Strong-coupling of quantum dots in microcavities
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Strong-coupling of quantum dots in microcavities
Fabrice P. Laussy, ∗ Elena del Valle, and Carlos Tejedor
Departamento de F´ısica Te´orica de la Materia Condensada,Universidad Aut´onoma de Madrid, Spain (Dated: August 5, 2018)
Abstract
We show that strong-coupling (SC) of light and matter as it is realized with quantum dots (QDs)in microcavities differs substantially from the paradigm of atoms in optical cavities. The type ofpumping used in semiconductors yields new criteria to achieve SC, with situations where the pumphinders, or on the contrary, favours it. We analyze one of the seminal experimental observation ofSC of a QD in a pillar microcavity [Reithmaier et al. , Nature (2004)] as an illustration of our mainstatements.
PACS numbers: 42.50.Ct, 78.67.Hc, 42.55.Sa, 32.70.Jz ∗ Electronic address: [email protected] strong-coupling (SC) regime occurs in a coupled system where the interac-tion strength overcomes the losses. A case of fundamental interest is that of light (photons)and matter (atoms, electrons, etc.), coupled by the electromagnetic force. That this cou-pling is usually so weak accounts for the tremendous success of quantum electrodynamics(QED), that affords all the required accuracy at the level of perturbation theory. By con-fining the emitter in a cavity, repeated interactions with the trapped photon(s) occur andstrong-coupling can thus be obtained, as was demonstrated in pioneering experiments withRydberg atoms migrating in optical cavities [1]. Every achievement with atoms becomesan objective for semiconductors, that offer unique advantages in terms of integration andscalability, but come with their disadvantages in the form of the overall complexity broughtby condensed matter over its fundamental elements. Strong-coupling was first reported insemiconductors with quantum wells in planar microcavities [2], that launched a new fieldinvestigating exotic phases of matter in condensed systems such as Bose-Einstein conden-sates [3] and superfluids [4]. There is not yet a general consensus that quantum physics rulesthese systems and that nonlinear optics could not equally or better account for the observedphenomena. A more exact analogue to the atomic case that sticks closer to the quantumregime is provided by zero dimensional (0D) heterostructures—quantum dots (QDs)—wherethe material excitations—the excitons—are fully quantised. SC in such systems was onlyrecently realized [5, 6, 7]. Fig. 1 reproduces one of these seminal contributions, by Reith-maier et al. [5], with QDs in micro-pillars. The most striking feature of strong coupling isthe splitting of the spectral shape when the system is at resonance: the line of the cavityand that of the emitter, both at the same frequency, do not superimpose but anticross witha splitting related to the coupling strength. The doublet observed in Fig. 1 at 21K, whenthe two modes are expected to be resonant, demonstrates that they are strongly-coupled,and this observation is the central result of Ref. [5] and of the related works [6, 7]. Inthe ample literature devoted to the description of this spectral shape [8, 9, 10, 11, 12], theseminal work of Ref. [8] paid little attention to the excitation (using a coherent state as aninitial condition) and neglected dissipation, while works such as [9, 11] addressed the caseof coherent pumping. Ref. [10] described spontaneous emission of an excited state, and thusthe initial state was fixed to be the excited state of the atom in an empty cavity. This wasrepeated in the theoretical work addressing the semiconductor case [12], where, however, amore complicated dynamics enters the picture. In practice, it is not possible to initialise2
IG. 1: Anticrossing of the the cavity (C) and exciton (X) photoluminescence lines as reported byReithmaier et al. in Nature (2004), demonstrating SC in their system. Energies are given in meV.The red lines are our superimposed fits with the best global fit parameters in the top left corner.Such a good agreement cannot be obtained neglecting the pump-induced decoherence. the cavity-emitter system in a semiconductor as it is in the atomic case, where atoms canbe singled out, manipulated and sent one by one into the cavity. Semiconductor QDs in acavity are typically excited far above resonance and electron-hole pairs relax incoherentlyto excite the QD in a continuous flow of excitations, establishing a steady state that washesout the coherent Rabi oscillations. Therefore, a Fock state as an initial state does not corre-spond to the experimental reality. Instead, the system is maintained in a mixed state withprobabilities p ( n ) to realize the n th excited state, that are imposed by the experimentalconfiguration.In this Letter, we provide the appropriate theoretical model to describe SC with 0Dsemiconductors. For the sake of illustration, we support our discussion with the resultsof Fig. 1, that our model reproduces with an excellent agreement. On the contrary, othermodels, with their particular initial condition [8, 10, 12] cannot account for these spectrabeyond the mere prediction of the line splitting. The shortcoming of downplaying the im-3ortance of the quantum state that is realized in the system owing to pumping, has as itsworst consequence a misunderstanding of the results, the most likely being the qualificationof weak-coupling (WC) for a system in SC that cannot be spectrally resolved because ofdecoherence-induced broadening of the lines. Being blind to the theory makes the trackfor SC in 0D semiconductors particularly difficult, involving a strong element of chance.Understanding the excitation scheme drastically reduces this element of hazard, as we shallsee below. Most importantly, our model unravels the physics behind the experimental re-sult, by spelling out which quantum state has been produced, by providing most-likelihoodestimators of the sample parameters, by distinguishing the Bose of Fermi-like character ofits excitations and by predicting results as the excitation is changed, most interestingly, asthe pump is increased and the system is brought into the nonlinear regime.We describe the system with a quantum dissipative master equation ∂ t ρ = L ρ for thedensity matrix ρ [13], with the Liouvillian L defined by its action on any operator O of thetensor product of the light and matter Hilbert spaces H a and H b : L O = i [ O, ω a a † a + ω b b † b + g ( a † b + ab † )] (1)+ X c = a,b (cid:16) γ c cOc † − c † cO ) + P c c † Oc − cc † O ) + h . c . (cid:17) , where g is the interaction strength between the cavity mode—with annihilation operator a atenergy ω a —and the material excitation—with operator b at energy ω b —and respective decayand pumping rates γ a,b and P a,b . An important experimental parameter is the detuningbetween the bare modes, ∆ = ω a − ω b , that can be tuned effectively by changing thetemperature. In our case where the interplay of pumping and dissipation establishes asteady state, the system is ergodic and the cavity emission spectrum follows from the Wiener-Khintchine theorem as S ( ω ) ∝ lim t →∞ ℜ R ∞ h a † ( t ) a ( t + τ ) i e iωτ dτ . According to the quantumregression theorem, a set of operators A { α } that satisfy Tr( A { α } L O ) = P { β } M { αβ } Tr( A { β } O )for all O ∈ H a ⊗ H b for some M { αβ } , yields the equations of motion for the two-timecorrelators as ∂ τ h O ( t ) A { α } ( t + τ ) i = P { β } M { αβ } h O ( t ) A { α } ( t + τ ) i . If b is a Bose operatorlike a (the photon operator), M is defined by M nmnm = − i ( nω a + mω b ) − n Γ a / − m Γ b / M nmn +1 ,m − = M mnm − ,n +1 = − igm and zero otherwise, where we defined as a shortcut the effective broadenings Γ a,b = γ a,b − P a,b . We also introduce the following notation: γ ± = ( γ a ± γ b ) / ± = (Γ a ± Γ b ) / . (2)4nstead of ad hoc initial conditions for the cavity population and off-diagonal coherence,such as those provided by the excited state of the QD [10, 12], we use the steady statevalues obtained by solving Tr( a † a L ρ ) = 0 and Tr( a † b L ρ ) = 0. For instance, the steady statecavity population n a = lim t →∞ h a † a i ( t ), reads: n a = g Γ + ( P a + P b ) + P a Γ b (Γ + ( ∆2 ) )4 g Γ + Γ a Γ b (Γ + ( ∆2 ) ) . (3)The equations are closed and the normalised photoluminescence (PL) spectrum S ( ω ) cantherefore be expressed analytically: S ( ω ) = ( L + L ) − ℜ ( C )( A − A ) − ℑ ( C )( L − L ) , (4)defined in terms of the Lorentzian L and dispersive A functions that characterise the emissionof the lower (1) and upper (2) eigenstates ( dressed states ): L ( ω ) = 12 π Γ + ± ℑ ( R )(Γ + ± ℑ ( R )) + ( ω − ( ω a − ∆2 ∓ ℜ ( R ))) , (5) A ( ω ) = 12 π ω − ( ω a − ∆2 ∓ ℜ ( R ))(Γ + ± ℑ ( R )) + ( ω − ( ω a − ∆2 ∓ ℜ ( R ))) , (6)where the complex coefficient C is defined as: C = 1 R " Γ − + i ∆2 + i g ( γ a P b − γ b P a )( i Γ + − ∆2 ) g Γ + ( P a + P b ) + P a Γ b (Γ + ( ∆2 ) )) (7)and the complex Rabi splitting as: R = r g − (cid:16) Γ − + i ∆2 (cid:17) . (8)The spectral shape given by formulas (4–8) is that of a coupled (SC and WC) system ina steady state maintained by incoherent pumping with rates P a,b , with the QD excitationobeying Bose statistics. This is a valid approximation when the QD is large (it becomes exactin the limit of a quantum well) or in any case when the number of excitations is vanishing.If the QD follows Fermi statistics, the analytical expression for the spectrum is lost. Wetherefore keep this case out of the present discussion. In the case of Ref. [5], both large QDsand low excitations were used, and we verified numerically that a Fermionic model is lessappropriate. The basic structure of eqn (4) is the same as in other descriptions of SC, suchas the decay of an initial state. Namely it consists of two peaks, each the sum of a Lorentzian5 IG. 2: ( a ) Regions of strong (blue) and weak (red) coupling at resonance in the space of pa-rameters ( γ a , P b ), with γ b ≈ . g and P a ≈ . γ a fitting the experiment of Reithmaier et al. [5],marked by a plain blue point. Region 1 exhibits line-splitting, while in the darker area 2, althoughstill in SC, the splitting cannot be resolved. The dashed vertical line marks the criterion for SC inabsence of pumping, giving rise to region 3 where SC is recovered (with line-splitting) thanks topumping and region 4 where it is lost because of it. In inset, the same but for P a = 0, in whichcase the line-splitting of [5] would not be resolved. ( b ) and ( c ): Spectra of emission with increasingexciton pumping P b marked by the hollow points in panel ( a ). For γ a = 2 g in ( b ), SC is retainedthroughout and made more visible. For the best fit parameter, γ a = 3 . g in ( c ), line-splitting is lostincreasing pumping, first because the line-splitting is not resolved (region 2 of ( a )) then becausethe system goes into WC (region 4 of ( a )). and of a dispersive part. The limit of vanishing pumping is formally equal to the particularcase of the decay of an initial state with independent initial populations, whose ratio is thesame as that of the pumping rates. The main result is to be found in the way incoherentpumping, even if it is small, affects this intrinsic structure of SC through eqns (7–8). This is6emonstrated by confronting the theory with the experiment. In Fig. 1, we have optimisedthe global nonlinear fit of the results from Ref. [5] with eqns (4–8). That is, the detuning( ω a and ω b ) and pumping rates ( P a and P b ) are the fitting parameter from one curve to theother, while g and γ a,b have been optimised but kept constant for all curves. We find anexcellent overall agreement, that instructs on many hidden details of the experiment.First, the model provides more reliable estimations of the fitting parameters than a directreading of the line-splitting at resonance or of the linewidths far from resonance: The best-fitting coupling constant is g = 61 µ eV. The value for γ a = 220 µ eV is consistent with theexperiment (the authors place it at 180 µ eV but from a Lorentzian fit of the 5K curve inthe assumption that the system is not strongly-coupled here, where our model shows thisto be a poor approximation), and the value for γ b , that is the most difficult to estimateexperimentally, is reasonable in the assumption of large QDs, as is the case of those thathave been used to benefit from their large coupling strength. Our point here is not to conductan accurate statistical analysis of this particular work but to show the excellent agreementthat is afforded by our model with one of the paradigmatic experiment in the field. Such agood global fit cannot be obtained without taking into account the effect of pumping, evenwhen it is small. More interestingly, it is necessary to include both the exciton pumping P b (expected from the experimental configuration) but also the cavity pumping P a . The latterrequirement comes from the fact that in such samples, there are numerous QDs weakly-coupled to the cavity in addition of the one that undergoes SC. Beyond this QD of interest,a whole population of “spectator” dots contributes an effective cavity pumping, which loomsup in the model as a nonzero P a . The fitting pumping rates vary slightly with detuning,as can be explained by the change in the effective coupling of both the strongly-coupleddot with the cavity (pumping tends to increase out of resonance) and the spectator QDsthat drift in energy with detuning. We find as best fit parameters at resonance P a ≈ . γ a and P b ≈ . γ b (the mean over all curves is ¯ P a ≈ . γ a and ¯ P b ≈ . γ b with rms deviationsof ≈ P a in an experiment with electronic pumping is supported bythe authors of [5] who observed a strong cavity emission with no QD at resonance. We shallsee in the following the considerable importance of this fact to explain the success of theirexperiment.From a fundamental point of view, our incoherent pumping model of SC not only fills in agap in extending the theory to the steady-state case where the excitation is not given (some-7imes arbitrarily) as an initial state, it also defines new criteria for SC. The conventionalone, from the condition that R be real at resonance, is, neglecting pumping: g > | γ − | . (9)With incoherent pumping, it becomes: g > | Γ − | . (10)The full extent of this new criterion can be appreciated in Fig. 2, where is displayed inshades of blue the region where R is real and in shades of red where it is pure imaginary.This corresponds respectively to oscillations or not in the time correlators and therefore tooscillations (SC) or damping (WC) of the fields. In white (delimited by the red frontier) is theregion where there is no steady state because of a too-high pumping. The dashed black linedelimits the conventional (without incoherent pumping) criterion, eqn (9). Regions 3 and 4show how pumping can make a qualitative difference. In region 3, given by | Γ − | < g < | γ − | ,SC is not expected according to eqn (9), but holds thanks to pumping, eqn (10) (in this case,thanks to cavity pumping P a ; in inset, P a is set to zero and this region has disappeared).In region 4, given by | γ − | < g < | Γ − | , where on the contrary SC is expected accordingto eqn (9), it is lost because of pumping. In the regions 1 and 5, the effect of the pumpis quantitative only, renormalising the broadening and splitting of the peaks, but is stillimportant to provide a numerical agreement with experimental data. Region 2 is that where,although in SC, only one peak is observed in the PL spectrum because of the broadeningof each peak being too important as compared to their splitting. The position where weestimate the result of Ref. [5] in this diagram validates that SC has indeed been observed inthis experiment. In inset, however, one sees that in the case where the cavity pumping P a isset to zero keeping all other parameters the same, the point falls in the dark region 2 where,although still in SC, the line-splitting cannot be resolved. Even if it is possible in principleto demonstrate SC through a finer analysis of the crossing of the lines, it is obviously lessappealing than a demonstration of their anticrossing. This is despite the fact that the caseof P a = 0 is equally, if not more, relevant as far as SC is concerned, as it correspondsto the case where only the QD is excited, whereas in the case of Fig. 1, it also relies oncavity photons. With the populations involved in the case of the best fit parameters thatwe propose— n a ≈ .
15 from eqn (3)—one can still read in Reithmaier et al. ’s experiment8 good vacuum
Rabi splitting, so the appearance of the line-splitting with P a is not due tothe photon-field intensity. Rather, the system is maintained in a quantum state that is morephoton-like in character, which is more prone to display line splitting in the cavity emissionwith the parameters of Fig. 1. One can indeed check that the PL spectrum without pumpingfor a strongly-coupled state prepared as a photon exhibits a line-splitting whereas the samesystem prepared as an exciton does not show it. This is the same principle that applieshere, with the nature of the state (photon-like or exciton-like) resolved self-consistently bypumping. In this sense, there is indeed an element of chance involved in the SC observation,as one sample can fall in or out of region 2 depending on whether or not the pumping schemeis forcing photon-like states.A natural experiment to build upon our results is to tune pumping. In our interpretation,it is straightforward experimentally to change P b , but it is not clear how P a would then beaffected, as it is due to the influence of the crowd of spectator QDs, not directly involved inSC. In Fig. 2( c ), we hold P a to its best fit parameters and vary P b in the best fit case γ a = 3 . g on panel ( c ), then for γ a = 2 g on panel ( b ), where the system is in SC for all possible valuesof P b . Spectra are displayed for the values of P b marked by the points in ( a ). Two verydifferent behaviours are observed for two systems varying slightly in one of their parameters.In one case ( b ), strong renormalisation of the linewidths and splitting results from Boseeffects in a system that retains SC throughout. In the other ( c ), line-splitting is lost andtransition towards WC then follows. At very high pumping, the model breaks down. Themost interesting possibility is that the QD becomes Fermi-like, in which case the annihilationoperator b should be a two-level system. Loss of line-splitting also results in this case witha smaller decrease in the linewidth than is observed here at moderate pumpings and asubsequent increase at higher pumpings, as the system reaches the self-quenching regime(when b is a Bose operator, linewidths tend to zero as the populations diverge when effectsassured to take place at high density, such as particles interaction, are neglected). A carefulstudy of pump-dependent PL can tell much about the underlying statistics of the excitonsand the precise location of one experiment in the SC diagram.In conclusion, we obtained a self-consistent analytical expression for the spectrum ofemission of a coupled light-matter system in a steady state maintained by a continuousincoherent pumping. Our formalism fully takes into account the effects of the incoherentpump, that, by randomising the arrival time of the excitation, averages out the Rabi oscil-9ations and through the interplay of pumping and decay rates, imposes a quantum steadystate that influences considerably the observed spectra. Close to its threshold, SC can belost or imposed by increasing pumping, and even when SC holds, the decoherence can hin-der line-splitting. Prospects for applications of SC with semiconductor heterostructures aregreat, provided that a quantitative understanding of the system can guide the advances nowthat the qualitative effects have been observed. Our model offers such a fine theoreticaldescription, while still staying at a fundamental level with a transparent physical interpre-tation. We showed that the experimental reports of SC can be fully explained taking intoaccount conjectures such as the influence of weakly-coupled dots, and other similar factorsthat can be better taken advantage of in the future.Authors are grateful to Dr. Sanvitto and Dr. Amo for helpful comments. Support bythe Spanish MEC under contracts QOIT Consolider CSD2006-0019, MAT2005-01388 andNAN2004-09109-C04-3 and by CAM under contract S-0505/ESP-0200 is acknowledged. [1] R. J. Thompson, G. Rempe, and H. J. Kimble, Phys. Rev. Lett. , 1132 (1992).[2] C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. , 3314 (1992).[3] J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun, J. M. J. Keeling, F. M.Marchetti, M. H. Szymanska, R. Andr´e, J. L. Staehli, et al., Nature , 409 (2006).[4] A. Amo, D. Sanvitto, D. Ballarini, F. P. Laussy, E. del Valle, M. D. Martin, A. Lemaitre,J. Bloch, D. N. Krizhanovskii, M. S. Skolnick, et al., arXiv:0711.1539 (2007).[5] J. P. Reithmaier, G. Sek, A. L¨offler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh,V. D. Kulakovskii, T. L. Reinecker, and A. Forchel, Nature , 197 (2004).[6] T. Yoshie, A. Scherer, J. Heindrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B.Shchekin, and D. G. Deppe, Nature , 200 (2004).[7] E. Peter, P. Senellart, D. Martrou, A. Lemaˆıtre, J. Hours, J. M. G´erard, and J. Bloch, Phys.Rev. Lett. , 067401 (2005).[8] J. J. Sanchez-Mondragon, N. B. Narozhny, and J. H. Eberly, Phys. Rev. Lett. , 550 (1983).[9] G. S. Agarwal and R. R. Puri, Phys. Rev. A , 1757 (1986).[10] H. J. Carmichael, R. J. Brecha, M. G. Raizen, H. J. Kimble, and P. R. Rice, Phys. Rev. A , 5516 (1989).
11] V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, Phys. Rev. B , 8774 (1994).[12] L. C. Andreani, G. Panzarini, and J.-M. G´erard, Phys. Rev. B , 13276 (1999).[13] H. J. Carmichael, Statistical methods in quantum optics 1 (Springer, 2002).(Springer, 2002).