arXiv:0708.2789v1 [cond-mat.other] 21 Aug 2007
Collective excitations of trapped one-dimensional dipolar quantum gases
P. Pedri,1S. De Palo,2E. Orignac,3R. Citro,4and M. L. Chiofalo5,6
1Laboratoire de Physique Th´ eorique e Mod` eles Statistiques, Universit´ e Paris-Sud, Orsay, France
2DEMOCRITOS INFM-CNR and Dipartimento di Fisica Teorica, Universit` a Trieste, Trieste, Italy
3Laboratoire de Physique de l’´Ecole Normale Sup´ erieure de Lyon, CNRS-UMR5672, Lyon, France
4Dipartimento di Fisica ”E. R. Caianiello” and CNISM, Universit` a degli Studi di Salerno, Salerno, Italy
5Classe di Scienze, INFN and CNISM, Scuola Normale Superiore, Pisa, Italy
6Centre´Emile Borel, Institut Henri Poincar´ e, Paris, France
We calculate the excitation modes of a 1D dipolar quantum gas confined in a harmonic trap with
frequency ω0and predict how the frequency of the breathing n = 2 mode characterizes the interaction
strength evolving from the Tonks-Girardeau value ω2 = 2ω0 to the quasi-ordered, super-strongly
interacting value ω2 =√5ω0. Our predictions are obtained within a hydrodynamic Luttinger-Liquid
theory after applying the Local Density Approximation to the equation of state for the homogeneous
dipolar gas, which are in turn determined from Reptation Quantum Monte Carlo simulations. They
are shown to be in quite accurate agreement with the results of a sum-rule approach. These effects
can be observed in current experiments, revealing the Luttinger-liquid nature of 1D dipolar Bose
PACS numbers:03.75.Kk, 03.75.Hh, 71.10.Pm, 02.70.Ss, 31.15.Ew
ceptually novel technological applications is the possi-
bility of reaching extreme quantum degeneracy under
controlled conditions. This is quite a remarkable prop-
erty of ultracold atomic gases, which is being evidenced
by the interdisciplinary contribution of quantum optics,
quantum information, condensed matter, atomic, and
fundamental physics. Extreme quantum limit can here
be obtained after lowering the temperature down to the
nanokelvin scale, and by tuning the atomic interactions
and the dimensionality. Dimensionality can indeed be re-
duced to two and one dimensions (1D) after using a vari-
ety of magnetic and optical techniques including optical
lattices [1, 2, 3, 4]. The interactions can also be manipu-
lated almost at will in their short-range part by means of
the Fano-Feshbach resonance mechanism [5, 6, 7, 8, 9].
In the case of atomic or molecular species with large mag-
netic or electric moments, proposals have been put for-
ward which predict the possibility of tuning the long-
range dipolar tail of the interaction [10, 11]. The ob-
servation of dipolar interactions  in atomic52Cr va-
pors with relatively large magnetic moment µ ≃ 6µB,
is especially promising for applications, since it opens
the way to the realization of e.g.
phases [13, 14] and the observation of spin-charge sep-
aration . Molecular dipolar crystals have been more
recently proposed as a realization of high fidelity quan-
tum memory for quantum computation . In a set of
landmark experiments [17, 18] and supported by simu-
lational work [19, 20], vapors of52Cr atoms have been
Bose-condensed . Yet, the effect of the dipolar in-
teraction has been largely enhanced after reducing the
strength of the short-range part .
The bottom line in the design of con-
Combination of 1D geometries with tunable interac-
tions may provide an easy access to enhanced quantum
correlations. Experimental realizations of the strongly
correlated Tonks-Girardeau (TG) gas  have already
been observed in atomic vapors with contact interac-
tions [24, 25]. Going beyond the TG regime under these
conditions is not an easy task . However, as we have
more recently demonstrated by a combined Reptation
Quantum Monte Carlo (RQMC) and bosonization ap-
proach , a homogeneous 1D Bose gas with dipolar
interactions is a very strongly correlated Luttinger liquid
with the parameter K < 1  at all densities nr0? 0.1
with r0being the range of the dipolar potential (see be-
low). The Luttinger liquid crosses over from a Tonks-
Girardeau gas setting in for nr0? 0.1 to a high-density
quasi-ordered state  which can be viewed as the ana-
logue of a Charge Density Wave. We have also discussed
how the use of polar molecules may provide access to
this quasi-ordered state, which from now on we refer to
as Dipolar Density Wave (DDW).
The knowledge of the interaction regime is a basic tool
for all conceivable applications. Since ongoing experi-
ments will be performed in confined geometry, one of the
best suited and controlled methods is to exploit the col-
lective excitations related to the discrete modes in the
harmonic trap , which has indeed been largely used
since the very first experiments [31, 32]. The study of the
collective modes can also be useful in view of the proposal
to implement the quantum memory, to reveal and inves-
tigate decoherence mechanisms possibly arising from the
coupling of internal and external degrees of freedom on
the DDW state .
In this Letter we predict for the first time the crossover
behavior of the collective modes of the trapped 1D dipo-
lar Bose gas by two different theoretical methods, namely
a sum-rule approach and a hydrodynamic Luttinger-
liquid model. The two methods share the application
of the Local Density Approximation to the equation of
state as determined from our RQMC simulational data,
and are shown to give results in quite remarkable agree-
The equation of state.-
ground-state energy per particle (within the statisti-
cal error) of the homegeneous dipolar Bose gas by re-
sorting to Reptation Quantum Monte Carlo simulations
(RQMC) . As described in more detail in Ref. ,
we consider N atoms or molecules of mass M and
permanent dipole moments arranged along a line in
the limit of negligible contact interaction  and po-
larized in the orthogonal direction.
defined in effective Rydberg units Ry∗= ?2/(2Mr2
r0≡ MCdd/(2π?2) the effective Bohr radius, Cdd= µ0µ2
or Cdd= d2/ǫ0the interaction stengths for magnetic µd
or electric d dipole moments respectively. The governing
dimensionless parameter is nr0, with n the number of
dipoles per unit length.
The RQMC data for the dependence of the energy per
particle ε on n are fitted by the expression ε(nr0)/Ry∗=
(π2/3)(nr0)2[1+d(nr0)g]−1, where a = 3.1(1), b = 3.2(2),
c = 4.3(4), d = 1.7(1), e = 3.503(4), f = 3.05(5),
and g = 0.34(4) with an overall χ2
nr0≪ 1, ε(nr0)/Ry∗∼ (π2/3)(nr0)2and for nr0≫ 1,
ε(nr0)/Ry∗∼ ζ(3)(nr0)3so that both the TG and DDW
limits are satisfied. The functional form for ε(n) here
provided can be used in further calculations. Here, we
use it to obtain the chemical potential µRQMC(n) =
(1 + n(∂/∂n))ε(n).
The Local Density Approximation.-
the effectively 1D dipolar gas is confined by a harmonic
potential V (x) = Mω2
dynamical behavior of the confined gas can be found from
hydrodynamic equations at temperature T = 0 using a
Local Density Approximation (LDA) to the equation of
state, in which the energy of the inhomogeneous system
is a local functional of the local density n(x) expressed
as the integral over x of the energy density of the homo-
geneous gas with density n(x). The validity of such an
approach is limited to dynamical effects in which the typ-
ical length over which n(x) varies is much larger than the
average interparticle distance in the axial direction. Fur-
thermore, the dynamical behavior in the radial (trans-
verse) plane must remain frozen. Under these conditions,
the ground-state density profile n(x) of the trapped dipo-
lar gas is obtained by plugging µRQMC(n) into the equa-
We first determine the
−(nr0)2?(∂2/∂x2) + (nr0)3?
0x2/2 in the axial direction. The
µ(n(x)) + V (x) = µ0,(1)
and solving it for µ0and n(x) with the condition that the
total number of particles N =?R
where the Thomas-Fermi radius R = (2µ0/(Mω2
−Rn(x)dx is conserved,
such that n(±R) = 0. The condition can be casted in the
form N(r0/aho)2= (µ0/Ry∗)1/2?1
x2)]dx, where aho= (?/Mω0)1/2is the harmonic oscilla-
tor length. This identifies N(r0/aho)2as the interaction
parameter driving the trapped dipolar gas from the TG
across the DDW regime.
The evolution of the calculated density profiles through
the crossover, shows the expected increase of the central
density n(0) with the chemical potential µ0, as well as a
steepening of the profiles at the trap edges. The calcu-
lated profiles agree with the analytical results expected
in the TG and in the DDW limits.
The breathing mode from a sum rule (SR) approach.-
We first determine the effect of the interactions on
the frequency of the lowest compressional (breathing)
mode of the trapped gas by a sum-rules approach. This
mode is coupled to the ground state by the operator
ˆ X2 =
in current experiments, as it is excited by modulat-
ing the trap frequency and observed by following the
time-evolution of the width of the cloud by conventional
breathing mode satisfies an inequality that can be de-
rived from the sum rule [34, 35, 36] ω2
where 2m1 = ?[X,[H,X]]? and 2m−1 is the static re-
sponse function. After some simplification, this yields
the upper bound:
i.This makes it the easiest to probe
The frequency ωB of the
where ?x2? = N−1?x2n(x)dx. A closed form expression
of ΩB in Eq. (2) can be obtained by means of a scaling
argument whenever µ(n) is of the form µ(n) = λnγ, re-
sulting in ΩB = ω0(2 + γ)1/2. In particular, in the TG
case with γ = 2 this yields ΩB= 2ω0, and in the DDW
case with γ = 3 it yields ΩB≤√5ω0.
For intermediate interaction strengths, we have to re-
sort to a numerical estimation of Eq. (2) using the LDA
density profile. The result is represented by the solid line
in Fig. 1, showing the smooth evolution of the breathing
mode frequency ωB from the TG to the DDW regimes.
Figure 1 represents an useful tool to identify the inter-
action regime of the trapped dipolar gas through one
of the best handled experimental probes available with
cold atomic (molecular) quantum gases [31, 32]. How-
ever, Eq. (2) in principle yields an upper bound for the
frequency of the lowest compressional mode. Moreover,
estimating the frequencies of the higher modes by the
sum-rule approach is not as simple. We thus switch to
an alternative, hydrodynamic, approach to compute their
The excitation modes from hydrodynamic Luttinger
equations.-In , we have shown that the low-energy
behavior of a homogeneous dipolar Bose gas is well de-
scribed by the Luttinger hamiltonian, and we have de-
FIG. 1: Squared frequency ω2
to the trap frequency ω0 vs.
N(r0/aho)2, as calculated from two models. Solid line: sum-
rule approach Eq. (2) (SR). Symbols: Luttinger-liquid hydro-
dynamics Eq. (6) (LL).
Bof the breathing mode scaled
the interaction parameter
termined the density-dependence of the velocity u and
Luttinger exponent K by combining bosonization and
RQMC techniques.The u and K obtained from the
RQMC structure factor were found to agree with those
extracted from the RQMC energy via the Luttinger rela-
tions uK = M−1πn and u/K = π−1∂nµ(n) embodying
Galilean invariance [37, 38]. We now assume that in a
slowly varying external trapping potential, the dipolar
Bose gas can be described by a Luttinger liquid hamilto-
where u(x) and K(x) now depend on position via the
LDA n(x).They are related through u(x)K(x) =
M−1πn(x) and u(x)/K(x) = π−1∂nµ(n)|n=n(x), extend-
ing the Luttinger-liquid relations to the weakly inhomo-
geneous system. In Eq. (3), φ and Π satisfy canonical
commutation relations [φ(x),Π(y)] = iδ(x − y), leading
to the equations of motion:
∂tφ(x,t) = πu(x)K(x)Π(x,t),(4)
π∂tΠ(x,t) = ∂x
These equations of motion must be complemented by
boundary conditions expressing that no current flows
across the edges, i. e. j(±R,t) = 0. Since δn = −∂xφ/π,
the continuity equation ∂t(δn) + ∂xj = 0 leads naturally
to j = ∂tφ/π, and allows us to rewrite the boundary
conditions as φ(−R) = φ0and φ(R) = φ1.
Eqs. (4)–(5) possess a stationary solution, ∂xφ ∝
K(x)/u(x). This solution actually describes the addition
of one particle to the system φ(x) → φ(x) − πg(x)/g(R)
where g(x) =?x
φ0 and φ1 are not independent, but related through
φ1− φ0 = −πN with N being the number of parti-
cles added to the system.
bosonization formula derived in the case of a homoge-
neous system . Alternatively, the static solution can
be derived by considering the effect of a perturbation
to the trapping potential . Combining Eqs. (4)–(5)
and using linearity to search for solutions of the form
φ(x,t) = φ0− πNg(x)/g(R) +?
with the boundary conditions ϕn(±R) = 0 for the dis-
crete Fourier components ϕn.
While Eq. (6) is cast in a form identical to the hydro-
dynamic equation for density excitations , we remark
that here it has been derived from HLLEq. (3). Thus, a
comparison of the measured excitation frequencies with
those predicted from (6) with our RQMC data, provides
a test of Luttinger-liquid behavior in trapped 1D dipolar
Bose gases within the validity of LDA.
In the case of a harmonic trapping, one of the eigen-
frequencies in Eq. (6) is obtained straightfowardly. In-
deed, substituting ϕ1(x) ∝ n(x) in Eq. (6), differentiat-
ing Eq. (1) with respect to x and using the Luttinger-
liquid relations, we find that ϕ1(x) is an eigenfunction
of (6) associated with the eigenvalue ω2
lar solution is simply the Kohn mode or sloshing mode
describing a center-of-mass oscillation. Indeed, it can be
recovered by expanding to first order in A the expression
φ(x,t) = φ(x−Acos(ω0t),0) which describes a center of
mass motion with rigid density. The eigenfrequencies are
exactly known also in the two asymptotic limits. Indeed,
insertion of µ(n) ∝ nγin Eqs. (1) and (6), yields solutions
of the form ϕn(x) = An(1 − x2/R2)1/γC(1/γ+1/2)
the associated eigenvalues being ω2
[36, 40], where C(α)
are Gegenbauer polynomials and An
normalization factors , Thus, at the two opposite TG
(γ = 2) and DDW (γ = 3) limits one finds respectively
ωn = n2ω2
ate densities, we have solved Eq. (6) using the Sledge
algorithm available online , after inserting as ingre-
dients the computed LDA density profiles from Eq. (1)
and of the analytical expression for ∂nµ obtained from
the RQMC fit, evaluated at the local density for differ-
ent values of the interaction parameter N(r0/aho)2. In
the numerical solution we have taken special care of the
finite mesh-size effects for best accuracy.
The values of the breathing frequency ωB/ω0obtained
from Eq. (6) are represented in Fig. 1 by the filled sym-
bols, and agree up to the second digit with the sum-
rule result.This was expected in the TG and DDW
regimes where, as already noticed in Ref. , an equal-
ity sign holds in Eq. (2). The eigenfrequencies of the
−R(K(y)/u(y))dy. A consequence is that
This form generalizes the
nϕn= n(x)∂x(∂nµ(n(x))∂xϕn). (6)
0. This particu-
0and ωn = n(3n − 1)ω2
0/2. For intermedi-
FIG. 2: (ωn/ω0)2vs. N(r0/aaho)2from Eq. (6). From bottom
to top: the modes with n = 2, 3, 4, and 5. Tick solid lines:
limiting values in the TG and DDW regimes at each n.
higher modes with n = 3, 4, and 5 are displayed in Fig. 2
as functions of N(r0/aho)2, showing the same smooth
crossover behavior between the two opposite TG and
DDW regimes. The exact frequencies in these asymp-
totic regimes are recovered by the numerical calculations.
evolution of the collective modes of a 1D dipolar Bose gas
through the crossover from the Tonks-Girardeau to the
Dipolar-Density-Wave regime. These modes, and espe-
cially the breathing mode can be excited and measured
by quite standard and reliable techniques. Our results
are relevant to experiments, where they can be used to
determine the interaction regime, and extended to inves-
tigate the occurrence of decoherence effects in quantum
applications . From the theoretical point of view, our
results confirm for the trapped gas the smooth crossover
from the TG to the DDW regimes
for weak inhomogeneity. The RQMC functional form of
the energy per particle that we explicitly provide in this
work, can be useful to investigate the issue, crucial for
applications, of the crystal phase stabilization by means
of e.g. a commensurate, though shallow, optical lattice.
MLC would like to thank the Institut Henri Poincar´ e -
Centre Emile Borel in Paris for hospitality and support.
This work was suported by the Minist` ere de la Recherche
(grant ACI Nanoscience 201), by the ANR (grants NT05-
42103 and 05-Nano-008-02) and by the IFRAF Institute.
In conclusion, we have predicted the
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