Emergence of phase coherence in a magnon Bose–Einstein condensate
Malte Koster, Matthias R. Schweizer, Timo B. Noack, Vitaliy I. Vasyuchka, Dmytro A. Bozhko, B. Hillebrands, Mathias Weiler, Alexander A. Serga, Georg von Freymann
Abstract The spontaneous emergence of coherence is a defining feature of multibody quantum systems, underlying phenomena from superconductivity to quantum information processing. Although Bose–Einstein condensates provide a unique setting for studying this process, direct observation of how a condensate acquires a coherent global phase has remained challenging. Here we provide evidence of spontaneous phase formation in a magnon Bose–Einstein condensate. Using a phase-referenced detection technique, we track the phase of the condensate coherent state relative to an external reference, revealing how an initially incoherent magnon gas thermalizes and undergoes a spontaneous transition into a coherent quantum state with a well-defined macroscopic phase. This observation provides evidence of spontaneous symmetry breaking in a quasiparticle condensate, confirming a central prediction of Bose–Einstein condensation theory that extends across diverse quantum systems.
Abstract The spontaneous emergence of coherence is a defining feature of multibody quantum systems, underlying phenomena from superconductivity to quantum information processing. Although Bose–Einstein condensates provide a unique setting for studying this process, direct observation of how a condensate acquires a coherent global phase has remained challenging. Here we provide evidence of spontaneous phase formation in a magnon Bose–Einstein condensate. Using a phase-referenced detection technique, we track the phase of the condensate coherent state relative to an external reference, revealing how an initially incoherent magnon gas thermalizes and undergoes a spontaneous transition into a coherent quantum state with a well-defined macroscopic phase. This observation provides evidence of spontaneous symmetry breaking in a quasiparticle condensate, confirming a central prediction of Bose–Einstein condensation theory that extends across diverse quantum systems.
According to quantum statistical mechanics, generic many-body systems decohere and thermalize regardless of their initial state. A recent experiment found a surprising exception to this rule, wherein an otherwise thermalizing system maintains its coherence only for a particular initial condition, a phenomenon dubbed ``quantum many-body scarring.'' Here, the authors propose and provide numerical evidence for a physical mechanism underlying the observed phenomena: Bose-Einstein condensation, wherein macroscopic numbers of quasiparticles occupy a single quantum state. While this phenomenon is well known at equilibrium, its emergence in a strongly interacting, far-from-equilibrium system is striking and unexpected. The physical principle explored in this work can be applied to other systems to reveal new examples of quantum many-body scars.
The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Mølmer [1]).
Phase transitions abound in the physical world, from the subatomic length scales of quark condensation to the decoupling forces in the early universe. In the Bose-Einstein condensation phase transition, a gas of trapped bosonic atoms is cooled to a critical temperature. Below this temperature, a macroscopic number of atoms suddenly starts to occupy a single quantum state; these atoms comprise the Bose-Einstein condensate (BEC). The dynamics of the BEC phase transition are the focus of this dissertation and the experiments described here have provided new information on the details of BEC formation. New theoretical developments are proving to be valuable tools for describing BEC phase transition dynamics and interpreting new experimental results. With their amenability to optical manipulation and probing along with the advent of new microscopic theories, BECs provide an important new avenue for gaining insight into the universal dynamics of phase transitions in general.Spontaneous symmetry breaking in the system's order parameter may be one result of cooling through a phase transition. A potential consequence of this is the spontaneous formation of topological defects, which in a BEC appear as vortices. We experimentally observed and characterized the spontaneous formation of vortices during BEC growth. We attribute vortex creation to coherence length limitations during the initial stages of the phase transition. Parallel to these experimental observations, theory collaborators have used the Stochastic Gross-Pitaevski Equation formalism to simulate the growth of a condensate from a thermal cloud. The experimental and theoretical statistical results of the spontaneous formation of vortex cores during the growth of the condensate are in good quantitative agreement with one another, supporting our understanding of the dynamics of the phase transition. We believe that our results are also qualitatively consistent with the Kibble-Zurek mechanism, a universal model for topological defect formation.Ultimately, our understanding of the dynamics of the BEC phase transition may lead to a broader understanding of phase transitions in general, and provide new insight into the development of coherence in numerous systems.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
The realisation of Bose-Einstein condensation under grand-canonical conditions has provided the experimental evidence for the simultaneous occurrence of macroscopic fluctuations and phase coherence of the condensate. The observation of these two features, against a consolidated tradition which wants the fluctuations to be pathological (grand-canonical catastrophe) and incompatible with spontaneous symmetry braking, calls for a comprehensive rethinking of the approach to the problem. In this paper we consider the uniform ideal gas in a box and we present an alternative conceptual framework. We show that the usually-employed Bogoliubov quasi-average construction fails to reproduce the broken-symmetry state. The observed features are accounted for by a different pattern of spontaneous symmetry breaking, characterised by condensation of fluctuations and long-range correlations of the order parameter.
A quantised spin wave – magnon – in magnetic films can undergo Bose-Einstein condensation (BEC) into two energetically degenerate lowest-energy quantum states with nonzero wave vectors ±k BEC . This corresponds to two interfering condensates forming spontaneously in momentum space. Brillouin Light Scattering studies for a microwavepumped film with submicrometer spatial resolution experimentally confirm the existence of the two wavefunctions and show that their interference results in a nonuniform ground state of the condensate with the density oscillating in space. Moreover, fork dislocations in the density fringes provide direct experimental evidence for the formation of pinned half-quantum vortices in the magnon condensate. The measured amplitude of the density oscillation implies the formation of a nonsymmetric state that corresponds to nonequal occupation of two energy minima. We discuss the experimental findings and consider the theory of magnon condensates, which includes, to leading order, the contribution from the noncondensed magnons. The effect of the noncondensed magnon cloud is to increase the contrast of the asymmetric state and to bring about the experimental measurements. Introduction Magnons are quasiparticles corresponding to quantised spin waves that describe the collective motion of spins [1]. In recent years, it has become possible to realise a BEC of magnons in two remarkably different systems: in superfluid phases of an isotope of helium – Helium-3 at ultralow temperatures [2, 3] and in ferrimagnetic insulators [4, 5, 6, 7]. In the normal, noncondensed state, these magnetic materials exhibit a magnetically ordered state with a gas of magnons whose phases are not correlated. However, once condensation occurs, spins develop phase coherence resulting in a common global frequency and phase of precession. At room temperature, ferrimagnetic insulators, such as yttrium-iron garnet (YIG) films, together with a combination of an in-plane magnetic field and microwave radiation, are required for the magnon condensation [4, 5, 6, 7]. Therefore, in analogy with exciton– polariton condensates, this provides yet another example of a BEC of quasiparticle excitations in a solid-state system. Experimental Realisation of a Bose-Einstein Condensate of Magnons in an Epitaxial YIG-film A room-temperature Bose-Einstein condensate of magnons was created in an epitaxial YIG-film using an experimental setup shown in Fig. 25.1a, which, in general, is similar to that used in our previous studies [4, 5]. Given that magnons are quasiparticle excitations, their number is not conserved and, therefore, the chemical potential is zero.