We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, pressure, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is found that in the spin-2 nematic phase there are special Bogoliubov modes that have gapless linear dispersion relations but do not belong to the NG modes.
We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, pressure, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is found that in the spin-2 nematic phase there are special Bogoliubov modes that have gapless linear dispersion relations but do not belong to the NG 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.
The interplay of quantum fluctuation through the Lee-Huang-Yang (LHY) correction and spin-orbit coupling can modify the stability of the ultradilute Bose-Bose mixtures and result in rich nonlinear physics. Here, under the action of spin-orbit coupling, LHY correction, and rotation, the ground state and vortex excitation of spin-orbit coupled Bose-Einstein condensates (BECs) with LHY correction trapped in a rotating potential are studied. Stable isotropic and anisotropic ground states with zero momentum (in the zero-momentum phase) and nonzero momentum (in the plane wave phase) are obtained. It is shown that anisotropic ground states are persistent near the phase boundary. Particularly, unpolarized and polarized vortex states are predicted. The phase transition condition, the effective mass and quadrupole surface mode describing the anisotropic (deformation) of the condensates, and the critical rotation frequency for vortex excitation are provided analytically. Owing to the action of spin-orbit coupling, the effective mass of the condensates is spatially dependent and asymmetric in two phases, resulting in distinct quadrupole modes, deformations, and vortex excitations in the two phases, especially when the LHY correction is present. The LHY correction promotes the system in the zero-momentum phase with unpolarized vortex states, while the spin-orbit coupling promotes the emergence of a plane wave phase with polarized vortex states. The phase transition and vortex excitation can be well manipulated by spin-orbit coupling, Raman coupling, LHY correction, and external potential.
We investigate a weakly interacting two-component Bose--Einstein condensate in the miscible regime in the presence of \emph{altermagnetism}, i.e., a collinear and globally compensated magnetic order that breaks spin-rotation symmetry while maintaining zero net magnetization. Within Bogoliubov theory, we derive the quasiparticle spectrum and coherence factors and show that altermagnetic order generically induces an angular dependence of the low-energy excitations. As a result, the sound velocity, momentum-resolved magnetization in the quantum depletion, and density--spin response functions acquire anisotropic components. We show that these anisotropic contributions vanish after angular averaging, consistent with the defining feature of altermagnetism: nontrivial local spin polarization without a global magnetization. Finally, we evaluate the Lee--Huang--Yang correction to the ground-state energy in the altermagnetic phase. Our results should be testable with ultracold-atom experiments in the foreseeable future.
Quantum fluctuations play a pivotal role in determining the ground state properties and phase transitions of ultra-cold bosonic systems. This study presents a comprehensive theoretical investigation of fluctuation effects beyond mean-field theory in dilute Bose gases, examining their impact on the superfluid–Mott insulator transition, quantum droplet formation, and critical phenomena. Beginning with the Gross–Pitaevskii mean-field framework , we systematically incorporate quantum corrections through Bogoliubov theory, yielding the Lee–Huang–Yang (LHY) energy correction scaling as . The quantum depletion quantifies the fraction of atoms driven from the condensate by zero-point fluctuations. For the Bose–Hubbard model , quantum Monte Carlo simulations establish the superfluid–Mott insulator critical point at , shifted 15% from the mean-field prediction. Critical exponents and confirm 3D XY universality. Quantum droplets in binary mixtures are stabilized by the repulsive LHY correction against mean-field collapse, with equilibrium densities matching experimental observations in K mixtures. Finite-size scaling analysis and the Ginzburg criterion delineate regimes where fluctuation effects dominate. All theoretical predictions are validated against precision measurements in ultracold atomic gases.
Quantum droplets have been realized in experiments on binary boson mixtures and dipolar Bose gases. In these systems, the mean-field energy of the Bose-Einstein condensation is attractive, and the repulsive Lee-Huang-Yang energy is crucial for stability. However, since the Lee-Huang-Yang term is not taken into account when constructing the Bogoliubov Hamiltonian, in the droplet regime one faces the problem of dynamically unstable phonon modes. In this work, we go beyond the Bogoliubov theory to study how the phonon mode is stabilized in the quantum droplet of a binary boson mixture. Similar to Beliaev's approach to a single-component Bose gas, we compute higher-order contributions to the self-energy of the boson propagator. We find that the interaction between spin and phonon excitations is the key for the phonon stability. We obtain the sound velocity which can be tested by measuring the superfluid critical velocity of the droplet in experiments. Beliaev damping of this quantum droplet is also discussed.
We investigate the formation, stability, and dynamics of solitons in a one-dimensional binary Bose-Einstein condensate under the action of the spin-orbit-coupling and Lee-Huang-Yang (LHY) correction to the underlying system of the Gross-Pitaevskii equations. We identify the semi-dipole (SD) family of solitons and thoroughly analyze its properties. The numerical analysis reveals intricate bifurcations, including transitions from real to complex-valued stationary wave functions of the SD solitons and norm-dependent dynamical instabilities. Stability maps in the plane of the solitons' norm and interaction strength exhibit areas of monostability, oscillatory behavior, and soliton splitting. Solitons with complex stationary wave functions emerge as ground states in broad parameter areas due to the effects of the LHY terms. The other soliton species, in the form of mixed modes (MMs), do not feature the compexification bifurcation. In the LHY-dominated regime, the SD and MM solitons exhibit identical values of the energy for the same norm. The results deepen the understanding of nonlinear matter-wave states and reveal multistable ones in quantum gases.