Galilean boosts and superfluidity of resonantly driven polariton fluids in the presence of an incoherent reservoir

In the last decade, fluids of exciton-polaritons have emerged as a most powerful platform where to investigate quantum hydrodynamics questions related to superfluidity Carusotto and Ciuti (2013). A key strength of these systems is that the speed of flow of the fluid can be directly controlled by varying the incidence angle of the pump driving laser, while defects can be engineered both optically and at the sample fabrication stage. These features have allowed a direct experimental implementation of the Landau criterion for superfluidity in terms of the density pattern generated by polariton flow past a static defect  Carusotto and Ciuti (2004). In particular, experimental observation of superfluid behaviours was reported Amo et al. (2009), as well as the hydrodynamic nucleation of vortices Nardin et al. (2011) and solitons Amo et al. (2011).

In addition to their intrinsic non-equilibrium character Carusotto and Ciuti (2013), a series of recent experiments has unveiled another novel feature of polariton fluids, namely the presence of a reservoir of incoherent excitations interacting with the polariton fluid and modifying its dynamical properties. While such a reservoir is naturally present under generic incoherent pumping schemes Wouters and Carusotto (2007); Bobrovska et al. (2014, 2017); Baboux et al. (2018), its presence was not expected a priori in coherent pump schemes and has been experimentally established in a series of recent works Sarkar et al. (2010); Walker et al. (2017); Stepanov et al. (2019).

More specifically, our contribution Stepanov et al. (2019) inferred the presence of the reservoir from an important modification of the dispersion of collective excitations in the fluid, in particular a significantly reduced sound speed. In order to get a deeper understanding of the role of the reservoir and, at the same time, reconcile our observations with previous works on polariton superflows past a static defect, a general theoretical study of the effect of Galilean transformations on polariton superfluidity is needed, so to identify the consequences of the privileged frame of reference set by the underlying semiconductor cavity structure. These conceptual issues are the subject of the present article.

The structure of the paper is the following. In Sec. II we review the standard theory of Galilean boosts in quantum mechanics, which formally represent a symmetry for the undriven polariton field within the parabolic approximation for the polariton dispersion; as a consequence, these transformations preserve the form of the generalized Gross-Pitaevskii equation (GPE) for driven-dissipative polariton fluids in the absence of a reservoir, provided one keeps into account the proper covariance law for the coherent pump term. The Galilean transformation is meant as a useful mathematical mapping to link the GPE dynamics of fluids driven with different wavevectors and entails that, for a given value of the polariton density and thus of the speed of sound, the perturbation induced by a defect only depends on its relative velocity with respect to the fluid. For instance, our mapping provides a useful link between the relevant situations of polaritons injected against a static defect and of a defect moving in a fluid at rest. However, since optics in materials is generally not Lorentz invariant and Fresnel drag effects Jackson (1975); Landau et al. (1984) take place, such a mapping does not describe a physical change of frame of reference. As next step in Sec. III we provide a reformulation of the Landau criterion for superfluidity in non-equilibrium fluids based on complex-valued wavevectors. In Sec. IV we move to the core of our work and we discuss the effect of the reservoir of incoherent excitations on the Bogoliubov dispersion of the collective excitations on top of a spatially homogeneous, coherently pumped polariton fluid. The fact that the incoherent excitations forming the reservoir, e.g. dark excitons, are physically bound to the semiconductor cavity structure, breaks the formal Galilean invariance of the undriven polariton field and has deep implications for the superfluidity properties of the polariton fluid, as highlighted in Sec. V. The critical relative velocity turns out to be different depending on whether a moving polariton fluid is hitting a defect at rest, or a moving defect is flowing through a polariton fluid at rest: indeed, in the former case, the critical velocity is related to the total blue-shift of polariton modes, while in the latter case, it receives the sole contribution of the coherent polariton component. Conclusions are finally drawn in Sec.VI.

The goal of this section is to illustrate a mathematical mapping that relates the GPE dynamics of polariton fluids driven with different pump wavevectors, in the absence of a incoherent reservoir. As a corollary, if the relative velocities and the polariton densities are the same, the two cases of polaritons injected against a static defect and of a defect moving in a fluid at rest turn out to be the same situation depicted in two different systems of coordinate. Because of its formal analogy, we call this transformation a Galilean boost, even though it does not correspond to a physical change of reference frame.

We start by considering the standard driven–dissipative Gross–Pitaevskii equation for a resonantly pumped polariton fluid (Carusotto and Ciuti, 2004, 2013),

$$i\partial_{t}\psi_{lab}(\mathbf{x},t)=\\ =\left(\omega_{0}-\frac{\hbar}{2m}\nabla^{2}+g|\psi_{lab}|^{2}+V_{ext}(\mathbf{x})-i\frac{\gamma}{2}\right)\psi_{lab}+\\ +F_{lab}(\mathbf{x},t)\,.$$

(1)

In the conservative part of the evolution, $\omega_{0}$ is the bottom of the lower polariton band and $m$ is the polariton mass in the parabolic band approximation, $V_{ext}(\mathbf{x})$ the static potential acting on the polaritons, $g$ quantifies the strength of polariton–polariton interactions. Concerning the driven-dissipative terms, $\gamma$ is the loss rate and the driving term $F_{lab}(\mathbf{x},t)$ is proportional to the spatio-temporal profile of the coherent laser amplitude. In particular, the cavity material determines the mass and interactions of the polaritons, but no reservoir of incoherent excitations is assumed to be present at this stage.

Now we develop the aforementioned formal Galilean boost and with some abuse of notation employ the terminology commonly adopted for physical Galilean transformations. For instance, when within this analogy we refer to the “frame of reference moving at velocity $\mathbf{v}_{G}$ with respect to the lab”, we have in mind the change of coordinates $\mathbf{y}=\mathbf{x}-\mathbf{v}_{G}t$ (the lab is just one chosen frame), while the time variable remains the same in the two coordinate systems. The chosen convention for the sign of $\mathbf{v}_{G}$ is such that a fluid moving at velocity $\mathbf{v}_{G}$ in the laboratory frame is seen as at rest in the boosted one. The Galilean transformation is given by the unitary operator

$$\hat{U}_{lab\to G}=e^{i\frac{\hat{\mathbf{p}}\mathbf{v}_{G}t-\hat{\mathbf{x}}m\mathbf{v}_{G}}{\hbar}}=e^{i\frac{\hat{\mathbf{p}}\mathbf{v}_{G}t}{\hbar}}e^{-i\frac{\hat{\mathbf{x}}m\mathbf{v}_{G}}{\hbar}}e^{+i\frac{m{v}_{G}^{2}t}{2\hbar}},$$

(2)

so that applied on the wavefunction reads

$$\psi_{G}(\mathbf{y})=[\hat{U}_{lab\to G}\psi_{lab}](\mathbf{y})=\\ =e^{-i\frac{m{\mathbf{y}}\mathbf{v}_{G}}{\hbar}}e^{-i\frac{m{v}_{G}^{2}t}{2\hbar}}\psi_{lab}(\mathbf{y}+\mathbf{v}_{G}t)\,.$$

(3)

Galilean invariance of the conservative part of the GPE evolution (1) is then guaranteed by the parabolic form of the kinetic energy according to elementary quantum mechanics Cohen-Tannoudji et al. (1991), so that in our terminology the (undriven) polariton field is Galilean invariant (instead, the covariance of the driving term is to be discussed in a moment). For usual polariton systems resulting from the strong coupling of a cavity photon mode to an excitonic transition this parabolic approximation is accurate for the typical flow speeds considered in the experiments Carusotto and Ciuti (2013).

Concerning the pump and loss terms, it is straightforward to see ¹¹1A possible derivation involves the Hamiltonian that generates the GPE (1); in particular the pump term reads $\hat{H}^{drive}_{lab}=\int d\mathbf{x}F_{lab}(\mathbf{x},t)\hat{\Psi}^{\dagger}(\mathbf{x})+{\textrm{h}.c.}$, which transforms as $\hat{H}^{drive}_{G}=\hat{U}_{lab\to G}\hat{H}^{drive}_{lab}\hat{U}^{\dagger}_{lab\to G}=\int d\mathbf{y}F_{lab}(\mathbf{y}+\mathbf{v}_{G}t,t)e^{-i\frac{m{\mathbf{y}}\mathbf{v}_{G}}{\hbar}}e^{-i\frac{m\mathbf{v}_{G}^{2}t}{2\hbar}}\hat{\Psi}^{\dagger}(\mathbf{y})+\textrm{h.c.}~{}$ that the dynamics in the boosted frame is described by the same GPE

$$i\partial_{t}\psi_{G}(\mathbf{y},t)=\\ =\left(\omega_{0}-\frac{\hbar}{2m}\nabla^{2}+g|\psi_{G}|^{2}+V_{ext}(\mathbf{y}+\mathbf{v}_{G}t)-i\frac{\gamma}{2}\right)\psi_{G}+\\ +F_{G}(\mathbf{y},t),$$

(4)

provided the pump term is covariantly transformed according to

$$F_{G}(\mathbf{y},t)=F_{lab}(\mathbf{y}+\mathbf{v}_{G}t,t)e^{-i\frac{m{\mathbf{y}}\mathbf{v}_{G}}{\hbar}}e^{-i\frac{m\mathbf{v}_{G}^{2}t}{2\hbar}}.$$

(5)

Note that this transformation involves a shift of the wavevector proportional to the velocity as well as an overall frequency shift ${m{v}_{G}^{2}t}/{(2\hbar)}$. The loss term remains unchanged thanks to the spatio-temporally local form that we have assumed from the outset.

Eqs. (1) and (4) describe the same dynamics, the observables in the two frames being linked by the usual Galilean prescriptions: $|\psi_{lab}(\mathbf{x},t)|^{2}=|\psi_{G}(\mathbf{y},t)|^{2}$ for the density and $\mathbf{v}^{flow}_{lab}(\mathbf{x},t)=\mathbf{v}^{flow}_{G}(\mathbf{y},t)+\mathbf{v}_{G}$ for the flow velocity, defined in terms of the wavefunction as usual as $\mathbf{v}^{flow}=\hbar\,\textrm{Im}[\psi^{*}\nabla\psi]/(2m\,|\psi|^{2})$.

Along these lines, it is natural to define for a plane-wave ²²2For finite pump spots the Galilean arguments still hold and p.e. they relate a stationary spot centered at wavevector $\mathbf{k}_{p}$ with a spot moving with velocity $-\hbar\mathbf{k}_{p}/m$ but at a given time centered at zero wavelength. The second situation is not so relevant for our discussion, so we always consider pump spots that can be considered spatially uniform (apart from a plane wave factor). coherent drive with frequency $\omega_{p}$ and wavevector $\mathbf{k}_{p}$ the detuning in the frame comoving with the fluid at $\mathbf{v}_{p}=\hbar\mathbf{k}_{p}/m$ as $\Delta_{p}=\omega_{p}-\hbar{k}_{p}^{2}/2m-\omega_{0}$. When $\Delta_{p}=gn_{0}$ or equivalently $\omega_{p}=\omega_{0}+\hbar{k}_{p}^{2}/2m+gn_{0}$, the fluid of density $n_{0}=|\psi_{0}|^{2}$ is characterized by a sonic dispersion with speed of sound $mc_{s}^{2}=\hbar\Delta_{p}=\hbar gn_{0}$. In what follows, most of the plots will refer to this most remarkable sonic case. Note that the overall phase factor in Eq. (5) is needed to ensure that the density and thus the linearity of the dispersion is independent of the Galilean frame.

Before proceeding, it is important to stress that the Galilean transformation discussed here is useful to mathematically relate the dynamics of fluids injected with different speed, but it does not correspond to a physical change of reference frame, for instance an experimentalist running parallel to the cavity mirrors. Indeed, for a medium with a refractive index different from unity, light propagation in a physically boosted Lorentz frame at velocity $\mathbf{v}_{L}$ is affected by the celebrated Fresnel drag effect Jackson (1975); Landau et al. (1984); Jones (1972); Artoni et al. (2001); Carusotto et al. (2003), which changes the dispersion relation to the same order in $v_{L}/c$ as the Doppler shift:

$$\omega^{\prime}(\mathbf{k}^{\prime})=\omega(\mathbf{k}^{\prime})+\left(\frac{1}{n_{cav}^{2}}-1\right)\mathbf{v}_{L}\cdot\mathbf{k}^{\prime}+...$$

(6)

where the apex refers to quantities measured in the Lorentz frame and $n_{cav}$ is the refractive index inside the cavity.

According to the Landau criterion Landau et al. (1981); Pitaevskii and Stringari (2016), a superfluid is able to flow without friction at speed $\mathbf{v}$ around a static defect until it is energetically favourable to create excitations in it, ie if $\omega(\mathbf{k})+\mathbf{k}\cdot\mathbf{v}\geq 0$, where $\omega(k)$ is the excitation dispersion of the fluid at rest. This provides the well-known expression for the critical velocity,

$$v_{c}=\min_{\mathbf{k}}\frac{\omega(\mathbf{k})}{k}.$$

(7)

For a weakly interacting fluid of bosons with contact interactions and dispersion $\omega(k)=c_{s}|k|$ this gives $v_{c}=c_{s}=\sqrt{\hbar\,g\,n_{0}/m}$ where $n_{0}=|\psi_{0}|^{2}$ is the particle density, $m$ is the mass and $g$ the interaction coupling strength.

Alternatively, a weak defect is able to move through a superfluid without friction if the dispersion $\omega(\mathbf{k})$ of elementary excitations in the latter has no intersection with a straight line $\omega=-\mathbf{v}\cdot\mathbf{k}$. For particles with a parabolic dispersion and local interactions, equivalence of the two points of view is ensured by the Galilean invariance.

This formulation applies well to superfluids of material particles with a long lifetime of the collective excitations, i.e. where the imaginary part $\textrm{Im}[\omega(\mathbf{k})]$ of the dispersion relation is much smaller than the real part $\textrm{Re}[\omega(\mathbf{k})]$ and can be neglected at long wavelength. However, subtleties arise in the case of driven-dissipative fluids, e.g. the polariton ones where the real and imaginary parts of the dispersion $\omega(\mathbf{k})$ may have comparable magnitudes. The crucial importance of this effect for incoherently pumped polariton condensates was first unveiled in Wouters and Carusotto (2010), where generalized forms of the Landau criterion for driven-dissipative systems were introduced in terms of real frequencies and complex momenta. In particular, for a polariton superfluid flowing against an obstacle it was shown that a pattern forms when the velocity of the fluid is larger than the critical velocity, thus showing clear superfluid-like features even in the presence of drive and dissipation. For the case of coherent pumping, a phenomenological way of assessing superfluidity is by computing the drag force, for which a pioneering discussion of the effect of drive and losses was reported in Berceanu et al. (2012). Notice that driven dissipative polariton fluids may not satisfy other definitions of superfluidity e.g. the one in terms of the transverse current-current response Juggins et al. (2018).

In spite of the complications arising from its driven-dissipative nature, the Galilean invariance argument stating that the critical (relative) speed depends on the density but not on the reference frame remains true for the polariton field, as a corollary of what shown in the previous section; importantly, this does not assume to work at the acoustic point. More precisely, in the approximation of an infinite uniform excitation spotNote (2), the pattern created by scattering against a defect (hence the superfluidity threshold) depends only on density and on the relative velocity between the defect and the fluid: the flow against a static defect and the defect moving through the fluid at rest correspond to the same dynamics viewed in two different Galilean frames. Having argued the validity of our next results in any reference frame, in this Section we turn to a formal study of the density pattern created when a polariton fluid is coherently excited into motion against a static obstacle by a monochromatic pump of frequency $\omega_{p}$ and in-plane wavevector $\mathbf{k}_{p}$ Note (2), with a special focus on the effects due to drive and dissipation.

In order to determine the density modulation pattern of the fluid flowing around a weak defect at rest, we adopt the method of Ref. Carusotto and Ciuti, 2004 and linearize the GPE (1) on top of the homogeneous solution at the pump’s wavevector $\mathbf{k}_{p}$ and frequency $\omega_{p}$, via the Ansatz

$$\psi(\mathbf{x},t)=e^{i(\mathbf{k}_{p}\cdot\mathbf{x}-\omega_{p}t)}\left[\psi_{0}+\int d\,\mathbf{k}\,\ \delta\psi_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{x}}\right]\,.$$

(8)

Importantly, since the defect produces a static perturbation on the fluid, the wavefunction (8) keeps a monochromatic form at the pump frequency $\omega_{p}$.

Inserting this Ansatz into the GPE (1) and expanding to lowest order in the defect’s potential $V_{\rm def}$, one finds for each Fourier component $\delta\psi_{\mathbf{k}}$

$$\left(\omega_{0}+\frac{\hbar(\mathbf{k}_{p}+\mathbf{k})^{2}}{2m}+2g|\psi_{0}|^{2}-\omega_{p}-i\frac{\gamma}{2}\right)\delta\psi_{\mathbf{k}}+\\ +\psi_{0}^{2}\,\delta\psi_{-\mathbf{k}}^{*}=-\delta V_{\rm def}({\mathbf{k}})\psi_{0}\,.$$

(9)

Then, by combining this equation with the complex conjugate one and solving the matrix inversion problem, we obtain for an arbitrary (real) potential

$$\delta\psi_{\mathbf{k}}=\chi_{{}_{V}}(\mathbf{k})\psi_{0}\delta V_{\rm def}(\mathbf{k})$$

(10)

where the response function to an external potential is defined as

$$\chi_{{}_{V}}(\mathbf{k})=[\mathscr{L}(\mathbf{k})]^{-1}_{11}-[\mathscr{L}(\mathbf{k})]^{-1}_{12}$$

(11)

and the $2\times 2$ Bogoliubov matrix reads

$$\mathscr{L}(\mathbf{k})=\\ \begin{pmatrix}\eta(\mathbf{k_{p}}+\mathbf{k})+gn_{0}-i{\gamma}/{2}&{g}{\psi_{0}}^{2}\\ -{g}{{\psi_{0}}^{*}}^{2}&-\eta(\mathbf{k_{p}}-\mathbf{k})-gn_{0}-i{\gamma}/{2}\end{pmatrix}\,.$$

(12)

Here, $\mathbf{k}=\mathbf{k}_{p}+\mathbf{k}$ is the (real-valued) total momentum and the detuning function

$$\eta(\mathbf{k})=\omega_{0}+\hbar k^{2}/2m+{g}n_{0}-\omega_{p}\,.$$

(13)

The perturbed wavefunction in real space is finally obtained from (10) by means of the Fourier transform (8). The Bogoliubov dispersion relation $\omega(\mathbf{k})$ corresponds to the eigenvalues of the matrix (12), i.e. the zeros of $\det\left[\omega-\mathscr{L}(\mathbf{k})\right]$.

Because of the presence of the imaginary loss terms in (9) and then (12), the frequency $\omega(\mathbf{k})$ has complex values for real values of the momentum $\mathbf{k}$. Physically, this corresponds to plane-wave excitations having a finite lifetime. In Fig. 1 we plot in red dashes the dispersion of an interacting polariton fluid at $\mathbf{k}_{p}=0$ and in the sonic regime. The underlying color plot shows the magnitude of the response $\delta\psi(\mathbf{k},\omega)$ to a monochromatic optical probe at $(\mathbf{k},\omega_{p}+\omega)$, which is proportional to the $(1,1)$ component of the susceptibility matrix, $\left.\left[{\omega-\mathscr{L}(\mathbf{k})}\right]^{-1}\right|_{11}$. The linewidth of the response along the frequency axis is set by the imaginary part of $\omega$, which in this case is flat and equal to $\gamma$.

While these complex frequency modes allow to study dynamical excitations, it was first noted in Wouters and Carusotto (2010) that the response of the steady state to static external perturbations is most conveniently characterized in terms of modes with a real frequency and a complex momentum. In classical electromagnetism Tait (1972), such waves naturally appear when dealing with monochromatic light incident on an absorbing medium. In our context, this point of view is implicitly assumed upon using the residue theorem to evaluate the Fourier integral (8): since the defect is at rest, it generates a static perturbation in the fluid at $\omega=0$, whose peak wavevectors $\mathbf{k}$ are determined by the zeros of $\det\mathscr{L}(\mathbf{k})$, i.e. the poles of $\chi_{{}_{V}}(\mathbf{k})$.

In the simplest case of a one-dimensional geometry and a delta-like potential at rest giving a momentum-independent $\delta V_{\rm def}(k)=\delta V_{\rm def}$, the position of the poles in the complex $k$ plane are shown in Fig. 2(a) for different values of the pump wavevector $k_{p}>0$ (that is, of the speed $\hbar k_{p}/m$ of the fluid) and a resonant laser frequency $\omega_{p}=\hbar k_{p}^{2}/(2m)+gn_{0}+\omega_{0}$ such that the Bogoliubov dispersion is gapless and has a sonic behaviour with a well-defined speed of sound $c_{s}$.

With the residue theorem technique, evaluation of the Fourier integral (8) for $x>0$ ($x<0$) only picks the poles in the upper (lower) complex half-plane. For the $x>0$ region, a single pole is present and this has a vanishing real part. It corresponds then to the exponentially decaying perturbation that is visible in Fig.2(b) in the $x>0$ downstream region. The faster the flow, the closer the pole to the real axis, so the slower the exponential decay.

The behaviour is richer in the $x<0$ upstream region: for small speeds $k_{p}$, the two poles have again a vanishing real part and the perturbation displays a monotonic decay. Around $\hbar k_{p}/mc_{s}\simeq 0.75$ the poles merge in the complex $k$ plane at a finite $\textrm{Im}[k]$ and then separate again along a direction parallel to the real axis. For sufficiently large speeds, their real part exceeds the imaginary one so that the perturbation in the fluid starts displaying a clear oscillatory character upstream of the defect.

As the association between the real part of the wavevector and the transferred momentum suggests, this change in behaviour is expected to result into a sharp change in the value of the drag force exerted by the moving fluid onto the defect, defined as Astrakharchik and Pitaevskii (2004); Wouters and Carusotto (2010); Berceanu et al. (2012)

$$F_{d}=-\int\,dx\,\nabla V_{\rm def}(x)\,|\psi(x)|^{2}\,.$$

(14)

A plot of $F_{d}$ as a function of the fluid speed is shown in Fig.2(d) for the sonic case and qualitatively agrees with this prediction.

In particular, the position of the threshold position is consistent with the naive Landau criterion based on comparing the flow speed with the speed of sound. The velocity-independent value of the friction force at high speeds is typical of one-dimensional superfluids and was first anticipated in Astrakharchik and Pitaevskii (2004) for conservative atomic systems. Finally, the smaller the loss rate $\gamma$, the sharper the transition from a frictionless superfluid behaviour at slow speeds to a finite friction force at fast speeds.

While this picture is qualitatively accurate, establishing a precise relation between the location of the threshold and the behaviour of the poles in the $k$-complex plane requires a bit more careful analysis. As one can see in the lower half-plane of panel (a), the $k$ vectors aquire a real part in fact at a smaller value $k_{p}\approx 0.75$ than the threshold that is visible in the force plot around $k_{p}\approx 1$. To explain this feature, one can see in panel (c) that the different curves of $\textrm{Re}[k]/\textrm{Im}[k]$ for different values of the loss rate cross at a single value close to $\sqrt{3}$ ³³3 Indeed, at $\hbar k_{p}\simeq mc_{s}\simeq 1$ the pole condition in terms of $z=k/k_{p}$ simply reads $z^{4}+iaz+a^{2}$, whose solution satisfy $|\textrm{Re}[z]/\textrm{Im}[z]|\to\sqrt{3}$ as $a\equiv\frac{\hbar\gamma}{mc_{s}^{2}}\to 0$. for a value of the pump wavevector $\hbar k_{p}\simeq mc_{s}\simeq 1$ that approximately corresponds to the threshold for the drag force shown in panel (d). This suggests that the threshold is not determined by the point where the $k$ vectors aquire a real part, but rather by the point when the real part exceeds (by a factor $\sqrt{3}$) the imaginary part.

The discussion in the previous sections was based on a driven-dissipative, yet purely coherent dynamics of the polariton fluid. Recent experimental works Sarkar et al. (2010); Walker et al. (2017); Stepanov et al. (2019) have suggested that an incoherent reservoir of excitations – most likely of dark-excitonic nature – is excited even under a coherent pump via non-radiative absorption processes.

As it was introduced in these works, the effect of the incoherent reservoir can be theoretically described by including the reservoir density $n_{R}(\mathbf{x})$ to the equations of motion,

	$\displaystyle i\partial_{t}\psi$	$\displaystyle=$	$\displaystyle\left(\omega_{0}-\frac{\hbar\nabla^{2}}{2m}+g|\psi|^{2}++g_{R}n_{R}-i\frac{\gamma}{2}\right)\psi+$		(15)
		$\displaystyle+$	$\displaystyle F(\mathbf{x},t),$
	$\displaystyle\partial_{t}n_{R}$	$\displaystyle=$	$\displaystyle-\gamma_{R}n_{R}+\gamma_{inc}|\psi|^{2}\,.$		(16)

Here, the decay of coherent polaritons into incoherent excitations occurs at a rate $\gamma_{inc}$ and the latter give a contribution $g_{R}n_{R}$ to the polariton blue-shift. The total decay of polaritons $\gamma$ thus includes a $\gamma_{inc}$ contribution, while the incoherent excitations decay at a rate $\gamma_{R}$.

At stationarity under a monochromatic pump at $\omega_{p}$, one has $\psi(\mathbf{x},t)=\psi_{0}\,\exp(-i\omega_{p}t)$ and, from (16), one gets a time-independent $n_{R}(\mathbf{x})=\frac{\gamma_{inc}}{\gamma_{R}}|\psi_{0}(\mathbf{x})|^{2}$. Reinjecting this expression into (15), one simply obtains a renormalized nonlinear coupling strength

$$g_{\rm eff}=g+\frac{\gamma_{inc}}{\gamma_{R}}g_{R}\,.$$

(17)

Except for the reinforced interactions and the consequently reinforced blue shift $\mu_{T}=g_{\rm eff}n_{0}=gn_{0}+g_{R}n_{R}$, the reservoir has thus no effect on the stationary state. The usual optical bistability and optical limiting behaviours are found depending on whether the laser frequency $\omega_{p}$ is blue- or red-detuned as compared to the polariton mode at $\mathbf{k}_{p}$.

Even more importantly for our purposes, superfluidity features the usual behaviours with a speed of sound defined by the total blue-shift as $mc_{s,T}^{2}=\mu_{T}=gn_{0}+g_{R}n_{R}$. Since this reasoning requires stationarity of both the polariton $\psi(\mathbf{x})$ and the reservoir $n_{R}(\mathbf{x})$ densities, this result only holds for static defects that do not induce time-dependent modulations to the fluid density, that is defects at rest in a (possibly moving) fluid. And, of course, these statements are only relevant if the fluid is indeed able to reach a dynamically stable steady state: as it was pointed out in Amelio and Carusotto (2019), the presence of a slow reservoir can in fact give rise to dynamical instabilities that destabilize the stationary state.

The physics gets even more intriguing as soon as one looks at the dynamics of the excitations on top of the fluid, as first noticed in Ref.Stepanov et al., 2019. In the homogeneous case under a plane-wave coherent pump of wavevector $\mathbf{k}_{p}$ and frequency $\omega_{p}$, the steady-state solution has the form $\psi_{0}(\mathbf{r},t)=\psi_{0}\exp[i(\mathbf{k_{p}}\cdot\mathbf{r}-\omega_{p}t)]$ and the Bogoliubov theory involves a 3$\times$3 matrix

$$\mathscr{L}(\mathbf{k})=\\ \begin{pmatrix}\eta(\mathbf{k}_{p}+\mathbf{k})+gn_{0}-i\frac{\gamma}{2}&{g}{\psi_{0}}^{2}&g_{R}\psi_{0}\\ -{g}{{\psi_{0}}^{*}}^{2}&-\eta(\mathbf{k}_{p}-\mathbf{k})-gn_{0}-i\frac{\gamma}{2}&-g_{R}\psi_{0}\\ i\gamma_{inc}\psi_{0}&i\gamma_{inc}{\psi_{0}}^{*}&-i\gamma_{R}\end{pmatrix},$$

(18)

where $\mathbf{k}$ is again the relative wavevector of the excitation on top of the moving fluid and the effective detuning function is now $\eta(\mathbf{k})=\omega_{0}+\hbar k^{2}/2m+{g}n_{0}+g_{R}n_{R}-\omega_{p}$. The first and second columns/rows of $\mathscr{L}(\mathbf{k})$ correspond to the polariton modulation $\delta\psi_{\mathbf{k}}$ and $\delta\psi^{*}_{-\mathbf{k}}$, while the third column/row corresponds to the modulation of the reservoir density $\delta n_{R}$.

The corresponding eigenvalue problem can be formulated in a physically trasparent way by defining a frequency–dependent effective coupling

$$g_{\rm eff}(\omega)=g+\frac{\gamma_{inc}}{-i\omega+\gamma_{R}}g_{R},$$

(19)

which allows to eliminate the reservoir by replacing $g$ with $g_{\rm eff}(\omega)$  ⁴⁴4Of course, the replacement must not be applied to the $g$ appearing in the definition of the detuning function $\eta$, which involves the stationary background., and thus reduce matrix (18) to an effective $2\times 2$ matrix involving the polaritons only. The eigenvalue equation for the collective mode dispersion then reads

$$\left(\omega-\frac{\hbar\mathbf{k}_{p}}{m}\cdot\mathbf{k}+i\frac{\gamma}{2}\right)^{2}=\eta(\tilde{k})\left[\eta(\tilde{k})+2g_{\rm eff}(\omega)n_{0}\right]$$

(20)

with $\tilde{k}=\sqrt{k_{p}^{2}+k^{2}}$. While this expression is formally nearly identical to the usual one (12), the $\omega$-dependence of the right–hand side has crucial consequences onto the dispersion of collective excitations. Of course, the usual Bogoliubov dispersion is recovered in the limit where high-$\omega$ perturbations are considered, so that $g_{\rm eff}$ recovers $g$. On the other hand, the static value (17) for $g_{\rm eff}$ is recovered for stationary perturbations at $\omega=0$.

In Sec.II we have seen that in the absence of reservoir, the generalized Gross-Pitaevskii equation (1) has specific invariance properties under Galilean boosts and in Sec.III we have shown that the superfluidity properties must be then the same in the two cases of a defect moving through a polariton fluid at rest and of a moving polariton fluid hitting a static defect: these two configurations represent in fact the same process seen in two different Galilean frames. Correspondingly, since the (not invariant but) covariant coherent pump does not explicitely enter the linearized Bogoliubov calculation, the complex-valued dispersion $\omega(\mathbf{k})$ simply gets Doppler-shifted $\omega(\mathbf{k})\rightarrow\omega(\mathbf{k})+\mathbf{k}\cdot\mathbf{v}$ when going to a reference frame moving at speed $\mathbf{v}$.

The situation is completely different in the presence of an incoherent reservoir, as described by the generalized dynamics of Eqs.(15-16). This latter, in fact, defines a privileged frame of reference linked to the underlying semiconductor cavity structure. Such a feature is visible by comparing the Bogoliubov spectra shown in the left and right panels of Fig.3: even though the total blue-shift is the same in the two cases, the dispersions are markedly different in both the real and the imaginary parts.

In Fig.4, we illustrate this breaking of Galilean invariance by looking at the effect of the incoherent reservoir on the density modulation pattern generated by a defect in motion through a fluid at rest. As we expected and explicitly verified by numerical integration of the Gross-Pitaevskii equation (15-16), a defect moving with constant velocity $\mathbf{v}_{d}$ in a fluid at rest with respect to the semiconductor substrate generates a pattern which is stationary in the frame of reference of the defect.

Therefore, within linear response to a shallow defect, it is possible to solve for the field perturbation in this frame by using the technique illustrated in Eq. (9). Since the reservoir equation in the defect frame ⁶⁶6Remember that first-order equations for scalar quantities require total time derivatives (also called convective derivatives) in order to be Galileo invariant, like the one for density in the incompressible Euler continuity equation. reads

$$\partial_{t}n_{R}=-\mathbf{v}_{d}\cdot\nabla n_{R}-\gamma_{R}n_{R}+\gamma_{inc}|\psi|^{2}\ ,$$

(31)

the $\omega=0$ condition discussed in Sec.III allows for elimination of the reservoir via a momentum-dependent effective coupling

$$g_{\rm eff}(\mathbf{k})=g+\frac{\gamma_{inc}}{-i\mathbf{v}_{d}\cdot\mathbf{k}+\gamma_{R}}g_{R}~{}.$$

(32)

Notice that this procedure of imposing $\omega=0$ in the defect frame can be equivalently implemented in the lab frame by solving for $\omega=\mathbf{v}_{d}\cdot\mathbf{k}$; this is proven by expressing the defect potential as $\delta V_{def}(\mathbf{x}-\mathbf{v}_{d}t)=\int d\mathbf{k}d\omega\ \delta(\omega-\mathbf{v}_{d}\cdot\mathbf{k})\ \delta V_{def}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}-i\omega t}$, and similarly for the Ansatz of the field and reservoir.

For a fully coherent polariton fluid in the absence of a reservoir, the Galilean invariance holds and the physics only depends on the relative velocity of the fluid and the defect. As a result, the left panels of Fig. 4 equivalently represent the two cases of a fluid flowing against a static defect or of a moving defect in a fluid at rest.

On the basis of the discussion in the previous sections, it is natural to expect that the situation be completely different in the presence of an incoherent reservoir, which sets a privileged reference frame linked to the semiconductor matrix. To start with, a pattern identical to the fully coherent case is found for a static defect via the renormalized coupling (17), as long as the total blueshift is the same and no dynamical instabilities develop (Amelio and Carusotto, 2019). Instead, when it is the defect to move in a polariton fluid at rest in the presence of a reservoir. the density modulation pattern is shown in the right panels of Fig. 4. These panels are plotted in the experimentally relevant $\gamma_{R}\ll\gamma$ regime for the same values of the speed $\mathbf{v}_{d}$ and the total interaction energy $\mu_{T}$ used in the left panels. It is apparent that the critical speed is strongly reduced, as expected from the Bogoliubov dispersion discussed in Sec.IV. Moreover, the shape of the density modulation profile shows a clear Mach cone of angle $2\alpha$ with $\sin\alpha\simeq c_{s,0}/v_{d}$.

A more quantitative insight on the critical speed can be obtained looking at the plot of the friction force as a function of the defect speed for a polariton fluid at rest shown in Fig.5. The force is evaluated using (14) under the assumption that the defect only interacts with the coherent polaritons. Both in the absence (blue line) and in the presence (orange) of the reservoir, the friction force displays a clear threshold behaviour, losses being as usual Berceanu et al. (2012) responsible for a smoothening of the threshold. In contrast to the 1D case of Fig.2, in the high-speed limit the force tends to the asymptotically linear dependence on $v_{d}$ predicted by Astrakharchik and Pitaevskii (2004).

As expected, the position of the threshold occurs at a markedly lower speed in the presence of the reservoir, at a value consistent with the effective speed of sound $c_{s,0}$. The fact that the critical speed is set by the effective high-$k$ speed of sound $c_{s,0}$ rather than by the low-$k$ value $c_{s}$ is physically understood by noting that the density modulation is peaked in $\mathbf{k}$-space at the intersection of the Bogoliubov dispersion with the $\omega=\mathbf{k}\cdot\mathbf{v}_{d}$ condition for the moving defect. A further confirmation of this statement can be found in Fig. 6.a of the Appendix, where we show the same plot for a faster reservoir for which the distinction between $c_{s,0}$ and $c_{s}$ is more evident.

The origin of the peculiar negative value $F_{d}<0$ found in the presence of the reservoir is illustrated in the panels (b,c) of Fig.5. A very slow defect excites quasi–resonantly the reservoir branch of the dispersion, leaving in its wake a reservoir depletion, which is partially refilled by the faster polaritons. This results in an excess of polaritons behind the defect and, thus, to a negative drag. Of course, the fact that the force tends to accelerate (rather than brake) the defect does not violate energy conservation, since we are dealing with a driven-dissipative system.

Coming back to the case of a defect at rest in a moving fluid, here the density modulation pattern is stationary in the frame of the semiconductor cavity structure, so the $\omega=0$ value of the effective interaction constant $g_{\rm eff}(\omega)$ is to be used. As we have discussed in the previous sections, this value recovers the interaction constant $g_{\rm eff}$ defined in (17) that enters the expression for the total blue shift $\mu_{T}$, so that the critical speed for superfluidity is set by $c_{s,T}$ such that $mc_{s,T}^{2}=\hbar\mu_{T}$. It is quite remarkable how this simple result holds independently of the relative magnitude of the polariton and reservoir contributions to this latter and of the details of the complex Bogoliubov dispersion in a moving fluid discussed in Sec.IV.2. The only requirement is that the flow is dynamically stable for the chosen pump parameters. This last subtle feature is the reason why the pioneering experiments in Amo et al. (2009) were in quantitative agreement with a theory that did not include the reservoir. For what concerns the dynamical experiments in Nardin et al. (2011), instead, the quantitative agreement with the reservoir-less theory was guaranteed by the fact that the experiments were performed using a short pulse of coherent pump light, so that the reservoir density did not have time to build up.

In this work we have reported a detailed theoretical study of the effect of a reservoir of incoherent excitations on the superfluidity properties of polariton fluids in planar microcavities. In the absence of a reservoir, a formal Galilean tranformation relates the two situations of a fluid flowing against a static defect and of a defect moving in a fluid at rest. As a result, the dispersions of the Bogoliubov excitations are related by a simple Doppler shift and the density modulation pattern are identical in the two cases, as it normally happens in Galilean invariant fluids of material particles in free space.

On the contrary, the presence of the reservoir fixes a privileged laboratory reference frame linked to the semiconductor cavity structure. This breaking of Galilean invariance is visible in the Bogoliubov dispersion of the collective excitations in the fluid and in the density modulation pattern generated by a defect: while the effective speed of sound probed by a defect at rest is univocally determined by the total blue shift of the polariton modes as in the experiments of Ref. Amo et al., 2009, the one probed by a moving defect is significantly smaller and mostly determined by the polariton contribution to the blue shift. This results is of crucial importance to reconcile the historical demonstrations of polariton superfluidity in Amo et al. (2009); Nardin et al. (2011) with the recent experiment in Stepanov et al. (2019).

Beyond the microcavity polariton systems on which this article is focused, our results can be straightforwardly applied to other physical realizations of fluids of light such as photons propagating in cavityless nonlinear optical media Carusotto (2014). While a sort of Galilean invariance along the transverse plane holds for instantaneous Kerr-like nonlinearities Fontaine et al. (2018), a strong breaking of Galilean invariance is in fact expected to occur when the optical nonlinearity has a thermal nature Vocke et al. (2015). This is a crucial feature that needs being duly taken into account when using quantum fluids of light as quantum simulators.

We are grateful to M. Wouters for stimulating exchanges; I.A. thanks L. Giacomelli for continuous discussions on Bogoliubov formalism. We acknowledge financial support from the European Union FET-Open grant “MIR-BOSE” (n. 737017), from the H2020-FETFLAG-2018-2020 project “PhoQuS” (n.820392), and from the Provincia Autonoma di Trento. All numerical calculations were performed using the Julia Programming Language Bezanson et al. (2012).