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arXiv:2108.04403 · Physical Review Research

Quantum fluctuations on top of a PT -symmetric Bose-Einstein condensate

Xiaoling Cui

Published Jan 21, 2022• 11 citations
arXivHTMLPDF

Abstract

We investigate the effects of quantum fluctuations in a parity-time $(\mathcal{PT})$ symmetric two-species Bose-Einstein condensate (BEC). It is found that the $\mathcal{PT}$ symmetry, though preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasiparticles under quantum fluctuations. The associated $\mathcal{PT}$-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy in spin channels and the strength of $\mathcal{PT}$ potential. In the $\mathcal{PT}$-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the gapless phonon mode in Hermitian case. Moreover, the presence of $\mathcal{PT}$ potential is found to enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of $\mathcal{PT}$ symmetry and interaction can be directly probed in cold atoms experiments, which shed light on related quantum phenomena in general $\mathcal{PT}$-symmetric systems.

DOI
10.1103/physrevresearch.4.013047

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We investigate the effects of quantum fluctuations in a parity-time (PT)(\mathcal{PT})(PT) symmetric two-species Bose-Einstein condensate (BEC). It is found that the PT\mathcal{PT}PT symmetry, though preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasiparticles under quantum fluctuations. The associated PT\mathcal{PT}PT-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy in spin channels and the strength of PT\mathcal{PT}PT potential. In the PT\mathcal{PT}PT-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the gapless phonon mode in Hermitian case. Moreover, the presence of PT\mathcal{PT}PT potential is found to enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of PT\mathcal{PT}PT symmetry and interaction can be directly probed in cold atoms experiments, which shed light on related quantum phenomena in general PT\mathcal{PT}PT-symmetric systems.
Interaction effect in a PT\mathcal{PT}PT-symmetric Bose-Einstein Condensate
We investigate the ground state and excitation properties of a weakly interacting Bose-Einstein Condensate(BEC) under a parity-time(PT\mathcal{PT}PT) symmetric potential with balanced gain and loss. It is shown that the PT\mathcal{PT}PT-symmetry, as preserved by the macroscopic condensate, can be spontaneously broken by its Bogoliubov quasi-particles. The associated PT\mathcal{PT}PT-breaking transitions in the Bogoliubov spectrum can be conveniently tuned by the interaction anisotropy of BEC and the strength of PT\mathcal{PT}PT potential. In the PT\mathcal{PT}PT-unbroken regime, the real Bogoliubov modes are generally gapped, in contrast to the Hermitian case with gapless phonon mode. Moreover, it is found that the presence of PT\mathcal{PT}PT potential can enhance the mean-field collapse and thereby intrigue the droplet formation after incorporating the repulsive force from quantum fluctuations. These remarkable interplay effects of PT\mathcal{PT}PT-symmetry and interaction can be directly probed in cold atoms experiments, which shed light on intriguing collective phenomena in other non-Hermitian systems with PT\mathcal{PT}PT-symmetry.
We employ mean-field, Bogoliubov and many-body theories to study critical fluctuations in the position and momentum of a Bose-Einstein condensate whose translation symmetry is spontaneously broken due to attractive interactions. In a homogeneous system, the many-body ground state of the symmetry-preserving Hamiltonian is very fragile against superposition of low-lying states, while the mean-field theory predicts a stable bright soliton which spontaneously breaks translation symmetry. We show that weak symmetry-breaking perturbations cause the translation-symmetric many-body ground state to cross over to a many-body bright soliton. We argue that the center-of-mass fluctuations in the soliton state arise primarily from the depletion of the condensate to translation modes. We develop an extended mean-field theory to analytically reproduce these results obtained by the exact diagonalization method.
We study the coherent nonlinear tunneling dynamics of a binary mixture of Bose-Einstein condensates in a double-well potential. We demonstrate the existence of a type of mode associated with the ``swapping'' of the two species in the two wells of the potential. In contrast to the symmetry-breaking macroscopic quantum self-trapping (MQST) solutions, the swapping modes correspond to the tunneling dynamics that preserves the symmetry of the double-well potential. As a consequence of two distinct types of broken-symmetry MQST phases where the two species localize in different potential wells or coexist in the same well, the corresponding symmetry-restoring swapping modes result in dynamics where the two species either avoid or chase each other. In view of the possibility to control the interaction between the species, the binary mixture offers a very robust system to observe these novel effects as well as the phenomena of Josephson oscillations and $\ensuremath{\pi}$ modes.
Due to the coexistence of phase coherence and magnetic order, Bose-Einstein condensates with internal degrees of freedom, i.e., spinor BECs, are considered to serve as an ideal table-top playground for the studies of various phenomena in different fields of physics.They have been widely used in studying spin textures and topological defects, coherent spin dynamics, quantum symmetry breaking, quantum phase transitions, etc.However, the theoretical framework used in these works is based on the mean-field theory in which the effects of quantum fluctuations are ignored.Despite the success of the mean-field theory in the description of spinor BECs, there are particular features whose account must rely on a beyond-mean-field theory as the effects of quantum fluctuations become significant.In this thesis, we develop the Beliaev theory, which is a Green's function approach beyond the Bogoliubov theory, for spinor BECs to study the effects of quantum fluctuations on the condensates' phase structures, elementary excitations, stabilities, and phase transitions.First, we point out that the Bogoliubov theory fails to capture the metastable states associated with first-order quantum phase transitions in spin-2 BECs.In fact, we show that these metastable states are induced by quantum fluctuations, and only by going to the next-order approximation, i.e., the Beliaev theory, can we show that they indeed appear around the phase boundaries.Besides the fluctuation-induced metastability, we find another important class of first-order quantum phase transitions in both spin-1 and spin-2 BECs.In this class, there is no metastable state to all orders of approximation since it is prohibited by a high symmetry of the Hamiltonian at the phase boundary, resulting in a characteristic flat energy landscape.Despite being first-order phase transitions, the flat energy landscape leads to the criticality in the dynamics of the condensate through these transitions in a manner similar to second-order phase transitions.Second, we succeed in deriving the analytic expression for the energy gap of the so-called quasi-Nambu-Goldstone (quasi-NG) modes, which are excitations not generated by spontaneous symmetry breaking, in the nematic phase of spin-2 BECs.Although quasi-NG modes are gapless at the mean-field level, we prove that with quantum corrections they acquire a nonzero energy gap.From the obtained magnitude of the energy gap, we can evaluate the critical temperature above which a topological defect such as a vortex of spin nematicity would decay by emitting thermally excited quasi-NG modes.We also study how the propagation of quasi-NG modes in space is affected by the particle-number density fluctuations of the condensate.Third, we calculate the damping rates of various types of quasiparticles in a spin-2 BEC including phonons, magnons, and quasi-NG modes.They actually have finite lifetime and decay via numerous channels of collision with the condensate atoms.Using either the Fermi's golden rule or the spinor Beliaev theory, we obtain the analytic expressions for the damping rates of phonons and magnons, from which the power-law dependence of the damping rates on the momentum is deduced.In contrast, the damping of quasi-NG modes is suppressed due to the energy conservation.
We investigate dipolar Bose-Einstein condensates in a complex external double-well potential that features combined parity and time-reversal symmetry. On the basis of the Gross-Pitaevskii equation we study the effects of the long-ranged anisotropic dipole-dipole interaction on ground and excited states by the use of a time-dependent variational approach. We show that the property of a similar nondipolar condensate of possessing real energy eigenvalues in certain parameter ranges is preserved despite the inclusion of this nonlinear interaction. Furthermore, we present states that break the PT\mathcal{PT}PT symmetry and investigate the stability of the distinct stationary solutions. In our dynamical simulations we reveal a complex stabilization mechanism for PT\mathcal{PT}PT-symmetric as well as for PT\mathcal{PT}PT-broken states which are, in principle, unstable with respect to small perturbations.
We discuss the collective excitations in a spatially inhomogeneous (cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature (T≪TλT \ll T_λT≪Tλ​). The main result is the dispersion relation for a (first) sound wave that is obtained by describing the perturbation as a Lagrangian coordinate. The dispersion curve is in good agreement with the Bogoliubov phonon spectrum ω=ckω=c kω=ck, where k=kθ=m/rk=k_θ=m/rk=kθ​=m/r, the wave number and c=[4πaρℏ2]1/2/Mc=[4 πa ρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M, the speed of first sound. Based on Bohm's quantum theory, a spontaneously broken gauge symmetry in a quantum fluid is discussed in terms of the quantum fluctuation-dissipation, from which it is shown that the symmetry breaking takes place at the free surface of BEC in an external field.