Lattice modulation spectroscopy of strongly interacting bosons in disordered and quasiperiodic optical lattices

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Lattice modulation spectroscopy of strongly interacting bosons in disordered and quasiperiodic optical lattices

ORSO, G., et al.

Abstract

We compute the absorption spectrum of strongly repulsive one-dimensional bosons in a disordered or quasiperiodic optical lattice. At commensurate filling, the particle-hole resonances of the Mott insulator are broadened as the disorder strength is increased. In the noncommensurate case, mapping the problem to the Anderson model allows us to study the Bose-glass phase. Surprisingly, we find that a perturbative treatment in both cases, weak and strong disorders, gives a good description at all frequencies. In particular, we find that the infrared-absorption rate in the thermodynamic limit is quadratic in frequency. This result is unexpected since for other quantities, like the conductivity in one-dimensional systems, perturbation theory is only applicable at high frequencies. We discuss applications to recent experiments on optical lattice systems and, in particular, the effect of the harmonic trap.

ORSO, G., et al . Lattice modulation spectroscopy of strongly interacting bosons in disordered and quasiperiodic optical lattices. Physical Review. A , 2009, vol. 80, no. 3

DOI : 10.1103/PhysRevA.80.033625

Available at:

http://archive-ouverte.unige.ch/unige:36009

Disclaimer: layout of this document may differ from the published version.

Lattice modulation spectroscopy of strongly interacting bosons in disordered and quasiperiodic optical lattices

G. Orso,¹A. Iucci,^2,3M. A. Cazalilla,^4,5and T. Giamarchi³

1Laboratoire Physique Théorique et Modèles Statistiques, Université Paris Sud, Bat. 100, 91405 Orsay Cedex, France

2Instituto de Física la Plata (IFLP)–CONICET and Departamento de Física, Universidad Nacional de La Plata, CC 67, 1900 La Plata, Argentina

3DPMC-MaNEP, University of Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva 4, Switzerland

4Centro de Física de Materiales (CSIC-UPV/EHU), Edificio Korta, Avenida de Tolosa 72, 20018 San Sebastián, Spain

5Donostia International Physics Center (DIPC), Manuel de Lardizábal 4, 20018 San Sebastián, Spain 共Received 21 July 2009; published 30 September 2009兲

We compute the absorption spectrum of strongly repulsive one-dimensional bosons in a disordered or quasiperiodic optical lattice. At commensurate filling, the particle-hole resonances of the Mott insulator are broadened as the disorder strength is increased. In the noncommensurate case, mapping the problem to the Anderson model allows us to study the Bose-glass phase. Surprisingly, we find that a perturbative treatment in both cases, weak and strong disorders, gives a good description at all frequencies. In particular, we find that the infrared-absorption rate in the thermodynamic limit is quadratic in frequency. This result is unexpected since for other quantities, like the conductivity in one-dimensional systems, perturbation theory is only applicable at high frequencies. We discuss applications to recent experiments on optical lattice systems and, in particular, the effect of the harmonic trap.

DOI:10.1103/PhysRevA.80.033625 PACS number共s兲: 03.75.Lm, 71.23.⫺k, 61.44.Fw I. INTRODUCTION

In recent years, developments in the field of ultracold atomic gases have considerably enlarged the possibilities for exploring the physics of strongly correlated systems关1兴. For instance, the study of quantum phase transitions, a subject of continuous theoretical interest, has been strongly stimulated by the experimental observation of the superfluid to Mott insulator transition using optical lattices 关2兴. Indeed, ultra- cold atom systems offer us an unprecedented control over the system parameters, which can allow us to ultimately under- stand the physics of very hard problems such as the phase diagram of the Hubbard model in two dimensions关3兴.

A particularly interesting and fertile arena is the study of disordered ultracold atoms. Random potentials can be intro- duced in a controlled way by laser beams generating speckle patterns关4–9兴or by loading in an optical lattice a mixture of two kinds of atoms, one heavy and one light. When the heavy atoms become randomly localized in the lattice, they will act as impurities for the lighter atoms 关10兴. Another available technique is to superimpose two optical lattices with incommensurate periodicities, thus, generating a quasi- periodic potential关11兴. The quasiperiodic lattices as well as the speckle patterns have been recently used in the experi- mental efforts to observe the effects of Anderson localization in dilute Bose gases expanding in highly elongated traps 关12,13兴. Since Anderson localization is a single-particle ef- fect, the next logical step is to study the interplay of disorder and interactions in strongly interacting ultracold atomic sys- tems 关14–17兴. The latter may be accessible by tuning inter- particle interactions using Feshbach resonances 关18兴 or by loading the atoms in sufficiently deep optical lattices 关9兴 and/or strongly confining them to low dimensions in tight traps关1兴.

In the context of the efforts described above, one of the experimental challenges is to observe in ultracold gases clear

signatures of the theoretically predicted transition from a su- perfluid to the Bose-glass phase 关19–21兴. A pioneering step in this direction was recently taken by the Florence group by using lattice modulation spectroscopy 关11兴. This technique consists in heating an ultracold gas loaded in an optical lat- tice by periodically modulating the depth of the lattice 关22–24兴. When perturbed in this way, the gas is driven out of equilibrium and absorbs energy. When the perturbation is switched off, and after rethermalization, the broadening of the momentum distribution around zero momentum is taken as a measure of the energy absorbed by the system during the lattice modulation 关22兴. On the theory side, for nondisor- dered lattices, the calculation of the energy absorption rate due to the lattice modulation was first performed analytically within linear-response theory by some of the present authors 关23兴. These results were confirmed and extended beyond lin- ear response using time-dependent density-matrix renormalization-group methods, both for the case of bosons 关24兴and fermions关25兴. More recently, for disordered optical lattices, the energy absorption rate has been numerically cal- culated using full diagonalization in small systems 关26,27兴. Other methods that have also been discussed in the literature for detecting signatures of the effect of disorder 共or quasip- eriodicity兲 on interacting boson systems in one dimension focus on the momentum distribution 关16,28兴, the expansion dynamics in quasiperiodic potentials关15兴, and the dipole os- cillations in the presence of defects关29兴.

Let us consider a one-dimensional disordered Bose gas described by the following Hamiltonian:

H= −J

兺

_j ^{共b†j+1bj+ H.c.兲+U}₂

兺

j

nj共nj− 1兲+

兺

j

共⑀j+V^ho_j 兲nj, 共1兲 where bj denotes the boson annihilation operator at sites j, n_j=b^†_jb_jbeing the local density. Here JandUare the usual PHYSICAL REVIEW A80, 033625共2009兲

1050-2947/2009/80共3兲/033625共10兲 033625-1 ©2009 The American Physical Society

parameters of the Bose-Hubbard model corresponding to the tunneling rate and the on-site repulsion 共U⬎0兲. The last term in the right-hand side of Eq. 共1兲accounts for the pres- ence of both the harmonic trapV_j^hoand the disorder potential

⑀j.

The distribution of the on-site energies ⑀j in Eq.共1兲de- pends on the specific choice of the random 共or pseudoran- dom兲 potential. In this work, we extensively compare two cases

共a兲⑀j uniformly distributed in关−⌬,⌬兴, 共2兲 共b兲⑀j=⌬cos共2␲^j␴兲, ␴ irrational, 共3兲 where ⌬ measures the strength of the disorder. From the experimental point of view, speckle patterns and lattice con- taining heavy atom impurities can be modeled by case 共a兲, whereas quasiperiodic potentials obtained by superimposing two optical potentials with incommensurate periodicities d₁/d₂=␴ are described by case共b兲.

In this work, we assume that the hopping amplitude Jin Eq. 共1兲 is modulated periodically in time according to J共t兲

=J+␦^Jcos␻^{t, where}␻is the modulation frequency. We cal- culate the energy absorption rate within linear-response theory, which is valid for weak lattice modulations␦JⰆ J.

We shall restrict ourselves to the strongly repulsive re- gime of Eq. 共1兲, where the on-site repulsion is large com- pared to both the hopping amplitude and the disorder strength 共UⰇJ,⌬兲. We first discuss systems in the thermo- dynamic limit by setting V^ho_j = 0. Effects of the harmonic trapping potential will be discussed later, in Sec.V. We also setប= 1 to simplify the notation.

In the thermodynamic limit, both the numberNof bosons and the length M of the chain diverge. The ratio␯^=N/M is instead finite and corresponds to the filling factor, i.e., the average number of bosons per lattice site. We must distin- guish between two physically different situations. Forincom- mensurate fillings␯⬍1, the system has gapless excitations.

Thus, a good approximation to the absorption rate at fre- quencies ␻ⰆU can be obtained by formally takingU→+⬁

and mapping the resulting hard-core bosons to noninteracting fermions using a Jordan-Wigner transformation 共see, e.g., Ref.关30兴兲. The resulting single-particle problem can be eas- ily solved for a given choice of ⑀i and V^ho and the energy absorption obtained. On the other hand, for unit filling 共␯

= 1兲, the ground state is a Mott insulator with a gap. In this case, the elementary excitations are particles and holes关23兴 and, in a homogeneous system, the absorption can only occur at frequencies␻⬃U. In the absence of disorder, the response of the system to the lattice modulation has been computed using degenerate perturbation theory within the subspace of particle and hole excitations关23兴. The same methods can be generalized to deal with disordered case, as shown below.

The resulting general picture of the energy absorption is depicted in Fig. 1. The low frequency 0⬍␻⬍W, where W

⬃max共4J, 2⌬兲is the effective bandwidth, is only present for incommensurate fillings,␯⬍1. A second distribution is cen- tered at frequency␻^=Uand comes from particle-hole exci- tations generating doubly occupied sites. The width of this absorption line is also given byW.

The paper is organized as follows. In Sec. II, we present the general formalism needed to calculate the absorption rate for incommensurate and unit filling. In Secs.IIIandIV, we present our analytical and numerical results obtained for the random and the quasiperiodic potential, respectively. In Sec.

V, we discuss the effects of a trapping potential. Finally in Sec. VI, we provide our conclusions. A derivation of the general formula关see Eq.共9兲兴for the absorption rate valid for weak disorder is given in the Appendix兲.

II. ENERGY ABSORPTION: LINEAR-RESPONSE THEORY

For weak perturbations, corresponding to␦JⰆ J, the en- ergy absorption rate E˙

␻ can be calculated using the linear- response theory. The general formula has been first derived in Ref.关23兴and in the presence of disorder it takes the form

E˙

␻=1 2␦^J0

2␻Im关−␹K共␻兲兴, 共4兲

where␹K共␻兲is the Fourier transform of theretarded corre- lation function ␹K共t兲= −i⌰共t兲具关K共t兲,K共0兲兴典 of the hopping operatorK= −兺j共b^†_j+1bj+ H.c.兲, being ⌰共t兲the step function.

In Eq. 共4兲, the bar means average over different disorder 共Sec. IV兲or quasiperiodic 共Sec. III兲realizations.

In general, the calculation of the correlation function in Eq. 共4兲 is a complicated many-body problem. However, in the limit of strong repulsion whereUⰇJ,⌬, calculations are considerably simplified by the fact that we can accurately restrict ourselves to work within a subspace of the total Hil- bert space, whose detailed structure depends on the filling. In the case of an incommensuratefilling共i.e., ␯⬍1兲, this sub- space can be described in terms of noninteracting fermion states共that is, Slater determinants兲. For commensurate filling 共i.e.,unit filling,␯= 1兲, we can restrict ourselves to the sub- space with one particle and one hole excitation, as described below.

A. Incommensurate filling

For filling ␯⬍1 and large on-site repulsionJ,⌬ⰆU, we take the hard-core limit U→+⬁. Bosons are then mapped ontononinteractingspinless fermions via the Jordan-Wigner transformation,

0 W

W

U ω

F

FIG. 1. Sketch of the full absorption spectrum at incommensu- rate filling. Here,W⬃max共4J, 2⌬兲is the effective bandwidth. The peak at ␻⬃U stems from particle-hole excitations, whereas the low-frequency absorption appears as a consequence of the forma- tion of a Bose glass at incommensurate filling.

cj= exp

冋

^i␲

^兺

^k=1j−1nk

册

^{bj, 共5兲}

where c_j satisfy fermionic commutation relations 兵c_j,c_j其= 0 and兵cj,c^†_j其= 1. Under the above transformation, the Hamil- tonian 共1兲is mapped onto the single-particle Hamiltonian

H⬘^{= −J}

兺

_j ^{共cj+1† cj+ H.c.兲+}

兺

_j ^{⑀jc†jcj, 共6兲}

where the on-site repulsionU has disappeared and the hop- ping operator K in Eq. 共4兲 is now given by K⬘^{= −兺j共c}j+1

† cj

+ H.c.兲. Notice that the mapping of observables that are non- local in space is far less trivial, an example is the momentum distribution studied in Ref. 关28兴 for disordered hard-core bosons in one dimension.

After some algebra, the absorbtion rate共4兲becomes E˙

␻= ␦^J02␲␻

2

兺

_␣,␤^{K␣␤关f共␧␣兲−f共␧␤兲兴␦共␻+␧␣−␧␤兲, 共7兲}

where the matrixK is defined as

K_␣␤=

冏 ^兺

^{j 关␺␣*共j+ 1兲␺␤共j兲+␺␣*共j兲␺␤共j+ 1兲兴}

冏

^{2. 共8兲}

In the previous expressions␧_␣and␺_␣共j兲are the eigenvalues and eigenfunctions of H⬘, respectively. In Eq. 共7兲, f共␧兲

=兵exp关共␧−␮兲/T兴+ 1其⁻¹ is the Fermi-Dirac distribution func- tion at a temperatureTand chemical potential␮. The latter is fixed by the normalization condition ␯⁼兺_␣f共␧_␣兲. At zero temperature, the only relevant processes in Eq. 共7兲 corre- spond to transitions from an occupied level 关with energy

␧_␣⬍␮共T= 0兲兴to an unoccupied level关␧_␤⬎␮共T= 0兲兴. In par- ticular, for unit filling the absorption 共7兲 vanishes, consis- tently, with the fact that a Mott insulator can only absorb at much higher frequencies ␻⬃U.

In the absence of disorder关i.e., for ⑀i= 0 in Eq.共6兲兴, the hopping modulation commutes with the HamiltonianH⬘^and, therefore, the absorption rate vanishes to all orders, even beyond linear response. In the above expression, this is re- flected in that, for⑀i= 0, the eigenstates ofH⬘ become plane waves, ␺k共j兲⬀e^ikj, with energy dispersion ⑀k= −2Jcosk 共k being the lattice momentum兲. Thus, the matrix共8兲is diago- nal, i.e., Kkk⬘= 4␦k,k⬘cos²k, which, together with the factor f共␧k兲−f共␧k⬘兲in Eq.共7兲makesE˙

␻ vanish.

At weak disorder共i.e.,JⰇ⌬兲, the absorption rate共7兲can be evaluated using the perturbation theory共the details can be found in the Appendix兲, which yields

E˙

␻=␦^J02␲␻

2M

兺

k,k⬘ 兩V_k−k⬘兩²

J² 关f共␧k兲−f共␧k⬘兲兴␦共␻⁺␧k−␧k⬘兲, 共9兲 whereVk=_冑¹_M兺_j=0^M−1e^ikj⑀j is the Fourier transform of the dis- order potential. Equation共9兲shows that the perturbation ex- pansion in disorder is well defined provided the Fourier transformVkis finite.

Let us finally consider the so-called atomic limit, which corresponds toJⰆ⌬共yet⌬ⰆU兲. In this limit, tunneling can be neglected and the eigenstates are given by ␺m共j兲=␦jm.

From Eq.共8兲, we findKjj⬘= 1 ifjandj⬘are nearest neighbor and zero otherwise. The absorption rate共7兲thus reduces to

E˙

␻= ␦^J0 2␲␻

2 _r=⫾1

兺 兺

j=0 L−1

关f共␧j兲−f共␧j+r兲兴␦共␻⁺␧j−␧j+r兲.

共10兲 In Sec. III, we shall explicitly compare the results obtained using exact numerical diagonalization with the above results obtained both in the limit of weak共9兲and strong共10兲disor- ders. Finally, it is important to emphasize that Eq. 共7兲does not account for particle-hole excitations, which are relevant at much higher frequencies␻⬃U. These excitations become particularly important at unit filling共␯^{= 1}兲, when the strongly repulsive Bose gas becomes a Mott insulator and the absorp- tion at low frequency ␻ⰆU predicted by Eq. 共7兲 vanishes because there are no empty sites 共i.e., holes兲 in the ground state. The contribution to the absorption from particle-hole excitations at unit filling will be discussed next.

B. Unit filling

We next turn our attention to thecommensuratecase with

␯= 1. In the strong-coupling regime UⰇJ,⌬, the system is deep in the Mott insulator phase, corresponding to exactly one particle per site. In Ref.关23兴, it was shown that for clean systems the absorption rate is zero at low frequencies and exhibits a narrow peak of width⬃Jcentered about␻^{=U. In} this section, we consider the broadening of such peak due to a disorder or quasiperiodic potential⑀i.

In order to obtain the energy absorption rate within linear response, we first use the spectral decomposition of the cor- relation function ␹K共␻兲in terms of the exact eigenstates of the unperturbed Hamiltonian. This yields the following ex- pression for the energy absorption:

E˙

␻=␦^J02␻␲

2

兺

_n ^{兩具⌿n兩K兩⌿0典兩2␦共␻+E0−En兲, 共11兲}

where 兩⌿n典 are the eigenstates of the original Hamiltonian 共1兲 with energies En, and 兩⌿0典=兩1 , 1 , . . . , 1典 is the ground state in the Fock representation corresponding to one boson per lattice site. The low-energy states are particle-hole exci- tations 兩␾共m,j兲典=_冑¹₂b_m^†b_m+j兩⌿0典 corresponding to double oc- cupation at sitemand an empty sitem+j. These excitations are all degenerate with energyUin the absence of tunneling and共i.e., for⌬=J= 0兲.

For finite values of ⌬ and J, the eigenstates ⌿n can be calculated using degenerate perturbation theory by writing 兩⌿n典=兺m,jf_m,j兩␾共m,j兲典, where the coefficients satisfy

兺

m⬘^,j⬘

具␾共m,j兲兩H兩␾共m⬘^,j⬘^兲典fm_⬘,j⬘=Ef_m,j. 共12兲

The above matrix element and the energy absorption can be computed by taking into account that 具␾共m,j兲兩K兩⌿0典=

−

冑

2␦j,⫾1, within the subspace containing just one particle and one hole. Hence, the matrix elements in Eq.共11兲corre- spond to 具⌿n兩K兩⌿0典= −兺_m,r=⫾1

冑

2f_m,r. Notice that a similar technique has been recently employed in Ref. 关31兴to calcu-

LATTICE MODULATION SPECTROSCOPY OF STRONGLY… PHYSICAL REVIEW A80, 033625共2009兲

033625-3

late the dynamic structure factor of a one-dimensional Mott insulator in a parabolic trap.

In the “atomic limit” where the tunneling can be ne- glected, Eq. 共11兲simplifies considerably. Localized particle- hole excitations become the exact eigenstates 兩⌿n典

=兩␾共m,j兲典with energyE₀+U+␧m−␧m+j, whereE₀=兺_i=1^M ␧iis the energy of the ground state for a given realization of ⑀i. The absorption rate共11兲then takes the form

E˙

␻=␦^J0 2␲␻

兺

r=⫾1

兺

m=1 M

␦共␻^−U+␧m−␧m+r兲, 共13兲

which can be readily evaluated numerically once the disorder potential is known. We emphasize that Eq.共13兲assumes that the system is a Mott insulator with exactly one atom per site.

This condition is easily satisfied in the strong-coupling re- gimeJ,⌬ⰆU, where the disorder only broadens the absorp- tion spectrum without affecting the nature of the ground state. A finite response at low frequency only occurs for stronger⌬⬃U, where the system is close to the Bose-glass transition. In this limit, however, the random 共or quasiperi- odic兲 potential changes the occupation numbers at different sites, creating holes and doublons, so the ground state is no longer given by the Fock state兩⌿0典and Eq.共13兲ceases to be valid.

III. RESULTS FOR A DISORDER POTENTIAL In this section, we assume that the on-site energy⑀iin Eq.

共1兲are random numbers uniformly distributed within the in- terval 关−⌬,⌬兴. The absorption rate 共4兲 can thus be conve- niently recast as

E˙

␻=M␦^J02F, 共14兲 whereFis a dimensionless function that can be numerically evaluated. The results for incommensurate and commensu- rate cases are described below.

A. Incommensurate filling

As stated above, the absorption rate is calculated numeri- cally at zero temperature starting from Eqs. 共7兲 and共8兲. In Fig. 2, we plot the frequency dependence of the response functionFfor differentfillings and increasing values of dis- order ⌬= 0.1J 共upper panel兲, ⌬=J 共central panel兲, and ⌬

= 5J共lower panel兲. Since, in the fermionic representation, the Hamiltonian of Eq. 共6兲 is particle-hole symmetric, the ab- sorption rate in Eq. 共7兲is unchanged under the transforma- tion ␯→1 −␯, so we restrict our discussion to fillings ␯ ⱕ1/2.

A noticeable feature of Fig.2is that the response at high frequencies isindependent of the filling factor. In this limit, the relevant processes contributing to the absorption mainly

0 0.0005 0.001 0.0015

0 1 2 3 4 5

F

ω/J

∆ = 0.1J ν= 0.1

ν= 0.2 ν= 0.3 ν= 0.4 ν= 0.5

0 0.05 0.1 0.15

0 1 2 3 4 5

F

ω/J

∆ =J ν= 0.1

ν= 0.2 ν= 0.3 ν= 0.4 ν= 0.5

0 0.2 0.4 0.6 0.8

0 5 10

F

ω/J

∆ = 5J ν= 0.1

ν= 0.2 ν= 0.3 ν= 0.4 ν= 0.5 (b)

(a)

(c)

FIG. 2.共Color online兲Energy absorption rate of hard-core bosons in arandompotential: the response functionF关see Eq.共14兲兴is plotted versus modulation frequency for different filling factors and increasing values of disorder strength:⌬= 0.1J共top panel兲,⌬= J, and⌬= 5J.

Calculations are done on a ring ofM= 500 lattice sites yielding negligible finite-size effects. The number of disorder realizations used was N_r= 500.

involve transitions from states far below the Fermi level关i.e., 共␧_␣Ⰶ␮兲兴into empty states far above it关i.e.,共⑀_␤Ⰷ␮兲兴. As a result, the Fermi-Dirac distributions in Eq.共7兲become irrel- evant, and thus any dependence on the value chemical po- tential ␮disappears.

However, at low frequencies, the absorption rate vanishes quadraticallywith frequency for any strength ⌬ of the dis- order. To understand this, let us expand in Eq.共7兲the distri- bution functions f共␧_␣兲−f共␧_␤兲⯝共␧_␣−␧_␤兲⳵_␧f共␧_␣兲. Taking into account that, at zero temperature,⳵_␧^f共␧兲= −␦共␮⁻␧兲, we find that F⯝C␻², where the constant

C=␲ 2

兺

␣⫽␤K_␣␤␦共␮⁻␧_␣兲␦共␮⁻␧_␤兲 共15兲 can be evaluated numerically. From Eq.共15兲, it can be seen that the constantCis nonzero provided there are nonvanish- ing matrix elements K_␣␤ connecting two states ␣ ^and␤ ^at the Fermi level␧_␣=␧_␤=␮^.

In the limit of weak disorder corresponding to ⌬ⰆJ, the absorption can be evaluated using Eq. 共9兲, where 兩Vk兩²

=⌬²/3M, as follows from the expression for the Fourier transform of the disorder potential V_k. Going to the thermo- dynamic limit and introducing the density of states ␳共␧兲

=共2␲^J

冑

1 −␧²/4J²兲⁻¹, we find F= ␻⌬²

24␲^J4

冕

a

b d⑀

冑

1 −⑀²/4J²

冑

1 −共⑀⁺␻兲²/4J², 共16兲 where a= max共−2J,␮⁻␻兲 and b= min共␮^{, 2J−}␻兲. Here the chemical potential ␮ is related to the filling factor by ␮=

−2Jcos␲␯. By expanding Eq. 共16兲 at low frequencies, we again obtain that the quadratic behavior discussed above

F=⌬²␲

6J²␳共␮兲²␻^{2= ⌬2} 24␲^J4

␻²

sin²共␲␯兲. 共17兲 It should be noticed that the right-hand side of Eq.共17兲 di- verges in the limit of vanishing lattice filling␯→0 because the density of states␳共␧兲has a Van Hove singularity at zero energy in one dimension. Thus, the low filling limit, the qua- dratic behavior of Eq.共17兲is only recovered at increasingly low frequencies, as shown in Fig.2 共upper panel兲.

In Fig. 3, we show a comparison, in the limit of weak disorder共⌬= 0.01J兲, of the numerical results共open symbols兲 obtained using exact diagonalization with the analytical ex- pression of Eq. 共16兲 共continuous lines兲. The agreement is indeed very good over the entire frequency range. In the inset, it is demonstrated that numerical results are consistent with the quadratic behavior of Eq.共17兲, expected at low fre- quencies.

On the other hand, in the opposite limit of strong disorder

⌬Ⰷ J, the tunneling can be neglected. In this limit, the ab- sorption rate can be evaluated directly from Eq.共10兲. Taking into account that the on-site energies⑀j at different sites are completely uncorrelated, we find

F=␲␻

冕

−⌬

␮¯

d⑀␳^¯共⑀兲

冕

␮¯

⌬

d⑀⬘^¯␳共⑀⬘兲␦共␻⁺⑀⁻⑀⬘兲, 共18兲 where␮^¯ and^¯␳ ^{are the}disorder-averagedchemical potential

and density of states, respectively. In the random potential, the latter is constant and given by^¯␳共⑀兲= 1/2⌬, and therefore

␮^¯=共2␯− 1兲⌬. Using Eq. 共18兲, the following low-frequency behavior is obtained:

F= ␲

4⌬²␻关min共␮^¯,⌬−␻兲− max共−⌬,␮^¯ −␻兲兴. 共19兲 This behavior is exhibited by the numerics, as shown in Fig.

2 共lower panel, see also discussion further below兲. Further- more, in the low-frequency limit共19兲, we again recover the quadratic behavior described above on general grounds

F= ␲

4⌬²␻^2. 共20兲

Notice however that the proportionality constant is now in- dependent of the filling factor. In Fig. 4, a more detailed comparison of the numerics with the analytical result of Eq.

0 1 2 3 4

ω/J

0 5e−06 1e−05 1.5e−05

2e−05 ν=0.5

ν=0.3

0 0.1 0.20 5e−08 1e−07

FIG. 3. 共Color online兲Comparison between numerics共symbols兲 and analytics 关Eq. 共9兲, solid line兴 for weak disorder. Here, ⌬/J

= 0.01 and we consider two filling factors ␯= 0.3 and␯= 0.5. The length of the chain isM= 2000 and the number of disorder realiza- tions isN_r= 2000. The inset is a zoom of the low-frequency regime, where the absorption rate is quadratic in frequency as given by Eq.

共17兲.

0 50 100 150 200

ω/J

0 0.2 0.4 0.6 0.8 1

F

ν=0.3 ν=0.5

0 200

0.03

FIG. 4. 共Color online兲Comparison between numerics共symbols兲 and analytics关solid line, Eq.共19兲兴forstrongdisorder. The value of the disorder is ⌬/J= 100 and the filling factors are␯= 0.3 and ␯

= 0.5. The length of the chain isM= 2000 and the number of disor- der realizations isN_r= 2000. In the inset, we compare our numerics with the quadratic expansion 共20兲 expected at low frequency. In both cases, the agreement is quite good.

LATTICE MODULATION SPECTROSCOPY OF STRONGLY… PHYSICAL REVIEW A80, 033625共2009兲

033625-5

共19兲for the limit of strong disorder is shown. In the inset, we also compare our numerical results with the quadratic behav- ior共20兲expected at low frequency. In both cases, the agree- ment is remarkably good.

B. Unit filling

The absorption rate for ␯= 1 is calculated numerically starting from Eqs.共11兲and共12兲. The result is shown in Fig.

5 as a function of frequency for different values of the dis- order strength⌬andJ= 0.01U.

In the absence of disorder共⌬= 0兲, the absorption rate can be evaluated analytically and the dimensionless functionFin Eq. 共14兲is given by关23兴

F=2␻

3J

冏

^sin

冋

^cos−1

冉

^␻6J−U

冊 册 ^冏

^{, for兩␻−U兩⬍6J,}

共21兲 and vanishes otherwise. The dashed line in Fig. 5 corre- sponds to our numerical result for⌬= 0, which fully agrees with the above formula.

As the strength of disorder is increased, we see from Fig.

5that the absorption spectrum becomes broader and progres- sively develops a trianglelike shape as the limitJⰆ⌬is ap- proached. This behavior can be obtained analytically starting from Eq.共13兲. In theJⰆ⌬limit, since the on-site energies at different lattice sites are uncorrelated, we obtain

F⯝ ␲␻

2⌬²

冕

_−⌬^{⌬ d⑀}

冕

_−⌬^{⌬ d⑀⬘␦共␻−U+⑀−⑀⬘兲, 共22兲}

where^¯␳共⑀兲= 1/2⌬ is the disorder-averaged density of states in the atomic limit introduced above. Upon integration, Eq.

共22兲yields

F⯝ ␲␻

2⌬²共2⌬−兩␻^−U兩兲, 共23兲 showing that the line shape of the absorption rate in the atomic limit is approximately triangular and vanishes at 兩␻

−U兩= 2⌬ forJⰆ⌬.

IV. QUASIPERIODIC POTENTIAL

In this section, we assume that the external potential in Eq. 共1兲 is ⑀j=⌬cos共2␲^j␴兲 关15–17兴, being ␴ an irrational number. This quasiperiodic potential distribution is realized experimentally by superimposing two different periodic po- tentials with incommensurate lattice periods关11兴. Differently from the disorder potential considered in the previous sec- tion, where all states are localized 共in the thermodynamic limit兲for an arbitrarily small amount of disorder, in the qua- siperiodic case there is a phase transition关32兴at⌬c= 2J: for

⌬⬍2Jall states areextended while for⌬⬎2Jall states are exponentiallylocalized.

As described below, we find that the absorption spectra of bosons in the quasiperiodic potential are remarkably differ- ent from the spectra described in Sec. III for the bosons moving on a disorder potential.

A. Incommensurate filling

For lattice fillings␯⬍1 共i.e., less than a boson per site兲, we calculate the absorption rate numerically starting from Eqs. 共7兲 and共8兲. We restrict to zero temperature and fix ␴

= 0.771 452 45, which is relevant to the experiments carried out by the Florence group.

In Fig. 6 we plot the frequency dependence of the re- sponse function 共14兲 for increasing values of the disorder strength ⌬. Compared to Fig. 2, we see that the absorption spectra in quasiperiodic potential exhibit a much richer struc- ture. Interestingly, the response function becomes very large 共and actually diverges兲 when ␻ matches special values.

Moreover, for a given filling factor, the absorption rate is nonzero only within a certain range of frequencies.

These features can again be understood in the limits of weak and at strong disorder, where results can be obtained analytically. In the limit ⌬Ⰶ J, the absorption rate can be calculated starting from Eq. 共9兲 and taking the thermody- namic limit. This yields共␧k= −2Jcosk兲

F=␲␻

2

冕

^2dk␲ dk⬘ 2␲

兩V共k−k⬘兲兩²

J² ␦共␻⁺␧k−␧k⬘兲

⫻⌰共␮−␧k兲⌰共␧k⬘−␮兲, 共24兲 where the integration over momenta is restricted to 关0 , 2␲兴. Using that the共modulus square of the兲Fourier transform of the disorder potential is兩V共k兲兩²= limM→⬁兩VM共k兲兩², where

兩VM共k兲兩²= ⌬²

4M

冏

^{eei共ik+k共k+k0兲0M兲− 1− 1+eei共ik−k共k−k0兲0M兲− 1− 1}

冏

^{2, 共25兲}

and k₀= 2␲␴^{, for M}→⬁, the right-hand side of Eq. 共25兲 becomes a series of delta functions

0.7 0.8 0.9 1 1.1 1.2 1.3

ω/U

0 20 40 60 80

F

∆=0U

∆=0.03U

∆=0.06U

∆=0.1U

J=0.01U

FIG. 5. 共Color online兲Energy absorption rate in the Mott insu- lator phase for hard-core bosons in a random potential: the response functionF关see Eq.共14兲兴is plotted versus modulation frequency for fixed J= 0.01U and increasing values of the disorder strength

⌬/U= 0共black dotted line兲, 0.03, 0.06, 0.1. The length of the chain is M= 60 and the number of disorder realizations is N_r= 200. A broadening of the absorption peak is observed for increasing disorder.

兩V共k兲兩²=␲⌬²

2 _n=−⬁

兺

^{+⬁ 关␦共k+k0+ 2␲n兲+␦共k−k0+ 2␲n兲兴. 共26兲}

Inserting Eq.共26兲into Eq.共24兲, we obtain F=␻␲⌬²

4J²

冕

0 2␲ dk

2␲␦共␻⁺␧k−␧k+k₀兲

⫻⌰共␮−␧k兲⌰共␧k+k₀−␮兲. 共27兲 The integral in Eq. 共27兲can be evaluated using the identity asink+bcosk=ccos共k+␥兲, where c=

冑

a²+b² and tan␥⁼

−a/b. Since a= sink₀ and b= 1 − cosk₀, we obtain c

= 2 sin共k0/2兲 and tan␥= −1/tan共k0/2兲. The zeroes of the ␦ function in Eq. 共27兲 occur at k=k_⫾= −␥⫾arccos共␻/2Jc兲.

Hence, we obtain F= ⌬²

4J²

␻/2

冑

共2Jc兲²−␻²

兺

r=⫾⌰共−␰r兲⌰共␻⁺␰r兲, 共28兲 where ␰r=␧k_r−␮^{= −2Jcosk}r−␮^{. Equation} 共28兲 shows that the absorption is finite only in a range of frequencies given by the conditions ␰r⬍0 and ␰r+␻⬎0, where ␰r itself de- pends on ␻ through the␻ dependence ofkr. Moreover, the response diverges for␻= 2Jc, provided this frequency value is allowed for a given filling factor. Since c= 2 sin␲␴

= 1.315 76, the divergence occurs at ␻= 2.631 5J, as found numerically and shown in Fig. 6共upper panel兲.

For a potential strength comparable to the tunneling rate, i.e., ⌬⬃ J, the absorption spectrum develops very sharp

peaks as shown in the central panel of Fig. 6. These peaks gradually disappear as⌬increases and becomes much larger than J. In this regime, however, the energy absorption spec- trum becomes indeed rather similar to the case of weak qua- siperiodic potential, as can seen by comparing the upper and lower panels of Fig.6. This peculiar effect can be explained analytically starting from Eq. 共10兲, which applies in the atomic limit. Introducing the variable y= 2␲␴n, in the ther- modynamic limit, we obtain

F=␻␲

冕

0 2␲ dy

2␲␦关␻⁺⌬cosy−⌬cos共y+ 2␲␴兲兴

⫻⌰共␮⁻⌬cosy兲⌰共␻⁺⌬cosy−␮兲. 共29兲 The integral in Eq.共29兲becomes the integral in Eq.共27兲after a change of variablek=y+␲. We thus obtain

F= ␻/2

冑

共⌬c兲²−␻²_r=⫾

兺

^{⌰共␮+⌬coskr兲⌰共␻−⌬coskr−␮兲,}

共30兲 showing that the behavior of the absorption rate at strong quasiperiodic potential can be obtained from Eq. 共28兲 by simply replacing 2Jwith⌬.

B. Unit filling

We have obtained the absorption spectrum at unit filling numerically using Eqs.共11兲 and共12兲, for the same value of

0 1 2 3 4

ω/J

0 0.001 0.002 0.003 0.004 0.005

F

ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.5

∆=0.1J

0 1 2 3 4 5

ω/J

0 0.2 0.4 0.6 0.8

F

ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.5

∆=J

0 4 8 12 16 20

ω/J

0 0.4 0.8 1.2 1.6 2

F

ν=0.1 ν=0.2 ν=0.3 ν=0.4 ν=0.5

∆=10J

(b) (a)

(c)

FIG. 6. 共Color online兲 Energy absorption rate of hard-core bosons in a quasiperiodic potential with␴= 0.771 452 45: the response functionF关see Eq.共14兲兴is plotted versus modulation frequency for different filling factors and increasing values of disorder strength:⌬

= 0.1J共top panel兲,⌬=J, and⌬= 10J. Calculations are done on a ring ofM= 1200 lattice sites yielding almost negligible finite-size effects.

The number of disorder realization isN_r= 500.

LATTICE MODULATION SPECTROSCOPY OF STRONGLY… PHYSICAL REVIEW A80, 033625共2009兲

033625-7

␴= 0.771 452 45. The result is plotted in Fig. 7 for fixed J/U= 0.01 and increasing values of the quasiperiodic poten- tial strength⌬. The dashed line corresponds to the clean case

⌬= 0, where the absorption rate is given by Eq.共21兲.

We see that the shape of the absorption spectrum changes considerably as⌬increases. For sufficiently weak quasiperi- odic potential⌬ⱗ3 J, the spectrum does not become broad, but instead satellite peaks appear on the sides of the central absorption feature. For stronger quasiperiodic potentials, the central peak disappears and the spectrum develops a two hump structure. To understand these features, let us again focus on the atomic limit, where the absorption rate can be obtained analytically using Eq.共13兲. Introducing the variable y= 2␲␴ⁿ and passing to the continuum limit, we obtain the result

F= 2␻␲

冕

0 2␲ dy

2␲␦关␻^−U+⌬cosy−⌬cos共y+ 2␲␴兲兴. 共31兲 The integral共31兲can be readily evaluated using the identity asiny+bcosy=ccos共y+␥兲, where tan␥^{= −a}/b and c

=

冑

a²+b². From Eq. 共31兲 we have that c= 2 sin␲␴ and, therefore,

F= 2␻

冑

共⌬c兲²−共␻⁻U兲², 共32兲 showing that the absorption rate diverges at the edge, where 兩␻⁻U兩=⌬c. This means that the shape of the absorption spectrum changes completely going from weak to strong quasiperiodic potential, as obtained numerically and shown in Fig. 7. Finally, in comparing Eqs. 共23兲and 共32兲, we see that in a quasiperiodic potential the absorption spectrum is 共at least in the atomic limit兲narrower because c⬍2.

V. EFFECTS OF A PARABOLIC TRAP

In this section, we discuss the effects of a harmonic trap- ping potential V共z兲=m␻ho2 z²/2 on the absorption spectrum.

Here,mis the atom mass and␻hois the trapping frequency.

In this case, V^ho_j in Eq. 共1兲 is nonzero and given by V_j^ho

=␣^ho共j−M/2兲², where␣^ho=m␻ho2 d²/2,dbeing the lattice pe- riod.

We have repeated the calculations of the absorption spec- trum including V_j^ho and the result is shown in Fig. 8. For a system of hard-core bosons in the absence of disorder or quasiperiodic potential, the trap favors the formation of a Mott insulator in the center surrounded by a superfluid re- gion at the trap edges. This gives rise to a finite absorption at low frequency共see inset in the upper panel兲, which is related to the creation of excitations at the edge of the trap. By contrast, in a uniform system of hard-core bosons, as we have described in Sec. II, the energy absorption vanishes to all orders because the hopping operatorKcommutes with the Hamiltonian.

Let us next consider the effect of a small amount of dis- order or a weak quasiperiodic potential. Clearly, the Mott

0.7 0.8 0.9 1 1.1 1.2 1.3

ω/U

0 20 40 60 80

F

∆=0 U

∆=0.03U

∆=0.06U

∆=0.1U

J=0.01U

FIG. 7. 共Color online兲Energy absorption in the Mott insulator phase for bosons in a quasiperiodic potential with ␴

= 0.771 452 45: the response function F 关see Eq.共14兲兴 is plotted versus modulation frequency for fixedJ= 0.01Uand increasing dis- order strength ⌬/U= 0共black dotted line兲, 0.03, 0.06, 0.1. Here, convergence is achieved for system size L= 70. As disorder in- creases, the response function broadens and changes convexity, as predicted by Eq.共32兲.

0 0.5 1 1.5 2 2.5

0 5 10 15

F

ω/J

0 5 10 15 20 250

0.5 1 1.5 2 2.5

F

ω/J 0

0.05 0.1 0.15 0.2

0 1 2 3 4

F

0 1 2 3 4 50

0.05 0.1 0.15 0.2

F

N= 100 N= 200 N= 300 N= 400 0

0.001 0.002 0.003 0.004 0.005

0 1 2 3 4

F

Quasi-periodic

0 1 2 3 4 50

0.001 0.002 0.003 0.004 0.005

F

Disordered

∆= 10J

∆=J

∆= 0.1J

0 0.002 0.004

0 1 2

F

∆= 10J

∆=J

∆= 0.1J N= 200

∆= 0

FIG. 8. 共Color online兲Energy absorption rate of disordered and quasidisordered hard-core bosons in a parabolic trap. The response functionFis plotted versus modulation frequency for different fill- ings. Upper panel: ⌬= 0.1J, middle panel: ⌬=J, and lower panel:

⌬= 10J. The system size isM= 500, the number of disorder realiza- tions is N_r= 4000, and ␣^ho= 2.88⫻10⁻⁴J. The inset in the upper panel shows the low-frequency absorption in the absence of disor- der, which is finite for trapped gases. As ⌬ increases, this sharp peak fragments out and is no longer visible on the scale of the bulk contribution.

insulator at the center of the trap cannot absorb energy at low frequency, so the only contribution comes from the outer shell, where the filling factor is less than unity. In particular, the low-frequency peak arising from edge excitations frag- ments in multiple peaks with little spectral weight compared to the response from the bulk discussed in Secs. IIIandIV. We also see in Fig.8 that the behavior of the absorption spectrum at frequencies close to the bandwidth crucially de- pends on whether the applied potential is truly random or quasiperiodic. Whereas the disordered case exhibits a smooth behavior in the absorption up to the bandwidth 4Jwhere it falls to zero, the quasiperiodic one shows a sharp peak lo- cated at the bandwidth 4Jc, which resembles the divergence found in the corresponding homogeneous case. Note that the position of this peak is almost independent on the number of atoms in the tube and, therefore, the peak should be visible in a realistic experimental situation, where an average over a multiple tube setup with variable filling is performed 关33兴.

The same conclusion applies to the system with strong dis- order or quasiperiodic potential as can be observed in the lower panel in Fig. 8. For⌬⬃J共Fig. 8, middle panel兲, the peak structure in the quasiperiodic case is more complex and, thus, the averaging procedure will produce some round- ing off of the peaks. Still, the absorption can be considerably larger than in the disordered case and this difference should be clearly visible.

VI. CONCLUSIONS

In conclusion, we have investigated the energy absorbed by a disordered strongly interacting Bose gas in the presence of periodically modulated optical lattices. For filling factor less than one, the absorption rate has been calculated exactly in the hard-core limit via the Bose-Fermi mapping. For com- mensurate filling, corresponding to one boson per lattice site, the gas is a Mott insulator and can only absorb energy at much higher frequency 共on the order of the repulsive inter- actionU兲. The disorder-induced broadening of the absorption spectrum has been calculated by restricting to the subspace of particle-hole excitations.

We have performed extensive calculations comparing two different sources of disorder: a random potential, which is relevant for current experiments based on speckle patterns, and a quasiperiodic potential, which is obtained by superim- posing two optical lattices with incommensurate periods.

Our results indicate that the response of the gas to the lattice modulation significantly depends on the chosen source of randomness.

ACKNOWLEDGMENTS

This work was partly supported by ANR under Grant No.

08-BLAN-0165-01 and by the Swiss National Fund under MaNEP and Division II. A.I. gratefully acknowledges finan- cial support from CONICET and UNLP. During the initial stage of this project, G.O. was also supported by the Euro- pean Commission under Contract No. EDUG-038970.

M.A.C. was supported by MEC 共Spain兲 under Grant No.

FIS2007-066711-C02-02 and CSIC 共Spain兲 through Grant

No. PIE 200760/007.

APPENDIX

In this appendix, we shall derive the asymptotic formula 共9兲based on the perturbation theory for weak disorder. For clarity, we rewrite the general expression共7兲

E˙

␻=␦^J02␲␻

2

兺

k,k⬘Kkk⬘关f共␧k兲−f共␧k⬘兲兴␦共␻⁺␧k−␧k⬘兲, 共A1兲 using new indiceskandk⬘. Moreover, we find convenient to introduce the matrix elements

Akk⬘=

兺

j

关␺kⴱ共j+ 1兲␺k⬘共j兲+␺kⴱ共j兲␺k⬘^{共j+ 1兲兴, 共A2兲}

so thatKk,k⬘=兩A_k,k⬘兩².

In the absence of disorder⌬= 0, the eigenstates are plane waves ␺k

0共n兲=e^ikn/

冑

L with energy ⑀k

0= −2Jcosk. Therefore, from Eq. 共A2兲, we find

A_kk⁰_⬘= 2␦kk⬘cosk, 共A3兲 showing that the matrixA isdiagonal in momentum space.

Since the matrixK_kk⁰_⬘= 4␦k,k⬘cos²kis also diagonal, the ab- sorption rate 共7兲vanishes.

For⌬Ⰶ J, we formally expand the right-hand side of Eq.

共A2兲 in powers of the disorder strength A_k,k⬘=A_kk⁰_⬘+A_kk¹_⬘ +A_kk²_⬘+O共⌬³兲, so the matrix Ktakes the form

Kk,k⬘=K_kk⁰_⬘+A_kk⁰_⬘共A_kk¹_⬘+A_k¹_⬘k+A_kk²_⬘+A_k²_⬘k兲+兩A_kk¹_⬘兩²+ O共⌬³兲.

共A4兲 Taking Eq.共A3兲into account, we see that the onlynondiago- nalterm appearing in the expansion共A4兲is兩A_kk¹_⬘兩², which is on the second order in⌬. This term can be readily evaluated from Eq.共A2兲by applying the first-order perturbation theory for the eigenstates

␺k=␺k 0+_q

兺

⫽k

具␺q0兩V兩␺k0典

⑀k0−⑀q0 ␺q

0. 共A5兲

After a simple algebra, we obtain A_kk¹_⬘=具␺k0兩V兩␺k⁰⬘典

⑀k 0−⑀k⁰⬘

2共cosk− cosk⬘^{兲, 共A6兲}

which is valid up to the linear order in⌬. Finally, by using the dispersion relation ␧k

0= −2Jcosk, Eq. 共A6兲 further sim- plifies yielding

A_kk¹_⬘=具␺k 0兩V兩␺k⁰⬘典

J . 共A7兲

Substituting Eq. 共A6兲 into Eq. 共A1兲 and replacing the eigenstates by their zero-order values ⑀k=⑀k

0, we recover the asymptotic formula共9兲.

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033625-9

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