Bose-Einstein condensates with balanced gain and loss beyond mean-field theory Von der Fakultät Mathematik und Physik der Universität Stuttgart zur Erlangung der Würde eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung Vorgelegt von Dennis Dast aus Stuttgart Hauptberichter: Prof. Dr. Günter Wunner Mitberichter: Jun.-Prof. Dr. Thomas Weiss Tag der mündlichen Prüfung: 09. Februar 2017 1. Institut für Theoretische Physik 2017 Inhaltsangabe Die meisten Arbeiten zu Bose-Einstein-Kondensaten mit ausgeglichenem Gewinn und Verlust wurden in der Mean-Field-Näherung unter Verwendung der nichthermiteschen PT -symmetrischen Gross-Pitaevskii-Gleichung durchgeführt. In diesen Systemen spielt allerdings der Austausch von Teilchen mit der Umgebung eine entscheidende Rolle, was im Allgemeinen zu Abweichungen vom Mean-Field-Verhalten führt. Es ist daher nicht im Voraus ersichtlich, dass die Mean-Field-Beschreibung hierfür geeignet ist. ZieldieserArbeitistdieFormulierungundUntersuchungeinerVielteilchenbeschreibung eines Bose-Einstein-Kondensats mit ausgeglichenem Gewinn und Verlust. Das wird durch eine Mastergleichung erreicht, die ein Doppelmuldenpotential beschreibt, in welchem das Einkoppeln von Teilchen in der einen und das Auskoppeln aus der anderen Mulde durch Lindblad-Superoperatoren beschrieben wird. Die Ein- und Auskoppelraten müssen auf geeignete Weise angepasst werden, um einen ausgeglichenen Gewinn und Verlust zu erzeugen. Es wird gezeigt, dass der Mean-Field-Grenzfall dieser Mastergleichung eine PT -symmetrische Gross-Pitaevskii-Gleichung liefert. Darüber hinaus besitzt die Mas- tergleichung die charakteristischen dynamischen Eigenschaften von PT -symmetrischen Systemen. Es gibt jedoch auch fundamentale Unterschiede zur Mean-Field-Beschreibung. So wird gezeigt, dass die Reinheit des Kondensats periodisch auf kleine Werte abfällt, aber dann nahezu vollständig wiederhergestellt wird, während die Teilchen zwischen den beiden Mulden oszillieren. Da in der Mean-Field-Beschreibung ein vollständig reines Kondensat angenommen wird, kann dieser Effekt nicht durch die Gross-Pitaevskii-Gleichung erfasst werden. Diese Reinheitsoszillationen haben einen direkten Einfluss auf den gemittelten Kontrast in einem Interferenzexperiment. Insbesondere zeigt sich, dass die Extrempunkte der Reinheit präzise gemessen werden können, da der Kontrast zu diesen Zeitpunkten nicht durch ein Ungleichgewicht in der Teilchenzahlverteilung reduziert wird. Um die Reinheitsoszillationen im Detail zu diskutieren, werden analytische Lösungen für die Dynamik ohne Wechselwirkung betrachtet und die Bogoliubov-Backreaction- Methode wird verwendet, um den Einfluss der kurzreichweitigen Wechselwirkung zu untersuchen. Ein zentrales Ergebnis ist, dass die Amplitude der Reinheitsoszillationen weder von der Teilchenzahl noch von der Wechselwirkungsstärke abhängt, sondern nahezu vollständig durch die Stärke der Ein- und Auskoppelprozesse festgelegt ist. Für größere Teilchenzahlen treten die starken Reinheitsoszillationen allerdings zu immer späteren Zeiten auf. Ohne Wechselwirkung würde das dazu führen, dass die Reinheitsoszillationen für realistische Teilchenzahlen nicht beobachtbar sind, aber durch das Anpassen der Wechselwirkungsstärke treten die starken Oszillationen wieder früher auf. iii Abstract Most of the work done in the field of Bose-Einstein condensates with balanced gain and loss has been performed in the mean-field approximation using the non-Hermitian PT -symmetric Gross-Pitaevskii equation. However, the exchange of particles with the environment plays a crucial role in such systems which in general leads to deviations from the mean-field behavior. Thus, it is not clear whether a mean-field approach is appropriate. It is the purpose of this work to formulate and study a many-particle description of a Bose-Einstein condensate with balanced gain and loss. This is achieved by using a quantum master equation describing a double well where the incoupling of particles in one well and the outcoupling from the other are implemented with Lindblad superoperators. The in- and outcoupling rates are adjusted in an appropriate manner such that balanced gain and loss is achieved. It is shown that the mean-field limit of this master equation yields a PT -symmetric Gross-Pitaevskii equation. Furthermore the master equation supports the characteristic dynamical properties of PT -symmetric systems. There are, however, fundamental differences compared with the mean-field description revealing a new generic feature of PT -symmetric Bose-Einstein condensates. It is shown that the purity of the condensate periodically drops to small values but then is nearly completely restored, when the particles oscillate in the double well. Since in the mean- field limit a completely pure condensate is assumed, this effect cannot be covered by the Gross-Pitaevskii equation. These purity oscillations have a direct impact on the average contrast in interference experiments. In particular it is found that the extrema of the purity can be precisely measured since the average contrast at these points is not reduced by an imbalance of the particle distribution. To gain a detailed understanding of the purity oscillations, analytic solutions for the dynamics in the non-interacting limit are presented and the Bogoliubov backreaction method is used to discuss the influence of the on-site interaction. A central result is that the strength of the purity revivals does neither depend on the amount of particles in the system nor the interaction strength, but is almost exclusively determined by the strength of the in- and outcoupling processes. However, the strong revivals are shifted towards longer times for larger particle numbers. Without interaction this would make the purity oscillations unobservable for a realistic particle number, but by adjusting the interaction strength the strong revivals again occur earlier. v Contents Inhaltsangabe iii Abstract v 1. Introduction 1 1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Bose-Einstein condensation 5 2.1. Ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2. Interacting Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3. Purity of a condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.1. Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.2. Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . 12 2.5. Many-particle product states . . . . . . . . . . . . . . . . . . . . . . . . . 14 3. PT symmetry 17 3.1. Properties of PT -symmetric systems . . . . . . . . . . . . . . . . . . . . 17 3.2. Nonlinear PT -symmetric systems . . . . . . . . . . . . . . . . . . . . . . 19 3.3. Bose-Einstein condensate in a PT -symmetric double well . . . . . . . . . 21 3.3.1. Linear two-mode system . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.2. Nonlinear two-mode system . . . . . . . . . . . . . . . . . . . . . 27 3.3.3. Spatially extended double well . . . . . . . . . . . . . . . . . . . . 29 4. Quantum master equation with balanced gain and loss 31 4.1. Quantum master equations in Lindblad form . . . . . . . . . . . . . . . . 31 4.2. Localized particle loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3. Localized particle gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.4. Two-mode master equation with balanced gain and loss . . . . . . . . . . 38 5. Numerical treatment 41 5.1. Quantum jump method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.2. Bogoliubov backreaction method . . . . . . . . . . . . . . . . . . . . . . 44 5.2.1. Bose-Hubbard chain with gain and loss . . . . . . . . . . . . . . . 45 vii Contents 5.2.2. Two-mode system with balanced gain and loss . . . . . . . . . . . 51 5.2.3. First- and second-order moments of pure states . . . . . . . . . . 54 6. Comparison with the mean-field limit 55 6.1. Mean-field limit of the master equation with balanced gain and loss . . . 55 6.2. Dynamical behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3. Dynamics on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . 62 7. Beyond mean-field theory 67 7.1. Representation of initial states . . . . . . . . . . . . . . . . . . . . . . . . 67 7.2. Purity oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.3. Contrast in interference experiments . . . . . . . . . . . . . . . . . . . . 71 7.4. Non-interacting limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4.1. Analytic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4.2. Purity oscillations in the oscillatory regime . . . . . . . . . . . . . 81 7.5. Accuracy of the Bogoliubov backreaction method . . . . . . . . . . . . . 84 7.6. Strength of the purity revivals . . . . . . . . . . . . . . . . . . . . . . . . 87 7.7. Eigenvectors of the single-particle density matrix . . . . . . . . . . . . . 93 8. Stationary states 95 8.1. PT -symmetric stationary states . . . . . . . . . . . . . . . . . . . . . . . 95 8.2. Non-oscillatory states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.2.1. Non-interacting limit . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.2.2. Non-oscillatory states in first order . . . . . . . . . . . . . . . . . 103 8.2.3. Non-oscillatory states in second order . . . . . . . . . . . . . . . . 105 8.3. Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.3.1. Non-interacting limit . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.3.2. Gain and loss in a single mode . . . . . . . . . . . . . . . . . . . . 110 8.3.3. Influence of the interaction . . . . . . . . . . . . . . . . . . . . . . 111 9. Summary and Outlook 119 A. Bose-Hubbard chain with gain and loss 127 A.1. Equations of motion of the first-order moments . . . . . . . . . . . . . . 127 A.2. Equations of motion of the covariances . . . . . . . . . . . . . . . . . . . 129 B. Purity envelope for large particle numbers 133 Bibliography 135 Zusammenfassung in deutscher Sprache 145 Danksagung 153 viii 1. Introduction 1.1. Motivation In 1998 Bender and Boettcher studied the Hamiltonian Hˆ = pˆ2 +xˆ2(ixˆ)κ with real κ numerically [1]. They found the remarkable result that for κ ≥ 0 the eigenvalue spectrum is real and positive although the Hamiltonian is in general non-Hermitian. This property was later proved in [2]. For κ < 0 complex conjugate pairs of eigenvalues were found. It was argued that these properties arise due to the PT symmetry of the Hamiltonian, ˆ [H,PT ] = 0, where P and T are the parity and time reversal operators defined as P: xˆ → −xˆ, pˆ→ −pˆ, and T : pˆ→ −pˆ, i → −i. In the regime where only real eigenvalues exist the eigenstates are PT symmetric, and thus PT symmetry is not spontaneously broken. In this case one can find an additional symmetry called C, which can be used to construct a positive definite inner product, with respect to which the time evolution is unitary [3]. In this sense PT symmetry can be interpreted as a complex generalization of Hermitian quantum mechanics. However, PT symmetry is neither necessary nor sufficient for the eigenvalue spectrum to be real. The eigenvalues of PT -symmetric Hamiltonians might be complex, and there exist Hamiltonians with a real spectrum that are not PT symmetric. This motivated Mostafazadeh to introduce the concept of pseudo-Hermiticity [4–6]. In particular it was shown that pseudo-Hermiticity is a necessary condition for a real eigenvalue spectrum. However, non-Hermitian quantum mechanics is not only used to formulate a funda- mental generalization of Hermitian quantum mechanics. Instead it also provides an elegant approach to solve problems that are much harder when using Hermitian quantum mechanics. Various applications, with a focus on resonance phenomena, of this formalism are reviewed in the book Non-Hermitian Quantum Mechanics by Moiseyev [7]. Interpreting PT symmetry as the effective description of an open system interacting with its environment has inspired various works finally leading to the first experimental realizations in optics [8, 9]. The optical experiments exploited the fact that the paraxial equation of diffraction for the electric field envelope has the same form as the Schrödinger equation where the time is replaced by the propagation distance and the potential is given by the refractive index [10–14]. Thus, the equivalent of a non-Hermitian potential is generated by a complex refractive index describing pumping or dissipation. In [9] PT symmetry was achieved by two coupled waveguides with loss in one waveguide and gain with equal strength in the other. In the following years further experiments in optical systems were performed [15–23] but a realization in a genuine quantum system is still 1 1. Introduction missing. However, this is necessary to discuss quantum effects in PT -symmetric systems. A promising candidate for the realization of a genuine PT -symmetric quantum system is a Bose-Einstein condensate. In the mean-field limit Bose-Einstein condensates are described by the Gross-Pitaevskii equation, which is a nonlinear Schrödinger equation. Imaginary potentials have a clear physical interpretation as the in- and outcoupling of particles depending on the sign of the imaginary part [24]. A PT -symmetric Bose- Einstein condensate can be created in a double-well potential with an influx of particles in one well and an outflux from the other. This system has been investigated using a double-δ potential [25, 26] and a spatially extended double well [27–30]. In these works stable stationary solutions, a rich dynamics, and a variety of bifurcation scenarios were found. Furthermore proposals for the realization of such a system exist by embedding the PT -symmetric double well in a larger Hermitian system [31, 32] or by using a coupling approach [33]. These studies were performed in the mean-field limit, thus, it was assumed that all particles are in the condensed phase, i.e., the single-particle density matrix is quantum mechanically pure. However, the purity of a condensate is reduced by both the coupling to the environment and the interaction of the particles [34]. In fact, studying the controlled coupling of Bose-Einstein condensates with the environment has led to various fascinating many-particle effects. For example it was shown in [35] that dissipation due to strong inelastic collisions can actually induce the correlations in a Bose-Einstein condensate. Furthermore, a two-mode system where dissipation and phase noise are taken into account can, if carefully prepared, yield a revival of the condensate’s purity before it eventually decays [36–38]. A strong growth of entanglement was found in a one-dimensional optical lattice by introducing a localized loss of atoms [39]. Again in a one-dimensional optical lattice with a lossy site it was demonstrated that a bistability exists if the dissipation has an appropriate strength [40]. Thus, it cannot be expected that PT -symmetric systems, in which the exchange of particles with the environment plays a crucial role, are appropriately described in the mean-field limit. To check this expectation it is necessary to carry out an analysis in the many-particle system. A possible many-particle description of PT -symmetric Bose-Einstein condensates has been previously investigated with a non-Hermitian Bose-Hubbard dimer [41, 42]. There, gain and loss were introduced as complex on-site energy contributions. However, the mean-field limit of such a system does not lead to the known Gross-Pitaevskii equation with complex potentials, but to an adapted equation, in which the nonlinear term is divided by the norm squared of the wave function. While this equation has the same normalized eigenstates as the Gross-Pitaevskii equation, the dynamical behavior, including the stability properties of the eigenstates, clearly differs [28, 29], and thus it is not the desired many-particle description. A different approach to open quantum systems are master equations in Lindblad form [43], which are well established to describe phase noise, feeding and depleting of a Bose-Einstein condensate [34, 44]. Recently it has been shown that the mean-field limit of a master equation, where the coherent dynamics is governed by a Bose-Hubbard 2