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arXiv:1301.3642 · Physical Review A

Fluctuation-induced and symmetry-prohibited metastabilities in spinor Bose-Einstein condensates

Nguyen Thanh Phuc, Yuki Kawaguchi, Masahito Ueda

Published Oct 21, 2013• 16 citations
arXivHTMLPDF

Abstract

Spinor Bose-Einstein condensates provide a unique example in which the Bogoliubov theory fails to describe the metastability associated with first-order quantum phase transitions. This problem is resolved by developing the spinor Beliaev theory which takes account of quantum fluctuations of the condensate. It is these fluctuations that generate terms of higher than the fourth order in the order-parameter field which are needed for the first-order phase transitions. Besides the conventional first-order phase transitions which are accompanied by metastable states, we find a class of first-order phase transitions which are not accompanied by metastable states. The absence of metastability in these phase transitions holds to all orders of approximation since the metastability is prohibited by the symmetry of the Hamiltonian at the phase boundary. Finally, the possibility of macroscopic quantum tunneling from a metastable state to the ground state is discussed.

DOI
10.1103/physreva.88.043629

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Spinor Bose-Einstein condensates provide a unique example in which the Bogoliubov theory fails to describe the metastability associated with first-order quantum phase transitions. This problem is resolved by developing the spinor Beliaev theory which takes account of quantum fluctuations of the condensate. It is these fluctuations that generate terms of higher than the fourth order in the order-parameter field which are needed for the first-order phase transitions. Besides the conventional first-order phase transitions which are accompanied by metastable states, we find a class of first-order phase transitions which are not accompanied by metastable states. The absence of metastability in these phase transitions holds to all orders of approximation since the metastability is prohibited by the symmetry of the Hamiltonian at the phase boundary. Finally, the possibility of macroscopic quantum tunneling from a metastable state to the ground state is discussed.
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.
Spinor Bose-Einstein condensates under external magnetic fields exhibit well-characterized spin domains of its ground state due to spin-dependent interactions. At low temperatures, collision-induced spin-mixing instabilities may promote the condensate to dwell into metastable states occurring near the phase boundaries. In this paper, we study theoretically the metastable spin-phase diagram of a spin-1 antiferromagnetic Bose-Einstein condensate at zero and finite temperatures. The approach makes use of Hartree-Fock theory and exploits the symmetry of the Hamiltonian and of the order parameters yielding a closed system of transcendental equations for the free energy, fully avoiding the use of self-consistency methods. Our results are consistent with recent experiments and allow us to explain qualitatively the different types of observed quench dynamics. In addition, we found that similar phenomena should occur in antiferromagnetic spinor condensates with a sudden change in the temperature. It is shown also that the increase in temperature induces a traceable shift of the ferromagnetic-polar transition boundary, behavior previously not noted by self-consistent mean-field calculations.
We study atomic number fluctuations in a spinor Bose–Einstein condensate and use it to characterize behaviors of dynamical quantum phase transitions. We observe that the quench dynamics of the atomic number fluctuations exhibit a nonanalytical change with respect to a parameter in the final Hamiltonian and demonstrate that the atomic number fluctuations can be used to detect both the ground state phase diagram and the highest energy state quantum phase transition. We further analyze the dynamics of atomic number fluctuations for finite magnetization and show that the quantum fluctuations could also be used to probe the phase transitions of the highest energy level and the ground state.
We generalize the concept of quantum phase transitions, which is conventionally defined for a ground state and usually applied in the thermodynamic limit, to one for metastable states in finite-size systems. In particular, we treat the one-dimensional Bose gas on a ring in the presence of both interactions and rotation. To support our study, we bring to bear mean-field theory, i.e., the nonlinear Schr\"odinger equation, and linear perturbation or Bogoliubov--de Gennes theory. Both methods give a consistent result in the weakly interacting regime: there exist two topologically distinct quantum phases. The first is the typical picture of superfluidity in a Bose-Einstein condensate on a ring: average angular momentum is quantized and the superflow is uniform. The second is where one or more dark solitons appear as stationary states, breaking the symmetry, the average angular momentum becomes a continuous quantity. The phase of the condensate can therefore be continuously wound and unwound.
It is argued that a continuous quantum phase transition between different ordered phases in spinor Bose-Einstein condensates predicted by the mean-field theory is vulnerable to quantum fluctuations. By analyzing Lee-Huang-Yang corrections in the condensate, we demonstrate that the so-called Coleman-Weinberg mechanism takes place in such a transition, that is, the transition becomes of the first order by quantum fluctuations. A jump to be expected in this first-order transition is induced by a correction from density fluctuations despite a transition between different magnetic properties with keeping condensation. We exemplify this with an experimentally relevant case and show that a measurement of a condensate depletion can be utilized to confirm the first-order transition.
It is argued that a continuous quantum phase transition between different ordered phases in spinor Bose-Einstein condensates predicted by the mean-field theory is vulnerable to quantum fluctuations. By analyzing Lee-Huang-Yang corrections in the condensate, we demonstrate that the so-called Coleman-Weinberg mechanism takes place in such a transition, that is, the transition becomes of the first order by quantum fluctuations. A jump to be expected in this first-order transition is induced by a correction from density fluctuations despite a transition between different magnetic properties with keeping condensation. We exemplify this with an experimentally relevant case and show that a measurement of a condensate depletion can be utilized to confirm the first-order transition.