Ultracold Dipolar Bosons In Optical Lattices · Ultracold Dipolar Bosons In Optical Lattices ... r¶ealis¶es dans la manipulation d’atomes ultra-froids permettant le pi¶egeage

Ultracold Dipolar Bosons

In Optical Lattices

Mathieu Alloing

Master 2 Sciences de la Matiere,ENS Lyon/Universite Lyon 1, France

Supervisor : Maciek Lewenstein

At : ICFO The Institute of Photonic SciencesMediterranean Technology Park

Av. del Canal Olmpic s/n08860 Castelldefels (Barcelona), Spain

13 April - 2 August 2009

2

Abstract :

Since the first observation of Bose-Einstein condensation, significant progress in the manipulationof ultra-cold atoms have been achieved leading to the possibility of trapping ultracold bosonic atomsin optical lattices. Such systems allow one a complete control on the physical parameters providingan efficient experimental tool for mimicking condensed matter process and for the observation ofexotic quantum phases. While, in the presence of single species interacting via on-site interaction,the phase diagram presents only superfluid or Mott insulating phases, for long-range interactions ormultiple species, exotic phases such as supersolid or supercounterfluid appear. This report thereforepropose an eventual implementation with dipolar Bosons in 2D optical lattices, where the dipolesare anti-polarized (up/down) perpendicularly to the 2D plane. A possible extended Bose-HubbardHamiltonian is presented with the corresponding phase diagrams as well as a description in terms ofdipole flipping or super-exchange.

Resume :

Depuis la premiere observation d’un condensat de Bose-Einstein, d’important progres ont eterealises dans la manipulation d’atomes ultra-froids permettant le piegeage d’atomes bosoniques ultra-froids dans des reseaux optiques. De tels systemes autorisent un controle complet des parametresphysiques fournissant un outil experimental tres performants pour la simulation des processus dela matiere condensee et l’observation de nouvelles phases quantiques. Alors que le diagramme dephase d’une espece unique interagissant via une interaction intra-site presente seulement des phasessuperfluide and isolant de Mott, des phases exotiques telles que la supersolidite ou la superfluiditea contre-courant apparaissent dans le cas d’interactions a longue portee ou de la presence d’especesmultiples. Ce rapport propose ainsi une eventuelle application avec des bosons dipolaires orientes versle haut et le bas dans des reseaux optiques a 2 dimensions. Une extension possible de l’hamiltonien deBose-Hubbard est presentee avec les diagrammes de phases correspondants ainsi qu’une descriptionen terme de retournement de dipole ou de super-echange.

CONTENTS

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1. Optical lattices and simple Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Optical lattices and experimental realizations . . . . . . . . . . . . . . . . . . . 71.1.2 Band structures and contact interaction . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Simple Bose-Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2. Extended Bose-Hubbard model : Up-Down dipoles . . . . . . . . . . . . . . . . . . . . . . . 192.1 Dipolar interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Mean-field perturbative approach . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.2 Gutzwiller Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3. Flipping excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 Rewriting of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Diagonal Hamiltonian and µ+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Phase diagrams with µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Mean-field perturbative approach . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3.2 Gutzwiller Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Appendix 39

A. Matrix expression of the mean-field perturbative method . . . . . . . . . . . . . . . . . . . . 41

4 Contents

0. INTRODUCTION

The initial observation of Bose-Einstein condensation in dilute atomic gases in 1995, seventy yearsafter its prediction, has heralded in a new era in the study of many body quantum physics, that hasbrought with it new challenges and opportunities.

But while in the early days of atomic BEC experiments, the main focus was to investigate con-densate properties of matter waves like coherence, emphasis has more recently shifted to stronglyinteracting systems, which are much more in line with present interests in theoretical condensed mat-ter physics. In particular, as first pointed out in the seminal work of Jaksch et al. [9], stronglyinteracting systems can be realized with cold atomic gases in optical lattices, i.e. periodic arrays ofmicrotraps generated by standing wave laser light fields. This leads to Hubbard type lattice models,where atomic physics provides a whole toolbox to engineer various types of Hamiltonians for 1D, 2Dand 3D Bose and Fermi systems which can be controlled by varying external field parameters in atime dependent way. In addition, atomic physics provides systematic ways of loading these latticeswith atoms. A prominent example is the Mott insulator-superfluid quantum phase transition withcold bosonic atoms, as first observed in the milestone experiment by I. Bloch et al. [2]. More generally,it is expected that cold atoms in optical lattices will be developed in the coming years as a generalquantum simulator of lattice models, allowing experimental insight into phase diagrams for certainclasses of (toy) models (such as high-Tc superconductivity) and the observation of novel quantumphases.

Following these ideas, in the last few years, several researchers have pointed out that the anisotropicdipole-dipole interaction, acting between particles having a permanent electric or magnetic dipole mo-ment, should lead to novel kinds of degenerate quantum gases in the strongly interacting regime.And indeed the outcomes of the theoretical studies on the subject have proved the existence of nu-merous original physical behaviors such as density wave, pairsupersolid, supercounterfluid quantumphases [14, 10] or the presence of metastable states [21].

But researches so far mainly concerned single component gas of dipolar bosons with repulsivedipolar interactions, the purpose of my internship was thus to implement in this framework an extendedBose-Hubbard model in a 2D optical lattice with two species (up and down dipoles) in order to haveboth attractive and repulsive interactions. We will see in this report that such a system supportssupersolid and supercounterfluid quantum phases. The longtime purpose of this kind of system is alsoto find a correspondence with the dipole picture of excitons in semiconductors which will be part ofmy PhD work. My internship was mainly theoretical which explains that experimental applicationswill not be discussed in details. However, based on works of B. Lev et al. [11, 12], Hund A moleculesseem to be appropriate for an implementation of this kind of system (further details will be given inthe conclusion).

The topic of my internship is thus the theoretical study of the phase transitions of ultracoldup/down dipolar bosons in 2D optical lattice and this report is organized as follows. First a presen-tation of optical lattices, then basic ideas and calculation methods used to obtain phase diagrams areapplied on simple Bose-Hubbard model. Chapter two gets onto the core of the subject with develop-ment and study of the extended Bose-Hubbard Hamiltonian. In chapter three will be presented analternative description of the system and a presentation of the results and the possible further studies.

6 0. Introduction

1. OPTICAL LATTICES AND SIMPLE BOSE-HUBBARD MODEL

In this chapter we will explain what is an optical lattice, how it is capable to trap neutral atomsand the way it is realized in the laboratory. We will see that the Bose-Hubbard Hamiltonian is a verygood description of a system of Bose particles in an optical lattice, providing the temperature of thesystem is small enough, and we will see the ideas and calculations used to study the different phasesof such Hamiltonian.

1.1 Optical lattices

1.1.1 Optical lattices and experimental realizations

The basic tool to create ultracold lattice gases are optical potentials. The presence of the oscillatingelectric field ~E(~r, t) of a laser on an atom creates a time dependent dipole moment ~d on the atom.When the field oscillations are far off resonance (i.e. they do not cause any real transition in theatom), the induced dipole moment follows the laser field oscillations [5, 3]

di =∑

j=x,y,z

αij(ωL)Ej(~r, t), (1.1)

where di is the corresponding component of ~d(i = x, y, z), ωL is the laser frequency,and αij(ωL) denotesthe matrix elements of the polarizability tensor. The polarizability depends in general on the laserfrequency, and on the energies of the non-resonant excited states of the atom. One of these states(with excitation energy, say, E0 = ~ω0) is usually much closer to the resonance than the others; insuch case the polarizability becomes inversely proportional to the laser detuning from the resonance,∆ = ωL − ω0. Electronic energy undergoes in this situation a shift, ∆E, which is nothing else than aAC-version of the standard quadratic Stark effect. The energy shift is proportional to :

∆E(~r) =∑

i,j=x,y,z

αij(ωL)〈Ej(~r, t)Ej(~r, t)〉 ∝ I(~r)/∆ (1.2)

where the bra-ket denotes the averaging of the product of electric fields over the fast optical oscillations,and I(~r) is the laser beam intensity. The consequences of the above simple formula are enormous.The atom feels an optical potential Vopt(~r) = ∆E(~r), that follows the spatial pattern of the laser fieldintensity. This is the basis for optical manipulations and trapping of atoms. Atoms are attracted tothe nodes or anti-nodes of the laser intensity for blue detuned (ωL < ω0) or red detuned (ωL > ω1).Within a two level model, an explicit form of the optical potential may be derived by using the rotatingwave approximation, which is a reasonable approximation provided that the detuning ∆ = ωL−ω0 ofthe laser field is small compared to the transition frequency itself |∆| ¿ ω0. With Γ as the decay rateif the excited state, one obtains for |∆| ¿ Γ [5]:

Vopt(~r) =3πc2

2ω30

Γ∆

I(~r) (1.3)

which is respectively attractive or repulsive for red (∆ < 0) or blue (∆ > 0) detuning.Then to create an optical lattice one needs to create a spatially periodic potential [5]. This can

be generated by overlapping two or more counter-propagating laser beams depending on the desiredgeometry. Due to the interferences between for instance two laser beams with a λ wavelength anoptical standing wave with period d2D = λ/2 (see figure 1.1) is formed, in which the atoms can betrapped.

8 1. Optical lattices and simple Bose-Hubbard model

The simplest possible periodic optical potential is formed by overlapping two counterpropagatingbeams. For a gaussian profile, this results in a trapping potential of the form :

Vopt(r, z) = −V0e−2r2

ω2(z) sin2(kz) (1.4)

where k = 2π/λ is the wave vector of the laser light, ω(z) the beam size and V0 being the maximumdepth of the lattice potential including the prefactor of Eq. (1.3). The system studied in this reportconcerns 2D optical lattices and there are different ways to achieve this specific geometry. On figure 1.1the simplest set-up is described by the case a) where two pairs of counterpropagating lasers with samewavelength and intensity on perpendicular directions are used to create a potential of the type :

Vopt(x, y) = V0

(sin2(kx) + sin2(ky)

)(1.5)

The problem of this configuration is that the trapping along the z-direction is not very efficientand then will require an additional trapping potential (a magnetic field for instance). Set-up b) andc) allow one to avoid additional trapping potential while preserving an effective 2D geometry, in factone obtains a set of parallel 2D lattices physically independent from each other. Indeed for caseb) the third pair of lasers have the same wavelength but higher intensity which creates a potentialbarrier that isolates from tunneling in the third direction (or this direction). This effectively createsa stack of 2D layers which are isolated from inter-layer tunneling of particles but not from inter-layerdipolar interactions due to its long range character. Another configuration is case c) where z-directionlasers are tilted with an angle θ what moves the 2D lattices away from each other with a spacing ofd1D = d2D/sin(θ/2) while the periodicity on the 2D lattice is still d2D = λ/2. This configuration hasthe advantages to suppress tunneling and dipolar interactions between layers.

There are numerous ways to create lattices with different dimensionality and characteristics (onecan also add extra lasers with other wavelengthes, orientations or intensities to superpose super-lattices to the original one creating pseudo random site offset) leading to a total freedom of simulatingcondensed matter potentials.

z

yx

d2D

x

y

y

z

a)

y

x

z

b)

x

z

yq

q

z

d1D

c)

Fig. 1.1: Optical lattices.

1.1. Optical lattices 9

1.1.2 Band structures and contact interaction

For deep optical lattice potential, the confinement on a single lattice site is approximately harmonic.The atoms are then tightly confined at a single lattice site, with trapping frequencies ω. The energy~ω = 2Er (V0/Er)

1/2 of local oscillations in the well is on the order of several recoil energies Er =~2k2/2m which is a natural measure of energy scales in optical lattice potentials. We consider in thefollowing the single particle eigenstates in an infinite 2D periodic lattice (case a)). In such potentialas (1.5), in the limit V0 À Er, each well supports a number of vibrational levels, separated by anenergy ~ω À Er. At low temperature, the atoms are restricted to lowest vibrational level at each site.Their kinetic energy is then frozen, except for the small tunneling amplitude to nearest-neighboringsites.

As in any periodic potential, the associated single-particle eigenstates are Bloch functions Ψn,~q(~r).They are characterized by a discrete band index n and a quasi-momentum ~q within the first Brillouin-zone of the reciprocal lattice [6] and [5]. Since Bloch functions are multiplied by a pure phase factorexp(i~r · ~R) upon translation by one of the lattice vector ~R, they can be extended over the whole lattice.An alternative single-particle basis, which is more useful for describing the hopping of particles amongthe discrete lattice sites~R, are the Wannier functions wn,~R(~r). They are connected with the Blochfunctions by a Fourier transform

Ψn,~q(~r) =∑

~R

wn,~R(~r)ei~r·~R (1.6)

on the lattice. The Wannier functions depend only on the relative distance ~r− ~R and, at least, for thelowest bands, they are centered around the lattice sites ~R.

By choosing a convenient normalization, they obey the orthonormality relation :∫

d2 ~rw∗n(~r − ~R

)wn′

(~r − ~R

)= δn,n′δ~R′, ~R (1.7)

for different bands n and sites ~R. Since the Wannier functions for all bands n and sites ~R form acomplete basis the (bosonic in this report) operator ψ(~r) which destroys a particle at an arbitrarypoint ~r can be expanded in the form :

ψ(~r) =∑

~R,n

wn

(~r − ~R

)a~R,n (1.8)

Provided the lattice is deep enough and for low enough temperatures, the system is in goodapproximation described by the lowest energy band, and in rewriting in terms of sites i of the latticeone obtains :

ψ(~r) =∑

i

wi (~r) ai (1.9)

with ai being the annihilation operator for one boson at site i in the Wannier function wi(~r) localizedat the bottom of lattice site i. We also have the basic commutation rules :

[ψ(~r), ψ(~r′)

]= 0[

ψ(~r), ψ†(~r′)]

= δ~r,~r′

[ai, aj ] = 0[ai, a

†j

]= δi,j

(1.10)

Another important point to consider before writing an Hamiltonian for the the system is theisotropic contact interaction that occurs when two or more atoms occupy the same lattice site. Atultra-low temperatures, scattering occurs only in the s-wave, therefore all partial waves are reflectedbut the s-wave [9, 1]. A consequence of this fact is that the real and complicated on-site potential(which at long distances is essentially given by Van der Waals attraction −C6/r6) can be replaced for


most purposes by a simple, isotropic and short-range model potential proportional to the scatteringlength a of the atoms. This contact interaction reads

Ucontact(~r) =4π~2

mδ(~r) ≡ gδ(~r) (1.11)

where m is the atomic mass and a the scattering length.

1.2 Simple Bose-Hubbard model

The purpose of this section is to introduce to the basic methods to obtain the Bose-HubbardHamiltonian and to study the different phases of the systems.

1.2.1 Hamiltonian

We consider a gas of interacting particles moving in a 2D optical lattice (by considering 2D wedon’t loose in mathematical generality). As explained previously, ultra-cold atoms interact via s-wavescattering, then the Hamiltonian of the system in second quantization reads [4, 9] :

H =∫

d~r ψ†(~r)[

p2

2m+ Vext(~r)

]ψ(~r) +

∫ ∫d~rd~r ′ ψ†(~r ′)ψ†(~r)

Ucontact

2ψ(~r)ψ(~r ′)

−µ

∫d~r ψ†(~r)ψ(~r) (1.12)

with µ being the chemical potential which, in the grand canonical ensemble, fixes the total num-ber of atoms1. Vext(~r) contains the optical potential and any other potentials which only involveone body interactions. The contact interaction part can be readily simplify with equation (1.11) ing2

∫d~r ψ†(~r)ψ†(~r)ψ(~r)ψ(~r).Now using equation (1.9) one obtains :

H =∑

i,j

∫d~r w∗i (~r)

[p2

2m+ Vext(~r)

]wj(~r)a

†i aj

+g

2

∑

i,j,k,l

∫d~r w∗i (~r)w

∗j (~r)wk(~r)wl(~r)a

†i a†j akal

−µ∑

i

a†i ai (1.13)

where the orthonormality of the Wannier function (see Eq. (1.7)) has been used to simplify the chemicalpotential part. The contact interaction part involves Wannier states in different lattice sites but canbe simplified using the assumption of deep optical lattice. In this case one can exclude the overlappingintegrals of the Wannier functions and only consider the summation term where i = j = k = l leadingto :

Hcontact =g

2

∑

i

∫d~r |wi(~r)|4a†i a†i aiai (1.14)

As the optical potential is the same at each site wi(~r) does not depend on i and then the integralcan be excluded from the summation. Moreover a†i a

†i aiai can be simplified in ni(ni−1) with ni = a†i ai

being the number of particles at site i operator. So finally,

Hcontact =U

2

∑

i

ni(ni − 1) , where U = g

∫d~r |wi(~r)|4 (1.15)

1 In the following calculation it is assumed that µ is the same at each site, this statement is no longer valid in the caseof particles of different kind or if a pseudo random site offset has been added to the system, for a further look on thissubject see [18].

1.2. Simple Bose-Hubbard model 11

The same process can be applied to the first part of the Hamiltonian except that in this case onlythe overlap beyond nearest neighbors can be neglected due to the action of p = −i~∇ and Vext(~r) andagain the system being translationary invariant

∫d~r w∗i (~r)

[p2

2m + Vext(~r)]wj(~r) does not depend on

particular i and j but is general for all the lattice. And defining

J = −∫

d~r w∗i (~r)[

p2

2m+ Vext(~r)

]wj(~r) (1.16)

one finally obtains the Bose-Hubbard Hamiltonian :

H = −J∑

〈i,j〉a†i aj +

∑

i

[U

2ni(ni − 1)− µni

](1.17)

where∑〈i,j〉 is the summation over nearest neighbors i and j and J > 0.

J

d2D

d2D

U

Fig. 1.2: Parameters for a simple Bose-Hubbard model in a 2D optical lattice.

One can observe (see figure 1.2) that this Hamiltonian decomposes itself in three very distinctiveparts : the first one describes the positive kinetic energy (∼ J) gained by a particle tunneling betweentwo nearest neighbor sites i and j, the second counts for the on-site energy (∼ U) and the lastone indicates the total number of particles. This Hamiltonian looks like the one extensively studiedin condensed matter physics, but with the important advantage that thanks to optical lattices theparameters J and U directly depend on the characteristics of the lasers used to create them. This,added to the possibility to tune the chemical potential µ by changing the number of trapped particles,allow an incredible freedom for experimental studies what explains the interest of such systems.

1.2.2 Phase diagram

When the tunneling is suppressed compared to on-site interactions, the Hamiltonian presents aquantum phase transition between a superfluid phase, characterized by large number fluctuationsat each lattice site (∀i, 〈ai〉 6= 0), and a Mott insulator phase where each lattice well is occupiedprecisely with an integer number of atoms (∀i, 〈ai〉 = 0). The nature of this phase transition andthe qualitative phase diagram can be inferred based on very simple arguments. At zero tunnelingJ = 0, the Hamiltonian becomes H =

∑i Hi where Hi = U

2 ni(ni − 1) − µni, which is separable anddiagonalizable. Then at each site, the energy per site is minimized by populating each lattice wellwith exactly n atoms.

∀i, Ei(ni) =U

2ni(ni − 1)− µni (1.18)

with ni being the energy at site i and equal to n in this case leading to ∀i, Ei = E(n) = U2 n(n−1)−µn.

Then energy considerations can tell which is the range of chemical potential µ at which the filling factorn is the most energetically convenient :


Ei(ni) < Ei(ni + 1)Ei(ni) > Ei(ni − 1)

}⇒ (ni − 1)U < µ < niU (1.19)

where Ei(ni + 1) and Ei(ni − 1) are respectively particle and hole excitations meaning adding orremoving a particle at site i. Since a particle plus hole excitation at J = 0 costs an energy ∆E = U ,the Mott is the lowest energy state at commensurate filling. For a finite J the energy cost to createan excitation decreases thanks to the kinetic energy favoring particle hopping. However, for largeinteractions and small tunneling, the gain in kinetic energy (∼ J) is not yet sufficient to overcome thecost in interaction energy (∼ U), which leads to the existence of Mott insulating phases in a finite regionof the µ-J plane, called a Mott lobe. For large enough tunneling, instead, particle hopping becomesenergetically favorable and the system becomes superfluid. The regions J vs. µ phase diagram wherethe Mott insulating state is the ground state are called Mott lobes.

Due to the finite energy cost required to add or remove one particle, the Mott phase is gapped and itis also incompressible, while in the superfluid regions the gap vanishes and the system is compressible.In Eq. (1.19) the boundaries of the Mott lobes at J = 0 have been identified. In order to find theshape of the lobes at finite J more sophisticated calculations are required. In particular there is noexact analytical method which allows one to calculate them. There are however some methods suchas mean-field approximation and perturbative expansion that can be used to provide further insighton the conditions required for the insulating lobes to exist and are the ones that will be presented inthe next parts.

Mean-field perturbative approach

The mean-field perturbative approach allows one to find the boundaries of the lobes in an semi-analytical way [1, 7]. The main approximation underlying this method is the mean-field approximation,starting from the Hamiltonian (1.17) one can define the order parameter 〈ai〉 = ϕi and proceed withthe following substitution :

ai = ai + ϕi , a†i = a†i + ϕ∗i (1.20)

obtaining for the hopping term

− J∑

〈i,j〉

[ϕ∗i

(aj − ϕj

2

)+ ϕj

(a†i −

ϕ∗i2

)+ a†i aj

](1.21)

where a†i aj is supposed to be small and then is neglected, this is indeed the case of Mott or deepsuperfluid. Based on this result, the full Hamiltonian can be rewritten as

HMF = H0 + HJ where (1.22)

H0 =∑

i

[U


]and (1.23)

HJ = −J∑

i

[ϕ∗i

(aj − ϕj

2

)+ ϕj

(a†i −

ϕ∗i2

)](1.24)

where ϕi =∑〈j〉i ϕj is the sum of the order parameters at sites neighboring to a given site i. Assuming

that the order parameters ϕi are small, we rewrite HJ at first order in the order parameter as :

HJ ' −J∑

i

[ϕ∗i ai + ϕia

†i

](1.25)

Then one can use this approximate expression in the calculation of the partition function Z =Tr

[e−β(H0+HJ )

]. Considering HJ as a small perturbation on H0, i.e. J ¿ U, µ, the partition function

can be written using a Dyson expansion on the exponential [18] :


Z ' Tr[e−βH0

]−

∫ β

0dτ Tr

[e−(β−τ)H0HJe−τH0

]+O(ϕ2) (1.26)

In the limit β →∞, corresponding to zero temperature, the partition function becomes :

Z ' Z0 = e−βE0 (1.27)

with E0 being the energy of the ground state. In the calculation of the order parameter instead, thefirst order term in the creation and destruction operators also contributes, in the form :

ϕk ' −e−βE0

∫ β

0dτ Tr

[ake

−(β−τ)H0HJe−τH0

]

= −e−βE0∑

|Φ〉

∫ β

0dτ 〈Φ|

[ake

−(β−τ)H0HJe−τH0

]|Φ〉 (1.28)

where the trace has been calculated on the basis of eigenfunctions of H0, namely H0|Φ〉 = E|Φ〉|Φ〉.From this equation, one can replace HJ by the expression (1.22).Some simplifications can be doneusing the facts that the expectation value of two destruction operators over eigenstates of H0 vanishesand that non local averages vanish in the mean field approximation (the only non vanishing term inthe summation is for i = k), what leads to :

ϕi = Jϕie−βE0

∑

|Φ〉

∫ β

0dτ 〈Φ|

[aie

−(β−τ)H0 a†ie−τH0

]|Φ〉 (1.29)

For a general state |Φ〉 one can calculate 〈Φ|[aie


]|Φ〉 :

〈Φ|[aie


]|Φ〉 =

(n|Φ〉i + 1

)e−βE|Φ+1,i〉e−τ(E|Φ〉−E|Φ+1,i〉), (1.30)

where |Φ+1, i〉 is the notation used for the state which is obtained by adding one particle to the state|Φ〉 at site i, and n

|Φ〉i = 〈Φ|ni|Φ〉 and E|Φ〉 = 〈Φ|H0|Φ〉 respectively represent the number of particles

at site i and the energy in the state |Φ〉.The integration of 〈Φ|

[aie


]|Φ〉 over τ in the range [0, β] and the multiplication

by eβE0 give :

(n|Φ〉i + 1

)[eβ(E0−E|Φ+1,i〉)

E|Φ〉 − E|Φ+1,i〉− eβ(E0−E|Φ〉)

E|Φ〉 − E|Φ+1,i〉

](1.31)

Since by definition of ground state E0 −E|Φ〉 ≤ 0 for all states |Φ〉, the precedent result convergesin the limit β →∞ to a non-zero result if and only if either

1. |Φ〉 is the ground state or

2. |Φ+1, i〉 is the ground state, i.e. |Φ〉 is the state obtained by removing one particle from groundstate at site i.

Consequently, only two terms contribute in the summation (1.29) : the one arising from the groundstate 〈GS|

[aie


]|GS〉 and the one arising from the states which are obtained from the

ground state by removing one particle 〈GS− 1, i|[aie


]|GS− 1, i〉. Hence, for a Mott

state with integer well occupation equal to ni, the order parameter reads :

ϕi = ϕiJ

[ni + 1

Uni − µ− ni

U(ni − 1)− µ

](1.32)


By defining EiP = Uni − µ and −Ei

H = U(ni − 1) − µ i.e. the energies required to add orremoving (creating a hole) a particle at site i (at J = 0) and using the obvious relations Ei

P > 0 andEi

H > 0) one can see the direct link with the range for the chemical potential deduced in Eq (1.19).The equation( 1.32) can easily be rewritten in term of matrixes and the solving essentially consistsin calculating a determinant (see appendix A for matrix rewritting). Thus this equation gives anexpression of µ as a function of J defining the boundaries of the Mott lobe for a given ensemble of{ni}i∈(Lattice). All configurations of ni don’t present a lobe, indeed a configuration has to fulfill theinequalities (1.19). For an extensive selection among random configurations this condition can berewritten as :

µmin < µ < µmax, where :µmin = maxi∈(Lattice) [U(ni − 1)] andµmax = mini∈(Lattice) [Uni]

(1.33)

A random research with these inequalities confirm the guessing of the previous part that for agiven inequality µmin < µ < µmax the Mott phase with a same identical integer filling factor n at allsites i is the ground state. An other way to state the condition for the existence of the lobes is thefollowing : given an insulating state (classical distribution of atoms in the lattice),this is stable in agiven range of chemical potential at J = 0 if the energy increases by adding or removing a particle atany site of the lattice.

μ/U

n = 1

n = 2

n = 4

n = 3

Hole

excitation

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

J/U

n = 0

Particle

excitation

Fig. 1.3: Phase diagram for a 2×2 lattice with U = 4 and for integers n = 0, 1, 2, 3, 4.

The figure 1.3 shows such Mott lobes. The particle (hole) excitation energy is the distance from theµ point to the upper (lower) boundary of the lobe. The lobe for n = 0 marks the inferior boundarybelow which the lattice can not sustain any particle. The expression of this expression can easilycalculate from Eq. (1.32) by writing n = 0 and considering that the system is uniform (meaning∀i, ϕi = ϕ and then ϕi = zϕ, z being the number of nearest neighbors, z = 4 for a 2D lattice)what leads to ϕ (1 + zJ/µ) = 0 and µ = −zJ . Outside of the lobes the order parameter qualitativelypredicts superfluid phase outside the Mott lobes (ϕ 6= 0) but does not provide with any furtherquantitatively details. However, in the framework of the mean-field approximation, the expressionfor the lobes boundaries is exact. However the prediction about the existence of the lobes i.e. thelobes boundaries at J = 0 are exact also beyond the mean-field approximation and confirmed by exactnumerical method [16, 3].

The method explained previously, even though very efficient to plot the lobes, presents some lacksnamely it does not allow one to find the ground state configuration (if there is one) of the ni for agiven couple (µ, J) ad it does not provide the value of the ϕi along the phase diagram. That is whythis method can be completed by another one : the equation of the movement within the mean-fieldapproximation.


Mean-field decoupling and Gutzwiller Ansatz

Starting from the two parts of the Hamiltonian (1.22) and (1.25), one can see that the Hamiltonianis separable HMF =

∑i H

MFi with

HMFi = −J

[ϕ∗i ai + ϕia

†i

]+

[U


](1.34)

This characteristic and the previously found ground states lead to the idea to use a trial wave-vectorin order to calculate directly the desired variables. This mean-field solution is completely equivalentto the one provided by the Gutzwiller Ansatz [1] and corresponds in writing the state of the systemas a product over the different lattice sites of single-site wave-functions :

|Φ(t)〉 =⊗

i

|Φi(t)〉 =⊗

i

∑m

f (i)m (t)|i,m〉 (1.35)

where |i, m〉 is the Fock state of m particles at site i. The coefficients f(i)m (t) are called Gutzwiller

coefficients and fulfill to the normalization relation :

〈Φ(t)|Φ(t)〉 = 〈Φi(t)|Φi(t)〉 =∑m

|f (i)m (t)|2 = 1 (1.36)

The solution obtained by the Gutzwiller Ansatz is intrinsically mean-field. On the other hand itallows to extract many important information and to study the dynamics of the system. Indeed, usinga time dependent variational approach on the Lagrangian (~ = 1 all along the report)

L =〈Φ(t)|Φ(t)〉 − 〈Φ(t)|Φ(t)〉

2i− 〈Φ(t)|HMF |Φ(t)〉 (1.37)

i.e. solving the Euler-Lagrange equations ∂L∂f

(i)∗m

− ddt

∂L∂f

(i)∗m

= 0, one obtains the dynamical equations

for the Gutzwiller coefficients :

idf

(i)m

dt= −J

[ϕi

√mf

(i)m−1 + ϕ∗i

√m + 1f

(i)m+1

]+

[U

2m(m− 1)− µm

]f (i)

m (1.38)

By solving those equations in imaginary time (τ = it), one can access the stationary states of thesystem which are “selected” since imaginary time is suppose to converge to the lowest energy states(due to dissipation). The type of phase given by the solution for a given couple (µ, J) can then bestudied by calculating the main variables : the value of the order parameter ϕi =

∑m f

(i)∗m f

(i)m+1

√m + 1

and the number of particles ni =∑

m m|f (i)m (t)|2 at site i. For instance if ∀i we have ϕi = ϕ 6= 0 then

we are in the superfluid phase, on the contrary if ∀i, ϕi = 0 we are in a Mott lobe and the {ni} showthe particle distribution. The procedure of the solving is the following one : for a given couple (µ, J),the Gutzwiller coefficients are set to random values,,then we run the simulation and the integrationis stopped when the difference of ϕi between two consecutive integration steps is small (typically weconsider the value 10−6 to be zero) then one can have a look at the obtained stationary state. Forinstance for a L× L lattice with U = 4 and L = 2 one can obtain (for comparison with the lobes seefigures 1.3 and 1.4):

For µ/U = 0.5 and J/U = 0.01 :

ni ϕi × 1012

1 1 0.6588 0.65881 1 0.6588 0.6588

where the values of ϕi ' 0 indicates that we are in a Mott lobe and the ni gives the distribution o theparticles. Another example : the point µ/U = 0.5 and J/U = 0.05 gives :

ni ϕi

1.0278 1.0278 0.4787 0.47871.0278 1.0278 0.4787 0.4787


and ϕi > 0 indicates that we are in the superfluid phase, in this case the values of the ni must beinterpreted as an expectation value.

The figure 1.4 shows the results of the plotting of the different values of ϕ(J, µ) =∑

i ϕi(J, µ)/L2.One can see the perfect matching of the lobes with the two methods : the average ϕ ' 0 inside theMott lobe and ϕ increases as one moves away from the lobes. Figure 1.5 shows the other interestingparameter namely the number of particles per site and the matching is again perfect and confirm theresults of a commensurate number of particles at each site in a specific lobe of the previous part.

Fig. 1.4: Phase diagram of a L × L lattice with U = 4 and L = 2 by plotting ϕ(J, µ) =∑

i ϕi(J, µ)/L2. Theshaded tones correspond to the different values of ϕ(J, µ) while the black lines to the lobes calculatedwith the mean-field perturbative approach (see figure 1.3)

μ/U

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.01 0.02 0.03 0.04 0.05-1

J/U

0.005 0.015 0.025 0.035 0.045 0

4

1.5

2.5

3.5

4.5

3

2

1

0.5

Fig. 1.5: Phase diagram for a L × L lattice with U = 4 and L = 2 by plotting n(J, µ) =∑

i ni(J, µ)/L2. Theshaded tones correspond to the different values of n(J, µ) while the black lines to the lobes calculatedwith the mean-field perturbative approach (see figure 1.3)

The solving of Eq. (1.35) in real time allows one to investigate the dynamics of the system inthe case of time-dependent Hamiltonian parameters and also to observe the oscillations of the systembetween two degenerate stationary states (such oscillations don’t appear in this precise case but canhappen for a degenerate ground state what will be the case with up-down dipoles). It is worth noticingthat the solving of Eq. (1.35) in imaginary time does not conserve the initial number of particles2 (set

2 This is due to the fact that on a Schro dinger equation picture the exponential of the Hamiltonian would no longer be


by the initial random values of the Gutzwiller coefficients), during the integration one can considerthat the system is open which is consistent with the grand canonical ensemble formalism.

The size of the lattice has never been discussed so far. In the case of the simple Bose-HubbardHamiltonian studied is this part, the size has no importance since the Hamiltonian is translationaryinvariant and then the lobes are the same for any size. For the same reasons the calculation of thispart are all based on square lattices L × L for programming convenience. These justifications will apriori no longer stand up for other system particularly when different kind of particles are set on thelattice (which will be the case with the model considered in section 2).

Furthermore considering the nearest neighbors hopping (and later the dipolar interaction) oneneed to define the boundary conditions. For most of the calculations done in this report the latticeis considered periodic. This important statement can be justified by considering that the equationsand the solutions only work in the core of a much bigger experimental lattice, limiting side effects.However, simulations have been performed during my stay with limited size lattices. It required toset a very high chemical potential on the edges to avoid any populating of the boundaries (what isconsistent with the usual application of an external harmonic potential on the lattice for a betterconfinement), and the results appeared to be not qualitatively different from the ones with periodicboundary conditions.

The basic ideas and methods of treatment having been explained, it is now time to introducedipolar interactions.

unitary and then would not disappear during the calculation of the expectation value of the number of particles operator(the eigenstates are no longer normalized during time evolution). On the contrary in real time solving the total numberof particles is conserved.


2. EXTENDED BOSE-HUBBARD MODEL : UP-DOWN DIPOLES

After the presentation of the main tools and ideas of Bose-Hubbard model and phase diagrams,this chapter presents a novel system that has not been studied before : an extended Bose-Hubbardmodel with up and down polarized dipoles.

2.1 Dipolar interaction

Atoms or molecules having a permanent dipole moment (either magnetic or electric) interact notonly via short-range potentials (i.e. contact interaction, see part 1.1.2), but also via the dipole-dipoleinteraction. The corresponding potential is

Udipole(~r) =Cdip

4πr3

[~p.~p ′ − 3

(~p.~r)(~p ′.~r)r2

](2.1)

r

p p ’

Fig. 2.1: Parameters of dipole-dipole interaction.

where ~p and ~p′ refer to the dipole moments (see figure 2.1)and Cdip respectively corresponds to µ0 or1/ε0 for magnetic or electric dipole moments.

In the following study, we will consider parallel along the z-axis and unitary dipoles. Howeverdipoles can be either up (~p = +~ez) or down (~p = −~ez). The Eq. (2.1) thus becomes

Udipole(r) = ±Cdip

4πr3= ±V

r3, (2.2)

what can be written in a more convenient way for the calculation of the Hamiltonian

U s,s′dipole(r) =

Vs,s′

r3, where s =↑, ↓ and Vs,s′ =

{+V if s = s′

−V else(2.3)

The dipolar interaction is a long-range interaction meaning that the potential can affect two dipolessome lattice sites apart, this question of the range will be discussed for the calculation of the Hamil-tonian.

2.2 Hamiltonian

The main issue to write the Hamiltonian is to take in account the two different dipoles that are nowon the lattice : up and down dipoles. From this picture one can think to the analogy with a spin-basedsystem to develop the Hamiltonian [17, 22, 19] and the main statement is to write dipole-dependentfield operators that is, starting from Eq. (1.9) :

ψs(~r) =∑

i

wi (~r) ai,s where s =↑, ↓ (2.4)

20 2. Extended Bose-Hubbard model : Up-Down dipoles

with the new commutation rules

[ψs(~r), ψs′(~r ′)

]= 0[

ψs(~r), ψ†s′(~r

′)]

= δ~r,~r′ .δs,s′[

ai,s, aj,s′]

= 0[ai,s, a

†j,s′

]= δi,j .δs,s′

(2.5)

With these new field operators one can derive the full Hamiltonian from Eq.( 1.12) in taking inaccount the dipolar interaction :

H =∫

d~r ψ†s(~r)[

p2

2m+ Vext(~r)

]ψs(~r) +

∫ ∫d~rd~r ′ ψ†s(~r

′)ψ†s′(~r)Ucontact

2ψs′(~r)ψs(~r ′)

−µs

∫d~r ψ†s(~r)ψs(~r) +

∫ ∫d~rd~r ′ ψ†s(~r

′)ψ†s′(~r)U s,s′

dipole (|~r − ~r ′|)2

ψs′(~r)ψs(~r ′) (2.6)

where the summation over indices s and s′ is implied and the chemical potential now depends onthe specie. The following is then very similar to the precedent calculation, except for the dipolarinteraction part. By expanding the field operators one obtains :

Hdipole =12

∑

s,s′Vs,s′

∑

i,j,k,l

a†i,sa†j,s′ ak,s′ al,s

∫ ∫d~rd~r ′ w∗i (~r

′)w∗j (~r)1

|~r − ~r ′|3 wk(~r)wl(~r ′) (2.7)

The summation over sites i, j, k and l can be separated in two terms for on-site and off-siteinteractions. For on-site interaction i = j = k = l and the on-site part gives

Hdip on−site =12

∑

s,s′Vs,s′

∑

i

a†i,sa†i,s′ ai,s′ ai,s

∫ ∫d~rd~r ′ w∗i (~r

′)w∗i (~r)1

|~r − ~r ′|3 wi(~r)wi(~r ′)

=Udip

2

∑

i,s

[ni,s(ni,s − 1)− ni,sni,s′ 6=s

](2.8)

where Udip =∫ ∫

d~rd~r ′w∗i (~r′)w∗i (~r)

1|~r − ~r ′|3 wi(~r)wi(~r ′) (2.9)

This expression has to be compared to the on-site interaction part rising from the contact interac-tion which contribution to the Hamiltonian is :

Hcontact =∫ ∫

d~rd~r′ψ†s(~r′)ψ†s′(~r)

Ucontact

2ψs′(~r)ψs(~r′)

=Ucontact

2

∑

i,s

[ni,s(ni,s − 1) + ni,sni,s′ 6=s

](2.10)

where Ucontact has the same expression than in Eq. (1.15). The on-site interaction is then given by twocontributions : one due to the Ucontact and the other to the on-site dipole-dipole interaction Udip. It canbe proved [1] that Udip strongly depends on the anisotropy of the wavefunction at the bottom of thelattice wells, for instance for vertically pointing dipoles, the on-site dipole-dipole interaction is mainlyattractive for cigar-shaped wells, vanishes for spherical wells and is mainly repulsive for pancake-shapedwells. The strength and the sign of Udip can thus be tuned by changing the anisotropy of the Wannierfunctions that is by modifying the lattice through the lasers. For simplification purposes we assume aspherical trapping at the bottom of the well leading to the statement Udip = 0.

Then the off-site interaction part of the dipole interaction has to be discretized to take in accountthe lattice sites. Since the dipolar interaction is of a long-range type, for computational reasons onehas to set a cut-off at a certain number of nearest neighbors. For most of the results of this report the

2.2. Hamiltonian 21

2 1 2

1 0 1

2 1 2

3

3

3

3 4

4

4

44

4

4

4

Fig. 2.2: Representation of the first four nearest neighbors in a 2D square optical lattice. For a lattice step setto 1, the distance to the 0th site are respectively 1,

√2, 2 and

√3 for the 1st, 2nd, 3rd and 4th nearest

neighbors.

choice was done to stop at the 1st nearest neighbor as the system already presents interesting behaviorat this range, but all programs allow range up to the 4th nearest neighbor (figure 2.2 presents thedifferent ranks of nearest neighbors).The integral (2.7), a part from the on-site contribution already discussed, has non zero contributionfor i = k and j = l, the summation then can be simplified as follows :

∑

i,j,k,l

=∑

~l

∑

〈i,j〉~l

.=∑

〈〈i,j〉〉~l

where ~l is the distance connecting the two optical lattice sites i and j and 〈i, j〉~l meaning i and j

nearest neighbors at distance ~l. With these notations Eq. (2.7) gives :

12

∑

s,s′Vs,s′

∑

〈〈i,j〉〉~lni,snj,s′

∫ ∫d~rd~r ′w∗i (~r

′)w∗j (~r)1l3

wj(~r)wi(~r ′) =12

∑

s,s′

∑

〈〈i,j〉〉~lV

~ls,s′ ni,snj,s′ (2.11)

where V~ls,s′ =

Vs,s′l3

∫ ∫d~rd~r ′w∗i (~r

′)w∗j (~r)wj(~r)wi(~r ′) is the strength of the dipolar interaction atdistance ~l (1/l3 can be taken out the integral since the integration mainly concern the size site whichis much smaller than inter-site distance for deep enough latticesite) and ni,s = a†i,sai,s is the number ofparticles at site i with dipolar index s operator. The extended Bose-Hubbard Hamiltonian for dipolaratoms is thus :

H = −J∑

〈i,j〉

∑s

a†i,saj,s −∑

i

∑s

µsni,s

+U

2

∑

i

∑s


]+

12

∑

s,s′

∑

〈〈i,j〉〉~lV

~ls,s′ ni,snj,s′ (2.12)

where the meaning of the different interactions is shown in Fig. 2.3. With this Hamiltonian it isnow possible to use the two methods derived above of the imaginary time evolution and mean-fieldperturbative approach to investigate the phases of the system.

The relative strength of on-site U and long range dipolar V interactions has never been discussedso far. In order to make the effects of long-range interactions observable it might be necessary to havea not too small ratio U/V that is why this will mostly be set equal to 20 in this report, this is notparticular value but this order of magnitude is often used in the literature [21]. It is worth noticingthat that this Hamiltonian does not allow the flipping process between ↑ and ↓ atoms.


JU

V-+d

2D

d2D

l

Fig. 2.3: Parameters for a extended dipolar Bose-Hubbard model in a 2D optical lattice.

2.3 Phase diagrams

2.3.1 Mean-field perturbative approach

Similarly to the precedent calculation, one can define the order parameter 〈ai,s〉 = ϕi,s, apply themean-field approximation to the Hamiltonian( 2.12) and separate it in diagonal and off-diagonal partsH = H0 + HJ :

H0 = −∑

i,s

µsni,s +U

2

∑

i,s


]+

12

∑

s,s′

∑

〈〈i,j〉〉~lV

~ls,s′ ni,snj,s′ (2.13)

HJ = −J∑

i,s

(ϕ∗i,sai,s + ϕi,sa

†i,s

)(2.14)

where ϕi,s =∑〈j〉i ϕj,s. The calculation for the order parameter is then straightforward as in Sec. 1.2.2,

and gives

ϕi,s = Jϕi,s

[ni,s + 1EP (i, s)

− ni,s

EH(i, s)

]where, (2.15)

EP (i, s) = U(ni,↑ + ni,↓)− µs +∑

〈〈j〉〉~l,i

∑

σ=↑,↓V

~ls,σnj,σ (2.16)

EH(i, s) = U(ni,↑ + ni,↓ − 1)− µs +∑

〈〈j〉〉~l,i

∑

σ=↑,↓V

~ls,σnj,σ (2.17)

EP (i, s) and EH(i, s) are respectively the energies to add or remove a particle with a dipole s at sitei. Now one can remark the differences with the simple Bose-Hubbard model. Indeed the energies havedifferent expression for the two kind of dipoles and we now have 4 different excitations. This leads toa problem for defining the ground state and the stability of a configuration, indeed a configuration canbe stable either for a ↑ or a ↓-hole/particle excitation or for both of them and not necessarily with thesame boundaries. The extensive research for stable configuration can be performed with conditionssimilar to the Eq. (2.18), that is :

µs,min < µs < µs,max, where :µs,min = maxi∈(Lattice)

[U(ni − 1) +

∑〈〈j〉〉~l,i

∑σ=↑,↓ V

~ls,σnj,σ

]and

µs,max = mini∈(Lattice)

[Uni +

∑〈〈j〉〉~l,i

∑σ=↑,↓ V

~ls,σnj,σ

] (2.18)

2.3. Phase diagrams 23

Another interesting quantity for the research of stable configurations and ground states is theenergy of a configuration as a function of the chemical potential that is the diagonal part of theHamiltonian (2.12) :

E = −∑

i

∑s

µsni,s +U

2

∑

i

∑s


]+

12

∑

s,s′

∑

〈〈i,j〉〉~lV

~ls,s′ni,snj,s′ (2.19)

with ni,s being the classical (integer inside a Mott lobe) number of particle s at site i. In the followingstudies we set the hypothesis that µ↑ = µ↓, this statement can be justified by the fact that as we willsee the stable configurations are mostly symmetrical in the number of both particles.

0 0.5 1 1.5 2 2.5-5

0

5

10

15

20

J/V

m/V

-5 0 5 10 15 20-80

-60

-40

-20

0

20

E

0 0.5 1 1.5 2 2.5-5

0

5

10

15

20

φ-

-5 0 5 10 15 20-80

-60

-40

-20

0

20

E

J/V

m/V

φ+

m/V

m/V

E/V

E/V

(a)

(b)

Fig. 2.4: Lobes and energies for 2 different configurations (α) and (β) for the two possible excitations : φ+ foradding/removing a ↑-atom to the 2 configurations and φ− for adding/removing a ↓-atom. Calculationsdone for a 2×2 periodic lattice with U = 20, V = 1, first rank of nearest neighbors and periodicboundaries.

For instance figure 2.4 shows the lobes for two distributions of up and down particles which areobtained one from the other by exchanging up with down particles and viceversa, both configurationshave lobes for the two excitations but not for the same chemical potential and with different shapesand the actual Mott region is then the intersection area of the two lobes (red with blue). We also lookfor the configurations which have the same Mott lobe for up-type of excitations and for down-typebecause these are expected to be more robust.

Figure 2.5 shows such configurations for the lowest chemical potentials, then one has to comparethe energies to find the ground states, for instance the (β) and (α) configurations have lower energiesthan the (γ) one (green lines on the energy diagram) so they are the ground states for this range ofchemical potential. The (γ) configuration is usually called “metastable” state since it lies inside theground state lobes and is then in competition with states of lower energies (more stable, for moredetails on metastable states see [21]).

One can observe that contrarily to the lobes of the simple Bose-Hubbard model the ground stateslobes of the extended model overlap on each other which is a sign that the excitations we are consideringare not the smallest of the system. In particular the particle/hole excitation is surely too big and wewill see that there exist a smaller type of excitation, namely the flip of a dipole. But first it is usefulto complete this study by solving the Gutzwiller equation.


0 0.5 1 1.5 2 2.5-10

0

10

20

30

φ+

J/V

m/V

-5 0 5 10 15 20 25 30 35-250

-200

-150

-100

-50

0

50

E

m/V

E/V

0 0.5 1 1.5 2 2.5-10

0

10

20

30

φ-

-5 0 5 10 15 20 25 30 35-250

-200

-150

-100

-50

0

50

E

m/V

J/V m/V

E/V

(a)

(b)

(g)

Fig. 2.5: Lobes and energies for configurations identical for both excitations : φ+ for adding/removing a ↑-atomto the 2 configurations and φ− for adding/removing a ↓-atom. Calculations done for a 2×2 latticewith U = 20, V = 1, first rank of nearest neighbors and periodic boundaries. Configurations (β) and(α) are called “checkerboard” and “double checkerboard” in analogy to the checkerboard and doublecheckerboard of the model with only up dipoles [21].The red line corresponds to the empty lattice (itis similar to the one on figure 1.3).

2.3.2 Gutzwiller Equation

As seen in part 1.2.2 the starting point to derive the Gutzwiller equation is to write the GutzwillerAnsatz. Using the analogy with a spin-based model [19] and the shape of the previous Ansatz thetrial wave-function is the following :

|Φ(t)〉 =⊗

i

|Φi(t)〉, where |Φi(t)〉 =∑

n↑,n↓

f (i)n↑,n↓(t)|n↑, ↑, i〉|n↓, ↓, i〉 (2.20)

where |n↑, ↑, i〉|n↓, ↓, i〉 are the Fock states of n↑ and n↓ particles at site i. It has to be noticedthat now the Gutzwiller coefficients depend on two indices and fulfill to normalization relation :∀i, ∑

n↑,n↓ |f(i)n↑,n↓(t)|2 = 1. One can then apply the same process to obtain the Gutzwiller equation :

idf

(i)n↑,n↓(t)

dt= −J

[ϕi,↑

√n↑f

(i)n↑−1,n↓(t) + ϕ∗i,↑

√n↑ + 1f

(i)n↑+1,n↓(t)

+ϕi,↓√

n↓f(i)n↑,n↓−1(t) + ϕ∗i,↓

√n↓ + 1f

(i)n↑,n↓+1(t)

]

+[U

2n↑(n↑ − 1) +

U

2n↓(n↓ − 1) + Un↑n↓ − µ↑n↑ − µ↓n↓

+(n↑ − n↓)∑

〈〈j〉〉~l,iV

~l (nj,↑ − nj,↓)

f (i)

n↑,n↓(t) (2.21)

where notations mean (with s =↑, ↓):

ϕi,s =∑

〈j〉iϕj,s (2.22)

ϕj,s = 〈Φ(t)|aj,s|Φ(t)〉 (2.23)

2.3. Phase diagrams 25

ϕj,s =

{ ∑n↑,n↓

√n↑ + 1f

(j)∗n↑,n↓f

(j)n↑+1,n↓ if s =↑

∑n↑,n↓

√n↓ + 1f

(j)∗n↑,n↓f

(j)n↑,n↓+1 if s =↓ (2.24)

∑

〈〈j〉〉~l,i= sum over j nearest neighbor of i at distance ~l (2.25)

V~l =

Cdip

4πr3

1l3

∫ ∫d~rd~r ′ w∗i (~r

′)w∗j (~r)wj(~r)wi(~r ′) (2.26)

nj,s = 〈Φ(t)|nj,s|Φ(t)〉 =∑

n↑,n↓

ns|f (j)n↑,n↓(t)|2 (2.27)

From this equation one can see that there are mainly 4 interesting variables to look at for studyingsystems with up and down dipoles : ϕi,↑ and ϕi,↓ for the Mott insulator/superfluid phases and ni,↑and ni,↓ for the distribution of the two species on the lattice. For reasons similar to the previous partthe following statement is done : µ↑ = µ↓. For instance the solving of the equation for a 4×4 latticewith U = 20, V = 1, µ = 1, J = 0.05, first nearest neighbors and periodic boundaries gives :

ni,↑ ni,↓0.0000 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 0.00001.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.00000.0000 1.0000 0.0000 1.0000 1.0000 0.0000 1.0000 0.00001.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 1.0000

ϕi,↑ × 107 ϕi,↓ × 107

0.1294 0.2992 0.0719 0.3110 0.1976 0.0545 0.1536 0.06760.2858 0.0625 0.1434 0.0669 0.0555 0.1230 0.0398 0.15560.1326 0.3072 0.0751 0.3243 0.2240 0.0609 0.1694 0.07400.5818 0.1375 0.3435 0.1420 0.0887 0.2197 0.0730 0.2739

As excepted we are in a Mott lobe (see phase diagram 2.5) and the {ni,s} configuration correspondsto the simple checkerboard found with the other method. As explained in the previous part we haveto implement a new model which would take in account the flipping excitation and this is the subjectof the next chapter.

Remark : The distribution of the particles in the ground state (checkerboard) is no longer uniform(contrary to the case of the simple Bose-Hubbard model), such states are sometimes called densitywave phases because they show a spatial modulation of the density. Starting from the checkerboardthe solving of the Gutzwiller equation in real time would show an oscillation between the two possiblecheckerboards.


3. FLIPPING EXCITATION

As seen previously the particle/hole excitation seems not being the smallest excitation allowed fora system of up-down dipolar particles. Inspired by the analogy with spin-based models a guessingwould be that this excitation could be the flipping of the dipoles. This model will now be developed.

3.1 Rewriting of the Hamiltonian

To account the flipping process we has to define a low-energy subspace L is spanned by all classicaldistributions of atoms in the lattice |α〉 =

⊗i |na

i , nbi〉 (where na

i and nbi are the number of a =↑ and

b =↓ particles at site i). Indeed one can check, starting from the checkerboards found in the previouschapter, that the ground states are simple and double checkerboard and that the state of (simplecheckerboard+one particle) is higher in energy than simple and double checkerboards. One otherthing is that there are two ways of viewing the flipping of the dipoles :

1. the first one is to consider it as the addition of one e.g up-dipole and the removal of one downin one site that is a particle+hole excitation,

2. the second one is the exchange of 2 different dipoles between two nearest neighbors lattice sites.This is at constant µ.

The description of these processes can not be based on standard mean-field theory, which accountsfor particle hopping through the replacement in the Hamiltonian of the single particle creation anddestruction operators by their expectation values. In fact, the physics of this exchange relies onsecond-order tunneling and takes place in the low-energy subspace where singe-particle hopping issuppressed. In this subspace, at the mean-field level, hopping completely vanishes. A successful wayto take in account second order tunneling is to write an effective Hamiltonian in the subspace offlipping and include tunneling through second order perturbation theory [10, 8]. The validity of theeffective Hamiltonian relies on the existence of a low-energy subspace well separated in energy fromthe subspace of virtual excitations, to which it is coupled via single particle hopping. In our case therelevant excited (virtual) subspaces are |γ(a)

ij 〉 and |γ(b)ij 〉, given by the states obtained from the states

|α〉 via a single particle hopping

|γ(a)ij 〉 =

a†j ai√na

i (naj + 1)

|α〉 ; |γ(b)ij 〉 =

b†j bi√nb

i(nbj + 1)

|α〉 (3.1)

with aj and bj being respectively the annihilation operators destructing a ↑-particle or a ↓-particle atsite j, the action of these operators is illustrated on figure 3.1. In the low-energy subspace, the matrixelements of the Hamiltonian in second order perturbation theory are given by [8, 15]

〈α|Heff |β〉 = 〈α|H0|β〉 − 12

∑

|γ〉〈α|HJ |γ〉〈γ|HJ |β〉

[1

Eγ − Eα+

1Eγ −Eβ

](3.2)

where |β〉, |α〉 ∈ L, |γ〉 ∈ E . H0, given by the interaction terms of the Hamiltonian (2.12), is diagonalon the states |α〉, and HJ , given by the single particle tunneling term HJ = −J

∑〈i,j〉

[a†i aj + b†i bj

],

is treated as a perturbation at second order, with Eα = 〈α|H0|α〉 being the unperturbed energies.The calculation of the matrix elements can be decomposed in two parts, the diagonal and off-diagonal

28 3. Flipping excitation

i j

| β

(b)| γij

i j

(a)| γji

i j

| α

i j

| α

i j

b bj i

+

b bj i

+

a ai j

+

a ai j

+

c ci j

+

Fig. 3.1: Action of the different operators on different substates to achieve the flipping (or the exchange).

elements and the difficulty mainly lies in the second order terms (for convenience the following calcu-lations are done for a dipolar interaction limited to the first rank of nearest neighbors but can easilybe expanded for longer range, then we set V

~l .= V ).

Diagonal terms |β〉 = |α〉 : By expanding HJ and after some calculations one obtains :

〈α|Heff |α〉 = Eα − J2∑

〈i,j〉

[na

j (nai + 1)

Eγaij− Eα

+nb

j(nbi + 1)

Eγbij− Eα

](3.3)

and{

Eγaij− Eα = U + V ∆(ji)

NN where ∆(ji)NN

.= naj − nb

j − nai + nb

i − 1

Eγbij− Eα = U + V ∆(ij)

NN where ∆(ij)NN

.= nbj − na

j − nai + nb

i − 1(3.4)

where as usual nbj =

∑〈k〉j nb

k. Then assuming that U À V we can expand the fractions :

1Eγa

ij−Eα

' 1U

[1− V

U ∆(ji)NN + ...

]

1E

γbij−Eα

' 1U

[1− V

U ∆(ij)NN + ...

] (3.5)

Off-diagonal elements : The expansion of the off-diagonal is a bit more tricky but the main stepsare essentially the same. One important step is the following statement :

〈α|a†i aj |γbij〉〈γb

ij |b†j bi|β〉= 〈α|b†j bi|γa

ji〉〈γaji|a†i aj |β〉

= 〈α|c†i cj |β〉(3.6)

where c†i = bia†i and cj = aj b

†j are the flipping operators. This statement can easily be understood by

looking the action of the operators on figure 3.1.Then summarizing the different parts one obtains the full Hamiltonian :

Heff = H0 − J2

U

∑

〈i,j〉

[na

j (nai + 1)

(1− V

U∆(ji)

NN + ...

)

+nbj(n

bi + 1)

(1− V

U∆(ij)

NN + ...

)]− 2J2

U

(1− V

U

)∑

〈i,j〉c†i cj (3.7)

3.1. Rewriting of the Hamiltonian 29

where

H0 = −∑

i,s

µsni,s +U

2

∑

i,s


]+

12

∑s

∑

〈〈i,j〉〉V

[ni,snj,s − ni,snj,s′ 6=s

](3.8)

As we are in the subspace of constant ν per site, it is a convenient choice to change the basis thatis defining new operators for counting the number of particles per sites :

νi =na

i + nbi

2and mi =

nai − nb

i

2(3.9)

The new basis is then {|mi, νi〉}i on which the action of the operators is :

c†i |mi, νi〉 =√

νi(νi + 1)−mi(mi + 1)|mi + 1, νi〉 (3.10)

ci|mi, νi〉 =√

νi(νi + 1)−mi(mi − 1)|mi − 1, νi〉 (3.11)

and the Hamiltonian becomes :

Heff = −µ−∑

i

mi +(

2V − 2J2

U−2J2V

U2

) ∑

〈i,j〉mimj

(2J2

U−2J2V

U2

) ∑

〈i,j〉c†i cj

+∑

i

[−µ+νi + Uνi(2νi − 1)]−(

2J2

U−2J2V

U2

) ∑

〈i,j〉νi(νj + 1)

+4J2V

U2

∑

〈i,j〉mj(mj − mi) (3.12)

where µ+ = µ↑ + µ↓, µ+ = µ↑ − µ↓ and the terms in red are the ones arising from the 1st order ofthe expansion of the fractions (3.5). Some remarks can be done about the new basis, for example forthe simple checkerboard ∀i, νi = ν = 1/2 and mi = −1/2 or 1/2 and this can be generalized : fora given νi, mi takes a value inside [−νi,−νi + 1, ..., νi − 1, νi]. This property is very similar to thespin-based model and a parallel can be established between the spin-operators and the ones definedin this report [19], that is :

m ↔ Sz

c† ↔ S+

c ↔ S−ν ↔ spin of the particle

Another remark is that at 0th and 1st orders the Hamiltonian 3.12 is separable in three independentpart Heff = H({νi})+H({mi})+Htunneling and moreover

[H({νi}), c†i

]= [H({νi}), ci] = 0 what means

that the results of the order parameter and Gutzwiller calculation are independent of the {νi}. The{νi} are then fixed by the H({νi}) and it is then interesting to study the diagonal HamiltonianH({νi}) + H({mi}).


3.2 Diagonal Hamiltonian and µ+

We consider the diagonal part of the Hamiltonian at 0th order (same study can easily be done at1st order)

H(0)diag = H(0)({νi}) + H(0)({mi})

=∑

i

[−µ+νi + Uνi(2νi − 1)]− 2J2

U

∑

〈i,j〉νi(νj + 1)

−µ−∑

i

mi +(

2V − 2J2

U

) ∑

〈i,j〉mimj (3.13)

for classical states |α〉 =∑

i |νi,mi〉 with ∀i, νi = ν. Then we study the energies gain by adding orremoving particle in all sites.

Adding particles The reached state by adding particles is

|α+〉 =∑

i

|νi + 1/2,mi ± 1/2〉 where{

mi + 1/2 if mi > 0mi − 1/2 if mi < 0

then one can calculate 〈α+|H(0)diag|α+〉−〈α|H(0)

diag|α〉 = E+α −Eα, by requiring the excitation to increase

the energy E+α − Eα ≥ 0 as in the standard calculation for the Mott lobes [18], we get

Nµ+ ≤ 4NUν − 4J2

UNz(ν + 3/4)− µ−∑

θi+ 4(V − J2

U)∑

〈i,j〉

(θimj +

14θiθj

)(3.14)

where N is the number of sites, z the number of nearest neighbors and θi = sign(mi). Some simplifi-cations can be done for the checkerboard (which is still the less energetic ground state). For instancefor a 2×2 lattice we have

θ =(

1 −1−1 1

)

what leads with Eq. (3.14) to :

µ+ ≤ 4Uν − 4J2

UNz(ν + 3/4)− 4(V − J2

U)z(ν + 1/4) (3.15)

Removing particles The same process can be applied for the state

|α−〉 =∑

i

|νi − 1/2,mi ∓ 1/2〉 where{

mi + 1/2 if mi < 0mi − 1/2 if mi > 0

and again requiring for a positive excitation E−α − Eα ≥ 0, we get gives for a 2×2 checkerboard

µ+ ≥ 4U(ν − 1/2)− 4J2

UNz(ν + 1/4)− 4(V − J2

U)z(ν − 1/4) (3.16)

With Eqs (3.15) and (3.16) one can plot the phase diagram for different values of ν as shown onfigure 3.2. The role of µ+ thus becomes obvious : it fixes the value of ν and the ground state that canbe reached (for a weak value of J). For a chosen µ+ the part of the Hamiltonian (3.12) which dependson ν becomes a constant and the remaining part can now be studied with the usual methods.

3.3. Phase diagrams with µ− 31

0 0.05 0.1 0.15 0.2 0.25 0.3

-0.2

1.4

3

4.6

6.2

J/U

m+/U

n= 0

= 1/2

= 1

= 3/2

n

n

n

n

= 2

Fig. 3.2: Domains of values of ν function of µ+ and J for a 2×2 lattice with U = 20, V = 1 and µ+ = 0.

3.3 Phase diagrams with µ−

Using the previous results it is now possible to limit the studies to the Hamiltonian

Heff = Htunneling + H({mi})

= −µ−∑

i

mi +(

2V − 2J2

U−2J2V

U2

)∑

〈i,j〉mimj

(2J2

U−2J2V

U2

)∑

〈i,j〉c†i cj+

4J2V

U2

∑

〈i,j〉mj(mj − mi)

3.3.1 Mean-field perturbative approach

Following the usual steps for this method we calculate the order parameter :

ψi = 〈ci〉

=2J2

U

(1−V

U

)ψie

βE0

∫ β

0dτ Tr

[cie

−(β−τ)H({mi})c†ie−τH({mi})

](3.17)

and after some calculations one obtains :

ψi =2J2

U

(1−V

U

)ψi

[νi(νi + 1)−mi(mi + 1)

EiP

+νi(νi + 1)−mi(mi − 1)

EiH

](3.18)

where

EiP = −µ− + 2

(2V − 2J2

U−2J2V

U2

)mi+

4J2V

U2

2zmi − 2

∑

〈k〉imk − z2

(3.19)

EiH = µ− − 2

(2V − 2J2

U−2J2V

U2

)mi−4J2V

U2

2zmi − 2

∑

〈k〉imk + z2

(3.20)

with z being the number of nearest neighbors and the red terms arising from 1st order. Here theindices P and H do not refer to a particle/hole excitation but to the two available flipping P ⇔↓→↑


and H ⇔↑→↓. Figure 3.3 shows the lobes obtained for simple, double and triple checkerboards at 0th

and 1st orders. One can see that 1st order lobes are bigger than the 0th ones what can introduce adoubt concerning the convergence of the second-order expansion but, even if further studies must bedone to check this, it is most probably the case.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1.5

-1

-0.5

0

0.5

1

1.5

J/U

m-/U

Simple Checkerboard order 0

Simple Checkerboard order 1

Double Checkerboard order 0

Double Checkerboard order 1

Triple Checkerboard order 0

Triple Checkerboard order 1

Fig. 3.3: Mott lobes for the first checkerboards at 0th and 1st orders for a 2×2 lattice with U = 20 and V = 1.

3.3.2 Gutzwiller Equation

As ν is more a parameter than a variable for a given µ+, the Gutzwiller Ansatz can very easily bewritten :

|Φ(t)〉 =⊗

i

ν∑m=−ν

f (i)m (t)|ν, m〉i where ∀i,

ν∑m=−ν

|f (i)m (t)|2 = 1 (3.21)

and one can derive the Gutzwiller equation corresponding to the Hamiltonian (3.17) :

idf

(i)m

dt=

−µ− +

(4V − 4J2

U−4J2V

U2+

8zJ2V

U2

)〈mi〉−16J2V

U2m− 8J2V

U2

∑

〈j〉i

∑

〈k〉j〈mk〉(1− δi,k)

mf (i)

m

−2J2

U

(1−V

U

) [√ν(ν + 1)−m(m− 1)ψif

(i)m−1 +

√ν(ν + 1)−m(m + 1)ψ∗i f

(i)m+1

](3.22)

where notations mean

〈mi〉 =ν∑

m=−ν

m|f (i)m |2 (3.23)

...i =∑

〈k〉i... (3.24)

ψi =ν∑

m=−ν

√ν(ν + 1)−m(m + 1)f (i)∗

m f(i)m+1 (3.25)

and with red terms arising from the 1st order expansion. Figure 3.4 shows the results of the solvingof the Gutzwiller equation with the usual variables at 0th order and for ν = 1/2 (first checkerboard).One can see the perfect matching between the lobes given by the two lobes and the confirmation that


ν = 1/2 corresponds to the simple checkerboard (n+ = n− = 1/2 inside the lobe). The last phasediagram also presents the super-solid phase which was found between the Mott lobe and the superfluidphase. Super-solid phase is characterized by non-vanishing order parameter coexisting with brokentranslational symmetry, namely a modulation of both density and order parameter on a scale largerthan the one of the lattice spacing [10]. For instance here are the values of n↑,i, n↓,i and ψi in thesuperfluid and supersolid phases for a solving of the Gutzwiller equation with parameters similar tofigure 3.4 but with L = 4 (to make broken translational symmetry more obvious) :

• Superfluid phase : µ−/U = 0.3 and J/U = 0.2

n↑,i n↓,i ψi

0.7500 0.7500 0.7500 0.7500 0.2500 0.2500 0.2500 0.2500 0.4330 0.4330 0.4330 0.4330

0.7500 0.7500 0.7500 0.7500 0.2500 0.2500 0.2500 0.2500 0.4330 0.4330 0.4330 0.4330

0.7500 0.7500 0.7500 0.7500 0.2500 0.2500 0.2500 0.2500 0.4330 0.4330 0.4330 0.4330

0.7500 0.7500 0.7500 0.7500 0.2500 0.2500 0.2500 0.2500 0.4330 0.4330 0.4330 0.4330

• Supersolid phase : µ−/U = 0.0155 and J/U = 0.1585

n↑,i n↓,i ψi

0.5175 0.5212 0.5175 0.5212 0.4825 0.4788 0.4825 0.4788 0.4997 0.4996 0.4997 0.4996

0.5212 0.5175 0.5212 0.5175 0.4788 0.4825 0.4788 0.4825 0.4996 0.4997 0.4996 0.4997

0.5175 0.5212 0.5175 0.5212 0.4825 0.4788 0.4825 0.4788 0.4997 0.4996 0.4997 0.4996

0.5212 0.5175 0.5212 0.5175 0.4788 0.4825 0.4788 0.4825 0.4996 0.4997 0.4996 0.4997

Supersolid phases are also present for double and triple checkerboard as shown on figure 3.5, itis worth noticing that a bigger ν (ν = 1 and 3/2 respectively for double and triple checkerboard)implies that the corresponding lobe of the ground state is more stable against flipping because thelobe gets wider. The extended Bose-Hubbard model with up/down dipoles thus appears to be a verygood candidate for the observation of supersolid phase. It is worth noticing that in the case of theflipping/exchange excitation considered in this chapter the superfluid phase where 〈ci〉 = 〈aib

†i 〉 6= 0

is usually known as supercounterfluid phase [13, 14] to indicate that the transports of the two specieshappen in opposite direction.

The new approach developed in this chapter, although quite successful, still presents some lacks.One of the most important is that all phase diagrams were plotted in terms of µ+ = µ↑ + µ↓ andµ− = µ↑ − µ↓. The use of these variables has two disadvantages : their physical meaning is notvery clear and they don’t allow one to plot the lobes on the usual way (that is lobes organizing on“floors” with an increasing chemical potential) making the comparison with other system not easy,particularly with the lobes found in chapter 2. Moreover the statement in chapter 2 was µ↑ = µ↓ whatwould lead with the flipping model to µ− = 0 and lobes reducing to lines. But I didn’t manage to finda clearer way to study these equations, it will require further studies. An other point which has notbeen checked in this study are the limits of validity of the effective Hamiltonian [10, 1], beyond whichthe virtual subspace introduced to account the flipping excitation looses its meaning. This could beachieve by selecting the dominant classical configurations from the Gutzwiller wavefunction for eachpoint of the phase diagram. Then one has to calculate for each of these configurations, the lobe withrespect to usual particle/hole excitations. If the system at this given point of the phase diagram turnsout to be stable against all dominant particle/hole excitations (in other words, if this point is insideall selected particle/hole lobes), the effective Hamiltonian is valid at that point as done in [10].


Fig. 3.4: Phase diagram for different variables namely n± =∑

i(ν±mi)/L2 and φ =∑

i ψi/L2 for a 2×2 latticewith U = 20, V = 1, ν = 1/2 (simple checkerboard), 0th order, first rank of nearest neighbors andperiodic boundaries. As usual the shaded tones correspond to the different values of φ and n± whilethe black lines to the simple checkerboard lobes calculated with the mean-field perturbative approach(see figure 3.3). The Super Solid domain diagram is based on the φ diagram with the white super-solidpoints added.


Double checkerboard Triple checkerboard

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

m- /U

J/U J/U

m- /U

Fig. 3.5: Mean-field perturbative lobes (black lines) and corresponding supersolid phase (green points) for doubleand triple checkerboards for a 2×2 lattice with U = 20, V = 1, 0th order, first rank of nearest neighborsand periodic boundaries.


4. CONCLUSION

As explained along the different chapters further studies are required to confirm the analysis andthe results of this report. Indeed major points like stability of the effective Hamiltonian, convergenceof the perturbative terms, physical understanding of the µ+ and µ− variables, phase transition char-acterization (with transition order and scaling theory see [18] for calculation on simple Bose-Hubbardmodel) and exact calculations (Monte-Carlo solving) are some of the necessary works which have stillto be done. However, considering the first results, mixtures of up/down dipoles in optical latticesappear to be a very promising system to observe original quantum phases like supersolid and super-counterfluid. And even more exciting are the many variations which can be added to the basic systemand could eventually induce very interesting physical effects, among them :

• pseudo-random site offset [18] which can be introduced by a site-dependent chemical potential,

• side effects and non-periodic optical lattices,

• introduction of more 2D-layers [10], up/down dipoles in 3D optical lattices and large 2D lattices,

• research for the presence of metastable states [21] which could appears inside the checkerboardlobes,

• dipole interactions with more nearest neighbors to study long-range effects,

• etc.

All these possibilities show that the study of dipolar systems in optical lattices is a wide-open fieldof research full of promises, in particular with the constant development of the experimental works onthe subject. About possible experimental realizations it has never discussed so far in this report whichparticles/atoms could be employed to implement such systems. From the different experimental worksabout dipoles in optical lattices it seems that the most promising particles could be heteronuclearHund A molecules [11, 12]. Indeed such molecules present two independent magnetic and electricdipoles and, thanks to this particularity, one of the dipoles could be used to trap the molecules inthe optical lattice while the other one could be driven by either electric or magnetic fields in order tocreate up and down dipoles.

Concerning my personal interest in this internship, I found really interesting to work on a systemwhich has never been studied before and thus to develop a new model with all the inevitable trial anderror, especially because part of my PhD work will concern this subject and so I will maybe have thepossibility to keep on working on the points I didn’t finish during my stay.

38 4. Conclusion

Thanks : I would like to thank the ICFO, Maciek for giving me the opportunity to effectuate aso interesting internship in such a good atmosphere and for allowing me to continue this experiencethrough my PhD. I warmly thank Hannes, Andre, Emmanuel, Nico, Amel, Antoine and Diego forenjoying my time at ICFO and Barcelona. I also thank Chiara and Francois for their very useful helpand advices during my stay. I would especially like to thank Christian for all his help and patience, itwas a real pleasure to work with you and I am glad to continue next year.

APPENDIX

A. MATRIX EXPRESSION OF THE MEAN-FIELD PERTURBATIVE METHOD

Starting from Eq. (1.32), one can define the following variables :

Φ .=

ϕ1

ϕ2

...ϕL2

N.= L2 × L2−matrix of nearest neighbors

Ci.= J

[ni + 1

Uni − µ− ni

U(ni − 1)− µ

]

C.=

C1

C2

...CL2

such that the L2 Eqs. (1.32) rewrite

IL2×L2Φ = (tC IL2×L2) N Φ⇔ [

IL2×L2 − (tCIL2×L2) N]Φ = 0

and by defining M = IL2×L2 − (tCIL2×L2) N , the system MΦ = 0 has non-zero solution only fordet(M) = 0.

42 A. Matrix expression of the mean-field perturbative method

BIBLIOGRAPHY

[1] C. Menotti, M. Lewenstein, T. Lahaye, T. Pfau, Dipolar interaction in ultra-cold atomic gases,cond-mat/07113422v1 (2007).

[2] M. Greiner, O. Mandel, T. Esslinger, T. W. Hnsch, I. Bloch, Quantum phase transition from asuperfluid to a Mott insulator in a gas of ultracold atoms, Nature, 415 39-44 (2002).

[3] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De), U. Sen, Ultracold atomicgases in optical lattices : mimicking condensed matter physeics and beyond, cond-mat/0606771v2(2006).

[4] D. Jaksch, P. Zoller, The cold atom Hubbard toolbox, cond-mat/0410614v1 (2004).

[5] I. Bloch, J. Dalibard, W. Zwerger, Many-body physics with ultracold gases, cond-mat/07043011v2(2007).

[6] N. W. Ashcroft, N. D. Mermin, Solid state physics, Brooks Cole (1976).

[7] S. Sachdev, Quantum Phase Transitions, Cambridge University Press (1999).

[8] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Atom-Photon interactions, Wiley-VCH(1998).

[9] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, P. Zoller, Cold bosonic atoms in optical lattices,Phy. Rev. Lett., 81 (15), 3108 (1998).

[10] C. Trefzeger, C. Menotti, M. Lewenstein, Pair-supersolid phase in a bilayer system of dipolarlattice bosons, cond-mat/0904.1552v1 (2009).

[11] B. C. Sawyer, B. L. Lev, E. R. Hudson, B. K. Stuhl, M. Lara, J. L. Bohn, J. Ye, Magnetoelectro-static trapping of ground state OH molecules, Phy. Rev. Lett., 98 253002 (2007).

[12] B. L. Lev, E. R. Meyer, E. R. Hudson, B. C. Sawyer, J. L. Bohn, J. Ye, OH hyperfine groundstate : from precision measurement to molecular qubits, Phy. Rev. A, 74 061402 (2006).

[13] A. Kuklov, N. Prokof’ev, B. Svistunov, Superfluid-superfluid phase transitions in two-componentsBose system, cond-mat/0305694v1 (2003).

[14] A. Kuklov, N. Prokof’ev, B. Svistunov, Commensurate two-component bosons in an optical lattice: ground state phase diagram, Phy. Rev. Lett., 92 5 050402 (2007).

[15] A. Kuklov, B. Svistunov, Counterflow superfluidity of two-species ultracold atoms in a commen-surate optical lattice, Phy. Rev. Lett., 90 10 (2003)

[16] R. T. Scalettar, G. G. Batrouni, G. T. Zimanyi, Localization in interacting, disordered, Bosesystems, Phys. Rev. Lett., 66 3144 (1991).

[17] J-S. Bernier, K. Sengupta, Y. B. Kim, Mott phases and superfluid-insulator transition of dipolarspin-three bosons in an optical lattice : implications for 52Cr atoms, cond-mat/0612044v1 (2006).

[18] M. P. A. Fisher, P. B. Weichman, G. Grinstein, D. S. Fisher Boson localization and the superfluid-insulator transitions, Phys. Rev. B, 40 1 (1989).

44 Bibliography

[19] A. Isacsson, M-C. Cha, K. Sengupta, S. M. Girvin Superfluid-insulator transitions of two-speciesbosons in an optical lattice, cond-mat/0508068v1 (2005).

[20] M.A. Baranov, Theoretical progress in many-body physics with ultracold dipolar gases, NaturePhysics, Physics Reports, 464 71111 (2008).

[21] C. Trefzeger, C. Menotti, M. Lewenstein, Ultracold dipolar gas in an optical lattice : the fate ofmetastable states, Phys. Rev. A, 78 043604 (2008).

[22] I. Carusotto, Y. Castin, An exact reformulation of the BoseHubbard model in terms of a stochasticGutzwiller ansatz, New Journal of Physics, 5 91.191.13 (2003).

Documents

Ultracold Dipolar Bosons In Optical Lattices · Ultracold Dipolar Bosons In Optical Lattices ... r¶ealis¶es dans la manipulation d’atomes ultra-froids permettant le pi¶egeage