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

We theoretically investigate how the presence of a reservoir of incoherent excitations affects the superfluidity properties of resonantly driven polariton fluids. While in the absence of reservoir the two cases of a defect moving in a fluid at rest and of a fluid flowing against a static defect are linked by a formal Galilean transformation, where the reservoir defines a privileged reference frame attached to the semiconductor structure and causes markedly different features between the two settings. The consequences on the critical velocity for superfluidity are highlighted and compared to experiments in resonantly driven excitons polaritons.

Figure 1

Frequency and momentum dependence of the (non-normalized) transmittivity of a coherently pumped cavity to a weak monocromatic probe of frequency $\omega$ (measured with respect to the pump frequency) and wave vector $\mathbf{k}$. The pump injects a coherent polariton fluid at ${\mathbf{k}}_p=0$ and no incoherent reservoir is assumed to be present. The pump intensity is adjusted to be at the resonant point at which ${\mathrm{\Delta }}_p=g{n}_0$ and the Bogoliubov dispersion is sonic and gapless. As usual [26], the transmission amplitude is set by the matrix element ${\left.{[\omega -{\mathcal{L}}_{\mathbf{k}}]}^{-1}\right|}_{11}$ with the Bogoliubov matrix ${\mathcal{L}}_{\mathbf{k}}$ being defined in Eq. (12). The green dashes (dots) indicate the particle (hole) branch of the Bogoliubov dispersion of the elementary excitations, while the cyan dashes indicate the bare polariton band at linear regime. Frequencies are measured in units of the interaction energy $g{n}_0$; lengths (momenta) in units of the (inverse) healing length $\xi ={[{\hslash }^2/\sqrt{mg{n}_0}]}^{1/2}$.

Figure 2

(a) Evolution of the four complex poles of the one-dimensional response function ${\chi }_{{}_V}(k\in \mathbb{C})$ for different values of the flow speed $k_p>0$. The residue theorem is to be applied to the upper (lower) half–plane for $x>0$ ($x<0$). (b) Spatial profile of the density modulation induced by the defect (located at $x=0$) for different values of $k_p$. More precisely, the renormalized density perturbation $\frac{g[n\left(x\right)-n_0]}{\delta V_{\mathrm{def}}}$ is reported. (c) Ratio $\left|\mathrm{Re}\right[k]/\mathrm{Im}[k\left]\right|$ between the real and imaginary part of the poles as a function of $k_p$ and for different dissipation rates; the horizontal dashed line indicates $\sqrt{3}$. (d) Drag force as a function of the flow speed for different loss rates $\gamma$. Across (a–d) the pump frequency is kept at the sonic resonance point ${\mathrm{\Delta }}_p=g{n}_0$, unless differently specified the damping is set to $\gamma /g{n}_0=0.2$, and no incoherent reservoir is present.

Figure 3

Dispersion of collective excitations in a polariton fluid at rest $k_p=0$ [left column, panels (a) and (b)] and in motion at $v_p=0.8c_{s,T}$ along the $x$ direction [right column, panels (c) and (d)] in the presence of an incoherent reservoir. Upper [(a) and (c)] panels show the real part of the dispersion, the lower [(b) and (d)] panels show the imaginary part. The total blueshift ${\mu }_T$ is the same in all panels and pumping is tuned at the resonance point such that ${\mathrm{\Delta }}_p={\mu }_T$. Other parameters: $\gamma /{\mu }_T=0.2, g_R=2g$, and ${\gamma }_R=2{\gamma }_{\mathrm{inc}}=0.08\gamma$ which means $c_{s,0}=c_{s,T}/\sqrt{2}$ and $c_s\simeq 0.9c_{s,0}$. Note that for a slightly larger ${\gamma }_{\mathrm{inc}}\simeq 0.05\gamma$ or for $v_p\simeq 0.9c_{s,T}$, the flow configuration in the right panels would become dynamically unstable. The different curves are colored according to their nature at large wave vector $k$. The dashed cyan lines in the upper panels indicate the low-$k$ sonic dispersions (25) and (30).

Figure 4

Density modulation induced by a moving defect in the absence (left) and in the presence (right) of an incoherent reservoir. The total blueshift ${\mu }_T$ is the same in all panels. The polariton fluid is at rest $k_p=0$ and the pump frequency is tuned at the resonant point ${\mathrm{\Delta }}_p={\mu }_T$. In the upper panels, the defect speed is chosen in the vicinity of the critical speed for superfluidity in the absence of incoherent reservoir, $v_d=c_{s,T}$. In the lower panels, the defect speed is larger $v_d=1.3c_{s,T}$. The dashed green lines in the lower panels indicate the Mach cone of angle $2\alpha$ expected from the chosen values of the flow $v_d$ and sound (27) speeds, $sin\alpha =c_{s,0}/v_d$. Reservoir parameters are close to the ones estimated in Ref. [12], $g_R=2g, {\gamma }_R=2{\gamma }_{\mathrm{inc}}=0.08\gamma$. For thse values, the contributions of the polaritons and the incoherent reservoir to the blueshift are equal, $g_R{n}_R=g{n}_0={\mu }_T/2$.

Figure 5

Critical speed for superfluidity in the presence of a reservoir, for a small and shallow defect moving in a polariton fluid pumped at the sonic point ${\mathrm{\Delta }}_p={\mu }_T$ and at rest $k_p=0$. (a) Drag force as a function of the defect velocity $v_d$ in the absence (blue) and in the presence (orange) of the incoherent reservoir. The force is here renormalized by the effective coupling $g_{\mathrm{eff}}{F}_d$, so to have a fair comparison of the two cases with and without reservoir. The vertical lines confirm that in the former case the critical speed is at $c_{s,T}$, while in the latter case it is at $c_{s,0}$. An explanation for the negative drag at small $v_d$ is provided in panels (b,c) where $v_d=0.02c_{s,T}$ is taken. Panel (b) shows the polariton-induced component to the blueshift ${g\left|\psi \left(x\right)\right|}^2$. Panel (c) shows the incoherent reservoir contribution $g_R{n}_R\left(x\right)$. The defect consists of a Gaussian perturbation indicated in the plot by the cyan circle of radius three times its width. The depletion of the (slow) reservoir density that it leaves behind it is partly filled by the (faster) polariton. Same parameters as in the previous figures, namely $\gamma =0.2{\mu }_T, g_R=2g, {\gamma }_R=2{\gamma }_{\mathrm{inc}}=0.08\gamma$, so that $c_{s,0}=c_{s,T}/\sqrt{2}$.

Figure 6

Upper: Dispersion of the collective excitations of a polariton fluid at the acoustic point and in the presence of reservoir with parameters tuned so to make clear the distinction between the three definitions of the speed of sound: $c_s$ is the slope of the dispersion at $k=0, c_{s,T}$ takes into account the total blueshift and it is correct in the adiabatic limit, $c_{s,0}$ is only due to the self-interaction of the coherent polariton fluid. The reservoir parameters are ${\gamma }_R=2{\gamma }_{\mathrm{inc}}=0.2\gamma$. For the parameters in this figure, ${\mu }_R=g{\left|{\psi }_0\right|}^2={\mu }_T/2, c_{s,0}=c_{s,T}/\sqrt{2}\approx 0.71c_{s,T}$, and $c_s\approx 0.41c_{s,T}$. In the lower panel, one sees that the speed $c_{s,0}$ turns out to be the critical speed when superfluidity is considered for a moving defect, e.g., by computing the drag force (solid orange line). For comparison, the blue line shows the drag force for a purely coherent fluid without reservoir at the same ${\mu }_T$.

Figure 7

From left to right: Color plots of $|{\chi }_{11}|,|{\chi }_{12}|,|{\chi }_{11}-{\chi }_{12}|$ as functions of $(k,\omega )$. From top to bottom: without reservoir and $k_p=0$, without reservoir and $v_p=1.1c_{s,T}$, with reservoir and $k_p=0$, and with reservoir and $v_p=1.1c_{s,T}$. The reservoir parameters are ${\gamma }_R=2{\gamma }_{\mathrm{inc}}=0.08\gamma$. In particular, looking at the last column, it is clear that having both ${\mathbf{k}}_p\ne 0$ and a reservoir allows for having different luminescence (as generated by phononic white noise) on the left and right particle branches, while the colorplot is only Doppler shifted (shear mapping) if the reservoir is absent.