Driven-Dissipative Landau Polaritons: Two Highly Nonlinearly-Coupled Quantum Harmonic Oscillators

Abstract

Landau levels (LLs) are the massively-degenerate discrete energy spectrum of a charged particle in a transverse magnetic field and lie at the heart of many intriguing phenomena such as the integer and fractional quantum Hall effects as well as quantized vortices. In this Letter, we consider coupling of LLs of a transversely driven, single charge-neutral particle in a synthetic gauge potential to a quantized field of an optical cavity -- a setting reminiscent of superradiant self-ordering setups in quantum gases. We uncover that this complex system can be surprisingly described in terms of two highly nonlinearly-coupled quantum harmonic oscillators, thus enabling a full quantum mechanical treatment. Light-matter coupling mixes the LLs and the superradiant photonic mode, leading to the formation of hybrid states referred to as "Landau polaritons". They inherit partially the degeneracy of the LLs and possess intriguing features such as non-zero light-matter entanglement and quadrature squeezing. Depending on the system parameters and the choice of initial state, the system exhibits diverse nonequilibrium quantum dynamics and multiple steady states, with distinct physical properties. This work lays the foundation for further investigating the novel, driven-dissipative Landau-polariton physics in quantum-gas--cavity-QED settings.

Figures (7)

• Figure 1: The lowest four energy bands $E_j$ as a function of the light-matter coupling $\eta$ for two different values of $k_cl_B=2$ (a) and $\sqrt{5}$ (b). In the zero-coupling limit in panel (a), the uncoupled LLs and the photon Fock states $\left| \Phi_{\ell,x_0} \right>\otimes\left| n \right>$ are indicated by a short form as $\left| \ell,n \right>$ with the $x_0$ degeneracy implied. The color map of the energy bands displays the average photon $\langle \hat{a}^\dag\hat{a}\rangle$ content of the corresponding states. The insets show the von Neumann entropy $S_{\rm vN}$ of the reduced system as a function of $\eta$ for the lowest band (composed of eight sub-bands). Absolute value of the overlap $|c_{\ell,x_0,n}^{(j)}|$ of the Landau polaritons $\left| \Psi_j \right>$ with the uncoupled states for $k_cl_B = \sqrt{5}$ and two different coupling strengths $\eta/\omega=0.1$ (c) and $1$ (d). The order of the uncoupled states is indicated in panel (c). For all the plots, $\Delta_c=-0.8\omega$ and $x_0=\pi l_B\{-1,-3/4,-1/2,\cdots,3/4\}$.
• Figure 2: Variance $\{\sigma^2_{{\mathcal{Q}}_{O}}, \sigma^2_{{\mathcal{P}}_{O}} \}$ of the atomic (a) and photonic (b) quadrature operators in the lowest band (consisting of eight sub-bands) as a function of the atom-photon coupling strength $\eta$ for $k_cl_B=\sqrt{5}$. While for the Landau oscillator the position quadrature is squeezed, $\sigma^2_{{\mathcal{Q}}_{b}}<1/4$ (i.e., below the dotted line), for the photonic oscillator the momentum quadrature is squeezed, $\sigma^2_{{\mathcal{P}}_{a}}<1/4$.
• Figure 3: The nonequilibrium dynamics of the system for $\eta=3\omega$, $\kappa=\omega$, $k_cl_B=1$, and three different values of $x_0/l_B=-\pi/2$ (a), $-3\pi/4$ (b), and $-\pi$ (c). The column (i) shows the quantum (solid blue) and semi-classical (dashed red) dynamics of the system in the phase space of the average photon vs. average LL-occupation number. The filled (crossed) dark-red circles represent the stable (unstable) semi-classical fixed points, while the filled dark-blue circles are the quantum steady states. The black crosses are the chosen initial product coherent states: $\left| 1.6 \right>\otimes\left| 1.6-0.15i \right>$ (a), $\left| 1.6+0.05i \right>\otimes\left| 1+0.2i \right>$ (b), and $\left| 0.1+0.05i \right>\otimes\left| 1.75-0.25i \right>$ (c). The column (ii) shows the lowest order correlation $\mathcal{C}_{ab}$ between the two oscillators as well as the statistical mixing $\text{Tr}(\hat{\rho}^2)$. The steady-state atomic and photonic distributions are depicted in the columns (iii) and (iv), respectively, with the Poisson $P_{\rm P}$ and thermal $P_{\rm th}$ distributions represented as references. The corresponding steady-state atomic and photonic $Q$ functions are shown in the insets, with the filled red (crossed white) circles representing the semi-classical stable (unstable) steady states.
• Figure A1: Schematic sketch of the system. A single atom inside an optical cavity is box trapped in the $x$-$y$ plane and pierced by a transverse external synthetic magnetic field $\boldsymbol{\mathcal{B}}={\mathcal{B}}\mathbf{e}_z$ along the $z$ direction. The atom is coupled to a single off-resonant longitudinal mode of the cavity with the strength $\mathcal{G}(\hat{x})=\mathcal{G}_0\cos(k_c \hat{x})$, and is further driven in the transverse $z$ direction by an off-resonance standing-wave laser with the amplitude $\Omega_0$. The laser is closely red-detuned from the cavity resonance.
• Figure A2: Comparison of the energy spectrum $E_j$ calculated from the Hamiltonians \ref{['eq:2D_L_H']}, \ref{['eq:1D_L_H']}, and \ref{['eq:coupled_QHO']} at the pump strength $\eta=\omega$ for two different values of $k_cl_B=2$ (a) and $\sqrt{5}\approx2.24$ (b), corresponding to cuts from Fig. \ref{['fig:energy_spectra']} in the main text. In the two-dimensional case of Eq. \ref{['eq:2D_L_H']}, a system of size $20l_B\times8l_B$ with 60 (8) grid points along the $x$ ($y$) direction is chosen. The system size along the $y$ direction (i.e., $8l_B$) and its discretization ensure that for the periodic boundary condition they yield $x_0=\hbar k_y/\mathcal{B}=\pi l_B\{-1,-3/4,-1/2,\cdots,3/4\}$ as used in Fig. \ref{['fig:energy_spectra']}. In the one-dimensional case of Eq. \ref{['eq:1D_L_H']}, 100 grid points are chosen for the same system size along the $x$ direction, $20l_B$. The photon cut-off is set to 15, 20, and 35, respectively, for diagonalizing the Hamiltonians \ref{['eq:2D_L_H']}, \ref{['eq:1D_L_H']}, and \ref{['eq:coupled_QHO']}; we have checked the validity of these cut-offs a posteriori. The cut-off for the atomic oscillator is set to 20 in the coupled harmonic oscillator model of Eq. \ref{['eq:coupled_QHO']}, as only a few LLLs are involved for the number of bands and parameters shown. The other parameters are the same as Fig. \ref{['fig:energy_spectra']}.