# Bogoliubov spectrum and the dynamic structure factor in a quasi-two-
dimensional spin-orbit coupled BEC

Inderpreet Kaur and Sankalpa Ghosh  Department of Physics, Indian Institute of
Technology Delhi, New Delhi-110016, India

###### Abstract

We compute the the Bogoliubov-de-Gennes excitation spectrum in a trapped two-
component spin-orbit-coupled (SOC) Bose-Einstein condensate (BEC) in quasi-
two-dimensions as a function of linear and angular momentum and analyse them.
The excitation spectrum exhibits a minima-like feature at finite momentum for
the immiscible SOC-BEC configuration. We augment these results by computing
the dynamic structure factor in the density and pseudo-spin sector, and
discuss its interesting features that can be experimentally measured through
Bragg spectroscopy of such ultra cold-condensate.

##  I Introduction

The low energy excitation spectrum of an ultra cold atomic Bose-Einstein
condensate (BEC) Stringari ; Pethick reveals a plethora of information about
the collective behaviour of such macroscopic quantum systems. For example, the
dispersion relation indicate the breakdown of superfluidity when an object
moves through such Bose-Einstein condensate (BEC) bdsup_theory ; bdsup_exp1 ;
bdsup_exp2 , the nature of the quasiparticle modes plays a vital role in
characterizing critical behavior near the phase transition Subir , and most
importantly it registers the response of such superfluid to external
fluctuations Roth  within the frame-work of perturbation theory. Realisation
of artificial light-induced spin-orbit coupling YJlin ; Spielman09 ; Lin09  in
such Bose-Einstein condensate (BEC) and has appended a new powerful tool to
the simulation toolbox Dalibardcoll  with such ultra cold atomic systems.
Naturally, an extensive insight into the novel properties of such systems can
be gained by looking into the behaviour of their collective excitations
Hall_dipole ; Hall_Zhai ; Rashba_collmodes ; RashbaZhai_CM ; sumrules_SOC ;
CM_Hydrodynamic ; dipole_exp .

Several studies on the effect of collective modes on various properties of
such SOC bosonic superfluid in different dimensions had been carried out
focussing on certain specific aspects. They include the study of mean-field
dynamics of such SOC BEC in terms of their collective modes in one and two
dimensionRashbaZhai_CM ; Rashba_collmodes , the role of dipole oscillation on
the collective behaviour of the system as it goes through a phase transition
sumrules_SOC ; dipole_exp , the change in the velocity of sound as a quasi-one
dimensional SOC condensate goes through a second-order phase transition
anisodynamics , study of collective modes under hydrodynamic approximation
anisodynamics ; CM_Hydrodynamic , and the existence of mean-field ground state
with exotic topology and its subsequent evolutionHall_Zhai . For a SOC-BEC
with one-dimensional spin-orbit coupling, the collective excitations have been
experimentally observed using Bragg spectroscopy, both in homogenous Peter
and trapped configurations rotmaxon . These theoretical and experimental
studies clearly points the necessity of detailed investigation of the full
spectrum of these SOC bosonic system in realistic trapping geometry for other
dimensions, particularly in two spatial dimension, and the subsequent
calculation of the dynamic structure factor using these excitations that can
be experimentally measured using the Bragg spectroscopy BS1 ; BS2 ; BS3 ; BS4
; BS5 .

Motivated by this, in this work, we investigate the excitation spectrum in a
trapped quasi-two-dimensional SOC-BEC for both miscible and immiscible
configurations and present an analysis on the nature of various quasiparticle
excitation modes exploring a wide range of excitation spectrum. We evaluated
the excitation spectrum both as a function of linear momentum as well as
angular momentum.We observe a minima-like dip at finite wave vector in the
excitation spectrum and observe smudged amplitudes of the quasiparticles near
this minima in the immiscible case. As we plot the spectrum as a function of
angular momentum, we see that the the first eigen value in the excitation
spectrum consists a non-zero value, compared to the other cases, such as the
single component (scalar) BEC, and, two component BEC without SOC.
Subsequently we study the consequences of the Bdg spectrum by computing the
dynamic structure factor in the linear-response regime, a quantity that can be
explored in experiments using Bragg spectroscopy. Accordingly, the outline of
the paper is as follows. In section II.1, we discuss briefly the model system
taken under consideration. In section II.2, we discuss the Bogoliubov-de-
Gennes (BdG) excitation spectrum for this configuration and characterize the
various quasiparticle excitations for various range of momenta’s and energy’s.
We also discuss the features of the low-lying excitations such as dipole and
quadrupole modes in this system. In section III, we compute the static and
dynamic structure factor using which one can experimentally probe excitations
in the trapped SOC-BEC.

##  II Model Hamiltonian for SOC BEC and their low energy collective
excitations

In this section we spell out the details of the methodology of computing the
excitation spectrum of a SOC-BEC. In the first part sec. II.1, we shall
introduce the the nature of spin-orbit coupling, the model Hamiltonian, and
the corresponding Gross-Piatevskii equation. In sec. II.2 we introduce the
details of the Bogoliubov de Gennes formalism applied to such SOC BEC.

###  II.1 Synthetic spin-orbit coupling for an ultra cold atom and the Gross-
Piteavskii equation

We will consider the spin-orbit coupled single-particle Hamiltonian,
possessing a non-abelian gauge potential of the form $\bm{A}$ :
$\it{m}(\eta\check{\sigma}_{y},\eta^{\prime}\check{\sigma}_{z},0$) brandon ,
given by:

$\displaystyle\hat{h}$ $\displaystyle=$
$\displaystyle\frac{\bm{\hat{p}}^{2}}{2m}\check{I}-\eta\hat{p}_{x}\check{\sigma}_{y}-\eta^{\prime}\hat{p}_{y}\check{\sigma}_{z}$
(1)

with $\bm{\hat{p}}=\\{\hat{p}_{x},\hat{p}_{y},\hat{p}_{z}\\}$;
$\check{\sigma}$ and $\check{I}$’s are Pauli and Identity matrices
respectively and $\eta$,$\eta^{\prime}$ are the strengths of spin-orbit
coupling (SOC). The interaction effects are incorporated through the two-body
mean field interaction term $1/2\sum_{\kappa\kappa^{\prime}}\int d{\bf
r}V^{\kappa\kappa^{\prime}}_{int}n_{\kappa}n_{\kappa^{\prime}}$ where
‘$n_{\kappa}$’ is the density of the ‘$\kappa$’-component,
$V^{\kappa\kappa^{\prime}}_{int}=4\pi\hbar^{2}a_{\kappa\kappa^{\prime}}/m$
correspond to the coupling constants between different spin channels and
$a_{\kappa\kappa^{\prime}}$ represents the $s$-wave scattering length.

The full Hamiltonian within the framework of Gross-Pitaevskii (GP) theory,
after projecting the single-particle Hamiltonian into the lower energy
subspace, satisfies the following spinorial time-dependent Gross-Pitaevskii
equation (see SOC_SBH  for details):

$\displaystyle i\hbar\frac{\partial\psi_{\kappa}}{\partial t}$
$\displaystyle=$
$\displaystyle\bigg{[}\frac{\hbar^{2}}{2m}(-i\partial_{x}-\kappa\frac{m\eta}{\hbar})^{2}-\frac{\hbar^{2}}{2m_{y}}\partial_{y}^{2}+V_{Trap}$
(2) +
$\displaystyle\frac{g+g_{+-}}{2}\bm{\psi^{\dagger}}\bm{\psi}+\frac{g-g_{+-}}{2}(\bm{\psi^{\dagger}}\check{\sigma}_{z}\bm{\psi})\kappa\bigg{]}\psi_{\kappa}$

where $\bm{\psi}=[\psi_{+}\hskip 5.69046pt\psi_{-}]^{T}$ and $\kappa=\pm$
labels the two components. Also,
$m_{y}=\frac{m}{(1-(\frac{\eta^{\prime}}{\eta})^{2})}$ is the effective mass
along the y- direction (with $\eta^{\prime}<\eta$) and,
$V_{Trap}=m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2})/2$ is the external
trapping potential with trapping frequencies $\omega_{x,y}$ along x and y
directions respectively. Here, we have considered tight confinement along the
z-direction and thus, frozen the dynamics along the z-axis Quasi2D . The
harmonic trap parameters considered in our simulations are
$\omega_{x}=\omega_{y}=2\pi\times$ 4.5 Hz ($=\omega_{\rho}$),
$\omega_{z}=2\pi\times$ 123 Hz. The inter and intraspecies interaction
strengths for this quasi-two-dimensional condensate are denoted by $g_{+-}$
and $g_{\kappa\kappa}$ respectively;
$g_{\kappa\kappa^{\prime}}=\frac{V^{\kappa\kappa^{\prime}}_{int}}{\sqrt{2\pi}a_{\perp}}$
where $a_{\perp}=\sqrt{\frac{\hbar}{m\omega_{z}}}$ is the transverse harmonic
oscillator length.

In Eq.(2), we have considered the intra- species interaction strengths to be
equal $g_{++}=g_{--}(=g$), which is a good approximation to the experimental
situation with 87Rb atoms g_equal  chosen in this work. At various places in
the paper, we compare our results to a single-component BEC Stringari ;
Pethick ; DM1  and a two-component BEC Stringari ; Pethick ; Myatt ; Gordan ;
Angom  as limiting cases of Eq.(2). It corresponds to substituting the
parameters $g_{+-}=0$, $m_{y}=m$, and $\eta=0$ in Eq.(2), for the single-
component BEC. For a two-component BEC we set in Eq.(2), $g_{+-}\neq 0$,
$m_{y}=m$, and $\eta=0$. Additionally we consider the miscible, i.e.,
$g_{+-}<g$ and immiscible configurations, i.e., $g_{+-}>g$ SOC_Angela ;
SOC_XIAO , for a given strength of the spin-orbit coupling. The condensate
components overlap each other in the miscible configuration, whereas they are
spatially-separated in the immiscible configuration.

###  II.2 Bogoliubov de-Gennes formalism

We numerically propagate the GP Eq.(2) in imaginary time to obtain the two-
component ground state wavefunction of the condensate,
$\bm{\psi_{0}}(x,y)=[\psi_{+,0}(x,y)\hskip 5.69046pt\psi_{-,0}(x,y)]^{T}$. The
ground state solution obtained from the GPE in this way, for various cases,
are shown in Fig. 16 and Fig. 17. The excitation spectrum and the nature of
the quasiparticle amplitudes in such SOC-BEC, are then obtained in the
framework of the Bogoliubov theory Stringari ; Pethick  by considering the
fluctuations over the ground state wavefunction, of the GP Eq.(2) as:

$\begin{bmatrix}\psi_{+}\\\ \psi_{-}\\\ \end{bmatrix}=e^{\frac{-i\mu
t}{\hbar}}\bigg{(}\begin{bmatrix}\psi_{+,0}\\\ \psi_{-,0}\\\
\end{bmatrix}+\sum_{j}\bigg{\\{}\begin{bmatrix}u_{+,j}\\\ u_{-,j}\\\
\end{bmatrix}e^{-i\omega_{j}t}+\begin{bmatrix}v^{*}_{+,j}\\\ v^{*}_{-,j}\\\
\end{bmatrix}e^{i\omega^{*}_{j}t}\bigg{\\}}\bigg{)}$

(3)

where $\mu$ is the chemical potential of the condensate and, $\omega_{j}$ =
$\epsilon_{j}/\hbar$ where $\epsilon_{j}$ is the energy corresponding to each
quasiparticle excitation.

Figure 1: (color online). Dispersion relation as a function of both $k_{rms}$
and ($\hat{L}_{z})_{rms}$ : (a) shows the single-component BEC case with
$g_{+-}=0$, (b) shows the dispersion relation for two-component BEC with
$g_{+-}=g$. (c-d) shows the dispersion relation for $\eta^{\prime}/\eta=0.4$
and $0.78$ respectively, with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$ and
$g_{+-}=g$. Inset of (d) shows the canonical and gauge contribution to the
$rms$ value of total angular momentum.

The index ‘$j$’ represents the sequence of the quasiparticle excitation.
‘$u_{\kappa,j}$’ and ‘$v_{\kappa,j}$’ are spatially dependent complex
functions denoting the Bogoliubov quasiparticle amplitudes corresponding to
the $j$th energy eigenstate and are normalized as,

$\displaystyle\int\int
dxdy\sum_{\kappa=\pm}[|u_{\kappa,j}(x,y)|^{2}-|v_{\kappa,j}(x,y)|^{2}]$
$\displaystyle=$ $\displaystyle 1$ (4)

Inserting $\psi_{\pm}({\bf r},t)$ using Eq.(3) into GP Eq. (2) and retaining
the fluctuations upto the linear order, we get the Bogoliubov-de-Genes (BdG)
equations for the considered SOC-BEC system:

$\displaystyle\begin{bmatrix}\bar{L}_{+}&g\psi_{+,0}^{2}&g_{+-}{\psi^{*}_{-,0}}{\psi_{+,0}}&g_{+-}{\psi_{-,0}}{\psi_{+,0}}\\\
-g{\psi^{*}_{+,0}}^{2}&\bar{\bar{L}}_{+}&-g_{+-}\psi^{*}_{-,0}\psi^{*}_{+,0}&-g_{+-}{\psi_{-,0}}\psi^{*}_{+,0}\\\
g_{+-}\psi^{*}_{+,0}\psi_{-,0}&g_{+-}\psi_{-,0}\psi_{+,0}&\bar{L}_{-}&g{\psi^{2}_{-,0}}\\\
-g_{+-}\psi^{*}_{-,0}\psi^{*}_{+,0}&-g_{+-}\psi_{+,0}\psi^{*}_{-,0}&-g{\psi^{*}_{-,0}}^{2}&\bar{\bar{L}}_{-}\\\
\end{bmatrix}\begin{bmatrix}u_{+,j}\\\ v_{+,j}\\\ u_{-,j}\\\ v_{-,j}\\\
\end{bmatrix}=\epsilon_{j}\begin{bmatrix}u_{+,j}\\\ v_{+,j}\\\ u_{-,j}\\\
v_{-,j}\\\ \end{bmatrix}$ (5)

where,

$\displaystyle\bar{L}_{+}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\hbar^{2}}{2m}[-\partial_{x}^{2}+\frac{m^{2}\eta^{2}}{\hbar^{2}}+2i\frac{m\eta}{\hbar}\partial_{x}]-\frac{\hbar^{2}}{2m_{y}}\partial_{y}^{2}+V_{Trap}\bigg{)}$
$\displaystyle+$
$\displaystyle(2g|\psi_{+,0}|^{2}+g_{+-}|\psi_{-,0}|^{2})-\mu_{+}$
$\displaystyle\bar{\bar{L}}_{+}$ $\displaystyle=$ $\displaystyle-\bar{L}_{+}$
$\displaystyle\bar{L}_{-}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\hbar^{2}}{2m}[-\partial_{x}^{2}+\frac{m^{2}\eta^{2}}{\hbar^{2}}-2i\frac{m\eta}{\hbar}\partial_{x}]-\frac{\hbar^{2}}{2m_{y}}\partial_{y}^{2}+V_{Trap}\bigg{)}$
$\displaystyle+$
$\displaystyle(2g|\psi_{-,0}|^{2}+g_{+-}|\psi_{+,0}|^{2})-\mu_{-}$
$\displaystyle\bar{\bar{L}}_{-}$ $\displaystyle=$ $\displaystyle-\bar{L}_{-}$

We diagonalise the above BdG matrix 5 numerically to find quasiparticle
excitation energy $\epsilon_{j}$, as well as the Bogoliubov quasiparticle
amplitudes {$u_{ij}$ and $v_{ij}$}. To this end we expand $u_{ij}$ and
$v_{ij}$ in terms of harmonic oscillator eigenfunction
$\phi_{n}(x),\phi_{n}(y)$ for such harmonically trapped SOC-BEC (details given
in Appendix A) as

$\displaystyle u_{+,j}(x,y)=\sum_{k,l=0}^{N_{b}}p_{jkl}\phi_{k}(x)\phi_{l}(y)$
$\displaystyle v_{+,j}(x,y)=\sum_{k,l=0}^{N_{b}}q_{jkl}\phi_{k}(x)\phi_{l}(y)$
$\displaystyle u_{-,j}(x,y)=\sum_{k,l=0}^{N_{b}}r_{jkl}\phi_{k}(x)\phi_{l}(y)$
$\displaystyle v_{-,j}(x,y)=\sum_{k,l=0}^{N_{b}}s_{jkl}\phi_{k}(x)\phi_{l}(y)$
(6)

where $p_{jkl}$, $q_{jkl}$, $r_{jkl}$ and $s_{jkl}$ are the coefficients of
the linear combination, and,
$\phi_{n}(x)=\frac{1}{\sqrt{2^{n}{n!}a_{\rho}\sqrt{\pi}}}e^{-x^{2}/2a_{\rho}^{2}}H_{n}(x/a_{\rho})$;
where $H_{n}$ is the $n$th order Hermite polynomial and
$a_{\rho}=\sqrt{\frac{\hbar}{m\omega_{\rho}}}$. Substituting the above
expression, after some straightforward algebra, the resulting equation can be
written as

$\displaystyle\begin{bmatrix}[A]&[B]&[C]&[D]\\\ [E]&[F]&[G]&[H]\\\
[I]&[J]&[K]&[L]\\\ [M]&[N]&[O]&[P]\\\
\end{bmatrix}=\epsilon\begin{bmatrix}[P]\\\ [Q]\\\ [R]\\\ [S]\\\
\end{bmatrix}$ (7)

where, the each matrix [.] is of the dimension
$(N_{b}+1)^{2}\times(N_{b}+1)^{2}$ and the elements of each of these matrices
are evaluated using the integrals defined in Appendix A. The resulting BdG
matrix in Eq.(7) has $4(N_{b}+1)\times 4(N_{b}+1)$ dimensions for equal number
of basis (=$N_{b}$) along x and y directions. The size of the BdG matrix
increases rapidly due the $\sim N_{b}^{2}$ scaling. We obtain the converged
eigenvalues (upto four digits after decimal) for $N_{b}$ = 50.

To construct the dispersion relation of the Bogoliubov modes, we adopt the
methodology used for the trapped configurations such as two-component BEC
Ticknor  and dipolar condensates Wilson ; Blakie1 ; Blakie2 ; Blakie3 . Since
there is no translational symmetry in such finite, trapped configuration, the
linear momentum is not a good quantum number. Thus each quasiparticle can be
linked to an $rms$ value of the linear momentum. Additionally the
irrotationality condition that is obeyed in single component BEC-superfluid
with no SOC coupling gets violated in SOC coupled two-component BEC superfluid
diffvort . However, the angular momentum operator $\hat{L}_{z}$ also does not
commutes with the Hamiltonian in Eq.(2). So, to see the consequence we
evaluate the $rms$ value of the z-component of angular momentum for each such
excitation ang . Accordingly we define

$\displaystyle k^{j}_{rms}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{\sum_{\kappa}\int d{\bf k}{\bf
k}^{2}[|u_{\kappa,j}({\bf k})|^{2}+|v_{\kappa,j}({\bf
k})|^{2}]}{\sum_{\kappa}\int d{\bf k}[|u_{\kappa,j}({\bf
k})|^{2}+|v_{\kappa,j}({\bf k})|^{2}]}\bigg{)}^{1/2}$ (8)

($\hat{L}^{j}_{z})_{rms}=\bigg{(}\frac{\sum_{\kappa}\int d{\bf
r}[u^{*}_{\kappa,j}({\bf r})\hat{L}^{2}_{z}u_{\kappa,j}({\bf
r})+v^{*}_{\kappa,j}({\bf r})\hat{L}_{z}^{2}v_{\kappa,j}({\bf
r})])}{\sum_{\kappa}\int d{\bf r}[|u_{\kappa,j}({\bf
r})|^{2}+|v_{\kappa,j}({\bf r})|^{2}]}\bigg{)}^{1/2}$

(9)

where $\kappa=+,-$ labels the two components. Here, in Eq.(8),
$u_{\kappa,j}({\bf k})=\mathcal{F}[u_{\kappa,j}({x,y})]$ and
$v_{\kappa,j}({\bf k})=\mathcal{F}[v_{\kappa,j}({x,y})]$ are the quasiparticle
amplitudes in the momentum space. Also, the angular momentum operator can be
written as $\hat{L}_{z}=\hat{L}^{c}_{z}+\hat{L}^{g}_{z}$. The effective
momentum along a particular direction (along x) in Eq.(2) gets modified due to
spin-orbit coupling; thus, the angular momentum operator also contains
additional terms apart from the canonical one. Here,
$\hat{L}^{c}_{z}=(x\hat{p}_{y}-y\hat{p}_{x})$ corresponds to the canonical
part, and $\hat{L}^{g}_{z}=\kappa$($\eta ym$) represents the spin-dependent
gauge part.

Figure 2: (color online). Quasi-particle amplitudes $u_{+}$ and $v_{+}$, for
the dipole (a,b) and quadrupole modes (c,d) is shown for the single-component
BEC with intra-species interaction strength $g_{+-}=0$. $u_{+}$ and $v_{+}$
are in units of $a_{\rho}^{-1}$.  Figure 3: (color online). Quasi-particle
amplitudes $u_{\pm}$ and $v_{\pm}$, for the dipole mode is shown for the case
with intra-species interaction strength $g_{+-}\neq 0$ and
$\eta^{\prime}/\eta=0.78$. $u_{+}$ and $v_{+}$ are in units of
$a_{\rho}^{-1}$.

The excitation spectrum as a function of $rms$ value of both the linear and
angular momentum is shown in Fig. 1. Fig. 1 (a) illustrate the excitation
spectrum for the single-component case, with intra-species interaction
strength $g_{+-}=0$, and Fig. 1 (b) shows the spectrum for the two-component
BEC with no SO coupling and $g_{+-}=g$. Fig. 1 (c-d) shows the excitation
spectrum in two-component SOC-BEC for lower and higher ratio of spin-orbit
coupling strengths $\eta^{\prime}/\eta$ = 0.4 and 0.78 respectively.

Figure 4: (color online). Quasi-particle amplitudes $u_{\pm}$ and $v_{\pm}$,
for the quadrupole mode is shown for the case with intra-species interaction
strength $g_{+-}=g$ and $\eta^{\prime}/\eta=0.78$. $u_{+}$ and $v_{+}$ are in
units of $a_{\rho}^{-1}$.

We start our discussion with the low-lying dipole and quadrupole excitations.
Collective dipole oscillation represents the center-of-mass motion of all the
atoms and are the excitations corresponding to the lowest finite energy modes
DM1 . In order to excite these modes, the trap can be suddenly displaced, in
such a way that the initial ground state of the BEC is now no longer the
ground state of the displaced trap. This results to collective excitation of
the atoms and thus, can be easily excited experimentally DM_exp . For the
first case shown in Fig. 1 (a), the dipole mode occurs at energy
$\epsilon=\hbar\omega_{\rho}$. This is also in consonance with the fact that
for a scalar condensate, the frequency of the dipole oscillation is just the
harmonic-trap frequency DM1 ; DM_rmp . However, for a SO-coupled condensate,
the dipole-oscillation frequency deviates from the trap frequency
RashbaZhai_CM ; dipole_exp ; sumrules_SOC ; diffvort , which is also
consistent with our simulations. For example, the energy at which the dipole
mode occur in SOC-BEC for strengths $\eta^{\prime}/\eta=0.78$ is
$E=\epsilon/\hbar\omega_{\rho}=0.3$, which is different from 1.

We next investigate the structure of the quasiparticle amplitudes to interpret
the physics behind the dispersion curve. For comparison, the quasiparticle
amplitudes {$u_{+},v_{+}$} of the dipole and quadrupole mode for the case of
single-component BEC are showed in Fig. 2 (a,b) and (c,d), respectively.
Examining the dipole mode and quadrupole mode behavior provides a signature of
the quantum phase transition and is studied in SOC-BEC experimentally DPT . We
here show the quasiparticle amplitudes for both components labeled through
$\pm$, {$u_{\pm},v_{\pm}$} for dipole and quadrupole modes, for one of the SOC
strengths $\eta^{\prime}/\eta=0.78$ with non-zero intra-species interaction
strength, in Fig. 3 (a-d) and Fig. 4 (a-d), respectively. In both these
figures, (a-b) shows the quasiparticle amplitudes for the ‘+’ component, and
(c-d) contains the quasiparticle amplitudes for the ‘-’ component.

Figure 5: (color online). Quasi-particle amplitudes $u_{+}(x,y)$ (in units of
$a_{\rho}^{-1}$) in the uncoupled scalar condensate (without SO coupling) with
$g_{+-}=0$, for few excitations marked in Fig. 1 (a), is shown. Here (a-b),
(c-e) and (f-h) shows the quasiparticle amplitude corresponding to the
excitations marked with  ,  and  respectively. In each set shown here,
quasiparticles begin with small values of $k_{rms}$ (at left) and move towards
large values of $k_{rms}$ (at right). The value of dimensionless energy
$\epsilon/(\hbar\omega_{\rho})$ is written at the left corner (at bottom) in
each figure. Each sub-figure is superimposed with the condensate density
contours (black lines) corresponding to the $`+^{\prime}$ component.

After discussing the low energy excitations, we will now examine the nature of
the excitations for higher energy values. For illustration, we evaluate the
quasiparticle amplitudes at dimensionless energies
$\epsilon/(\hbar\omega_{\rho})$ near to 1 (low), 7 (intermediate) and 14
(high), marked with  ,  and  respectively. The three energy ranges are chosen
to examine the whole range of energies in the dispersion (Fig. 1) exhibited by
the cases under consideration.

For each energy range chosen, we now discuss the effect of different $k_{rms}$
values on the quasiparticle amplitudes. We choose a particular excitation, at
a given energy, each from the rightmost and leftmost point, from the
dispersion curve in Fig. 1, and one in between these two. The leftmost
excitation (with smaller $k_{rms}$) that lies nearly on the linear part of the
dispersion, is phonon-like. The rightmost one (with larger $k_{rms}$)
corresponds to the surface excitation DM1 , as we shall see the quasiparticle
amplitude gets distributed over the boundary of the condensate only for such
modes.

Again for comparison, we discuss the case of a scalar condensate, whose
quasiparticle amplitudes ($u_{+}$) are displayed, in Fig. 5. We will analyze
the radial and angular nodes in various examples of quasiparticle excitations.
The radial nodes are the nodes existing in each quasiparticle excitation mode
along any cross-section from the origin (${\bf r}=0$), i.e., nodes along
$r=\sqrt{(x^{2}+y^{2})}$ counts $(n_{r})$ except the zero at large $r$. The
number of azimuthal nodes ($n_{\phi}$) is equal to twice the maxima (or
minima) number along the outermost boundary of the excitation amplitude
containing both. Fig. 5(a-b) shows two quasiparticles with energy,
$\epsilon/(\hbar\omega_{\rho})$ = 1. These degenerate modes are marked, in
Fig. 1 (a) with red squares. Both of these have one radial node and two
angular nodes.

In Fig. 5 (c-e), we show quasiparticles with energy,
$\epsilon/(\hbar\omega_{\rho})$ $\sim$ 7 for three illustrative values of
$k_{rms}$: low (c), intermediate (d) and high (e) value. The excitation at
lower value of $k_{rms}$ lies nearly on the linear part of the dispersion
curve, has three radial nodes and 10 angular nodes. The intermediate $k_{rms}$
value at the same value of energy has two radial nodes and 16 angular nodes,
shown in Fig. 5 (d). Whereas, larger $k_{rms}$ value at the same value of
energy has only one radial node and 22 angular nodes. Therefore, roughly at
same range of energy, the quasiparticle amplitude gains azimuthal nodes at the
cost of loss in radial nodes. This is also evident from Fig. 1 (a), where the
quasiparticle excitation mode with smaller $k_{rms}$ has lower value of
($\hat{L}_{z})_{rms}$ and the quasiparticle excitation mode with larger
$k_{rms}$ value has higher ($\hat{L}_{z})_{rms}$ value. The observation holds
for other considered cases also.

Additionally, Fig. 1 illustrates that for a scalar BEC and a two-component
BEC, the lowest quasiparticle excitation has zero $rms$ value of angular
momentum. However, for the SOC case, the gauge part of the $rms$ value of
angular momentum ($L_{z}^{g}$) results in contributing non-zero angular
momentum to the lowest-energy excitation mode, shown in Fig. 1 (d) (inset) for
one of the SOC strengths. It occurs due to the modification of the effective
momentum along a particular direction [Eq. (2)], subjected to variation in
spin-orbit coupling parameters, which results in an anisotropic nature of the
SOC-BEC’s velocity profile. In turn, this leads to violation of
irrotationality condition on the velocity fields in the SOC- BEC superfluid,
which is in contrast to the case of scalar BEC superfluid diffvort . Hence,
the lowest energy excitation in SOC-BEC, as a consequence, have a finite $rms$
value of angular momentum.

Figure 6: (color online). Quasiparticle amplitudes $u_{+}(x,y)$ (in units of
$a_{\rho}^{-1}$) with moderate ratio of SO coupling strengths
$\eta^{\prime}/\eta=0.4$ with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$, for few
excitations marked in Fig. 1 (c), is shown. Here (a-c), (d-f) and (g-i) shows
the quasiparticle amplitude corresponding to the excitations marked with  ,
and  respectively. In each set shown here, quasiparticles begin with small
values of $k_{rms}$ (at left) and move towards large values of $k_{rms}$ (at
right). The value of dimensionless energy $\epsilon/(\hbar\omega_{\rho})$ is
written at the left corner (at bottom) in each figure.

Figure 7: (color online). Quasi-particle amplitudes $u_{+}(x,y)$ (in units of
$a_{\rho}^{-1}$) with SO coupling strengths ratio $\eta^{\prime}/\eta=0.78$
with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$, for few excitations marked in Fig.
1 (d), is shown. Here (a-c), (d-f) and (g-i) shows the quasiparticle amplitude
corresponding to the excitations marked with  ,  and  respectively . In each
set shown here, quasiparticles begin with small values of $k_{rms}$ (at left)
and move towards large values of $k_{rms}$ (at right). The value of
dimensionless energy $\epsilon/(\hbar\omega_{\rho})$ is written at the left
corner (at bottom) in each figure.

Figure 8: (color online). Excitation spectrum for various SOC strengths
$\eta=1.5,1.8,2$ (in units of $\sqrt{\hbar\omega_{\rho}/m}$) with
$\eta^{\prime}/\eta=0.78$, in the miscible (a) and immiscible (b) cases, are
shown. (b1)-(b2) respectively shows the magnified part of the dispersion near
the minima, in miscible configuration, concerning $k_{rms}$ and
$(L_{z})_{rms}$ separately. Some points on the outermost branch of the
dispersion, viewed w.r.t $k_{rms}$, are marked (using squares). The
corresponding points are marked in (b2) also, and dashed lines guide the eye
joining these.

Also, at the same range of energy value, but with increasing value of
$k_{rms}$ the quasiparticle amplitudes move towards the outermost boundary of
the condensate and away from the center of the trap. To illustrate this we
have superposed the quasiparticle amplitude for a particular component
‘$u_{+}(x,y)$’ with the contour lines of the corresponding condensate density
i.e, for the ‘+’-component $|\psi_{+}|^{2}$. It is shown in Fig. 5 (c-h). At
relatively larger values of $k_{rms}$ shown in Fig. 5 (e,h), the excitation
amplitude lies at the outermost edge/contour of the condensate and thus,
corresponds to the surface mode. It shows the coexistence of surface and
phonon-like modes is observed at the similar range of energy. The observation
is also consistent for the energy value $\sim$ 14 at low, intermediate and
high value of $k_{rms}$. (f) has 8 radial nodes and 4 angular nodes; (g) has 3
radial nodes and 28 angular nodes; and (h) has only a single radial node but
36 angular nodes.

Figure 9: (color online). Quasiparticle amplitudes for modes near the minima:
(a) displays the excitation spectrum for miscible case from Fig. 8 (b), for
ratio of SOC strengths $\eta^{\prime}/\eta=0.78$ with $\eta=1.8$ (in units of
$\sqrt{\hbar\omega_{\rho}/m}$). Inset shows the magnified view near the
minima. (b-e) respectively shows the quasiparticle amplitude $u_{+}(x,y)$ for
a few modes in (a), labelled as (i-iv) with marker ‘+’.

After discussing the nature of the quasiparticle amplitudes of the various
excitations in a scalar condensate, we next analyze the nature of these in our
SO coupled configuration. Fig. 6-7 shows the quasiparticle amplitude
$u_{+}(x,y)$ at different energy values for SOC strengths
$\eta^{\prime}/\eta=0.4,0.78$, respectively. Table 1 contains the details
about the number of radial nodes and angular nodes in the quasiparticle
amplitude for the cases considered. Some points of differences and
similarities between SOC and scalar condensate, based on Table 1, are
discussed in the following. Firstly, in a SOC-BEC, at low $k_{rms}$ and
considered energy values, the quasiparticle excitations are aligned along a
specific direction. It is along x-direction in the case
$\eta^{\prime}/\eta=0.78$, shown in Fig.7(d,g). It is along y-direction in the
case $\eta^{\prime}/\eta=0.4$, shown in Fig. 6(d,g). Whereas, for the case of
scalar condensate, the distribution of amplitude is uniform and symmetric,
shown in Fig. 5(c,f). The anisotropy in quasiparticle amplitudes in SOC-BEC is
present, although the trap is isotropic. It occurs due to the different
effective masses and the effective momentum along x and y directions in Eq.
(2). Also, in a SOC-BEC, the number of nodes is direction-dependent and is not
constant. As an example, consider the case illustrated in Fig. 6(e) for one of
the SOC-strengths ratios at $\epsilon/\hbar\omega_{\rho}=7$. The number of
radial nodes at intermediate $k_{rms}$ consists of either 3 or 4 radial nodes
depending on the direction chosen.

Furthermore, at high $k_{rms}$, the number of angular nodes for
$\epsilon/\hbar\omega_{\rho}=7,14$ increases in the order: single component,
$\eta^{\prime}/\eta=0.78$, $\eta^{\prime}/\eta=0.4$. At the same value of
$\epsilon/\hbar\omega_{\rho}=7,14$ but at low $k_{rms}$, the number of angular
nodes is only 2 for both $\eta^{\prime}/\eta=0.78,0.4$ and is largest for
single-component BEC. Also, the number of radial nodes for
$\epsilon/\hbar\omega_{\rho}=1,7,14$, at high $k_{rms}$, is mostly 1 for all
the cases. The number of radial nodes for $\epsilon/\hbar\omega_{\rho}=7,14$,
at low $k_{rms}$, is greater in SOC cases than the single-component BEC.

$n_{r}$ | Scalar | $\eta^{\prime}/\eta=0.4$ | $\eta^{\prime}/\eta=0.78$
---|---|---|---
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 1 | 1 | (1/2,2,1) | (1,1,1)
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 7 | (3,2,1) | (6,3/4,1/2) | (5,3,1)
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 14 | (8,3,1) | (9,3/4,1) | (9,4,1)

$n_{\phi}$ | Scalar | $\eta^{\prime}/\eta=0.4$ | $\eta^{\prime}/\eta=0.78$
---|---|---|---
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 1 | 2 | (10,2,8) | (6,4,6)
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 7 | (10,16,22) | (2,10,30) | (2,14,26)
$\epsilon/\hbar\omega_{\rho}$ $\sim$ 14 | (4,28,36) | (2,30,46) | (2,26,42)

Table 1:  The number of radial and angular nodes $n_{r}$ and $n_{\phi}$
respectively are listed in order
($k^{low}_{rms},k^{med}_{rms},k^{high}_{rms}$) for various cases.

Fig. 8(a-b) shows the dispersion relation in SOC-BEC for various SOC strengths
$\eta=1.5,1.8,2$ (in units of $\sqrt{\hbar\omega_{\rho}/m}$) with
$\eta^{\prime}/\eta=0.78$ in miscible, i.e., $g_{+-}<g$ and immiscible
configurations, i.e., $g_{+-}>g$ respectively.

The excitation spectrum in the immiscible system [Fig. 8(b)] contains a
minima-like feature at finite $k_{rms}$. A few outermost points (branch) of
the dispersion, viewed against $k_{rms}$, are marked with squares. The
minima’s location gets right-shifted towards a higher value of $k_{rms}$ with
an increase in the magnitude of $\eta$, shown in Fig. 8(b1). We also show the
magnified part of the dispersion from Fig. 8(b) concerning $(L_{z})_{rms}$ in
Fig. 8(b2). The points corresponding to (b1) are marked in (b2) also. Fig.
8(b2) also contains minima at finite $rms$ value of angular momentum,
demonstrating that the lowest energy excitation have a non-zero, finite
angular momentum, that increases with an increase in $\eta$’s magnitude.

We next investigate the nature of the quasiparticle modes near the minima. To
this purpose, we consider the dispersion relation from Fig. 8(b1) concerning
$k_{rms}$ only. It is shown, in Fig. 9(a), for the parameters $g<$ $g_{+-}$
and $\eta^{\prime}/\eta=0.78$ with $\eta=1.8\sqrt{\hbar\omega_{\rho}/m}$. Fig.
9(b-e) shows the quasiparticle amplitude $u_{+}(x,y)$ for the modes marked
with ‘+’ in (a), for four illustrative modes labeled as (i-iv). The mode
marked with ‘(ii)’ identifies the minima location and has four angular nodes.
Fig. 9(b) with the label (i), has eight angular nodes. Both (d-e) marked with
(iii-iv), respectively, have six angular nodes. Thus, the considered case for
investigating quasiparticle modes near the minima has no radial nodes but only
angular nodes. Here, the number of the angular nodes is least at the minima
location, with the largest on its left. Moreover, the effect of unbalanced
interaction strengths, i.e., $g_{+-}\neq g$ also impacts the spread of the
quasiparticle amplitudes which become smeared spatially, illustrated in Fig.
9(b-e).

##  III Structure Factor

Using the excitations spectrum of a trapped SOC-BEC computed in the previous
section, we can now calculate the dynamic structure factor (DSF), that can be
measured using a robust experimental tool, Bragg spectroscopy BS1 ; BS2 ; BS3
; BS4 ; BS5 . DSF yields the information on the spectrum of collective
excitations, which can be explored at low momentum transfer BS3 . It also
provides knowledge about the momentum distribution through which the behavior
of the system at high momentum transfer can be characterized, where the
response is dominated by single-particle effects BS1 . Its investigation has
provided a crucial understanding of physics in superfluid 4He Griffin .
Specially, it has facilitated the measurement of the roton spectrumHespectrum
, pair-distribution function and condensate fraction available from neutron
scattering experiments Griffin2 , in this system.

Figure 10: (color online). (a) shows the dynamic structure factor for single-
component BEC with respect to $k_{x}(=k)$ and $\omega$. (b-c) shows the
dynamic structure factor for density $S_{D}(k_{x},k_{y}=0,\omega)$ and spin
$S_{S}(k_{x},k_{y}=0,\omega)$ in two-component BEC. (d-e) shows these for SOC
case for strengths $\eta^{\prime}/\eta=0.78$ with
$\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$. Figure 11: (color online). (a) shows
the dynamic structure factor for single-component BEC with respect to
$k_{x}(=k)$ and $\omega$. (b-c) shows the dynamic structure factor for density
$S_{D}(k_{x},k_{y}=0,\omega)$ and spin $S_{S}(k_{x},k_{y}=0,\omega)$ in two-
component BEC. (d-e) shows these for SOC case for strengths
$\eta^{\prime}/\eta=0.78$ with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$.  Figure
12: (color online). (a,b) shows the density and spin dynamic structure factor
in a SOC-BEC for strengths $\eta^{\prime}/\eta=0.78$ with
$\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$ at $k_{y}=a_{\rho}^{-1}$. The
corresponding results at $k_{y}=4a_{\rho}^{-1}$ are shown (c,d) respectively.

Figure 13: (color online). (a-b) respectively shows the density
$S_{D}(k_{x},k_{y}=0)$ and spin static structure factor $S_{S}(k_{x},k_{y}=0)$
for various listed cases.  Figure 14: (color online). (a-b) shows contribution
of the first three excited states (j=1,2,3) to the total density and spin
static structure factor in SOC-BEC, respectively, for
$\eta^{\prime}/\eta=0.78$; (c-d) shows the same for the two-component BEC.

Figure 15: (color online): (a) displays the Fourier transform of the
fluctuations in the total density relative to the total density, for several
cases mentioned. (b) shows the Fourier transform of the spin-density
fluctuations.

In the experiment, the condensate is impinged by a Bragg pulse, with the help
of two laser beams having wavevectors ${\bf k_{1}}$ and ${\bf k_{2}}$ and a
frequency difference $\omega$. The following Hamiltonian describes the
resulting external perturbation term:

$\displaystyle\hat{H}_{pert}(t)=\lambda\hat{G}_{\bf k}^{\dagger}e^{-i\omega
t}e^{\gamma t}+H.c.$ (10)

where $\lambda$ is the strength of the Bragg potential, ${\bf k}={\bf
k_{1}}-{\bf k_{2}}$ is the momentum transferred by the Bragg pulse to the
condensate and the factor $e^{\gamma t}$ in Eq.(10), with $\gamma\rightarrow
0^{+}$, ensures that the system at initial time ($t=-\infty$) is governed by
the unperturbed Hamiltonian. Directly after perturbing the system, the DSF
relative to the operator $\hat{G}_{\bf k}$ Stringari  can be probed:

$\displaystyle S({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\sum_{j}Z_{j}\delta(\hbar\omega-\hbar\omega_{j0})$ (11)

The quantity $Z_{j}=|\bra{j}\hat{G}_{\bf k}\ket{0}|^{2}$ is known as the
strength of the operator $\hat{G}_{\bf k}$ with respect to state $\ket{j}$ and
$\hbar\omega_{j0}=\epsilon_{j}-\epsilon_{0}$. Here $\ket{0}$ ($\ket{j}$)
correspond to the ground (excited) state with the energy $\epsilon_{0}$
($\epsilon_{j}$). For the two-component SOC-BEC, the DSF corresponding to the
total density $\hat{\rho}({\bf r})=\hat{\bm{\psi}}^{\dagger}({\bf
r})\hat{\bm{\psi}}({\bf r})$ and spin density $\hat{s}_{z}({\bf
r})=\hat{\bm{\psi}}^{\dagger}({\bf r})\hat{\sigma}_{z}\hat{\bm{\psi}}({\bf
r})$ can be also calculated from the BdG excitations computed in previous
section, where $\hat{\bm{\psi}}({\bf r})$ is defined in Eq.(21). Here, the
operator $\hat{G}_{\bf k}$, corresponding to the total density is
$\hat{\rho}_{\bf k}=\int d{\bf r}\hat{\bm{\psi}}^{\dagger}({\bf
r})\hat{\bm{\psi}}({\bf r})e^{i{\bf k}\cdot{\bf r}}$ and, the operator
corresponding to the spin density is $\hat{s}_{z_{\bf k}}=\int d{\bf
r}\hat{\bm{\psi}}^{\dagger}({\bf r})\hat{\sigma}_{z}\hat{\bm{\psi}}({\bf
r})e^{i{\bf k}\cdot{\bf r}}$, where $\hat{\bm{\psi}}(\bf
r)=[\hat{\psi}_{+}(\bf r)\hskip 5.69046pt\hat{\psi}_{-}(\bf r)]^{T}$.

The strength’s $Z_{j}$ in Eq.(11) can be simplified and written in terms of
the qausiparticle amplitudes (details in Appendix B.2), resulting DSF
corresponding to total density (spin density), labelled with subscript
‘D’(‘S’), given as:

$S_{D}(k_{x},k_{y},\omega)=\sum_{j}\big{|}\int
dxdye^{i(k_{x}x+k_{y}y)}\bra{0}\hat{\rho}({\bf
r})\ket{j}\big{|}^{2}\delta(\hbar\omega-\hbar\omega_{j})$

$S_{S}(k_{x},k_{y},\omega)=\sum_{j}\big{|}\int
dxdye^{i(k_{x}x+k_{y}y)}\bra{0}\hat{s}_{z}({\bf
r})\ket{j}\big{|}^{2}\delta(\hbar\omega-\hbar\omega_{j})$

(12)

The DSF integrated over $S(k_{x},k_{y},\omega)$ over frequency domain, gives
the static structure factor,

$S(k_{x},k_{y})=\frac{1}{N}\int d\omega S(k_{x},k_{y},\omega)$

We start by discussing the analytical results for the structure factor in a
uniform scalar and two-component BEC. The dispersions in these two cases are
respectively given by

$\displaystyle\epsilon_{B}$ $\displaystyle=$
$\displaystyle\hbar\sqrt{c^{2}k^{2}+(\hbar k^{2}/2m)^{2}}$ (13)
$\displaystyle\epsilon_{B}^{D,S}$ $\displaystyle=$
$\displaystyle\hbar\sqrt{c_{D,S}^{2}k^{2}+(\hbar k^{2}/2m)^{2}}$ (14)

In a uniform scalar BEC, the sum in Eq.(11) is expended by only one mode. The
static structure factor in this system is $S({\bf k})=\epsilon_{0}({\bf
k})/\epsilon_{B}({\bf k})$ ($Feynman$ relation) Feynman , where
$\epsilon_{0}({\bf k})$ is the dispersion of non-interacting bosons. In the
limit $k\rightarrow 0$, $S({\bf k})$ = $\hbar k/2mc$ where $c=\sqrt{gn/m}$ is
the Bogoliubov sound velocity. This can be generalised in a uniform two-
component BEC using the dispersion relation Timmermann  given in Eq. (14). The
static factor in such configuration for $k\rightarrow 0$ gives $S_{D,S}({\bf
k})$ = $\hbar k/2mc_{D,S}$, where $c_{D,S}$ represents the sound velocity
corresponding to density and spin-density, respectively. Thus, as
$k\rightarrow 0$, the static structure factors in both configurations vary
linearly with wavevector. However, the static structure factor approaches
unity at large momentum BS2 ; BS3  in both of the systems, which is the value
of the static structure factor for any momentum in an uncorrelated non-
interacting atoms.

To benchmark our computation for the trapped SOC-BEC, we now evaluate the
dynamic and the static structure factor in the trapped single and two-
component BEC by considering the suitable limit of the excitation energy and
the quasiparticle amplitudes determined in section II.2. Substituting
parameters $g_{+-}=0$, $m_{y}=m$, and $\eta=0$ in Eq.(2) we get the results
for trapped scalar BEC. The dynamic structure factor $S(k_{x},k_{y}=0,\omega)$
for this case is shown Fig.11(a). To get a clear insight into its behavior, we
integrate the dynamic structure factor $S(k_{x},k_{y},\omega)$ over the
frequency domain and compute the static structure factor. The evaluated
structure factor is illustrated, with a bold (blue) line in Fig. 13(a,b)
(Here, $S_{D}=S_{S}=S$). The substitution of parameters $g_{+-}\neq 0$,
$m_{y}=m$, and $\eta=0$ in Eq.(2) corresponds to the case of a trapped two-
component BEC, discussed in Ticknor2 . In the case of two-component BEC, the
density and spin dynamic structure factor are represented, in Fig. 11 (b-c),
respectively. The density static structure factor’s general behavior is
identical to that of the single-component BEC, illustrated in Fig. 13(a).

The magnitude of $S_{D}(k_{x},k_{y})$ in two-component BEC is relatively lower
than that of the single-component BEC at most $k_{x}$ values for the uniform
and trapped case. Similarly $S_{S}(k_{x},k_{y})$ has a higher magnitude
relative to the corresponding single-component static structure factor. It
reaches the plateau of the unit static structure factor rapidly, shown in Fig.
13(b). Thus, finite interspecies interaction strength diminishes the density
static structure factor in two-component BEC, whereas it intensifies the spin
structure factor, in agreement with results of Ticknor2 . Again this can be
understood from $S({\bf k})=\epsilon_{0}({\bf k})/\epsilon_{B}({\bf k})$. For
a finite $g_{+-}$, the effective interaction strength ($g+g_{+-}$) increases
in the total density part [refer to the first term in the second line of
Eq.(2)], leading to increment in the energy of the excitation relative to the
single-component case with $g_{+-}=0$ and thus decreasing the static structure
factor. Simultaneously, the effective interaction strength ($g-g_{+-}$) is
reduced in the spin-density term [refer to the second term in the second line
of Eq.(2)], resulting in the lower energy and increasing the static structure
factor relative to $S_{D}$ in two-component BEC.

However, the $Feynman$ relation discussed above is not generally applicable to
uniform BEC’s with spin-orbit coupling FR_SOCBEC . The relation is not
satisfied in the whole momentum space, and applicability depends on the type
of ground state of SOC-BEC FR_SOCBEC2 . Therefore, we will utilize Eq. (12)
only to compute and discuss the dynamic structure factor features in trapped
SOC-BEC.

Figure 16: (color online). (a-b) respectively shows the ground state
corresponding to the total density and spin-density (in units of
$a_{\rho}^{-2}$) in miscible SOC configuration ($g_{+-}<g$) for
$\eta^{\prime}/\eta=0.78$ with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$. The
dynamic structure factor for density $S_{D}(k_{x},k_{y}=0,\omega)$ and spin
$S_{S}(k_{x},k_{y}=0,\omega)$ in this case is shown in (c-d) respectively.
(e-h) shows the corresponding results for $g_{+-}>g$.  Figure 17: (color
online). (a-b) respectively shows the ground state corresponding to the total
density and spin-density (in units of $a_{\rho}^{-2}$) in miscible SOC
configuration ($g_{+-}<g$) for $\eta^{\prime}/\eta=0.78$ with
$\eta=2\sqrt{\hbar\omega_{\rho}/m}$. The dynamic structure factor for density
$S_{D}(k_{x},k_{y}=0,\omega)$ and spin $S_{S}(k_{x},k_{y}=0,\omega)$ for this
case is shown in (c-d) respectively. (e-h) shows the corresponding results for
$g_{+-}>g$.

Coming to the SOC-BEC with all interaction strengths equal, namely $g_{+-}=g$,
the $S_{D,S}(k_{x},k_{y}=0,\omega)$ is shown for $\eta^{\prime}/\eta=0.78$
with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$ in Fig. 11(d-e), respectively. The
figures illustrate a relatively broader non-vanishing frequency response at a
given wavevector and vice-versa, compared to the case without SOC. The
corresponding static structure factors are shown in Fig. 13 (a-b),
respectively. In contrast to the single-component and two-component BEC, both
the static structure factors are finite in the limit of $k_{x}\rightarrow$ 0\.
For fixed $k_{y}=a_{\rho}^{-1},4a_{\rho}^{-1}$, the dynamic structure factor
corresponding to the density and spin dynamic structure factor is shown in
Fig. 12 (a-b) and (c-d) respectively.

We also illustrate the contribution of the first three excited states,
corresponding to j =1,2,3 from Eq.(11), individually to the total density and
spin structure factor in Fig. 14(a-b), respectively. And, compare them with
the same for a two-component BEC in Fig. 14(c-d). In SOC-BEC, a finite
contribution develops in both the static factors $j=3$ onwards, as
$k_{x}\rightarrow 0$. However, in two-component BEC, these terms’ contribution
to $S_{D,S}(k_{x},k_{y})$ as $k_{x}\rightarrow 0$ is negligible, illustrated
in Fig. 14(c-d), respectively.

Furthermore, the $S_{D}(k_{x},k_{y})$ in SOC-BEC consists a peak at
$k_{x}\rightarrow 0$ followed by a minima and a maxima as $k_{x}$ increases,
illustrated in Fig. 13(a). Also, for $S_{S}(k_{x},k_{y})$, we observe a peak
at finite $k_{x}$, illustrated in Fig. 13(b). The maxima in the
$S_{D}(k_{x},k_{y})$ is due to the behavior of the fluctuations in the ground
state wave function. The density and spin fluctuation in terms of
quasiparticle amplitudes is respectively, given as,

$\displaystyle\delta\rho({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{j}[|u_{+,j}+v^{*}_{+,j}|^{2}+|u_{-,j}+v^{*}_{-,j}|^{2}]$
(15) $\displaystyle\delta s_{z}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{j}[|u_{+,j}+v^{*}_{+,j}|^{2}-|u_{-,j}+v^{*}_{-,j}|^{2}]$
(16)

The density static structure factor’s maxima coincides approximately with the
location of the maximum in the fluctuations in the density relative to the
total density, in the momentum space, i.e.,
$\delta_{\rho}(k_{x},k_{y}=0)/\rho(k_{x},k_{y}=0)$, illustrated in Fig. 15 (a)
corresponding to the total density. Fig. 15 (b), shows the Fourier transform
of the fluctuations, $(\delta s_{z})_{k}$ in the ground state corresponding to
the spin-density alongwith $k_{y}=0$. It does not have clear maxima/minima
like the previous one [Fig. 15 (a)]. However, $S_{S}$ shown in Fig. 13 (b) for
$\eta^{\prime}/\eta=0.78$ has a distinct peak whose location coincides with
the peak in Fig.15 (b) for the corresponding case.

The spin-orbit coupling further enhances the structure factor’s amplitude for
both the density and spin static structure factor, shown in Fig. 13 (a-b),
respectively. Also, the peak’s amplitude at non-zero $k_{x}$ is relatively
large in the $S_{S}(k_{x},k_{y})$ than in $S_{D}(k_{x},k_{y})$. Similar to the
case without SOC, the effective interaction strength gets reduced in the spin-
density term which further leads to lower excitation energy and enhancing the
spin static structure factor compared to the density structure factor.

The above discussion confines to the balanced inter-component and intra-
component interaction strengths, i.e., $g_{\pm}=g$. Based on the interaction
strengths, the characterisation of the ground state in an interacting gas
depends on the minima in the single-particle dispersion brandon ; SOC_SBH ,
which is either a single-well (miscible, $g<g_{\pm}$ ) or double-well
(immiscible, $g>g_{\pm}$) type. The ground state of our system has two
possible phases: zero-momentum phase and plane-wave phase Li ; Rev . In the
miscible regime, SOC-BEC condenses in the zero-momentum state and has a
extremely small value ($\sim$ 0) of spin-density, apparent from the magnitude
of randomly fluctuating spin-density, shown in Fig. 16-17(b). Whereas in the
immiscible regime, SOC-BEC chooses one of the two-wells as the ground state
and is in the plane-wave phase. In this regime, the spin-density is finite-
valued Rev , illustrated in Fig. 16-17(f). Another typical ground state phase
in SOC-BEC literature is the stripe phase Li , where the BEC remains in a
superposition state of both well. However, the stripe phase is not achievable
in our system because there is no off-diagonal coupling term in Eq. (2).

We next discuss the density and spin dynamic structure factor and the
corresponding static structure factors in SOC-BEC, in such miscible and and
immiscible configuration. For both these configurations, we have shown total
density and spin-density in the ground state with the respective dynamic
factors in Fig. 16-17, for $\eta^{\prime}/\eta=0.78$ with $\eta=0.6,1.8$
(units of $\sqrt{\hbar\omega_{\rho}/m}$), respectively.

In the case of $g_{+-}<g$, the density and spin dynamic structure factor
$S_{D,S}(k_{x},k_{y}=0,\omega)$ for SOC strengths $\eta^{\prime}/\eta=0.78$
with $\eta=0.6\sqrt{\hbar\omega_{\rho}/m}$ is shown in Fig. 16 (c-d)
respectively, and for $g_{+-}>g$ in Fig. 16 (g-h) respectively. For a higher
value of SOC strength $\eta=2\sqrt{\hbar\omega_{\rho}/m}$ with
$\eta^{\prime}/\eta=0.78$, the corresponding results appear in Fig. 17. At a
given SOC-strength and $g_{+-}<g$, the DSF for both total density and spin-
density are symmetric about $k_{x}$ = 0, i.e.,
$S_{D,S}(k_{x},k_{y}=0,\omega)=S_{D,S}(-k_{x},k_{y}=0,\omega)$. In the
opposite case, $g_{+-}>g$, the energy spectrum possesses a minima-like
feature, as discussed earlier. Also, SOC-BEC chooses either of the two wells
as the ground state through the spontaneous symmetry breaking mechanism, which
consequently affects the nature of the DSF anisodynamics . The symmetry under
the exchange of $k_{x}$ to $-k_{x}$ is now not respected. The DSF exhibits
asymmetric character about $k_{x}$ = 0 in both total density and spin-density,
i.e., $S_{D,S}(k_{x},k_{y}=0,\omega)\neq S_{D,S}(-k_{x},k_{y}=0,\omega)$,
illustrated respectively in Fig. 16-17(g-h).

Figure 18: (color online).Static structure factor for density
$S_{D}(k_{x},k_{y}=0)$ and spin $S_{S}(k_{x},k_{y}=0)$ in the SOC-BEC case for
ratio SOC strengths $\eta^{\prime}/\eta=0.78$ with $\eta=0.6,2$ (in units of
$\sqrt{\hbar\omega_{\rho}/m}$), for $g_{+-}<g$ is shown in (a-b) respectively.
(c-d) shows the corresponding results for $g_{+-}>g$. Here, the results have
been normalized by the maximum value of the corresponding static structure
factor.

Fig. 18 displays the static structure factor in both miscible and immiscible
SOC-BEC. In the miscible case, for a ratio of SOC strengths
$\eta^{\prime}/\eta=0.78$ with values of $\eta$ listed in Fig. 18, the density
and spin-density static structure factor is shown in (a-b), respectively;
(c-d) shows the corresponding results for the immiscible case. The features
discussed previously for balanced interaction strengths, e.g., non-zero static
structure factor as $k_{x}\rightarrow 0$ and the magnitude of static structure
factor approaching unity at large wave vector, remain valid.

The static structure factors in Fig. 18 are normalized by their maximum value,
i.e., $S_{D,S}/\text{max}\\{S_{D,S}\\}$, to show the general features such as
peaks, maxima/minima, and not magnitude. The behavior of both the density and
spin static structure factors for smaller $\eta$ is similar. It develops a
peak at finite $k_{x}$ and then reaches a constant value. However, the
increase in the value of SOC strength $\eta$ results in maxima-minima
variation in density and spin static structure factors for
$\eta=2\sqrt{\hbar\omega_{\rho}/m}$. Thus, the structure factor’s character is
strongly affected by the variation in the interaction strengths and the SOC
strengths.

##  IV Conclusion

Within the BdG theory framework, we analyse the excitation spectrum in a
trapped quasi-two-dimensional two-component SOC-BEC for both balanced and
unbalanced interaction strengths. For comparison, we also computed the
excitation spectrum in configurations without spin-orbit coupling, i.e., in
the single-component and two-component BEC. Apart from identifying low-energy
excitations such as dipole, quadrupole modes in SOC-BEC, and single-component
BEC, we examined various excitation modes probing a range of energy spectrum.
We particularly demonstrate how the SOC results in the anisotropic
quasiparticle amplitudes even in an isotropic trapping potential. We also
demonstrated that the gauge part of the total angular momentum is responsible
for granting non-zero angular momentum to the lowest-energy excitation in SOC-
BEC.

Furthermore, the immiscible SOC-BEC configuration possesses a minima-like
feature in the excitation spectrum. The quasiparticle amplitudes near the
minima exhibit smudged-asymmetrical nature due to the imbalanced interaction
strengths. We evaluate the dynamic structure factor to understand the response
of all these configurations to the external perturbation, within the BdG
framework. Results in section III display the strong impact of SOC strengths
and the interaction strengths on the the dynamic and static structure factor.
The results presented can be studied beyond the mean-field theory by taking
account of quantum fluctuations, e.g., using the truncated Wigner
approximation. Thus, enabling the evaluation of structure factor in SOC-BEC at
finite temperature. Moreover, Bragg spectroscopy allows one to tune the
momentum transfer over a wide range, and various properties, e.g., coherence
BS1 , and vortices vort1 ; vort2  can be probed in such a configuration.

We also show that the lowest energy excitation has a non-zero angular momentum
arising from the gauge part. This may lead to the study the Einstein-de Haas
effect EDH ; EDH_CM ; EDH_BEC  in such system (magnetization present in the
initial system can cause mechanical rotation in it). We can tune the SOC
parameters that change the non-Abelian gauge potential and hence the effective
magnetic field. The magnetic field variation can lead to finite polarisation
in the system causing rigid rotation in this system. These may lead to
interesting theoretical and experimental work in future.

##  V ACKNOWLEDGMENTS

The work is supported by a BRNS (DAE, Govt. of India) Grant No.
21/07/2015-BRNS/35041 (DAE SRC Outstanding Investigator scheme).

##  Appendix A Details of Basis chosen and Integrals used in section II.2

Consider the following part of $\bar{L}_{+}$ from Eq. (5),

$\displaystyle\bigg{[}\frac{1}{2m}(p_{x}+m\eta)^{2}+\frac{1}{2}m\omega^{2}x^{2}\bigg{]}+\bigg{[}\frac{1}{2m_{y}}p_{y}^{2}+\frac{1}{2}m\omega^{2}y^{2}\bigg{]}$

The above can be rewritten as:

$\displaystyle\bigg{[}P_{1}^{2}+X_{1}^{2}\bigg{]}+\bigg{[}P_{2}^{2}+X_{2}^{2}\bigg{]}$

where, $P_{1}=\frac{1}{\sqrt{2m}}(p_{x}+m\eta)$,
$P_{2}=\frac{p_{y}}{\sqrt{2m_{y}}}$, $X_{1}=\sqrt{\frac{m}{2}}\omega x$ and
$X_{2}=\sqrt{\frac{m}{2}}\omega y$. Defining the operators:

$\displaystyle a_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\hbar\omega}}(X_{1}+iP_{1}),a_{2}^{\prime}=\frac{1}{\sqrt{\hbar\omega}}({\frac{m}{m_{y}}})^{1/4}(X_{2}+iP_{2})$

We get,

$\displaystyle X_{1}^{2}+P_{1}^{2}$ $\displaystyle=$
$\displaystyle\bigg{[}a_{1}^{\prime\dagger}a_{1}^{\prime}+\frac{1}{2}\bigg{]}\hbar\omega,$
$\displaystyle X_{2}^{2}+P_{2}^{2}$ $\displaystyle=$
$\displaystyle\bigg{[}a_{2}^{\prime\dagger}a_{2}^{\prime}+\frac{1}{2}\bigg{]}\hbar\omega\sqrt{\frac{m}{m_{y}}}$

Using these operators, first part of $\bar{L}_{+}$ (or $\bar{\bar{L}}_{+})$
becomes diagonal in the chosen basis with energy

$E(K,L)=(K+m_{R}L+\frac{1}{2}[1+m_{R}])\hbar\omega$

where $m_{R}=\sqrt{m/m_{y}}$ and K,L =0,1,…$N_{b}$.

We next provide the explicit form of the integrals used for computing the
matrix elements of Eq.(7) in the following:

$\displaystyle A_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[\bar{L}_{+}]\phi_{k}\phi_{l}$
$\displaystyle B_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[g_{2D}{\psi^{2}_{+,0}}]\phi_{k}\phi_{l}$
$\displaystyle C_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[g_{2D}\psi_{+,0}{\psi^{*}_{-,0}}]\phi_{k}\phi_{l}$
$\displaystyle D_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[g_{2D}\psi_{+,0}\psi_{-,0}]\phi_{k}\phi_{l}$
$\displaystyle E_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[-g_{2D}{\psi^{*}_{+,0}}^{2}]\phi_{k}\phi_{l}$
$\displaystyle G_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[-g_{2D}{\psi^{*}_{+,0}}{\psi^{*}_{-,0}}]\phi_{k}\phi_{l}$
$\displaystyle H_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[-g_{2D}{\psi^{*}_{+,0}}\psi_{-,0}]\phi_{k}\phi_{l}$
$\displaystyle K_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[\bar{L}_{-}]\phi_{k}\phi_{l}$
$\displaystyle L_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[g_{2D}\psi^{2}_{-,0}]\phi_{k}\phi_{l}$
$\displaystyle O_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$
$\displaystyle\int\int
dxdy\phi^{*}_{k^{\prime}}\phi^{*}_{l^{\prime}}[-g_{2D}{\psi^{*}_{-,0}}^{2}]\phi_{k}\phi_{l}$
$\displaystyle F_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$ $\displaystyle-
A_{k^{\prime}l^{\prime}kl},\hskip
2.84544ptI_{k^{\prime}l^{\prime}kl}=-H_{k^{\prime}l^{\prime}kl},\hskip
2.84544ptJ_{k^{\prime}l^{\prime}kl}=D_{k^{\prime}l^{\prime}kl}$ $\displaystyle
M_{k^{\prime}l^{\prime}kl}$ $\displaystyle=$ $\displaystyle
G_{k^{\prime}l^{\prime}kl},\hskip
2.84544ptN_{k^{\prime}l^{\prime}kl}=-C_{k^{\prime}l^{\prime}kl},\hskip
2.84544ptP_{k^{\prime}l^{\prime}kl}=-K_{k^{\prime}l^{\prime}kl}$

##  Appendix B Details of Derivation from Section III

###  B.1 Dynamic Structure factor in Lehmann representation

The dynamic correlation of a density fluctuation at time t=0 and one at time
‘t’ is given as

$\displaystyle S_{D}({\bf k},t)$ $\displaystyle=$
$\displaystyle\bra{0}\hat{\rho}_{\bf k}(t)\hat{\rho}_{\bf k}(0)\ket{0}$ (17)

Introducing a complete set of eigenstates of the Hamiltonian $\ket{j}$ and
using the time evolution of the density fluctuation i.e., $\hat{\rho}_{\bf
k}(t)\ket{0}=e^{i\hat{H}_{eff}t}\hat{\rho}_{\bf
k}e^{-i\hat{H}_{eff}t}\ket{0}$, Eq.(17) becomes

$\displaystyle S_{D}({\bf k},t)$ $\displaystyle=$
$\displaystyle\sum_{j}\bra{0}e^{i\hat{H}_{eff}t}\hat{\rho}_{\bf
k}e^{-i\hat{H}_{eff}t}\ket{j}\bra{j}\hat{\rho}_{\bf k}\ket{0}$ (18)
$\displaystyle=$ $\displaystyle\sum_{j}|\bra{0}\hat{\rho}_{\bf
k}\ket{j}|^{2}e^{-i(\epsilon_{j}-\epsilon_{0})t}$

where, $\hat{H}_{eff}$ is the effective Hamiltonian used for obtaining Eq.(2)
and is given as

$\displaystyle\hat{H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2m}(\hat{p}_{x}\check{I}-\check{\sigma}_{z}m\eta)^{2}+\frac{\hat{p}_{y}^{2}}{2m_{y}}\check{I}$
(19) $\displaystyle+$
$\displaystyle\bigg{[}V_{Trap}+g_{2D}\bigg{(}|\psi_{+}|^{2}+|\psi_{-}|^{2}\bigg{)}\bigg{]}\check{I}$

Taking the Fourier transform of Eq.(18), we get

$\displaystyle S_{D}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\sum_{j}|\bra{0}\hat{\rho}_{\bf
k}\ket{j}|^{2}\delta(\omega-(\epsilon_{j}-\epsilon_{0})/\hbar)$ (20)

The above expression is written in Lehmann representation Dpines . Similarly,
the DSF for spin-density can be computed.

###  B.2 Details of matrix element in Eq.(12)

In this part of Appendix, we will compute the matrix element
$\bra{0}\hat{\rho}({\bf r})\ket{j}$. To this purpose, we define the field
operators corresponding to each component as,

$\displaystyle\hat{\bm{\psi}}({\bf r})=e^{\frac{-i\mu
t}{\hbar}}\bigg{(}\hat{\bm{\psi}}_{0}({\bf
r})+\sum_{j}\bigg{\\{}{\bm{u}}_{j}({\bf
r})\hat{b}_{j}e^{-i\omega_{j}t}+{\bm{v}}^{*}_{j}({\bf
r})\hat{b}^{\dagger}_{j}e^{i\omega^{*}_{j}t}\bigg{\\}}\bigg{)}$ (21)

where, $\hat{\psi}({\bf r})=[\hat{\psi}_{+}({\bf r})\hskip
5.69046pt\hat{\psi}_{-}({\bf r})]^{T}$, $\hat{\psi}_{0}({\bf
r})=[\hat{\psi}_{+,0}({\bf r})\hskip 5.69046pt\hat{\psi}_{-,0}({\bf r})]^{T}$,
${\bm{u}}_{j}({\bf r})=[u_{+,j}({\bf r})\hskip 5.69046ptu_{-,j}({\bf r})]^{T}$
and, ${\bm{v}}_{j}({\bf r})=[v_{+,j}({\bf r})\hskip 5.69046ptv_{-,j}({\bf
r})]^{T}$.

The density operator corresponding to each component is given as,

$\displaystyle|\hat{\psi}_{+}({\bf r})|^{2}$ $\displaystyle=$
$\displaystyle|\hat{\psi}_{+,0}({\bf r})|^{2}$ $\displaystyle+$
$\displaystyle\sum_{j}\hat{b}_{j}e^{-i\omega_{j}t}[u_{+j}(\bf
r)\psi^{*}_{+,0}({\bf r})+v_{+j}(\bf r)\psi_{+,0}({\bf r})]+c.c.$
$\displaystyle|\hat{\psi}_{-}({\bf r})|^{2}$ $\displaystyle=$
$\displaystyle|\hat{\psi}_{-,0}({\bf r})|^{2}$ $\displaystyle+$
$\displaystyle\sum_{j}\hat{b}_{j}e^{-i\omega_{j}t}[u_{-j}(\bf
r)\psi^{*}_{-,0}({\bf r})+v_{-j}(\bf r)\psi_{-,0}({\bf r})]+c.c.$

The matrix element $\bra{0}\hat{\rho}({\bf r})\ket{j}=\bra{0}[\hat{\rho}({\bf
r}),\hat{b}^{\dagger}_{j}]\ket{0}$ as $\hat{b}^{\dagger}_{j}\ket{0}=\ket{j}$.
Therefore, the matrix element corresponding to operators for total density
$\hat{\rho}({\bf r})=|\hat{\psi}_{+}({\bf r})|^{2}+|\hat{\psi}_{-}({\bf
r})|^{2}$ and spin-density $\hat{s}_{z}({\bf r})=|\hat{\psi}_{+}({\bf
r})|^{2}-|\hat{\psi}_{-}({\bf r})|^{2}$ can be expressed as,

$\displaystyle\bra{0}\hat{\rho}({\bf r})\ket{j}$ $\displaystyle=$
$\displaystyle e^{-i\omega_{j}t}\big{\\{}[u_{+j}(\bf r)\psi^{*}_{+,0}({\bf
r})+v_{+j}(\bf r)\psi_{+,0}({\bf r})]$ $\displaystyle+$
$\displaystyle[u_{-j}(\bf r)\psi^{*}_{-,0}({\bf r})+v_{-j}(\bf
r)\psi_{-,0}({\bf r})]\big{\\}}$ $\displaystyle\bra{0}\hat{s}_{z}({\bf
r})\ket{j}$ $\displaystyle=$ $\displaystyle
e^{-i\omega_{j}t}\big{\\{}[u_{+j}(\bf r)\psi^{*}_{+,0}({\bf r})+v_{+j}(\bf
r)\psi_{+,0}({\bf r})]$ $\displaystyle-$ $\displaystyle[u_{-j}(\bf
r)\psi^{*}_{-,0}({\bf r})+v_{-j}(\bf r)\psi_{-,0}({\bf r})]\big{\\}}$

## References

  * (1) L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity (Clarendon, Oxford, UK, 2016).
  * (2) C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases (Cambridge university press, 2002).
  * (3) T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis, and B. P. Anderson, Observation of Vortex Dipoles in an Oblate Bose-Einstein Condensate, Phys. Rev. Lett. 104, 160401 (2010).
  * (4) T. Frisch, Y. Pomeau, and S. Rica, Transition to dissipation in a model of superflow T. Frisch, Y. Pomeau, and S. Rica, Phys. Rev. Lett. 69, 1644 (1992).
  * (5) C. Raman, M. Kohl, R. Onofrio, D.S. Durfee, C.E. Kuklewicz, Z. Hadzibabic, and W. Ketterle, Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas, Phys. Rev. Lett. 83, 2502 (1999).
  * (6) S. Sachdev, Quantum criticality: competing ground states in low dimensions,  Science 288, 475–480 (2000).
  * (7) R. Roth and K. Burnett, Dynamic structure factor of ultracold Bose and Fermi gases in optical lattices, J. Phys. B 37, 3893–3907 (2004).
  * (8) Y.-J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V. Porto, and I. B. Spielman, Synthetic magnetic fields for ultracold neutral atoms,  Nature (London) 462, 628 (2009).
  * (9) I. B. Spielman, Raman processes and effective gauge potentials, Phys. Rev. A. 79, 063613 (2009).
  * (10) Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose–Einstein Condensate in a Uniform Light-Induced Vector Potential, Phys. Rev. Lett. 102, 130401 (2009).
  * (11) J. Dalibard, F. Gerbier, G. Juzeliūnas, and P. Öhberg, Colloquium: Artificial gauge potentials for neutral atoms, Rev. Mod. Phys. 83, 1523 (2011).
  * (12) E. van der Bijl and R. A. Duine, Anomalous Hall Conductivity from the Dipole Mode of Spin-Orbit-Coupled Cold-Atom Systems, Phys. Rev. Lett. 107, 195302 (2011).
  * (13) Y. Zhang, L. Mao, and C. Zhang, Mean-Field Dynamics of Spin-Orbit Coupled Bose-Einstein Condensates, Phys. Rev. Lett. 108, 035302 (2012).
  * (14) B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Half-quantum vortex state in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 85, 023606 (2012).
  * (15) Z. Chen and H. Zhai, Collective-mode dynamics in a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 86, 041604(R) (2012).
  * (16) Y. Li, G. I. Martone, and S. Stringari, Sum rules, dipole oscillation and spin polarizability of a spin-orbit coupled quantum gas, Europhys. Lett. 99, 56008 (2012).
  * (17) J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du, B. Yan, G.-S. Pan, B. Zhao, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Collective Dipole Oscillations of a Spin-Orbit Coupled Bose-Einstein Condensate, Phys. Rev. Lett. 109, 115301 (2012).
  * (18) W. Zheng and Z. Li, Collective modes of a spin-orbit-coupled Bose-Einstein condensate: A hydrodynamic approach, Phys. Rev. A 85, 053607 (2012).
  * (19) G. I. Martone, Y. Li, L. P. Pitaevskii, and S. Stringari, Anisotropic dynamics of a spin-orbit-coupled Bose-Einstein condensate, Phys. Rev. A 86, 063621(2012).
  * (20) M. A. Khamehchi, Y. Zhang, C. Hamner, T. Busch, and P. Engels, Measurement of collective excitations in a spin–orbit-coupled Bose–Einstein condensate, Phys. Rev. A 90, 063624 (2014).
  * (21) Si-Cong Ji, Long Zhang, Xiao-Tian Xu, Zhan Wu, Youjin Deng, Shuai Chen, and Jian-Wei Pan, Softening of Roton and Phonon Modes in a Bose-Einstein Condensate with Spin-Orbit Coupling, Phys. Rev. Lett. 114, 105301(2015).
  * (22) J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and W. Ketterle, Bragg Spectroscopy of a Bose-Einstein Condensate, Phys. Rev. Lett. 82, 4569(1999).
  * (23) D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D. E. Pritchard, and W. Ketterle, Excitation of Phonons in a Bose-Einstein Condensate by Light Scattering, Phys. Rev. Lett. 83, 2876(1999).
  * (24) J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Excitation Spectrum of a Bose-Einstein Condensate, Phys. Rev. Lett. 88, 120407 (2002).
  * (25) J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo, Bragg Spectroscopy of the Multibranch Bogoliubov Spectrum of Elongated Bose-Einstein Condensates, Phys. Rev. Lett. 90, 060404 (2003).
  * (26) C. Tozzo and F. Dalfovo, Bogoliubov spectrum and Bragg spectroscopy of elongated Bose?Einstein condensates, New J. Phys. 5, 54 (2003).
  * (27) Tudor D. Stanescu, Brandon Anderson, and Victor Galitski, Spin-orbit coupled Bose-Einstein condensates, Phys. Rev. A 78, 023616 (2008).
  * (28) Inderpreet Kaur and Sankalpa Ghosh, (2+1) -dimensional sonic black hole from a spin-orbit-coupled Bose-Einstein condensate and its analog Hawking radiation, Phys. Rev. A 102, 023314 (2020).
  * (29) D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Bose-Einstein Condensation in Quasi-2D Trapped Gases, Phys. Rev. Lett. 84, 2551 (2000).
  * (30) Y.-J. Lin, K. J-Garcia, and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature (London) 471, 83 (2011).
  * (31) S. Stringari, Collective Excitations of a Trapped Bose-Condensed Gas, Phys. Rev. Lett. 77, 2360 (1996).
  * (32) C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Production of Two Overlapping Bose-Einstein Condensates by Sympathetic Cooling, Phys. Rev. Lett. 78, 586 (1997).
  * (33) D. Gordon and C. M. Savage, Excitation spectrum and instability of a two-species Bose-Einstein condensate, Phys. Rev. A 58, 1440 (1998).
  * (34) K. Suthar and D. Angom, Optical-lattice-influenced geometry of quasi-two-dimensional binary condensates and quasiparticle spectra, PRA 93, 063608 (2016).
  * (35) Angela C. White, Yongping Zhang, and Thomas Busch, Odd-petal-number states and persistent flows in spin-orbit-coupled Bose-Einstein condensates, Phys. Rev. A 95, 041604(R) (2017).
  * (36) X. F. Zhang, M. Kato, W. Han, S. G. Zhang and H. Saito1, Spin-orbit-coupled Bose-Einstein condensates held under a toroidal trap, Phys. Rev. A 95, 033620 (2017).
  * (37) C. Ticknor, Dispersion relation and excitation character of a two-component Bose-Einstein condensate, PRA 89, 053601 (2014).
  * (38) R. M. Wilson, S. Ronen, and J. L. Bohn, Critical Superfluid Velocity in a Trapped Dipolar Gas, Phys. Rev. Lett. 104, 094501 (2010).
  * (39) P. B. Blakie, D. Baillie, and R. N. Bisset, Depletion and fluctuations of a trapped dipolar Bose-Einstein condensate in the roton regime, Phys. Rev. A 88, 013638 (2013).
  * (40) R. N. Bisset, D. Baillie, and P. B. Blakie, Roton excitations in a trapped dipolar Bose-Einstein condensate,  Phys. Rev. A 88, 043606 (2013).
  * (41) R. N. Bisset and P. B. Blakie, Fingerprinting Rotons in a Dipolar Condensate: Super-Poissonian Peak in the Atom-Number Fluctuations, Phys. Rev. Lett. 110, 265302 (2013).
  * (42) S. Stringari, Diffused Vorticity and Moment of Inertia of a Spin-Orbit Coupled Bose-Einstein Condensate, Phys. Rev. Lett. 118, 145302.
  * (43) A. D. Martin and P. B. Blakie, Stability and structure of an anisotropically trapped dipolar Bose-Einstein condensate: Angular and linear rotons, Phys. Rev. A 86, 053623 (2012).
  * (44) D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews, and W. Ketterle, Collisionless and Hydrodynamic Excitations of a Bose–Einstein Condensate, Phys. Rev. Lett. 81, 500 (1998).
  * (45) F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  * (46) Chris Hamner, Chunlei Qu, Yongping Zhang, JiaJia Chang, Ming Gong, Chuanwei Zhang and Peter Engels, Dicke-type phase transition in a spin-orbit-coupled Bose–Einstein condensate, Nature Communications 5, 4023 (2014) .
  * (47) A. Griffin, Excitations in a Bose-Condensed Liquid, No. 4 in Cambridge Studies in Low-Temperature Physics (Cambridge University Press, New York, 1993.
  * (48) H. Palevski, K. Otnes, and K. E. Larson, Phys. Rev. 112, 11 (1958); D. G. Henshaw and A. D. B. Woods, ibid. 121, 1266 (1961).
  * (49) P. Sokol in Bose-Einstein Condensation, edited by A. Griffin, D. W. Snooke, and S. Stringari (Cambridge University Press, Cambridge, 1995), p. 51.
  * (50) R. P. Feynman, Atomic Theory of the Two-Fluid Model of Liquid Helium, Phys. Rev. 94, 262 (1954).
  * (51) Timmermans, E. Phase separation in Bose-Einstein condensates, Phys. Rev. Lett. 81, 5718 (1998).
  * (52) R. N. Bisset, P. G. Kevrekidis and C. Ticknor, Enhanced quantum spin fluctuations in a binary Bose-Einstein condensate, PRA 97, 023602 (2018).
  * (53) P. S. He, R. Liao, and W. M. Liu, Feynman relation of Bose-Einstein condensates with spin-orbit coupling, Phys. Rev. A 86, 043632(2012).
  * (54) T. Wu and R. Liao, Bose-Einstein condensates in the presence of Weyl spin- orbit coupling, New J. Phys. 19, 013008 (2017).
  * (55) Y. Li, G. I. Martone, and S. Stringari, Spin-orbit-coupled Bose-Einstein condensate,Annual Review of Cold Atoms and Molecules (World Scientific, Singapore, 2015), Vol. 3, pp. 201– 250\.
  * (56) Y. Zhang, M. E. Mossman, T. Busch, P. Engels, C. Zhang, Properties of spin–orbit-coupled Bose–Einstein condensates, Front. Phys. 11, 118103 (2016).
  * (57) P. B. Blakie and R. J. Ballagh, Spatially Selective Bragg Scattering: A Signature for Vortices in Bose-Einstein Condensates, Phys. Rev. Lett. 86, 3930 (2001).
  * (58) S. R. Muniz, D. S. Naik, and C. Raman, Bragg spectroscopy of vortex lattices in Bose-Einstein condensates, Phys. Rev. A 73, 041605(R) (2001).
  * (59) A. Einstein and W. J. de Haas, Verh. Dtsch. Phys. Ges. 17, 152 (1915).
  * (60) L. Zhang and Q. Niu, Angular Momentum of Phonons and the Einstein–de Haas Effect, PRL 112, 085503 (2014).
  * (61) Y. Kawaguchi, H. Saito, and M. Ueda, Einstein–de Haas Effect in Dipolar Bose-Einstein Condensates, PRL 96, 080405 (2006).
  * (62) D. Pines and P. Nozieres, The theory of quantum fluids, Vol. I (W. A. Benjamin, New York, 1966).