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The Dicke Quantum Phase Transition with a Superfluid Gas in an Optical Cavity

Kristian Baumann, Christine Guerlin, Ferdinand Brennecke, Tilman Esslinger

arXiv:0912.3261v3quant-phcond-mat.quant-gas

TL;DR

The paper addresses whether the Dicke quantum phase transition can be realized and observed in a controlled quantum gas with long-range interactions. It couples a Bose-Einstein condensate to a driven optical cavity and maps the resulting transition to the Dicke Hamiltonian. The experiment observes self-organization into a phase identified as a supersolid, with a phase boundary quantitatively agreeing with the Dicke model.

  • Problem

    Quantum gases need controlled realizations linking collective matter-light interactions, quantum phase transitions, and long-range interactions.

  • Method

    The authors drive a Bose-Einstein condensate inside an ultrahigh-finesse cavity and use two-photon pump-cavity processes to implement an effective Dicke Hamiltonian.

  • Results

    The experiment realizes a second-order dynamical Dicke transition with spontaneous Ising-type symmetry breaking, self-organization, and a phase boundary in quantitative agreement with mean-field theory.

  • Takeaways & Limitations

    The organized phase combines density modulation and phase coherence and can therefore be regarded as a supersolid with cavity-mediated long-range interactions.

  • Takeaways & Limitations

    The realization is dynamical because external driving and cavity loss make it an open-system version of the Dicke transition.

Abstract

from arXiv · show

A phase transition describes the sudden change of state in a physical system, such as the transition between a fluid and a solid. Quantum gases provide the opportunity to establish a direct link between experiment and generic models which capture the underlying physics. A fundamental concept to describe the collective matter-light interaction is the Dicke model which has been predicted to show an intriguing quantum phase transition. Here we realize the Dicke quantum phase transition in an open system formed by a Bose-Einstein condensate coupled to an optical cavity, and observe the emergence of a self-organized supersolid phase. The phase transition is driven by infinitely long-ranged interactions between the condensed atoms. These are induced by two-photon processes involving the cavity mode and a pump field. We show that the phase transition is described by the Dicke Hamiltonian, including counter-rotating coupling terms, and that the supersolid phase is associated with a spontaneously broken spatial symmetry. The boundary of the phase transition is mapped out in quantitative agreement with the Dicke model. The work opens the field of quantum gases with long-ranged interactions, and provides access to novel quantum phases.

INTRODUCTION

The work targets quantum phase transitions and long-range interacting quantum gases by realizing the Dicke transition with a Bose-Einstein condensate in an optical cavity. It observes self-organization into a supersolid phase with spontaneously broken spatial symmetry.

  • INTRODUCTION: Optical cavities provide a route to infinitely long-range atom-atom forces mediated by the cavity field.This approach complements proposed routes based on dipolar interactions.
  • INTRODUCTION: The Dicke model describes infinitely coordinated spins coupled to one electromagnetic mode and predicts a transition to a superradiant phase above sufficient coupling.The model provides a generic framework for connecting collective matter-light interactions with quantum phase transitions.
  • INTRODUCTION: The experiment realizes the Dicke quantum phase transition in an open cavity system and observes self-organization of a Bose-Einstein condensate.The two-level system is represented by two momentum states coupled through the cavity field.
  • INTRODUCTION: The transition spontaneously breaks a spatial symmetry of the pump-cavity lattice and produces macroscopic occupation of higher-order momentum and cavity modes.The resulting density wave and off-diagonal long-range order identify the organized phase as a supersolid.

THEORETICAL DESCRIPTION AND THE DICKE MODEL

The cavity-pumped condensate is modeled through dynamically generated optical potentials and a two-mode atomic description. Its light-scattering interaction is equivalent to the Dicke Hamiltonian, while positive feedback drives self-organization above a critical coupling.

  • THEORETICAL DESCRIPTION AND THE DICKE MODEL: Adiabatic elimination of the excited atomic state yields kinetic, pump-lattice, pump-cavity scattering, and cavity-field contributions to the effective Hamiltonian.The elimination is justified for large pump-atom detuning, and the pump creates a standing-wave potential along z.
  • THEORETICAL DESCRIPTION AND THE DICKE MODEL: Above a critical two-photon Rabi frequency, positive feedback attracts atoms toward one checkerboard sublattice and increases cavity scattering.The process reaches a steady state when potential-energy gain balances kinetic and collisional energy costs.
  • THEORETICAL DESCRIPTION AND THE DICKE MODEL: A two-mode atomic description maps the zero-momentum state to a symmetric higher-momentum state through two balanced Raman channels.The channels include pump absorption followed by cavity emission and the reverse cavity-absorption pathway, together with reverse processes.
  • THEORETICAL DESCRIPTION AND THE DICKE MODEL: The resulting interaction is exactly the Dicke interaction, including counter-rotating terms, with a transition at λcr = √ω0ω/2.At self-organization, both the cavity field and atomic polarization acquire macroscopic occupations.
  • THEORETICAL DESCRIPTION AND THE DICKE MODEL: The driven cavity realization is dynamical because external pumping and cavity loss prevent it from being the original closed Dicke transition.Cavity output nevertheless enables in-situ monitoring of the transition and extraction of system properties.

EXPERIMENTAL DESCRIPTION

The experiment places nearly pure rubidium Bose-Einstein condensates inside an ultrahigh-finesse cavity and drives them transversely with a red-detuned standing-wave laser. Cavity output and atomic expansion imaging monitor the light and momentum distributions.

  • EXPERIMENTAL DESCRIPTION: The setup uses typically 10^5 87Rb atoms in a crossed-beam dipole trap centered inside an ultrahigh-finesse Fabry-Perot cavity.Atoms occupy the |F, mF⟩ = |1, −1⟩ hyperfine ground state.
  • EXPERIMENTAL DESCRIPTION: A linearly polarized standing-wave pump beam perpendicular to the cavity axis is red-detuned by 4.3 nm from the atomic D2 line.The pump-atom detuning exceeds the atomic linewidth by more than five orders of magnitude, supporting a coherent-scattering description.
  • EXPERIMENTAL DESCRIPTION: The cavity operates in the strong dispersive-coupling regime, with the maximum collective dispersive shift equal to 6.5 times the cavity decay rate κ = 2π × 1.3 MHz.The shift is NU0 for the full atomic ensemble.
  • EXPERIMENTAL DESCRIPTION: Calibrated photon counting measures intracavity light intensity, while absorption imaging after ballistic expansion infers the atomic momentum distribution.The two measurements provide complementary optical and motional signatures of self-organization.

OBSERVING THE PHASE TRANSITION

Increasing pump power triggers an abrupt cavity-field buildup and momentum redistribution, revealing self-organization into a phase-coherent checkerboard state. The organized phase can evolve into a normal crystalline phase at high lattice depth, while coherence returns as the lattice is lowered.

  • Onset of self-organization: At a critical pump power, the intracavity photon number rises abruptly and momentum components appear at (px, pz) = (±ℏk, ±ℏk).The new components directly indicate density modulation on one checkerboard sublattice.
  • Supersolid character: The self-organized gas has checkerboard density order while retaining off-diagonal long-range order, consistent with a supersolid.The coherence length extends over almost the full atomic ensemble.
  • Steady organized phase: After crossing the transition, the system reaches a steady organized state, with photon-number decay attributed to atom loss and residual heating.The reported spontaneous-scattering rate is Γsc = 3.7 /s, and the sequence produces 30% overall atom loss.
  • Evolution with lattice depth: A checkerboard lattice depth of 22 Er confines atoms into tubes, where suppressed tunnelling causes tube dephasing and reduced interference contrast.The corresponding single-site trapping frequencies are 19 kHz and 30 kHz along x and z.
  • Evolution with lattice depth: When the lattice depth is reduced, phase coherence between tubes is restored, and turning off the pump retrieves an almost pure BEC.The coherence recovery is observed as the intracavity photon number and lattice depth decrease.

MAPPING OUT THE PHASE DIAGRAM

The phase boundary is mapped by varying pump-cavity detuning and pump power, revealing a sharp transition that agrees quantitatively with a mean-field calculation. Cavity-resonance shifts can additionally interrupt the feedback and produce oscillations between organized and non-organized states.

  • Phase boundary: A sharp phase boundary appears across a wide range of pump-cavity detunings and agrees very well with the theoretical mean-field model.The measurement uses two-dimensional intracavity photon-number traces while increasing pump power for different detunings.
  • Dicke-model scaling: For large negative detuning, the critical pump power scales linearly with the effective cavity frequency ω = −∆c + U0B0, as expected from the Dicke model.The critical coupling has no real solution for ω < 0.
  • Dicke-model scaling: Approaching the dispersively shifted cavity resonance from below increases scattering into the cavity and the intracavity photon number.Above the shifted resonance at U0B0 = −2π × 3.5 MHz, almost no light scattering is observed.
  • Beyond the Dicke model: Checkerboard ordering changes the atom-cavity spatial overlap and dynamically shifts the cavity resonance beyond the basic Dicke model.This shift produces a frustrated regime for U0N > ∆c > U0B0.
  • Beyond the Dicke model: When self-organization brings the atom-cavity system into resonance with the pump, feedback is interrupted and the system oscillates between organized and non-organized phases.The oscillatory behavior is observed in intracavity photon-number traces.

CONCLUSIONS AND OUTLOOK

The experiment realizes a second-order dynamical quantum phase transition in a driven condensate-cavity system and links self-organization to the open-system Dicke transition. It identifies phase-coherent crystalline regimes while exposing the system's distance from the Hamiltonian-dominated limit.

  • Conclusions: The driven condensate-cavity system realizes a second-order dynamical quantum phase transition with spontaneous Ising-type symmetry breaking.The transition is accompanied by self-organization of the superfluid atoms.
  • Conclusions: The organized light-atom crystal can retain phase coherence and therefore qualify as a supersolid.The process is shown to be equivalent to the Dicke quantum phase transition in an open system.
  • Outlook: For a very cold classical gas, the corresponding phase boundary is predicted to scale with temperature and be driven by classical rather than quantum density fluctuations.This prediction contrasts with the fluctuation mechanism described for the presented quantum-gas experiment.
  • Outlook: The collective critical interaction reaches only λcr/κ = 0.2, leaving the experiments below the regime where Hamiltonian dynamics dominates cavity losses.The paper presents stronger interaction relative to loss as an outlook for studying critical Dicke-model dynamics.

EXPERIMENTAL DETAILS

The experiment uses nearly pure ^87Rb condensates positioned inside a high-finesse cavity and transversely pumped for mode matching. Cavity output is polarization-resolved and deliberately attenuated to extend the photon-detection dynamic range.

  • Condensate and cavity: Nearly pure ^87Rb BECs are prepared in a crossed-beam dipole trap with frequencies 2π × (252, 48, 238) Hz and radii (3.2, 16.6, 3.3) µm.The cavity TEM00 mode has a 25 µm waist, and the cavity finesse is 3.4 × 10^5.
  • Pump configuration: A weak stabilization lattice contributes less than 0.35 Er of lattice depth.The stabilization beam is referenced to the transverse pump laser.
  • Pump configuration: The pump beam has waist radii (29, 53) µm, wavelength 784.5 nm, and polarization optimized for cavity scattering.Pump powers reported in the text refer to the incoming beam, with a 20% systematic uncertainty in the atomic intensity.
  • Photon detection: Cavity leakage is monitored with two single-photon counting modules, one for each circular polarization.The nominal detection efficiency is about 5%, but it is reduced tenfold in these experiments to avoid detector saturation.

MAPPING TO THE DICKE HAMILTONIAN

The two-mode description maps the BEC–cavity system onto the Dicke Hamiltonian, with self-organization corresponding to the normal-to-superradiant transition and breaking parity symmetry.

  • MAPPING TO THE DICKE HAMILTONIAN: Self-organization is equivalent to a dynamical normal-to-superradiant quantum phase transition in the transversely pumped BEC–cavity system.The mapping uses the two-mode expansion and connects the onset of organization to the Dicke transition.
  • MAPPING TO THE DICKE HAMILTONIAN: The atomic field is reduced to two modes: the non-organized condensate state ψ0 and a momentum-carrying state ψ1 coupled by photon scattering.The second mode contains additional ±ℏk momentum components along the cavity and pump directions.
  • MAPPING TO THE DICKE HAMILTONIAN: Expanding the field operator in these modes introduces bosonic operators c0 and c1 and collective spin operators, identifying the atomic modes with two-level systems.The total atom number is N = c0†c0 + c1†c1.
  • MAPPING TO THE DICKE HAMILTONIAN: Apart from the dispersive term, the resulting Hamiltonian is the Dicke Hamiltonian for N two-level systems coupled to a bosonic cavity mode.The transition frequency is ω0 = 2ωr, while the cavity-mode frequency is ω = −∆c + NU0/2.
  • MAPPING TO THE DICKE HAMILTONIAN: The transition spontaneously breaks Dicke-model parity, producing atomic order on either even or odd checkerboard sites and two cavity–pump relative phases separated by π.The two possible signs of the order parameter correspond to the two spatial arrangements.

A MEAN-FIELD DESCRIPTION

A mean-field stability analysis incorporates experimental confinement, cavity and pump profiles, and collisional interactions to derive the self-organization threshold and connect it quantitatively to Dicke-model coupling.

  • A MEAN-FIELD DESCRIPTION: The quantitative threshold is derived by stability analysis of the coupled BEC–cavity system using a generalized Gross–Pitaevskii equation.The analysis includes the external trap, finite cavity and pump transverse sizes, and atom–atom collisions.
  • A MEAN-FIELD DESCRIPTION: The critical pump strength ηcr is obtained by linearizing around the non-organized state with a two-mode ansatz and identifying a dynamical instability.The ansatz is ψ = ψ0(1 + ϵφcφp) with ϵ ≪ 1.
  • A MEAN-FIELD DESCRIPTION: The threshold uses an effective number of maximally scattering atoms, a dispersively shifted cavity detuning, and an interaction energy that accounts for the mean-field dispersion shift.These quantities are defined through overlaps with the non-organized condensate and the cavity and pump mode profiles.
  • A MEAN-FIELD DESCRIPTION: Identifying ω = −˜∆c, ω0 = 2ωr + 4Eint/ℏ, and λcr = ηcr/√Neff makes the stability result agree with the Dicke-model critical coupling including cavity decay.This provides the quantitative connection between the mean-field threshold and the Dicke prediction.
  • A MEAN-FIELD DESCRIPTION: The plotted phase boundary is calculated from the threshold equation using a Thomas–Fermi approximation for the condensate wave function in the crossed-beam dipole trap.The approximation is used to obtain the dashed phase-boundary curve in Fig. 5a.
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