Landau Polariton Dispersion

• Landau polariton dispersion is defined as the collective excitation spectrum in driven–dissipative polaritonic fluids, showing phonon-like linear behavior at low momentum and ghost (negative-energy) branches.
• High-resolution probe spectroscopy maps precise energy–momentum relationships, verifying Bogoliubov-type predictions and quantifying parameters like the speed of sound and critical superfluid velocity.
• The interaction with a dark exciton reservoir renormalizes polariton dispersions and induces modulational instability precursors, deepening insights into non-equilibrium quantum hydrodynamics.

Landau polariton dispersion refers to the collective excitation spectrum of quantum fluids of polaritons, particularly those realized in semiconductor microcavities, where strong light–matter coupling creates mixed photon–exciton quasiparticles. When such systems are driven and dissipative, their excitation spectrum reveals rich Landau-type features: phonon-like linear dispersions at low momentum, the presence of a critical superfluid velocity governed by Landau’s criterion, the emergence of ghost (negative-energy) branches, the renormalization of dispersions by dark exciton reservoirs, and the onset of dynamical instabilities. High-resolution probe spectroscopy now enables the direct mapping of these dispersions, providing stringent validation of theoretical predictions and deep insight into quantum hydrodynamics in polaritonic systems.

1. Driven–Dissipative Bogoliubov Polariton Dispersion

In a homogeneous lower-polariton fluid continuously pumped at frequency $\omega_0$ and detuning $\delta = \omega_0 - \omega_{LP}$, the excitation spectrum can be derived by linearizing the generalized complex Gross–Pitaevskii equation—accounting for polariton–polariton interactions ($gn$), finite decay rate ($\gamma$), and the driven–dissipative nature of the system. The resulting Bogoliubov-type dispersion reads: $$\hbar\,\omega_B(k) = \pm\sqrt{[\epsilon_k + gn - \delta]^2 - (gn)^2} - i\frac{\hbar\gamma}{2}$$ where

$$\epsilon_k = \frac{\hbar^2 k^2}{2m_{LP}}, \quad gn = \text{interaction-induced blueshift}, \quad \delta = \text{laser detuning}, \quad \gamma = \text{polariton linewidth}.$$

At the bistability turning point ($\delta = gn$), the real part simplifies to the standard equilibrium Bogoliubov form: $$\hbar\omega_{B,\pm}(k) = \pm\sqrt{\epsilon_k(\epsilon_k + 2gn)} \equiv E_\pm(k)$$ The $+$ branch corresponds to the normal (positive energy) excitations, while the $-$ branch represents the "ghost" branch (negative excitation energies). Explicitly,

$$E_\pm(k) = \pm\sqrt{\frac{\hbar^2 k^2}{2m_{LP}}\left(\frac{\hbar^2 k^2}{2m_{LP}} + 2gn\right)}$$

which is fully symmetric under $k \mapsto -k$ and dominated by linear ("phonon-like") behavior at small $k$.

2. Landau Critical Velocity and Superfluidity

Landau’s criterion for superfluidity establishes a threshold flow velocity $v_c$ below which no excitations can be generated by impurities: $v_c = \min_k \frac{E_+(k)}{\hbar k}$ For $E_+(k)\simeq \hbar c_s k$ at low $k$, the minimum is reached as $k \rightarrow 0$, yielding

$v_c = c_s$

The speed of sound is thus

$c_s = \sqrt{\frac{\hbar gn}{m_{LP}}}$

For the complete Bogoliubov dispersion, $E_+(k)/k$ increases at large $k$, maintaining this minimum at $k = 0$. High-resolution dispersion measurements can directly test this relationship and extract $c_s$ experimentally.

3. High-Resolution Experimental Measurement of the Dispersion

Claude et al. implemented MHz-linewidth heterodyne-coherent probe spectroscopy combined with DMD-based momentum filtering, achieving ultrahigh resolution ($$\Delta k \approx 5\times 10^{-4}\;\mu\mathrm{m}^{-1}$$, $\Delta E \approx 4$ neV) and enabling precise mapping of both upper (normal) and lower (ghost) branches of the collective excitation spectrum near the turning point. Representative measured data for energies (indexed relative to the pump frequency, in meV) are:

$k$ ($\mu\mathrm{m}^{-1}$)	$E_+(k)$ (meV)	$E_-(k)$ (meV)
0.10	+0.012	-0.012
0.20	+0.032	-0.032
0.40	+0.068	-0.062
0.60	+0.148	-0.132

Fitting the low-$k$ region of $E_+(k)$ to the linear form yields an experimental determination of $c_s$. Systematic measurements as a function of detuning $\delta$ show that the measured speed of sound $c_s^r$ follows: $$c_s^r \simeq \alpha \sqrt{\frac{\hbar \delta}{m_{LP}}}$$ with $m_{LP} = 5.5\times 10^{-35}\;\mathrm{kg}$ and average $\alpha \approx 0.59$. This scaling is consistent with the reservoir-renormalized prediction arising from coupled light–matter–reservoir dynamics.

4. Role of the Dark Exciton Reservoir

The experimentally observed renormalization of spectral features—especially the reduction in $c_s$—requires inclusion of a dark, nonradiative excitonic reservoir of density $n_r$, interacting with bright polaritons via coupling constant $g_r$. The modification manifests by shifting the pump detuning as $\delta \rightarrow \delta - g_r n_r$ in the real energy shift term, while leaving the polariton–polariton interaction $gn$ unchanged. The critical point is given by $\delta = gn + g_r n_r$, and the effective sound speed becomes: $$c_s^r = \alpha\,\sqrt{\frac{\hbar \delta}{m_{LP}}}, \quad g_r n_r = (1 - \alpha^2)\delta$$ Empirically, $g_r n_r \simeq 0.65\,\delta$, i.e., a substantial fraction of the total blueshift comes from the dark reservoir. For $\delta = 0.20$ meV, $gn \simeq 0.12$ meV and $g_r n_r \simeq 0.08$ meV.

5. Ghost Branches and Modulation Instability Precursors

The “ghost” branch—corresponding to the negative-energy solution $E_-(k)$—is a hallmark of collective many-body physics and is directly observed in the spectral response. On the lower branch of the bistability loop (for $gn < \delta/3$), the argument of the square root in the Bogoliubov formula becomes negative over a finite $k$-interval, indicating the parametric onset of dynamical (modulational) instability. Nonetheless, the net imaginary part $\mathrm{Im}[\omega_B]$ retains a negative value due to losses, so the excitation remains formally stable—yet clear spectroscopic precursors emerge:

• Flat, narrow plateaus in transmission or FWM spectra, fixed at $\omega_0$, whose linewidth is sub-$\gamma$, occur where $\mathrm{Im}[\omega_B]\rightarrow 0$.
• Asymmetric response in direct versus four-wave-mixing signals points to spatial density modulation near the instability threshold.

These features denote the departure from the simple Landau superfluid picture and indicate the presence of complex non-equilibrium hydrodynamics, where the dissipative stabilization coexists with local instability precursors.

6. Synthesis and Significance

The Landau polariton dispersion framework, as tested and validated via high-resolution experiments, critically establishes:

• The strict validity of the driven–dissipative Bogoliubov theory at the bistability turning point, with the canonical form $E_\pm(k)=\pm\sqrt{\epsilon_k(\epsilon_k+2gn)}$;
• Direct confirmation that the measured Landau critical velocity coincides with the speed of sound determined from the low-energy slope;
• Quantitative observation of both normal and ghost branches;
• Explicit measurement of the reduction of $c_s$ due to interaction with a dark exciton reservoir, with a consistent estimation of $g_r n_r$ and the scaling factor $\alpha$;
• Spectroscopic evidence for the onset of modulational instability, providing quantitative markers for the breakdown of simple fluid hydrodynamics in driven-dissipative quantum fluids of light.

These developments open avenues for the use of polariton fluids as model platforms for quantum hydrodynamics, the paper of exotic superfluidity in non-equilibrium settings, and the precision investigation of nonlinear light–matter interactions dominated by both bright and dark degrees of freedom.