Spin noise of exciton-polaritons in microcavities
We develop a theory of spin fluctuations of exciton-polaritons in a semiconductor microcavity under the non-resonant unpolarized pumping. It is shown that the corresponding spin noise is sensitive to the scattering rates in the system, occupation of the ground state, statistics of polaritons, and interactions. The spin noise spectrum drastically narrows in the polariton lasing regime due to formation of a polariton condensate, while its shape can become non-Lorentzian owing to interaction-induced spin decoherence.
pacs:72.25.Rb, 72.70.+m, 71.36.+c, 03.75.Kk
Introduction. Quantum microcavity is a system where a semiconductor quantum well is placed between the Bragg mirrors making it possible to achieve the strong coupling between the light and matter. In this system the energy is coherently transferred back and forth between the photon trapped in the microcavity and the exciton, the elementary excitation of the semiconductor. First observed by Weisbuch et al. Weisbuch et al. (1992), the strong coupling results in the formation of mixed light-matter quasiparticles, exciton-polaritons, extensively studied since then Kavokin et al. (2011); Timofeev and Sanvitto (2012). The exciton-polaritons combine an extremely small effective mass, inherited from the photon, and strong interactions between themselves and with the environment due to the excitonic fraction in this quasiparticle. These quasiparticles are at the origin of several fascinating phenomena, which primarily include polariton lasing (5): macroscopic accumulation of polaritons in a single quantum state (non-equilibrium condensate) accompanied by spontaneous emission of coherent light by this state.
In semiconductor microcavities exciton-polaritons are characterized by the spin projections onto the structure growth axis corresponding to the right or left circular polarization of the photon and to the same spin component of the exciton. Superpositions of states give rise to the linear or elliptical polarization of exciton-polaritons. Polariton lasers represent a model bosonic system where the polarization of light emitted by a microcavity directly corresponds to the spin state quasiparticles. This allows studying the polariton spin dynamics by optical methods Martin et al. (2002). A great amount of prominent spin-related effects has been realized in microcavities, including self-induced Larmor precession Krizhanovskii et al. (2006); Solnyshkov et al. (2007), linear polarization inversion Kavokin et al. (2005a), optical spin Hall effect Kavokin et al. (2005b); Leyder et al. (2007); Kammann et al. (2012), spin Meissner effect Rubo et al. (2006); Larionov et al. (2010); Walker et al. (2011), spin multistability Gippius et al. (2007); Paraïso et al. (2010), etc., see Refs. Kavokin et al. (2011); Shelykh et al. (2010) for reviews. Hence, quantum microcavities became a solid state playground to study interacting and, generally, non-equilibrium Bose systems.
In polariton lasers, the spontaneous symmetry breaking results in the appearance of a stochastic vector polarization including circular polarization related with the spin of a polariton condensate Baumberg et al. (2008); Read et al. (2009). The Stokes vector (pseudospin) of a condensate may be pinned to one of the crystal axes (21), due to the structure anisotropy. In this case the time-averaged polarization of emission of polariton lasers is defined by pinning, while it momentary value fluctuates. The spin fluctuations give rise to the “spin noise” which may be studied by optical measurements. The spin fluctuations, being inherent to any system at equilibrium or not, were first observed in 1981 in Na vapor Aleksandrov and Zapasskii (1981). The studies of spin fluctuations has become an important part of spintronics. The measurements of spin noise provide a crucial information on spin dynamics of carriers, excitons, and nuclei, including spin precession and relaxation rates and statistics of spin fluctuations. These characteristics are hardly accessible otherwise Crooker et al. (2004); Mueller et al. (2010); Crooker et al. (2010); Aleksandrov and Zapasskii (2011); Dahbashi et al. (2012); Li et al. (2012).
Here we study theoretically the spin noise spectra of polariton lasers both below and above the laser threshold and demonstrate that it is extremely sensitive to the mean occupation number of the condensate and to the statistics of polaritons. Similar approach can be used for a variety of systems where condensates of spin polarized bosons interact with an incoherent reservoir. An appropriate example is given by magnon condensates in crystals, either strongly driven by periodic fields (29) or imposed to sufficiently strong static magnetic field at a low temperature (30). Here the noise is seen as fluctuations in the sample magnetization. The finite-temperature Bose-Einstein condensates of atoms in optical traps with the synthetic spin-orbit coupling represent another important example of direct applicability of our approach (31).
Model. The dynamics of the exciton-polariton spin doublet can be described with the pseudospin approach where the density matrix of polariton state with the wavevector can be written as . Here is the unit matrix, is a pseudovector composed of Pauli matrices, is the spin-average occupancy of the state , and is the pseudospin of polaritons in this state. In what follows we focus on the dynamics of the ground state, corresponding to , treat all other states as a reservoir, and omit subscript in the notations. The pseudospin components , where is the Cartesian index corresponding to the polarization of emission: gives its circular polarization degree, while and give the linear polarization degree in the axes frames and rotated by with respect to each other. Kavokin et al. (2011) The dynamics of the occupation number and the polariton pseudospin is governed by the set of kinetic equations Shelykh et al. (2010)
where is the lifetime of polaritons in the ground state, is the effective magnetic field arising from anisotropy of the system and from polariton-polariton interactions, its explicit form will be given below. The scalar, , and pseudovector, , collision integrals describe arrival and departure of the particles into the ground state, see Refs. Shelykh et al. (2010); Krizhanovskii et al. (2006) for general expressions. Here we adopt their simplest form
Here and are out- and in-scattering rates, related with the presence of the reservoir, as schematically illustrated in Fig. 1(a). In particular, is proportional to the occupation of the reservoir and it is determined by the pumping rate. The term proportional to describes the stimulated transitions due to the bosonic statistics of quasiparticles. In the collision integral for the polariton spin, , is the spin relaxation time and is the steady-state spin induced by the effective field . If polariton exchange with reservoir is efficient, one can introduce the effective temperature of the polariton system. The steady-state spin is for and for . In what follows, unless otherwise specified, we consider the case of high temperatures, where , and the occupancy of the polariton ground state given by the balance of in- and out- scattering processes is:
where the condition should hold (32). The straightforward generalization of our approach to account for the entire ensemble of polaritons following Refs. Shelykh et al. (2010); Krizhanovskii et al. (2006) using the full density matrix is not expected to yield qualitatively different results.
The fluctuations of the condensate occupation number , and pseudospin are described by the correlation functions, namely: and , where the angular brackets denote the averaging over the time for a given shift . According to the general theory of fluctuations Gantsevich et al. (1979); Landau and Lifshitz (1981); Ivchenko (1974) their correlation functions obey the same set of kinetic equations for or dependence as fluctuating quantities (36). The solution of Eqs. (1) and (2) in the absence of effective magnetic fields and interactions (), results in the exponential time-decay of correlations and isotropic spin fluctuations:
where is the Kronecker -symbol, single time correlators (mean square fluctuations) , will be found below, while the particle-spin correlations vanish and will be disregarded. We introduced the correlation times , according to
Equations (6) and (7) clearly show that the particle number and spin fluctuations of exciton-polaritons decay exponentially with time and the correlation time of the particle number fluctuations is , while the spin fluctuations vanish faster, at . The spin fluctuation spectra defined as are Lorentzian:
with the half-width half maximum determined by the inverse spin correlation time. It follows from Eq. (7) that the correlation time of fluctuations is strongly enhanced and the spin noise spectrum is strongly narrowed, if , i.e. where the ground state is macroscopically occupied. In particular, if and which corresponds to the negligible spin-flip in the ground state and negligible depletion of the condensate due to the scattering of polaritons back to the reservoir, we have for the correlation time of particle and spin fluctuations . The life-time of exciton-polaritons in the state-of-the-art structures varies from ps to ps with the corresponding maximum being between and , yielding in the range of ns … s. Hence, the spin noise frequencies range from MHz to GHz for the macroscopically occupied ground state. The typical spin noise spectra and temporal dependence of the correlators in Fig. 1(b),(c) make the drastic effect of the ground state occupation clear. The several orders of magnitude enhancement of the spin correlation time and narrowing of the noise spectrum is a general bosonic effect. Indeed, any fluctuation is amplified by the bosonic stimulation, factor in Eq. (3).
The mean square of the particle and pseudospin fluctuations can be found using the master equation approach Timofeev and Sanvitto (2012). In the absence of interactions and effective magnetic fields, the system is spin-isotropic and can be described by the independent occupations of two orthogonal spin states, and , described by the same distribution functions . Using these distribution functions one can express the mean square fluctuations, that is correlators and , as
where corresponds to isotropic fluctuations, and is the second order coherence of a single state. In particular, corresponds to the thermal statistics (37), where particle and spin square fluctuations are and grow quadratically with the ground state occupation. This situation is realized for equilibrium Bose gas Landau and Lifshitz (2000) or if the ground state feedback on reservoir is negligible. By contrast, corresponds to the coherent statistics, where the mean square fluctuations are suppressed, being . In the limit of low occupancy, in agreement with the theory of classical gas. Moreover, the statistics of spin fluctuations can be determined, as a convolution of distribution functions and . In the limit we obtain:
The full statistics of polariton condensates can be determined by numerical integration of Langevin equations for the condensate wavefunctions, as it was done, e.g. in Ref. Read et al. (2009) for stochastic polarization under pulsed excitation. Here we resort to an analytical approach based on kinetic equations treating phenomenologically.
Role of effective magnetic fields. We begin the discussion of the effective magnetic fields with the case of an anisotropic system where the polariton doublet is split into the pair of states, linearly polarized along and axes. In such a case the vector determines the anisotropic splitting Klopotowski et al. (2006); Krizhanovskii et al. (2006); Amo et al. (2009b); Glazov and Golub (2010), and we assume that the effective temperature of the system exceeds the anisotropic splitting to neglect the steady spin polarization. The and rotate, while the dynamics of remains purely dissipational. As a result, the fluctuations become anisotropic with nonzero spectrum components (c.f. Ref. Glazov and Ivchenko (2012))
and , , is given by Eq. (9), and by Eq. (7). Figure 1(d) shows the spin noise spectra calculated with the allowance for the anisotropic splitting for different occupations of the ground state. It is seen that the single peak is transformed into the two peak structure even for a very small value of taken in our calculation. This is because the spin correlation time depends strongly on the ground state occupation: for small pumping rates and small the product and the splitting is not visible, however, for larger pumping becomes comparable or larger than 1, making the anisotropic splitting resolvable, as demonstrated in Fig. 1(d). If the temperature is so low that , the polaritons are predominantly polarized along the effective field direction . In this case, , , while the mean square fluctuations take the form , . Moreover, the non-trivial single time correlation appears: . Spin noise spectrum in this case can be obtained in a similar way. Equation (12) holds here, albeit with different mean square fluctuations found above, in the numerators.
Now let us consider the effect of polariton-polariton interactions on the spin noise. As follows from multiple experimental and theoretical works Kavokin et al. (2005a); Glazov et al. (2009); Rubo et al. (2006); Larionov et al. (2010); Walker et al. (2011); Paraïso et al. (2010) these interactions are strongly spin-anisotropic: the exciton-polaritons with the same pseudospin components, i.e. with the same circular polarizations, repel each other efficiently due to the exchange interaction of electrons/holes with the same spin, while the polaritons with opposite circular polarizations can weakly attract each other, the latter is neglected. Hence, the interactions create an effective fluctuating field , directed along the axis. Its magnitude ( is the interaction constant) is related to the fluctuations of polaritons pseudospin component. This effective field has two important effects on polariton spin dynamics and spin noise: (i) it induces precession of the pseudospin around the axis, known as self-induced Larmor precession Krizhanovskii et al. (2006); Solnyshkov et al. (2007), and (ii) it suppresses fluctuations of the pseudospin component, favoring linear polarization of the macrooccupied state Rubo et al. (2006); Laussy et al. (2006).
It is instructive to start the analysis with the spin precession effect assuming that the effective temperature is high enough, . In addition, we neglect below the anisotropic splitting (). Clearly, the effective field induces dephasing of the in-plane pseudospin components. It follows from Eq. (2) that
Here the upper (lower) sign corresponds to , correlations between and in-plane components are disregarded, since the field does not couple them, and can be considered as an independent parameter whose fluctuations are given by Eqs. (6), (8). The set of linear Eqs (14) can be readily solved as:
with meaning the averaging over the fluctuations of . The treatment of the general case is beyond the scope of this work, here we consider two limiting cases: the regimes of fast and slow fluctuations, respectively.
If , the interaction induced effective field changes much faster than the psuedospin rotates. This case corresponds to the motional narrowing and
In this limit the spin fluctuations decay exponentially, resulting in the Lorentzian spectrum of spin noise. This spectrum is, however, anisotropic: the width of and is larger than that of because the interaction-induced field does not affect .
In the opposite limit, where the fluctuations of spin component are slow enough and the in-plane spin components make several oscillations during the correlation time of , i.e. , the fluctuations of and of the effective field can be assumed frozen. In this case, the spin dephasing takes place on the timescale of . The particular dependence of the correlator is determined by statistics of the condensate (45). For we obtain for two important limiting cases of coherent and thermal statistics ():
where the dephasing rates are: and . In this limit, the temporal dependence of the spin fluctuations is directly related to the ground state statistics: Gaussian fluctuations of described by in Eq. (10) result in the Gaussian decay of the in-plane spin components, while the sharply-peaked in Eq. (11) for thermal statistics results in the slow power-law decay of the fluctuations due to high probability of small , corresponding to low precession rates. As a result, the noise spectrum of the in-plane pseudospin components deviates strongly from Lorentzian. Doing Fourier transform of Eqs. (17), (18) we obtain:
The calculated non-analytical dependence of the spin fluctuations for the thermal statistics at is related to the power-law temporal decay of the fluctuations. The typical temporal dependence of the in-plane pseudospin correlators and noise power spectra are plotted in Fig. 2(a) and (b), respectively. Figure clearly shows different qualitative behavior of the noise of the in-plane pseudospin components for different statistics of the polaritons. We stress that the drastic difference of the decoherence times and noise spectral widths for coherent and thermal statistics is related to different dependences of on the ground state occupancy, , see Eq. (9).
Note, that if is comparable with or larger than the effective temperature of the system, the fluctuations of the pseudospin component would be suppressed. This effect can be modeled by multiplying the distribution function of in Eqs. (10) and (11) by the Boltzmann factor for the probability of thermal fluctuations Rubo et al. (2006). As a result, in the limit of we obtain . In this case, the dephasing rate, which determines the spin noise spectral width, can be estimated as .
Conclusions. We have developed an analytical theory of spin fluctuations of polaritons in microcavities in the lasing regime and demonstrated that the spin noise spectra, being a fundamental property of any spin system, qualitatively depend on the occupation numbers, statistics, and interactions between the particles. Various regimes of spin noise have been identified. Thus, spin noise spectroscopy allows one to study a large variety of spin-related properties of bosonic systems in a single experiment. Experimental verification of these predictions can be done by Fourier spectroscopy of Kerr or Faraday rotation. Extension of this model to other spin-polarized bosonic systems is straightforward.
Acknowledgments. The financial support from the Russian Ministry of Education and Science (contract No. 11.G34.31.0067 with SPbSU and leading scientist A. V. Kavokin), RFBR, RF President Grants NSh-5442.2012.2 and 2901.2012.2, EU project POLAPHEN is acknowledged. EYS was supported by the University of Basque Country UPV/EHU program UFI 11/55, Spanish MEC (FIS2012-36673-C03-01), and “Grupos Consolidados UPV/EHU del Gobierno Vasco” (IT-472-10).
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