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Quantum simulations of the early universe
B. Opanchuk1, R. Polkinghorne1, O. Fialko2, J. Brand2, P. D.
Drummond1∗1Centre for Atom Optics and Ultrafast Spectroscopy,
Swinburne University of Technology, Melbourne 3122, Australia
and2Institute of Advanced Studies, Massey University, Albany, New
Zealand
A procedure is described whereby a linearly coupled spinor Bose
condensate can be used as aphysically accessible quantum simulator
of the early universe. In particular, an experiment to gen-erate an
analog of an unstable vacuum in a relativistic scalar field theory
is proposed. This is relatedto quantum theories of the inflationary
phase of the early universe. There is an unstable vacuumsector
whose dynamics correspond to the quantum sine-Gordon equations in
one, two or three spacedimensions. Numerical simulations of the
expected behavior are reported using a truncated Wignerphase-space
method, giving evidence for the dynamical formation of complex
spatial clusters. Pre-liminary results showing the dependence on
coupling strength, condensate size and dimensionalityare
obtained.
I. INTRODUCTION
The use of quantum simulation as a route to betterunderstanding
of complex quantum dynamics has muchto recommend it. Conceptually,
this creates an analogquantum computer. In quantum simulations, a
table-top experiment is carried out to mimic a more com-plex
quantum system which we would like to understand.This method has
been used to treat, for example, modelsof condensed matter
phase-transitions [1]. Such an ap-proach can be complemented by the
use of approximatecomputer simulations. A numerical simulation can
thenbe verified and tested in the table-top experiment,
whileallowing a wider variety of parameters to be treated.
But why stop at condensed matter: why not modelthe entire
universe? This seems presumptuous, and pos-sibly is. The entire
universe will never be shoehornedinto a table-top experiment with
every complexity in-tact. Nevertheless, cosmologists today often
use quan-tum field theory models to describe the early
universe.This can be traced back to the pioneering works of
Higgsand colleagues [2–5], studying the origins of mass, and
toColeman’s groundbreaking studies on unstable quantumvacuum decay
[6, 7].
A combination of these approaches, together with theinclusion of
general relativity, leads to the current in-flationary universe
scenario [8–13]. Inflation provides apossible explanation for the
origin of structure in theuniverse. Can we model at least part of
this picture ofthe early universe in a laboratory? The simplest
modelfor the scalar inflaton field φ(x) is described by the
La-grangian
L = 12∂µφ∂
µφ− V (φ), (1.1)
where V (φ) is the potential down which the scalar fieldrolls or
decays. In this paper, we show how to real-ize the above
relativistic scalar field model in a coupled
∗ [email protected]
Bose condensate, and model the fate of the scalar field ofthe
early universe. We note that there have been earlierproposals using
somewhat different, albeit related tech-niques [14, 15].
While the potential V (φ) is specific, depending on themodel of
inflation under consideration, coupled Bose con-densates provide a
situation with V (φ) ∝ − cos(φ). Inthe spirit of inflation, we
consider a cold, unstable vac-uum with φ = π as the initial
condition. The vacuumgradually decays [9–11], to produce a hot
universe, re-plete with dynamical clumping into random
structures.This quantum dynamical “universe on a table-top”
exper-iment is probed by means of an internal Rabi rotation,and
imaged.
Relativistic quantum field theory is usually tested
ex-perimentally at large accelerators. However, energies athigh
energy particle accelerators like CERN are not highenough for these
field theories, although observationalevidence for the Higgs boson
[16, 17] is thought to pro-vide evidence for the low energy sector
of scalar quantumfields in the universe. Instead of accelerating
particles tolight speed, we propose, essentially, to slow the
velocityof light down to atomic speeds. This allows us to
studynovel phenomena outside of CERN’s limits. A quantumsimulation
of interacting relativistic fields in two, threeor four space-time
dimensions has the useful feature thatit allows us to access
quantum physics that we cannot doexperiments on by any other
means.
We note some recent experiments have explored rele-vant physics
with ultra-cold atoms, demonstrating thatour proposal is indeed
feasible. These show that thephysics of interest here is very close
to realization. Theinvestigations that are similar to our proposal
includelong time-scale interferometry with a two-level BEC [18],and
the study of the thermalization of a BEC [19]. Morerecent
experiments have realized a flat trap or “can” fora BEC in three
dimensions [20]. In principle, simula-tions of this type might
eventually produce results thatcan predict the early universe
temperature maps [21] pro-duced recently by the Planck telescope
surveys, or verylarge-scale density inhomogeneities. Here we have a
morecautious goal, of simply being able to demonstrate quan-
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tum simulations of an unstable quantum vacuum in arelativistic
quantum field theory.
Our starting point is a trapped Bose condensate withtwo internal
quantum levels. The levels are coupled by amicrowave field, and the
atoms interact through S-wavescattering. We assume zero
temperatures, homogeneouscouplings and perfect microwave phase
stability. Whiledepartures from such perfection can be treated, and
theserequirements can be relaxed, we will study the ideal casehere.
We show that by introducing a simple π phaseshift into the coupling
field, it is possible to generatean experimentally accessible model
of an unstable rela-tivistic vacuum. A quantum simulation of the
decay ofthis unstable vacuum then provides the simplest of
earlyuniverse models. The relevant field variable is the rela-tive
phase of the two condensates. In a regime of smallcoupling, the
relative phase obeys the sine-Gordon equa-tion [22, 23], a popular
model for a relativistic scalar fieldtheory where the field
potential V (φ) is a cosine.
This paper demonstrates the feasibility of a class ofexperiments
which can simulate fundamental aspects ofmodels of inflation. Such
experiments, as well as theoret-ical calculations and numerical
simulation, should followa step-by-step programme that starts with
the simplestpossible theories. In the early stages, naturally, one
willlearn a lot about how to extract meaningful data
fromexperiments and the adequacy of approximate numeri-cal
techniques but not much about the universe. Herewe illustrate our
approach by a numerical simulation ofquantum effects, using a
truncated Wigner approxima-tion described later. The results show
the importanceof a hybrid approach, combining numerical and
experi-mental techniques. Each method involves different phys-ical
approximations. By comparing them, one can hopeto come to an
improved understanding of how quantumfluctuations behave in the
nonlinear regime, where exactquantum field predictions are hard to
come by. As theprogramme is extended to include additional
quantumfields and the effects of gravity one can hope to connectthe
results of experiments with astronomical observationsand refine our
models of the universe.
In general, the longer term fate of vacuum fluctuationsin the
nonlinear regime is of great fundamental inter-est. It is not known
how to calculate this exactly, dueto quantum complexity and the
failure of perturbationtheory. We speculate that these nonlinear
effects may berelated to very large scale inhomogeneities in the
massdistribution of the universe. This in turn could providethe
basis for observational tests of inflationary universetheories
beyond those available now. Unlike some mod-ern models of inflation
[9–11], our system can supportdomain walls of relative phase [24].
For experimentalstudies, these provide an easily measured signature
ofvacuum decay, and may have cosmological implications.
The paper is organized as follows. In Section (II) wedescribe
the general theory of coupled Bose fields, asfound in Bose-Einstein
condensate (BEC) experimentson ultra-cold atomic gases at
nanoKelvin temperatures.
In Section (III), we treat the different types of classi-cal
vacuum that exist, and the density-phase representa-tion. In
Section (IV) we transform to the sine-GordonLagrangian, and
demonstrate how this can behave as arelativistic field theory with
an unstable vacuum. In Sec-tion (V) we carry out detailed, though
approximate, nu-merical simulations of the original field theory,
to indicatewhat to expect in an experiment. Finally, in Section
(VI)we discuss the conclusions.
II. COUPLED BOSE FIELDS
Our first task then, is to slow down light: to a fewcentimeters
per second if possible. How is this to beachieved? For the purpose,
we use quasi-particles withdispersion relations equivalent to
relativistic equations,in a coupled ultra-cold Bose condensate
(BEC). To un-derstand this, let us consider the dynamical
equationsfor a coupled condensate; a D-dimensional Bose gas withtwo
spin components that are linearly coupled by an ex-ternal microwave
field. This system obeys the followingnon-relativistic Heisenberg
equation:
i~∂tΨ1 ={− ~
2
2m∇2x + g1Ψ
†1Ψ1 + gcΨ
†2Ψ2
}Ψ1 − νΨ2,
i~∂tΨ2 ={− ~
2
2m∇2x + g2Ψ
†2Ψ2 + gcΨ
†1Ψ1
}Ψ2 − νΨ1.
(2.1)
Here, ∇2x represents the D-dimensional Laplacian op-erator, and
g1, gc are the D−dimensional coupling con-stants, which are assumed
positive. These have knownrelations with the measured scattering
length. Thecoupling-constant is renormalizable at large
momentumcutoff, although our simulations are in the regime of
lowmomentum cutoff, which makes renormalization unnec-essary. The
field commutators are:[
Ψ1 (x) ,Ψ†1 (x
′)]
= δD (x− x′) (2.2)
We use spinor notation, introducing Ψ = (Ψ1,Ψ2)Tand write the
energy functional as W =
´dDxw, where
the energy density w(x) is
w =~2
2m∇xΨ† · ∇xΨ− νΨ†σxΨ +
gs2
: (Ψ†Ψ)2 : +
+gsa2
: (Ψ†σzΨ)2 : +ga : (Ψ
†Ψ)(Ψ†σzΨ) : .
(2.3)
Here we have defined 2gs = 12 (g1+g2)+gc, 4ga = g1−g2,and 2gsa =
12 (g1 + g2)− gc. From now on, for simplicitywe will assume the
case of symmetric self coupling g1 =g2 ≡ g , so that ga = 0. In the
case where gc = g,we further find gsa = 0. It is important for our
purposes
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that the cross-term gc have a different strength to the
self-term gi. This is generally achievable in ultra-cold
atomicphysics, depending on the atomic species and
particularFeshbach resonance used for magnetically tunable
cases.The fully symmetric case with gc = g is less interesting,and
we will assume that gc = 0 for definiteness in laternumerical
examples.
We recover the Heisenberg equation from i∂tψi =δW/δψ∗i .
Alternatively, we will also recover these equa-tions from a
stationary action principle δS = 0 with
S =
ˆdt dDx [R{Ψ†i~∂tΨ} − w] =
ˆdt dDx [LB − w] .
(2.4)
The coupling field ν is supplied by an external microwavesource,
and couples the hyperfine levels in the atomic con-densate
together. In experimental realizations [18], thishas an adjustable
amplitude and phase, good homogene-ity, and a long coherence
time.
Before proceeding further, we rescale the equationsinto natural
units. The time and distance scale is cho-sen so that mean-field
frequency shifts are of order unity,as are the corresponding
Laplacian terms. For typicalatomic densities per Bose field of n =
N/V , where N isthe particle number of a given species, and V the
volume,this leads to the choice τ = t/t0, and ζ = x/x0, where:
t0 = ~/gnx0 = ~/
√gnm . (2.5)
Scaling the fields so that they correspond to densitiesin the
new fields, i.e. ψ = ΨxD/20 , the resulting dimen-sionless field
equations in the symmetric case are
i∂τψ1 =
{−1
2∇2 + γψ†1ψ1 + γcψ
†2ψ2
}ψ1 − ν̃ψ2,
i∂τψ2 =
{−1
2∇2 + γψ†1ψ2 + γcψ
†1ψ1
}ψ2 − ν̃ψ1. (2.6)
Here, ∇ represents derivatives with respect to ζ, γ =1/(nxD0 ),
γc = gc/(gnxD0 ), and ν̃ = ν/(gn). The dimen-sionless field
commutators are:[
ψ1 (ζ) , ψ†1 (ζ′)]
= δD (ζ − ζ′) . (2.7)
In one dimension, γ2 = γLL ≡ mg/(~2n), whichis the famous
Lieb-Liniger parameter [25] of the one-dimensional quantum Bose
gas.
III. STABLE AND UNSTABLE VACUA
We now consider a semi-classical approach. We defineψ̃1 as a
classical mean field, and we linearize around theclassical
equilibrium solutions. Vacuum solutions withconstant fields, apart
from an oscillating phase, are eas-ily found from Eq. (2.6) as ψ̃01
=
√ñ = ±ψ̃02 . An overall
phase is chosen to make ψ̃01 positive and ñ = nxD0 is
thedimensionless density. The different signs of ψ̃02 corre-spond
to two inequivalent vacua. In order to study theirstability
properties and elementary excitations, we in-vestigate small
oscillations around the vacuum solutions.This generalizes the
analysis presented in Ref. [26] to in-clude the cross-coupling with
γc.
A. Linearized solutions
Writing ψ̃j = [ψ̃0j+(ujkei(k̃ζ−ω̃τ)+vj∗k e
−i(k̃ζ−ω̃τ))]e−iµ̃τ ,where µ̃ is the vacuum chemical potential,
substitutinginto Eq. (2.6) and keeping linear terms we obtain
thesecular equation det[B − ω̃] = 0 with
B =
H0 γñ ∓ν̃ + γcñ γcñ−γñ −H0 −γcñ ±ν̃ − γcñ∓ν̃ + γcñ γcñ
H0 γñ−γcñ ±ν̃ − γcñ −γñ −H0
,(3.1)
where H0 = − 12 k̃2 + (2γ + γc)ñ− µ̃ and consistency de-
mands that 2γsñ = µ̃± ν̃, where we define 2γs = γ + γc.The sign
"+" corresponds to the in-phase and "-" corre-sponds to the
out-of-phase vacua. We find two indepen-dent solution branches of
the secular equation. The firstone is gapless
ω̃1 =
√1
2k̃2(
1
2k̃2 + 4γsñ
), (3.2)
with eigenvector ∼ (1,−1, 1,−1)T for small k, which cor-responds
to the phase fluctuations of the fields δ arg ψ̃1 =δ arg ψ̃2 ∝
sin(k̃ζ − ω̃1τ). This leads to a sound wavewith sound speed ṽBog
=
√2γsñ. Solutions with real
frequencies indicate stable, propagating waves of
smallamplitude, whereas imaginary roots indicate an instabil-ity.
Excitations along this branch are always stable when2γs ≡ γ + γc
> 0.
The other branch is gapped, since
ω̃2 =
√(1
2k̃2 ± 2ν̃
)(1
2k̃2 + 4γsañ+ 2ν̃
), (3.3)
where 2γsa = γ − γc. The nonlinearity parameter in thissecond
branch differs from the first branch and can betuned independently
due to the presence of the cross-coupling γc. The eigenvector
corresponding to ω2 is∼ (1,−1,−1, 1)T for small k and ν, which
corresponds toan excitation of relative phase fluctuations of the
fieldsδ arg ψ̃1 = −δ arg ψ̃2 ∝ sin(k̃ζ − ω̃2τ). It is interesting
tonote that the two branches are decoupled in the linear
ap-proximation. We will be particularly concerned with therelative
phase dynamics, which has characteristic prop-erties of the
elementary excitations of a relativistic fieldtheory. Indeed,
expanding to second order in k̃ we maywrite ω̃22 = k̃2c̃22 +
m̃22c̃42, which is a relativistic dispersion
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relation with a light speed of c̃22 = 2γsañ±2ν̃ and rest
en-ergy m̃22c̃42 = 4ν̃(2γsañ± ν̃). The relative phase dynamicswill
analyzed be more rigorously in the next section.
The second branch also supports unstable modes andgoverns the
dynamics of the unstable vacuum. We areparticularly interested in
the situation where both γs andγsa are positive. In this case the
second branch becomesunstable if the linear coupling is such that
±ν̃ < 0, withthe unstable modes satisfying −4γsañ ∓ 2ν̃ <
k̃2/2 <∓2ν̃. This occurs either with the fields having the
samesign and ν̃ < 0, or with the fields having the oppositesign
and ν̃ > 0.
Thus, in a sufficiently large system where k is continu-ous, one
of the two possible vacua is stable and the otherone unstable. The
classical field dynamics resulting fromthe unstable vacuum has been
discussed for some specialcases in Refs. [27, 28] (see also Ref.
[26]). In this pa-per we are going to simulate the decay from the
unstablevacuum due to quantum fluctuations. The
characteristiclength scale of the decay modes is larger than π/
√|ν̃|,
which will become the largest length scale in the systemin the
interesting regime where |ν̃| is small. The timescales of the
unstable mode become large in the sine-Gordon regime. This is
compatible with the followinginflation requirement: When the scalar
field rolls downthe potential hill very slowly compared to the
expansionof the universe, inflation occurs.
B. Density-phase representation
We will continue to consider these equations as clas-sical, or
mean-field equations. We use this procedure togive us a better
understanding of how the dynamics canbe reduced to those of a
relativistic scalar field. Once wehave made this transformation, we
will use Lagrangianquantization methods to construct a low-energy
effectivequantum field theory for the elementary excitations.
We parametrize the spinor as follows:
ψ̃1 = uei(φs+φa)/2 cos(θ)
ψ̃2 = uei(φs−φa)/2 sin(θ), (3.4)
by introducing the density mixing angle θ and total den-sity u2,
the relative phase φa and total phase φs. Thefirst part of the
dimensionless Lagrange density becomes
L̃B =t0x0~LB = R{ψ̃†i∂τ ψ̃}
=u2
2∂τφs +
u2
2cos(2θ)∂τφa. (3.5)
Writing the energy density as w̃ = K̃ + Ṽ, the kinetic
energy part becomes
K̃ =12∇ψ̃† · ∇ψ̃
=1
2
{(∇u)2 + u2(∇θ)2 +
[(∇φs)2 + (∇φa)2
] u24
+∇φs · ∇φau2
4cos(2θ)
}. (3.6)
For the potential part we find
V =− ν̃u2 cosφa sin(2θ) +γs2u4
+γsa4u4[1 + cos(4θ)], (3.7)
where 2γs = γ+γc and 2γsa = γ−γc. The potential is afunction of
the three real field variables u, θ, and φa. Acut for constant u is
shown in Fig. 1. Note:
• As long as γsa ≥ 0, there is a valley at θ = π/4corresponding
to a vacuum with equal density inthe two fields. This expresses a
well-known con-dition for the stability of two coupled Bose
fields,g1 + g2 ≥ 2gc, where only the total particle num-ber is
conserved. This condition is weaker thanthe miscibility condition
for binary mixtures with-out interconversion [29], g1g2 > g2c ,
and turnsinto it when g1 = g2. The vacuum is locatedat (θ = π/4, φa
= 0). An equivalent point is(θ = 3π/4, φa = π). Working at fixed
particle num-ber N , we find for the dimensionless total densityof
the vacuum u2 = 2ñ ≡ 2nxD0 = γ−1.
• As θ terms primarily couple to the relative phaseφa, there are
dynamical solutions following the bot-tom of the θ valley with φa =
0 and θ = π/4corresponding to solutions of the type ψ1(x, t) =ψ2(x,
t). This is just the dynamics of a singleD-dimensional Bose field
involving compressionalwaves (Bogoliubov phonons) and dark
solitons.
• The point (θ = π/4, φa = π) or, equivalently, (θ =3π/4, φa =
0) is an unstable point.
• Excitations of the relative phase φa have low energycost when
|ν̃| � γsau2, i.e. in the regime of smallcoupling and strong
nonlinearity. This is the mainfocus of this paper.
We can see from Fig. 1 that within each large canyonin θ-space,
there is a secondary feature where a hill andvalley form in the
φ−direction. We wish to focus onthis relative phase feature. It is
able to form a quantumfield with a relativistic dispersion
relation, albeit with aneffective “light” velocity of very much
smaller size than c.
C. Domain walls
Domain walls of relative phase appear as a connec-tion between
two degenerate vacua along a θ valley.
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Figure 1. Plotted is V for ν̃ = 0.1, u = 1, γsa = 0.5 as a
func-tion of θ and φa. At θ = 3π/4 a saddle point occurs,
whichcorresponds to π phase jump between the two components. Itis a
known and dynamically unstable stationary state of theGP equations
[26, 27].
The corresponding solution were discussed in the con-text of
three-dimensional two-component BECs with in-ternal coupling in
Ref. [24]. In the context of one di-mensional coupled BECs they are
also known as atomicJosephson vortices or rotational fluxons, since
they arerelated to circular atomic currents, and were discussedin
Refs. [22, 30–32]. As features of the relative phase,they only
exist at sufficiently small linear coupling ν̃ <1/3 [30]. They
are unstable and can decay if a mecha-nism for dissipation of
energy is provided unless ν̃ . 0.14,where they become local energy
minima with topologi-cal stability [33]. In this regime, they
acquire the typicalproperties of topological solitons, or kinks, in
the sine-Gordon equation. By tuning the coupling parameter ν̃,one
can thus move between regimes of different stabilityproperties of
domain walls. The regime dominated bysine-Gordon physics is found
for small ν̃.
IV. SINE-GORDON REGIME
Aiming at the dynamics of the relative phase φa, wesimplify the
action and corresponding equations of mo-tion by assuming that all
dynamics takes place in a θvalley and that both θ and u only have
small devia-tions from their respective equilibrium values. We
conse-quently write θ = π/4+y and expand the action to lead-ing
orders in y. Using cos(2θ) = −2y, sin(2θ) = 1− 2y2,
cos(4θ) = 8y2 − 1 we obtain
Ṽ = ν̃u2 cosφa(1− 2y2)−γs2u4 − 2γsau4y2
2K̃ = (∇u)2 + u2(∇y)2 +{
(∇φs)2 + (∇φa)2} u2
4
−∇φs∇φau2
2y
L̃B =u2
2∂τφs − u2y∂τφa. (4.1a)
Low energy dynamics is possible in two sectors: com-pressional
waves involving u and φs on the one hand, anddynamics of the
relative phase involving φa and y on theother. Keeping only the
terms that are relevant for thelatter, the Lagrangian density
reads
L̃ = −u2(∂τφa)y −u2
8(∇φa)2 + ν̃u2 cos(φa)− 2γsau4y2.
(4.2)
Variation with respect to the fields gives the Euler-Lagrange
equation
∂L∂f−∇ ∂L
∂∇f− ∂τ
∂L∂fτ
= 0, (4.3)
which implies that∂τφa + 4γsau2y = 0. This allows us toreplace y
in the Lagrangian to yield
L̃ = 18γsa
(∂τφa)2 − ñ
4(∇φa)2 + 2ν̃ñ cos(φa). (4.4)
This is the D-dimensional sine-Gordon Lagrangian.Now we need to
extract parameters. The critical speed
c̃ is found by dividing the ∇ pre-factor by the ∂τ pre-factor
and taking the square root to yield c̃ =
√2γsañ.
The critical velocity thus matches the Bogoliubov speedof sound
ṽbog =
√2γsñ only in the case of vanishing
cross coupling γc, where γsa = γs. We note that so farwe have
used classical arguments. However, our originalequations are also
valid as quantum field equations. If wequantize the Lagrangian, we
accordingly have a quantumsine-Gordon equation, which describes the
physics at suf-ficiently low real temperatures.
On quantizing the Lagrangian, Eq. (4.4), the canonicalmomentum
field is:
π =∂τφa4γsa
,
with corresponding canonical commutators of:[φa (ζ) , π
(ζ′)]
= iδD(ζ − ζ′
).
The sine-Gordon action can be brought into Lorenz-invariant form
by rescaling time as ζ0 = c̃τ to read
S = ~ˆdD+1ζ
1
2
{(∂ζ0φ
′)2 − (∇φ′)2}
+α̃
β2cos(βφ′),
(4.5)
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with the rescaled field φa = βφ′ and β2 = 2√
2γsaγ, α̃ =4ν̃. Here, β2 is the universal dimensionless
parameterof the quantum sine-Gordon equation. In one dimen-sion and
the case of vanishing cross coupling γc = 0 and2γsa = γ, we obtain
β2 = 2γ = 2
√γLL, which again links
to the Lieb-Liniger coupling parameter γLL = mg/(~2n).The
stationary action principle leads to the (classical)
sine-Gordon equation
∇2φa − ∂ζ0ζ0φa + α̃ sinφa = 0. (4.6)
Note that the original field φa reappears and the parame-ter β
completely drops out of the classical equation. Thisis a
relativistic field equation with a dimensionless speedof light of
unity, as in all sensible relativistic field theo-ries.
Alternatively, using atomic units, we consider c̃ asthe effective
light-speed.
The parameter α = α̃x−20 is connected to the char-acteristic
length scale of the sine-Gordon equation. Inphysical units
`SG =1√α
=~
2√νm
. (4.7)
Remarkably, this length scale depends on the tunnel cou-pling
but is independent of the nonlinearity. It is thesecond
characteristic length scale besides the GP healinglength x0 of Eq.
(2.5). It sets the spatial scales of thedomain walls in the
sine-Gordon regime.
The parameter α̃ is also connected to the restmass of the
elementary excitations of the sine-Gordonequation. Indeed, small
amplitude oscillations ∼exp−i(k̃ζ − ω̃sGτ) of the vacuum have a
relativistic dis-persion relation of ω̃2sG = k̃
2c̃2 + α̃c̃2, with a rest massof m̃sG =
√α̃/c̃ and a rest energy of m̃2sGc̃
4 = 8ν̃γsañ.Note that the sine-Gordon light speed c̃ and rest
massm̃sG agree to leading order in ν̃ with the
correspondingparameters of the gapped second branch of the
Bogoli-ubov spectrum of the stable vacuum (3.3). The relativis-tic
sine-Gordon equation thus describes asymptotically asector of
excitations of the vacuum that, for small ampli-tude, completely
decouple from the compressional wavesof the Bogoliubov sound.
In the course of inflation tiny quantum fluctuationsare
important: they form the primordial seeds for allstructure created
in the later Universe. We now turn tocomputational quantum
simulations of these equations.
V. COMPUTATIONAL SIMULATIONS
To investigate the feasibility and likely behavior ofthese
experiments, we have carried out a number of nu-merical simulations
of the expected dynamics. We simu-lated the full coupled Bose
fields, using atomic densities,confinement parameters and S-wave
scattering lengthsgenerally similar to those employed in recent
experimen-tal studies with ultra-cold atoms. We employed an
accu-rate fourth-order interaction-picture algorithm. Careful
monitoring of step-size errors was needed to ensure accu-rate
results.
The theoretical method used is a truncated, proba-bilistic
version of the Wigner-Moyal [34, 35] phase-spacerepresentation
[36–40]. This is a probabilistic phase-space method for quantum
fields, that correctly simulatessymmetrically-ordered quantum
dynamics in the limit oflarge numbers of atoms per mode. It has
been shownto give results in agreement with quantum-limited
pho-tonic and BEC experiments [41, 42], provided the
largeatom-number restriction is satisfied in the experiment.
Other methods of this type are also possible. One ex-ample is
the positive-P representation [43, 44]. In thisnormal-ordered
approach, there is no truncation. Un-like the truncated Wigner
method, no systematic errorsoccur at small occupation numbers [45].
However, withcurrent algorithms, the statistical sampling error
growsin time for long time-scale nonlinear problems withoutdamping.
This rules out the positive-P approach for thepresent simulations,
until better algorithms are found.
Although the resulting equations are similar to
theGross-Pitaevskii (GP) equations, the initial noise termshave a
precise meaning from quantum mechanics. Theycorrespond to the exact
vacuum noise needed to generatesymmetrically ordered operator
moments that includethe quantum fluctuations of the initial quantum
state.The truncated Wigner method therefore includes quan-tum terms
correct to order 1/n, which are omitted in theGP approach.
Additional stochastic terms are neededif there is decoherence or
atomic absorption [40]. How-ever, in these preliminary studies,
such effects will beomitted. We also omit the cross-coupling
between spinsfor simplicity. Following the dimensional notation
intro-duced previously, except that our c-number fields are
nowstochastic representations of quantum fields, we obtain:
i∂τ ψ̃1 =
{−1
2∇ζ2 + γ
∣∣∣ψ̃1∣∣∣2} ψ̃1 − ν̃ψ̃2,i∂τ ψ̃2 =
{−1
2∇ζ2 + γ
∣∣∣ψ̃2∣∣∣2} ψ̃2 − ν̃ψ̃1. (5.1)For the truncated Wigner
calculations, with an initial
coherent state of ψ̄(ζ), so that ψ̃(ζ) = ψ̄(ζ) + ∆ψ̃(ζ),
onewould have:〈
∆ψ̃(ζ)∆ψ̃∗(ζ ′)〉
=1
2δ (ζ − ζ ′) .
We emphasize here that our assumption of an initial co-herent
state is certainly not the only one possible. Itcan be replaced by
other quantum states. For example,a finite temperature ground-state
followed by a suddenjump in phase is probably closer to the way
that exper-iments will take place. Nevertheless, in the absence of
agood model for the quantum state of the early universe,a coherent
state is as reasonable a choice as any other.
Experimental state preparation is a complex issue, anddepends on
such details as the magnitude of the mi-crowave coupling during
cooling. Our initial state cor-responds approximately to one in
which the fields are
-
7
0 5 10 15 20 25τ
−0.50
−0.25
0.00
0.25
0.50J
Figure 2. Decay of 1D quantum field coherence J , as an
indi-cator of Sine-Gordon unstable dynamics. γ = 0.1. Couplings,ν̃
= 0.001, 0.01, 0.1, 1 (blue solid, red dashed, green dash-dotted,
yellow dotted). Condensate size L = 80, 256 spatialgrid points,
10000 time steps, 100 ensembles, opposite initialphases to give an
unstable vacuum.
cooled to a very weakly interacting ground state, witha coherent
coupling of ν < 0, so that the lowest energystate has fields
with an opposite sign. Then the inter-actions are increased to a
large value, and the couplingphase is reversed to ν > 0, in
order to create an unstablevacuum. Naturally, other models are
possible, includinga strongly interacting initial state at finite
temperature.These will be treated elsewhere.
A. Dependence on couplings
We firstly investigate the dependence of the system onthe
coupling strength. This is a crucial part of the cal-culation, as
it creates the potential minimum such thatthe relative phase is
able to be treated as an independentscalar field. The measurable
quantities in an experimentare the densities obtained after
interfering the two hy-perfine field amplitudes [18]. These are the
quantities:
n± =1
2
〈(ψ̂1 ± ψ̂2
)† (ψ̂1 ± ψ̂2
)〉(5.2)
They correspond simply to the atomic densities aftera rapid Rabi
rotation of π/2 is performed to allow theatomic fields to
interfere. Substituting the mean fields ofEq. (3.4) into Eq. (5.2),
these measurable quantities are
n± = u2
[1
2± 1
2cosφa +O
(y2)].
At the vacuum phase, φa = 0, the atoms are all in theeven mode,
with n+ = u2 and n− = 0. At the unstableequilibrium, φa = π, n+ =
0, and n− = u2. We plot therelative visibility, which is a
quadrature-like measure ofthe phase, namely [46]:
J =
´dDζ (n+ − n−)
4N≈ 1
2cosφa .
0 5 10 15 20 25τ
−0.50
−0.25
0.00
0.25
0.50
J
Figure 3. Decay of average 1D quantum field coherence J
fordifferent condensate sizes, to indicate the effects of
bound-aries. Condensate size L = 10, 20, 40, 80 (blue solid,
reddashed, green dash-dotted, yellow dotted), spatial grid sizes32,
64, 128, 256, 10000 time steps, 100 ensembles, γ = 0.1.Coupling ν̃
= 0. Opposite initial phases.
We see in Fig. 2, that for very weak couplings, there isa
relative phase decay induced purely by quantum phasediffusion. This
is the opposite to sine-Gordon physics,and occurs because the two
condensates are uncoupled,and experience an increasingly random
relative phase.For stronger couplings, with ν̃ ≈ 0.1, one can see
the dis-tinctive rapid decay of an unstable vacuum, which is
thesignature of sine-Gordon physics with this initial condi-tion.
The slight oscillation near ν̃ ≈ 0.1 may indicatesome form of
collective instability of coupled domains,and deserves further
investigation.
B. Effects of condensate size
Although experimentally one would use a finite trap,we simulate
a system with periodic boundary conditionsfor simplicity. To some
extent, this allows us to ignorethe detrimental effects of
boundaries, and hence to allowa small computer simulation mimic a
larger experimentalcondensate. However, this is not entirely the
case.
As shown in Fig. 3, the finite coherence length of thequantum
fluctuations has an effect on the decay statis-tics. Initially, the
condensate size has no effect, as the co-herence length is very
small, and certainly much smallerthan the condensate size as long
as it is larger than ahealing length. For increasing times, if ν̃ =
0, the co-herence length increases until it reaches the size of
thecondensate itself. At this stage, the decay accelerates.For very
large condensates, a boundary-independent be-havior is found. We
note that this effect is strongest inthe limit of zero coupling ν̃,
which is plotted here. Atlarger couplings the formation of domain
walls limits thegrowth of coherence.
-
8
0 10 20 30 40 50 60 70τ
403020100
10203040
z
0.50.40.30.20.1
0.00.10.20.30.40.5
j
Figure 4. Plot of 1D space-time dynamics of the local
visibil-ity j for a single trajectory, showing contour plot vs time
andspace. Coupling ν̃ = 0.1, condensate size 80, 256 spatial
gridpoints, γ = 0.1, 10000 time steps.
C. Single trajectory examples
Quantum mechanics only predicts ensemble averages.However, it is
now common to use quantum theory topredict the behavior of the
universe, which is only a sin-gle ensemble member. Of course, this
experiment is noteasy to repeat, especially as the observer is part
of it.In the laboratory, one can obtain ensemble averages
byrepeating the state preparation. Even so, the densitypatterns and
evolution in a single ensemble member isinstructive. We expect the
Wigner ensemble membersto have a characteristic behavior like
individual labora-tory runs, and perhaps to the universe itself, if
currenttheories are correct.
In the one-dimensional case, this is shown in Fig. 4.This
corresponds to the largest of the condensates plot-ted in Fig. 3,
and it is visually clear that the densitycorrelation length is
still substantially smaller than thecondensate size throughout the
time evolution. The localfringe visibility is defined in general
as:
j =n+ − n−
2 (n+ + n−)≈ 1
2cosφa .
Since the total density is nearly constant through-out, we used
the excellent approximation of j ≈(n+ − n−) /(2ū2) for simplicity
of calculation and plot-ting.
D. Two-dimensional behavior
The universe is certainly not one-dimensional. Any-one might
reasonably question the applicability of a one-dimensional
simulation, even though this is the mostamenable to theoretical
treatment. In the laboratory, theavailability of engineered traps
means that dimensional-ity can be readily adjusted to obtain
artificial universesof one, two or three space dimensions.
0 5 10 15 20 25τ
−0.50
−0.25
0.00
0.25
0.50
J
Figure 5. Plot of 2D quantum field coherence J , as an
indi-cator of sine-Gordon unstable dynamics. γ = 0.1. Couplings,ν̃
= 0.001, 0.01, 0.1, 1 (blue solid, red dashed, green dash-dotted,
yellow dotted). Condensate size L = 80, 256 × 256spatial grid
points, 4000 time steps, 200 ensembles, oppositeinitial phases.
The average dynamics of the sine-Gordon field dependon
dimensionality as shown in Fig. 5, which shows thatthe
two-dimensional averages differ from Fig. 2, althoughthere are
qualitative similarities. However, Fig. 6 is muchmore interesting,
demonstrating clearly the growth ofcomplex spatial structures.
These are two-dimensionalcross-sections, taken at dimensionless
times of τ = 5,τ =10 andτ = 20. These single-trajectory plots show
howthe early, relatively high momentum quantum noise pat-terns have
decayed into coherent spatial domains with atypical spatial
coherence length of order ζ̄ = 10 spatialunits at τ = 10, and start
to evaporate at later times. Itremains to be investigated whether
this later evaporationis an artifact of the coupled BEC system, or
a genuinesine-Gordon feature.
E. Three-dimensional behavior
Recent experiments have shown the possibility of ex-ploring a
condensate in a “can” or cylindrical trap. Thisis a flat trap in
three-dimensions, with reflecting bound-ary conditions. While our
simulations do not have thistype of boundary, it is feasible to
carry out large three-dimensional calculations relatively quickly
using GPUhardware [42]. In Fig. 7, we show a slice through a
con-densate, in which one coordinate is held fixed, while theother
two are varied. This is a single ensemble mem-ber, and the other
parameters are similar to the two-dimensional case, Fig. 6.
The resulting spatial geometry of clusters of hot andcold matter
is distinctly different to the two-dimensionalcase, with greater
connectedness, as larger coherentstructures are formed over the
same time-scale. Here wehave plotted the real spatial image of a
two-dimensionalslice, to show domain-like clustering effects. This
demon-
-
9
40 30 20 10 0 10 20 30 40y
403020100
10203040
x
0.50.40.30.20.1
0.00.10.20.30.40.5
j
τ=5 τ=10 τ=20
Figure 6. Plot of 2D spacial clustering of the local visibility
j for a single trajectory, showing contour plot vs space at
finaltime. Coupling ν̃ = 0.1, condensate size 80× 80, 128× 128
spatial grid points, γ = 0.1, 5000 time steps, opposite initial
phases,time τ = 10.
40 30 20 10 0 10 20 30 40y
403020100
10203040
x
0.50.40.30.20.1
0.00.10.20.30.40.5
j
τ=5 τ=10 τ=20
Figure 7. Plot of 3D spacial clustering geometry of the local
visibility j for single trajectory, showing contour plot vs
space,sliced through a single z-coordinate. Coupling ν̃ = 0.1,
condensate size 80×80×80, 128×128×128 spatial grid points, γ =
0.1,5000 time steps, opposite initial phases. Plotted: a slice
orthogonal to x-axis at τ = 10.
strates that very large-scale clustering effects occur evenin
the absence of explicit gravitational interactions, forour
model.
Rapid progress in imaging technology means that itis now
possible to image such 2D slices in the experi-ment [47], although
it is technologically more difficultthan typical BEC experimental
techniques which producea column density image of an expanded
condensate.
VI. SUMMARY
The general behavior and unstable dynamics of an ef-fective
quantum sine-Gordon field using a two-mode BECwith linear coupling
has been clearly demonstrated. Ex-perimentally, for the scenario
outlined here, it is impor-
tant to use an atomic species whose cross-coupling dif-fers from
its self-coupling. This largely rules out the mostcommonly used
atomic species, 87Rb, which has very sim-ilar self and cross
couplings apart from a small regionnear a Feshbach resonance. A
Feshbach-type experimentis still possible, although losses are
greater in this regime.Apart from this case, there are many species
which areboth able to be evaporatively cooled and have
multiplehyperfine levels. For example,39K can be used. It has
asizable difference of g1g2 − g2c [48, 49], and it is thereforemore
suitable for realization of our proposal.
Our results will be extended to a more complete anal-ysis of the
numerical simulations elsewhere, in which wewill analyze in greater
detail the effects of different typesof state preparation. However,
it is clear that in prin-ciple the use of ultra-cold quantum gases
as quantumsimulators is not restricted to condensed matter and
non-
-
10
relativistic analogies. Simulating interacting
relativisticquantum fields in one, two or three dimensions
appearscompletely feasible, both experimentally and
computa-tionally. An experimental implementation would
throwmuch-needed light on the question of how accurate ournumerical
approximations are for these parameter values.
While we have focused on the practical issues of howto carry out
the experiment and corresponding numericalsimulations, these issues
pale somewhat in comparisonto another issue. We hope our proposal
will stimulateinterest in this fundamental question, which is:
• how do we carry out measurements on the quantumuniverse, when
we are part of it?
ACKNOWLEDGEMENTS
The authors acknowledge helpful discussions with An-drei
Sidorov, Tim Langen and Uli Zülicke. OF andJB were supported by the
Marsden Fund (contractNo. MAU0910). PDD was funded by the
Australian Re-search Council.
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Quantum simulations of the early universeAbstractI
IntroductionII Coupled Bose fieldsIII Stable and unstable vacuaA
Linearized solutionsB Density-phase representationC Domain
walls
IV Sine-Gordon regimeV Computational SimulationsA Dependence on
couplingsB Effects of condensate sizeC Single trajectory examplesD
Two-dimensional behaviorE Three-dimensional behavior
VI Summary Acknowledgements References
