Quantum coherence in momentum space of light-matter condensates
C. Antón, G. Tosi, M. D. Martín, Z. Hatzopoulos, G. Konstantinidis, P. S. Eldridge, P. G. Savvidis, C. Tejedor, L. Viña
QQuantum coherence in momentum space of light-matter condensates
C. Ant´on,
1, 2
G. Tosi, M. D. Mart´ın,
1, 2
Z. Hatzopoulos,
3, 4
G. Konstantinidis, P. S. Eldridge, P. G. Savvidis,
3, 5
C. Tejedor,
6, 2, 7 and L. Vi˜na
1, 2, 7, ∗ Departamento de F´ısica de Materiales, Universidad Aut´onoma de Madrid, Madrid 28049, Spain Instituto de Ciencia de Materiales “Nicol´as Cabrera”,Universidad Aut´onoma de Madrid, Madrid 28049, Spain FORTH-IESL, P.O. Box 1385, 71110 Heraklion, Crete, Greece Department of Physics, University of Crete, 71003 Heraklion, Crete, Greece Department of Materials Science and Technology,University of Crete, 71003 Heraklion, Crete, Greece Departamento de F´ısica Te´orica de la Materia Condensada,Universidad Aut´onoma de Madrid, Madrid 28049, Spain Instituto de F´ısica de la Materia Condensada, Universidad Aut´onoma de Madrid, Madrid 28049, Spain (Dated: October 29, 2018)We show that the use of momentum-space optical interferometry, which avoids any spatial overlapbetween two parts of a macroscopic quantum state, presents a unique way to study coherencephenomena in polariton condensates. In this way, we address the longstanding question in quantummechanics: “Do two components of a condensate, which have never seen each other, possess adefinitive phase?” [P. W. Anderson,
Basic Notions of Condensed Matter Physics (Benjamin, 1984)].A positive answer to this question is experimentally obtained here for light-matter condensates,created under precise symmetry conditions, in semiconductor microcavities taking advantage of thedirect relation between the angle of emission and the in-plane momentum of polaritons.
I. INTRODUCTION
Cold atoms and exciton-polaritons in semiconduc-tor microcavities are systems where their capability toconstitute Bose-Einstein condensates (BECs) has beendemonstrated in recent years . These BECs, due totheir dual wave-particle nature, share many propertieswith classical waves as, for instance, interference phe-nomena , which are crucial to gain insight into theirundulatory character . One of the main differences be-tween atomic and polariton condensates resides in theparticles lifetime: the finite lifetime of polaritons, in con-trast with the infinite one of atoms, can be regarded as acomplication. But making virtue of necessity, a short life-time also implies a significant advantage: polaritons havea mixed exciton-photon character , their lifetime beingdetermined by the escape of their photonic componentout of the cavity. These photons are easily measuredeither in real- (near field spectroscopy) or momentum-space (far field spectroscopy) , rendering full informa-tion about the polariton BECs wave-function and, in par-ticular, about its coherence . Our goal is to profit fromthese measurements in momentum space to experimen-tally investigate something far from accessible in atomiccondensates: the interference in momentum space pro-duced by the correlation between two components of acondensate, which are, and have always been, spatiallyseparated. Understanding coherence is important for alarge number of disciplines spanning from classic opticsto quantum information science and optical signal pro-cessing .Pitaevskii and Stringari made a theoretical proposalto investigate experimentally these interference effects inmomentum space via the measurement of their dynamic structure factor . In related experiments, coherence be-tween two spatially separated atomic BECs has been in-directly obtained using stimulated light scattering .In this work we perform a direct measurement of thiscorrelation in polariton BECs, which moving in a sym-metrical potential landscape, acquire a common relativephase, obtaining a positive answer to Anderson’s ques-tion , which opens new perspectives in the field ofmulti-component condensates. II. EXPERIMENTAL RESULTS ANDDISCUSSION
We confront this task in a quasi one-dimensional (1D)system made of a high-quality AlGaAs-based microcav-ity, where 20 × µ m ridges have been sculpted. Thesample, kept at 10 K, is excited with 2 ps-long light pulsesfrom a Ti:Al O laser. In order to create polaritons intwo separated spatial regions, the laser beam is split intwo, named A and B , impinging simultaneously at po-sitions distanced by d AB = 70 µ m. Additional experi-mental details are described in the Supplementary infor-mation . A crucial issue when optically creating polari-tons is the excess energy of the excitation laser. Thereare two well explored alternatives: non-resonant excita-tion at very high energies and strictly resonant excita-tion . The latter situation generally produces macro-scopic polariton states with a phase inherited from thatof the laser, unless special care is taken in the experi-ments . The former case is appropriate to avoid phaseheritage, but it does not provide the momentum distribu-tion, shown below, required for our experiments. In orderto avoid these difficulties, we opt for a different alterna- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug tive, depicted in Fig. 1(a): the laser beams excite thesample at the energy of bare excitons and k x ∼
0. Thebroad bands between 1.542 and 1.548 eV corresponds toexcitonic emission bands; the sub-bands below 1.542 eVare the confined lower polariton branches. After energyrelaxation, polariton condensates are created in a processthat involves a non-reversible dressing of the excitons andtherefore an erasure of the laser phase . Above a given FIG. 1. (a) Sketch of the excitation and relaxation processesto form propagating polariton wave packets ( WP s) on a back-ground showing the energy vs. k x emission obtained undernon-resonant, low power excitation conditions. The grey el-lipse depicts the excitation laser at 1.545 eV and k x ∼
0. Thedashed lines indicate the energy relaxation of excitons intopolariton WP s. Polariton WP s, propagating with k x ≈ ± . µ m − (slightly displaced for the sake of clarity), are depictedwith circles, coded in colors explained in (b). The emission in-tensity is coded in a logarithmic, false color scale. (b) Sketchin real space of the experimental configuration. A laser beamis split into two arms, A and B , distanced by d . They cre-ate four propagating polariton WP s, coded in different colors, n A , (magenta, blue) and n B , (red, green) moving along the x axis of a microcavity ridge in the direction depicted by thearrows. pump intensity threshold, polaritons with k x ∼ ± k x (Fig. 1(a)). Assketched in Fig 1(b), this procedure results in the forma-tion of four propagating polariton wave packets ( WP s).We label the macroscopic state of the WP s as ψ A , ψ A , ψ B , ψ B , where the superscript refers to the excitationbeam, the subscript 1(2) is for WP s initially moving tothe left (right), i.e. with k x < k x > V A and V B ), associatedto a blue-shifted dispersion relation, coming from carrier-carrier repulsive interactions . The densities of the po-lariton WP s are given by n A,Bj = (cid:12)(cid:12)(cid:12) ψ A,Bj (cid:12)(cid:12)(cid:12) , j = 1 , WP s created by A have never been together with thosegenerated by B , as sketched in Fig. 1(b). However, WP swith the same subscript j are in the same quantumstate . Using the capability of measuring directly inmomentum space, a unique condition only achievable inlight-matter condensates, we can assess whether or not WP s ψ A and ψ B (or ψ A and ψ B ) are correlated to eachother, being components of the same condensate. Thetwo WP s propagating to the left are described by a com- mon macroscopic order parameterΨ coh ( x ) = ψ A ( x ) + e iφ ψ B ( x ) , (1)while those propagating to the right are described byΨ coh ( x ) = e iφ ψ A ( x ) + ψ B ( x ) . (2)The phases are chosen to have inversion symmetry withrespect to x = 0, because in our experiments we tune theintensities of the two lasers in order to get a symmetricalpotential V ( x ) = V ( − x ). In that respect, our conden-sates are related to each other through the symmetry ofthe excitation process.Furthermore, our potential landscape renders an equalmotion for ψ Aj and ψ Bj , i.e. equal momenta | ( k x ) Aj | = | ( k x ) Bj | = k x . These are precisely the suitable condi-tions to observe coherence between two components spa-tially separated by d , i.e. ψ Aj ( x − d/
2) = ψ Bj ( x + d/
2) = ψ ( x ), of a given condensate Ψ cohj . This coherence canbe observed in k -space as we discuss now.For the sake of clarity, we focus in the following discus-sion only on the left-propagating WP s. The correspond-ing order parameter in k -space can be written as:Ψ coh ( k x ) = ψ A ( k x ) + e iφ ψ B ( k x ) = e − ik x d/ ψ ( k x ) + e i ( φ + k x d/ ψ ( k x ) (3)with ψ ( k x ) being the Fourier transform of ψ ( x ) .This yields a momentum distribution n coh ( k x ) = (cid:12)(cid:12) Ψ coh ( k x ) (cid:12)(cid:12) = 2 [1 + cos ( k x d + φ )] | ψ ( k x ) | . (4)The coherence between the two components produces in-terference fringes with a period∆ k x = 2 π/d. (5)Our aim is to observe the existence of interferences in k -space coming from this macroscopic two-component con-densate. Far-field detection allows the direct measure-ment of momentum distributions, i.e. it gives a directdetermination of the existence, and the period, of theseinterference fringes. It must be taken also into accountthat the measured total polariton density is formed bya condensed population, n coh , coexisting with a thermalone , therefore the interference patterns visibility, ν , islower than 1 (see Supplemental Material ).Our most important result is shown in Fig. 2(b): weindeed observe the interference fringes in k -space, de-scribed by Eq. 4, directly in the polariton emission. Thiscertifies the correctness of our hypothesis that each cou-ple of WP s ( ψ Aj , ψ Bj ) constitutes a two component con-densate. Figure 2(a) shows the actual evolution in timeof the four WP s schematically depicted in Fig. 1(a): ourresults clearly demonstrate that the distance d betweenthe two components of each condensate remains constant FIG. 2. (a) Emission in real space, along the x axis of the ridge, versus time. Gray circles at x = ± µ m indicate the spatiallocation of the A and B laser beams; the trajectories of the four WP s, n A , n A , n B and n B , are indicated by the dashed arrows.(b) Momentum space emission, along k x , versus time. The grey circle indicates that the laser beams, A and B , excite the ridgeat k x ∼
0. The dashed, black arrows indicate the acceleration of the condensates n coh and n coh , as well as the deceleration ofthe WP s n B and n A . Intensity is coded in a normalized, logarithmic false color scale. with time during the first ∼
70 ps ( d = d AB ), as evi-denced by the dashed parallel arrows. Figure 2(a) con-tains also interesting real-space interferences when WP s ψ A and ψ B overlap in real space at 66 ps that we shalldiscuss in more detail below. A peculiarity of our exper-iments is that we observe the dynamics of the coherence;this allows us to determine that the two components ofthe condensate are phase locked since there is not anydrift in the interference patterns.As readily seen in Fig. 2(b), an initial acceleration ofthe four WP s, from rest, k x = 0, to k x = ± . µ m − during the first 40 ps, is followed by a uniform motiontaking place from 40 ps to 70 ps. The interference pat-tern of each condensate is observed until ∼
75 ps, instantat which ψ A and ψ B disappear from the sample regionimaged in the experiments. Then WP s ψ B and ψ A areprogressively slowed by the presence of the barriers at theexcitation spots ( V A / V B halts ψ B / ψ A ). When these two WP s, which are the components of two different conden-sates Ψ coh and Ψ coh , are stopped (at ∼
100 ps) anotherinterference appears in k -space, but now at k x = 0 asit corresponds to WP s at rest. This means that thesetwo condensates also interfere with each other, being re-markable that Ψ coh and Ψ coh still preserve some kindof mutual coherence, supporting the functional form ofEqs. (1) and (2). For longer times, the two WP s moveagain, as can be observed in Figs. 2(a,b), becoming moredifficult to track their trajectories.Note that our measurements are performed averagingover millions of shots of the pulsed laser, therefore if φ were a phase determined by the projection involved inthe measurement process , it would take a randomvalue in each realization. Then, averaging over all thepossible results, the interference pattern would not beobserved. However, as a consequence of the symmetry V ( x ) = V ( − x ) of the potential, the whole state of thefour WP s, Ψ, is symmetric, both in real- and momentum-space. The continuity in k -space of the wave-function(Ψ( k x )) and of its derivative ( ∂ Ψ( k x ) /∂k x ) sets the rela-tive phase φ and makes the experimental realizations con-tribute constructively to the observed interference pat-terns. In other words, the spatial symmetry involved inthe buildup of the condensates determines the relativephase φ . In this sense, they are not independent fromeach other although they have never before coincided inreal space.Further insight into the quantum coherence is ob-tained by analyzing in detail the interferences occurringin momentum- and real-space. Accordingly, we presentin Fig. 3 two-dimensional maps of the polariton emis-sion at three consecutive, relevant times . We focus onthe correspondence between the period of the interfer-ence patterns in each space (real and momentum) andthe separation between the WP s in the complementaryspace. Figure 3(a) shows the momentum distribution n ( k x , k y ), 35 ps after the impinging of the laser beamson the sample. The coherence of each Ψ cohj is observedby the conspicuous interference patterns, n cohj , centeredat k x = ± . µ m − . In both cases, the fringes periodamounts to ∆ k x = 0 . µ m − that, according toEq. 5, should correspond to a distance between WP sof d = 71(4) µ m. This is in good agreement with theexperimental distance seen in Fig. 3(b): the two com-ponents of each condensate, n Aj and n Bj , are separatedby d (cid:39) µ m (see dashed arrows). Our findings are fur-ther supported by the Fourier transform map of n ( k x , k y )shown in Fig. 3(c): a well-defined Fourier component at∆ X = d = 70 µ m is obtained, in accordance with theseparation directly observed in real space. FIG. 3. (a) Momentum distribution n ( k ), at 35 ps after the excitation, showing the condensates n coh / n coh at k x = ∓ . µ m − , respectively. (b) Corresponding n ( r ) distribution showing WP s n A , n A , n B and n B . (c) Fourier transform of n ( k ),obtaining a frequency at ∆ X = d = 70 µ m. (d) Momentum distribution n ( k ) at 66 ps showing n B and n A at k x = ∓ . µ m − ,respectively. (e) Real space distribution n ( r ) showing the interferences of n at x = 0, created by the overlapping in realspace of ψ B and ψ A . White dashed rectangle marks the region of interest where the interference occurs. (f) Fourier transformrestricted to the region of interest in n ( r ), showing a frequency at ∆ K x = κ = 3 . µ m − . (g) Momentum distribution n ( k ) at108 ps, showing the interferences n at k x ∼
0. (h) Corresponding n ( r ) distribution showing n B and n A . (i) Fourier transformof n ( k ), obtaining a frequency at ∆ X = d = 60 µ m. Intensities in the false color scales for momentum, real and Fourierspaces are normalized to unity. The tilt in all panels originates from the orientation of the ridge with respect to the entranceslit of the spectrometer. The white dashed arrows mark the distances in real- and momentum-space between WP s. The fullarrows show these distances in the corresponding Fourier transform. Supplementary Video S1/S2 shows the time evolution ofthe emission in real/momentum space . Coherence in real space have been profusely studiedin cold atoms , excitons and polariton conden-sates . Our experiments also show interferences inreal space between two condensates, similar to those re-ported in atomic BECs . This is shown in Fig. 3(e)at 66 ps when WP s ψ A and ψ B meet each other at x ∼
0. The appearance of interference fringes in realspace, n , signals unambiguously to coherence betweenthese two WP s. Since real and momentum spaces are re-ciprocal to each other, equivalent results for the interfer-ence patterns are expected. The complementary expres-sion in real space to Eq. 5 reads now ∆ x = 2 π/κ , where∆ x is the period of the fringes and κ the difference inmomentum of the propagating WP s. The experimentalperiod of the fringes, seen in the dashed-rectangle areain Fig. 3(e), ∆ x = 1 . µ m, should correspond to κ = ( k x ) A − ( k x ) B = 3 . µ m − . This is again borneout by our results, as shown in Fig. 3(d), where the emis-sion in k -space shows clearly that WP s ψ A and ψ B arecounter-propagating with k x = ± . µ m − , respectively.Figure 3(f) shows the Fourier transform of n in the re-gion enclosed by the rectangle in Fig. 3(e). It reveals a strong ∆ K x Fourier component at 3.1 µ m − , in fullagreement with the value of κ displayed in Fig. 3(d). Letus also emphasize that WP s first meet in real space at66 ps, while interferences in momentum space are seenas early as ∼
10 ps demonstrating that the phase lockingoccurs before the WP s spatially overlap.The third result that we present corresponds to thearrival at 108 ps of ψ A and ψ B to the excitation re-gions B and A , respectively. Here, they run into the hillsof the photogenerated potentials V B and V A that elasti-cally convert their kinetic energy into potential energy .They slow down, halting, providing a new separation be-tween WP s n A and n B , d ∼ µ m (see Fig. 3(h)).Their emission in momentum space, arising from k x ∼ k x = 0 . µ m − ( n , see Fig. 3(g)). Once again, Eq. 5 predictsa separation d = 60(4) µ m between n A and n B , asobserved in the experiments. For completeness, we alsoshow in Fig. 3(i) the Fourier transform map of the densitythat exhibits an emerging component at ∆ X = d = 60 µ m. Further insight into this scaling behavior, relatingdistances in real space between WP s with the fringesperiod in momentum space, is presented in the Supple-mentary information . III. CONCLUSIONS
In summary, the convenience of monitoring the evolu-tion of exciton-polaritons in semiconductor microcavites,through the detection of emitted light, makes this systeman ideal platform to study quantum coherence propertiesin real- as well as in momentum-space. Profiting fromthis fact, we have demonstrated the existence of quantumremote coherence between spatially separated polaritoncondensates whose phase is determined by the symmetryof the excitation conditions and therefore is constant ineach realization of our multi-shot experiments. This is- sue is related to the superposition principle in quantummechanics and it is crucial to understand how mutualcoherence is acquired.
IV. ACKNOWLEDGEMENTS
We thank D. Steel and J.J. Baumberg for a crit-ical reading of the manuscript. C.A. acknowledgefinancial support from a Spanish FPU scholarship.P.G.S. acknowledges Greek GSRT program “ARIS-TEIA” (1978) for financial support. The work waspartially supported by the Spanish MEC MAT2011-22997, CAM (S-2009/ESP-1503) and FP7 ITN’s “Cler-mont4” (235114), “Spin-optronics” (237252) and “IN-DEX” (289968) projects. ∗ [email protected] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys.Rev. Lett. , 3969 (1995). J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P.Jeanbrun, J. M. J. Keeling, F. M. Marchetti, M. H.Szyma´nska, R. Andr´e, J. L. Staehli, V. Savona, P. B. Lit-tlewood, B. Deveaud, and L. S. Dang, Nature , (2006). M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S.Durfee, D. M. Kurn, and W. Ketterle, Science , 637(1997). D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A.Cornell, Phys. Rev. Lett. , 1543 (1998). T. Esslinger, I. Bloch, and T. W. H¨ansch, J. Mod. Opt. , 2725 (2000). I. Bloch, Nature Phys. , 23 (2005). M. Born and E. Wolf,
Principles of Optics: Electromag-netic Theory of Propagation, Interference and Diffractionof Light (Cambridge University Press, 2000). Z. Ficek and S. Swain,
Quantum Interference and Coher-ence. Theory and Experiments (Springer, 2005). A. Kavokin, Nature Photon. , 591 (2013). L. Novotny,
Principles of Nano-Optics (Cambridge Uni-versity Press, 2006). L. Mandel and E. Wolf,
Optical Coherence and QuantumOptics (Cambridge University Press, 1995). G. J. Pryde, Nature Photon. , 461 (2008). L. Pitaevskii and S. Stringari, Phys. Rev. Lett. , 4237(1999). M. Saba, T. A. Pasquini, C. Sanner, Y. Shin, W. Ketterle,and D. E. Pritchard, Science , 1945 (2005). Y. Shin, G. B. Jo, M. Saba, T. A. Pasquini, W. Ketterle,and D. E. Pritchard, Phys. Rev. Lett. , 170402 (2005). P. W. Anderson,
Basic Notions of Condensed MatterPhysics (Benjamin, Menlo Park, California, 1984). Y. Castin and J. Dalibard, Phys. Rev. A , 4330 (1997). A. J. Leggett,
Quantum Liquids (Oxford University Press,Oxford, 2006). L. P. Pitaevskii and S. Stringari,
Bose-Einstein Condensa-tion (Oxford University Press, 2003). See Supplemental Material athttp://link.aps.org/supplemental/XXX for further details. A. Amo, J. Lefr`ere, S. Pigeon, C. Adrados, C. Ciuti, I.Carusotto, R. Houdr´e, E. Giacobino, and A. Bramati, Na-ture Phys. , 805 (2009). A. Amo, S. Pigeon, D. Sanvitto, V. G. Sala, R. Hivet,I. Carusotto, F. Pisanello, G. Lem´enager, R. Houdr´e, E.Giacobino, C. Ciuti, and A. Bramati, Science , 1167(2011). E. Wertz, L. Ferrier, D. D. Solnyshkov, R. Johne, D. San-vitto, A. Lemaitre, I. Sagnes, R. Grousson, A. V. Kavokin,P. Senellart, G. Malpuech, and J. Bloch, Nature Phys. ,860 (2010). Since all WP have the same spreading, we employ the usualterminology simply labeling each WP by its central value k x . E. del Valle, D. Sanvitto, A. Amo, F. P. Laussy, R. Andr´e,C. Tejedor, and L. Vi˜na, Phys. Rev. Lett. , 096404(2009). Since our samples are not strictly one-dimensional, we pro-vide full images of the emission in two-dimensional spaces. S. S. Hodgman, R. G. Dall, A. G. Manning, K. G. H.Baldwin, and A. G. Truscott, Science , 1046 (2011). D. Snoke, Science , 1368 (2002). A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler,L. V. Butov, A. V. Kavokin, K. L. Campman, and A. C.Gossard, Nature , 584 (2012). R. Balili, V. Hartwell, D. Snoke, L. Pfeiffer, and K. West,Science , 1007 (2007). G. Roumpos, M. Lohse, W. H. Nitsche, J. Keeling, M. H.Szymanska, P. B. Littlewood, A. Loffler, S. Hofling, L.Worschech, A. Forchel, and Y. Yamamoto, Proc. Natl.Acad. Sci. USA , 6467 (2012). F. Manni, K. G. Lagoudakis, R. Andre, M. Wouters, andB. Deveaud, Phys. Rev. Lett. , 150409 (2012). A. Rahimi-Iman, A. V. Chernenko, J. Fischer, S. Brod-beck, M. Amthor, C. Schneider, A. Forchel, S. H¨ofling,S. Reitzenstein, and M. Kamp, Phys. Rev. B , 155308(2012). R. Spano, J. Cuadra, G. Tosi, C. Anton, C. A. Lingg,D. Sanvitto, M. D. Martin, L. Vina, P. R. Eastham, M.van der Poel, and J. M. Hvam, New J. Phys. , 075018(2012). C. Ant´on, T. C. H. Liew, G. Tosi, M. D. Mart´ın, T. Gao, Z.Hatzopoulos, P. S. Eldridge, P. G. Savvidis, and L. Vi˜na,Phys. Rev. B88