Polarized Nonequilibrium Bose-Einstein Condensates of Spinor ...

PHYSICAL REVIEW LETTERS

PRL 105, 256401 (2010)

week ending 17 DECEMBER 2010

Polarized Nonequilibrium Bose-Einstein Condensates of Spinor Exciton Polaritons in a Magnetic Field A. V. Larionov,1 V. D. Kulakovskii,1 S. Ho¨fling,2 C. Schneider,2 L. Worschech,2 and A. Forchel2 1

2

Institute of Solid State Physics, RAS, Chernogolovka, 142432 Russia Technische Physik, Physikalisches Institut and Wilhelm Conrad Ro¨ntgen Research Center for Complex Material Systems, Universita¨t Wu¨rzburg, D-97074 Wu¨rzburg, Germany (Received 9 August 2010; published 14 December 2010) The effect of a magnetic field on a spinor exciton-polariton condensate has been investigated. A quenching of a polariton Zeeman splitting and an elliptical polarization of the condensate have been observed at low magnetic fields B < 2 T. The effects are attributed to a competition between the magnetic field induced circular polarization buildup and the spin-anisotropic polariton-polariton interaction which favors a linear polarization. The sign of the circular polarization of the condensate emission at B < 3 T is negative, suggesting that a dynamic condensation in the excited spin state rather than the ground spin state takes place in this magnetic field range. From about 2T on, the Zeeman splitting opens and from then on the slope of the circular polarization degree changes its sign. For magnetic fields larger than the 3 T, the upper spin state occupation is energetically suppressed and circularly polarized condensation takes place in the ground state. DOI: 10.1103/PhysRevLett.105.256401

PACS numbers: 71.36.+c, 42.55.Sa, 42.65.Pc

Bose-Einstein condensation (BEC) and polariton lasing in the dissipative system of exciton polaritons in microcavities (MCs) have been the subject of numerous investigations in recent years [1,2]. Because of a very light effective mass ( 105 the free electron mass), the condensation of polaritons can occur at high temperatures [1,3,4]. On the other hand, due to their short radiative lifetime (on the order of a few ps) the polaritons never achieve thermal equilibrium with the crystal lattice. To date, there are several sources of experimental evidence for macroscopic occupation numbers of single quantum states by polariton-polariton scattering in resonantly excited MCs [5–9], and more recently also in nonresonantly excited CdTe and GaAs MCs [1,2,10]. Strictly speaking, being two-dimensional objects, quantum-well (QW) exciton polaritons can form a spontaneous BEC only nonresonantly excited if they are confined in a natural [1] or artificial [2,11,12] potential trap. An important feature of exciton polaritons is a polarization degree of freedom which is related to the exciton spin states. Because of the effects of exchange, the interparticle interactions are strongly spin anisotropic: particles with similar spin projections strongly repel, while there is weak attraction between the polaritons with opposite spins. Consequently, polariton condensates behave differently from spinless atomic condensates even in the thermodynamic limit. The energy of a polariton system is minimized at equal numbers of left and right circularly polarized polaritons, resulting in linear polarization for polariton systems in a condensed state. Large linear polarization degrees of light emitted by polariton condensates have been reported in several references [1,13,14]. The orientation of the polarization plane is found to be pinned to one of the crystal axes. This indicates that the linear 0031-9007=10=105(25)=256401(4)

polarization is favored by the polariton state splitting caused by MC anisotropy rather than by spontaneous symmetry breaking of gauge invariance [15,16] in the condensed state [1,17]. In fact, polariton condensates in GaAs samples with small anisotropy have shown only small degrees of linear polarization of 0:15 [18]. Despite being of major physical interest [17,19–21], the behavior of spinor polariton condensates in a magnetic field has not yet been experimentally explored, which is the topic of the present work. The properties of polarized condensates are expected to differ substantially from those of spinless bosons, especially when a magnetic field is applied [16,21,22]. For the noninteracting condensate, the tendency to occupy the lowest available energy state, would lead to a redshift of the chemical potential when the magnetic field is raised, and the condensate would become increasingly circularly polarized. One of the expected intriguing polarization-related phenomena in systems in thermodynamic equilibrium with a spin-anisotropic polaritonpolariton interaction is the full paramagnetic screening, also known as spin-Meissner effect [21]. It appears due to a competition between the magnetic field trying to make the polariton system circularly polarized and the polaritonpolariton interaction favoring a linear polarization. In weak magnetic fields, this competition was predicted to result in the formation of an elliptically polarized condensate and the complete suppression of the Zeeman splitting [21]. In this Letter we present magneto-optical investigations of spinor exciton-polariton condensates in GaAs based QW MCs with a high Q factor. The main advantage of the high finesse MCs is a strong suppression of the bottleneck effect in the relaxation of the lower polaritons (LPs) due to an increased LP lifetime. Near-field photoluminescence measurements show that the LP condensation occurs in

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potential traps with 5 to 15 m size. The condensates in the largest potential traps at negative detuning ( ¼ EC EX & 8 meV) exhibit a very narrow emission line with a full width at half maximum (FWHM) less than 0.2 meV and a linear polarization degree at zero magnetic field lin ðB ¼ 0Þ > 0:9. Here, EC and EX are the energies of cavity and exciton modes at wave vector k ¼ 0. The experiment in Faraday configuration has revealed elliptically polarized emission of the condensate and quenching of the Zeeman splitting at small B. At a critically large field, the Zeeman splitting is recovered as predicted recently [16,21,22]. At the same time, the observed polarization properties of our condensate at small B turn out to be qualitatively different from the predicted ones for the case of thermodynamic equilibrium: The condensate shows negative instead of positive circular polarization at B < Bcr . Furthermore, the emission is dominated by the excited spin state in a small range of B above Bcr , which indicates that a nonequilibrium dynamic LP condensation occurs. The MC sample grown by molecular beam epitaxy consists of four 7 nm thick GaAs QWs, separated by 4 nm AlAs barriers, which are centered in a half- thick AlAs cavity. A large number of Al0:2 Ga0:8 As=AlAs distributed Bragg reflector pairs of 32 (36) have been grown as the top (bottom) mirror, respectively, to minimize polariton leakage by the photonic component. The MC Q factor and the Rabi splitting of the exciton-cavity modes are 7000 and 10:5 meV, respectively. The sample was placed into an optical thermostat with the temperature control at T ¼ 6–30 K or into a magneto-optical cryostat at T ¼ 2 K with a superconducting solenoid. An excitation was produced by a pulsed laser ( ¼ 503:2 nm, pulse duration 7 ns) at a repetition rate of 3 kHz. The excitation spot has a diameter of 50 m. Figure 1 shows LP emission spectra recorded normal to the MC plane with an angle resolution of 1 under an above band gap excitation with 7-ns long pulses in a wide range of excitation densities P. The LP emission line changes

very weakly until P 50 kW=cm2 . In this excitation range, its intensity increases slightly superlinearly with P. In the narrow range of P ¼ 50 to 55 kW=cm2 the line FWHM decreases from 0.6 meV to 0:2 meV and shifts to higher energies by 0.3 meV. A further increase in P to 95 kW=cm2 causes more than 2 orders of magnitude increase in the emission intensity, a monotonic increase in FWHM from 0.2 to 0.3 meV, and a blueshift of 0.1 meV. A thresholdlike increase in the emission intensity is observed in a small range of 15 m inside the excited spot with a total diameter of 50 m. The emission is directed normal to the MC plane (k ¼ 0). This behavior of LP emission is characteristic of Bose-Einstein condensation of LPs in k space [1,2]. Figure 1 shows that the LP emission prior to condensation is unpolarized. Nevertheless, the emission from the condensed phase has a well pronounced linear polarization. The maximum lin is observed at P slightly above the critical one, Pthr , for the LP condensation when the condensate emission line FWHM U1 > 0. The chemical potential at B ¼ 0 is 0 ¼ ðU0 U1 Þn. In a weak magnetic field, when jgB Bj 2nU1 , does not depend on B. Instead, ¼ 0 and Sz ¼ gB B=4U1 as can be derived from Eq. (1) [21]. The condensate state in this case is elliptically polarized with circ ¼ 2Sz =n ¼ gB B=2nU1 . Thus, the magnetic field in this regime changes the polarization degree of the emitted light but not its free energy. The redshift of the LP energy due to Zeeman effect is then exactly compensated by increase of the blueshift due to condensate polarization. With increasing B above Bcr ¼ 2nU1 =gB , the condensate at T ¼ 0 should become fully circularly polarized. From there on, the energy of photons emitted by the condensate should decrease linearly with increasing B as no additional interaction mediated compensation occurs anymore, and consequently the normal Zeeman effect should be observed in photoluminescence. Let us now compare theoretical predictions with the experiment in more detail. Figures 2 and 3 show that the experiment reproduces (i) the predicted quenching of Zeeman splitting at small B, (ii) its opening above Bcr , (iii) elliptical polarization of condensate emission with a monotonic decrease in lin below Bcr , and (iv) circular polarizations of Zeeman components at high B. In addition, the relative experimental intensity of the þ component at high B decreases much quicker than expðEZ =kTLP Þ, as it is expected for LP condensation. The value expðEZ =kTLP Þ calculated at B ¼ 4:25 T with the experimental value EZ ¼ 0:062 meV exceeds 0.7 even under the assumption that TLP ¼ Tbath ¼ 2 K, whereas the experimental ratio is as low as 0:3. Note that the degree of circular polarization of the exciton reservoir at large k does not exceed 0.1. Hence, the strong condensate polarization at B > Bcr is a direct result of LP condensation. Note, however, that several of the experimentally observed condensate emission features are in qualitative contradiction with the model describing condensation in thermal equilibrium. The most important discrepancy is the fact that circ does not follow a predicted monotonic growth from 0 to 1 with increased B. Instead, it is seen in Figs. 2 and 3(a) that circ becomes negative at low B and increases in its magnitude up to B ¼ Bcr when the Zeeman splitting is recovered. At B ¼ Bcr , when circ reaches its minimum value and changes the sign of its slope, lin is still as high as 0.25. At higher B the negative circular polarization becomes more positive, circ changes sign and grows quickly reaching 0:44 0:2 at B ¼ 4:25 T. Note that no sign reversal is observed in Zeeman splitting of LPs prior to condensation. The latter is in agreement

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with earlier measurements of the exciton g factor in narrow (7 nm-wide) GaAs QWs [23] The negative sign of circ in the condensate emission at B < Bcr can be explained by assuming that the dominated contribution in the condensate originates from the excited rather than the ground state Zeeman LP level. That can occur, for example, due to a somewhat higher occupation of the excited level prior to the condensation threshold because of a higher rate of the LP decay compared to the rate of energy relaxation at small k vectors. Such dynamic rather than equilibrium polariton condensation into excited states has previously been observed in metal-grating modulated MCs [24] and in pillar MCs [12,25]. A similar effect of nonequilibrium LP condensation in the upper state has been observed as well in our studies of MC pillars fabricated from the investigated planar MC where an enhanced splitting of x and y components has been reached due to a rectangular shape of pillars. The assumption of excited LP state condensation is also supported by the fact that the intensity of the upper spin component at small Zeeman splitting (B ¼ 2–3 T) exceeds that of the lower one as is well seen in Fig. 2. The higher intensity of the lower Zeeman component which is characteristic to condensation in thermal equilibrium is observed only at B > 3 T when EZ exceeds 30 eV. Only beginning from this field, the behavior of relative intensities of the upper and lower Zeeman components approaches the expected one for condensates in thermal equilibrium. Thus at B & Bcr , one has to go beyond the scope of the equilibrium model of Ref. [21], and our observations suggest that the condensation starts at the upper (L2) rather than lower (L1) LP spin level. The quenching of the Zeeman splitting in the condensate at small B in this case is possible only if the condensation of LPs at the L2 level causes its lowering relative to the empty level L1 until their convergence is reached. That is impossible in condensates of cylindrical symmetry where the population of a circularly polarized state results in its blueshift relative to the empty level [21]. Note, however, that the investigated LP trap has a lowered symmetry resulting in the splitting of k ¼ 0 LP level into two linearly polarized ones with the gap EL2 EL1 ¼ below condensation threshold. The population of the upper level (L2) with a total concentration of the LPs n should result in the decrease of the gap till the convergence of the populated upper level with the empty lower one [26] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EL2 ðnÞ EL1 ð0Þ ¼ 2 2U1 n: (2) The convergence of the levels persists at low magnetic fields until gB B is markedly smaller than . Note that the convergence of the upper populated level with the empty lower one is a necessary and sufficient condition for a quenching of Zeeman splitting in the condensate and a reverse sign of its polarization degree, observed in the experiment at B < 2 T.


In conclusion, magneto-optical studies of the spinor LP condensate revealed a quenching of a LP Zeeman splitting and an elliptical polarization of the condensate emission characteristic to the spin-Meissner effect expected in this condensate [21]. At the same time, the sign of the circular polarization of the emission from elliptically polarized condensate turns out to correspond to the excited rather than ground LP spin state. The revealed sign reversal is tentatively connected to an earlier condensation in the nonequilibrium system in the excited LP spin state. Our magneto-optical results show that nonequlilibrium condensation effects are important in the present MC and further investigations are called for to fully explore the rich spinor polariton condensate physics in magnetic fields. We thank S. S. Gavrilov, M. M. Glazov, N. A. Gippius, A. V. Kavokin, S. G. Tikhodeev, and V. B. Timofeev for fruitful discussions. Financial support of this work by the Russian Foundation for Basic Research, the Russian Academy of Sciences, and the State of Bavaria is gratefully acknowledged.

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