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Collective excitations of trapped one-dimensional dipolar quantum gases


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Collective excitations of trapped one-dimensional dipolar quantum gases

P. Pedri,1S. De Palo,2E. Orignac,3R. Citro,4 and M. L. Chiofalo5,6

1Laboratoire de Physique Théorique e Modèles Statistiques, Université Paris-Sud, Orsay, France 2

DEMOCRITOS INFM-CNR and Dipartimento di Fisica Teorica, Università Trieste, Trieste, Italy


Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS-UMR5672, Lyon, France


Dipartimento di Fisica “E. R. Caianiello” and CNISM, Università degli Studi di Salerno, Salerno, Italy


INFN Dipartimento di Matematica, Facoltà di Farmacia, Università di Pisa, Pisa, Italy

6Centre Émile Borel, Institut Henri Poincaré, Paris, France

共Received 4 August 2007; published 29 January 2008兲

We calculate the excitation modes of a one-dimensional共1D兲 dipolar quantum gas confined in a harmonic trap with frequency␻0and predict how the frequency of the breathing n = 2 mode characterizes the interaction

strength evolving from the Tonks-Girardeau value ␻2= 2␻0 to the quasi-ordered, superstrongly interacting value␻2=

5␻0. 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 agree-ment with the results of a sum-rule approach. These effects can be observed in current experiagree-ments, revealing the Luttinger-liquid nature of 1D dipolar Bose gases.

DOI:10.1103/PhysRevA.77.015601 PACS number共s兲: 03.75.Kk, 03.75.Hh, 71.10.Pm, 02.70.Ss

The bottom line in the design of conceptually novel tech-nological applications is the possibility of reaching extreme quantum degeneracy under controlled conditions. This is quite a remarkable property of ultracold atomic gases, which is being evidenced by the interdisciplinary contribution of quantum optics, quantum information, condensed matter, atomic, and fundamental physics. The 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共1D兲 after using a variety of magnetic and optical techniques including optical lattices 关1–4兴. The interactions can also be manipulated almost at

will in their short-range part by means of the Fano-Feshbach resonance mechanism 关5–9兴. In the case of atomic or

mo-lecular species with large magnetic or electric moments, pro-posals have been put forward which predict the possibility of tuning the long-range dipolar tail of the interaction关10,11兴.

The observation of dipolar interactions 关12兴 in atomic 52Cr vapors with relatively large magnetic moment ␮⯝6␮B is

especially promising for applications, since it opens the way to the realization of, e.g., different quantum phases关13,14兴

and the observation of spin-charge separation关15兴.

Molecu-lar dipoMolecu-lar crystals have been more recently proposed as a realization of high-fidelity quantum memory for quantum computation关16兴. In a set of landmark experiments 关17,18兴

and supported by simulational work关19,20兴, vapors of 52Cr atoms have been Bose condensed关21兴. Yet the effect of the

dipolar interaction has been largely enhanced after reducing the strength of the short-range part关22兴.

The combination of 1D geometries with tunable interac-tions may provide an easy access to enhanced quantum cor-relations. Experimental realizations of the strongly correlated Tonks-Girardeau共TG兲 gas 关23兴 have already been observed

in atomic vapors with contact interactions 关24,25兴. Going

beyond the TG regime under these conditions is not an easy task关26兴. However, as we have more recently demonstrated

by a combined reptation quantum Monte Carlo共RQMC兲 and bosonization approach 关27兴, a homogeneous 1D Bose gas

with dipolar interactions is a very strongly correlated Lut-tinger liquid with the parameter K⬍1 关28兴 at all densities

nr0ⲏ0.1 with r0being the range of the dipolar potential共see below兲. The Luttinger liquid crosses over from a Tonks-Girardeau gas setting in for nr0ⱗ0.1 to a high-density quasi-ordered state 关29兴 which can be viewed as the analog 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 a dipolar density wave共DDW兲.

Knowledge of the interaction regime is a basic tool for all conceivable applications. Since ongoing experiments will be performed in confined geometry, one of the best suited and controlled methods is to exploit the collective excitations related to the discrete modes in the harmonic trap 关30兴,

which has indeed been largely used since the very first ex-periments 关31,32兴. The study of the collective modes can

also be useful in view of the proposal to implement the quan-tum memory, to reveal and investigate decoherence mecha-nisms possibly arising from the coupling of internal and ex-ternal degrees of freedom on the DDW state关16兴.

In this Brief Report we predict the crossover behavior of the collective modes of the trapped 1D dipolar Bose gas by two different theoretical methods: namely, a sum-rule ap-proach and a hydrodynamic Luttinger-liquid model. The two methods share the application of the local density approxi-mation共LDA兲 to the equation of state as determined from our RQMC simulational data and are shown to give results in quite remarkable agreement.

We first determine the ground-state energy per particle 共within the statistical error兲 of the homogeneous dipolar Bose gas by resorting to RQMC simulations 关33兴. As

de-scribed in more detail in Ref.关27兴, we consider N atoms or

molecules of mass M and permanent dipole moments ar-ranged along a line in the limit of negligible contact interac-tion 关22兴 and polarized in the orthogonal direction. The PHYSICAL REVIEW A 77, 015601共2008兲


Hamiltonian H =关−共nr0兲2



i⬍j兩xi− xj兩−3兴 is

defined in effective Rydberg units Ryⴱ=ប2/共2Mr02兲 with r0⬅MCdd/共2␲ប2兲 the effective Bohr radius, Cdd=␮0␮d2 or

Cdd= d2/⑀0the interaction strengths for magnetic␮dor

elec-tric d dipole moments, respectively. The governing dimen-sionless parameter is nr0, with n the number of dipoles per unit length.

The RQMC data for the dependence of the energy per particle␧ on n are fitted by the expression

␧共nr0兲/Ry=共3兲共nr0兲4+ a共nr0兲e+ b共nr0兲f+ c共nr0兲2+g ⫻关1 + nr0兴−1+2/3兲共nr0兲2关1 + d共nr0兲g−1, where a = 3.1共1兲, b = 3.2共2兲, c = 4.3共4兲, d = 1.7共1兲, e = 3.503共4兲, f =3.05共5兲, and g=0.34共4兲 with an overallred2 ⯝5. For nr01, ␧共nr0兲/Ryⴱ⬃共␲2/3兲共nr0兲2, and for

nr01, ␧共nr0兲/Ryⴱ⬃␨共3兲共nr0兲3 so that both the TG and DDW limits are satisfied. The functional form for ␧共n兲 here provided can be used in further calculations. Here, we use it to obtain the chemical potential ␮RQMC共n兲 =关1+n共⳵/⳵n兲兴␧共n兲.

In experiments, the effectively 1D dipolar gas is confined by a harmonic potential V共x兲=M␻02x2/2 in the axial direc-tion. The dynamical behavior of the confined gas can be found from hydrodynamic equations at temperature T = 0 us-ing a LDA to the equation of state, in which the energy of the inhomogeneous system is a local functional of the local sity n共x兲 expressed as the integral over x of the energy den-sity of the homogeneous gas with denden-sity n共x兲. The validity of such an approach is limited to dynamical effects in which the typical length over which n共x兲 varies is much larger than the average interparticle distance in the axial direction. Fur-thermore, the dynamical behavior in the radial 共transverse兲 plane must remain frozen. Under these conditions, the ground-state density profile n共x兲 of the trapped dipolar gas is obtained by plugging␮RQMC共n兲 into the equation

„n共x兲… + V共x兲 =␮0 共1兲

and solving it for␮0and n共x兲 with the condition that the total number of particles N =兰−R

R n共x兲dx be conserved, where the

Thomas-Fermi radius R =关2␮0/共M02兲兴1/2 is such that n共⫾R兲=0. The condition can be cast in the form N共r0/aho兲2=

0/Ryⴱ兲1/2兰−1 1

r0␮−1关共␮0/Ryⴱ兲共1−x2兲兴dx, where aho=共ប/M␻0兲1/2is the harmonic oscillator length. This iden-tifies N共r0/aho兲2 as 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 den-sity n共0兲 with the chemical potential ␮0, as well as a steep-ening of the profiles at the trap edges. The calculated profiles agree with the analytical results expected in the TG and DDW limits.

We first determine the effect of the interactions on the frequency of the lowest compressional共breathing兲 mode of the trapped gas by a sum-rule approach. This mode is coupled to the ground state by the operator Xˆ2=兺i=1


xi2. This

makes it the easiest to probe in current experiments, as is excited by modulating the trap frequency and observed by following the time evolution of the width of the cloud by conventional density-imaging techniques. The frequency␻B

of the breathing mode satisfies an inequality that can be de-rived from the sum rule 关34–36兴 ␻B2ⱕm1/m−1, where 2m1=具关X2,关H,X2兴兴典 and 2m−1is the static response function. After some simplification, this yields the upper bound

B2ⱕ ⍀B2= − 2



2, 共2兲

where具x2典=N−1兰x2n共x兲dx. A closed-form expression of ⍀B

in Eq.共2兲 can be obtained by means of a scaling argument

whenever ␮共n兲 is of the form共n兲=␭n␥, resulting in ⍀B=␻0共2+␥兲1/2. In particular, in the TG case with␥= 2 this

yields ⍀B= 2␻0, and in the DDW case with ␥= 3 it yields ⍀B=


For intermediate interaction strengths, we have to resort to a numerical estimation of Eq.共2兲 using the LDA density

profile. The result is represented by the solid line in Fig.1, showing the smooth evolution of the breathing-mode fre-quency␻B from the TG to the DDW regimes. Figure1

rep-resents a useful tool to identify the interaction regime of the trapped dipolar gas through one of the best handled experi-mental probes available with cold atomic共molecular兲 quan-tum gases 关31,32兴. However, Eq. 共2兲 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 frequencies.

In关27兴, we have shown that the low-energy behavior of a

homogeneous dipolar Bose gas is well described by the Lut-tinger Hamiltonian, and we have determined the density de-pendence 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

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 N(r0/aho)2 4 4.2 4.4 4.6 4.8 5 (ω Β /ω0 ) 2 LL SR

FIG. 1. Squared frequency␻B2 of the breathing mode scaled to the trap frequency ␻0 vs the interaction parameter N共r0/aho兲2, as

calculated from two models. Solid line: sum-rule approach, Eq.共2兲

共SR兲. Symbols: Luttinger-liquid hydrodynamics, Eq. 共6兲 共LL兲.



Luttinger relations uK = M−1n and u/K=−1


em-bodying Galilean invariance关37,38兴. We now assume that in

a slowly varying external trapping potential, the dipolar Bose gas can be described by a Luttinger liquid Hamiltonian


−R R dx 2␲

u共x兲K共x兲共␲⌸兲 2+ u共x兲 K共x兲共⳵x␾兲2

, 共3兲

where u共x兲 and K共x兲 now depend on position via the LDA n共x兲. They are related through u共x兲K共x兲=M−1n共x兲 and u共x兲/K共x兲=兩␲−1

n共n兲兩n=n共x兲, extending the Luttinger-liquid

relations to the weakly inhomogeneous system. In Eq.共3兲,␾

and⌸ satisfy canonical commutation relations 关␾共x兲,⌸共y兲兴 = i共x−y兲, leading to the equations of motion

t共x,t兲 =u共x兲K共x兲⌸共x,t兲, 共4兲

␲⳵t⌸共x,t兲 =x



. 共5兲

These equations of motion must be complemented by bound-ary conditions expressing that no current flows across the edges—i.e., j共⫾R,t兲=0. Sincen = −x␾/␲, the continuity

equation ⳵t共␦n兲+xj = 0 leads naturally to j =t␾/␲ and

al-lows us to rewrite the boundary conditions as␾共−R兲=␾0and ␾共R兲=␾1关39兴.

Equations 共4兲 and 共5兲 possess a stationary solution

x⬀K共x兲/u共x兲. This solution actually describes the addition

of one particle to the system ␾共x兲→共x兲−g共x兲/g共R兲 where g共x兲=兰−R

x 关K共y兲/u共y兲兴dy. A consequence is that

0and ␾1 are not independent, but related through ␾1−␾0= −␲N with N being the number of particles added to the system. This form generalizes the bosonization formula derived in the case of a homogeneous system 关37兴. Alternatively, the

static solution can be derived by considering the effect of a perturbation to the trapping potential关40兴. Combining Eqs.

共4兲 and 共5兲 and using linearity to search for solutions of the

form␾共x,t兲=0−␲Ng共x兲/g共R兲+兺nⱖ1␸n共x兲eintwe find

− Mn2␸n= n共x兲x关⳵n„n共x兲…xn兴, 共6兲

with the boundary conditions ␸n共⫾R兲=0 for the discrete

Fourier components␸n.

While Eq.共6兲 is cast in a form identical to the

hydrody-namic equation for density excitations关36兴, we remark that

here it has been derived from HLL, Eq.共3兲. Thus, a

compari-son of the measured excitation frequencies with those pre-dicted from 共6兲 with our RQMC data provides a test of

Luttinger-liquid behavior in trapped 1D dipolar Bose gases within the validity of the LDA.

In the case of a harmonic trapping, one of the eigenfre-quencies in Eq. 共6兲 is obtained straightforwardly. Indeed,

substituting␸1共x兲⬀n共x兲 into Eq. 共6兲, differentiating Eq. 共1兲

with respect to x, and using the Luttinger-liquid relations, we find that␸1共x兲 is an eigenfunction of 共6兲 associated with the

eigenvalue ␻02. This particular solution is simply the Kohn mode or sloshing mode describing a center-of-mass oscilla-tion. Indeed, it can be recovered by expanding to first order in A the expression共x,t兲=(x − A cos共␻0t兲,0) which de-scribes a center-of-mass motion with rigid density. The

eigenfrequencies are exactly known also in the two asymptotic limits. Indeed, insertion of ␮共n兲⬀n␥ into Eqs. 共1兲 and 共6兲 yields solutions of the form ␸n共x兲

= An共1−x2/R2兲1/␥Cn

共1/␥+1/2兲共x/R兲, the associated eigenvalues being␻n2=␻02n关2+共n−1兲␥兴/2 关36,40兴, where Cn

共␣兲are Gegen-bauer polynomials and Annormalization factors 关41兴. Thus,

at the two opposite TG共␥= 2兲 and DDW 共␥= 3兲 limits one finds, respectively,␻n= n2␻0


and␻n= n共3n−1兲␻0

2/2. For in-termediate densities, we have solved Eq. 共6兲 using the

SLEDGE algorithm available online 关42兴, after inserting as ingredients the computed LDA density profiles from Eq.共1兲

and the analytical expression for ⳵n␮ obtained from the

RQMC fit, evaluated at the local density for different values of the interaction parameter N共r0/aho兲2. In the numerical so-lution we have taken special care of the finite-mesh-size ef-fects for best accuracy.

The values of the breathing frequency ␻B/␻0 obtained from Eq.共6兲 are represented in Fig. 1 by the solid symbols 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.关36兴, an equality sign holds in Eq. 共2兲.

The eigenfrequencies of the higher modes with n = 3, 4, and 5 are displayed in Fig. 2 as functions of N共r0/aho兲2, showing the same smooth crossover behavior between the two oppo-site TG and DDW regimes. The exact frequencies in these asymptotic regimes are recovered by the numerical calcula-tions.

In conclusion, we have predicted the evolution of the col-lective modes of a 1D dipolar Bose gas through the cross-over from the Tonks-Girardeau to the dipolar-density-wave regime. These modes and especially the breathing mode can be excited and measured by quite standard and reliable tech-niques. Our results are relevant to experiments, where they can be used to determine the interaction regime and extended to investigate the occurrence of decoherence effects in quan-tum applications关16兴. From the theoretical point of view, our

results confirm for the trapped gas the smooth crossover from the TG to the DDW regimes关27兴, as expected for weak

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 N(r0/aho)2 0 10 20 30 40 (ω n /ω 0 ) 2 FIG. 2. 共␻n/␻0兲2vs N共r

0/aaho兲2from Eq.共6兲. 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.



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 commensu-rate, though shallow, optical lattice.

M.L.C. would like to thank the Institut Henri Poincaré– Centre Emile Borel in Paris for hospitality and support. This work was supported by the Ministère de la Recherche共Grant No. ACI Nanoscience 201兲, by the ANR 共Grants Nos. NT05-42103 and 05-Nano-008-02兲, and by the IFRAF Institute.

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Si assuma che le condizioni del metallo siano tali che il numero degli elettroni che riescono ad abbandonarlo nell’unit` a di tempo sia molto pi` u piccolo del numero totale

Keywords and Phrases: quasilinear ordinary differential equation, Minkowski-curvature, Dirichlet boundary conditions, positive solution, existence, multiplicity, critical point

The definition of the Dirac operator perturbed by a concentrated nonlinearity is given and it is shown how to split the nonlinear flow into the sum of the free flow plus a