Observing Gibbons-Hawking quantum radiation in linearly expanding cigar-shaped Bose-Einstein condensates
Petr O. Fedichev$^{1,2}$ and Uwe R. Fischer$^1$
We propose an experimental scheme to observe the Gibbons-Hawking effect in the acoustic analog of a 1+1-dimensional de Sitter universe, produced in an %linearly expanding, cigar-shaped Bose-Einstein condensate. It is shown that a %n effective two-level %atomic system created at the center of the trap, an atomic quantum dot interacting with phonons, observes a thermal Bose distribution at the de Sitter temperature. %It is shown that a concrete physical realization of a detector for de %Sitter quantum radiation is an atomic quantum dot, %a two-level system for a single atom, %created by an optical potential well. %, in which %at most one atom of a hyperfine component %of the gas %different from that forming the condensate is trapped. %, which is placed at the center of the condensate. %and proved that transitions between the levels, %induced by the quantum radiation,
NEWPRLcylinderhawking.tex
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%\draft
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%\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
\title{Observing Gibbons-Hawking quantum radiation 
in linearly expanding cigar-shaped Bose-Einstein condensates}
\title{Observing thermal quantum radiation 
in linearly expanding cigar-shaped Bose-Einstein condensates}
\title{Observing quantum radiation from acoustic horizons 
in linearly expanding cigar-shaped Bose-Einstein condensates}
\title{Hawking Radiation from Sonic de Sitter %Quasiparticle
Horizons in Expanding \\ Bose-Einstein-Condensed Gases}
\title{Gibbons-Hawking Radiation from Sonic de Sitter %Quasiparticle
Horizons in Expanding \\ Bose-Einstein-Condensed Gases}
\title{Gibbons-Hawking Effect and Sonic de Sitter %Quasiparticle
Horizons in Expanding \\ Bose-Einstein-Condensed Gases}
\title{Gibbons-Hawking Effect in the Sonic de Sitter 
Space-Time \\ of an Expanding Bose-Einstein-Condensed Gas}
\author{Petr O. Fedichev$^{1,2}$ and Uwe R. Fischer$^1$}
\affiliation{$^{1}$Leopold-Franzens-Universit\"{a}t Innsbruck, 
Institut f\"{u}r Theoretische Physik, Technikerstrasse 25, 
A-6020 Innsbruck, Austria \\
$^2$Russian Research Center Kurchatov Institute, 
Kurchatov Square, 123182 Moscow, Russia}



\begin{abstract} 
We propose an experimental scheme to observe the Gibbons-Hawking effect 
in the acoustic analog
of a  1+1-dimensional de Sitter universe, produced in an %linearly 
expanding, 
cigar-shaped Bose-Einstein condensate. 
It is shown that a %n effective 
two-level %atomic 
system created at the center of the trap,
an atomic quantum dot interacting with phonons, 
observes a thermal Bose distribution at the de Sitter temperature. 


%It is shown that a concrete physical realization of a detector for de
%Sitter quantum radiation is an atomic quantum dot,  
%a two-level system for a single atom, 
%created by an optical potential well.
%, in which 
%at most one atom of a hyperfine component %of the gas 
%different from that forming the condensate is trapped. 
%, which is placed at the center of the condensate.
%and proved that transitions between the levels, 
%induced by the quantum radiation, 
\end{abstract}
%\pacs{03.75.Kk, 03.75.Lm; cond-mat/0304342}  
\maketitle
%\def\d{{\mathrm{d}}}

%{\em Introduction.}-- 
Cosmology in the early universe 
is a branch of physics which is, for all too obvious reasons, 
removed far from experiment. To use a concise 
formulation of the primary dilemma which cosmology faces, 
there exists nothing like the 
possibility  of ``reproducing'' experiments, 
because there has in fact ``only ever been one experiment, still running, and 
we are latecomers watching from the back'' \cite{pickett}. 
While there can be  no truly experimental cosmology, 
as regards in particular the 
reproduction of the (presumed) extreme conditions
which prevailed in the early stages of the universe, one might look 
for analogous phenomena in condensed matter experiments, 
which {\em are} indeed reproducible, and  
can be done at energy scales smaller by 
many orders of magnitude \cite{Grisha}. 



An archetypical model of cosmology is the de Sitter universe, in which 
space is essentially empty, %and flat 
and the curvature of space-time is 
due to a nonvanishing cosmological constant $\Lambda$ 
\cite{deSitter}. The de Sitter space-time is used in inflationary models 
of the cosmos \cite{Linde}, displays a cosmological horizon,  
and is associated with a thermal spectrum of particles    
at a temperature $T_{\rm dS} \propto \sqrt{\Lambda}$ \cite{Gibbons}. 

Atomic Bose-Einstein condensates (BECs) \cite{Anglin}
have emerged as one of the most suitable condensed matter
systems for the simulation of quantum phenomena in effective space-times 
\cite{SlowLight,Garay,CSM,Leonhardt,Ourselves,BLV}. 
In the present study, we investigate an effective 1+1--dimensional (1+1D)
de Sitter
universe and the associated thermal phonon spectrum of the 
Gibbons-Hawking type \cite{Gibbons}. 
%in an expanding, strongly elongated condensate. 
We examine the propagation of axial low-energy modes in 
expanding, strongly elongated condensates, and find  
that the spectrum consists of one massless (phonon) mode 
and a sequence of massive excitations, moving in an 
effective curved 1+1D space-time. 
It is demonstrated that a
1+1D version of the de Sitter metric \cite{deSitter} can be 
realized by the massless phonon mode 
in a linearly expanding Bose-Einstein condensate with 
constant particle interaction. 
%close to the center of the cloud.  
We show that an atomic quantum dot (AQD) \cite{AQD},  
placed at the center of the cloud, can be used to measure the 
Gibbons-Hawking quantum process. The detector's coupling to the superfluid is 
constructed in such a way that the natural time interval of the 
detector is equal to the time interval in the de Sitter metric. Therefore, the
detector inside the expanding condensate is capable of measuring a thermal 
state at the de Sitter temperature.
The Gibbons-Hawking effect is a curved space-time generalization 
%analogous to 
of the Unruh-Davies effect, in which a constantly accelerated 
detector in vacuum responds as if it were placed in a thermal 
bath with temperature proportional to its acceleration 
\cite{unruh76,BirrellDavies}. 
Therefore, our proposal %ed scheme 
represents  
%in a more general sense, 
a means to confirm the observer dependence of the %effective 
particle content of a quantum field in curved space-time \cite{fulling}. 




Our approach is based on the by now 
well established identity of the action of a massless scalar 
field propagating on a curved space-time background in $D+1$ dimensions
and the action of the phase fluctuations $\Phi$  
in a moving inhomogeneous superfluid \cite{unruh,Matt,PGRiemann}
(we set $\hbar = m =1$, where $m$ is the mass of a superfluid particle): 
\begin{eqnarray}
S  & = & \int d^{D+1}x %dt  %r d\phi\, %dz %\, dr 
\frac{1}{2g} \left[ -\left(\dot \Phi -{\bm v} 
\cdot \nabla \Phi\right)^2 + c^2  (\nabla \Phi)^2 \right] \nonumber\\
& \equiv & \frac12 \int d^{D+1}x % dt dz 
\sqrt{-{\sf g}} {\sf g}^{\mu\nu} 
\partial_\mu \Phi \partial_\nu \Phi \,. \label{action}
\end{eqnarray}
Here, $\bm v ({\bm x},t) $ is the superfluid background 
velocity, $c({\bm x},t) =\sqrt{g\rho_0}$ the velocity of sound, % with 
$\rho_0({\bm x},t)  $ the background density, 
and $g=4\pi a_s$ is the coupling constant describing the short-range 
interaction between the particles in the superfluid; $a_s$ is the 
$s$-wave scattering length.
In the second line of (\ref{action}), the conventional 
hydrodynamic action is identified with the action 
of a minimally coupled scalar field in an effective curved space-time.  
In the following, we derive the effective 1+1D action 
of the form (\ref{action}) for the lowest 
axial excitations of an elongated condensate from the full 3D dynamics
of hydrodynamic excitations of the cigar-shaped condensate, and identify the 
metric tensor ${\sf g}_{\mu\nu}$. 

%{\em Hydrodynamics in an expanding cigar-shaped BEC.}--

We consider a large condensate confined in a strongly 
anisotropic harmonic trap, characterized by the frequencies 
$\omega_\parallel$ and $\omega_{\perp}$ in the axial and radial directions, 
respectively ($\omega_{\perp}\gg \omega_\parallel$). 
The initial condensate density 
is given by the usual Thomas-Fermi (TF) expression  
$%\begin{equation}
|\Psi_{\rm TF}|^2 %=\rho_0(r,z)
=\rho_m (1-{r^2}/{R_{\perp}^2}-{z^2}/{R_\parallel^2}). 
$ %\end{equation} 
Here, $\rho_m$ is the maximum initial density %(in the center of the cloud) 
and the squared TF radii are $R^2_\parallel= 2\mu/\omega_\parallel^2 $ 
and $R^2_{\perp}=2\mu/\omega_{\perp}^2$, where 
$\mu=\rho_m g$ is the (initial) chemical  potential.
%from a short reminder of the so-called scaling solution for an
%expanding, cigar-shaped Bose-Einstein condensate, containing 
%a large number of particles.  
%($N\sim 10^7-10^8$ in current experiments). 
According to \cite{Scaling}, 
%for highly anisotropic expansion,
%in the Thomas-Fermi limit, 
the temporal evolution of a Bose-Einstein-condensed atom 
cloud under variation of the trapping frequencies is 
described by the scaling condensate wave function
\begin{equation}
\Psi \left(r/{b_\perp}, z/b \right)= %\frac{
\left( %\frac{|\Psi_{\rm TF}|^2 (r/b_\perp,z/b)}{b_\perp^2 b} 
\rho_0  +\delta\rho \right)^{1/2} 
%\left(\frac{\textstyle r}{\textstyle b_\perp},
%\frac {\textstyle z}{\textstyle b} \right)}{b^{1/2}b_\perp} 
\exp\left[i (\Phi_0 +\Phi) 
%\exp\left[-i \int g \rho ({0},t) dt+ i\frac{\dot b z^2}{2b}  
%+ i\frac{\dot b_\perp r^2}{2b_\perp} 
\right] . \label{Wavefunction}
\end{equation}
Here, $\Phi_0 =-\int g \rho_0 ({0},t) dt +{\dot b z^2}/{2b}  
+{\dot b_\perp r^2}/{2b_\perp}$ is  the background phase, where  
$b_{\perp}(t)$ and $b(t)$ are the scaling parameters describing the condensate
expansion in the radial ($\hat r$) and axial ($\hat z$) directions, 
cf. Fig.\,\ref{Fig1}. The mean-field condensate density 
$\rho_0=|\Psi_{\rm TF}|^2 (r/b_\perp,z/b)/(b_\perp^2 b)$ 
contains the {scaling volume} $b_\perp^2 b$, and 
$\delta \rho$ are the fluctuations of the density around mean-field.
% around the scaled mean-field condensate density. 
%The superfluid velocity is the gradient of the condensate  phase.
%,  in Eq. (\ref{Wavefunction}), 
% and the density $\rho =|\Psi|^2$.
%, \begin{equation}
%{\bm v}=\frac{\dot b_{\perp}}{b_{\perp}} r {\bm e}_r 
%+\frac{\dot b}{b}z {\bm e}_z.\label{scaledv}
%\end{equation}

\vspace*{-1.5em}
%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[bt]
\psfrag{zH}{$z_{\rm H}$}
\psfrag{-zH}{$-z_{\rm H}$}
\psfrag{b}{\large $b\propto t$}
\psfrag{bperp}{\large $b_\perp\propto \sqrt t$}
\psfrag{Atomic}{Atomic}
\psfrag{Quantum Dot}{Quantum Dot}
\vspace*{1em}
\centerline{\epsfig{file=ellipse.eps,width=0.44\textwidth}}
\caption{\label{Fig1} Expansion of a cigar-shaped Bose-Einstein 
condensate.
%, linear in time along the axial, and scaling with $\sqrt t$ in the
%  radial direction. 
The stationary horizon surfaces 
are located at $\pm z_{\rm H}$, respectively. 
%The thermal spectrum of quasiparticles, due to 
%the Gibbons-Hawking process, is measured 
%by an ``Atomic Quantum Dot'' (AQD) 
%at the center of the condensate.
The thick dark lines represent lasers creating an optical potential 
well in the center.} %, which hosts the Atomic Quantum Dot.}
% created by an optical well for a different hyperfine species, 
%, where oscillations of a radial vortex are excited  by the radiation.}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{-1.5em}

The elementary excitations above the ground state of the BEC in the limit 
$\omega_{\perp}/\omega_\parallel \rightarrow \infty$ 
%sufficiently elongated 
%(formally infinitely extended), condensate 
%(i.e., in the limit $\omega_{\perp}/\omega_\parallel \rightarrow \infty$)  
were first studied in \cite{Zaremba}.
The description of the modes is based 
on an adiabatic separation for the axial and
longitudinal variables of the phase fluctuation field: 
\begin{equation}
\label{varsep}
\Phi (r,z,t) = \sum_n \phi_n(r) \chi_n(z,t), 
\end{equation}
where $\phi_n(r)$ is the radial wavefunction characterized by the 
quantum number $n$ (we consider only zero angular momentum modes). 
For long wavelength axial  excitations,  
the dispersion relation reads \cite{Zaremba} 
\begin{equation}\label{zarembadisprel}
\epsilon^2_{nk} = 2\omega_{\perp}^2 n (n+1)+c_0^2 k^2, 
%\frac{\omega_{\perp}^2}{4}(kR_{\perp})^2 
\end{equation}
where $c_0 =\sqrt{\mu/2} $ and $k$ is the axial wavenumber 
($kR_\perp \ll 1$).
%The Eqs. (\ref{varsep}) and (\ref{zarembadisprel}) can be 
%generalized for an expanding condensate: 
In the action (\ref{action}), 
we use Eq.\,(\ref{varsep}), with the rescaled radial wavefunction 
$\phi_n\equiv\phi_n(r/b_\perp)$,
and integrate over the radial coordinate. 
We then find the following effective action for the
axial modes of a given radial quantum number $n$, 
\begin{eqnarray}
S_n  &= & \int dt  dz \frac{ b_\perp^2 C_n}{2g}
\left[ -\left(\dot \chi_n -{v}_z \partial_z \chi_n\right)^2 
\right. \nonumber\\& & \qquad  \qquad \quad \qquad\left. 
+ \bar c^2_n (\partial_z \chi_n)^2 + M^2_n \chi_n^2 \right].
\label{1daction}
\end{eqnarray}
Here, the common factor $C_n(z)$ is given by  
$%\begin{equation}
b_\perp^2 C_n(z)=\int_{r<r_m} d^2r \phi^2_n , %\left(r/{b_\perp}\right).
$ %\end{equation}
%The integration limits are fixed by the $z$ dependent 
%radial size of the cigar 
%
the averaged speed of sound is 
$%\begin{equation}
\bar c_n^2(z,t)= {g}{C_n}^{-1}b_\perp^{-2} 
\int_{r<r_m} d^2r \rho_0 %(r/b_\perp,z)
\phi^2_n %\left(r/{b_\perp}\right),
$, %\end{equation}
and the effective mass term
$%\begin{equation}
M_n^2(z,t)={g}{C_n }^{-1} b_\perp^{-2} 
\int_{r<r_m} d^2r \rho_0 %(r/b_\perp,z)
[\partial_r\phi_n %\left(r/{b_\perp}\right)
]^2$; the radial cigar size is given by 
$r^2_m=R_{\perp}^2b^2_{\perp}(1-z^2/R_\parallel^2b^2)$. 
The phonon branch of the excitations corresponds to the $n=0$ 
solution of Eq.\,(\ref{zarembadisprel}), for which 
the  radial wavefunction $\phi_0=$ const. 
%does not depend on the radial variable $r$ 
\cite{Zaremba}; hence the mass term $M_0=0$ and 
$%\begin{equation}
C_0(0)=\pi R_{\perp}^2 %b^2_{\perp}
%C_0(z)=\pi R_{\perp}^2b^2_{\perp}\left(1-{z^2}/{b^2 R_\parallel^2}\right)
%= \frac{\bar c^2(z)}{c_0^2} \pi R_\perp^2 b_\perp^4 b, 
$, %\end{equation}
 $\bar c (0,t) \equiv \bar c_{n=0} (0,t) = c_0 /(b_\perp b^{1/2}).
%\bar c^2(z)=({c^2_0}/{b_{\perp}^2 b}) %(1-{z^2}/{b^2 R_\parallel^2}).
$ 

The action (\ref{1daction}) can be identified with the action of a minimally
coupled scalar field in 1+1D, close to the center of the condensate. 
%according to Eq.\,(\ref{action}).
%Remarkably, such an identification is possible only 
%$z=0$. 
The line element 
%tensor %close to the center 
reads \cite{unruh,Matt}
\begin{equation}
%{\sf g}_{\mu\nu}= A_c \left( \begin{array}{cc} -(c^2-v_z^2) & -v_z \\ -v_z & 1
%\end{array} \right), \label{gdown} 
%\end{equation}
ds^2 = A_c \left[-(c^2 -v_z^2)dt^2 -2v_z dt dz +dz^2 \right], 
\end{equation} 
where $A_c$ is some arbitrary (space and time dependent) 
conformal factor and we set $c=\bar c (z=0,t)$.
The actions (\ref{action}) and (\ref{1daction}) can be made consistent 
if we renormalize the phase field according to $\Phi = Z \tilde \Phi $ 
{\em and}  require that $Z^2 b_\perp^2 C_0 (0)/g =   1/\bar c(0) $, leading to
\begin{equation}
%\label{ZC0} \\%\nonumber \\%
%= \sqrt{-{\sf g}} \left( \begin{array}{cc} -1 & -v \\ -v & c^2 -v^2 
%\end{array} \right)
%{\sf g}^{\mu\nu}
%\end{equation} 
%holds. Eq.\,(\ref{ZC0}) leads to  
%\begin{equation}
%\Longleftrightarrow 
B\equiv 
\frac{b_\perp}{b^{1/2}} =  8 \sqrt{\frac{\pi}2} %\frac{1}{\sqrt 2 \pi } 
\frac1{Z^2} 
(\rho_m a_s^3)^{1/2} %(\rho_m g^3)^{1/2} 
\left(\frac{\omega_\perp}\mu\right)^2 = {\rm const}. 
\label{cond1}
\end{equation}
%\end{equation} 
The constant factor $Z$ in the above relations  
does not influence the equation of motion, and
corresponds  to a renormalization of the field amplitude,
$\Phi = Z \tilde \Phi $, which  determines the coupling of 
$\tilde \Phi$ to a detector \cite{No3}.
%According to the above relation, 
%We see that we have to impose that the expansion %of the cigar 
%in the perpendicularly proceeds like the square root of the axial expansion.

We require that the space-time metric is, in appropriately chosen
coordinates, identical to that of a 1+1D de Sitter universe.  
We first apply the transformation $c_0 d\tilde t = c(t) dt$,  
%to the line element defined by (\ref{gdown}),  
connecting the laboratory time $t$ to the time variable $\tilde t$.
Defining $v_z/c = \sqrt \Lambda z = (B \dot b/c_0) z$ 
(note that the ``dot'' on $b$ and other quantities always refers
to ordinary laboratory time),
this results, up to the conformal factor $A_c$, in the line element  
%\begin{eqnarray}
$d s^2 =-c_0^2(1- \Lambda z^2)d \tilde t^2 -2c_0 z \sqrt \Lambda d\tilde t 
dz+dz^2$.
%\end{eqnarray}
We then employ a second transformation 
%\begin{equation} 
$c_0 d\tau= c_0 {d\tilde t} +z\sqrt \Lambda dz/(1-\Lambda z^2)$, with a
time independent $\Lambda$, 
to obtain the 1+1D de Sitter metric
\cite{deSitter,Gibbons}
\begin{equation}
d s^2  = -c_0^2 \left(1-\Lambda  z^2 \right)  
d \tau^2 + \left({1-\Lambda z^2 }\right)^{-1} 
d z^2\,.
\label{deSitterlineelement}
\end{equation}
The quantity $\Lambda = B^2\dot b^2/c_0^2$ is a ``cosmological  
constant,'' provided $\dot b$ does not depend on $t$. 
%As a result, t
The transformation between $t$ and the de Sitter time $\tau$, 
at a given coordinate $z$, then involves the exponential ``acceleration,'' 
$%\begin{equation}
{t}/{t_0} = \exp[B \dot b \tau], % \label{trafo} 
$ %\end{equation} 
where $t_0 \sim 2\pi /\omega_\parallel $ 
is set by the initial conditions. 
%for the scaling variables $b$ and $b_\perp$.
The (conformally invariant) temperature associated with the effective 
metric (\ref{deSitterlineelement}) is the Gibbons-Hawking 
temperature \cite{Gibbons} 
\begin{eqnarray}
T_{\rm dS} %= \frac{\hbar c}{2\pi } \sqrt{\frac\Lambda 3} 
& = &  \frac{c_0}{2\pi } \sqrt{\Lambda} = \frac B{2\pi}  {\dot b}  
%%\nonumber\\ 
%%& = &  \frac{\mu}{2\pi} \sqrt{\frac{\Lambda}{2\mu}} 
%\nonumber\\ & = &  \frac{1}{4\pi } \sqrt{\frac{\Lambda}{\xi_0^2}} 
%& = &  \frac1\pi \frac{a_s}{\xi_0} \frac{\dot b}{\sqrt b b_\perp}
%\frac1{Z^2} \left(\frac{\omega_\perp}\mu\right)^2
\, ,\label{TdS}
\end{eqnarray}
%where $\xi_0^{-1} = \sqrt{2\mu}$ is the coherence length as defined by the
%chemical potential. 
and the %stationary 
horizon(s) are located at 
$
z= \pm z_{\rm H}= \pm {R_\parallel} ({\omega^2_\parallel}/
{2\mu\Lambda})^{1/2} . 
$
Combining the latter relation with (\ref{TdS}), %and (\ref{rH}), 
we see that $z_{\rm H}/R_\parallel \ll 1$, 
if $T_{\rm dS} \gg \omega_\parallel /4\pi $. 
%The statioThe stationary horizons are 
%located well inside the expanding cloud, so that 
We are thus justified 
in neglecting the $z$ dependence in $C_0$ and $\bar c$ to  describe   
the physics inside the horizon surfaces. 
%Therefore, the  
%de Sitter temperature needs to be at least of the order of
%$\omega_\parallel$ for the horizon location to be well inside the cloud. 
%Though there is no metric ``behind'' the horizon, i.e. at large $z$, 
%this should not affect the low-energy behavior of the quantum vacuum.

According to Eq.\,(\ref{1daction}), the equation of motion 
$\delta S_0 /\delta \chi_0=0$  is given, at constant $B$, by
\begin{equation}
B^2 b^2 \frac d{dt} \left( b^2 \frac d{dt}\chi_0 \right) 
- \frac1{C_0 (z_b)}\partial_{z_b} \left[{\bar c^2(z_b) C_0 (z_b)}
\partial_{z_b} \chi_0\right]  \label{chiEqmotion}
= 0,
\end{equation}
where $z_b = z/b$ is the scaling coordinate. Apart from the  
factor $C_0(z_b)$, stemming from averaging over the perpendicular direction,
this equation corresponds to the hydrodynamic equation of phase
fluctuations in inhomogeneous superfluids \cite{Stringari}.
%, written in scaling variables.   


At $t\rightarrow -\infty$, the condensate is in equilibrium and the
quantum vacuum phase fluctuations close to the center of the condensate can 
be written in the following form 
\begin{equation}
\hat\Phi \equiv {\hat \chi_0} %|_{t-> -\infty} 
= \sum_k \sqrt{\frac{g}{4C_0(0) R_\parallel \epsilon_{0k}}}
\,\hat a_{k} e^{ %\left[
-i\epsilon_{0k} t + ikz }% \right]
+ {\rm H.c.} ,
\end{equation}
where $\hat a_k,\hat a^\dagger_k$ are the annihilation and creation 
operators of a phonon. The intial quantum state of phonons, which we call the
``adiabatic vacuum,'' is the ground state 
of the superfluid and is annihilated by the operators 
$\hat a_k$ (this definition of vacuum and excitations  
corresponds to the ``adiabatic basis'' of \cite{BirrellDavies}). 
%With these initial conditions, 
Assuming that the expansion is switched on adiabatically,  
the density fluctuation operator, which is conjugate to 
$\hat\chi_0$, then reads
\begin{eqnarray}
\delta\hat\rho & = & \sum_k
\sqrt{\frac{\epsilon_{0k}}{4C_0(0) R_\parallel g}} 
\nonumber\\& & %\hspace*{-1em} 
\times 
\frac  d{dt}  \left\{ 
\hat a_{k} \exp \! \left[
-i\int^t\! \frac{dt' \epsilon_{0k}}{Bb^2}+ ik z_b \right]\right\} \!
+ {\rm H.c.} \label{deltarho} 
%{\hat a}^\dagger_{k} \exp\left[i\int \frac{dt \epsilon_{0k}}{Bb^2}- ikz
%\right\} 
\end{eqnarray}
The above equation 
%Eqs. (\ref{chi0}) and (\ref{deltarho}) 
completely characterizes 
the $n=0$ evolution of the condensate density fluctuations.
%{\em The 1+1D de Sitter metric in the condensate center.}--
%{\em Detecting the thermal spectrum.}--
%According to (\ref{trafo}), the de Sitter time $\tau$ is exponentially 
%different from laboratory time. 

The particle content of a quantum field state in curved space-time
depends on the observer \cite{fulling}. 
%To detect the de Sitter quantum radiation, one 
%has to set up a detector which measures frequencies in units of the inverse
%de Sitter time $\tau$, rather than in units of the inverse laboratory time 
%$t$.
%While the only ``physical'' time of our system is the lab time $t$, 
%the fact that particle definition 
%in curved space-times is generally not unique \cite{fulling}, forces us 
%{\em also in our effective theory which has a preferred time, $t$}, 
%to choose a detector which is ``tuned'' correctly to the 
%de Sitter co-ordinate time interval in the effective space-time 
%(\ref{deSitterlineelement}). 
We now show that the de Sitter time interval 
$d\tau=dt/bB=dt/b^{1/2}b_\perp$ 
can be effectively measured by an {\em Atomic Quantum Dot} \cite{AQD}.
Here, the AQD is an effective
two-level system with a time-dependent level splitting (see Fig.\,\ref{Fig2}).
It can be made in a gas of atoms possessing  two hyperfine
ground states $\alpha$ and $\beta$. The atoms in the state $\alpha$ 
represent the superfluid cigar,
and are used to model the expanding universe. 
The AQD itself is formed by trapping atoms in the state $\beta$ in a 
tightly confining optical potential
$V_{\rm opt}$. The interaction of atoms in the two internal levels is
described by a set of coupling parameters $g_{cd} = 
4\pi a_{cd}$ ($c,d =\{\alpha,\beta\}$),
where $a_{cd}$ are the $s$-wave scattering
lengths characterizing short-range intra- and inter-species collisions;
$g_{\alpha\alpha}\equiv g$, $a_{\alpha\alpha} \equiv a_s$.
The on-site repulsion between the atoms $\beta$ 
in the dot is $U\sim g_{\beta \beta}/l^{3}$,
where $l$ is the characteristic size of the ground state wavefunction
of atoms $\beta$ localized in $V_{\rm opt}$. In the following we consider the collisional
blockade limit of large $U>0$, where only one atom of type
$\beta$ can be trapped in the dot. This assumes that $U$ is larger
than all other relevant frequency scales in the dynamics of both
the AQD and the expanding superfluid. (The value of  
$g_{\beta\beta}$ may be increased using Feshbach resonances \cite{feshbach}.)
As a result, the appropriate collective coordinate of the AQD 
is modeled by a %n effective 
pseudo-spin-$1/2$, with 
spin-up/spin-down state corresponding to occupation by a single/no atom
in state $\beta$.


%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[t]
\psfrag{U}{\large $U$}
\psfrag{Omega}{\large $\Omega$}
\psfrag{Vopt}{\large $V_{\rm opt}$}
\psfrag{l}{\large $l$}
\psfrag{alpha}{\large $|\alpha \rangle$}
\psfrag{beta}{\large $|\beta \rangle$}
\psfrag{Delta}{\large $\Delta$}
\vspace*{-8em}
\centerline{\epsfig{file=trial5.eps,width=0.44\textwidth}}
\caption{\label{Fig2} 
Level scheme of the ``Atomic Quantum Dot,'' which is embedded in 
the superfluid cigar, and created by an optical well for atoms
of a hyperfine species different from that of the condensate. 
Double occupation of the dot is prevented by a 
collisional blockade mechanism.}
%by the very large ``on-site'' interaction energy $U$, which is 
%proportional to the coupling constant of the atoms 
%trapped by the optical well $V_{\rm opt}$ and the 
%inverse volume $l^{-3}$ occupied by the quantum dot wave function.}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{-2em}

Conversion of atoms between the %hyperfine
states $\alpha$ and $\beta$ is caused by a 
laser-driven transition characterized by Rabi frequency
$\Omega$ and detuning $\Delta$. The Hamiltonian of the AQD interacting
with the superfluid is 
\begin{eqnarray}
H_{\rm AQD} & = & 
\left\{-\Delta+g_{\alpha\beta}[\rho_0(0,t)+\delta\rho(0,t)]\right\}
\frac{\left(1+\sigma_{z}\right)}2\label{HAQD}\\
&  & +\Omega \left(\rho_0(0,t) l^{3}\right)^{1/2} 
\exp[i(\Phi+\Phi_0)]\sigma_{+}+ {\rm H.c.},  \nonumber
\end{eqnarray}
where we use Pauli matrix notation, %for pseudo-spin-$1/2$, 
i.e. $\sigma_+ = \sigma_x + i\sigma_y$.
% and in the second line the total density is replaced
%by its mean-field value. 
We introduce the wavefunction of the AQD in the form 
$\psi=\psi_\beta\exp[i(\Phi+\Phi_0)]|\beta\rangle+\psi_\alpha|\alpha\rangle$.
%,$ where  $|\pm > = (|\alpha\rangle \pm  |\beta \rangle)/\sqrt 2$. 
Using the scaling expression (\ref{Wavefunction}) for the condensate
wave function, we find
\begin{eqnarray}
i\frac{d\psi_\beta}{d\tau} & = & \left\{bB[-\Delta
+\rho_0 (g_{\alpha\beta}-g)]+\delta V \right\}\psi_\beta 
%\nonumber\\&  &
+\frac{\omega_{0}}2 \psi_\alpha, \nonumber\\
i\frac{d\psi_\alpha}{d\tau} & = &  \frac{\omega_{0}}2 \psi_\beta,
\end{eqnarray}
where $\tau$ is the \emph{de Sitter time}, and 
$\omega_{0}=2\Omega(\rho_m l^{3})^{1/2}$ is independent of $\tau$.
The perturbation potential 
\begin{equation}
\delta V (\tau) = (g_{\alpha\beta} -g)  B b(\tau)
\delta\rho({0},\tau) \label{deltaV}
\end{equation} 
provides the coupling between the AQD and the expanding superfluid
(if $g_{\alpha\beta}\simeq g$, higher order terms in the density 
fluctuations have to be taken into account 
in the Rabi term of (\ref{HAQD})).  
We suggest to operate the detector 
at $\Delta (t) =(g_{\alpha\beta}-g)\rho_{0}(0,t)=
(g_{\alpha\beta}-g)\rho_m/(\dot b^2 B^2 t^2)$,
so that in zeroth order in $\delta V$ we obtain an effective two-level
system with the level splitting $\omega_{0}$, which 
plays the role of a frequency standard of the
detector. 
By adjusting the laser intensity contained in 
the Rabi frequency $\Omega$, one can change 
$\omega_0$, and thus probe the response of the detector for 
various frequencies. 

The detector response can be calculated using the two-point correlation
function of the coupling operator $\delta \hat V$. The probabilities for 
the detector to ``click,'' corresponding to 
excitation ($P_+$) and de-excitation ($P_-$) of the detector, 
per unit of de Sitter time, are given by \cite{unruh76,BirrellDavies}:
\begin{equation}
\frac{dP_{\pm}}{d\tau}=\lim_{{\cal T}\rightarrow\infty}
\frac{1}{\cal T}\int^{\cal T}\!\!\!\int^{\cal T} 
\! d\tau d\tau^{\prime}\langle\delta \hat V(\tau)\delta \hat V(\tau^{\prime})
\rangle 
e^{\mp i\omega_{0}(\tau-\tau^{\prime})}.
\label{detectorresponse}
\end{equation}
Using the explicit form of the detector-to-field coupling (\ref{deltaV}),
we obtain 
\begin{equation}
\langle\delta \hat V(\tau)\delta \hat V(\tau^{\prime})\rangle
\propto { (\rho_m a_s^3)^{1/2}} \left(\frac{\omega_\perp}\mu\right)^2
\frac{B^2(g_{\alpha\beta}/g-1)^2 T_{\rm dS}^{2}}{\sinh^{2}
[\pi T_{\rm dS}(\tau-\tau^{\prime})]}.\label{correlator}
\end{equation}
This %vacuum 
correlation function is proportional to one
characterizing a thermal phonon state 
at a temperature $T_{\rm dS}$ in a condensate at rest. This means that our 
de Sitter detector
at the center of the condensate responds to the scaling vacuum 
as if it were placed in a thermal bath at a temperature $T_{\rm dS}$. 
Indeed, substituting the
correlator (\ref{correlator}) into Eq.\,(\ref{detectorresponse}) we
find that the quantities $dP_{\pm}/d\tau$ at late times 
%$\tau \gg 1/T_{\rm dS}$
are time independent, and satisfy detailed balance conditions 
corresponding to thermal
equilibrium at the de Sitter temperature given in Eq.\,(\ref{TdS}):
\begin{equation}
\frac{dP_{+}/d\tau}{dP_{-}/d\tau} = \frac{n_{\rm B}}{1+n_{\rm B}}, 
\label{occup}
\end{equation} 
where $n_{\rm B}=(\exp[\omega_{0}/T_{\rm dS}]-1)^{-1}$ 
is the Bose distribution function.
We stress that the AQD observes a thermal spectrum, even though 
the quantum state of the fluctuations is adiabatically connected 
to the initial vacuum state, i.e., no excitations in the adiabatic 
basis characterized by the $\hat a_k$ are created 
(note that the time interval of the adiabatic vacuum %in Eq.\,(\ref{deltarho})
is $dt / B b^2 = d\tau/b$).
%no frequency mixing occurs). 
The latter process %would 
corresponds to ``cosmological'' quasiparticle production 
\cite{Ourselves,BLV}, which 
%characterized by the energy scale $t_0^{-1} \sim \omega_\parallel /2\pi 
%\ll T_{\rm dS}$; 
can be neglected in our situation.



%{\em Conclusion.}--
Our experimental proposal consists in the following steps: 
(a) Preparation of a large TF condensate at very low temperatures. 
(b) Introduction of an AQD at the center (respectively an 
array of AQDs to increase the signal), which can be used
to determine the initial temperature of the cloud.
(c) Linear axial expansion of the condensate according to (\ref{cond1}).
The Rabi oscillations of the AQD can be used to monitor the
thermal distribution $n_{\rm B}$ in (\ref{occup}). % of the incoming phonons. 
Since $Z^2\propto (\rho_m a_s^3)^{1/2} 
\left({\omega_\perp}/\mu\right)^2$ and $P_\pm \propto Z^2$ \cite{No3}, 
the initial condensate has to be quite 
dense and close to the quasi-1D r\'egime.
%While $b$ increases linearly with time, 
%according to (\ref{cond1}), 
%$b_\perp$ simultaneously scales with $\sqrt t$ (Fig.\,\ref{Fig1}). 
%However, %from Eq.\,(\ref{Wavefunction}), we see that 
The particle density throughout the cloud decreases like $t^{-2}$. 
Therefore, the rate of three-body losses %dramatically 
quickly decreases during expansion, and %hence 
comparatively long observation times are feasible. 
%In addition, we mention that 
%three-body losses can be strongly suppressed \cite{Petrov},
%if a similar experiment is performed in a superfluid Fermi gas. 

The AQD as a detector 
is suitable to measure the de Sitter time interval,
because it couples linearly to the square root of the  
density of the superfluid gas. In principle, one can construct detectors
coupling to different powers of density or superfluid velocity, and 
then more generally experimentally 
study the non-uniqueness of the particle content of various
quantum states in curved space-time \cite{fulling}.
For example, %a Josephson type interaction analogous to (\ref{HAQD}), 
outcoupling pairs of atoms by photoassociation 
is a means to set up a detector which has $d\tau/dt \propto \rho_0$.  
Finally, we note that while in this paper we have studied only the 
$n=0$ massless axial phonon modes,  %expanding 
strongly elongated condensates can also be used 
to study the evolution of massive bosonic excitations.
% in curved space-times.
%, with the position dependent mass given in (\ref{Mass}). 
%The latter fact, t
Together with a natural Planck scale 
$E_{\rm Planck} \sim \mu$, this provides the opportunity 
%opens up the possibility 
to investigate, on a laboratory scale, 
the influence of finite quasiparticle mass 
and the trans-Planckian spectrum on the 
propagation of relativistic quantum fields in curved space-time.
%phenomenon Gibbons-Hawking effect. 

%{\em Acknowledgments.}--
We thank E.\,A. Cornell for an inspiring discussion on the 
experimental feasibility of our theoretical ideas. 
We acknowledge helpful discussions with 
R. Parentani, R. Sch\"utzhold,  G.\,E. Volovik, 
P. Zoller,  and A. Recati. 
P.\,O.\,F. has been supported by the Austrian %Science Foundation 
FWF and the 
Russian %Foundation for Basic Research 
RFRR, 
and U.\,R.\,F. by the FWF.
We gratefully acknowledge support from the ESF Programme 
``Cosmology in the Laboratory.''


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\end{document}   

The feat of reproducing certain features of the early universe
in the laboratory now seems closer than ever before. Due to the 
realization of the fact that phonons or generally ``relativistic'' 
quasiparticles, propagating in a nonrelativistic fluid,  
experience an effective curved space-time \cite{unruh,Matt,PGRiemann}, 
various exotic phenomena of the physics of quantum fields in 
the curved space-times of gravity are getting within reach of 
experiment \cite{Grisha}.


An archetypical model of cosmology is the de Sitter universe, in which 
space is essentially empty and flat and the curvature of space-time is 
due to a nonvanishing cosmological constant %$\Lambda$ 
\cite{weinberggravity}. 
This space-time displays a cosmological horizon, associated with 
which there is black body radiation 
at a temperature 
%\begin{equation}
$T_{\rm dS} = \frac{\hbar c}{2\pi}\sqrt{\Lambda /3}\,,$
%\end{equation} 
in analogy to the Hawking process of particle creation at the horizon  
of black holes \cite{Gibbons}. 

Indeed, the {contravariant} 1+1D metric may generally be written as 
\cite{Matt}
\begin{equation}
{\sf g}^{\mu\nu}= A  \left( \begin{array}{cc} -1 & -v_z \\ -v_z & c^2 -v_z^2 
\end{array} \right),
\end{equation}
where $A$ is some arbitrary conformal factor and $c=\bar c (z=0)$.
Inverting this expression to get the covariant metric, we obtain
\begin{equation}
{\sf g}_{\mu\nu}= (Ac^2)^{-1}  \left( \begin{array}{cc} -(c^2-v_z^2) & -v_z \\ -v_z & 1
\end{array} \right). \label{gdown} 
\end{equation}
The term $\sqrt{-{\sf g}} {\sf g}^{\mu\nu}$ contained in the action
(\ref{action}) 
therefore gives the familiar conformal invariance in a 1+1D space-time, 
i.e., the conformal factor $A$ drops out from the action.

Since both $C_0$ and $\bar c$ depend on $z$,  
an effective space-time metric for the axial phonons can be obtained 
only close to the center of the cigar-shaped condensate cloud. 
This is related to our averaging over the physical
perpendicular direction, and does not arise if the excitations are 
considered in the full D-dimensional situation, where the identification
is possible globally. 

%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[hbt]
\psfrag{zH}{\large $z_{\rm H}$}
\psfrag{-zH}{\large $-z_{\rm H}$}
\psfrag{tau}{\large $\tau$}
\psfrag{deltaz}{\large $\delta z$}
\psfrag{z}{\large $z$}
\centerline{\epsfig{file=fig3.eps,width=0.42\textwidth}}
\caption{\label{Fig3} 
Typical trajectories of phonons ($ds^2=0$)
in the de Sitter metric (\ref{deSitterlineelement}). 
The path taken by phonon I, which emanates at early 
times from the horizon, is described  in the text. 
The path taken by phonon II, which approaches the horizon 
surface at $-z_{\rm H}$ at late times $\tau$, does not lead to an 
excitation of the detector placed at $z=0$.}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%

We now show that $J$ is proportional to the total time 
of observation, so that in fact the probability per unit time
is a finite quantity \cite{Menshikov}. 
At late times, the detector measures phonon quanta coming
from close to the horizon, at a distance
$\delta z = z_{\rm H}-z \ll z_{\rm H} =\Lambda^{-1/2}$. 
The trajectory of such a phonon in the coordinates of the de Sitter metric 
(\ref{deSitterlineelement}), at late times $\tau$, is given by 
(cf.\,Fig.\,\ref{Fig3})
\begin{equation}
\ln \left[\frac{z_{\rm H}}{\delta z}\right] = 2\sqrt\Lambda c_0 \tau.
\end{equation}
This implies that the central AQD detector %, at large $\tau$,  
measures quanta which originated at the horizon with the large 
shifted frequency
\begin{equation}
\epsilon_{0k} = \frac{\omega_0}{\sqrt 2 \Lambda^{1/4}\delta z^{1/2}} =
\frac{\omega_0}{\sqrt 2}\exp[c_0\tau \sqrt\Lambda].
\end{equation}
Making use of the above equation, 
we rewrite the summation over $k$ in (\ref{J}) as an integral over
detector time:
\begin{eqnarray}
\sum_k \frac{\omega_0}{2\epsilon_{0k}} &  = &
\frac{2R_\parallel\omega_0/2}{\pi \bar c(0)}\int \frac{d\epsilon_{0k}}{\epsilon_{0k}}
= \frac{2R_\parallel\omega_0/2}{\pi} \sqrt\Lambda c_0d\tau \nonumber\\
& = &  \frac{R_\parallel\omega_0}{\pi \bar c (0)} B\dot b \tau \,.
\end{eqnarray} 
Therefore, $J\propto\tau$, 
and the probabilities per unit detector time (de Sitter time) read
\begin{eqnarray}
\frac{d P_{\pm}}{d\tau} 
& = & \left(\frac{1-B}{B} \right)^2  
\frac{g \omega_0/2}{C_0(0) \bar c (0)} \left\{
 \begin{array}{c} n_{\rm B} %\left(\frac{2\omega_0}{T_{\rm dS}}\right)
\\
1+n_{\rm B}%\left(\frac{\omega_0}{T_{\rm dS}}\right)
\end{array} ,
\right. %\!\!\! .
\end{eqnarray}
and are finite. 
In laboratory time, the transition probabilities evolve according to 
\begin{eqnarray}
P_{\pm} (t) 
& = & P_0 
\frac{\omega_0/2}{T_{\rm dS}} \ln \left[\frac t{t_0} \right] 
\left\{ \begin{array}{c} n_{\rm B} %\left(\frac{\omega_0}{T_{\rm dS}}\right)
\\
1+n_{\rm B}%\left(\frac{\omega_0}{T_{\rm dS}}\right)
\end{array} ,
\right. %\!\!\! .
\end{eqnarray}
where, from relation (\ref{ZC0}), $P_0 = Z^2 [(1-B)/{B}]^2 $.  
We see %from relation (\ref{ZC0})
that the detector response is, as it should be, proportional to $Z^2$, 
the square of the renormalization factor of the phase field.
The absorption and emission coefficients ${d P_{\pm}}/{d\tau} $
satisfy Einsteinian relations. Therefore, the detector approaches
thermal equilibrium at a temperature $T_{\rm dS}$ on a time scale of order
$Z^{-2} (\omega_0/2)^{-1}$. Since $Z^2 \propto 
(\rho_m a_s^3)^{1/2} \left({\omega_\perp}/\mu\right)^2$, larger, 
not-too-dilute condensates with $\omega_\perp \sim \mu $ 
(i.e., close to the quasi-1D r\'egime)
are most suitable to observe de Sitter radiation. 

the solution of (\ref{chiEqmotion}) is  
\begin{eqnarray}
\hat \chi_0 & = &
\left(\frac{g}{4C_0(0) R_\parallel \epsilon_{0k}}\right)^{1/2}
%
\nonumber\\& & \quad \times %\left( 
\hat a_{k} \exp\left[
-i\int_0^t \frac{dt' \epsilon_{0k}}{Bb^2}+ ikz \right]
+ {\rm H.c.} 
%{\hat a}^\dagger_{k} \exp\left[i\int \frac{dt \epsilon_{0k}}{Bb^2}- ikz
%\right) 
\label{chi0}
\end{eqnarray}

Calculating the integrals, we obtain 
\begin{eqnarray}
P_{\pm} & = & J \left(\frac{1-B}{B} \right)^2  
\frac{g\pi}{2B \dot b R_\parallel C_0(0)} 
\left\{
 \begin{array}{c} n_{\rm B}%\left(\frac{\omega_0}{T_{\rm dS}} \right)
\\
1+n_{\rm B} %\left(\frac{\omega_0}{T_{\rm dS}}\right)
\end{array} ,\right.
\end{eqnarray}
where the (formally divergent) sum
\begin{equation}
J= \sum_k \frac{\omega_0}{\epsilon_{0k}}, \label{J}
\end{equation}
and the factors 

The coupling operator $\delta \hat V$ causes transitions between the 
dressed detector states $|+\rangle$ and $| -\rangle$ 
and thus measures the quantum state of the phonons. 
We consider the detector response to fluctuations 
of $\hat \Psi$, by going beyond mean-field and 
using a perturbation theory in $\delta V$.
There are two physically different situations. The detector 
is either at $t=0$ in its ground state (optical well empty), 
or in its excited state (one atom in optical well).
We define $P_+$ and $P_-$ to be the probabilities that 
at $t=+\infty $ the detector is excited respectively de-excited.
%We demonstrate that the detector indeed measures a thermal Bose 
%distribution at a temperature $T_{\rm dS}$. 
Using second order perturbation 
theory in $\delta V$, we find that 
the transition probabilities for the detector may be written 
%(``$+$'' corresponds to excitation, ``$-$'' to de-excitation):
\begin{eqnarray}
P_{\pm} & = & \frac{g\epsilon_{0k}}{4R_\parallel C_0(0)} 
\left(\frac{1-B}{B} \right)^2 \left|T_\pm\right|^2 ,
\end{eqnarray} 
where the absolute square of the transition matrix element is given by 
\begin{eqnarray}
|T_\pm|^2 &= & \left|\int_{0}^\infty \frac{d\tau}{b(\tau)} 
\exp\left[\pm i\epsilon_{0k}\int_0^\tau 
\frac{d\tau'}b - i \omega_0\tau \right]\right|^2 .
\end{eqnarray} 

It is convenient to study the AQD response to the condensate
fluctuations in the Lagrangian formalism.
The detector Lagrangian takes the form 
\begin{eqnarray}
L_{\rm AQD} & = & 
i %\left(\frac d{dt}
\dot {\eta^*} %\right) 
\eta -\left[-\Delta+\bar g (\rho(0) +\delta\rho)
\right]\eta^* \eta  - \Omega \sqrt{\rho(0) l^3} %\times 
\nonumber\\
& &
\times \left( %\exp[-i\phi(0)]
\exp\left[-i \int_0^t g \rho (0,t') dt' - i\delta\phi\right]
\eta^*
%\right.\nonumber\\ & & \qquad \qquad \qquad\qquad\left.
+ {\rm H.c.} \!\!\right)\!\!.
\label{coupling}\nonumber\\
\end{eqnarray}
Here, $\Delta$ is the detuning of the laser light 
from resonance $\Delta$, $\rho (0)\equiv \rho(z=0)$ 
is the central mean-field part of the bath density, and $l$ is the size of the 
AQD ground state wave function. 
The detector variable $\eta$ is an anticommuting Grassmann variable
representing the effective spin degree of freedom of the AQD.  
The second line represents the coupling of the AQD to the 
surrounding superfluid, where $\delta\phi$ and $\delta\rho$
are the fluctuating parts of the condensate phase and density 
at $z=0$, respectively.
The laser intensity and the effective transition matrix element combine
into the Rabi frequency $\Omega$. Finally, the quantities $g$ and $\bar g$
are the coupling constants describing interactions between two
atoms in the surrounding superfluid, and 
between atoms in the superfluid and the atom in the AQD, respectively.

To simplify (\ref{coupling}), we use the canonical transformation 
\begin{eqnarray}
\eta & \rightarrow &  
\bar \eta \exp\left[-i \int_0^t g \rho (0,t') dt' - 
i\delta\phi\right]. \label{etatrafo} 
\end{eqnarray}
The above transformation amounts 
to absorbing the superfluid's chemical potential and 
the fluctuating phase  $\delta\phi$ into the wave function of the AQD, and
does not change the occupation numbers of the two AQD states.  
The transformation (\ref{etatrafo}) gives the detector Lagrangian the form 
\begin{eqnarray}
L_{\rm AQD} & = &  i %\left( %\frac d{dt} 
\dot {{\bar \eta}^*} %\right) 
\bar \eta 
- \Omega \sqrt{\rho(0) l^3}
\left(\bar \eta + \bar \eta^*\right) \nonumber\\
& & 
-\left[-\Delta+(\bar g-g)\rho (0)
+\bar g \delta \rho - %\frac d{dt}
\delta\dot \phi
\right] \bar \eta^* \bar \eta\,. \label{Lbareta} %\nonumber
%\nonumber\\& & 
\end{eqnarray}
The laser coupling (second term in the first line)
scales as $1/b^{1/2} b_\perp$, and hence like the de Sitter time interval.
The AQD can then be employed as a detector in de Sitter time, provided 
we are at zero detuning,
$\Delta=0$, and the intra- and interspecies interaction 
are equal, $\bar g=g$, leading to a vanishing of the first 
two terms in the square brackets of (\ref{Lbareta}). 
Introducing the wave function of the AQD, $\psi
= c_1 |1\rangle +c_0)|0\rangle$, 
%, where $|0\rangle$ is its ground 
%and $|1\rangle = \eta^\dagger |0\rangle$ its excited state. 
and defining the ``effective Rabi frequency''
$\omega_0=\Omega l^{3/2}\sqrt{\rho(0)}/2$,  we obtain a  
simple set of coupled equations for the AQD amplitudes
\begin{equation}
i\frac{d c_{1}}{d\tau}=(\omega_0/2) c_{0} + \delta V c_1,\,\,\,\,\qquad 
i\frac{d c_{0}}{d\tau}=(\omega_0/2) c_{1},\label{deSitterEvol}
\end{equation}
where $\tau$ {\em is the de Sitter time.} 
We have thus shown that the detector Eqs.\,(\ref{deSitterEvol}) are  
natural evolution equations in de Sitter time $\tau$.
The coupling of the AQD to fluctuations 
in the superfluid are described by the potential
\begin{equation} 
\delta V = b\,\frac{g(1-B)}{B}\delta \rho(0,t).
\end{equation}
Neglecting the fluctuations in the superfluid,
the level separation implied by (\ref{deSitterEvol})
is $\omega_{0}$, and the eigenfunctions
are $|\pm\rangle= (|0\rangle \pm |1\rangle)/\sqrt 2$. 
The quantity $\omega_0$ therefore 
plays the role of a frequency standard of the
detector. By adjusting the value of the laser intensity, one can change 
$\omega_0$ and therefore probe the response of the detector for 
various frequencies. 

The AQD is created by 
an optical potential and 
represents an effective two-level system, 
in which at most one atom of a hyperfine component of the gas 
different from that forming the condensate is trapped. 
We will denote the empty and filled states of the AQD as 
$|0\rangle$ and $|1\rangle$, respectively.  
They can be connected by  laser light, 
cf. Fig.\,\ref{Fig2}.


\begin{equation}
n_{\rm B} = \frac{1}{\exp[\omega_0/T_{\rm dS}]-1}
\end{equation} 
are Bose distribution functions at the de 
Sitter temperature (\ref{TdS}).

\[
\langle\delta V(\tau)\delta V(\tau^{\prime})\rangle=\frac{g^{2}(B-1)^{2}b(\tau)b(\tau^{\prime})}{B^{2}}\langle\delta\rho(\tau)\delta\rho(\tau^{\prime})\rangle.\]
The remaining correlator can be calculated using the scaling solution
for the density fluctuations:

In our cylindrical 3D trap, 
we have for the initial central density 
%\begin{equation} 
$\rho_m = %\mu/ g = 
\left({6 N \omega_\perp^2 \omega_\parallel}/{\sqrt 8 \pi g^{3/2}}\right)^{2/5}$.
%\end{equation} 
The condition that the TF condition be valid implies that 
$\mu\gg\omega_\parallel,\omega_{\perp}$. 

We conclude that an expansion of the condensate in $z$ direction, with 
a constant rate faster than the oscillation frequency in that direction, 
gives an effective space-time characterized by the de Sitter temperature
$T_{\rm dS}$, and is associated with a quasiparticle
de Sitter horizon close to the center of the trap.  
%and hence a thermal quantum radiation effect of the 
%Gibbons-Hawking type \cite{Gibbons}.
%
%Therefore, if we were to 
%consider initially a dense BEC, with $(\rho_m a_s^3)^{1/2}\gtrsim O(1)$ and 
%a large number of particles tightly bound by the trapping, while a 
%description in terms of the scaling ansatz initially may not describe the 
%system, the gas quickly becomes dilute and the formalism we present applies. 

The measured quanta can then, and only then, 
be accurately interpreted to be particles coming from a Gibbons-Hawking 
type process taking place at a horizon which has 
a {constant} de Sitter temperature (\ref{TdS}). 

In contrast, the method used to create a de Sitter universe
in \cite{Ourselves} is to expand the gas exponentially, and adjust the 
coupling $g$ accordingly ($\propto b^3$ for a 3D cloud). While feasible 
in principle, this is experimentally delicate to achieve: 
Increasing the interaction increases dramatically three-body losses 
as well, whose total rate is scaling like $g^4 \rho^2$. 

According to (\ref{Wavefunction}), t
The condensate density evolves as
\begin{equation}
\rho(r,z,t)=\frac{\rho_0\left({r^2}/{b_{\perp}^2},{z^2}/{b^2}\right)}
{b_{\perp}^2b}. \label{scaleddensity} 
\end{equation}
%, where $g$=4\pi a_s$, 
%and $a_s$ is the
%scattering length characterizing the particle 
%collisions in the (dilute) BEC. 

We have provided an experimental proposal to measure quantum radiation with
a thermal distribution.
% emanating from sonic de Sitter horizons in 
%linearly expanding Bose-Einstein-condensed gases. 
The proposal takes advantage
of the fact that existing condensates, trapped in 
anisotropic magnetic traps, 
%, without changes in the scattering length of the gas, 
can be used to simulate a 1+1D de Sitter universe close to 
their center. 

Due to the realization of the fact that phonons or generally ``relativistic'' 
quasiparticles, propagating in a nonrelativistic fluid,  
experience an effective curved space-time \cite{unruh,Matt,PGRiemann}, 
various exotic quantum field theoretical phenomena in 
the curved space-times of gravity \cite{BirrellDavies}, may be realized
in the laboratory.



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