# Probing low-energy Lorentz violation from high-energy modified dispersion in
dipolar Bose-Einstein condensates

Zehua Tian<EMAIL_ADDRESS>Hefei National Laboratory for Physical Sciences
at the Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  CAS Key Laboratory of Microscale
Magnetic Resonance, University of Science and Technology of China, Hefei
230026, China  Synergetic Innovation Center of Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Jiangfeng Du<EMAIL_ADDRESS>Hefei National Laboratory for Physical Sciences
at the Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei 230026, China  CAS Key Laboratory of Microscale
Magnetic Resonance, University of Science and Technology of China, Hefei
230026, China  Synergetic Innovation Center of Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China

###### Abstract

We theoretically propose an experimentally viable scheme to use an impurity
atom in a dipolar Bose-Einstein condensate (BEC), in order to probe analogue
low-energy Lorentz violation from the modified dispersion at high energies as
suggested by quantum theories of gravity. We show that the density
fluctuations in the dipolar BEC possess a Lorentz-violating Bogoliubov
spectrum
$\omega_{\mathbf{k}}=c_{0}|\mathbf{k}|f(c_{0}|\mathbf{k}|/M_{\star})$, with
recovery of approximate Lorentz invariance (LI) at energy scales much below
$M_{\star}$. When $f$ is adjusted to dip below unity somewhere, the impurity,
analogously dipole coupled to the density fluctuations, experiences analogue
drastic Lorentz violation at arbitrarily low energies, reproducing the same
responds of Unruh-DeWitt detector to Lorentz-violating quantum fields. Being a
fundamentally quantum mechanical device, our quantum fluid platform provides
an experimentally realizable test field to verify whether the effective low
energy theory can reveal unexpected imprints of the theory’s high energy
structure, in quantum field theory.

##  I Introduction

Lorentz invariance (LI) is one of the fundamental symmetries of relativity,
however, and may be challenged at sufficiently high energies suggested by many
theories of quantum gravity Amelino-Camelia (2013). As a phenomenal embodiment
of quantum gravity, any discovery of Lorentz violation thus would be an
important signal of beyond standard model physics. Testing LI has recently
given rise to widespread interest in diverse physical areas, spanning atomic
physics, nuclear physics, high-energy physics, relativity, and astrophysics,
see Ref. Colladay and Kostelecký (1997, 1998); Mattingly (2005); Kostelecký
and Russell (2011), and references therein.

In the LI violation theory, there may be an explicit energy scale, such as the
Planck energy $M_{\ast}$, which characterizes the violation and at energies
much below which the LI is preserved approximately. The Planck energy, about
$10^{19}~{}\text{GeV}$, is much larger than any currently experimentally
accessible energy scales, e.g., $10^{11}~{}\text{GeV}$ for the trans-GZK
cosmic rays that is the highest known energy of particles Mattingly (2005). In
this regard, direct observation of Planck scale Lorentz violation seems
impossible in any experiments. Fortunately, strong Planck scale Lorentz
violation could yield a small amount of violation at much lower energies, and
thus one can expect that the effective low-energy theory, for example in
quantum field theory, can reveal unexpected imprints of the theory’s high-
energy structure Mattingly (2005); Collins  _et al._ (2004); Polchinski
(2012).

Recently, it has been found that the transition behavior of low-energy Unruh-
DeWitt detectors Unruh (1976); Dewitt (1979); Birrell and Davies (1982);
Takagi (1986); Crispino  _et al._ (2008); Hu  _et al._ (2012) is ultra-
sensitive to high-energy effects, such as polymer quantum field theories
Kajuri (2016); Kajuri and Sardar (2018); Husain and Louko (2016); Louko and
Upton (2018). It has also been shown that the attainable sensitivity to
nonlocal field theories is expected to outperform that of LHC experiments by
many orders of magnitude Belenchia  _et al._ (2016). Therefore, this feature
of Unurh-DeWitt detectors may have highly potential application for the
falsifiability of theoretical proposals, e.g., quantum gravity Husain and
Louko (2016); Louko and Upton (2018); Belenchia  _et al._ (2016). However,
there is still no concrete experimental setup as a precursor to verify whether
the Unurh-DeWitt detector indeed will work as expected before its future
realistic application.

In this paper, we aim at closing this gap and propose to study the relevant
physics with an experimentally accessible platform consisting of a dipolar BEC
Baranov (2008) and an immersed impurity Recati  _et al._ (2005); Fedichev and
Fischer (2003). From the perspective of analog, the density fluctuations in
the dipolar BEC possessing a roton spectrum Ronen  _et al._ (2007); Tian  _et
al._ (2018); Chä and Fischer (2017); Fischer (2006); Santos  _et al._ (2003),
due to the dipole-dipole interaction (DDI) between atoms (see below for a
detailed discussion), are modeled as Lorentz-violating quantum fields. The
impurity, analogously dipole coupled to the density fluctuations in the
condensate, is modeled as an Unruh-DeWitt detector coupling to the Lorentz-
violating quantum fields. This specific quantum simulator in principle allows
us to test a possible manifestation of the effects caused by high energy
structure in the low-energy quantum detection—drastic low-energy Lorentz
violation in the impurity modeled as Unruh-DeWitt detector—as anticipated in
Ref. Husain and Louko (2016).

Our paper is constructed as follows. In Sec. II we simply introduce our
model—dipolar Bose-Einstein condensate, and show how the Lorentz-violating
quantum fields are simulated with the density fluctuation of the condensate.
In Sec. III we show how to simulate the Unruh-DeWitt detector with an impurity
immersed in the dipolar BEC. In Sec. IV we study the spontaneous excitation of
the detector with inertial trajectory. Experimental feasibility of the
relevant simulation within the current technologies of dipolar BEC is
discussed in Sec. V. Finally, a summary of the main results of our work is
present in Sec. VI.

##  II Simulating Lorentz-violating quantum fields in dipolar Bose-Einstein
condensate

Let us begin with an interacting Bose gas comprising atoms or molecules of
mass $m$, whose Lagrangian density is given by $(\hbar=1)$

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{i}{2}(\Psi^{\ast}\partial_{t}\Psi-\partial_{t}\Psi^{\ast}\Psi)-\frac{1}{2m}|\nabla\Psi|^{2}-V_{\text{ext}}|\Psi|^{2}$
(1)
$\displaystyle-\frac{1}{2}|\Psi|^{2}\int\,d^{3}\mathbf{R}^{\prime}\,V_{\text{int}}(\mathbf{R}-\mathbf{R}^{\prime})|\Psi(\mathbf{R}^{\prime})|^{2},$

where $\mathbf{R}=(\mathbf{r},z)$ are spatial three-dimensional coordinates.
The system is trapped by an external potential of the form
$V_{\text{ext}}(\mathbf{R})=m\omega^{2}\mathbf{r}^{2}/2+m\omega^{2}_{z}z^{2}/2$.
We will assume that over the whole time evolution the gas is strongly confined
in the $z$ direction, with aspect ratio $\kappa=\omega_{z}/\omega\gg 1$. The
two-body interaction contains two terms

$\displaystyle
V_{\text{int}}(\mathbf{R}-\mathbf{R}^{\prime})=g_{c}\delta^{3}(\mathbf{R}-\mathbf{R}^{\prime})+V_{\text{dd}}(\mathbf{R}-\mathbf{R}^{\prime}),$
(2)

where $g_{c}$ is the contact interaction coupling, and

$\displaystyle
V_{\text{dd}}(\mathbf{R}-\mathbf{R}^{\prime})=\frac{3g_{d}}{4\pi}\frac{[1-3(z-z^{\prime})^{2}/|\mathbf{R}-\mathbf{R}^{\prime}|^{2}]}{|\mathbf{R}-\mathbf{R}^{\prime}|^{3}}$
(3)

describes the dipolar interaction with coupling constant $g_{d}$. Here the
dipoles have been assumed to be polarized along the $z$ direction (or
perpendicular to the $x$-$y$ plane) by an external field. In addition, $g_{c}$
and $g_{d}$ can be controlled as required in the experiment. Let us note that
it is the interaction between atoms or molecules (including both the contact
and dipolar interaction) that results in the Lorentz-violating quasiparticle
spectrum, similar to the case in Ref. Edwards and Kostelecký (2018), as shown
in the following. To guarantee the stability in the DDI-dominated regime
Fischer (2006), we impose that the system remains sufficiently close to the
quasi-two-dimensional (quasi-2D) regime during the whole physical process
considered. We thus assume that in the $z$ direction, the condensate density
has a Gaussian form,
$\rho_{z}(z)=(\pi\,d^{2}_{z})^{-1/2}\exp[-z^{2}/d^{2}_{z}]$, with
$d_{z}=\sqrt{1/m\omega_{z}}$. Integrating out the $z$ dependence, we can
obtain the effective quasi-2D interaction,
$V^{\text{2D}}_{\text{int}}(\mathbf{r}-\mathbf{r}^{\prime})=\int\,dzdz^{\prime}\,V_{\text{int}}(\mathbf{R}-\mathbf{R}^{\prime})\rho_{z}(z)\rho_{z}(z^{\prime})$
Chä and Fischer (2017).

In the 2D case, we decompose the 2D field operator as

$\displaystyle\hat{\psi}=\psi_{0}(1+\hat{\phi})$ (4)

with $\psi_{0}=\sqrt{\rho_{0}}e^{i\theta_{0}}$, where
$|\psi_{0}(\mathbf{r})|^{2}=\rho_{0}\simeq\mathrm{const}$ represents the 2D
condensate density, and $\hat{\phi}$ describes the perturbations (excitations)
on the top of the condensate. The Bogoliubov-de Gennes equation for the
fluctuation field $\hat{\phi}$ reads Castin (2001); Tian  _et al._ (2018)

$\displaystyle i\partial_{t}\hat{\phi}$ $\displaystyle=$
$\displaystyle-\frac{1}{2m}\nabla^{2}_{\mathbf{r}}\hat{\phi}+\rho_{0}\int\,d^{2}\mathbf{r}^{\prime}\,V^{\text{2D}}_{\text{int,
0}}(\mathbf{r}-\mathbf{r}^{\prime})$ (5)
$\displaystyle\times\big{[}\hat{\phi}(\mathbf{r}^{\prime})+\hat{\phi}^{\dagger}(\mathbf{r}^{\prime})\big{]},$

where the condensate has been assumed to be static, i.e.,
$\mathbf{v}=\frac{1}{m}\nabla_{\mathbf{r}}\theta_{0}=0$. By solving Eq. (5),
we can write the density fluctuations in Heisenberg representation as

$\displaystyle\delta\hat{\rho}(t,\mathbf{r})$ $\displaystyle\simeq$
$\displaystyle\rho_{0}(\hat{\phi}+\hat{\phi}^{\dagger})$ (6) $\displaystyle=$
$\displaystyle\sqrt{\rho_{0}}\int[d\mathbf{k}/(2\pi)^{2}](u_{\mathbf{k}}+v_{\mathbf{k}})\times[\hat{b}_{\mathbf{k}}(t)e^{i\mathbf{k}\cdot\mathbf{r}}$
$\displaystyle+\hat{b}^{\dagger}_{\mathbf{k}}(t)e^{-i\mathbf{k}\cdot\mathbf{r}}],$

where the Bogoliubov quasiparticle operators
$\hat{b}_{\mathbf{k}}(t)=\hat{b}_{\mathbf{k}}\,e^{-i\omega_{\mathbf{k}}\,t}$
satisfy the usual Bose commutation rules
$[\hat{b}_{\mathbf{k}},\hat{b}_{\mathbf{k}^{\prime}}^{\dagger}]=(2\pi)^{2}\delta^{2}(\mathbf{k}-\mathbf{k}^{\prime})$.
Bogoliubov parameters are given by

$\displaystyle u_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle(\sqrt{\mathcal{H}_{\mathbf{k}}}+\sqrt{\mathcal{H}_{\mathbf{k}}+2\mathcal{A}_{\mathbf{k}}})/2(\mathcal{H}^{2}_{\mathbf{k}}+2\mathcal{H}_{\mathbf{k}}\mathcal{A}_{\mathbf{k}})^{1/4},$
$\displaystyle v_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle(\sqrt{\mathcal{H}_{\mathbf{k}}}-\sqrt{\mathcal{H}_{\mathbf{k}}+2\mathcal{A}_{\mathbf{k}}})/2(\mathcal{H}^{2}_{\mathbf{k}}+2\mathcal{H}_{\mathbf{k}}\mathcal{A}_{\mathbf{k}})^{1/4},$
(7)

and the quasiparticle frequency
$\omega_{\mathbf{k}}=\sqrt{\mathcal{H}^{2}_{\mathbf{k}}+2\mathcal{H}_{\mathbf{k}}\mathcal{A}_{\mathbf{k}}}$
with $\mathcal{H}_{\mathbf{k}}=k^{2}/2m$ and
$\mathcal{A}_{\mathbf{k}}=\rho_{0}V^{\text{2D}}_{\text{int, 0}}(k)$ Tian  _et
al._ (2018). Here $k=|\mathbf{k}|$ and the Fourier transformation of the
effective quasi-2D interaction is given by Fischer (2006)

$\displaystyle V^{\text{2D}}_{\text{int,
0}}(k)=g^{\text{eff}}_{0}(1-\frac{3R}{2}kd_{z}w[\frac{kd_{z}}{\sqrt{2}}]),$
(8)

with $w[x]=\exp[x^{2}](1-\operatorname{erf}[x])$, an effective contact
coupling $g^{\text{eff}}_{0}=\frac{1}{\sqrt{2\pi}d_{z}}(g_{c}+2g_{d})$, and
the dimensionless ratio is defined as

$\displaystyle R=\sqrt{\pi/2}/(1+g_{c}/2g_{d}).$ (9)

Note that the parameter $R$ could be tunable via Feshbach resonance Courteille
_et al._ (1998); Inouye  _et al._ (1998) and rotating polarizing field
Giovanazzi  _et al._ (2002). It ranges from $R=0$ (when
$g_{d}/g_{c}\rightarrow 0$, i.e., contact dominance), to $R=\sqrt{\pi/2}$
(when $g_{d}/g_{c}\rightarrow\infty$, i.e., DDI dominance).

Figure 1: (Color online) The dimensionless function $f$ shown in (10) as a
function of $c_{0}|\mathbf{k}|/M_{\ast}$. LI is the Lorentz invariant case
where $f=1$. $R=0$ denotes the contact interaction case, where $f$ is
independent of $A$. For DDI dominance, $R=\sqrt{\pi/2}$, $f$ dips below 1 for
an interval of $c_{0}|\mathbf{k}|/M_{\ast}$. Note that $f$ becomes negative
when $A>A_{c}=3.4454$, which means the spectrum of quasiparticle becomes
unstable.

The density fluctuations shown in Eq. (6) closely resemble a Lorentz-violating
scalar field with an explicit dispersion relation given by

$\displaystyle\omega_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
c_{0}|\mathbf{k}|\sqrt{1-\frac{3R}{2}\sqrt{A}\frac{c_{0}|\mathbf{k}|}{M_{\ast}}w\bigg{[}\sqrt{\frac{A}{2}}\frac{c_{0}|\mathbf{k}|}{{M_{\ast}}}\bigg{]}+\frac{1}{4}\frac{c^{2}_{0}|\mathbf{k}|^{2}}{M^{2}_{\ast}}}$
(10) $\displaystyle=$ $\displaystyle
c_{0}|\mathbf{k}|f(c_{0}|\mathbf{k}|/M_{\ast}),$

where $c_{0}=\sqrt{g^{\text{eff}}_{0}\rho_{0}/m}$ is the speed of sound,
another dimensionless parameter

$\displaystyle A=g^{\text{eff}}_{0}\rho_{0}/\omega_{z}$ (11)

represents the effective chemical potential as measured relative to the
transverse trapping, and

$\displaystyle M_{\ast}=mc^{2}_{0}$ (12)

is the analog energy scale of Lorentz violation. In the DDI dominance regime
$A$ should be assumed to be not larger than the critical value $A_{c}=3.4454$,
since beyond which the spectrum of quasiparticle becomes unstable Tian  _et
al._ (2018); Fischer (2006); Chä and Fischer (2017). Therefore, to keep the
stability of the spectrum of quasiparticle in the DDI dominance regime,
$A\leq\,A_{c}$ will be taken throughout the whole paper and this condition
requires that the frequency of the confinement along the $z$-direction
satisfies $\omega_{z}\geq\frac{2mg^{2}_{d}\rho^{2}_{0}}{\pi\,A^{2}_{c}}$. The
dispersion relation (10) is approximately Lorentz invariant
$(f(c_{0}|\mathbf{k}|/M_{\ast})\simeq 1)$ for $c_{0}|\mathbf{k}|/M_{\ast}\ll
1$. By appropriately setting the relevant parameters $A$ and $R$, the
dispersion could be analogously superluminal
$(f(c_{0}|\mathbf{k}|/M_{\ast})>1)$ and subluminal
$(f(c_{0}|\mathbf{k}|/M_{\ast})<1)$. In Fig. 1, we plot the function
$f(c_{0}|\mathbf{k}|/M_{\ast})$ shown in (10) to see how the LI is violated in
the dispersion. For the DDI dominance, $R=\sqrt{\pi/2}$, the analogous
subluminal spectrum develops a roton minimum for sufficiently large $A$, and
the LI is strongly broken near $c_{0}|\mathbf{k}|/M_{\ast}\simeq 0.9$ Tian
_et al._ (2018). Alternatively, $f(c_{0}|\mathbf{k}|/M_{\ast})$ could dip
below $1$ for an interval of $c_{0}|\mathbf{k}|/M_{\ast}$. This feature will
be very helpful for us to investigate in quantum field theory, how to reveal
unexpected imprints of the theory’s high energy structure with well-tested low
energy detection in the following.

##  III Unruh-DeWitt detector model in dipolar BEC

Figure 2: (Color online) Schematic of the moving (with speed parameter $v$)
impurity with effective internal frequency $\Omega$ immersed in a quasi-2D
dipolar condensate (in purple). Here the dipoles of atoms or molecules have
been assumed to be polarized along the $z$ direction or perpendicular to the
$x$-$y$ plane (in blue).

In quantum field theory, usually the field is probed with a linearly coupled
two-level (1 and 2) Unruh-DeWitt detector Unruh (1976); Dewitt (1979); Birrell
and Davies (1982); Takagi (1986); Crispino  _et al._ (2008); Hu  _et al._
(2012). This detector model captures the essential features of the light-
matter interaction when angular momentum interchange is negligible Martín-
Martínez  _et al._ (2013); Alhambra  _et al._ (2014). Inspired by the seminal
atomic quantum dot idea introduced in Refs. Recati  _et al._ (2005); Fedichev
and Fischer (2003), we model an impurity consisting of a two-level ($1$ and
$2$) atom as the Unruh-DeWitt detector, and assume the impurity is immersed in
the quasi-2D dipolar BEC discussed above (see Fig. 2). The impurity’s motion
is assumed to be externally imposed by a tightly confining and relatively
moving trap potential, so that we can focus only on its internal degrees of
freedom.

In this detector model, a monochromatic external electromagnetic field at
frequency $\omega_{L}$, which is close to resonance with the $1\rightarrow 2$
transition $\omega_{L}\simeq\omega_{21}$, is applied to illuminate the
impurity, and correspondingly the Rabi frequency is $\Omega$. We can write the
system Hamiltonian in the form

$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle H_{C}+H_{A}(t)=\sum_{\bf
k\neq 0}\omega_{\bf k}\hat{b}^{\dagger}_{\bf k}\hat{b}_{\bf k}$ (13)
$\displaystyle+\omega_{21}|2\rangle\langle
2|-\bigg{(}\frac{\Omega}{2}e^{-i\omega_{L}t}|2\rangle\langle
1|+\mathrm{H.c.}\bigg{)}$
$\displaystyle+\sum_{s}g_{s}\hat{\rho}(\mathbf{r}_{A}(t))|s\rangle\langle\,s|,$

where the last term is the collisional coupling between the impurity and Bose
gas.
$\hat{\rho}(\mathbf{r}_{A})=\hat{\psi}^{\dagger}(\mathbf{r}_{A})\hat{\psi}(\mathbf{r}_{A})$
denotes the field density operator of the atomic Bose gas, and
$\mathbf{r}_{A}(t)$ is the time-dependent impurity position. $g_{s}$ are the
interaction constant between the impurity in $s=1,2$ state and the condensate.
In the rotating frame, the detector’s Hamiltonian including its interaction
with the Bose gas can be rewritten as

$\displaystyle H_{A}(t)$ $\displaystyle=$
$\displaystyle\omega_{21}|2\rangle\langle
2|-\frac{1}{2}\omega_{L}(|2\rangle\langle 2|-|1\rangle\langle
1|)-\frac{1}{2}\Omega(|2\rangle\langle 1|$ (14)
$\displaystyle+\mathrm{H.c.})+\sum_{s}g_{s}\hat{\rho}(\mathbf{r}_{A}(t))|s\rangle\langle\,s|.$

Then, using the rotated $|g,e\rangle=(1/\sqrt{2})(|1\rangle\pm|2\rangle)$
basis and defining $g_{\pm}=\frac{1}{2}(g_{1}\pm\,g_{2})$, we can further
rewrite the Hamiltonian (14) as

$\displaystyle H_{A}(t)$
$\displaystyle=\frac{\Omega}{2}\sigma_{z}+\frac{\delta}{2}\sigma_{x}+\hat{\rho}(\mathbf{r}_{A}(t))[g_{+}+g_{-}\sigma_{x}],$
(15)

where $\sigma_{z}$ and $\sigma_{x}$ are the conventional Pauli matrices, and
$\delta=\omega_{L}-\omega_{21}$ is the detuning. In this rotated basis, the
Rabi frequency $\Omega$ determines the splitting between the $|g,e\rangle$
states, while the detuning $\delta$ gives a coupling term.

Let us note the last term of the above Hamiltonian (15) denotes the
interaction between the impurity and Bose gas. This interaction contains two
terms: the first one proportional to $g_{+}$ is similar to the reminiscent
coupling of a charged particle to an electric field, while the other resembles
a standard electric-dipole coupling mediated by a coupling constant $g_{-}$.
By suitably choosing the internal atomic states and properly tuning the
interaction constants (e.g., via Feshbach resonances Courteille  _et al._
(1998); Inouye  _et al._ (1998)), the first term could be cancelled as a
result of $g_{+}=0$ Zaccanti  _et al._ (2006); Simoni  _et al._ (2008); Chin
_et al._ (2010), behaving like the analog charge neutrality. Furthermore, the
atomic density operator $\hat{\rho}(\mathbf{r})$, as shown above, can be split
into its average value $\rho_{0}$ and small fluctuations
$\delta\hat{\rho}(\mathbf{r})$ in (6). With the suitable detuning $\delta$
between driving frequency and the impurity’s internal level space, one can
exactly compensate the coupling to the average density,
$\delta/2+g_{-}\rho_{0}=0$ Marino  _et al._ (2017, 2020). Under all these
assumptions, the impurity’s Hamiltonian including the its interaction with the
condensate can finally be written as

$\displaystyle H_{A}(t)$
$\displaystyle=\frac{\Omega}{2}\sigma_{z}+g_{-}\sigma_{x}\delta\hat{\rho}(\mathbf{r}_{A}).$
(16)

The coupling of the impurity to the condensate in the Hamiltonian (16) has the
analogous form $g_{-}\sigma_{x}\delta\hat{\rho}(\mathbf{r}_{A})$ of a two-
level atom dipole coupled to the quantum Lorentz-violating scalar field at its
position $\mathbf{r}_{A}$.

##  IV Spontaneous excitation of inertial detector

(a) $\Omega/M_{\ast}=0.3$ (b) $\Omega/M_{\ast}=0.1$ (c) $\Omega/M_{\ast}=0.01$

Figure 3: (Color online) Transition rate shown in (20) for excitations
$(\Omega/M_{\ast}>0)$ vs $\beta$ for different effective chemical potential
$A$, in units of $\frac{\rho_{0}M_{\ast}\,g^{2}_{-}}{(2\pi)^{3}c^{-2}_{0}}$.
Here we assume that the DDI dominance case ($R=\sqrt{\pi/2}$) is valid.

Consider the above impurity detector with the Hamiltonian given in (16), we
assume that it is pointlike and moves on the worldline $(t(\tau),{\bf
r}(\tau))$, where $\tau$ is the proper time. In the detector’s frame, the
coupling between the impurity and the density fluctuations of the condensate
(or Bogoliubov field) in the interaction picture can be rewritten as

$\displaystyle
H_{int}(\tau)=g_{-}\big{(}e^{i\Omega\tau}\sigma_{+}+e^{-i\Omega\tau}\sigma_{-}\big{)}\delta\hat{\rho}(t(\tau),\mathbf{r}_{A}(\tau)).$
(17)

We take the initial state of joint system (the impurity and the density
fluctuations) before the interaction to be $|0\rangle\otimes\,|g\rangle$. We
pay attention to the probability for the detector to be excited, i.e., at
state $|e\rangle$, as a result of the interaction. Since here the coupling
constant $g_{-}$ is treated as a small parameter, it enables this probability
to be calculated to first order in perturbation theory in $g_{-}$ Birrell and
Davies (1982); Takagi (1986); Crispino  _et al._ (2008); Hu  _et al._ (2012);
Louko and Satz (2008); Satz (2007); Schlicht (2004),

$\displaystyle P(\Omega)$ $\displaystyle=$
$\displaystyle\big{|}\langle\Psi\,e|ig_{-}\int^{\infty}_{-\infty}d\tau\big{(}e^{i\Omega\tau}\sigma_{+}+e^{-i\Omega\tau}\sigma_{-}\big{)}$
(18)
$\displaystyle\otimes\delta\hat{\rho}(t(\tau),\mathbf{r}_{A}(\tau))|g\,0\rangle\big{|}^{2}$
$\displaystyle=$ $\displaystyle
g_{-}^{2}\int^{\infty}_{-\infty}d\tau^{\prime}\int^{\infty}_{-\infty}d\tau^{\prime\prime}e^{-i\Omega(\tau^{\prime}-\tau^{\prime\prime})}W(\tau^{\prime},\tau^{\prime\prime}),$

where $|\Psi\rangle$ is quantum state of the Bogoliubov field satisfying
$|\Psi\rangle\langle\Psi|+|0\rangle\langle 0|={\bf 1}$, and
$W(\tau^{\prime},\tau^{\prime\prime})=\langle
0|\delta\hat{\rho}(t(\tau^{\prime}),\mathbf{r}_{A}(\tau^{\prime}))\delta\hat{\rho}(t(\tau^{\prime\prime}),\mathbf{r}_{A}(\tau^{\prime\prime}))|0\rangle$
denotes the Wightman function of the Bogoliubov field. We can consider
situations in which the Wightman function is stationary, i.e.,
$W(\tau^{\prime},\tau^{\prime\prime})=W(\tau^{\prime}-\tau^{\prime\prime})$.
It implies that the correlations between two events depend only on the
difference of times between them. Then we may convert $P(\Omega)$ into the
transition rate per unit time, as done in Refs. Louko and Satz (2008); Satz
(2007); Schlicht (2004). After some straight transformations and calculations
Louko and Satz (2008); Satz (2007); Schlicht (2004), the transition rate per
unit time finally is given by

$\displaystyle\mathcal{F}(\Omega)=g^{2}_{-}\int^{\infty}_{-\infty}dse^{-i\Omega\,s}W(s,0),$
(19)

where $s=\tau^{\prime}-\tau^{\prime\prime}$. We will use Eq. (19) in the
following study.

We consider an inertial detector with the trajectory given by
$(t(\tau),\mathbf{r}(\tau))=(\tau,c_{0}\tau\tanh\beta,0)$, where
$c_{0}\tanh\beta$ denotes the velocity with respect to the distinguished
inertial frame, and for simplicity is set to be positive. Note that the
detector’s velocity has been assumed to be smaller than the sound speed,
$c_{0}$. Assuming both the impurity and the Bogoliubov field initially in the
ground state and working in the frame comoving with the impurity, we obtain
the detector’s transition rate by substituting Eq. (6) into Eq. (19),

$\displaystyle\mathcal{F}(\Omega)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\infty}ds\int\,d\mathbf{k}\frac{\rho_{0}g^{2}_{-}}{(2\pi)^{4}}\frac{\mathcal{H}_{\mathbf{k}}}{\omega_{\mathbf{k}}}e^{-i(\Omega+\omega_{\mathbf{k}}-c_{0}k_{x}\tanh\beta)s}$
(20) $\displaystyle=$
$\displaystyle\frac{\rho_{0}M_{\ast}\,g^{2}_{-}}{(2\pi)^{3}c^{-2}_{0}}\int^{\infty}_{0}dg\frac{g^{2}}{f(g)}$
$\displaystyle\times\frac{\Theta\big{(}g\tanh\beta-|\Omega/M_{\ast}+gf(g)|\big{)}}{\sqrt{g^{2}\tanh^{2}\beta-\big{(}\Omega/M_{\ast}+gf(g)\big{)}^{2}}},$

where $g=c_{0}|\mathbf{k}|/M_{\ast}$, $\Omega$ denotes the effective energy
space between the impurity’s two levels, and $\Theta$ is the Heaviside
function. In the low-speed limit, i.e., when $\beta\ll 1$, we have
$\tanh\beta\thickapprox\beta$, then the effective-energy expression in Eq.
(20) $\Omega+\omega_{\bf k}-c_{0}k_{x}\tanh\beta$ is approximate to
$\Omega+\omega_{\bf k}-c_{0}k_{x}\beta$, and the detector’s transition rate
could be given by

$\displaystyle\mathcal{F}(\Omega)$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}M_{\ast}\,g^{2}_{-}}{(2\pi)^{3}c^{-2}_{0}}\int^{\infty}_{0}dg\frac{g^{2}}{f(g)}$
(21)
$\displaystyle\times\frac{\Theta\big{(}g\beta-|\Omega/M_{\ast}+gf(g)|\big{)}}{\sqrt{g^{2}\beta^{2}-\big{(}\Omega/M_{\ast}+gf(g)\big{)}^{2}}}.$

In Eq. (20) the crucial issue, resulting from the definition of Heaviside
function, is the behavior of the argument of $\Theta$: Under what condition is
the argument of $\Theta$ positive for at least some interval of $g$? If
$f(g)\geq 1$ for all $g$, the argument of $\Theta$ clearly is always negative
for all positive $\Omega$, and thus the corresponding transition rate
$\mathcal{F}(\Omega)$ vanishes. The vanishing transition rate suggests the
detector does not become spontaneously excited. Note that $f(g)=1$ corresponds
to the usual massless scalar field (Lorentz-invaiant field) case, thus the
Lorentz-invariant vacuum field induces no spontaneous excitations in the
inertial detector Birrell and Davies (1982).

An interesting case is when $f(g)$ dips somewhere below unity Husain and Louko
(2016); Louko and Upton (2018). Concretely, we assume $f_{c}=\inf\,f$, and
$0<f_{c}<1$ (see more details in Fig. 1). In this scenario, the detector
behaves quite differently for rapidities below and above the critical value
$\beta_{c}=\operatorname{arctanh}(f_{c})$. When $0<\beta<\beta_{c}$, we see
from the argument of $\Theta$ in (20) that $\mathcal{F}(\Omega)$ vanishes for
$\Omega>0$. It implies that the detector remains unexcited. This result is
consistent with that for the uniformly moving detector coupled to a massless
scalar field Birrell and Davies (1982). However, when $\beta>\beta_{c}$, the
argument of $\Theta$ in (20) keeps positive for $0<\Omega/M_{\ast}<\sup_{g\geq
0}g[\tanh\beta-f(g)]$, and thus $\mathcal{F}(\Omega)$ does not vanish. It
suggests that the detector gets spontaneously excited, at arbitrarily small
positive $\Omega$. This result is quite different from that of the uniformly
moving detector coupled to the massless scalar field, which never gets
spontaneously excited Birrell and Davies (1982). Besides, when $\Omega<0$, no
matter the rapidities are below or above the critical value, the detector has
a nonvanishing deexcitation rate that depends on the rapidities.

We give a brief summary here: The Bogoliubov spectrum in (10) violates Lorentz
invariance, but is approximately Lorentz invariant as
$c_{0}|\mathbf{k}|/M_{\ast}$ approaches to zero. Furthermore, its smooth
positive-valued function $f$ could dip somewhere below unity, satisfying
$0<f_{c}<1$ with $f_{c}=\inf\,f$. Then, the broken LI, strongly at the energy
scale $M_{\ast}$, may induce the inertial Unruh Dewitt detector with rapidity
$\beta>\beta_{c}=\operatorname{arctanh}(f_{c})$ in the preferred frame to
experience spontaneous excitations and deexcitations at arbitrarily low
$|\Omega|$. The anticipation in Ref. Husain and Louko (2016) could be verified
in the analogue gravity system proposed here.

In Fig. 3, we take the dispersion relation (10) in the DDI domination regime
($R=\sqrt{\pi/2}$) and plot the transition rate $\mathcal{F}(\Omega)$ in (20),
as a function of $\beta$. Clearly, the transition rate remains vanishing in
the beginning, then suddenly becomes a maximal value when the rapidity $\beta$
exceeds the critical $\beta_{c}$, and decays with the increase of the
rapidity. Moreover, in the DDI domination regime how the critical rapidity
$\beta_{c}$ changes with the effective chemical potential $A$ is shown in Fig.
4. Since $f_{c}$ in our setup could approach to zero when working in the DDI
dominance regime for sufficiently large $A$ (see Fig. 1), correspondingly, the
critical rapidity $\beta_{c}=\operatorname{arctanh}(f_{c})$ beyond which the
detector gets spontaneously excited approaches to zero as well. It suggests
that even for a quite small rapidity, one can still observe the spontaneous
excitations of inertial detector as a result of the broken LI of quantum
fields. This ultralow-speed demand could reduce the experimental challenge.

Figure 4: (Color online) The critical rapidity $\beta_{c}$ as a function of
the effective chemical potential $A$ for the DDI dominance case
($R=\sqrt{\pi/2}$).

##  V Experimental implementation

So far, the experimental realization of magnetic DDI-dominated condensate
Baranov (2008) has been reported with different atoms Lahaye  _et al._ (2007);
Aikawa  _et al._ (2012); Lu  _et al._ (2011). Besides, the realization of BEC
made up of molecules with permanent electric dipoles Quéméner and Julienne
(2012) is now at the forefront of ongoing research; cf., e.g., Refs. Park  _et
al._ (2015); Guo  _et al._ (2016); Rvachov  _et al._ (2017); De Marco  _et
al._ (2019); Son  _et al._ (2020). As a result of the DDI between atoms, the
excitation spectrum of dipolar BEC displays a deviation from the Lorentz
invariant one, and even displays a roton minimum when the BEC is in the DDI-
dominated regime Tian  _et al._ (2018); Chä and Fischer (2017); Fischer
(2006); Santos  _et al._ (2003); Ronen  _et al._ (2007). Experimental
observation of roton modes in ultracold dipolar quantum gases has been
reported recently Chomaz  _et al._ (2018); Kadau  _et al._ (2016); Natale  _et
al._ (2019); Petter  _et al._ (2019). The density fluctuations in the dipolar
BEC, as shown above, thus could furnish a quantum simulation of the quantum
field with broken LI, within current experimental reach. In addition, great
advances in high-precision measurements of correlation functions Hung  _et
al._ (2013); Hodgman  _et al._ (2011); Steinhauer (2016); Muñoz de Nova  _et
al._ (2019) pave the way to explore the possible extraordinary propagation of
quantum field due to the broken LI.

The hybrid systems Tomza  _et al._ (2019), consisting of trapped ions and
ultracold atomic systems, has recently emerged as a new platform for
fundamental research in quantum physics. In experiment, a trapped single ion
Zipkes  _et al._ (2010) and single electron Balewski  _et al._ (2013) coupled
to a BEC have been implemented. Moreover, the impurities’ dynamics can be used
to probe the density profile of BEC Schmid  _et al._ (2010); Scelle  _et al._
(2013); Spethmann  _et al._ (2012); Schmidt  _et al._ (2018). By designing the
external trap potential which tightly confines the impurity and moves
uniformly, one can impose the comoving motion on the impurity. A quasiparticle
formed by a mobile impurity interacting with a surrounding BEC, was observed
in two parallel experiments Hu  _et al._ (2016); Jørgensen  _et al._ (2016),
using different physical systems and techniques. Thus, it is possible in
principle, to design a hybrid system combining a impurity and a dipolar BEC,
in order to explore the dynamics of the impurity with the roton modes due to
DDI between atoms. This proposed cold atom setup could be as the quantum
simulator of the broken LI physics, as discussed above.

##  VI Conclusions and discussions

In this paper, we have proposed a cold atom setup where an external potential
trapped impurity immersed in a dipolar BEC and coupled to its density
fluctuations serves as a quantum simulator of a two-level atom dipole coupled
to the quantum scalar field with broken LI. We have shown that the Bogoliubov
spectrum in the dipolar BEC deviates from the Lorentz invariant one, and the
deviation could be tunable via Feshbach resonance and rotating polarizing
field. Due to the broken LI, the inertial impurity modeled as a Unruh-DeWitt
detector can be spontaneously excited when its rapidity exceeds the
criticality $\beta_{c}=\operatorname{arctanh}(f_{c})$. Interestingly, the
critical rapidity in our scenario can be controlled and even can approach to
zero if the ratio between the atomic contact interaction and DDI is
appropriate. Therefore, our proposal can experimentally demonstrate that the
effective low energy theory can reveal unexpected imprints of the theory’s
high energy structure, in quantum field theory.

Note that the detector initially prepared at its ground state could get
excited when its velocity exceeds the critical one as discussed above, and the
critical velocity is smaller than the speed of sound as a consequence of the
spectrum function $f(c_{0}|\mathbf{k}|/M_{\ast})<1$. During the excitation
process the moving detector could also simultaneously create the excitations
(or quasiparticles) in the dipolar BEC, which have the spectrum as shown in
Eq. (10). Therefore, one can also indirectly observe the created
quasiparticles in the dipolar BEC by checking if the detector becomes excited,
or by measuring the energy of the detector with the similar experimental
methods in Refs. Hu  _et al._ (2016); Jørgensen  _et al._ (2016).

Our proposed quantum fluid platform has potential as a quantum simulator
Georgescu  _et al._ (2014) of quantum field theories: the tunability of the
impurity’s motion, of the sound speed, of the Bogoliubov spectrum form, of the
condensate’s dynamics and geometry, allows us in the experimentally accessible
regime, to explore open questions concerning Unruh effect Retzker  _et al._
(2008), and why its robustness to high energy modifications of the dispersion
relation Carballo-Rubio  _et al._ (2019) behaves differently from that of its
equivalence principle dual—Hawking effect Unruh (1995). In addition, the
impurity could also be used as a detector to explore the intricate many-body
correlations between analogue high-energy quasiparticles in the dipolar BEC
Tian  _et al._ (2018) due to the DDI between atoms.

###### Acknowledgements.

This work was supported by the National Key R&D Program of China (Grant No.
2018YFA0306600), the CAS (Grants No. GJJSTD20170001 and No. QYZDY-SSW-SLH004),
and Anhui Initiative in Quantum Information Technologies (Grant No.
AHY050000). ZT was supported by the National Natural Science Foundation of
China under Grant No. 11905218, and the CAS Key Laboratory for Research in
Galaxies and Cosmology, Chinese Academy of Science (No. 18010203).

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