Search

The long-range and anisotropic nature of the dipolar interaction provides the so-called supersolid phases in Bose-Einstein condensates (BECs) in an optical lattice. However, in a certain area of dipole interaction parameters, BECs can form into a droplet. In this paper, in order to qualitatively understand the droplet formations, we propose a toy model that allows us to estimate the size and shape of droplets in dipolar Bose-Hubbard system in the optical lattice. We compare results of the toy model with numerical solutions of the mean-field calculation., 7 pages, 3 figures

The phase difference of the macroscopic wave function is a unique structure of the soliton in an atomic Bose--Einstein condensate (BEC). However, experiments on ultracold atoms so far have observed the valley of the density profile to study the dynamics of solitons. We propose a method to observe the phase difference of a soliton in a BEC by using an interference technique with Raman and rf pulses. We introduce a phase jump factor, which is an indicator to measure the phase difference between two points. It is demonstrated by using the projected Gross--Pitaevskii equation that an interference density ratio, the density ratio of two-component BECs after the Raman and rf pulses, reproduces the phase jump factor well. This technique will become an alternative method to study the decay and breakdown of a phase imprinted soliton in atomic BECs., Comment: 10 pages, 9 figures

We investigate dipole modes in a trapped Bose--Fermi mixture gas in the normal phase, composed of single-species bosons and single-species fermions with $s$-wave scattering. In the extremely low temperature regime, Bose--Einstein statistics and Fermi--Dirac statistics may give rise to an interesting temperature dependence of collective modes. Applying the moment method to the linearized Boltzmann equation, we study the transition of the dipole modes between the hydrodynamic regime and the collisionless regime., Comment: 8 pages, 1 figure

We study phase-imprinted solitons of ultracold bosons in an optical lattice with a harmonic trap, which shows the superfluid (SF) and Mott-insulator (MI) shell structures. The earlier study [Konstantin V. Krutitsky, J. Larson, and M. Lewenstein, Phys. Rev. A 82, 033618 (2010).] reported three types of phase-imprinted solitons in the Bose-Hubbard model: in-phase soliton, out-of-phase soliton, and wavelet. In this paper, we uncover the dynamical phase diagram of these phase-imprinted solitons, and find another type of the phase-imprinted soliton namely, the hybrid soliton. In the harmonically trapped system, the solitonic excitations created at the SF core cannot penetrate into the outer SF shell. This repulsion at the surface of the outer SF shell can be cured by inposing a repulsive potential at the center of the trap. These results can be interpreted as a kind of the impedance matching of excitations in BECs in terms of the effective chemical potentials or the local particle numbers in the shell, and the analogous results can be observed also in the sound wave created by the local single-shot pulse potential.

The Gibbs partition function is an important quantity in describing statistical properties of a system in thermodynamic equilibrium. There are several proposals to calculate the partition functions on near-term quantum computers. However, the existing schemes require many copies of the Gibbs states to perform an extrapolation for the calculation of the partition function, which could be costly performed on the near-term quantum computers. We propose a scheme to calculate the Gibbs function with the imaginary time evolution. After preparing Gibbs states with different temperatures by using the imaginary time evolution, we measure the overlap between them on a quantum circuit, which allows us to calculate the Gibbs partition function. Our scheme requires only 2N qubits to calculate the Gibbs function of N qubits.

Variational quantum algorithms (VQAs) have been considered to be useful applications of noisy intermediate-scale quantum (NISQ) devices. Typically, in the VQAs, a parametrized ansatz circuit is used to generate a trial wave function, and the parameters are optimized to minimize a cost function. On the other hand, blind quantum computing (BQC) has been studied in order to provide the quantum algorithm with security by using cloud networks. A client with a limited ability to perform quantum operations hopes to have access to a quantum computer of a server, and BQC allows the client to use the server's computer without leakage of the client's information (such as input, running quantum algorithms, and output) to the server. However, BQC is designed for fault-tolerant quantum computing, and this requires many ancillary qubits, which may not be suitable for NISQ devices. Here, we propose an efficient way to implement the NISQ computing with guaranteed security for the client. In our architecture, only N+ 1 qubits are required, under an assumption that the form of ansatzes is known to the server, where N denotes the necessary number of the qubits in the original NISQ algorithms. The client only performs single-qubit measurements on an ancillary qubit sent from the server, and the measurement angles can specify the parameters for the ansatzes of the NISQ algorithms. No-signaling principle guarantees that neither parameters chosen by the client nor the outputs of the algorithm are leaked to the server. This work paves the way for new applications of NISQ devices.

We study relaxation dynamics of dark soliton, created by a phase-imprinted method, in a two-dimensional trapped Bose-Einstein condensate at non-zero temperatures by using the projected Gross-Pitaevskii equation. At absolute zero temperature, a dark soliton is known to decay with a snake instability. At non-zero temperature, as we expected, we find that this snake instability cannot be clearly seen as in the absolute zero temperature case because of the presence of thermal fluctuations. However, we find that the decay rate, the half width of the overlap integral with respect to the phase-imprinted initial state, shows a power low decay as a function of the energy and finally remains a non-zero value., Comment: 5pages, 4figures

In order to model and evaluate large-scale quantum systems, e.g. quantum computer and quantum annealer, it is necessary to quantify the ``quantumness" of such systems. In this paper, we discuss the dimensionless combinations of basic parameters of large, partially quantum coherent systems, which could be used to characterize their degree of quantumness. Based on analytical and numerical calculations, we suggest one such number for a system of qubits undergoing adiabatic evolution, i.e., the accessibility index. Applying it to the case of D-Wave One superconducting quantum annealing device, we find that its operation as described falls well within the quantum domain., Comment: 14 pages, 2 figures

Boltzmann machine is a powerful tool for modeling probability distributions that govern the training data. A thermal equilibrium state is typically used for Boltzmann machine learning to obtain a suitable probability distribution. The Boltzmann machine learning consists of calculating the gradient of the loss function given in terms of the thermal average, which is the most time consuming procedure. Here, we propose a method to implement the Boltzmann machine learning by using Noisy Intermediate-Scale Quantum (NISQ) devices. We prepare an initial pure state that contains all possible computational basis states with the same amplitude, and apply a variational imaginary time simulation. Readout of the state after the evolution in the computational basis approximates the probability distribution of the thermal equilibrium state that is used for the Boltzmann machine learning. We actually perform the numerical simulations of our scheme and confirm that the Boltzmann machine learning works well by our scheme., 14 pages, 11 figures

Quantum annealing is an innovative idea and method for avoiding the increase of the calculation cost of the combinatorial optimization problem. Since the combinatorial optimization problems are ubiquitous, quantum annealing machine with high efficiency and scalability will give an immeasurable impact on many fields. However, the conventional quantum annealing machine may not have a high success probability for finding the solution because the energy gap closes exponentially as a function of the system size. To propose an idea for finding high success probability is one of the most important issues. Here we show that a degenerate two-level system provides the higher success probability than the conventional spin-1/2 model in a weak longitudinal magnetic field region. The physics behind this is that the quantum annealing in this model can be reduced into that in the spin-1/2 model, where the effective longitudinal magnetic field may open the energy gap, which suppresses the Landau--Zener tunneling providing leakage of the ground state. We also present the success probability of the $\Lambda$-type system, which may show the higher success probability than the conventional spin-1/2 model., Comment: 13 pages, 12 figures

Dipole oscillation is studied in a normal phase of a trapped Bose--Fermi-mixture gas composed of single-species bosons and single-species fermions. Applying the moment method to the linearized Boltzmann equation, we derive a closed set of equations of motion for the center-of-mass position and momentum of both components. By solving the coupled equations, we reveal the behavior of dipole modes in the transition between the collisionless regime and the hydrodynamic regime. We find that two oscillating modes in the collisionless regime have distinct fates in the hydrodynamic regime: one collisionless mode shows a crossover to a hydrodynamic in-phase mode, and the other collisionless mode shows a transition to two purely damped modes. The temperature dependence of these dipole modes are also discussed., 10 pages, 5 figures

We investigate a quantum spatial search problem on fractal lattices, such as Sierpinski carpets and Menger sponges. In earlier numerical studies of the Sierpinski gasket, the Sierpinski tetrahedron, and the Sierpinski carpet, conjectures have been proposed for the scaling of a quantum spatial search problem finding a specific target, which is given in terms of the characteristic quantities of a fractal geometry. We find that our simulation results for extended Sierpinski carpets and Menger sponges support the conjecture for the ${\it optimal}$ number of the oracle calls, where the exponent is given by $1/2$ for $d_{\rm s} > 2$ and the inverse of the spectral dimension $d_{\rm s}$ for $d_{\rm s} < 2$. We also propose a scaling hypothesis for the ${\it effective}$ number of the oracle calls defined by the ratio of the ${\it optimal}$ number of oracle calls to a square root of the maximum finding probability. The form of the scaling hypothesis for extended Sierpinski carpets is very similar but slightly different from the earlier conjecture for the Sierpinski gasket, the Sierpinski tetrahedron, and the conventional Sierpinski carpet., 7 pages, 7 figures

This paper discusses exact relations in Bose--Einstein condensates (BECs), starting from basic properties of an ideal Bose gas. In particular, focused on are the Hugenholtz--Pines relation, Nepomnyashchii--Nepomnyashchii identity, and identities for the density response function. After introducing these exact relations, a few approaches of many-body approximations are discussed, which satisfy the exact relations in BECs. This paper will serve as a bridge between theories on exact relations and those on approximations in BECs., 10 pages

We investigate collective excitations of density fluctuations and a dynamic density structure factor in a mixture of Bose and Fermi gases in a normal phase. With decreasing temperature, we find that the frequency of the collective excitation deviates from that of the hydrodynamic sound mode. Even at temperature much lower than the Fermi temperature, the collective mode frequency does not reach the collisionless limit analogous to zero sound in a Fermi gas, because of collisions between bosons and fermions.

We investigate a quantum spatial search problem on a fractal lattice. A recent study for the Sierpinski gasket and tetrahedron made a conjecture that the dynamics of the search on a fractal lattice is determined by spectral dimension. We tackle this problem for the Sierpinski carpet, and our simulation result supports the conjecture. We also propose a scaling hypothesis of oracle calls for the quantum amplitude amplification., 6 pages, 5 figures

We investigate multiparticle excitation effect on a collective density excitation as well as a single-particle excitation in a weakly interacting Bose--Einstein condensate (BEC). We find that although the weakly interacting BEC offers weak multiparticle excitation spectrum at low temperatures, this multiparticle excitation effect may not remain hidden, but emerges as bimodality in the density response function through the single-particle excitation. Identification of spectra in the BEC between the single-particle excitation and the density excitation is also assessed at nonzero temperatures, which has been known to be unique nature in the BEC at absolute zero temperature., 6 pages, 4 figures

The Hugenholtz-Pines (HP) theorem is derived for Bose-Einstein condensates (BECs) with internal degrees of freedom. The low-energy Ward-Takahashi identity is provided in the system with the linear and quadratic symmetry breaking terms. This identity serves to organize the HP theorem for multicomponent BECs, such as the binary BEC as well as the spin-$f$ spinor BEC in the presence of a magnetic field with broken U$(1)$$\times$SO$(3)$ symmetry. The experimental method based on the Stern-Gerlach experiment is proposed for studying the Ward-Takahashi identity., Comment: 14 pages, 1 figure

We study nonlinear mixing effects among quadrupole modes and scissors modes in a harmonically trapped Bose-Einstein condensate. Using a perturbative technique in conjunction with a variational approach with a Gaussian trial wave function for the Gross-Pitaevskii equation, we find that mode mixing selectively occurs. Our perturbative approach is useful in gaining qualitative understanding of the recent experiment [Yamazaki et al., J. Phys. Soc. Japan 84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well as a modulation phenomenon of the low-lying quadrupole mode by the high-lying quadrupole mode frequency. Within the second-order treatment of the nonlinear mode coupling terms, our approach predicts all the spectral peaks obtained by the numerical simulation of the Gross-Pitaevskii equation., Comment: 9pages, 5figures

We study mass and population imbalance effect on density (in-phase) and spin (out-of-phase) collective modes in a two-component normal Fermi gas. By calculating eigenmodes of the linearized Boltzmann equation as well as the density/spin dynamic structure factor, we show that mass and population imbalance effects offer a variety of collective mode crossover behaviors from collisionless to hydrodynamic regimes. The mass imbalance effect shifts the crossover regime to the higher-temperature, and a significant peak of the spin dynamic structure factor emerges only in the collisionless regime. This is in contrast to the case of mass and population balanced normal Fermi gases, where the spin dynamic response is always absent. Although the population imbalance effect does not shift the crossover regime, the spin dynamic structure factor survives both in the collisionless and hydrodynamic regimes., 9 pages, 6 figures

We propose a new criterion of stability or instability of one-directional superflow of Bose condensates. In d-dimensional condensates with steady superflow, spectral function \(\rho(\mathbf {r},\omega)=\sum_{l}|\langle l|\hat{n}(\mathbf{r})|\mathrm{g}\rangle|^{2}\delta (\omega-E_{l}+E_{\mathrm{g}})\) of the local density \(\hat{n}(\mathbf{r})\) at the critical velocity behaves as ρ(r,ω)∝ωβ with β

We study tunneling properties of collective excitations in spin-1 Bose-Einstein condensates. In the absence of magnetic fields, the total transmission in the long wavelength limit occurs in all kinds of excitations but the quadrupolar spin mode in the ferromagnetic state. The quadrupolar spin mode alone shows the total reflection. A difference between those excitations comes from whether the wavefunction of an excitation corresponds to that of the condensate in the long wavelength limit. The correspondence results in the total transmission as in the spinless BEC., Comment: 6 pages, 5 figures

We present a Green's function formalism for an interacting Bose-Einstein condensate (BEC) satisfying the two required conditions: (i) the infrared-divergent longitudinal susceptibility with respect to the BEC order parameter, and (ii) the Nepomnyashchii-Nepomnyashchii identity stating the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. These conditions cannot be described by the ordinary mean-field Bogoliubov theory, the many-body $T$-matrix theory, as well as the random-phase approximation with the vertex correction. In this paper, we show that these required conditions can be satisfied, when we divide many-body corrections into singular and non-singular parts, and separately treat them as different self-energy corrections. The resulting Green's function may be viewed as an extension of the Popov's hydrodynamic theory to the region at finite temperatures. Our results would be useful in constructing a consistent theory of BECs satisfying various required conditions, beyond the mean-field level., Comment: 31 pages, 9 figures

We compare many-body theories describing fluctuation corrections to the mean-field theory in a weakly interacting Bose-condensed gas. Using a generalized random-phase approximation, we include both density fluctuations and fluctuations in the particle-particle scattering channel in a consistent manner. We also separately examine effects of the fluctuations within the framework of the random-phase approximation. Effects of fluctuations in the particle-particle scattering channel are also separately examined by using the many-body $T$-matrix approximation. We assess these approximations with respect to the transition temperature ${T}_{\mathrm{c}}$, the order of phase transition, as well as the so-called Nepomnyashchii-Nepomnyashchii identity, which states the vanishing off-diagonal self-energy in the low-energy and low-momentum limit. Since the construction of a consistent theory for interacting bosons which satisfies various required conditions is a long-standing problem in cold atom physics, our results would be useful for this important challenge.

We study a stability criterion hypothesis for superfluids in terms of the the local density spectral function $I_n (r, \omega)$ applicable both to homogeneous and inhomogeneous systems. We evaluate the local density spectral function in the presence of a one-dimensional repulsive/attractive external potential within the Bogoliubov theory using solutions of the tunneling problem. We also evaluate the local density spectral function using an orthogonal basis, and calculate the autocorrelation function $C_n (r,t)$. When superfluids flow below a threshold, we find that in the $d$-dimensional system, $I_n (r, \omega) \propto \omega^{d}$ in the low-energy regime and $C_n (r, t) \propto 1/t^{d+1}$ in the long-time regime hold. When superfluids flow with the critical current, on the other hand, we find $I_n (r, \omega) \propto \omega^{\beta}$ in the low-energy regime and $C_n (r,t) \propto 1/t^{\beta+1}$ in the long-time regime with $\beta < d$. These results support the stability criterion hypothesis recently proposed., Comment: 18 pages, 10 figures

We investigate the tunneling properties of collective excitations through a domain wall in the ferromagnetic phase of a spin-1 spinor Bose--Einstein condensate. Within the mean-field theory at T=0, we show that the transverse spin wave undergoes perfect reflection in the low-energy limit. This reflection property differs considerably from that of a domain wall in a Heisenberg ferromagnet where spin-wave excitations exhibit perfect transmission at arbitrary energy. When the Bogoliubov mode is scattered from this domain wall soliton, the transmission and reflection coefficients exhibit pronounced non-monotonicity. In particular, we find perfect reflection of the Bogoliubov mode at energies where bound states appear. This is in stark contrast to the perfect transmission of the Bogoliubov mode with arbitrary energy through a dark soliton in a scalar Bose--Einstein condensate., Comment: 10 pages, 7 figures

We investigate tunneling properties of collective modes in the polar phase of a spin-1 spinor Bose-Einstein condensate. This spinor BEC state has two kinds of gapless modes, i.e., Bogoliubov mode and spin-wave. Within the framework of the mean-field theory at T=0, we show that these Goldstone modes exhibit the perfect transmission in the low-energy limit. Their anomalous tunneling behaviors still hold in the presence of superflow, except in the critical current state. In the critical current state, while the tunneling of Bogoliubov mode is accompanied by finite reflection, the spin-wave still exhibit the perfect transmission, unless the strengths of a spin-dependent and spin-independent interactions take the same value., 8 pages, 3 figures

We investigate tunneling properties of collective excitations in the ferromagnetic phase of a spin-1 spinor Bose-Einstein condensate (BEC). In addition to the Bogoliubov mode, this superfluid phase has two spin excitations, namely, the gapless transverse spin wave and the quadrupolar mode with a finite excitation gap. In the mean-field theory at T=0, we examine how these collective modes tunnel through a barrier potential that couples to the local density of particles. In the presence of supercurrent with a finite momentum $q$, while the Bogoliubov mode shows the so-called anomalous tunneling behavior (which is characterized by perfect transmission) in the low energy limit, the transverse spin-wave transmits perfectly only when the momentum $k$ of this mode coincides with $\pm q$. At $k=\pm q$, the wave function of this spin wave has the same form as the condensate wave function in the current carrying state, so that the mechanism of this perfect transmission is found to be the same as tunneling of supercurrent. Using this fact, the perfect transmission of the spin wave is proved for a generic barrier potential. We show that such perfect transmission does not occur in the quadrupolar mode. Further, we consider the effects of potentials breaking U(1) and spin rotation symmetries on the transmission properties of excitations. Our results would be useful for understanding excitation properties of spinor BECs, as well as the anomalous tunneling phenomenon in Bose superfluids., Comment: 28pages, 7figures

We investigate tunneling of excitations across a potential barrier separating two spin-1 Bose-Einstein condensates. Using the mean-field theory at the absolute zero temperature, we determine transmission coefficients of excitations in the saturated magnetization state and unsaturated magnetization states. All excitations except the quadrupolar spin mode in the saturated magnetization state show the anomalous tunneling phenomenon characterized as perfect tunneling in the low momentum limit through a potential barrier. The quadrupolar spin mode in the saturated magnetization state, whose spectrum is massive, shows total reflection. We discuss properties common between excitations showing the anomalous tunneling phenomenon. Excitations showing perfect tunneling have gapless spectrum in the absence of the magnetic field, and their wave functions in the low energy limit are the same as the condensate wave function., Comment: 45 pages, 16 figures

We present the simplest theory of perfect tunneling of an excitation in a Bose-Einstein condensate (BEC) through an impurity potential with an arbitrary shape in the low-momentum limit. That is for the transverse spin wave in the ferromagnetic phase of a spin-1 BEC. This mode obeys a Schr\"odinger-type equation; yet, effects of the potential on its transmission coefficient $T$ and on its scattering cross section $sigma$ vanish in that limit. The order parameter determines $T$, and the momentum $p$-dependence of $sigma$ exhibits a Rayleigh scattering type ($sigma propto p^{4}$). These properties are common between two types of Nambu-Goldstone modes: this spin wave and the Bogoliubov mode., Comment: 5 pages

The dynamic structure factor of a normal Fermi gas is investigated by using the moment method for the Boltzmann equation. We determine the spectral function at finite temperatures over the full range of crossover from the collisionless regime to the hydrodynamic regime. We find that the Brillouin peak in the dynamic structure factor exhibits a smooth crossover from zero to first sound as functions of temperature and interaction strength. The dynamic structure factor obtained using the moment method also exhibits a definite Rayleigh peak ({omega}{approx}0), which is a characteristic of the hydrodynamic regime. We compare the dynamic structure factor obtained by the moment method with that obtained from the hydrodynamic equations.

We propose a stability criterion of superfluids in condensed Bose-Einstein systems, which incorporates the spectral function or the autocorrelation function of the local density. Within the Gross-Pitaevskii-Bogoliubov theory, we demonstrate the validity of our criterion for the soliton-emission instability, with use of explicit forms of zero modes of the Bogoliubov equation and a dynamical scaling near the saddle-node bifurcation. We also show that the criterion is applicable to the Landau phonon instability and the Landau roton instability within the single-mode approximation.

On the basis of a moment method, general solutions of a linearized Boltzmann equation for a normal Fermi system are investigated. In particular, we study the sound velocities and damping rates as functions of the temperature and the coupling constant. In the extreme limits of collisionless and hydrodynamic regimes, eigenfrequency of sound mode obtained from the moment equations reproduces the well-known results of zero sound and first sound. In addition, the moment method can describe crossover between those extreme limits at finite temperatures. Solutions of the moment equations also involve a thermal diffusion mode. From solutions of these equations, we discuss excitation spectra corresponding to the particle-hole continuum as well as collective excitations. We also discuss a collective mode in a weak coupling case., 38 pages, 5 figures

We investigate transmission and reflection of Bose-Einstein-condensate excitations in the low-energy limit across a potential barrier separating two condensates with different densities. The Bogoliubov excitation in the low-energy limit has an incident angle where the perfect transmission occurs. This condition corresponds to Brewster's law for the electromagnetic wave. The total internal reflection of the Bogoliubov excitation is found to occur at a large incident angle in the low-energy limit. The anomalous tunneling named by Kagan et al. [Yu. Kagan et al., Phys. Rev. Lett. 90, 130402 (2003)] can be understood in terms of the impedance matching. In the case of the normal incidence, comparison with the results in Tomonaga-Luttinger liquids is made.

We determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in the harmonic trap by using the local density approximation. The adiabatic phase diagram shows the fermionic condensate fraction composed of condensed molecules and Cooper pair atoms. The key idea of our work is conservation of entropy through the adiabatic process, extending the study of Williams et al. [Williams et al., New J. Phys. 6, 123 (2004)] for an ideal gas mixture to include the resonant interaction in a mean-field theory. We also calculate the molecular conversion efficiency as a function of initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field., 13 pages, 8 figures. In press, "Journal of the Physical Society of Japan", Vol.76, No.6

We investigate tunneling properties of collective spin-wave excitations in the polar state of a spin-1 spinor Bose-Einstein condensate. Within the mean-field theory at T = 0, we show that when the condensate is in the critical supercurrent state, the spin wave mode exhibits perfect transmission through a nonmagnetic potential barrier in the low energy limit, unless the strength of a spin-independent interaction co equals that of a spin-dependent interaction co Such an anomalous tunneling behavior is absent in the case of a magnetic barrier. We also clarify a scaling law of the transmission probability as a function of the mode energy.

The ferromagnetic spin wave (FSW) in classical Heisenberg chain exhibits the perfect transmission in the long-wavelength limit in the transmission-reflection problem with an inhomogeneity of exchange integral. In the presence of local magnetic field, on the other hand, FSW undergoes the perfect reflection in the long-wavelength limit. This difference in the long-wavelength limit is attributed to the symmetry property of the scatterers; it is crucial whether the potential preserves or breaks the spin rotation symmetry. Our result implies that the anomalous tunneling (i.e., perfect transmission in the low-energy limit) found both in scalar and spinor BECs is not specific to gapless modes in superfluids but is a common property shared with generic Nambu-Goldstone modes in the presence of a symmetry-preserving potential scatterer.

We investigate transmission and reflection of Bose-Einstein condensate excitations in the low-energy limit across a potential barrier separating two condensates with different densities. Bogoliubov excitation in the low-energy limit has an incident angle where perfect transmission occurs. This condition corresponds to the Brewster's law for the electromagnetic wave. There also exists the total internal reflection of the Bogoliubov excitation in the low-energy limit. In the case of the normal incidence, our result in the low-energy limit is consistent with a result for weakly interacting one-dimensional Bose gases treated as Tomonaga-Luttinger liquids.