Dicke quantum phase transition with a superfluid gas in an optical cavity

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A phase transition describes the sudden change of state of a physical system, such as melting or freezing. Quantum gases provide the opportunity to establish a direct link between experiments and generic models that capture the underlying physics. The Dicke model describes a collective matter–light interaction and 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-range interactions between the condensed atoms, 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. Our results should facilitate studies of quantum gases with long-range interactions and provide access to novel quantum phases.

At a glance


  1. Concept of the experiment.
    Figure 1: Concept of the experiment.

    A BEC is placed inside an optical cavity and driven using a standing-wave pump laser oriented along the z axis (vertically in the figure). The frequency of the pump laser is far red-detuned with respect to the atomic transition line but close detuned from a particular cavity mode. Correspondingly, the atoms coherently scatter pump light into the cavity mode with a phase depending on their position within the combined pump–cavity mode profile. a, For a homogeneous atomic density distribution along the cavity axis, the build-up of a coherent cavity field is suppressed as a result of destructive interference of the individual scatterers. SPCM, single-photon counting module. b, Above a critical pump power, Pcr, the atoms self-organize onto either the even or odd sites of a chequerboard pattern, thereby maximizing cooperative scattering into the cavity. This dynamical quantum phase transition is triggered by quantum fluctuations in the condensate density. It is accompanied by spontaneous symmetry breaking both in the atomic density and in the relative phase between the pump and cavity fields. c, Geometry of the chequerboard pattern. The intensity maxima of the pump and cavity fields are depicted by the horizontal and vertical lines, respectively; λp, pump wavelength.

  2. Analogy to the Dicke model.
    Figure 2: Analogy to the Dicke model.

    In an atomic two-mode picture, the pumped BEC–cavity system is equivalent to the Dicke model including counter-rotating interaction terms. a, Light scattering between the pump field and the cavity mode induces two balanced Raman channels between |px,pzright fence = |0,0right fence, the atomic zero-momentum state, and |± k,± kright fence, the symmetric superposition of states with an additional unit of photon momentum along the x and z directions. Primes indicate electronically excited momentum states. b, The two excitation paths (dashed and solid) corresponding to the two Raman channels are illustrated in a momentum diagram.

  3. Observation of the phase transition.
    Figure 3: Observation of the phase transition.

    a–d, The pump power (dashed) is gradually increased while the mean intracavity photon number (solid; 20-μs bins) is monitored. After the sudden release of the atomic cloud and its subsequent ballistic expansion for 6ms, absorption images (clipped equally in atomic density) are made for pump powers corresponding to lattice depths of 2.6Er (b), 7.0Er (c) and 8.8Er (d). Self-organization is manifested by an abrupt build-up of the cavity field accompanied by the formation of momentum components at (px,pz) = (± k,± k) (d). The weak momentum components at (0,±2 k) (c) result from loading the atoms into the one-dimensional standing-wave potential of the pump laser. The pump–cavity detuning was Δc = -2π×14.9(2)MHz and the atom number was N = 1.5(3)×105 (parentheses show uncertainty in last digit).

  4. Steady state in the self-organized phase.
    Figure 4: Steady state in the self-organized phase.

    a, Pump power sequence (dashed) and recorded mean intracavity photon number (solid; 20-μs bins). After crossing the transition point, at 9ms, the system reaches a steady state within the self-organized phase. The slow decrease in photon number is due to atom loss (see text). The short-time fluctuations are due to detection shot noise. b–d, Absorption images are made at different times in the self-organized phase: after 3ms (b), after 7ms (c) and after returning the pump power to zero (d). The pump–cavity detuning was Δc = -2π×6.3(2)MHz and the atom number was N = 0.7(1)×105.

  5. Phase diagram.
    Figure 5: Phase diagram.

    a, The pump power is increased to 1.3mW over 10ms for different values of the pump-cavity detuning, Δc. The recorded mean intracavity photon number, , is displayed (colour scale) as a function of pump power (and corresponding pump lattice depth) and pump–cavity detuning, Δc. A sharp phase boundary is observed over a wide range of Δc values; this boundary is in very good agreement with a theoretical mean-field model (dashed curve). The dispersively shifted cavity resonance for the non-organized atom cloud is marked by the arrow on the vertical axis. b, c, Typical traces showing the intracavity photon number for different pump–cavity detunings: Δc = -2π×23.0(2)MHz, 20-μs bins (b); Δc = -2π×4.0(2)MHz, 10-μs bins (c). The atom number was N = 1.0(2)×105. In the detuning range -2π×7MHzΔc-2π×21MHz, the pump power ramp was interrupted at 540μW. Therefore, no photon data was taken in the area of a under the insets.


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  1. Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland

    • Kristian Baumann,
    • Christine Guerlin,
    • Ferdinand Brennecke &
    • Tilman Esslinger
  2. Present address: Thales Research and Technology, Campus Polytechnique, 1 Avenue Augustin Fresnel, F-91767 Palaiseau, France.

    • Christine Guerlin


The data was taken and analysed by K.B. and C.G. The theoretical analysis was mainly performed by F.B. and T.E. The relation to the Dicke model was realized by F.B. The experimental concept was developed by T.E. All authors contributed extensively to the discussion of the results as well as to the preparation of the manuscript.

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