Critical velocity in resonantly driven polariton superfluids
CCritical velocity in resonantly driven polaritonsuperfluids
Simon PigeonLaboratoire Kastler Brossel, Sorbonne Université, CNRS,ENS-PSL Research University, Collège de France, 4 placeJussieu, 75252 Paris, FranceAmandine AftalionÉcole des Hautes Études en Sciences Sociales, Centre d’Analyseet de Mathématique Sociales, UMR-8557, 54 boulevard Raspail,Paris, France.June 12, 2020
Abstract
We study the necessary condition under which a resonantly drivenexciton polariton superfluid flowing against an obstacle can generateturbulence. The value of the critical velocity is well estimated by thetransition from elliptic to hyperbolic of an operator following ideas de-veloped by Frisch, Pomeau, Rica [1] for a superfluid flow around anobstacle, though the nature of equations governing the polariton super-fluid is quite different. We find analytical estimates depending on thepump amplitude and on the pump energy detuning, quite consistentwith our numerical computations.
Keywords— vortex nucleation, superfluidity, exciton polariton a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Introduction
Since the discovery of Helium in 1937, superfluidity has attracted some of thegreatest minds of our time. After nearly one century of studies, this phenomenondoes not stop puzzling our understanding of matter. First observed in liquid Helium[2, 3], superfluidity has been studied in details more recently in atomic condensates[4, 5, 6, 7]. In analogy with the rotating bucket experiment in Helium [8], quantizedvortices have been observed in rotating one component [9, 10] and two componentcondensates [11]. Superfluid physics has now spread far beyond the field of atomicphysics and is used to describe the behavior of a large variety of system, fromnon-linear optical system [12, 13] to neutron star [14] or bird flocks [15].In this paper, we address the issue of the existence of a dissipationless flowinduced by the motion of a macroscopic object in a superfluid. The nucleationof vortices corresponds to the breakdown of this dissipationless phenomenon. Aclassical experiment on superfluid Helium consists in flowing Helium around anobstacle. If the velocity of the flow at infinity is sufficiently small, the flow isstationary and dissipationless, as opposed to what happens in a normal fluid. Onthe other hand, beyond a critical velocity, the flow becomes time dependent andvortices are emitted periodically from the north and south pole of the obstacle.Numerical simulations illustrating this behaviour have been performed by Frisch,Pomeau, Rica [1]: a pair of vortices is emitted and is flowing behind the obstacle,while the next pair is being formed on the boundary of the obstacle. In Ref. [1],the authors have also computed the critical velocity for the nucleation of vortices.Other related works, that we will describe below, include [16, 17, 18, 19, 20].The absence of dissipation at low velocity can be explained by the existence of astationary solution to some two-dimensional nonlinear Schrödinger equation. Thesuperfluid velocity is given at any point in the flow by the gradient of the phase ofthe wave function: if the wave function does not vanish, then the velocity is welldefined everywhere. The vortices are points where the wave function vanishes andaround which the circulation of the velocity is quantized.Following these theoretical works, an experiment was conducted at MIT byRaman et al. [6], (see also [5, 7]) in Bose Einstein condensates, to study there theexistence of a dissipationless flow. Instead of a macroscopic object, the obstacleis a blue detuned laser beam. The condensate is fixed and the obstacle is stirredin the condensate. Similar features to Helium are observed, namely the evidenceof a critical velocity for the onset of dissipation. The energy release is measuredas a function of the velocity of the stirrer: if the velocity is small, the flow isalmost dissipationless and the drag on the obstacle is very small, while above acritical value of the velocity, the flow becomes dissipative. Numerical simulationshave been performed by [21] for the 3D problem corresponding to the experiment, elating the increase in energy dissipation to vortex nucleation.Among the systems where superfluidity is observed, exciton-polariton fluidshave attracted significant attention as to their ease of control and manipulationthanks to their dual light-matter nature. Exciton-polariton fluids are compositebosons resulting from the strong coupling between the excitonic resonance of asemiconductor quantum well and the microcavity electromagnetic field [22]. Inparticular, the experimental study of a polariton field flowing past an obstacle,and the observation of quantized vortices in the wake of the obstacle has been thesubject of many papers [23, 24, 25]. This superfluid and turbulent behaviours havebeen the topic of quite a few theoretical papers [26, 27, 28, 29] and this is at the coreof our study. Mixing advantageously the low effective mass of cavity light with thestrong inter-particle interaction between matter excitation such as semiconductorexcitons, these systems have shown recently superfluid behavior even until roomtemperature [30].In many circumstances such as the one considered here, it is not necessary towork with the pair of equations of motions for the photonic and excitonic fields andone can restrict to a single classical field describing the lower polariton field. Thissimplified description is generally legitimate provided the Rabi frequency is muchlarger than all other energy scales of the problem, namely the kinetic and inter-action energies, the pump detuning, and the loss rates γ [22]. Contrary to atomicsuperfluids, polariton superfluids are driven-dissipative fluids. To compensate theirshort lifetime (of the order of tens of picoseconds), the system must be continuouslypumped. Here, we consider continuous and quasi-resonant pumping of frequency ω p and amplitude F . One of the interests of this technique to create a polaritonfluid, is that it allows the creation of a flowing fluid. If the laser beam is slightlytilted with respect to the cavity plan, the polariton fluid generated within the planof the cavity will carry a finite momentum k p . Consequently, in a polariton fluid,contrary to what happens in a cold-atomic ensemble, the obstacle is fixed and thefluid is moving at a speed far from the obstacle which is v ∞ = (cid:126) | k p | /m . This yieldsthe following generalized Gross-Pitaevskii equation for the polariton field ψ in thepump rotating frame: i (cid:126) ∂ t ψ ( x , t ) = (cid:18) − (cid:126) m ∇ − ∆ − i γ V ( x ) + g | ψ ( x , t ) | (cid:19) ψ ( x , t ) + F e i k p . x (1)where m and g are respectively the polariton effective mass and the interactionstrength. Contrary to atomic superfluids, no confinement potential is needed. Theother parameters are directly linked to the driven-dissipative nature of polaritons: γ is the decay rate of polaritons, ∆ the energy detuning between the driving field (cid:126) ω p and the polariton eigenenergy, F the coherent driving field and k p the drivingfield momentum. The potential V ( x ) is an added repulsive potential modelling the bstacle, which is therefore equal to 0 outside the obstacle.When varying the driving field F , the polariton density undergoes an S-likedependency with a bistable regime, presenting a low-density regime and a high-density regime [23, 22]. In the low-density regime the interaction term in Equation(1) can be neglected to lead to a standard linear system. On the contrary, inthe high-density regime, the interaction term cannot be neglected and leads tothe appearance of a superfluid behavior. In the first experiments, it was thoughtthat the driving inhibits the formation of vortices. Therefore, in order to observethe nucleation of vortices past an obstacle, the fluid was released from the drivingpresence either temporally [26, 25] or spatially [24]. A detailed numerical studiedrecently revealed a more subtle situation [28]: indeed, the driving field tends toinhibit the formation of turbulence, however, it can be reduced enough to release itsconstraints and allows the formation of vortices on the edge of the obstacle. Fine-tuning of the driving amplitude F eventually allows passing from a dissipationlesssuperfluid to a turbulent one [28] without having to remove it. This has beenachieved experimentally very recently [31].Whereas in atomic superfluids, the Mach number M = v ∞ /c s (where c s = (cid:112) g | ψ | /m is the fluid speed of sound), is the only parameter controlling the tran-sition from dissipating energy via vortex emission to dissipationless, in the presentdriven-dissipative scenario, the pump field amplitude plays a crucial role. In thiswork, we will focus on this phenomenon and disentangle the role played by thepump amplitude F and pump detuning ∆ in this transition from turbulent to anon-turbulent superfluid.We will perform numerical simulations of Equation (1) such as in Figure 1which illustrates the vortex nucleation behaviour and will be detailed below. Butwe will also perform an analytical approach of the transition behaviour. As mentioned above, this problem of onset of dissipation in a superfluid was firstaddressed by Frisch, Pomeau and Rica [1]. They have studied the case where theobstacle is a small disk in the frame where the obstacle is fixed. The nonlinearSchrödinger equation studied in [1] can be rewritten using the hydrodynamic for-mulation, where ρ is the density and which allows to identify ∇ φ with a velocity: ∂ρ∂t + ∇ · ( ρ ∇ φ ) = 0 (2) ∂φ∂t = ∆ √ ρ √ ρ − ρ + c s + 12 v ∞ − |∇ φ | , (3) k p = 0 . µ m − , m = 1 meV.ps .µ m − , and in units of (cid:126) = 0 . meV.ps, (cid:126) F = 0 . meV, (cid:126) ∆ = 0 . meV, (cid:126) γ = 0 . meV, (cid:126) g = 0 . meV; simulation parameters: × grid, dt = 0 . ps. 5 here the mass m is set to unity, c s is the sound speed, v ∞ the flow velocity atinfinity. They look for stationary solutions and assume that the quantum pressureterm ∆ ρ/ √ ρ is negligible, which is a kind of long wave approximation, and leadsto the following problem ∇ · ( ρ ∇ φ ) = 0 , ρ = c s + 12 v ∞ − |∇ φ | , (4)with boundary conditions ∂φ/∂n = 0 on the obstacle, ρ → c s , and ∇ φ → v ∞ atinfinity. Note that the second equation in (4) is a Bernouilli law for this problem.The system (4) has the same formulation as that of a stationary irrotational flow ofa compressible fluid about an obstacle. Mathematically, the existence of solutionsfor such a related subsonic problem and the non existence for large velocity atinfinity is proved by [32, 33] using a fixed point theorem. In one dimension, in Ref.[16], the saddle node bifurcation is analyzed and the critical velocity is related tothe spatial variation of the potential representing the obstacle.Let v = ∇ φ . From [1], Equation (4) goes from elliptic to hyperbolic when ∂ v ( vρ ( v )) = 0 , with ρ ( v ) = c s + 12 v ∞ − v . (5)This yields that the transition first takes place when c s + 12 v ∞ − v = 0 . (6)As explained in Ref. [1], this first happens at the point of maximum of v , whichis at the north and south pole of the obstacle. It is therefore crucial to estimatethis maximal velocity. Several papers deal with this question [17, 34, 19]. Themaximal velocity occurs at the equator of an object and is / v ∞ for a sphere and v ∞ for a cylinder. Nevertheless for a compressible fluid, this equatorial velocityis slightly larger due to pressure effects. In Ref. [19], an asymptotic expansion interms of the Mach number is made. Further studies about this critical velocityin particular including scaling laws and the bifurcation diagram can be found inRef. [17]. What happens beyond this critical velocity and the transition to anEuler-Tricomi equation has been studied in Ref. [20]. In Ref. [1], it is assumedthat the maximal velocity is v ∞ which yields from Equation (6) v ∞ = (cid:114) c s for the onset of dissipation.Similarly to Ref. [1], in our case of a polariton superfluid, Equation (1) can berewritten in terms of the phase and amplitude using the Madelung transform so hat ψ = √ ne iθ which yields, after setting m to 1, (cid:126) ∂ t n = − (cid:126) ∇ . ( n ∇ θ ) − n (cid:18) γ − F √ n sin( k p . x − θ ) (cid:19) (7) (cid:126) ∂ t θ = (cid:126) (cid:18) √ n ∇ √ n − ( ∇ θ ) (cid:19) + ∆ − V − gn − F √ n cos( k p . x − θ ) (8)We assume the flow to be directed along the x axis so that k p = ( k p , with k p ≥ and v = (cid:126) ∇ θ is the flow speed. With respect to the Gross-Pitaevskii equationdescribing atomic Bose-Einstein condensates (2)-(3), Equations (7)-(8) include ad-ditional terms to account for the driven-dissipative nature of the polariton gas,namely a loss rate proportional to γ and the coherent pumping proportional to F .To determine whether the fluid can remain superfluid, we will follow the sameapproach as in Ref. [1]. It consists in determining when the continuity equation,Equation (7), in its stationary version, goes from elliptic to hyperbolic. This onlydepends on the structure of the operator ∇ . ( n ∇ θ ) and not the right hand sideterm of the same equation. Therefore, we need to determine how the density n is connected to the flow velocity v = (cid:126) ∇ θ . To do this, we use Equation (8) inits stationary version, away from the obstacle where V = 0 , and we neglect thequantum pressure term. This yields v + F √ n cos( k p x − θ ) + gn − ∆ = 0 . (9)We point out that this equation is a polynomial of degree 3 in √ n . We both want to simulate Equation (1) and see numerically the change of behaviouron the one hand, and use the hydrodynamic formulation and the change of theoperator from elliptic to hyperbolic to find a critical pump amplitude consistentwith the numerics.
In the simulations, we observe two types of behaviours: for fixed pump detuning ∆ , and small pump amplitude F , vortices are emitted periodically from the northand south poles as in [1] as illustrated in Figure 1. On the contrary, for large pumpamplitude F , the solution is superfluid and there is a stationary flow as illustratedin Figure 2b. So for each ∆ , there is a critical value of the pump amplitude F c wherethe transition goes from a solution emitting vortices to a superfluid solution. Close n/n ∞ for two different pump powers: in a. (cid:126) F =0 . meV and in b. (cid:126) F = 0 . meV. The red circle indicate the positionof the potential barrier. The color scale is identical for both images. Physicalparameters: k p = 0 . µ m − , m = 1 meV.ps .µ m − , (cid:126) ∆ = 0 . meV, (cid:126) γ = 0 . meV, (cid:126) g = 0 . meV; simulation parameters: × grid, dt = 0 . ps. to the transition, the period of emission gets very large. The difference with Heliumor Bose Einstein condensates is that instead of prescribing a velocity at infinity, itis the pump amplitude which is prescribed and characterizes the behaviour.We simulate Equation (1) using a split-step method on a grid made of × pixels corresponding to × µ m . We use periodic boundary conditions andtailor the pump profile at the edge of the grid to avoid undesired propagation ofdensity modulation. The time step is . ps and we start with an empty system.Then the pump amplitude is slowly raised up to about ten times the pump ampli-tude of interest. Finally, the amplitude is decreased to reach the elected one, whichis then kept constant. This initialization procedure lasting about 200 ps allows usto prepare the system in the upper part of the bistable regime, corresponding tothe point where vortex nucleation was numerically observed [28].In Figure 2 we represent the polariton field density around an obstacle (indi-cated by a red circle) at a given time frame. The only difference between Figures2.a and b is the pump amplitude F which varies by less than 10%. As visible, forlow enough pump amplitude a pair of vortex has formed at the edge of the obstacle(2.a) whereas for a slightly higher pump amplitude no vortex forms (2.b). Bothimages correspond to the same time frame. o evaluate numerically the critical pump amplitude F c (and its dependencieswith respect to ∆ ), for each ∆ , we perform a large set of simulation at fixed F .For each one, while running we evaluate the formation of vortices at the edge ofthe defect and stop the simulation if one is found (low-density core distinguishablefrom the obstacle). We re-run the simulation slightly increasing F . If no vortexforms on the edge of the defect after a time long enough (0.5 ns), the simulation isre-run with a slightly lower F . The program stops after 8 iterations given accurateenough estimation of the critical pump amplitude F c . Notice that the closer we areto F c from below, the slower the nucleation of vortices. Consequently, a significantincrease of precision on the determination of F c will have required an importantcomputational effort not necessary given the result obtained.It is noticeable that despite the important change of the detuning ∆ , the crit-ical driving amplitude F c only slightly varies around . / (cid:126) . Since, it is almostconstant, this provides a new relation between the parameters of the problem. In-deed, from Equation (9), at infinity, v ∞ = (cid:126) k p /m the asymptotic fluid speed farfrom the obstacle, which is fixed by the driving field momentum k p , we find F √ n ∞ lim ∞ cos( k p x − θ ) = v ∞ − ∆ + gn ∞ (10)and from Equation (7), γ F √ n ∞ lim ∞ sin( k p x − θ ) . Adding the two, we obtain F √ n ∞ = (cid:114) γ v ∞ − ∆ + gn ∞ ) . (11)Therefore, when F = F c , it becomes a constant, and we check numerically thatEquation (11) holds. Now we want to estimate F c analytically. As in [1], the operator in Equation (7)changes type when ∂ v ( n ( v ) v ) = 0 (12)where v = | (cid:126) ∇ θ | . In our case, from Equation (9), the equation providing n ( v ) isnot simply quadratic in v but is given as a polynomial of order 3 in (cid:112) n ( v ) : gn ( v ) + v − ∆ + F (cid:112) n ( v ) cos( k p x − θ ) = 0 . (13) nstead of solving it, we prefer to differentiate it with respect to v to find (cid:32) g − F (cid:112) n ( v ) n ( v ) cos( k p x − θ ) (cid:33) n (cid:48) ( v ) + v = 0 . (14)We identify n (cid:48) ( v ) from Equation (14) and plug this into Equation (12) to find thatat the point of maximal velocity, the cosine is equal to − and v max = gn + F √ n . (15)We point out that we find something consistent with [25] who characterizes thetransition when the Mach number v max / √ gn is one. In our case, v max / √ gn = (cid:112) F/ gn √ n which is close to 1, and equal to 1 when the driving is zero as in[25].We replace F/ √ n from Equation (13) in Equation (15). Therefore, we findthat at the critical case, the cosine is equal to − , and gn ( v max ) − ∆3 − v max (16)and changes sign at the transition. This is quite consistent with our numericalcomputations because indeed the critical value of F c corresponds to the case wherethis function changes sign. For instance, for (cid:126) ∆ = 0 . , the function is negative for (cid:126) F = 0 . and positive for (cid:126) F = 0 . coinciding with the change of behaviorreported in Figure 2. In (15), we can also decide to replace v max from Equation(13) and we find at the critical case gn max F c √ n max . (17)This is verified by our numerical simulations leading to an error of order − to − . In fact, in this equation, the term in F c / √ n is of lower order leading to anapproximation gn max ∼ (18)which is quite consistent with our numerics.We point out that in our case the Mach number is large (from 0.9 to 1.6),therefore the approximation v max ∼ v ∞ (2 + 7 / ∗ M ) of Ref. [19] is not correct.Nevertheless, we always have that v max /v ∞ is not too far from 2.5 µ m.ps − . Wehave plotted in Figure 3 the Mach number at infinity vs ∆ , since it is very difficultto compute it precisely close to the obstacle. M vs ∆ . We point out that M = v ∞ / √ gn ∞ , v ∞ isa constant in the simulations and the density is proportional to ∆ . We analyze the driven-dissipative nature of a polariton superfluid, in particular theeffect of the pump amplitude and pump detuning. As they are varied, the solutiongoes from a superfluid solution to a solution emitting vortices. We can characterizeanalytically the change of behaviour and onset of turbulence by Equation (15),(16) or (17). The relation between the parameters that we derive is consistentwith the numerical simulations. In particular, we find that the maximal density isproportional to the pump detuning at the critical transition.
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