Superfluid Flow of Polaron Polaritons above Landau’s Critical Velocity

We develop a theory for the interaction of light with superfluid optical media, describing the motion of quantum impurities that are created and dragged through the liquid by propagating photons. It is well known that a mobile impurity suffers dissipation due to phonon emission as soon as it moves faster than the speed of sound in the superfluid—Landau’s critical velocity. Surprisingly we find that in the present hybrid light-matter setting, polaritonic impurities can be protected against environmental decoherence and be allowed to propagate well above the Landau velocity without jeopardizing the superfluid response of the medium.

Figure 1

(a) Illustration of a propagating photon generating a dark-state polaron polariton via impurity interactions. (b) The incident photon can form a dark-state polariton by coupling the atomic ground state $|b⟩$ to an excited state $|e⟩$ with a detuning $\mathrm{\Delta }$ and a coupling strength $g\sqrt{n}$, determined by the atomic density $n$. The state $|e⟩$ decays radiatively with rate $\gamma$ and is coupled to a stable impurity state $|c⟩$ via a classical control field with Rabi frequency $\mathrm{\Omega }$. Panels (c) and (d) show the decay rate $\mathrm{\Gamma }$ of the formed polaron polariton in units of $t_B=\xi /\sqrt{2}{c}_s$, determined by the coherence length, $\xi$, and the speed of sound, $c_s$, of the superfluid. (c) $\mathrm{\Gamma }$ as a function of the impurity speed $v$ and the polariton group velocity $v_{\mathrm{g}}$, varied through the density, $n\simeq 2\times {10}^{14}(\mathrm{red})$, $0.8\times {10}^{14}(\mathrm{blue})$, $0.3\times {10}^{14}{\mathrm{cm}}^{-3}(\mathrm{green})$, for $\mathrm{\Omega }/\gamma =2$ and $\mathrm{\Delta }/\gamma =-200$. The damping of the bare polaron is shown by the black lines. (d) $\mathrm{\Gamma }$ as a function of $\mathrm{\Omega }$ for $n\simeq 0.8\times {10}^{14}{\mathrm{cm}}^{-3}$ and $\mathrm{\Delta }/\mathrm{\Omega }=-300$, revealing the emergence of a critical field $\mathrm{\Omega }/\gamma \sim 3$. All calculations are performed for the $D_1$ transition of ultracold ${}^{23}\mathrm{Na}$ atoms and an impurity scattering length of $a=500a_0$, in units of the Bohr radius $a_0$.

Figure 2

Polariton dispersion curves in the absence of atomic interactions for $\mathrm{\Delta }=-200\gamma$, $\mathrm{\Omega }=2\gamma$, and $n=0.5\times {10}^{14}{\mathrm{cm}}^{-3}$. (a) Incoming photons generate dark-state polaritons (black solid line) with an approximate linear dispersion, ${ϵ}_{\mathbf{p}}\simeq v_{\mathrm{g}}(p-p_0)+(\mathbf{p}-{\mathbf{k}}_{\mathrm{cl}}{)}^2/2m$, around $p-p_0\approx 0$ (black dotted line) that facilitates low-loss form stable photon propagation with the slow-light group velocity $v_{\mathrm{g}}$. Atomic collisions with the surrounding condensate cause a typical momentum change of $\mathrm{\Delta }p\sim 1/\xi$ well outside this EIT regime, indicated by the vertical grey bar. The dark state is thereby broken apart by any atomic collision event, and scatters into the photon-free hybridized states $|\pm ⟩$, with the indicated energies ${ϵ}_{\mathbf{p}}^{(\pm )}$, shown by the orange and blue dashed lines in panels (a) and (b). This characteristic scattering process leads to the ansatz Eq. (4) for the polaron polariton. As illustrated in panel (b), the energy of the state $|-⟩$ is typically so far removed that it does not contribute significantly to the emerging polaron-polariton quasiparticle and its self-energy, Eq. (6). Panel (c) shows the same dispersion curves on an expanded momentum scale, revealing the quadratic contribution from the atomic kinetic energy and the light shift induced by the classical control field.

Figure 3

Pulse propagation through a condensate of ${}^{23}\mathrm{Na}$ atoms with a density of $n=2.6\times {10}^{14}{\mathrm{cm}}^{-3}$ and an impurity scattering length of $a=0.1\xi$. The dynamics of a bare impurity wave packet (blue lines) suffers strong damping due to the supersonic motion of the formed polaron with an initial velocity of $3\mathrm{m}/\mathrm{s}\gg c_s$. In contrast, the red lines show the asymptotically undamped motion of a polaron polariton with $\mathrm{\Delta }=-200\gamma$ and an identical initial group velocity of $v_{\mathrm{g}}=3\mathrm{m}/\mathrm{s}$, corresponding to a near-unity impurity fraction of $1-v_{\mathrm{g}}/c=0.99999999$. Polaron formation eventually leads to a slight lowering of the group velocity [41].