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arXiv:1309.0424 · Physical Review A

Spontaneous symmetry breaking in spinor Bose-Einstein condensates

M. Scherer, B. Lücke, J. Peise, O. Topic, Grebremedh Gebreyesus, F. Deuretzbacher, Wolfgang A. Ertmer, Luis Santos, Carsten Klempt, J. Arlt

Published Nov 19, 2013• 14 citations
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Abstract

We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state ${m}_{F}=0$ ($\left|0\right\ensuremath{\rangle}$) can be understood by approximating the effective trapping potential for the state $\left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle}$ with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.

DOI
10.1103/physreva.88.053624

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We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0{m}_{F}=0mF​=0 (\left|0\right\ensuremath{\rangle}) can be understood by approximating the effective trapping potential for the state \left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle} with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.
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.
We consider the statics and dynamics of F = 1 spinor Bose–Einstein condensates (BECs) confined in double-well potentials. We use a two-mode Galerkin-type quasi-analytical approximation to describe the stationary states of the system. This way, we are able to obtain not only earlier results based on the single-mode approximation (SMA) frequently used in studies of spinor BECs, but also additional modes that involve either two or all three spinor components of the F = 1 spinor BEC. The results based on this Galerkin-type decomposition are in good agreement with the analysis of the full system. We subsequently analyze the stability of these multi-component states, as well as their dynamics when we find them to be unstable. The instabilities of the symmetric or anti-symmetric states exhibit symmetry-breaking and recurrent asymmetric patterns. Our results yield qualitatively similar bifurcation diagrams both for polar (such as 23 Na) and ferromagnetic (such as 87 Rb) spinor BECs.
Summary form only given. The experimental discovery of Bose-Einstein condensation in dilute alkali atomic gases has revolutionized the field of macroscopic quantum coherence. Further, fundamental issues like symmetry breaking, collapse and revival of macroscopic wavefunctions can perhaps now be tested in an experimentally controllable way. A further step in the control and study of Bose-Einstein condensates (BEC) has been the creation of spinor condensates-BEC in different spin states-through optical confinement and has led to considerable theoretical activity.
Since the realization of Bose-Einstein condensation in alkali gases in 1995, Bose-Einstein condensates (BEC) have developed into a versatile subject of research. A particularly intriguing variant, extending the extraordinary properties of BEC as quantum gases to the spin as a degree of freedom, are spinor BEC in optical traps. Spinor BEC allow the study of magnetism in quantum gases, merging these two fascinating branches of physics. After the pioneering experiments in 1998, the past few years have seen a remarkable burst of activity on the topic of spinor BEC. In this context, the present thesis constitutes a significant contribution to the current understanding of spinor BEC physics. The present thesis deals with the coherent dynamics of spinor condensates, and with the mechanisms leading to decoherence and damping. Building on a highly optimized apparatus for the production of large spinor BEC in an optical trap, experimental methods used for the preparation and manipulation of the spin state and the control of environmental parameters had to be developed and sophisticated. Parallel to the experimental aspects, a better understanding of spinor BEC dynamics, including mathematical modeling and intuitive pictures, had to be developed. The techniques of radio-frequency induced Rabi and Ramsey oscillations have been generalized to spin-1 and spin-2 systems and, applied to spinor BEC, provide insight into the mechanisms and time scales of decoherence. Inherent spin-mixing dynamics is studied using a particular initial state, a fully transversely magnetized state, which combines ease of preparation and sensitivity to interaction effects. As a central result of this thesis, a fundamental resonance phenomenon in the spin-mixing dynamics of both ferromagnetic and anti-ferromagnetic spinor BEC is demonstrated for the first time. This resonance is a result of competition between the quadratic Zeeman effect and the spindependent interaction energy, and is analogous to nonlinear phase-matching in optical four-wave mixing. The single-mode approximation (SMA), neglecting any spatial variations of the spin state, is shown to reproduce the experimentally observed population dynamics in great detail. However, its validity is limited by thermalization processes and spatial structure formation – both of these effects are analyzed in detail. In the context of the present thesis, analytical solutions of the SMA equations of spin dynamics could be obtained for the first time. The particularly intriguing phenomenon of spatial structure formation is further studied with spin-2 BEC in a single-beam optical trap providing a 1D geometry. Beautiful, almost regular spin patterns are observed to emerge from a homogeneous initial state and to decay into chaos. Spontaneous pattern formation due to a dynamical instability as well as residual magnetic field gradients are discussed as the origin of the observed structure. The experimental observation of structure formation is complemented by a numerical analysis of the single-mode dynamics of spin-2 BEC, uncovering clear evidence of deterministic chaos in this nonlinear complex system. Moreover, the present work introduces experimental tools that will be useful for future studies. A Raman laser system allows to manipulate the spin state of a BEC locally, e.g. in order to prepare solitons. Using spatially incoherent light and short pulses is discussed as a way of avoiding interference fringes in absorption imaging of condensates. Alternatively or additionally, absorption images may be post-processed on a computer to eliminate fringes. The greatly improved understanding of single-mode spinor dynamics, to which the work at hand has significantly contributed, provides a firm basis for tackling the intriguing questions raised by current and future research on spinor BEC. Seit der Realisierung von Bose-Einstein Kondensaten in Alkaligasen im Jahr 1995 haben diese sich zu vielseitigen Forschungsobjekten entwickelt. Ein besonders faszinierender Zweig sind Spinor- Kondensate in optischen Fallen, die die ausergewohnlichen Quantengas-Eigenschaften von Bose- Einstein Kondensaten auf den Spin als Freiheitsgrad ausdehnen. Spinor-Kondensate ermoglichen es, Magnetismus in Quantengasen zu erforschen, und verbinden so diese zwei ausgesprochen fruchtbaren Gebiete der Physik. Seit den ersten grundlegenden Experimenten 1998 ist das Gebiet erst in den letzten Jahren in bemerkenswerter Weise wieder aufgelebt. In diesem Zusammenhang leistet die vorliegende Arbeit einen bedeutenden Beitrag zum gegenwartigen Verstandnis der Physik der Spinor-Kondensate. Die vorliegende Arbeit behandelt koharente Dynamik in Spinor-Kondensaten und ihre Grenzen, Dekoharenz und Dampfung. Aufbauend auf einer hochentwickelten Apparatur zur Erzeugung groser Spinor-Kondensate in einer optischen Falle, wurden experimentelle Methoden zur Praparation und Manipulation des Spinzustands sowie zur Kontrolle der Umgebungsparameter entwickelt und verfeinert. Neben den technischen Aspekten wurde insbesondere auch das Verstandnis der Spindynamik, deren mathematische Beschreibung ebenso wie intuitive Vorstellungen, erheblich weiterentwickelt. Als ein wichtiger Punkt wurde die Methode der Rabi- und Ramsey-Oszillationen mittels Anregung durch hochfrequenteMagnetfelder auf Spin-1 und Spin-2 Systeme verallgmeinert. Angewendet auf Spinor-Kondensate ergeben sich daraus Ruckschlusse auf Dekoharenzmechanismen und deren Zeitskalen. Inharente Spindynamik wurde anhand eines besonderen, vollstandig transversal magnetisierten Anfangszustands untersucht, der robust in der Praparation ist und empfindlich auf Wechselwirkungseffekte reagiert. Ein zentrales Ergebnis der vorliegenden Arbeit ist die erstmalige Beobachtung einer Spindynamik- Resonanz sowohl in ferro- als auch in antiferromagnetischen Kondensaten. Die Resonanz kann auf die Konkurrenz von quadratischem Zeeman-Effekt und spinabhangiger Wechselwirkung zuruckgefuhrt werden; sie ist aquivalent zu nichtlinearer Phasenanpassung bei optischem Vierwellenmischen. Weiterhin wurde gezeigt, dass die Einmoden-Naherung, obwohl sie jegliche raumliche Abhangigkeit des Spins vernachlassigt, die experimentellen Ergebnisse detailliert wiederspiegelt. Ihre Gultigkeit ist jedoch durch Thermalisierung und raumliche Strukturbildung begrenzt, und beide Prozesse werden umfassend analysiert. Im Rahmen dieser Arbeit konnten erstmals auch analytische Losungen der Spindynamik in Einmoden-Naherung gefunden werden. Das besonders interessante Phanomen der raumlichen Strukturbildung wird anhand von Spin-2 Kondensaten in einer eindimensionalen optische Falle weiter untersucht. Die entstehenden beinahe regelmasigen Spin-Muster sind asthetisch faszinierend, und zerfallen in chaotische Strukturen. Als Ursachen der Strukturbildung werden spontane Musterbildung infolge einer dynamischen Instabilitat sowie Rest-Inhomogenitaten im Magnetfeld diskutiert. Der experimentelle Befund der Strukturbildung wird durch eine numerische Analyse der Einmoden- Spindynamik in Spin-2 Kondensaten erganzt, die eindeutige Anzeichen far deterministisches Chaos in diesem nichtlinearen komplexen System ergibt. Daruberhinaus werden in dieser Arbeit experimentelle Techniken vorgestellt, die in zukunftigen Untersuchungen zum Einsatz kommen werden. Ein Raman-Lasersystem ermoglicht die lokale Manipulation des Spinzustands eines ausgedehnten Kondensats, beispielsweise um Solitonen zu praparieren. Die Nutzung von raumlich inkoharentem Licht und kurzen Pulsen zur interferenzfreien Abbildung von Bose-Einstein Kondensaten wird diskutiert; alternativ oder erganzend dazu besteht die Moglichkeit, Absorptionsbilder rechnerisch zu verarbeiten, um Interferenzen zu eliminieren. Das heute bedeutend weiter entwickelte Verstandnis der Einmoden-Spindynamik, zu dem die vorliegende Arbeit einen wesentlichen Beitrag liefert, stellt eine gute Grundlage dar um die faszinierenden Fragen aktueller und zukunftiger Forschung an Spinor-kondensaten anzugehen.
Parametric amplification of quantum fluctuations constitutes a fundamental mechanism for spontaneous symmetry breaking. In our experiments, a spinor condensate acts as a parametric amplifier of spin modes, resulting in a twofold spontaneous breaking of spatial and spin symmetry in the amplified clouds. Our experiments permit a precise analysis of the amplification in specific spatial Bessel-like modes, allowing for the detailed understanding of the double symmetry breaking. On resonances that create vortex-antivortex superpositions, we show that the cylindrical spatial symmetry is spontaneously broken, but phase squeezing prevents spin-symmetry breaking. If, however, nondegenerate spin modes contribute to the amplification, quantum interferences lead to spin-dependent density profiles and hence spontaneously formed patterns in the longitudinal magnetization.
The properties of an F=1F=1F=1 spinor Bose-Einstein condensate trapped in a double-well potential are discussed using both a mean-field two-mode approach and a two-site Bose-Hubbard Hamiltonian. We focus on the region of phase space in which spin effects lead to symmetry breaking of the system, favoring the spatial localization of the condensate in one well. To model this transition we derive, using perturbation theory, an effective Hamiltonian that describes N/2N/2N/2 spin singlets confined in a double-well potential.