Loop parametric scattering of cavity polaritons
LLoop parametric scattering of cavity polaritons
S. S. Gavrilov
Institute of Solid State Physics RAS, 142432 Chernogolovka, Russia andNational Research University Higher School of Economics, 101000 Moscow, Russia (Dated: January 2, 2021)Within the framework of the mean-field approximation, a coherently excited two-dimensional system ofweakly repulsive bosons is predicted to show a giant loop scattering when the rotational symmetry is reduced.The considered process combines (i) the parametric decay of the driven condensate into di ff erent k -states and(ii) their massive back scattering owing to spontaneous synchronization of several four-wave mixing channels.The hybridization of the direct and inverse scattering processes, which are di ff erent and thus do not balance eachother, makes the condensate oscillate under constant one-mode excitation. In particular, the amplitude of a po-lariton condensate excited by a resonant electromagnetic wave in a uniform polygonal GaAs-based microcavityis expected to oscillate in the sub-THz frequency domain. I. INTRODUCTION
Two-dimensional cavity polaritons are a result of exciton-photon coupling in layered heterostructures [1–3]. Beingcomposite bosons, they exhibit two kinds of coherent states,one of which is similar to Bose-Einstein condensates (BECs)formed with decreasing temperature [4], whereas the other ap-pears when a resonant electromagnetic wave excites polari-tons directly [5]. Both kinds of coherent states are character-ized by a mean-field amplitude ψ ( r , t ) obeying a generalizedwave equation i (cid:126) ∂ψ∂ t = (cid:2) E ( r , − i (cid:126) ∇ ) − i γ + V ψ ∗ ψ (cid:3) ψ + f ( r , t ) (1)(spin / polarization degrees of freedom are disregarded). If thepumping force f and decay rate γ are zero, Eq. (1) is reducedto the Gross-Pitaevskii equation for equilibrium BECs. Simi-lar to atomic gases, cavity polaritons combine repulsive inter-action ( V >
0) and positive mass in the vicinity of the ground-state level E g = E ( k =
0) [2, 3].Under plane-wave pumping (cid:2) e. g., f ( r , t ) = ¯ f e i ( k p r − E p t / (cid:126) ) (cid:3) ,the condensate has the same wave vector and frequency as thepump wave, provided that γ > E p is not too far fromresonance. The forced oscillation of ψ results in deep qual-itative changes of the Bogolyubov excitation spectrum ˜ E ( k )compared to equilibrium systems [6, 7]. In particular, the ex-citations around k p = E p is equalto E g + V | ψ k = ( ¯ f ) | for a given pump amplitude ¯ f [8]. Besides,as ¯ f and | ψ k p ( ¯ f ) | are increased, the sign of Im ˜ E may reverseat some k = k (cid:48) , which means the instability of the conden-sate against two-particle scattering ( k p , k p ) → ( k (cid:48) , k p − k (cid:48) ),often leading to a strong redistribution of polaritons in the k space. For instance, the break-up of the condensate excitedwith a nonzero k p near the inflection point of E ( | k | ) is knownto result in macroscopic occupation of two modes k ≈ k ≈ k p [9, 10]. Such processes, which attracted much in-terest in the early 2000’s, were firstly understood by analogywith optical parametric oscillators (OPOs) [11, 12] in whichan external pump beam splits into a pair of plane waves, con-ventionally referred to as “signal” and “idler”. This analogyis not perfect because the polariton system never comes to astate with only three nonempty wave modes [13–15]. Never-theless, the condensate induced by coherent pumping usually remains the most populated mode that governs all signals andidlers excited owing to the parametric scattering [16–19].Here, we report an unusual manifestation of the parametricscattering, which is expected to occur in wide (tens of µ m) andspatially uniform samples of a polygonal shape. We show thata reduced rotational symmetry leads to a population inversionsuch that a number of scattered modes get noticeably strongerthan the pumped mode ( k p = E p > E g ). Consequently,new two-particle interaction processes come into play whichare di ff erent from the direct break-up of a pumped condensateinto signals and idlers; furthermore, the pumped mode can it-self act as a parametric signal. By virtue of symmetry, severalprocesses of such kind synchronize, share the same target state k =
0, and thus yield a massive back scattering of polaritons.It is important that the direct and inverse scattering e ff ects donot cancel each other but constitute a unified loop interactionprocess. The common signal of the back scattering arises nearthe ground-state energy level E g rather than at the pump level E p . As a result, the pumped mode has two energy peaks andits amplitude | ψ k = | oscillates at frequency ∼ ( E p − E g ) / (cid:126) . Atthe same time, several scattered modes with k (cid:44) II. BLOWUP AND POPULATION INVERSION
Let us recall several basic facts about an interplay betweenthe parametric scattering and bistability in a homogeneous andisotropic polariton system. It is known that the one-mode de-pendence of | ψ k p | on | f | has an ‘S’-shaped form [Fig. 1(a)] a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Figure 1. (a) One-mode response of a driven condensate; dashed lineindicates unstable solutions; D = E p − E g = γ is the pump detuningfrom the resonance at k p = f (cid:63) = (cid:112) ( γ/ V ) (cid:2) ( D − γ ) + γ (cid:3) [20].(b) Dispersion law, position of the pumped mode, and scheme of theparametric scattering. The scattering wave number k (cid:63) ≈ . µ m − . as long as D ≡ E p − E ( k p ) > √ γ [7, 26, 27]. When | ψ k p | issmall, the repulsive interaction of polaritons ( V >
0) involvesa blueshift of their resonance energy E ( k p ) + V | ψ k p | towardsthe pump level E p , resulting in a superlinear increase of | ψ k p | as a function of | f | throughout the lower branch of solutions.On the upper branch, by contrast, the dependence of | ψ k p | on | f | is sublinear, because the e ff ective resonance energy has ex-ceeded E p and shifts still farther as | f | increases. The segmentwith a negative slope consists of unstable solutions.Notice that a sizable portion of the lower branch can alsobe unstable because of an intermode scattering [28] such asshown in Fig. 1(b) for the case of k p =
0. The imaginary partof the energy ˜ E ( k ) of elementary excitations changes its signfor some k = k (cid:63) at a certain threshold point | f | = f (cid:63) , result-ing in a spontaneous growth of | ψ k (cid:63) | . Specifically, this occurswhen blueshift V | ψ k p | exceeds γ , whereas at the end of thelower branch of one-mode solutions the blueshift would havereached a much greater value of D / γ/ D → γ one can estimate k (cid:63) directly froman unshifted dispersion law E ( k ) taking into account energyand momentum conservation [Fig. 1(b)]. If k p =
0, all k (cid:63) lie on a ring-shaped intersection of the renormalized energysurface ˜ E ( k ) and pump level E = E p [20].Since the scattering threshold f (cid:63) is less than the bistabil-ity turning point, the question arises of what exactly happensto the system when the pump amplitude slightly exceeds f (cid:63) .Proceeding from certain analogies in laser physics, one mightexpect a second-order phase transition with a continuous am-plification of scattered modes upon increasing f , which isindeed quite a common behavior of dissipative systems inwhich max k Im ˜ E ( k ) smoothly changes its sign in a criticalpoint [29]. However, the answer is di ff erent and counter-intuitive: the parametric break-up of the driven mode is ac-companied by a growth rather than decrease of its own am-plitude | ψ k p | [20]. This is possible despite the “conservative”kind of the | ψ | interaction, because the system is open. As aresult, the total | ψ | grows spontaneously even at constant | f | until the blueshift cancels the pump detuning. Such a processshows a hyperbolic time dependence with a latency period thattends to infinity for | f | → f (cid:63) + I s = (cid:80) k (cid:44) k p | ψ k | overcomes I p = | ψ k p | . When D /γ (cid:38)
10 and | f | is close to the threshold, I s can be several times greater than I p during a short period justbefore the jump to the upper branch. In a uniform system, thegrowth of I s results in the one-mode instability of the pumpedmode (so that its lower-energy state disappears [20]), thus,the population inversion cannot be permanent. In the specialcase of k p =
0, the parametric scattering turns into a purelytransient process that mediates the jump to the upper stabilitybranch and, in particular, reduces the corresponding thresholdat the cost of a potentially lengthy latency period.The key idea of this work is that the population inversioncan be stabilized owing to a reduced symmetry. Let us con-sider a square quantum well (mesa) with a side of L ≈ µ mand indefinitely high energy barrier at the boundary. The mainparameters are γ = .
01 meV, D = . k p = k = m is two times larger than the photon mass m ph = (cid:15) E / c , where E = . (cid:15) = . E ( k ), yet it does not play a significantrole at small D . The interaction constant V only determines f (cid:63) ∝ V − / and can be chosen arbitrarily.Figure 2(a) shows a steady-state dependence of an average V | ψ | on | f | . Compared to Fig. 1(a), it contains three ratherthan two stability branches as well as several isolated pointsrepresenting transient states. As expected, (i) the response Figure 2. (a) Steady-state response to excitation in a square mi-crocavity mesa. Parameters are indicated in the main text. Circlesrepresent independent solutions. Intensities | ψ ( r , t ) | are averaged inspace (over the mesa) and time (over 0.1 ns) after a 1 ns long periodof increasing | f | from zero to a given amplitude and extra 2 ns longperiod allotted for the establishment of a particular solution at a fixed | f | . (b, c) explicit spatial dependences for two solutions. is linear for | f | → V | ψ | / D (cid:38) | f | ≈ . f (cid:63) , which is not surprising since the presence of sharp po-tential walls involves the Rayleigh scattering into di ff erent k -states with ˜ E ( k ) = E p ; as a result, the polariton density (cid:104)| ψ | (cid:105) is higher than in a flat cavity at the same f unless the systemhas arrived at the upper branch where all scattering channelsare closed. It is remarkable, however, that the response be-comes nonlinear already at V (cid:104)| ψ | (cid:105) (cid:46) − D , i. e., far belowthe parametric scattering threshold, owing to the fact that thesystem is strongly inhomogeneous and has several short-rangeareas with large | ψ | . As seen in Fig. 2(b), the maximum V | ψ | equals 0 . D = γ , which is twice greater than the threshold,whereas the average V | ψ | is still less than 0 . D .In turn, the strong inhomogeneity is explained by a reducedrotational symmetry, owing to which the Rayleigh scatteringhas certain preferred directions matching the system geome-try. If a square mesa is oriented along the x and y axes, the pre-ferred k -states are ( ± k R ,
0) and (0 , ± k R ), where k R ≈ √ mD / (cid:126) .Indeed, the reflection of each of these waves from a potentialwall yields a twin wave with the inverse k , whereas all otherwaves would eventually scatter into many modes and lose co-herence. The filling of the geometrically preferred states re-veals itself in formation of a semi-periodic standing-wave pat-tern whose sharpness appears to be as high as in Fig. 2(b) evenat | f | →
0. Since k (cid:63) = k R for k p =
0, the already dominant k -states and the corresponding real-space lattice get amplifiedparametrically upon increasing | f | . Thus, the onset of theparametric scattering takes place in a number of small spotsrather than in the whole system at once.The short-range parametric instability has been studied ear-lier by focusing the pump into a 2 µ m spot on the sample [22].In that case one could no longer distinguish the pumped andscattered k -states and analyze their interaction in the spiritof Ref. [20]. Nonetheless, the bistability e ff ect as well asthreshold-like parametric scattering were observed. It wasfound that, in contrast to the case of one-mode excitation, theonset of the instability does not end up with a jump to the up-per branch in a finite range of pump powers. The blowup re-mains unfinished, because the growth of | ψ | at the spot centermakes polaritons more intensively spread out of the paramet-rically unstable area, which is equivalent to additional energylosses. Such a system permanently remains in a state withmany-mode instability; as a result, it exhibits strong quantumnoise and spontaneous pattern formation [22].The middle branch seen in Fig. 2(a) at 0 . (cid:46) | f | / f (cid:63) (cid:46) . k -statessuppress the field at the places of their negative interference,which makes the system inhomogeneous and prevents its con-tinuous transition to the uppermost branch where the field is,by contrast, spatially uniform [Fig. 2(c)]. Figure 3. Dynamics of the intensities of the pumped (a, b) andscattered (c, d) modes. The pump power | f | is linearly increased in1 ns from 0 to ∼ . f (cid:63) . (b) and (d) represent an established solutionon a more detailed time scale. III. MACROSCOPIC LOOP INTERACTION
Let us now turn to a detailed analysis of one characteristicsolution close to the beginning of the middle branch. Figure 3explicitly shows the intensities of the pumped ( I p ) and scat-tered ( I s ) modes depending on time for | f | / f (cid:63) ≈ .
22. Tech-nically speaking, I p is summed over k x , y = ± . µ m − inorder to take account of mode broadening in a confined systemand, accordingly, I s is summed over the rest of the k -space.The pump is turned on slowly (in 1 ns) to illustrate the onsetof the instability. Initially, we have I p ∝ | f | and I s ∝ I p (the Rayleigh scattering is linear). The increase of | f | beyondthe parametric threshold leads to a significant increase of I s ,which is typical of the first stage of blowup when much energyis transferred into the system of scattered modes whereas thepumped mode is increased only slightly [20]. However, aswe have argued previously, this process does not involve theentire system, so that the accumulated I s is still insu ffi cient fortriggering a global one-mode instability of k =
0. As a result,by t ≈ . I s / I p (cid:38) t ≈ . . ff ect isnaturally explained by the filling of a new coherent mode withthe same k = k p = ff erent frequency. The oscillationperiod T ≈
12 ps approximately matches the inverse pumpdetuning h / D ≈
10 ps, which means that the new mode islocated slightly above the ground polariton state.Notice that the emergence of a new coherent polariton statewith k = k p and E < E p is a very uncommon phenomenon.Being somewhat analogous to dynamical condensation [23], itis hardly expected under coherent driving, because such states Figure 4. (a) Scheme of the loop parametric scattering. (b–d) Thespectra of the main k -states at the stage of regular oscillations. cannot be excited via scattering from the pumped mode [8].If k p =
0, the direct scattering leads only to the states with˜ E ( k ) = E p , whereas all other two-particle processes are usu-ally weak and do not reach the threshold of the parametricamplification. However, the population inversion makes someof the indirect interaction channels particularly strong.The diagram in Fig. 4(a) represents the interaction processresulting in the filling of the k = E = E g . Specif-ically, each adjacent pair of the geometrically preferred states,e. g., k = ( ± k (cid:63) ,
0) and k = (0 , ± k (cid:63) ), scatters into (0 ,
0) and k + k = ( ± k (cid:63) , ± k (cid:63) ), which is consistent with energy conser-vation in the vicinity of the ground state where the polaritondispersion law is nearly parabolic. The overall scheme com-prises a number of two-particle interactions, however, each ofthe k (cid:63) states acts as a source in two scattering processes atonce, whereas the k = ff erent interaction channels.Figures 4(b)–(d) show the spectra of the main k -states en-gaged in the loop scattering, which are obtained by the Fouriertransform of ψ k ( t ) over 2 ns at the stage of regular oscilla-tions. As expected, the k = E = E p and E (cid:38) E g whose intensities are nearly the same,in agreement with the fact that I p ( t ) drops down to zero atthe oscillation minima. The geometrically preferred modes k = k (cid:63) [Fig. 4(c)] have, by contrast, only one strong peak at E = E p and thus appear to be particularly steady. The “idlers”with k = √ k (cid:63) [Fig. 4(d)] show the peak at E (cid:46) E p − E g whose total intensity (summed over 4 modes) nearly equalsthe intensity of the driven mode at E = E p . Notice that allspectral lines are almost unbroadened, which is indicative oftheir “parametric” nature and precise synchronization of therespective k -states.The explicit real- and momentum-space distributions of | ψ | are shown in Fig. 5 for two time instants which are nearly halfa period apart; the corresponding continuous evolution is dis-played by a separate video file [31]. In the k -space, both in-stances look very similar except the oscillating mode k = I p Figure 5. Typical real-space (a, b) and momentum-space (c, d)field patterns at the stage of regular oscillations for two time instantsseparated by 7 ps (approximately half of a period). The full evolutioncan be seen in the Supplemental Material [31]. has to be accompanied by a redistribution of | ψ | in the realspace. When I p is increased, the 2 π/ k (cid:63) -periodic spatial latticeexhibits a ∼ π/ I p also reveals itself in equally fastswitches between two sublattices. Notice as well that the max-imum of V | ψ ( r , t ) | , which is attained at the center of Fig. 5(b),is no greater than 0 . D and, therefore, the upper-branch statesare not yet feasible even at a single point.It is remarkable that rapid changes of I p at the regular stagehardly a ff ect I s , which might seem untypical for a system withstrong parametric instability of the pumped mode. The ob-served steadiness of I s results in a constant rate of back scatter-ing and, eventually, in a fully regular character of spatiotem-poral oscillations. In turn, it stems from the population in-version: since the pumped mode gives a comparatively smallaverage contribution to the blueshift, I s varies on the scale ofthe polariton lifetime τ = (cid:126) /γ that largely exceeds the oscil-lation period T (cid:38) h / D as long as γ (cid:28) D . The increase in (cid:104) I p (cid:105) / (cid:104) I s (cid:105) would lead to an increased oscillation amplitude of I s and, thus, increased volatility of the back scattering. As aresult, the system experiences a transition to dynamical chaos.For the series of solutions represented in Fig. 2(a), this occursin the last quarter of the middle branch where ratio (cid:104) I p (cid:105) / (cid:104) I s (cid:105) becomes nearly two times greater compared to the beginningof the same branch. Analysis of chaotic solutions is beyondthe scope of this paper; we only notice that chaos makes thetransition to the upper branch partially accidental, so that thetwo branches overlap within a finite range of pump powersaround 0 . f (cid:63) .The length of the middle branch depends on the system size.The necessary condition for the feasibility of the permanentpopulation inversion is a strong inhomogeneity achieved al-ready at | f | → k -states must be sig-nificantly stronger than all other modes with k (cid:44)
0. This con-dition is met when the system size L is comparable to the dis-tance traveled by the k (cid:63) polaritons on the scale of their life-time. When that is the case, all k -states become highly dis-sipative through multiple reflections from the potential walls,except for the “preferred” states which run perpendicular toone of them and form standing waves (generally speaking,this is true for all regular polygons of a reasonably small or-der). Thus, an increase of the Q -factor allows one to extendthe range of the system sizes suitable to achieve the popula-tion inversion and macroscopic loop scattering. The increaseof Q also conforms to the other assumption that γ (cid:28) D . IV. DISCUSSION
Up to date, the parametric scattering of cavity polaritonsis often thought to be just a macroscopic e ff ect of the two-particle interaction with well-defined “source” and “target” k -states. It was found, however, that for a greater D /γ theparametric scattering is an essentially collective process thatsuccessively involves many modes with di ff erent wave num-bers | k | even when the pump amplitude is arbitrarily close tothe threshold [8, 20, 21]. The most unstable many-mode statesexhibit the population inversion. As a rule, they are transientand only mediate the jump to the upper stability branch (for k p =
0) or to a well-developed OPO regime with strong signaland idler modes. The aim of this work was to find a way tomake the population inversion persistent under constant driv-ing conditions in a spatially extended polariton system. Wehave found it possible in the presence of a reduced rotationalsymmetry. The asymmetric (e. g., polygonal) systems showthe macroscopic loop interaction in which several k -states si-multaneously act as the “sources” and “targets” of the para-metric scattering.The considered phase transition is somewhat analogous tothe dynamical condensation [23]. Namely, some part of theenergy that has been transferred to the system of scatteredmodes during the first stage of blowup finds its way back to the k = k (cid:63) states constituting a dynam-ical reservoir that feeds the second condensate are themselvesmacroscopically coherent. When that states are su ffi ciently strong, they become parametrically unstable with respect tothe back scattering. At the same time, an induced synchro-nization of several back scattering channels is a more com-plex process that leads us far beyond the scope of two-particleinteractions. The greater the polygon order, the greater thenumber of comparatively strong modes that become synchro-nized. Calculations show that even a purely circular shape ofthe microcavity does not prevent the population inversion andmassive back scattering which in this case proceed throughthe spontaneous breakdown of the ring symmetry and resultin chaotic dynamics. The statistical and, possibly, quantumaspects of this problem have yet to be studied.The self-pulsation of a polariton fluid under coherent driv-ing is also a remarkable process that is usually prevented bythe dissipative nature of polaritons. Even when the groundstate is split into two spin or Josephson sublevels, both com-ponents of the condensate have the same “forced” energy E p and thus do not oscillate at | f | → | f | → ∞ . How-ever, the combination of the ground-state splitting and nonlin-earities does result in regular or chaotic oscillations in certainparticular cases [24, 25, 32, 33]. When V is nonzero and thespin splitting significantly exceeds γ , all one-mode states areforbidden in a finite range of | f | , resulting in chaos, dipolarnetworks, chimera states, and spontaneously formed vorticeseven for a purely uniform polariton system pumped by a planewave [24, 34, 35]. These phenomena are underlaid by a di ff er-ent kind of loop scattering leading to simultaneous occupationof many k -states [8, 36]; however, it presupposes linear cou-pling of opposite spins. By contrast, the system considered inthe current work is e ff ectively scalar, implying that all polari-tons have just the same spin, which corresponds to the case ofcircularly polarized excitation [37–39]. In contrast to anotherrecent study, in which the second mode comes into play ow-ing to size quantization in a µ m-sized micropillar and therebyalso causes oscillations [25], our system has a large spatialextent, so that the respective mode splitting is fairly negligi-ble. In other words, our system is one-component in spite ofa finite L and the second condensate appears at k = | f | or more complex shape of a microcavity result in newcollective phenomena that call for investigation. ACKNOWLEDGMENTS
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