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.
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.
Higgs and Goldstone modes manifest as fluctuations in the order parameter of system, offering insights into its phase transitions and symmetry properties. Exploring the dynamics of these collective excitations in a Rydberg atoms system advances various branches of condensed matter, particle physics, and cosmology. Here, we report an experimental signature of Higgs and Goldstone modes in a U(1) symmetry-broken Rydberg atomic gases. By constructing two probe fields to excite atoms, we observe the distinct phase and amplitude fluctuations of Rydberg atoms collective excitations under the particle-hole symmetry. Due to the van der Waals interactions between the Rydberg atoms, we detect a symmetric variance spectrum divided by the divergent regime and phase boundary, capturing the full dynamics of the additional Higgs and Goldstone modes. Studying the Higgs and Goldstone modes in Rydberg atoms allows us to explore fundamental aspects of quantum phase transitions and symmetry breaking phenomena, while leveraging the unique properties of these highly interacting systems to uncover new physics and potential applications in quantum simulation.
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.
A supersolid is a quantum phase of matter that demonstrates both solid and superfluid properties.In combination with gaseous and liquid superfluids it forms the triad of quantum phases analogous to the classical states of matter: solid, liquid and gas.Despite efforts in a number of systemsmost prominently helium-experimental verification of supersolidity has remained elusive.We report on the realization and characterization of a supersolid.A phase transition from a gaseous Bose-Einstein condensate to the supersolid crystal is induced by photon-mediated interactions among the atoms that are a result of symmetric coupling with the modes of two optical cavities.At the critical point, continuous translational symmetry is broken along one direction and a periodic density pattern with a coherent wave function forms.We verify both the structure formation and superfluidity by performing matterwave interference of the many-body wavefunction.The phase transition from the superfluid to the supersolid is marked by the onset of light fields in both optical cavities, whose amplitudes are equivalent to the real and imaginary part of the continuous symmetry.This unique real-time observable permits extensive studies of the supersolid state and the phase transition.We demonstrate a high ground-state degeneracy and observe the associated Higgs and Goldstone modes across the critical point.Their dynamics are visible as amplitude and phase oscillation in the cavity fields, facilitating studies on their decay and response.Approaching the phase transition, we examine the critical fluctuations of the density distribution in the vicinity of the critical point by permanently measuring the photon occupation of both cavities.The continuous symmetry is the result of symmetry enhancement from two competing order parameters with underlying discrete symmetries.In order to explore the crossover region between continuous and discrete symmetry breaking, we study the vicinity of the symmetry-enhanced region by creating a slight asymmetry in the two coupling strengths.We observe an increasing anisotropy in the distribution of the broken symmetry, accompanied by a rising mass of the Goldstone mode.The experiments were performed with a new apparatus that was constructed in the context of this thesis.The setup is optimized for a fast, robust preparation sequence of the quantum gas.We introduce a new method to transport ultracold atoms with focus-tunable lenses, which is superior to previously established methods in terms of flexibility, stability, costs and space requirements.Furthermore, the vacuum system features a conceptually new design including a transfer system that permits one to readily exchange the cavity setup.This establishes our experiment as a versatile quantum simulator to engineer atom-light interactions in various optical cavity environments.v ZUSAMMENFASSUNG Ein Suprasolid ist ein Quantenzustand der gleichzeitig feste und flüssige Eigenschaften aufweist.Gemeinsam mit gasförmigen und flüssigen Suprafluiden bildet er eine Dreiheit an Quantenzuständen in Analogie zu den klassischen Aggregatzuständen Gas, Flüssigkeit und Festkörper.Trotz Bestrebungen in verschiedenen Systemen -allen voran Helium -konnte ein Suprasolid noch nicht experimentell nachgewiesen werden.In dieser Arbeit wird die erste Herstellung und Charakterisierung eines solchen Suprasolids vorgestellt.Der Phasenübergang von einem gasförmigen Bose-Einstein-Kondensat zu dem suprasoliden Kristall wird von einer atomaren Wechselwirkung verursacht, die von Photonen aus zwei optischen Resonatoren getragen wird.Am kritischen Punkt bildet sich ein periodisches Dichtemuster in der kohärenten atomaren Wellenfunktion, das die kontinuierliche Translationssymmetrie des Kondensats entlang einer Richtung bricht.Wir weisen sowohl die Strukturbildung als auch die Suprafluidität durch Materiewellen-Interferenz der Vielteilchen-Wellenfunktion nach.Der Phasenübergang vom Suprafluid zum Suprasolid ist durch die Enstehung von Lichtfeldern in beiden Resonatoren gekennzeichnet, deren Amplituden dem realen und imaginären Teil der kontinuierlichen Symmetrie entsprechen.Diese einzigartige Echtzeit-Observable lässt weitreichende Studien des suprasoliden Zustands und des Phasenübergangs zu.Wir weisen eine hohe Entartung des Grundzustands nach und beobachten die entsprechenden Higgs-und Goldstone-Moden über den kritischen Punkt hinweg.Ihre Dynamik ist in den Resonatormoden als Amplituden-und Phasenoszillation sichtbar, was Studien zu ihrem Zerfall und Antwortverhalten ermöglicht.Nah am Phasenübergang untersuchen wir die kritischen Fluktuationen der Dichtemodulation, indem wir die Photonenbesetzung beider Resonatoren messen.Die kontinuierliche Symmetrie ist das Ergebnis einer Symmetrie-Erweiterung aus zwei konkurrierenden Ordnungsparametern mit zugrundeliegenden diskreten Symmetrien.Um den Übergang von der kontinuierlichen zur diskreten Symmetrie zu erforschen, studieren wir die Umgebung der symmetrie-erweiterten Region indem wir eine geringfügige Asymmetrie in die Kopplungsstärken einführen.Wir beobachten eine zunehmende Anisotropie in der Verteilung der gebrochenen Symmetrie, begleitet von einer ansteigenden Masse der Goldstone-Mode.Die Experimente wurden mit einer neuen Apparatur durchgeführt, die im Rahmen dieser Arbeit errichtet wurde.Der Aufbau ist auf eine schnelle und robuste Präparationssequenz des Quantengases ausgerichtet.Wir führen eine neue Transportmethode für ultrakalte Atome ein, die auf Linsen mit verstellbarer Brennweite basiert und bisherigen Methoden bezüglich Flexibilität, Stabilität, Kosten und Platzanforderungen überlegen ist.Ferner zeichnet sich das Vakuumsystem durch einen konzeptionell neuen Ansatz mit einem Transfersystem aus, das es erlaubt den Resonator-Aufbau ohne Weiteres auszutauschen.Unser Experiment schafft dadurch einen vielseitigen Quantensimulator für konstruierte Atom-Licht-Wechselwirkungen in verschiedenen Resonator-Umgebungen.
Quantum Simulation
The concept of broken symmetry has relevance across all branches of physics, including particle and condensed matter physics. In many cases, the shape of the potential energy resembles a Mexican hat, and the symmetry of the system is broken when it chooses a particular spot along the “trough” of the hat. Out of this minimum-energy state, the system can undergo collective excitations either along the trough or perpendicular to it. Leonard et al. detected these so-called Goldstone and Higgs modes in a supersolid Bose-condensed atomic gas held in two crossed optical cavities. By monitoring the dynamics of the light field in each cavity, the oscillations of the order parameter associated with both modes were observed in real time.
Science , this issue p. [1415][1]
[1]: /lookup/doi/10.1126/science.aan2608
We realize the mesoscopic Bose-Hubbard model at unit-filling on a two-dimensional array of transmon qubits and selectively excite collective modes associated with amplitude or phase fluctuations of the superfluid order parameter. We identify these modes respectively as the Higgs and Goldstone modes using complementary local observables, extracted from measuring the number of particles on each site. Modulation of the coupling between sites excites the Higgs mode, which we detect via the density of doubly occupied sites. In contrast, modulation of the local chemical potential with an appropriately chosen spatial profile excites the lowest-frequency Goldstone mode, which we detect via changes in the local density. The two modes have different symmetries and do not couple to one another. By varying the system geometry, we characterize the impact of finite system size on the evolution of both modes. In the thermodynamic limit, decay into a continuum of lower-energy excitations can obscure the Higgs response. In our mesoscopic system, however, the finite system size leads to spectral gaps and allows us to detect the Higgs mode as a coherent, long-lived excitation. Our work demonstrates how finite quantum simulators can isolate and probe physical phenomena that become difficult to resolve in the thermodynamic, large-scale limit.