorable. This mode competition behavior is well described by the coupled Boltzmann (rate) equation analysis as shown below (Fig. S4 to Fig. S6). The transition from the bottle-neck condensate to the k=0 state condensate at higher pump levels is a well established experimental fact in a homogeneous system 4,41,4, and the phenomenon we reported here is a similar dynamical process. This cascade cooling process through π-state to zero-state is a direct manifestation of the band structure as shown in Fig. 3. www.nature.com/nature 8
Here again, the simple optical coupling model cannot describe the observed threshold and mode competition behavior. Modeling We analyze the mode competition using rate equations consisting of three states: (1) inphase zero-state, () antiphase π-state, (3) exciton-like polaritons injected by the pump laser. As described above, we assume that (a) initially pumped exciton-polaritons (state 3) can relax toward π-state (state ) or zero-state (state 1), or leak out of the cavity, (b) state can relax toward state 1 or leak out of the cavity, and (c) state 1 can only leak out of the cavity. This simple model well approximates the progressive relaxation from high to low in-plane momentum (or energy). The coupled rate equations with population n 1, n and n 3 respectively for state 1,, and 3 can be approximated as: dn 3 dt dn dt dn 1 () t () t () t dt ( ) ( ) = Γn Γ n n + 1 Γ n n + 1, 3 3 3 3 31 3 1 ( ) ( ) = Γ n +Γ n n + 1 Γ n n + 1, 3 3 1 1 ( ) ( ) = Γ n +Γ n n + 1 +Γ n n + 1. 1 1 31 3 1 1 1 Γ i is the radiative decay rate of state i, while Γ ij is the transition rate from i state to state j. Assuming instantaneous initial pumping, the above coupled rate equations can be solved with the initial condition, ( ) ( ) ( ) n 0, n 0, n 0 = 0,0, n p 1 3, where p n is the initially pumped excitonlike LP of high in-plane wavenumber in state 3. The time-integrated LP emission intensity from state i is proportional to Γ n () t dt. Because the polariton relaxation proceeds from high to low i i in-plane wavenumber, the transition rate Γ 31 is negligible as compared to Γ 3. State 3 is an exciton-like state with long lifetime, thus the decay rate Γ 3 is small as compared to Γ 1 and Γ. Assuming reasonable numerical parameters: Γ 3 = 300 1, Γ1 Γ = 10 1, Γ =, Γ =, 1 5 1 3 0 1 Γ 31 = 100 1 (ps -1 ), we obtain qualitative agreement with the experimental results (Fig. S5 and S6). With increasing pumping, the π-state appears first then is surpassed by the dominant zero-state, consistent with experimental observation. The time-resolved LP emission from zero-state and π- state are shown in Fig. S6. www.nature.com/nature 9
The simple rate equations with parameters fixed for all pumping rates describe the observed mode competition. This unusual mode competition stems essentially from the inequality Γ Γ, Γ for cascade cooling of an array of coupled polariton condensates. This condition 31 3 1 would not hold if zero-state and π-state are induced by simple optical coupling or diffraction by metal strips. Figure S4 Schematic of the relaxation and mode competition of an array of exciton-polariton condensates. Figure S5 Integrated LP emission intensities of zero-state and π-state versus pump intensity. a, Experimental results obtained from the area of the cross-section taken near energy peaks of zero-state and π-state (see Fig. 4 for selected cross-section profiles). The unit of the x-axis is pump power normalized with respect to 0 mw. b, Simulation results based on the parameters and rate equations are given in the text. To quantitatively www.nature.com/nature 10
compare the experimental and simulation results, accurate determination of occupation number is necessary. Nevertheless, the mode competition of these two distinct zero-state and π-state are described well by the simple rate equation model. Figure S6 Dynamics of the zero-state and π-state. a, LP dynamics measured by a streak camera system. The black dash curve in a is the system response measured with scattered laser pulse from the sample. The zerostate appears ~10 ps after the π-state near the threshold. b, Simulation: All fitting parameters are the same as used for Fig. S5 as described in the text. www.nature.com/nature 11
Supplementary references 30. Bagnato, V. & Kleppner, D. Bose-Einstein condensation in low dimensional traps. Phys. Rev. A 44, 7439-7441 (1991). 31. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two dimensional systems. J. Phys. C 6, 1181-103 (1973). 3. Lasher, G. Coherent phonon states and long-range order in two-dimensional Bose systems. Phys. Rev. 17, 4-9 (1968). 33. Fisher, D. S. & Hohenberg, P. C. Dilute Bose gas in two dimensions. Phys. Rev. B 37, 4936-4943 (1988). 34. Berenzinskii, V. L. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group. 1. Classical systems. Sov. Phys. JETP 3, 493 (1971). 35. Berenzinskii, V. L. Destruction of long-range Order in one-dimensional and twodimensional systems possessing a continuous symmetry group.. Quantum systems. Sov. Phys. JETP 34, 610 (197). 36. Deng, H., Solomon, G. S., Hey, R. & Yamamoto, Y. Spatial coherence of a polariton condensate. ArXiv:cond-mat/0703508 (007). 37. Goodman, J. W. Statistical Optics (Wiley, New York, 1985). 38. Born, M. & Wolf, E. Principles of Optics (Cambridge University Press, Cambridge, 1999). 39. Huang, K. Statistical Mechanics (Wiley, New York, 1987). 40. Huang, R. et al. Exciton-polariton lasing and amplification based on exciton-exciton scattering in CdTe microcavity quantum wells. Phys. Rev. B 65, 165314 (00). 41. Tartakovskii, A. I. et al. Relaxation bottleneck and its suppression in semiconductor microcavities. Phys. Rev. B 6, R83-R86 (000). 4. Porras, D. et al. Polariton dynamics and Bose-Einstein condensation in semiconductor microcavities. Phys. Rev. B 66, 085304 (00). www.nature.com/nature 1