Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Collective excitations and supersolid behavior of bosonic atoms inside two crossed optical cavities

Johannes Lang, Francesco Piazza, Zwerge, W.

Published Dec 12, 2017• 21 citations
OpenAlex

Abstract

Wediscuss the nature of symmetry breaking and the associated collective excitations for a system of bosons coupled to the electromagnetic field of two optical cavities. For the specific configuration realized in a recent experiment at ETH [1, 2], we show that, in absence of direct intercavity scattering and for parameters chosen such that the atoms couple symmetrically to both cavities, the system possesses an approximate U(1) symmetry which holds asymptotically for vanishing cavity field intensity. It corresponds to the invariance with respect to redistributing the total intensity I = I-1 + I-2 between the two cavities. The spontaneous breaking of this symmetry gives rise to a broken continuous translation-invariance for the atoms, creating a supersolid-like order in the presence of a Bose-Einstein condensate. In particular, we show that atom-mediated scattering between the two cavities, which favors the state with equal light intensities I-1 = I-2 and reduces the symmetry to Z(2) circle times Z(2), gives rise to a finite value similar to root I of the effective Goldstone mass. For strong atom driving, this low energy mode is clearly separated from an effective Higgs excitation associated with changes of the total intensity I. In addition, we compute the spectral distribution of the cavity light field and show that both the Higgs and Goldstone mode acquire a finite lifetime due to Landau damping at non-zero temperature.

Related papers

Collective excitations and supersolid behavior of bosonic atoms inside two crossed optical cavities

J Lang, F Piazza, W. Zwerger

We discuss the nature of symmetry breaking and the associated collective excitations for a system of bosons coupled to the electromagnetic field of two optical cavities. For the specific configuration realized in a recent experiment at ETH [ 1 , 2 ], we show that, in absence of direct intercavity scattering and for parameters chosen such that the atoms couple symmetrically to both cavities, the system possesses an approximate U (1) symmetry which holds asymptotically for vanishing cavity field intensity. It corresponds to the invariance with respect to redistributing the total intensity between the two cavities. The spontaneous breaking of this symmetry gives rise to a broken continuous translation-invariance for the atoms, creating a supersolid-like order in the presence of a Bose–Einstein condensate. In particular, we show that atom-mediated scattering between the two cavities, which favors the state with equal light intensities and reduces the symmetry to , gives rise to a finite value of the effective Goldstone mass. For strong atom driving, this low energy mode is clearly separated from an effective Higgs excitation associated with changes of the total intensity I . In addition, we compute the spectral distribution of the cavity light field and show that both the Higgs and Goldstone mode acquire a finite lifetime due to Landau damping at non-zero temperature.

arXiv:1707.00017•New Journal of Physics•Nov 17, 2017•27 citations

Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas

Julian Léonard, Andrea Morales, Philip Zupancic +2 more

Higgs and Goldstone modes are collective excitations of the amplitude and phase of an order parameter that is related to the breaking of a continuous symmetry. We directly studied these modes in a supersolid quantum gas created by coupling a Bose-Einstein condensate to two optical cavities, whose field amplitudes form the real and imaginary parts of a U(1)-symmetric order parameter. Monitoring the cavity fields in real time allowed us to observe the dynamics of the associated Higgs and Goldstone modes and revealed their amplitude and phase nature. We used a spectroscopic method to measure their frequencies, and we gave a tunable mass to the Goldstone mode by exploring the crossover between continuous and discrete symmetry. Our experiments link spectroscopic measurements to the theoretical concept of Higgs and Goldstone modes.

arXiv:1704.05803•Science•Dec 14, 2017•203 citations

Experimental Realization of The Dicke Quantum Phase Transition

Kristian Baumann

We report on the first experimental realization of the Dicke quantum phase transition realized in the quantum motion of a Bose–Einstein condensate coupled to an optical cavity. Conceptually, the transition is driven by cavity-mediated long-range interactions, giving rise to the emergence of a self-organized supersolid phase. The Dicke phase transition, predicted in 1973, has not been demonstrated experimentally before this work, both due to fundamental and technological reasons. These challenges have been overcome in the present thesis by employing atomic momentum states of a Bose-Einstein condensate, which are coupled via two-photon Raman transitions involving a cavity photon and a free-space pump photon. This open-system implementation of the Dicke model allows to tune all relevant parameters and offers a unique detection scheme to monitor the manybody system in real time. We demonstrate that the phase transition is accompanied by a macroscopically occupied cavity field and a striking change in the atomic momentum distribution, due to spontaneous self-organization of the atomic density on a checkerboard lattice. The boundary of the transition is mapped out by scanning two parameters of the Dicke model, to reveal a phase diagram in close agreement with the model description. Two different ordered configurations are allowed in the superradiant phase, giving rise to the concept of spontaneous symmetry breaking at the phase transition. We experimentally distinguish the symmetry-broken states and study the origin of the symmetry-breaking process. The finite spatial extension of our system induces a small symmetry-breaking field which changes randomly on each experimental realization. The influence of this field is studied and shown to diminish upon dynamically crossing the transition point with increasing transition rates.

•Jan 1, 2011•2 citations

Emergence of continuous rotational symmetries in ultracold atoms coupled to optical cavities

E. I. Rodríguez Chiacchio, Andreas Nunnenkamp

We investigate the physics of a gas of ultracold atoms coupled to three single-mode optical cavities and transversely pumped with a laser. Recent work has demonstrated that, for two cavities, the ${\mathbb{Z}}_{2}$ symmetries of each cavity can be combined into a global U(1) symmetry [J. L\'eonard et al., Nature (London) 543, 87 (2017)]. Here we show that when adding an extra cavity mode, the low-energy description of this system can additionally exhibit an SO(3) rotational symmetry which can be spontaneously broken. This leads to a superradiant phase transition in all the cavities simultaneously, and the appearance of Goldstone and amplitude modes in the excitation spectrum. We determine the phase diagram of the system, which shows the emergence and breaking of the continuous symmetries and displays first- and second-order phase transitions. We also obtain the excitation spectrum for each phase and discuss the atomic self-organized structures that emerge in the different superradiant phases. We argue that coupling the atoms equally to $n$ different modes will in general generate a global $\text{SO}(n)$ symmetry if the mode frequencies can be tuned to the same value.

arXiv:1805.04392•Physical Review A•Aug 15, 2018•13 citations

Driven-Dissipative Supersolid in a Ring Cavity

Farokh Mivehvar, Stefan Ostermann, Francesco Piazza +1 more

Supersolids are characterized by the counterintuitive coexistence of superfluid and crystalline order. Here we study a supersolid phase emerging in the steady state of a driven-dissipative system. We consider a transversely pumped Bose-Einstein condensate trapped along the axis of a ring cavity and coherently coupled to a pair of degenerate counterpropagating cavity modes. Above a threshold pump strength the interference of photons scattered into the two cavity modes results in an emergent superradiant lattice, which spontaneously breaks the continuous translational symmetry towards a periodic atomic pattern. The crystalline steady state inherits the superfluidity of the Bose-Einstein condensate, thus exhibiting genuine properties of a supersolid. A gapless collective Goldstone mode correspondingly appears in the superradiant phase, which can be nondestructively monitored via the relative phase of the two cavity modes on the cavity output. Despite cavity-photon losses the Goldstone mode remains undamped, indicating the robustness of the supersolid phase.

arXiv:1801.00756•Physical Review Letters•Mar 23, 2018•57 citations

Controllable interatomic interaction mediated by diffractive coupling in a cavity

Ivor Krešić

Photon-mediated interaction can be used for simulating complex many-body phenomena with ultracold atoms coupled to electromagnetic modes of an optical resonator. We study theoretically a method of producing controllable interatomic interaction mediated by forward-diffracted photons circulating inside a ring cavity. One example of such a system is the three-mode cavity, where an on-axis mode can coexist with two diffracted sidebands. We demonstrate how the self-organized stripe states of a Bose-Einstein condensate (BEC) occurring in this cavity geometry can exhibit supersolid properties, due to spontaneous breaking of the Hamiltonian's continuous translational symmetry. A numerical study of the collective excitation spectrum of these states demonstrates the existence of massless and finite-gap excitations, which are identified as phase (Goldstone) and amplitude (Higgs) atomic density modes. We further demonstrate how judicious Fourier filtering of intracavity light can be used to engineer the effective atom-atom interaction profile for many cavity modes. The numerical results in this configuration show the existence of droplet-array and single-droplet BEC states for commensurate and incommensurate cavity modes, respectively. Diffractive coupling in a cavity is thereby introduced as an alternative route towards tailoring the photon-mediated interaction of ultracold atoms. Spatial features of the self-organized optical potentials can here be tuned to scales several times larger than the pump laser wavelength such that the corresponding atomic density distributions could be imaged and manipulated using low-numerical-aperture optics. These calculations and insights pave the way towards quantum simulation of exotic nonequilibrium many-body physics with condensates in a cavity.

arXiv:2407.10690•Physical Review A•Aug 2, 2024•0 citations
Wediscuss the nature of symmetry breaking and the associated collective excitations for a system of bosons coupled to the electromagnetic field of two optical cavities. For the specific configuration realized in a recent experiment at ETH [1, 2], we show that, in absence of direct intercavity scattering and for parameters chosen such that the atoms couple symmetrically to both cavities, the system possesses an approximate U(1) symmetry which holds asymptotically for vanishing cavity field intensity. It corresponds to the invariance with respect to redistributing the total intensity I = I-1 + I-2 between the two cavities. The spontaneous breaking of this symmetry gives rise to a broken continuous translation-invariance for the atoms, creating a supersolid-like order in the presence of a Bose-Einstein condensate. In particular, we show that atom-mediated scattering between the two cavities, which favors the state with equal light intensities I-1 = I-2 and reduces the symmetry to Z(2) circle times Z(2), gives rise to a finite value similar to root I of the effective Goldstone mass. For strong atom driving, this low energy mode is clearly separated from an effective Higgs excitation associated with changes of the total intensity I. In addition, we compute the spectral distribution of the cavity light field and show that both the Higgs and Goldstone mode acquire a finite lifetime due to Landau damping at non-zero temperature.
We discuss the nature of symmetry breaking and the associated collective excitations for a system of bosons coupled to the electromagnetic field of two optical cavities. For the specific configuration realized in a recent experiment at ETH [ 1 , 2 ], we show that, in absence of direct intercavity scattering and for parameters chosen such that the atoms couple symmetrically to both cavities, the system possesses an approximate U (1) symmetry which holds asymptotically for vanishing cavity field intensity. It corresponds to the invariance with respect to redistributing the total intensity between the two cavities. The spontaneous breaking of this symmetry gives rise to a broken continuous translation-invariance for the atoms, creating a supersolid-like order in the presence of a Bose–Einstein condensate. In particular, we show that atom-mediated scattering between the two cavities, which favors the state with equal light intensities and reduces the symmetry to , gives rise to a finite value of the effective Goldstone mass. For strong atom driving, this low energy mode is clearly separated from an effective Higgs excitation associated with changes of the total intensity I . In addition, we compute the spectral distribution of the cavity light field and show that both the Higgs and Goldstone mode acquire a finite lifetime due to Landau damping at non-zero temperature.
Higgs and Goldstone modes are collective excitations of the amplitude and phase of an order parameter that is related to the breaking of a continuous symmetry. We directly studied these modes in a supersolid quantum gas created by coupling a Bose-Einstein condensate to two optical cavities, whose field amplitudes form the real and imaginary parts of a U(1)-symmetric order parameter. Monitoring the cavity fields in real time allowed us to observe the dynamics of the associated Higgs and Goldstone modes and revealed their amplitude and phase nature. We used a spectroscopic method to measure their frequencies, and we gave a tunable mass to the Goldstone mode by exploring the crossover between continuous and discrete symmetry. Our experiments link spectroscopic measurements to the theoretical concept of Higgs and Goldstone modes.
We report on the first experimental realization of the Dicke quantum phase transition realized in the quantum motion of a Bose–Einstein condensate coupled to an optical cavity. Conceptually, the transition is driven by cavity-mediated long-range interactions, giving rise to the emergence of a self-organized supersolid phase. The Dicke phase transition, predicted in 1973, has not been demonstrated experimentally before this work, both due to fundamental and technological reasons. These challenges have been overcome in the present thesis by employing atomic momentum states of a Bose-Einstein condensate, which are coupled via two-photon Raman transitions involving a cavity photon and a free-space pump photon. This open-system implementation of the Dicke model allows to tune all relevant parameters and offers a unique detection scheme to monitor the manybody system in real time. We demonstrate that the phase transition is accompanied by a macroscopically occupied cavity field and a striking change in the atomic momentum distribution, due to spontaneous self-organization of the atomic density on a checkerboard lattice. The boundary of the transition is mapped out by scanning two parameters of the Dicke model, to reveal a phase diagram in close agreement with the model description. Two different ordered configurations are allowed in the superradiant phase, giving rise to the concept of spontaneous symmetry breaking at the phase transition. We experimentally distinguish the symmetry-broken states and study the origin of the symmetry-breaking process. The finite spatial extension of our system induces a small symmetry-breaking field which changes randomly on each experimental realization. The influence of this field is studied and shown to diminish upon dynamically crossing the transition point with increasing transition rates.
We investigate the physics of a gas of ultracold atoms coupled to three single-mode optical cavities and transversely pumped with a laser. Recent work has demonstrated that, for two cavities, the Z2{\mathbb{Z}}_{2}Z2​ symmetries of each cavity can be combined into a global U(1) symmetry [J. L\'eonard et al., Nature (London) 543, 87 (2017)]. Here we show that when adding an extra cavity mode, the low-energy description of this system can additionally exhibit an SO(3) rotational symmetry which can be spontaneously broken. This leads to a superradiant phase transition in all the cavities simultaneously, and the appearance of Goldstone and amplitude modes in the excitation spectrum. We determine the phase diagram of the system, which shows the emergence and breaking of the continuous symmetries and displays first- and second-order phase transitions. We also obtain the excitation spectrum for each phase and discuss the atomic self-organized structures that emerge in the different superradiant phases. We argue that coupling the atoms equally to nnn different modes will in general generate a global SO(n)\text{SO}(n)SO(n) symmetry if the mode frequencies can be tuned to the same value.
Supersolids are characterized by the counterintuitive coexistence of superfluid and crystalline order. Here we study a supersolid phase emerging in the steady state of a driven-dissipative system. We consider a transversely pumped Bose-Einstein condensate trapped along the axis of a ring cavity and coherently coupled to a pair of degenerate counterpropagating cavity modes. Above a threshold pump strength the interference of photons scattered into the two cavity modes results in an emergent superradiant lattice, which spontaneously breaks the continuous translational symmetry towards a periodic atomic pattern. The crystalline steady state inherits the superfluidity of the Bose-Einstein condensate, thus exhibiting genuine properties of a supersolid. A gapless collective Goldstone mode correspondingly appears in the superradiant phase, which can be nondestructively monitored via the relative phase of the two cavity modes on the cavity output. Despite cavity-photon losses the Goldstone mode remains undamped, indicating the robustness of the supersolid phase.
Photon-mediated interaction can be used for simulating complex many-body phenomena with ultracold atoms coupled to electromagnetic modes of an optical resonator. We study theoretically a method of producing controllable interatomic interaction mediated by forward-diffracted photons circulating inside a ring cavity. One example of such a system is the three-mode cavity, where an on-axis mode can coexist with two diffracted sidebands. We demonstrate how the self-organized stripe states of a Bose-Einstein condensate (BEC) occurring in this cavity geometry can exhibit supersolid properties, due to spontaneous breaking of the Hamiltonian's continuous translational symmetry. A numerical study of the collective excitation spectrum of these states demonstrates the existence of massless and finite-gap excitations, which are identified as phase (Goldstone) and amplitude (Higgs) atomic density modes. We further demonstrate how judicious Fourier filtering of intracavity light can be used to engineer the effective atom-atom interaction profile for many cavity modes. The numerical results in this configuration show the existence of droplet-array and single-droplet BEC states for commensurate and incommensurate cavity modes, respectively. Diffractive coupling in a cavity is thereby introduced as an alternative route towards tailoring the photon-mediated interaction of ultracold atoms. Spatial features of the self-organized optical potentials can here be tuned to scales several times larger than the pump laser wavelength such that the corresponding atomic density distributions could be imaged and manipulated using low-numerical-aperture optics. These calculations and insights pave the way towards quantum simulation of exotic nonequilibrium many-body physics with condensates in a cavity.