Table Of ContentDynamics of correlations in a quasi-2D dipolar Bose gas following a quantum quench
Stefan S. Natu,1,2 L. Campanello,2 and S. Das Sarma1,2
1Joint Quantum Institute and Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
2Condensed Matter Theory Center, Physics Department,
University of Maryland, College Park, Maryland 20742-4111, USA
We study the evolution of correlations in a quasi-2D dipolar gas driven out-of-equilibrium by a
4
sudden ramp of the interactions. On short timescales, roton-like excitations coherently oscillate in
1
andoutofthecondensate,givingrisetopronouncedfeaturesinthetime-evolutionofthemomentum
0
distribution, excited fraction and the density-density correlation function. The evolution of these
2
correlation functions following a quench can thus be used to probe the spectrum of the dipolar
t gas. We also findthat density fluctuationsinduced by thepresence of rotons following thequench,
c
O dramaticallyslowsdowntherateofspreadingofcorrelationsinthesystem: neartherotoninstability,
correlations take infinitely long to build up and show deviations from light-cone likebehavior.
7
1
I. INTRODUCTION 0.6 Νho
] L(cid:144)k Eq
as Recent advances in ultra-cold atomic and molecular HE0 1 2 tΝho=1
g gases have greatly expanded the potential of these sys- nk0.3 kHunitsof 2(cid:144)lzL tΝho=5
- tΝ =7
t tems as tools for studying many-body physics [1]. For ho
n
a example, the realization of quantum degenerate gases 0
u with large magnetic dipole moments [2–6], ongoing ef- 0 1 2 3
q fortstotrapandcoolpolarmolecules[7–10],experiments
kHunitsof 2(cid:144)l L
. on Rydberg atoms [11], trapped ions [12, 13], and atoms z
t
a in high finesse optical cavities [14] have opened up the FIG. 1: (Color Online) Top: Evolution of theradial momen-
m
possibility of realizing ultra-cold atomic systems with tumdistributionnk inaquasi-2Ddipolargas(lz isthewidth
- long-range interactions, such as the anisotropic dipole- of the cloud in the direction perpendicular to the 2D plane),
d
dipole interaction [15–18]. Concurrently, better control following a sudden ramp of the dipolar interaction. Momen-
n
o over experimental parameters and high resolution imag- tumdistributiondevelopsapeakatthewave-vector(blackar-
row) corresponding to the roton minimum in the inset. The
c ing techniques have introduced new probes for exploring
[ many-body physics, notable among which is the ability period of the oscillations is proportional to the roton gap.
Black curve is the the corresponding equilibrium feature for
to study the dynamics of correlations following a non-
2 the same interaction where no prominent roton signature is
v adiabatic ramp (quench) of system parameters [19–27]. found. Time (t) is given in unitsof 1/(νho =2~2/mlz2).
5 Remarkably, these quench experiments provide a wealth
3 of information about the low energy properties of the
7 underlying system such as the nature of the excitations,
0 correlationfunctions, following a sudden switching on of
and the manner in which correlations develop in a sys-
. the dipolar interactions. Our main result, summarized
1 tem. Here we study the evolution of correlations in a
in Fig. 1, shows that these correlations develop strik-
0 quasi-2D dipolar gas following a sudden quench of the
ing features that are directly related to the underlying
4
interaction strength, finding non-trivial dynamics even
1 roton minimum in the dispersion. This is because on
in this weakly interacting system.
: timescales short compared to the collision time, excita-
v
A novel property of quasi-2D dipolar superfluids is tions are coherent, and the momentum distribution de-
i
X that in addition to the phonon-like mode at low ener- velops oscillations at different k, at a frequency equal to
ar ffgoeiaerstsuuErffieksc∼aienrcotkltyo,nst-thlrieoknelogmwdoiedpneoelr(agErykine=txec∆riatca+ttiioo~nn2(sks[p2−e8c,ktr2r)u92m]/.2Tmalhs∗oe) toahsticevileelnaeetfferegacyttonofcetcahurelrysextwhchieteasrtaeibmoynetEehnkee.raAgmyspt(lhEiteukrd∼oetoo∆nf)o,esxacciicltloaaottpiiooennrss-
existence of roton-like excitations generally implies that isenhanced. Themomentumdistribution,whichisread-
thesystemhasatendencytowardsdevelopingcrystalline ilymeasuredfollowingtime-of-flight,canthereforeprobe
order,andthesofteningoftherotonmodeisoftenaroute the roton gap.
towards realizing correlated states of matter such as su-
In addition to providing information about collective
persolidsandWignercrystalsinBoseandFermisystems
excitations [37], quench experiments raise fundamental
[30–36]. Observingthismodethereforeconstitutesanes-
questionsaboutthenon-equilibriumdynamicsofisolated
sentialfirststeptowardsthestudyofmanybodyphysics
quantum systems [23, 24, 38]. One such question con-
in dipolar systems.
cernsthe manner inwhichcorrelationsbuild upbetween
Here we investigate how the roton mode is revealed initiallyuncorrelatedregionsonshortandlongtimesfol-
in the spatio-temporal evolution of one and two-body lowingaquench. Forlatticesystemsinteractingviashort
2
rangeforces,LiebandRobinsonshowedthatcorrelations F(x) = 2 3√πx Erfc(x)ex2, where Erfc(x) is the com-
−
evolveinalight-conemanner[24,39–41],whichhasbeen plimentary error function.
recently confirmed both experimentally and numerically The Hamiltonian for a uniform quasi-2D dipolar Bose
[24]. However, questions remain about continuum and gas,wherethedipolesarealignedalongthez axisreads:
−
lattice systems interacting with long range interactions. 1
Recent work on trapped ion chains have provided evi- H= (ǫk−µ)a†kak+ 2Ω V(q)a†p+qa†k qakap (1)
dence for the breakdownof the light-cone picture in sys- Xk pXqk −
tems withlong rangeinteractions[42–44]. We showthat whereǫ =~2k2/2m, a is the bosonicannhilationoper-
k k
for the continuum system, for weak dipolar interactions, ator at momentum k and time t, Ω is the area, and µ is
correlations exhibit some features analogous to a light-
the chemical potential.
cone. However the dynamics show marked deviations
We consider the evolution of the momentum distribu-
from light-cone evolution for strong dipolar interactions:
correlations take longer to build up, and one cannot as- ltaiotinonnkfu(tn)ct=iohna:†kg((t2))a(kr(,tt))i,=and tehieqdreρns(itty)ρ-den(s0i)ty,wcohrerree-
sociate a characteristic velocity with their spread. This q · h q −q i
is due to the oscillation of the system between a weakly ρq(t) = ka†k+q(t)ak(t), foPllowing a sudden quench in
interactingsuperfluid,andacorrelatedstatewithlocally the dimePnsionless interaction parameter g˜ = gd/g. The
ordered domains. former can be readily probed in time-of-flight [49], while
Thispaperisorganizedasfollows: InSecIIwepresent the latter can be studied using high resolution imaging
the time-dependent Bogoliubov approach for studying [23,24],Braggspectroscopy[50]ornoisecorrelations[51].
the evolution of correlations in weakly interacting quasi-
2D dipolar gases and discuss our results in Sec III. In
A. Time-dependent Bogoliubov ansatz
Sec IV, we compare our results with those of a fully self-
consistent theory, recently developed by Lin and Radzi-
hovsky [45], where the condensate fraction is considered In quasi-2D at zero temperature, there is a true Bose
as a time-dependent variable. In Sec V, we discuss the condensate [52], and we can model the dynamics us-
relevance of our work to ongoing experiments on dipolar ing a time-dependent Bogoliubov approach. This the-
Bose gases, and summarize our results in Sec VI. ory is valid for weak interactions, which are governed
by the small parameter a/l 1. Throughout we refer
z
≪
to “weak” dipolar interactions as the regime where the
underlying dispersion does not have a roton feature and
II. THEORY
“strong”dipolarinteractionsastheregimewhereitdoes.
Itisimportanttoemphasizethatthephysicsoftheroton
We consider a quasi-2D dipolar Bose gas of mass m, mode inquasi-2Ddipolargasesisnot afeature ofstrong
at zero temperature, confined in a harmonic potential density correlations (as in He-4 [53]), rather one of geo-
of the form U(z) = 1mω2z2, and free in the x y metric confinement [28]. Hence the condition a/l 1
(transverse)directions.2Furtzhermore,weassumethat−the is satisfied even for “strong” dipolar interactions,zw≪here
dipoles arepolarizedalongthe z direction,whichyields the roton mode is present.
a dipolar interaction potential −of the form Vdip(R) = We set the density of condensate atoms n0 = ak=0 2,
c|Rod2s|3(θ(1) −=3z/coRs2(.θ)A),ddwihtieornealdlyi,stthheerediispoaleshmorotm-reanntg,eainnd- annodteswrtihtee abko=6so0(nti)c =anunikh(itl)abtkio+n vok∗p(et)rba†−tokr, wfohrertehhebknodine--
teraction po|ten|tial, which we model as Vc(R) = gδ(R), condensed atoms [54]. The bk operators have no time
where g = 4π~2a/m, and a is the 3D s-wave scattering dependence, and are formally treated as small. Substi-
length. Effects of finite temperature (T) are small pro- tuting the expression for ak=0 into Eq. 1, and discard-
vided T ∆ [46]. ing all terms cubic or highe6r order in bk, we arrive at
≪
We model the quasi-2D gas by making the Gaussian u (t = 0) = 1 1+ ǫk+Vi(k)n0 and v (t = 0) =
ansatz: n3D(R={r,z})=n(r)Ψ(z)= √1πl2n(r)e−z2/l2z, k r2(cid:16) Eki (cid:17) k
where r and z is the radial and the axiazl co-ordinate sgn(V (k)) 1 ǫk+Vi(k)n0 1 , where sgn(x) denotes
respectively, and l is a variational parameter which − i r2(cid:16) Eki − (cid:17)
z
the signofthe argument,andEi = ǫ (ǫ +2V (k)n ),
can be determined by solving the Gross-Pitaevskiiequa- k k k i 0
tion in the axial direction [47]. When the confining where the index i, denotes the inpitial state. At fu-
ture times, these coherence factors u (t) and v (t) ac-
potential greatly exceeds the interaction energy, l k k
z
~/mω , while for weak axial confinement, l →ζ, quire complex values, but will always satisfy uk(t)2
z z | | −
∼ v (t)2 =1.
pthe healing length of the condensate. Integrating out | k |
For now we assume that the condensate is static, in
the z-direction, the Fourier transform of the resulting
other words, n n. The advantage of this somewhat
quasi-2D interaction potential reads [47, 48]: V(k) = 0
≈
simplified approximationis thatit allowsus to make an-
1 g + g F klz , where k = k2+k2 is the
√2πlz(cid:16) d (cid:16)√2(cid:17)(cid:17) q x y alytic predictions for the long time behavior of the ex-
magnitude of the radial momentum, g = 2πd2, and cited fraction. Later on we present a fully self-consistent
d 3
3
0.5
Bogoliubov theory which takes into account the time-
dependence of the condensate density via the relation 0.4
n = n0(t)+ knk, where n is the total density, which 0.3
is a constantPof motion. This allows us to put better Žnex0.2
quantitative bounds on the validity of our theory for ex-
periments on quasi-2D dipoles. 0.1
The equations of motion for uk and vk are obtained 0.0
fromtheHeisenbergequationsofmotionfora ,andread 0 2 4 6 8 10 12 14
k
[54]: tΝho
i∂t(cid:18)uvkk((tt))(cid:19)=(cid:18)−ABkk −BAkk (cid:19)(cid:18)uvkk((00))(cid:19) (2) ΝHL(cid:144)Ekho0112....5050 ggggŽŽŽŽ====2221...135 DΝ(cid:144)ho000...246æ æ æ æ æ æ æ æ æ æ æ æ
where we have introduced the functions Ak = ǫk + æ
Vf(k)n0, Bk = Vf(k)n0 and Ekf = A2k−Bk2 is the Bo- 0.00.0 0.5 1.0 1.5 2.0 02.4 2.5 Ž 2.6 2.7
goliubovdispersion. Asweconsiderpasuddenquench,the kHunitsof 2(cid:144)lzL g
evolution of uk and vk depends only on the final Hamil- FIG.2: (ColorOnline)Top: Coherentevolutionofthedimen-
tonian parameters, denoted by the label f. Physically, sionless excited fraction n˜ex(t) = lz2nex/2 for different values
the time-dependent Bogoliubov approximation amounts of g˜f = gd/g. To compare the scale on the vertical axis, the
to coherent oscillations of quasi-particles in and out of total density is nlz2 ∼ 3 for typical densities and axial trap-
the condensate at a frequency proportionalto the quasi- pingpotentials. Frombottomtotop: g˜f =1(thinblack),2.1
particle energy Ef. Absent collisions, excitations at dif- (dotted green), 2.3 (dashed blue), 2.5 (thick red). Bottom-
k left: energy-momentum dispersion curves for the values of
ferent momenta evolve independently of one another.
g˜ in Top. Bottom-Right: Data shows numerically extracted
f
We note that Eq. 2 is the bosonic analog of the time-
values of the roton gap from the oscillations in the top plot
dependent Bogoliubov deGennes equations, which de- (seesolidredcurve),solidlineistherotongapextractedfrom
scribe the dynamics of Cooper pairs following an inter- theenergy momentum dispersion. For g˜<2.3 thedispersion
action quench in a Fermi gas [55]. does not havea sharp roton-like feature.
The evolution of the momentum distribution is given
by n (t) = v (t)2. We also define the excited frac-
k k
tion as nex(|t) =| dk nk(t). The density-density netic dipole-dipole interaction via a rapidly varying pla-
correlation functionRtakes the form g(2)(r,t) = n2 + nar magnetic field in addition to the static field along z
0
n eikr 2v (t)2+u (t)v (t)+u (t)v (t) [56]. The [3, 62]. As discussedby Giovanazziet al. [62], this latter
0 k · (cid:16) | k | ∗k k k k∗ (cid:17) protocolyieldsaneffective(time-averaged)dipolarinter-
firsPt term is the correlation between the condensate action of the form V (R) = d2 (1 3cos2(θ))α(φ),
atoms, while the second term involves correlations be- h dip i R3 −
tween the condensed and non-condensed atoms. Terms where α(φ) = (3cos2(φ) 1)/2,| w|here φ is the tilting
−
which involve correlations between the non-condensed angle. Bycontrollingthe angleφ,thedipolarinteraction
atoms alone (quartic in u s and v s) are negligible on can be completely switched off and turned on. Hence-
k k
length scales r & ζ [54, 57, 58]. We define a dimen- forth, we keep g fixed (gi =gf), and produce a non-zero
sionless density-density correlation function g˜(2)(t) = g˜f by suddenly switching on the dipolar interaction gd,
g(2)(t)l2/2n . and study the subsequent evolution of correlations. The
z 0
finalvalue ofg is chosensuchthatthe quasi-2Ddipolar
f
gas is mechanically stable (Ef 0 for all k) [47].
k ≥
III. RESULTS In Fig. 1, we plot the temporal evolution of the mo-
mentumdistributionn (t)foraquenchtostrongdipolar
k
Wenowpresenttheresultsfortheevolutionofthemo- interactions (g˜f = 2.65). In addition to the usual diver-
mentum distribution and the density-density correlation gence near k = 0, which is associated with Bose con-
function followinga quench using the non-self-consistent densation,themomentumdistributiondevelopsasecond
versionoftheBogoliubovtheorypresentedabove. Inthe peakassociatedwithroton-likeexcitations. Furthermore
next section, we will go beyond this approximation by (andsomewhatsurprisingly),atthe peak maximum, the
making the condensate density time-dependent. occupationofrotonsissubstantiallylargerthantheequi-
Although we can model arbitrary quenches, through- librium (zerotemperature)occupationofthesemodesat
out we consider an initial state prepared in equilibrium the same interaction strength.
at zero temperature, with c˜2 = g n /ν = 1, where InFig.2(top) we plotthe excitedfractionobtainedby
i 0 ho
ν = 2~2/ml2 and g˜ = g /g = 0. The relevant param- integrating the momentum distribution over all k. For
ho z i d
eter governing the dynamics is the ratio of the strength weak dipolar interactions, the excited fraction rapidly
of the dipolar and contact interactions after the quench saturatestoaconstantvaluewhereasforstrongerdipolar
g˜ =g /g . Experimentallythisparametercanbevar- interactions, it develops oscillations which become pro-
f d,f f
ied by either tuning the contact interaction or the mag- nounced as g˜ is increased.
f
4
Physically, this is understood as follows: after the directly accessed from a sudden quench experiment.
quench, the system responds by populating modes at
different wave-vectors. For a non-dipolar gas, as quasi-
particles oscillate in and out of the condensate, “fast” 0 0
quasi-particles rapidly dephase relative to one another, r 2(cid:144)lz=4 r 2(cid:144)lz=4
leaving only the “slow” modes, namely the phonons. 0
T(gh+e egnde)nrg0y, asncadlecosnepseaqruateinntglyt,htehesloexwciatenddffraascttimonodseastuis- ŽHL-2HLg00 r 2(cid:144)lz=3 ŽHL-2HLg00 r 2(cid:144)lz=3
ratAesstthoeitdsipaosylamrpinttoetricacvtaiolunesotrnentigmthesisτin=cr1e/a(sged+(ggdr)enen0., ŽHL2HLgt0 r 2(cid:144)lz=2 ŽHL2HLgt0 r 2(cid:144)lz=2
dottedcurveinFig2(top)),theexcitedfractiondevelops
0
undamped oscillations. This is due to the fact that for r 2(cid:144)lz=1 r 2(cid:144)lz=1
thisinteractionstrength,thedispersiondevelopsabroad
0 2 4 6 8 10 0 2 4 6 8 10
flat shoulder (see green (dotted) dispersion curve). As a
tHunitsof1(cid:144)Ν L tHunitsof1(cid:144)Ν L
ho ho
result,manymodesoscillateatthesamefrequency,again
sa(gehpivqoUpiuunepilagvdoranesrlriesfnoeifstrultyrtloathhg˜reugfenre)pdrrtaehiimncnetucpeerrrexseadoaccristtoieintsodocgnifltlrshatathtercieetorninoogsntdtohiispnnsho.omnlwaeirxsn.ipmiTrnouthnmeirosauwbcntrhciooeiacndhd,, (cid:144)2rlz1246800 2 4 6 8 ΝH(cid:144)Lvunitsofl2zho01122.....826248ææææ0æ.5ææææ1ææŽævææ101æ...583æ2æææææægŽ2æfæææææ2æ.æ52æ.5
weakly damped oscillations. This is because for strong tdipHunitsof1(cid:144)ΝhoL gŽf
dipolarinteractions,there aretwodistinct“slow”modes
FIG. 3: Top: Density-density correlation function for differ-
in the system: phonons and rotons. These modes have ent values of r, for a quench to g˜ = 1 (left) and g˜ = 2.6
f f
different dispersions, hence different density of states (right) respectively. Arrows indicate the feature whose dis-
(DOS): the phonon DOS n(ǫ) = k dk/dǫ vanishes as persion we study in the bottom plot for different final values
ǫ 0, whereas the roton DOS reads n(ǫ) = m∗(1 + of g˜f. (Bottom-Left) Temporal location of the dip feature
k /→ 2m ǫ ∆), which diverges as 1/ ǫ ∆. For (arrow) in the density-density correlation (tdip) for different
r ∗| − | | − | values of r and different interactions: Black g˜ = 1, Green
quepnches to strong dipolar interactions, tphe larger DOS f
(dashed)g˜ =2,Blue(dotted)g˜ =2.2,Red(dashed-dotted)
for roton-like modes implies that the system preferen- f f
g˜ = 2.6. (Bottom-Right) Data points show the velocity of
tiallyoccupiesrotonsfollowingthequench. Furthermore, f
spreading of correlations in a quasi-2D gas for quenches to
unlike phonon modes which occupy a range of momenta
differentg˜ . Insetisazoom-inforstrongdipolarinteractions
(0 < ~k < 2m(g+gd)n0), all the roton modes have (v˜=v/(lzfνho/√2)),thesolidlineisthecharacteristicvelocity
nearly the spame energy (Eroton ∼ ∆). This leads to a of roton-like modes v=2∆/~kr for different g˜f.
cooperative amplification of the oscillations in the mo-
mentum distribution. Not surprisingly, the timescale for
We now turn to the evolution of the density-density
these oscillations is roughly τ = 1/∆. For quenches
roton correlationfunction[23,24,42,43],whichreads: g˜(2)(t)
to strong dipolar interactions, where τroton > τ, the g˜(2)(0) = 1 dkkJ (kr)sin(Eft)2ǫ Bfǫk+gin0, wher−e
timescale for the fast modes to dephase relative to one −π 0 k k k EiEf
another, the excited fraction also develops oscillations. J is the BesseRl function of the first kind. k k
0
These can be readily probed in time-of-flight [27]. The In Fig. 3 we plot g(2)(r,t) for different values of r fol-
dynamicsoftherotonmodesinthisbosonic dipolargasis lowing a quench to two different values of g˜ , starting
f
analogousto that of Cooper pairs in an attractive Fermi from g˜ = 0. For weak dipolar interactions (g˜ . 1),
f
gas, driven out of equilibrium following an interaction following rapid oscillation on very short timescales, the
quench [55]. In that case, the coherent oscillation of density-density correlation function develops a dip fea-
Cooper pairs leads to oscillations in the superconduct- ture which disperses to later times as r increases, and
ing pairing gap. then rapidly relaxes to its long time value [54, 65].
At long times, the excited fraction calculated using a For strong dipolar interactions, density correlations dis-
saddle point approximation takes the approximate an- play persistent oscillations, with a frequency largely in-
alytic form n (t ) cos(2∆t + φ)/∆˜2√η t ν , dependent of r. The rapid oscillations at very short
ex ho
→ ∞ ∝
where η = m/m , ∆˜ = ∆/ν , and φ is an arbitrary times stem from the propagation of very fast particles
∗ ho
constant phase factor. The dynamics are described by (Ek k2,klz 1), produced after the quench, and are
∼ ≫
damped oscillations at a frequency equal to twice the absentinsystemswithaboundedspectrum(arisingfrom
roton gap, and an envelope which decays algebraically a lattice) [24, 54, 65].
as 1/√t. Indeed, the roton gap extracted from fit- On longer timescales, correlations develop a dip fea-
ting the numerical data for n (t) to this long time ture,associatedwithlowenergyexcitations. Thetempo-
ex
asymptotic formula is in near perfect agreement with rallocationofthedipfeature(t )inthedensity-density
dip
the gap obtained directly from the dispersion relation correlations (Fig. 3 (bottom-left)) shows a linear depen-
E = ǫ (ǫ +2V (k)n ). Hence the roton gap can be dencewithratsmallg˜ . Theslopeofthecurvesatlarge
k k k f 0 f
p
5
r, plotted on the bottom-right panel, is the characteris- which is a constant of motion, and n = dk n (t), as
ex k
ticvelocityofspreadingofcorrelations. Absentdamping defined in the main text. R
processes, quasi-particles propagate ballistically, hence
0.5
correlationsdisplaylight-conebehavioratlargedistances
[39,40,54],whichavelocityv ∂Ek/∂k . Atsmallg˜f, 0.4
∼h i
we expand the dispersion as E c˜2(1+2g˜ )k, and
k f
approximate the Bessel function J≈0(px) cos(x/√2), to Žnex0.3
≈
obtain a velocity v 2lzνho, which is indeed what we 0.2
≈
findnumericallybytakingtheslopeoftheblackcurvein
0.1
(Fig. 3 (bottom-left)).
Increasing dipolar interactions decreases the velocity, 0 2 4 6 8 10 12 14
asrepulsiveinteractionslowertheaveragegroupvelocity tΝ
ho
ofthequasi-particlesinaquasi-2Ddipolargas. Nonethe-
0.7
less, for g˜ 2.5, correlations still spread in a light-cone
≤ 0.6
manner, and one can associate a characteristic velocity
0.5
with their spread. For strong dipolar interactions, this x
velocity approaches v 2∆/~k . Žne0.4
→ r 0.3
Forquenchestoevenstrongerdipolarinteractions,the
0.2
dispersion of the dip develops a non-trivial step-like fea-
0.1
ture. The width of the step growswith g˜ , becoming in-
f 0 2 4 6 8 10 12 14
finitelywideattherotoninstabilitythreshold,indicating
tΝ
thatneartherotoninstability,correlationstakeinfinitely ho
long to build up. Physically, this can be understood as FIG. 4: (Color Online) (Top) Coherent evolution of the ex-
follows: following the quench to strong dipolar interac- cited fraction n˜ex(t) for different values of η at g˜f = gd/g =
2.5, obtained by self-consistently solving for the condensate
tions,thesystemdevelopslocallyordereddomains,which
fraction using Eq. 3. The red (solid, thick) curve is the non-
are otherwise uncorrelated with one another. These do-
self consistent result (assuming n0 n), also shown in Fig. 2
mains prevent correlations from spreading through the ≈
ofthemaintext. Thegreen(dashed),blue(dotted)andblack
system. The timescale over which these domains appear (thin) curves correspond to η = 0.05,0.1,0.25 respectively.
and disappear is linked to the oscillations in the roton (Bottom) Same as top with g˜ =2.6.
f
occupation number, which is set by τ . Correlations
roton
spread rapidly when the domains are absent, and slowly
As in Sec III, we consider a quench from an initially
when the domains are present, giving rise to the step-
non-dipolar gas from g˜ =g /g =0, to some finite value
like feature. Near the roton-instability threshold, τ i d
roton g˜ >0,keepingthecontactinteractionparametergfixed.
diverges, and the domains become extremely long lived, f
At each time step (t) following the evolution, the con-
preventing correlations from spreading altogether.
densate fraction is varied according to Eq. 3, which is
then used to calculate u (t + δt) and v (t + δt) from
k k
Eq. 2. The updated excited fraction is used to obtain a
IV. SELF-CONSISTENT BOGOLIUBOV
new condensate fraction, and the procedure is repeated.
APPROACH
This procedure becomes exact in the limit of vanishing
δtandwehaveverifiedthatthe time-steps(δt)aresmall
In this section we discuss in detail the limitations of enough that effects of finite step-size are negligible over
the approximation of a static condensate, employed to the timescales we study.
obtain the results presented in Sec IV. This approach is The relevant parameter that controls the strength of
validprovidedthatnex n0 atall times duringtheevo- the excited fraction is η = 2/nl2, which we vary, along
≪ z
lution. While this condition is met for weak interactions with the quench amplitude g˜ . Eq. 3 then becomes:
f
or short times, it is dramatically violated very near the
roton instability threshold, and a better modeling of the n˜ (t)=1 η n˜ (t), (4)
0 ex
−
condensate is required.
In this section we compare our “static” treatment of where n˜0 denotes the condensate fraction and n˜ex is the
thetime-dependentBogoliubovapproximation[54]ofSec dimensionless excited fraction, which is plotted in Fig. 4
III to a fully self-consistent Bogoliubov treatment of the for a quench to g˜f = 2.5. Large η implies the gas is
dynamics,developedby LinandRadzihovski[45], where becoming more 2D-like.
weletthecondensatedensityfluctuateintimeaccording Integrating the momentum distribution in Fig. 4, we
to: find that the dynamics of the excited fraction is qualita-
tively similartoFig.2,evenwhenthecondensatefraction
n (t)=n n (t). (3) is assumed to be a dynamical variable. Quantitatively
0 ex
−
however,the amplitude and frequency of the oscillations
Heren is the condensatefraction, n is the totaldensity, decreases as η is increased. Although small oscillations
0
6
are present even for η & 0.3, the amplitude of the oscil- tongap,withouttheneedforsophisticatedspectroscopy.
lationsmaybedifficulttodetectexperimentally. Forthe Using a time-depedent Bogoliubov ansatz, we have
black (thin, solid) curve,the oscillation frequency differs shown that this mode can indeed be identified follow-
from that shown in the red (thick, solid) curve by 10%. ing a quench, and the roton gap energy can be mea-
Recall that the oscillation frequency for the red (thick, sured from simple time-of-flight measurements. To en-
solid) curve is equal to twice the equilibrium roton gap sure that this is not an artifact of our approximations,
at g˜=2.5. Increasing the quench amplitude (g˜ ) causes we have presented a fully self-consistent theory, which
f
the oscillations in the excited fraction to become anhar- allows the condensate density to fluctuate and identified
monic,andforlargerg˜ values,onehastogotosmallerη the parameter regimes where the non-self consistent re-
f
to recover quantitative agreement with the non-self con- sults are expected to be qualitatively and quantitatively
sistent results. The anharmonicity is to be expected as accurate. We predict that provided the inter-particle
the different k modes are now coupled to one another spacing n 1/2/l 0.4, the roton gap can be probed
− z
≤
via the time-dependent condensate fraction. Nonethe- using a quench to within 10%. While the quantitative
less, even at g˜ = 2.6, shown in the bottom panel, the magnitudeoftheeffectwepredictiscontingentuponex-
dynamics retains the qualitative features obtained from perimental realities, quench dynamics can complement
the simple theory presented in the main text. other probes of excitations in dipolar gases [68, 69].
The parameter η is experimentally tunable either by An important complication in most experiments is
tuningthedensityortheaxialtrappingpotential. Based the presence of weak harmonic confinement. In a trap,
on our findings above, quench dynamics can probe de- the low lying collective modes are density oscillations,
tails of the underlying equilibrium dispersion, even for which occur on frequencies proportional to ω , the ra-
r
strong quenches (g˜ 2.5), provided that η 0.3. De- dial trap frequency. The physics we describe here oc-
∼ ≤
spite the seemingly small value of this parameter, this curs on timescales τ = 1/∆ ms τ 50ms.
roton trap
implies that the inter-particle spacing n−1/2/lz 0.4. On timescales much shorter than∼the≪trap per∼iod, the
≤
These conditions can be readily achieved in experiments overall condensate density profile (which requires global
evenformoderatedensities(n 108 109cm−2),fortyp- mass transport)is relatively unaffected, anda localden-
∼ −
ical trapping potentials of ω /2π 100 1000Hz [46]. sity approximation can be used. Using this approxima-
z
∼ −
On even longer timescales, beyond mean-field correc- tion, we have verified that numerically that the oscilla-
tions such as collisions between quasi-particles, and be- tions in n persist for a few cycles in τ . On longer
ex roton
tweenquasi-particlesandcondensateatoms(Landauand times, comparable to the trap period, collective density
Beliaev damping [59]), which are not captured by our modes will wash out the oscillations in n , and a better
ex
self-consistent approach will endow the excitations with modelingofthecondensatedynamicsandtheinteraction
a finite lifetime (τ ), anddrive the systemtowardsther- between the condensed and non-condensed clouds is re-
el
malequilibrium. AtlowtemperaturesT µ,∆,Beliaev quired. While such a theory can be performed, should
≪
processesdominate,andtheirratescanbeestimatedper- experiments demand quantitative refinement, it is be-
turbativelyinpowersofa/l . Forquasi-2Ddipolargases yond the scope of the present paper. However, owing
z
however,these ratesarestronglysuppressedforphonons to the separation of timescales between τ τ in
roton trap
≪
and rotons, owing to the nature of the quasi-2D disper- typical experiments, the oscillations we find here should
sion [60, 61]. be observable, without complications arising from trap
effects. Furthermore, nearly homogeneous Bose Einstein
condensates have been experimentally realized by using
V. DISCUSSION box shaped traps [63].
We now discuss the relevance of our work to ongoing
experiments on quasi-2D dipolar gases. Experiments so VI. SUMMARY AND CONCLUSIONS
far have not been able to decisively verify the existence
of the roton mode, because the signatures of this mode In this work we have studied the non-equilibrium dy-
in correlation functions are rather weak [67]. Here we namicsofaquasi-2DdipolarBosegas,followingasudden
have proposed the use of a sudden quench to amplify quench in the interaction strength. Remarkably we find
these signatures. The physical reason for this is clear: thatquenchdynamicscanprobetheexcitationspectrum
on short timescales, excitations coherent oscillate in and of the dipolar gas, which otherwise in equilibrium, dis-
outofthe condensateandthe factthatrotonexcitations playsno pronouncedspectralsignatureoftheunderlying
all oscillate with nearly the same frequency will lead to roton mode. Physically, the reason for this is that on
an amplification of the signal, similar to what occurs in timescales t τ τ ,τ , excitations coherently
roton trap el
∼ ≪
drivenFermisystems. Inthisrespect,thebosonicdipolar oscillate in and out of the condensate. As roton excita-
gasbehaveslikeafermionicsystem. Themainadvantage tions all oscillate at nearly the same frequency ∆, this
of our approach is its simplicity; the momentum distri- leads to a cooperative enhancement of the signal in cor-
bution following a quench can be readily measured in relation functions. In this respect, our bosonic system
ultra-cold systems, and direct provides access to the ro- displays an interesting parallel with a Fermi gas with
7
attractive interactions, where the quench to attractive VII. ACKNOWLEDGEMENTS
interactions leads to an enhancement of the supercon-
ducting gap [55, 70–73]. It will be extremely interest-
ing to study whether the non-equilibrium enhancement It is a pleasure to thank Benjamin Lev, Kristian Bau-
ofrotons couldbe a route to realizing (albeit metastable mann, Mingwu Lu and Rajdeep Sensarma for insight-
[74,75]),quantumcrystallinephasesofmatter. Wehope ful discussions during the completion of the work, and
that our work will not only motivate experiments to use Ryan Wilson for his careful reading of the manuscript.
quantum quenches to look for roton physics, but also SN would like to thank V. Galitski for pointing him to
further theoretical studies on the nature of spreading of Ref. [55]. SN is supported by JQI-NSF-PFC, AFOSR-
correlations in dipolar systems. MURI, and ARO-MURI.
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