Critical velocity in resonantly driven polariton superfluids

Highlights

• Superfluid breakdown in resonantly pumped polariton fluids.

• Condition for elliptic to hyperbolic transition leading to vortex nuleation.

• Impact of pump detuning and pump amplitude on superfluidity of polaritons.

Abstract

We study the necessary condition under which a resonantly driven exciton–polariton superfluid flowing against an obstacle can generate turbulence. The value of the critical velocity is well estimated by the transition from elliptic to hyperbolic of an operator following ideas developed by Frisch et al. (1992) for a superfluid flow around an obstacle, though the nature of equations governing the polariton superfluid is quite different. We find analytical estimates depending on the pump amplitude and on the pump energy detuning, quite consistent with our numerical computations.

Introduction

Since the discovery of Helium in 1937, superfluidity has attracted some of the greatest minds of our time. After nearly one century of studies, this phenomenon does not stop puzzling our understanding of matter. First observed in liquid Helium [1], [2], superfluidity has been studied in detail more recently in atomic condensates [3], [4], [5], [6]. In analogy with the rotating bucket experiment in Helium [7], quantized vortices have been observed in rotating one component [8], [9] and two component condensates [10]. Superfluid physics has now spread far beyond the field of atomic physics and is used to describe the behaviour of a large variety of system, from non-linear optical system [11], [12] to neutron star [13] or bird flocks [14].

In this paper, we address the issue of the existence of a dissipationless flow induced by the motion of a macroscopic object in a superfluid. The nucleation of vortices corresponds to the breakdown of this dissipationless phenomenon. A classical experiment on superfluid Helium consists in flowing Helium around an obstacle. If the velocity of the flow at infinity is sufficiently small, the flow is stationary and dissipationless, as opposed to what happens in a normal fluid. On the other hand, beyond a critical velocity, the flow becomes time dependent and vortices are emitted periodically from the north and south poles of the obstacle. Numerical simulations illustrating this behaviour have been performed by Frisch, Pomeau, Rica [15]: 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. [15], 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 a stationary solution to some two-dimensional nonlinear Schrödinger equation. The superfluid velocity is given at any point in the flow by the gradient of the phase of the wave function: if the wave function does not vanish, then the velocity is well defined everywhere. The vortices are points where the wave function vanishes and around which the circulation of the velocity is quantized.

Following these theoretical works, an experiment was conducted at MIT by Raman et al. [5], (see also [4], [6]) in Bose–Einstein condensates, to study there the existence of a dissipationless flow. Instead of a macroscopic object, the obstacle is a blue detuned laser beam. The condensate is fixed and the obstacle is stirred in the condensate. Similar features to Helium are observed, namely the evidence of a critical velocity for the onset of dissipation. The energy release is measured as a function of the velocity of the stirrer: if the velocity is small, the flow is almost dissipationless and the drag on the obstacle is very small, while above a critical value of the velocity, the flow becomes dissipative. Numerical simulations have been performed by [21] for the 3D problem corresponding to the experiment, relating the increase in energy dissipation to vortex nucleation.

Among the systems where superfluidity is observed, exciton–polariton fluids have attracted significant attention as to their ease of control and manipulation thanks to their dual light-matter nature. Exciton–polariton fluids are composite bosons resulting from the strong coupling between the excitonic resonance of a semiconductor quantum well and the microcavity electromagnetic field [22]. In particular, 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 the subject of many papers [23], [24], [25]. This superfluid and turbulent behaviours have been the topic of quite a few theoretical papers [26], [27], [28], [29] and this is at the core of our study. Mixing advantageously the low effective mass of cavity light with the strong inter-particle interaction between matter excitation such as semiconductor excitons, these systems have shown recently superfluid behaviour even until room temperature [30].

In many circumstances such as the one considered here, it is not necessary to work with the pair of equations of motions for the photonic and excitonic fields and one can restrict to a single classical field describing the lower polariton field. This simplified description is generally legitimate provided the Rabi frequency is much larger than all other energy scales of the problem, namely the kinetic and interaction energies, the pump detuning, and the loss rates $\gamma$ [22]. Contrary to atomic superfluids, polariton superfluids are driven-dissipative fluids. To compensate their short lifetime (of the order of tens of picoseconds), the system must be continuously pumped. Here, we consider continuous and quasi-resonant pumping of frequency ${\omega }_p$ and amplitude $F$. One of the interests of this technique to create a polariton fluid, is that it allows the creation of a flowing fluid. If the laser beam is slightly tilted with respect to the cavity plan, the polariton fluid generated within the plan of the cavity will carry a finite momentum ${\mathbf{k}}_p$. Consequently, in a polariton fluid, contrary to what happens in a cold-atomic ensemble, the obstacle is fixed and the fluid is moving at a speed far from the obstacle which is $v_{\infty }=ħ\left|{\mathbf{k}}_p\right|∕m$. This yields the following generalized Gross–Pitaevskii equation (GPE) or Nonlinear Schrödinger equation (NLS) for the polariton field $\psi$ in the pump rotating frame:

$$iħ{\partial }_t\psi (\mathbf{x},t)=\left(-\frac{{ħ}^2}{2m}{\nabla }^2-\Delta -i\frac{\gamma }{2}+V\left(\mathbf{x}\right)+g{\left|\psi (\mathbf{x},t)\right|}^2\right)\times \psi (\mathbf{x},t)+F{e}^{i{\mathbf{k}}_p.\mathbf{x}}$$

where $m$ is the polariton effective mass set to 1 and $g$ is the interaction strength. Contrary to atomic superfluids, no confinement potential is needed. The other parameters are directly linked to the driven-dissipative nature of polaritons: $\gamma$ is the decay rate of polaritons, $\Delta$ the energy detuning between the driving field $ħ{\omega }_p$ and the polariton eigenenergy, $F$ the coherent driving field and ${\mathbf{k}}_p$ the driving field momentum. The potential $V\left(\mathbf{x}\right)$ is an added repulsive potential modelling the obstacle, which is therefore equal to 0 outside the obstacle.

When varying the driving field $F$, the polariton density undergoes an S-like dependency with a bistable regime, presenting a low-density regime and a high-density regime [22], [23]. In the low-density regime the interaction term in Eq. (1) can be neglected to lead to a standard linear system. On the contrary, in the high-density regime, the interaction term cannot be neglected and leads to the appearance of a superfluid behaviour. In the first experiments, it was thought that the driving inhibits the formation of vortices. Therefore, in order to observe the nucleation of vortices past an obstacle, the fluid was released from the driving presence either temporally [25], [26] or spatially [24]. A detailed numerical studied recently revealed a more subtle situation [28]: indeed, the driving field tends to inhibit the formation of turbulence, however, it can be reduced enough to release its constraints and allows the formation of vortices on the edge of the obstacle. Fine-tuning of the driving amplitude $F$ eventually allows passing from a dissipationless superfluid to a turbulent one [28] without having to remove it. This has been achieved experimentally very recently [31].

Whereas in atomic superfluids, the Mach number $M=v_{\infty }∕c_s$ (where $c_s=\sqrt{g{\left|\psi \right|}^2∕m}$ is the fluid speed of sound), is the only parameter controlling the transition from dissipating energy via vortex emission to dissipationless, in the present driven-dissipative scenario, the pump field amplitude plays a crucial role. In this work, we will focus on this phenomenon and disentangle the role played by the pump amplitude $F$ and pump detuning $\Delta$ in this transition from turbulent to a non-turbulent superfluid.

We will perform numerical simulations of Eq. (1) such as in Fig. 1 which illustrates the vortex nucleation behaviour and will be detailed below. But we will also perform an analytical approach of the transition behaviour.