Zero-mode operators associated with a spontaneous breakdown of internal symmetry are often suppressed and their quantum fluctuations are neglected owing to the absence of a definite criterion for determining the ground state and infrared divergences in physical quantities. Because the zero-mode fluctuations for systems with finite-size and spatially inhomogeneity cannot be neglected, we have proposed a novel treatment of the zero-mode for finite systems, which helps determine the unique ground state and evaluate the physical quantities without divergence. Following the proposed treatment, we consider trapped systems with two-component Bose-Einstein condensates. The results obtained demonstrate that the interaction strength between the two components considerably affect the dynamical properties, illustrating the importance of the zero-mode for finite systems.

Observed well-developed $\alpha$ cluster states in $^{16}$O, located above
the four $\alpha$ threshold, are investigated from the viewpoint of
Bose-Einstein condensation of $\alpha$ clusters by using a field-theoretical
superfluid cluster model in which the order parameter is defined. The
experimental energy levels are reproduced well for the first time by
calculation. In particular, the observed 16.7 MeV $0_7^+$ and 18.8 MeV $0_8^+$
states with low-excitation energies from the threshold are found to be
understood as a manifestation of the states of the Nambu-Goldstone zero-mode
operators, associated with the spontaneous symmetry breaking of the global
phase, which is caused by the Bose-Einstein condensation of the vacuum 15.1 MeV
$0^+_6$ state with a dilute well-developed $\alpha$ cluster structure just
above the threshold. This gives evidence of the existence of the Bose-Einstein
condensate of $\alpha$ clusters in $^{16}$O. It is found that the emergence of
the energy level structure with a well-developed $\alpha$ cluster structure
above the threshold is robust, almost independently of the condensation rate of
$\alpha$ clusters under significant condensation rate. The finding of the
mechanism why the level structure that is similar to $^{12}$C emerges above the
four $\alpha$ threshold in $^{16}$O reinforces the concept of Bose-Einstein
condensation of $\alpha$ clusters in addition to $^{12}$C.

We propose a new protocol to examine many-polaron properties in a cold atom
experiment. Initially, polaronic clouds are prepared around the opposite edges
of a majority gas cloud. After time evolution, the collision of two clouds
exhibits various polaronic effects. To see how {\it collective} properties of
many polarons with mediated interactions appear in the case in which the
impurity and majority gases are composed of mass-balanced fermions with
different spin components, we perform a nonlinear hydrodynamic simulation for
collisional dynamics of two Fermi polaronic clouds. We found that the dynamics
is governed by the impurity Fermi pressure, polaron energy, and multi-polaron
correlations. In particular, shock waves occur in such a way as to reflect the
many-body properties of polarons through the first sound of minority clouds.
Our idea is applicable to other systems such as Bose polarons as well as
mass-imbalanced mixtures.

We investigate how the presence of a localized impurity in a Bose-Einstein
condensate of trapped cold atoms that interact with each other weakly and
repulsively affects the profile of the condensed and excited components at zero
temperature. By solving the Gross-Pitaevskii and Bogoliubov-de Gennes
equations, we find that an impurity-boson contact attraction (repulsion) causes
both components to change in spatial structure in such a way as to be enhanced
(suppressed) around the impurity, while slightly declining (growing) in a far
region from the impurity. Such behavior of the quantum depletion of the
condensate can be understood by decomposing the impurity-induced change in the
profile of the excited component with respect to the radial and azimuthal
quantum number. A significant role of the centrifugal potential and the "hole"
excitation level is thus clarified.

Bose–Einstein condensation of α clusters in light and medium-heavy nuclei is studied in the frame of the field-theoretical superfluid cluster model. The order parameter of the phase transition from the Wigner phase to the Nambu–Goldstone phase is a superfluid amplitude, square of the moduli of which is the superfluid density distribution. The zero-mode operators due to the spontaneous symmetry breaking of the global phase in the finite number of α clusters are rigorously treated. The theory is systematically applied to Nα nuclei from 12C to 52Fe at various condensation rates. In 12C it is found that the energy levels of the gas-like well-developed α cluster states above the Hoyle state are reproduced well in agreement with experiment for realistic condensation rates of α clusters. The electric E2 and E0 transitions are calculated and found to be sensitive to the condensation rates. The profound raison d'être of the α cluster gas-like states above the Hoyle state, whose structure has been interpreted geometrically in the nuclear models without the order parameter such as the cluster models or ab initio calculations, is revealed. It is found that, in addition to the Bogoliubov–de Gennes vibrational mode states, collective states of the zero-mode operators appear systematically at low excitation energies from the Nα
threshold energy. These collective states, which are new-type soft modes in nuclei due to the Bose–Einstein condensation of the α clusters, emerge systematically in light– and medium-heavy–mass regions and are also located at high excitation energies from the ground state in contrast to the traditional concept of soft mode in the low-excitation-energy region.

© 2016 American Physical Society.An effective field theory of α-cluster condensation is formulated as a spontaneously broken symmetry in quantum field theory to understand the raison d'être and the nature of the Hoyle and α-cluster states in C12. The Nambu-Goldstone and Higgs mode operators in infinite systems are replaced with a pair of canonical operators whose Hamiltonian gives rise to discrete energy states in addition to the Bogoliubov-de Gennes excited states. The calculations reproduce well the experimental spectrum of the α-cluster states. The existence of the Nambu-Goldstone-Higgs states is demonstrated and crucial. The γ-decay transitions are also obtained.

１２Cの3アルファ粒子構造のホイル状態がボーズ凝縮状態であり、南部ゴールドストーン、ヒッグスモードの状態が実在すること、２＋及び４＋状態は回転バンド状態ではなく、凝縮体の振動状態であることを示した。

To formulate the zero modes in a finite-size system with spontaneous breakdown of symmetries in quantum field theory is not trivial, for in the naive Bogoliubov theory, one encounters difficulties such as phase diffusion, the absence of a definite criterion for determining the ground state, and infrared divergences. An interacting zero mode formulation that has been proposed for systems with a single zero mode to avoid these difficulties is extended to general systems with multiple zero modes. It naturally and definitely gives the interactions among the quantized zero modes, the consequences of which can be observed experimentally. In this paper, as a typical example, we consider an atomic Bose-Einstein condensed system with a dark soliton that contains two zero modes corresponding to the spontaneous breakdown of the U(1) gauge and translational symmetries. Then we evaluate the standard deviations of the zero mode operators and see how the mutual interaction between the two zero modes affects them.

A complex eigenvalue in the Bogoliubov-de Gennes equations for a stationary Bose-Einstein condensate in the ultracold atomic system indicates the dynamical instability of the system. We also have the modes with zero eigenvalues for the condensate, called the zero modes, which originate from the spontaneous breakdown of symmetries. Although the zero modes are suppressed in many theoretical analyses, we take account of them in this paper and argue that a zero mode can change into one with a pure imaginary eigenvalue by applying a symmetry breaking external perturbation potential. This emergence of a pure imaginary mode adds a new type of scenario of dynamical instability to that characterized by the complex eigenvalue of the usual excitation modes. For illustration, we deal with two one-dimensional homogeneous Bose-Einstein condensate systems with a single dark soliton under a respective perturbation potential, breaking the invariance under translation, to derive pure imaginary modes. (C) 2014 Elsevier Inc. All rights reserved.

The unperturbed Hamiltonian for the Bose-Einstein condensate, which includes not only the first and second powers of the zero mode operators but also the higher ones, is proposed to determine a unique and stationary vacuum at zero temperature. From the standpoint of quantum field theory, it is done in a consistent manner that the canonical commutation relation of the field operator is kept. In this formulation, the condensate phase does not diffuse and is robust against the quantum fluctuation of the zero mode. The standard deviation for the phase operator depends on the condensed atom number with the exponent of -1/3, which is universal for both homogeneous and inhomogeneous systems.

Investigating the Bogoliubov–de Gennes equation, widely used in the analysis of Bose–Einstein condensate in neutral atomic gases, we reveal some aspect of the interplay between the zero and complex modes. The complex mode implies that the system is dynamically unstable. The zero modes originate from the spontaneous symmetry breakdown, but their roles are not clear because they cause the infrared divergence. In this paper, we deal with the system of a dark soliton in Bose–Einstein condensate which has two zero modes corresponding to the spontaneously broken U(1) gauge and translational symmetries, and show that the zero and its adjoint modes, associated with the spontaneous breakdown of translational symmetry, turn into pure imaginary modes under theperturbation which explicitly breaks translational symmetry.