Alexandre Dauphin
Bruxelles, mai 2011
Université Libre de Bruxelles
1
Faculté des Sciences
Alexandre Dauphin
Bruxelles, mai 2011
Université Libre de Bruxelles
Service de Physique des Systèmes Complexes et Mécanique Statistique
Alexandre Dauphin Bruxelles, mai 2011
Mémoire présenté en vue de l’obtention du grade légal de maître en Sciences physiques
1 Service de Physique des Systèmes Complexes et Mécanique Statistique
Alexandre Dauphin Bruxelles, mai 2011
Mémoire présenté en vue de l’obtention du grade légal de maître en Sciences physiques
1
Cold atom quantum simulation of
topological phases of matter
A thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics.
May 2015
Under the supervision of Pierre Gaspard and Miguel ´Angel Mart´ın-Delgado
First of all, I am grateful to my supervisors Pierre Gaspard and Miguel Angel Martin Del-gado for their support and encouragement throughout these four years. Their advices and insights were always constructive and useful. I enjoyed the stimulating discussions about physics we had.
I also want to thank Nathan Goldman and Markus M ¨uller for their help. I learned a great deal from them. It was a pleasure to collaborate with them all along this thesis developing ideas and writing papers.
I am also grateful to the great physicists I collaborated with: M. Lewenstein, J. Dalibard, I. Spielman, F. Gerbier, P. Zoller, P. Massignan, A. Celi, L. Tarruell and D.-T. Tran. I am thankful for the opportunity to visit labs and groups. In particular, I want to mention the very interesting and stimulating visit of the labs of Ian Spielman at the NIST. More recently, at IFCO, I visited with great interest the labs of Leticia Tarruell and also had two very constructive visits to the group of Maciej Lewenstein.
Moreover, I want to thank my colleagues in Brussels of the group Complex Systems and Statistical Mechanics and my colleagues in Madrid of the Department of Theoreti-cal Physics 1. I want to thank particularly Pierre De Buyl for his great help at beginning of the thesis when I was struggling programming languages.
I had a lot of support from my close friends in Brussels: Laurent, Colin, Thi-Tien, Lu-cien, Simon, Manu, Sylvie, Olmo, Blagoje, Antoine and Jonathan. I also met great friends in Madrid: Alex, Yadira, Jenifer, Edgar, Samu, Marco, Viviana, Jose, Oscar, Laura, Alvaro, Emma, David, Davide, Jose Alberto, Joserra, Marion and Karla. I often spent quality time with them.
Contents 5
1 Introduction 9
2 Introduction to topological quantum insulating phases 15
2.1 Introduction . . . 15
2.2 Properties of the bulk: periodic boundary conditions . . . 16
2.2.1 Periodic boundary conditions and Harper’s equation . . . 16
2.2.2 Hall conductivity and relation to the topology . . . 18
2.2.3 Fiber bundles and topological invariants . . . 20
2.2.4 Lattice gauge theory and topological invariant . . . 22
2.3 Cylinder geometry and edge states . . . 23
2.3.1 Introduction . . . 23
2.3.2 Energy spectrum and chiral edge states . . . 24
2.4 Box geometry . . . 26
3 Basic aspects of cold atom physics 29 3.1 Introduction . . . 29
3.2 Light-matter interaction: the dipole potential . . . 30
3.3 Cooling the atomic gas . . . 31
3.4 Cold gases: Bose-Einstein condensates and Fermi gases . . . 33
3.5 Optical lattices and artificial gauge fields . . . 34
4 Topological Mott insulator 39 4.1 Introduction . . . 39
4.2 Spinless fermions on a hexagonal lattice and the Haldane model . . . 41
4.3 The interacting model . . . 45
4.4 Mean-field analysis . . . 45
4.4.1 Mean-field treatment of the nearest-neighbor interaction . . . 45
4.4.2 Mean-field treatment of the next-to-nearest-neighbor interaction . . 47
4.4.3 Complete Hamiltonian in the mean-field approximation . . . 49
4.6 Free energy study . . . 54
4.7 Chern number characterization . . . 55
4.8 Phase coexistence . . . 58
4.9 Quantum simulation with Rydberg atoms in optical lattices . . . 60
4.9.1 Introduction . . . 60
4.9.2 Rydberg atoms . . . 60
4.9.3 Rydberg dressing scheme and effective long-range interactions . . . 63
4.9.4 Experimental feasibility, Rydberg states and laser parameters for a quantum simulation . . . 67
4.10 Conclusions . . . 68
5 Detection of topological insulating phases in cold atom experiments 71 5.1 Introduction . . . 71
5.2 Extracting the Chern number from the dynamics of a Fermi gas . . . 72
5.2.1 Group velocity of a single state in the Fourier space . . . 72
5.2.2 Group velocity of the lowest fermionic filled band and position of the center of mass of the atomic cloud . . . 73
5.2.3 Measurement of the Chern number from the dynamics of the atomic cloud . . . 74
5.2.4 The model . . . 75
5.2.5 Time evolution of the atomic cloud . . . 77
5.2.6 Results of the simulations . . . 78
5.2.7 Summary of the first method . . . 90
5.3 Imaging the evolution of topological edge states . . . 90
5.3.1 The method . . . 91
5.3.2 Dynamics of topological edge states: case of the flat band . . . 92
5.3.3 Dynamics of topological edge states: case of the dispersive band . . . 93
5.3.4 Summary of the second method . . . 98
5.4 Conclusions . . . 98
6 Efficient algorithm to compute the Berry conductivity 101 6.1 Introduction . . . 101
6.2 Generalized Berry conductivity . . . 103
6.3 Construction and properties of the algorithm . . . 105
6.4 Calculation of the Berry curvature and Chern number: efficient algorithm . 107 6.4.1 Discretization of the TKNN Formula for a two-band model . . . 108
6.4.2 FHS formula for a two-band model . . . 109
6.4.3 Comparison between the algorithms . . . 110
6.5 Methods to compute the statistical error . . . 113
6.6 Practical application of the algorithm . . . 118
6.6.1 The Haldane model . . . 118
6.6.2 The Hofstadter model . . . 128
6.6.3 The BHZ model . . . 130
7 Characterization of topological quantum Hall phases in non-periodic structures133
7.1 Introduction . . . 133
7.2 Real-space Chern invariant . . . 134
7.2.1 Real-space Chern invariant for a crystal . . . 134
7.2.2 Real-space Chern invariant for a non-periodic system . . . 137
7.2.3 Real-space Chern invariant for a tight-binding system . . . 137
7.3 Characterising the topology of the Hofstadter model . . . 138
7.3.1 Open boundary conditions . . . 138
7.3.2 Adding disorder . . . 141
7.4 Characterizing the topology of a quasicrystal . . . 143
7.4.1 Introduction . . . 143
7.4.2 The model . . . 144
7.4.3 Local Chern number for quasicrystals . . . 147
7.4.4 Numerical results . . . 147
7.4.5 Probing the chiral edge states . . . 149
7.5 Conclusions . . . 152
8 Conclusions and perspectives 153 Appendices 155 A Topological Mott insulator in the honeycomb lattice 157 A.1 Details on the Brillouin zone and domain of integration . . . 157
A.2 Numerical calculation of the Chern numbers . . . 159
B Extracting the Chern Number from the dynamics of a Fermi gas 163 B.1 Group velocity of a single state of quasi-momentumk in the Fourier space . 163 B.2 Gauge Transformation of the Hamiltonian with an external force . . . 164
B.3 Group velocity of a ground state ofN fermions . . . 164
B.4 Velocity of the center of mass of the first filled band . . . 165
B.5 Velocity of the center of mass of the first partially filled band . . . 166
B.6 Bloch oscillations and dynamics at arbitrary times: flat bands vs dispersive bands . . . 166
B.7 The Chern number measurement using smooth confinements . . . 167
B.7.1 Releasing the confinement in both directions . . . 167
B.7.2 Releasing the confinement along the transverse direction only . . . . 169
Introduction
The study of phases of matter is fundamental in physics. Most phases of matter are characterized by the symmetries they break. In this context, Landau theory, the “standard model” of phase transitions, associates a local order parameter to each symmetry breaking [1] and allows the description of remarkable phases such as Bose-Einstein condensates [2, 3], superconductivity [4], superfluidity [5] and many others.However, Landau theory fails to describe topological phases, which constitute a new paradigm. Here, the order of the system is characterized by a global property, called topo-logical invariant. Topotopo-logical invariants classify the systems in terms of their topology [6]: Two objects are considered to belong to the same topological class if there exists a con-tinuous transformation which deforms the first object into the second object. This global property protects the phases against local perturbations such as disorder or interactions. Complementary to the fundamental interest in the new paradigm, topological phases have also fascinating, potential applications in quantum computation: The non-Abelian statistics of quasi-particles appearing in topological superconductors, such as Majorana fermions, or in fractional Chern insulators, can be used for topological quantum process-ing [7].
We here focus on topological insulating phases [8, 9]. Topological insulators are bulk insulators with conducting surface states. Discovered in the 80s, the integer quantum Hall effect [10] constitutes the first example of such kind of systems. When a magnetic field is applied on a 2D semi-conductor at very low temperature, the Hall conductivity is quantized in the energy gaps of the bulk [10]. In fact, this quantization is related to a topological invariant of the bulk, the total Chern number of the occupied energy bands [11]. The system is thus very robust against perturbations: As long as the gap is open, the system remains in the same class of topology and the Hall conductivity is stable and quantized in the energy gap. Finally, for a system with boundaries, the Hall current is transported by edge states, protected by the topology of the bulk [12], a mechanism often called the bulk-edge correspondence.
considered by Haldane in the end of the 80s [13], which have nontrivial topology but with no net magnetic flux through the unit cell of the lattice. In 2005, Kane and Mele realized that such property can be found in graphene for each spin [14, 15]: the spin-orbit cou-pling is generating the nontrivial topology. This model would exhibit the 2D quantum spin Hall effect, a time-reversal topological insulator. However, in the case of graphene, the effect turned out to be too small to be observed experimentally [16, 17]. A year later, the quantum spin Hall effect was proposed in quantum nanowells [18] and realized ex-perimentally in 2007 [19]. A generalization of this result was theoretically proposed in 3D [20] and realized experimentally [21].
Note that, here, in the quantum spin Hall effect, the Hall conductivity is null, due to the time-reversal symmetry. Nevertheless, the Hall conductivity for each spin is quan-tized and the transport is performed by topologically protected and spin-dependent edge states [18]: On each edge, two spins are transporting the current with opposite veloci-ties. One can already see that the choice of the topological invariant is depending on the discrete symmetries protected in the system. In this regard, the topological invariant as-sociated with the quantum spin Hall effect is aZ2topological invariant [18]. In particular,
if there are no interactions between spins such as Rashba interactions, theZ2topological
invariant is proportional to the difference of the spin conductivities.
A complete classification of the topological insulators and topological superconductors without interactions, the celebrated “periodic table” [22], has been carried out in terms of the discrete symmetries that are protected in the system: time reversal, particle-hole and sublattice symmetry. This classification associates aZ or Z2 topological invariant to
the model. Two models in the same discrete symmetry class with different values of the topological invariant are topologically inequivalent.
Let us finally mention that the “periodic” table is not the end of the story. Recent works pointed out extensions of the “periodic table” such as topological crystalline insulators [23], topological Kondo insulators [24], topological Weyl semimetals [25], Floquet topo-logical insulators [26] or symmetry-protected phases at finite temperature [27]. As illus-trated by these recent developments, research in topological phases of matter has become a highly active and rich field of physics.
First, laser cooling enables reaching very low temperatures [32], where atomic gases be-gin to have purely quantum behavior depending on the particle statistics. These sophisti-cated techniques were applied in seminal experiments where Bose Einstein condensates [33, 34] and degenerate Fermi gases [35] were achieved.
Furthermore, light can be used to trap particles [32], e.g. in optical tweezers, to gen-erate optical potentials or optical lattices. These tools allow one to simulate solid state crystalline physics in a very versatile environment with the control of the hopping, the lattice geometry, the interactions, and other relevant system parameters [29, 30]. These very controllable tools have been used, to realize, for example, the quantum simulation of the Bose-Hubbard model and study the transition between the superfluid and the Mott insulating regimes [36].
Finally, the simulation of the artificial gauge fields is also feasible. Indeed, cold atoms are neutral atoms and do not feel the magnetic field like charged particles. There are how-ever show-everal proposals to engineer artificial magnetic fields such as laser-assisted tunnel-ing, shaktunnel-ing, rotations of the atomic cloud [30, 37] and which enable to reach very high effective magnetic fields. Using such methods, several experiments have recently simu-lated the physics of the Hall effect [38], the spin Hall effect [39], the integer quantum Hall effect for bosons [40, 41, 42, 43], the Haldane model for fermions [44].
This thesis addresses three mains topics related the quantum simulation of topological insulators with cold atoms.
The second topic of this thesis concerns the detection of topological insulating phases in cold atom experiments. In condensed matter physics, topological order is generally measured in transport experiments [8, 9]. However, in cold atoms, transport experiments are very demanding [48, 49]. There is thus a need for methods to characterize topological properties in cold atom experiments. Two manifestations of the topology can be used to characterize the topology: the presence of chiral conducting edge states and the topolog-ical invariant of the bulk.
The third topic of this thesis concerns the development of tools to characterize the topology in systems beyond the standard insulating phases. In this thesis we present two studies in this direction: The first refers to the characterization of the topology for a sys-tem where the Fermi energy lies within an energy band. In this case, the Hall conductivity is no longer quantized and there is no direct relation with the topological invariant. It is still possible to characterize the topological contribution to the topology, called the Berry conductivity [50]. However, the computation of this quantity is nontrivial and there is the need for an efficient algorithm. The second one refers to the characterization of the topology of non-periodic systems and in particular quasicrystals. Here, since there is no periodicity, the bulk-edge correspondence is not obvious and techniques should be de-veloped to characterize the topological properties.
This thesis is organized as follows:
In Chapter 2, we briefly review the basic notions of topological insulators by treating the integer quantum Hall effect. We start by studying the properties of the bulk: the quantiza-tion of the Hall conductivity and its relaquantiza-tion to the topological invariant, the lattice gauge theory and its relation to the continuous gauge theory. Next, we discuss the bulk-edge correspondence in the cylindrical geometry. Finally, we characterize the energy spectrum of a finite size system.
In Chapter 3, we briefly discuss some basics of cold atom physics. We first start by pre-senting some cooling methods to reach very low temperatures. We then review the possi-ble quantum phase transitions at these temperatures, namely Bose Einstein condensate for bosons and degenerate Fermi gas for fermions. Next, we discuss the trapping of atoms with light and in particular optical potentials and optical lattices. Finally, we discuss the generation of artificial gauge fields and sketch the laser-assisted tunneling method intro-duced by Jaksch and Zoller in Ref. [51].
atomic cloud with the topological invariant in the presence of a constant force and there-fore provides a measure of the Chern number. The second one allows one to see by direct imaging the propagation of the topological edge states. The two methods are comple-mentary since they measure two different characteristics of topological insulators.
In Chapter 6, we introduce an efficient algorithm to compute the Berry conductivity, the topological contribution to the Hall conductivity. We show that the algorithm is effi-cient, gauge invariant and provides results with controllable error bars. Next, we bench-mark it on several physically relevant models, discuss the convergence of the algorithm, and compute upper error bounds.
In Chapter 7, we review the real-space Chern invariant introduced by Bianco and Resta in Ref. [52]. The latter allows one to characterize the topology of non-periodic systems. Next, we benchmark it on the integer quantum Hall effect with and without disorder to study the robustness of the method. Finally, we apply the method to characterize the topology of a quasicrystal in a constant magnetic field.
The original results that are presented in this thesis have been published in the follow-ing articles:
1. Rydberg-Atom Quantum Simulation and Chern Number Characterization of a Topo-logical Mott Insulator,
A. Dauphin, M. M ¨uller and M. A. Martin-Delgado, Phys. Rev. A 86, 053618 (2012).
2. Direct imaging of topological edge states in cold-atom systems ,
N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller and I. B. Spielman,
PNAS 110, 6736 (2013).
3. Extracting the Chern number from the dynamics of a Fermi gas: Implementing a quantum Hall bar for cold atoms,
A. Dauphin and N. Goldman, Phys. Rev. Lett. 111, 135302 (2013).
Highlights in Nature Physics by I. Georgescu [Nat. Phys. 9, 692 (2013)]. 4. Efficient algorithm to compute the Berry conductivity,
A. Dauphin, M. M ¨uller and M. A. Martin-Delgado, New J. Phys. 16, 073016 (2014) .
5. Topological Hofstadter Insulators in a Two-Dimensional Quasicrystal , D.-T. Tran, A. Dauphin, N. Goldman and P. Gaspard
Introduction to topological quantum insulating phases
2.1
Introduction
In this chapter, we review the properties of the integer quantum Hall effect (IQHE) [10], a hallmark example of a topologically protected system. We discuss the quantization of the Hall conductivity [11] and its relation to conducting edge states protected by topology [12]. A tight-binding model qualitatively describes the properties of the phenomenon: the Hofstadter model, where spinless fermions on a two-dimensional square lattice are subject to a uniform perpendicular magnetic fieldB [53]. Interestingly, the Hofstadter model can also be realized in cold atom simulations. in fact, it has been recently realized experimentally in particular with bosonic atoms [41, 42, 43].
B
Jei✓jk
Figure 2.1: The Hofstadter model is described by a tight-binding Hamiltonian of spinless fermions on a two-dimensional square lattice under a uniform magnetic fieldB perpen-dicular to the lattice. a is the lattice spacing and Φ the magnetic flux through each unit cell.
Let us consider a tight-binding Hamiltonian of spinless fermions on a square lattice: ˆ
H = JX
hj,ki
eiθjkcˆ†
where the sum is performed iver the nearest-neighbor links,J is the hopping amplitude, ˆ
c†j creates a fermion at the lattice siterj, andexp (iθjk) refers to the Peierls phase factor
due to the magnetic field associated to the vector potentialA [53]:
θjk = θ(rj, rk) = e ~ Z rk rj A· dl, (2.2)
In order to study the transport properties of the model, we write the Schr¨odinger equation forψm,n := ψ(ma1x+ na1y). In the Landau gauge A(r) = B rx1y, the Peierls phase factor
is equal to:
θjk = 2π for nj = nk,
θjk = 2π i φ mj(nk− nj) for mj = mk, (2.3)
(2.4) whereφ = Φ/φ0is the number of magnetic flux quanta expressed in terms of the
mag-netic fluxΦ = Ba2and the magnetic quantumφ
0= h/e. One then finds
Eψm,n = Jψm,n+1+ Jψm,n−1+ Je2πiφmψm,n+1+ Je−2πiφmψm,n−1. (2.5)
The solutions of this equation depend on the choice of the boundary conditions. We first study a system with periodic boundary conditions to characterize the properties of the bulk and then we proceed to geometries with boundaries.
2.2
Properties of the bulk: periodic boundary conditions
In this section, we introduce the periodic boundary conditions and derive the cele-brated Harper equation [54]. We also review the quantization of the Hall conductivity in the linear response theory and discuss its relation to a topological invariant, the Chern number.
2.2.1 Periodic boundary conditions and Harper’s equation
Let us consider a system ofNxsites in thex-direction and Nysites in they-direction. We
impose periodic boundary conditions on the Hamiltonian. The periodicity of the Peierls phase factor at the boundaries restricts the value ofNx in the Landau gauge. Ifφ = p/q,
the Peierls phase factor has a periodicity ofq sites in the x-direction. We thus work with the magnetic cell, composed ofq sites in the x-direction and one site in the y-direction, and we apply the Bloch theorem [55]: the solutions of the Hamiltonian are the Bloch states
whereuk(r) is periodic on the supercell and is expressed in terms of the quasimomentum
k of the Brillouin zone (BZ). The resulting Hamiltonian is given by: ˆ
H = X
k∈BZ
Ψ†kHˆkΨk, (2.7)
whereΨk= (c1,k, . . . , cn,k) are defined by Ψk†|0i = |uki and ˆHkis aq× q matrix:
ˆ Hk= J f (ky, 1) eikx e−ikx e−ikx . .. . .. . .. eikx eikx e−ikx f (k y, q) , (2.8) withf (ky, m) = 2 cos (ky+ 2πφm).
The Harper equation is the resulting eigenvalue problem for eachk:
Ekuk(m) = Jeikxuk(m + 1) + Je−ikxuk(m− 1) + Jf(ky, m)uk(m), (2.9) ⇡/5 ⇡/5 kxa kya ⇡ ⇡ 0 3 E/J 3
Figure 2.2: Energy spectrum of the Hofstadter model forp = 1, q = 5. The system presents four energy gaps.
2.2.2 Hall conductivity and relation to the topology
One can use the Kubo formula [56] to compute the Hall conductivityσH := σyx:
σH = 1 V Z ∞ 0 eiωtdt Z β 0 dλh ˆJy(−iλ~) ˆJx(t)ieq, (2.10)
where ˆJ(t) = eiHtJeˆ −iHtis the current operator in the interaction representation,ω is the
frequency of the electric field,V the volume, and β = 1/(kBT ) andh ˆJµieq = tr(ˆρeqJˆµ) is
written in terms of the density operatorρˆeq.
When the system is at zero temperature and the Fermi energy is in an energy gap, the Hall conductivity takes the form [11]:
σH =− e2 h X Ei<EF,Ej>EF 1 2π Z BZ d2k2 Im[hu i k|∂kxHk|u j kihu j k|∂kyHk|u i ki] (Ei(k)− Ej(k))2 , (2.11)
where the sum is involving the occupied states with energyEiand the empty bands with
energyEj. This equation is called the Thouless-Kohmoto-Nightingale-den Nijs (TKNN)
formula and can be written only in terms of the occupied states using the closure relation:
σH = e2 h X n∈occ. bands 1 2πi Z BZ d2k h∂u n k ∂kx| ∂un k ∂kyi − h ∂un k ∂ky| ∂un k ∂kxi . (2.12)
The last equation highlights the proportionality between the Hall conductivity and the sum over the occupied bands of a quantity, called the Chern number Cα of the bandα.
The Chern number is expressed as the surface integral over the whole Brillouin zone of a function of the eigenvectors of the bandα. Interestingly, the Chern number
Cα = 1 2πi Z BZ d2k h∂u n k ∂kx| ∂un k ∂kyi − h ∂un k ∂ky| ∂un k ∂kxi (2.13) is in fact a topological invariant of the bandα. The notion of topology and its relation to gauge theory will be reviewed in the next section. When the Fermi energy lies in an energy gap, the Hall conductivity is quantized and is proportional to the sum of the Chern number of the occupied bands.
Let us emphasize that since it is a topological property of the energy band, the quan-tization of the Hall conductivity is robust against perturbations. To be more precise, the Chern numberCαis robust as long as the energy bandα does not touch another energy
band (gap closing).
Figure 2.3 represents the Hall conductivity forp = 1, q = 5. As expected, when the Fermi energy lies in an energy gap,σH is proportional to the sum of the Chern numbers of the
0 2 2 3 3 0 EF/J C1= 1 C2= 1 C3= 4 C4= 1 C5= 1 a. b. H/(e2/h)
Figure 2.3: a. Chern numbers associated with the different energy bands forp = 1, q = 5.
b. Plot of the Hall conductivity (in red) in terms of the Fermi energyEF. ForEF in an
energy gap,σH is proportional to the sum of the Chern numbers of the bands belowEF.
4 0 4 0 1 E/J H =e 2/h H = e2 /h H =e 2/h H = e2 /h
Figure 2.4: The Hofstadter butterfly [53]: Energy spectrum (projection) of the Hofstadter model in terms of the magnetic fluxφ = p/q. The figure displays a fractal structure with energy gaps. The Hall conductivityσH is quantized in the energy gaps. The value of the
Figure 2.4 shows the celebrated Hofstadter butterfly [53]. The energy spectrum is pro-jected on a one-dimensional line and is presented in terms of the number of magnetic flux quantaφ = p/q. The figure depicts a fractal structure with energy gaps. If EF lies in
an energy gap, the Hall conductivity is quantized - as we have seen before - and remains constant as long as the gap is open.
Finally, it is worth to mention that when the Fermi energy lies in an energy band, Eq. (2.12) is no longer valid. In fact, the value of the Hall conductivity is not an integer but there is still a contribution of the topology, called the Berry conductivity. This will be discussed in Chapter 6.
2.2.3 Fiber bundles and topological invariants Adiabatic motion and Berry phase
The Berry phase [57, 6, 50] appears in the adiabatic motion of a particle. In quantum mechanics, the wave function is defined up to phase factor. Usually, the global phase factor plays no role in the physics of the problem. However, if the Hamiltonian depends on slowly varying parameters, the phase factor can play an important role, as discussed here below.
Let us consider a Hamiltonian H(R) depending on the parameters Ri(t), 1 ≤ i ≤
Nparam. At each timet, the Hamiltonian has N instantaneous eigenvectors|n(R)i labelled
byn and satisfying:
H(R)|n(R)i = En(R)|n(R)i. (2.14)
Now, we consider the time evolution of the state|n(R)i. At some time t, the state has the form:
|ψ(t)i = eiγ(t)e−iRt 0En(R(t
0))dt0
|n(R)i, (2.15)
where the second factor is the dynamical phase factor. To find the concrete expression of the first expression, we insert it in the Schr¨odinger equationi∂tψ(t) = H(R)|ψ(t)i, and
project it on the state|n(R)i to find
γn= Z
Γ
ihn(R)|∂R|n(R)i · dR, (2.16)
whereΓ is the path followed by R in the parameter space. We define the Berry connection as
An(R) :=hn(R)|∂
One notes thatAn is not a gauge invariant quantity and, therefore γn(t) is not gauge
invariant in general. One can thus argue that the phase can be canceled by a suitable gauge transformation. However, in the case of a closed pathR(T ) = R(0), γn is gauge
invariant and cannot be removed by a gauge transformation. In this case, one calls it the Berry phase or the geometric phase. We finally apply the Stokes theorem:
γn= i I
Γ
( ¯∇R× An)· dS, (2.18)
where the integral is performed on the surface encircled by the closed path Γ, and we define the Berry curvature:
Fn(R) = ¯∇R× An(R) (2.19)
which is a gauge invariant quantity.
Let us now consider the Hamiltonian Hk := H(k) defined in the preceding section.
Here, the set of parameters are the components of the quasi-momentumk. Since we are dealing with two-dimensional systems, the Berry connection and Berry curvature vector take the form:
An(k) = i(hun k|∂kxu n ki1x+hunk|∂kyu n ki1y), Fnxy(k) = (∂kyA n x− ∂kxA n y)1z. (2.20)
We finally define the Berry curvatureFn
xy = ∂kyA
n
x − ∂kxA
n
y. The role of the Berry phase
in Eq. (2.12) becomes clearer: the integrand is the Berry curvature and the integral is the Berry phase. We will now review the Chern number.
Chern numbers
From the viewpoint of differential geometry [6], the energy bands of the Hamiltonian Hk can be seen as a vector bundle. The connection on the vector bundle is the Berry
connection and the curvature is the Berry curvature. To be precise, we should define the curvature as a two-formF acting on the fiber bundle:
F = (∂µhR|)(∂ν|Ri)dRµ∧ dRν. (2.21)
We can now define the Chern character
ch(F) := Tr exp iF 2π =X j 1 j!Tr iF 2π j . (2.22)
By definition, thej-th Chern character chjis thej-th term of the sum in Eq. (2.22):
In the case of a two-dimensional manifold associated with the vector bundle, the only non-zero Chern character is the first Chern character:
ch1(F) :=
i
2πTr(F). (2.24)
In the particular case of the HamiltonianHk, the Chern character of the bandn is
writ-ten as: chn1 = i 2π I BZ Fnxy· dS, (2.25)
and is thus proportional to the Berry phase of the whole Brillouin zone. One can show that the Chern character is an integer [6].
To be consistent with the literature, we define the first Chern number asCn
1 := −ch n 1.
Looking back at Eq. (2.12), we can now write:
σH = e2 h X n∈occ. bands C1n. (2.26)
In the rest of the thesis, the term Chern number refers to the first Chern number since we will be dealing only with two-dimensional systems, where higher Chern numbers are null.
2.2.4 Lattice gauge theory and topological invariant
In the preceding section, we introduced the Chern number for a two-dimensional man-ifold. In some cases, such as the two-band models [58], the Chern number can be com-puted analytically but it is not the case in general. In this regard, ones needs a numer-ical method to compute efficitently the Chern numbers. The naive idea is to discretize Eq. (2.12): working on a grid, we replace the derivatives by finite differences. However, this method is not gauge invariant and not efficient, as we will illustrate for an example in Chapter 6. An efficient algorithm was introduced by Fukui, Hastugai, and Suzuki to compute the Chern number [59]. The latter is based on a lattice gauge theory formulation [60, 61, 37]. We now review the lattice gauge theory for the two-dimensional torus.
Let us consider the two-torus of coordinatesR and the phase γndefined in Eq. (2.16)
and associated with then-th energy band. The latter can be seen as a continuous gauge theory on the two-torus. Let us choose a discretization: a square lattice on the torus com-posed ofNx× Nyplaquettes (this result can be generalized to any type of polygons and to
higher dimensions [61]). Now, we define the link variable between the nearest-neighbors i and j:
Uijn = eiγn(Ri,Rj), (2.27)
whereγnis computed on the line betweenR
The link variable can be interpreted as the phase acquired during the parallel transport between sitesi and j. We define the lattice Berry curvature associated with the plaquette Plof indexl as the phase acquired through the parallel transport around the plaquette in
the anti-clockwise order:
Fn
xy,l = log (UkmUmoUopUpk), (2.28)
where the logarithm is taken on the principal branch andk, m, o, p are the four vertices of the plaquette in the anti-clockwise order .
An important point is to find the relation between the continuous gauge theory and the lattice gauge theory. In fact, one can show that when the continuous Berry curvature satisfies the following admissibility condition [61, 59] for every plaquettePlof the lattice,
1≤ l ≤ NxNy: | I Pl Fn xy(R)dS| < π, (2.29)
the lattice gauge field theory corresponding to the continuous gauge field theory is unique and satisfies the relation:
iFn xy(l) =
I
Pl
Fxy(R)dS, (2.30)
where the surface integral is taken over the plaquettePl.
A direct corollary of the last result is that the Chern number can be written in terms of the lattice Berry curvature:
Cn=
NXxNy
l=1
Fn
xy,l. (2.31)
This last expression provides a gauge invariant method to compute the Chern number. In practice, Eq. (2.29) cannot be verified explicitly since we cannot compute in general analytically the continuous Berry curvature but it gives a reasonable criterion for the con-vergence of the grid: if|Fn
xy(l)| < π and not too close to π, the algorithm should converge.
In practice, the algorithm converges extremely fast.
2.3
Cylinder geometry and edge states
2.3.1 Introduction
In the last section, we saw that the topological properties appear in the quantization of the Hall conductivity when the Fermi energyEF lies in an energy gap of the bulk
geometry. In this case, new states appear in the energy gap. In fact, as we will see, these states are edge states protected by the topology and they transport the current.
2.3.2 Energy spectrum and chiral edge states
3 3 0 EF 0 ⇡ 2⇡ E/J a. b. c. d.
Figure 2.5: Left: Projection of the energy spectrum on the planekx = 0 in the Hofstadter
model forφ = 1/5 with periodic boundary conditions in the x- and y-directions (torus). Right: Energy spectrum of the Hofstadter model with periodic boundary conditions iny and hard wall potentials along thex-direction, ψ(0, y) = ψ(100, y) = 0: (cylinder of size L = 100). The system presents new states in the energy gaps of the bulk.
Starting with Eq. (2.5), we impose periodic boundary conditions in they-direction and a hard wall boundary potential atx = 0 and x = L in the x-direction- i.e. ψ(0, y) = ψ(L, y) = 0 [12]. We then apply the Bloch theorem in the y-direction and find the Hamiltonian:
ˆ H = X ky∈B.Z. Ψ†k y ˆ HkyΨky, (2.32)
whereHkyis a tridiagonal matrix:
We diagonalize the matrix Hamiltonian and obtain the energy spectrum in the first Bril-louin zone. Figure 2.5 shows the energy spectrum forφ = 1/5 in both geometries. We clearly see the correspondence of the energy bands of the bulk spectrum between the two geometries. Interestingly, in the cylinder geometry, there are new states in the en-ergy gaps of the bulk spectrum. These states can be characterized analytically using the transfer matrix formalism [12]. We here present the numerical results.
0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0 x 100 | |2 0 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0 100 0 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0 x 100 0 a. b. c. d. 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 0 x 100 x
Figure 2.6: Density of the wave functions of the four states crossing the Fermi energy in Fig. 2.5. These states are localized on the edges: two on the left-hand edge with a positive group velocity and two on the right-hand edge with a negative group velocity. The number of edge branches on each edge is equal to the Hall conductivityσH.
We choose the Fermi energy to be in the second energy gap: the Fermi energy is crossing four states (see Fig. 2.5). Figure 2.6 depicts the spatial density of these four states: two are localized on the left-hand edge and two are localized on the right-hand edge. We then compute their group velocity usingvg ∝ ∂kyE: the two states on the left-hand edge have
a positive velocity and the two on the right-hand edge have a negative velocity. We thus find the presence of chiral edge states transporting the current at the edges of the cylinder. Finally, the number of edge branches in each energy gap is equal to the Hall conductivity we have computed in Subsec. 2.2.2 (See Ref. [12] for a detailed discussion). These chiral edge states are thus protected by the topology.
2.4
Box geometry
Vconf=1
Figure 2.7: Finite size system ofL× L sites. At the boundaries, the confining potential is Vconf =∞.
In this section, we discuss the case of the box geometry, a finite size system ofL×L sites with a hard-wall confinement potentialVconf =∞ at the boundary of the system (see Fig. 2.7).
Here, the periodicity is no longer given and solving the eigenvalue problem of Eq. (2.5) amounts to the exact diagonalization of aL2×L2matrix. The results of the diagonalization
are shown in Fig. 2.8a where we have diagonalized the Hamiltonian of a square lattice of size40 × 40 sites (1600 sites). Since there is no periodicity, there is no appropriate quantum number. The eigenstates are thus labelled by the labelλ. Figure 2.8b presents the density of states (D.O.S.) of the energy spectrum: the energy spectrum is characterized by two regions with a low D.O.S. These low D.O.S. regions are corresponding to the energy gap of the bulk spectrum we reviewed in Sec. 2.2.
Figure 2.9 represents the real-space density of one eigenstate in the low density region, namelyψ545(x) associated to the energy E545=−1.06J. The state is localized at the edges
of the box and is thus an edge state. The same analysis can be done for the other states of the low-density regions and one can conclude that the low-density regions are populated by edge states. One can show that these edge states are chiral and have a non-zero local current [62]. We will come back to the chirality of the topological edge states in Sec. 5.3 and use this property to propose a detection method of the dynamics of the topological edge states.
3 0 3 0 1600 D.O.S. E/J a. b.
Figure 2.8: a. Energy spectrum forφ = 1/3 of a system of size 40× 40 sites. λ is the label of the different states. b. Density of states (D.O.S.) of the energy spectrum. The system exhibits two low D.O.S. regions, corresponding in energy to the energy gaps of the bulk spectrum. 0 40 x/a y/a 0.01 E545= 1.06J | 545(x)|2 0 40 0
Figure 2.9: Density|ψλ|2of the stateλ = 545 associated with the energy Eλ =−1.06J (in
the energy gap). The state is localized at the edges of the box. The low D.O.S. regions are found to be populated by edge states.
4 0 4
0 1
E/J D.O.S.
Figure 2.10: Density of states of the energy spectrum in terms of the magnetic flux. The figure displays low D.O.S. regions and can be compared with Fig. 2.4: the low D.O.S. re-gions correspond to the energy gaps of the system with periodic boundary conditions.
Basic aspects of cold atom physics
3.1
Introduction
The control of the light-matter interaction plays a fundamental role in quantum simu-lation with cold atoms. The continuous improvement of lasers since the 60s has opened a plethora of possibilities for the manipulation of atoms, allowing the control of the internal states and external dynamics of atoms [63, 64].
At the beginning of the 80s, physicists realized that lasers could be used to cool atomic gases and reach very low temperatures [32, 65]. For example, Doppler cooling allows one to cool atomic gases to temperatures of about100 µK, whereas evaporative cooling, one can even reach temperatures on the order of100 nK. At these temperatures, the dynam-ics of gases is no longer described by the Maxwell-Boltzmann distribution since the de Broglie wavelength of the atoms starts to be on the same order of magnitude as the in-teratomic distance. In this regime, the gases show exotic quantum behaviors. In this case, the statistics becomes important: a bosonic gas becomes a Bose Einstein conden-sate (BEC) [34, 33] and a fermionic gas becomes a degenerate Fermi gas [35]. Note that here the notions of fermionic or bosonic refer to composite particles. A composite boson (resp. fermion) is composed of an even (resp. odd) number of fermions (electrons and nucleons).
Finally, the coupling between internal states allows one to generate background Abelian and non-Abelian gauge fields and thereby opens a road to the study of the topological phases in optical lattices. This extends the simulation toolbox beyond Bose and Fermi-Hubbard models, allowing the quantum simulation of phases of matter in the presence of gauge fields.
In this chapter, we
1. briefly review the light-matter interaction, introduce the dipole potential, and the notion of dressed states;
2. review several cooling methods that, if combined, allow one to reach tempera-tures on the order of100 nK;
3. discuss the experimental realization of BEC and degenerate Fermi gases;
4. briefly outline the working principle of optical lattices and outline one mecha-nism to engineer artificial gauge fields.
3.2
Light-matter interaction: the dipole potential
We here briefly review the dipolar coupling for a two-level atom and introduce the con-cept of dressed states. Several reviews treat this topic in detail (see e.g. Grimm et al. [32], Goldman et al. [37]). |gi |ei ⌦ |gi |ei ⌦ a. b. |g0i = |gi ⌦|ei E = ⌦ 2 |e0i = |ei + ⌦|gi
Figure 3.1: a. Coupling between the internal states|gi and |ei. Ω is the Rabi frequency and∆ is the detuning. b. For Ω ∆, the ground and excited states are coupled far off-resonantly. As a consequence, the ground state undergoes an energy shift of∆E' −Ω2/∆
Let us consider an electric fieldE(r, t). The atom interacts with light through the dipolar coupling term ˆVD =− ˆD· E(r, t), where ˆD is the dipolar operator. This term describes the
coupling of the two internal states|gi and |ei. In the co-rotating frame and in the rotating wave approximation, the dipolar coupling takes the form (see also Fig. 3.1a):
H = ∆|eihe| + Ω(|gihe| + |eihg|), (3.1) whereΩ is the Rabi frequency and is proportional to the amplitude of the electric field E, and∆ is the detuning, which accounts for the difference between the laser frequency and the atomic transition frequency. The energy spectrum can be found easily:
E±= ∆ 2 ± ∆ 2 s 1 + Ω2 (∆/2)2. (3.2)
For|Ω| |∆| (strongly off-resonant regime), the energy spectrum and the eigenstates are given to the lowest order in|Ω/∆| by:
E−=− Ω2 ∆,|g 0 i = |gi − Ω ∆|ei, (3.3) E+= ∆ + Ω2 ∆,|e 0 i = |ei + Ω ∆|gi. (3.4)
These states are called dressed states. For example, let us consider an atom initially in the state|gi and adiabatically turn on the laser beam, as shown in Fig. 3.1b. The new ground state is|g0i with an energy shift −Ω2/∆ depending on the Rabi frequency and the detuning
and with a small admixture of the excited state (since|Ω/∆| 1).
3.3
Cooling the atomic gas
Bose Einstein condensates or degenerate Fermi gases are phases of matter appearing at very low temperatures, on the order of100 nK. Initially, the atomic gas has a typical temperature on the order of300 K [33], and methods are thus needed to cool the atomic gas to very low temperatures. Here, we briefly review a selection of methods that can be combined to reach the low temperatures required to enter the quantum regime. Several articles provide a more complete and detailed overview of these and other cooling meth-ods [32, 65, 67].
First, Doppler cooling [32, 65, 67] allows one to reach temperatures of the order of 100 µK. Here, a standing wave, composed of two counterpropagating laser beams, is cou-pling atoms slightly resonantly (|∆| Ω) between the ground state |gi and the excited state|ei with a Rabi frequency Ω and a detuning ∆. The coupling between the two states is inducing an average radiative forceF+(resp. F−) for the left (resp. right) laser beam.
If the atoms are at rest, the total average radiative force is vanishing: F = F++ F− = 0
is proportional tov and ∆ [65]. This effect is a consequence of the Doppler effect: atoms see different effective detuning and will thus ”feel” different radiative forces. In particular, when∆ < 0, the force is opposite to the velocity and is thus decelerating the atoms, which leads to lower temperatures. The lower temperature limit that one can reach with this cooling method is determined by the finite lifetimeΓ−1 of the excited state|ei, yielding the Doppler cooling limitkBT =~Γ/2, which is typically on the order of 100 µK [68, 67].
|g1i |g2i
|r1i |r2i |r3i |r4i
a. b.
|g1i |g2i
Figure 3.2: a. Optical pumping: Depending on the light polarization, only certain transi-tions are allowed. Here|g2i can be excited to |r3i. Nevertheless, |r3i can decay to |g1i and
|g2i. In fact, the decay to |g2i is favoured, giving rise to a pumping from |g1i to |g2i. The
inverse mechanism appears for other polarisation. b. Sketch of the Sisyphus effect: An atom initially in|g2i feels the green potential. If it has sufficiently kinetic energy, it can
climb up the potential barrier. From There, the polarization has changed and the atom in|g2i is pumped to |g1i. From there, the atom again climbs up the blue potential barrier.
Thus, the atom reduces iteratively its kinetic energy.
Surprisingly, the temperatures achieved in the first experiments have been observed to be lower than the one predicted by the Doppler cooling [67]. In fact, for alkali gases, the simplified picture of only two internal coupled levels is no longer valid. There are typi-cally two degenerate ground states|g1,2i and four degenerate excited states |r1,2,3,4i. The
polarization of light is giving rise to an optical pumping, as sketched in Fig. 3.2a: the po-larization of light allows the excitation of the state|g1i to the state |r3i. However, the latter
decays partially into the state|g2i. There is thus a pumping from |g1i to |g2i. Note that the
inverse can appear for the opposite polarization. Furthermore, the interaction with light induces opposite spatially-dependent energy shifts of the two ground states, generating an optical potential, as sketched in Fig. 3.2. The combination of the two mechanisms gives rise to the Sisyphus effect : Consider an atom in state|g2i, as sketched in Fig 3.2b. If
change of polarization, the atom is pumped into|g1i, from where it again starts to climb
up the potential barrier. The atom reduces therefore little by little its kinetic energy. The system reaches temperatures typically on the order of1 µK [67].
Evaporative cooling allows one to reach even lower temperatures. The idea relies on trapping particles with low velocities and letting the others leave the trap. The system thus reaches lower temperatures at the cost of reducing the number of particles. The gas is first cooled with the Doppler cooling and trapped in a magneto-optical trap (MOT), which confines the atomic cloud. Using a radio frequency field, atoms in a initially given internal state are coupled to another electronic state, in which they are not trapped in the MOT. The radio frequency is chosen in such a way that only atoms with a high enough velocity undergo this transition and thereby are effectively ejected from the trap. By repeating this procedure for lower and lower velocities, one selects only atoms with low velocity, allowing one to reach temperatures of the order of100 nK [32].
Finally, we note that evaporative cooling is not efficient for fermions [69]: the Pauli ex-clusion principle suppresses elastic collisions between fermions. A good ratio between the elastic and inelastic collisions is needed to have an efficient evaporative cooling. This problem can solved if, instead of dealing with a single-species Fermi gas, the system is composed of a mixture of bosons and fermions. The evaporative cooling is performed on the bosons and the collisions between bosons and fermions sympathetically cool the Fermi gas, allowing one to reach temperatures typically on the order of300 nK [69].
3.4
Cold gases: Bose-Einstein condensates and Fermi gases
At very low temperatures, quantum properties and particle statistics determine the be-havior of atomic gases. Here, we review the Bose-Einstein condensate (BEC) and the de-generate Fermi gas.
In the case of bosons, the critical temperature where quantum properties start to play a role can be estimated by comparing the typical distance between atomsd∼ n1/3with the
de Broglie wavelength given byλdB = h/√2πmkBT , where kBis the Boltzmann constant,
m the mass of the atoms, and T the temperature. For λdB d, the system is well
de-scribed by the Maxwell-Boltzmann distribution. However, onceλdB ∼ d, the wavelength of the atoms is comparable to the distance between atoms, and a phase transition to the BEC sets in. The critical temperature is therefore proportional toT ∝ n2/3 and inversely
proportional to the massm of the atoms. The critical temperature can vary from 100 nK to1 µK, depending on the nature the gas, shape of the trap, etc. A BEC was realized exper-imentally for the first time in 1995 [33, 34]. Figure 3.3 depicts the first BEC in cold atoms. A gas of rubidium 87 atoms is cooled and trapped in a MOT at very low temperatures. The figure presents the velocity distribution for three temperatures. IfT > Tc, the velocity
Figure 3.3: First BEC (rubidium 87) realized experimentally. The velocity distribution is presented for three different temperatures (decreasing from left to right): T1 > Tc,T2 =
Tc= 170 nK and T3< Tc. The atomic cloud is condensing in the fundamental state of the
system. Figure is included by permission of the NIST Ref. [70] (Cover of Science of Ref. [33]).
momentum peak forms at the momentum of the fundamental state, characteristic for the BEC. ForT < Tc, the momentum peak becomes more and more pronounced.
In the case of fermions, due to the Pauli exclusion principle, at very low temperatures, fermions fill the states of the trap up to the Fermi energy. The quantum regime sets in for a critical temperatureTF ∝ n2/3(note that the dependence inn is the same for bosons).
ForT > TF, the gas can be treated as a classical gas. ForT < TF, the gas is a degenerate
Fermi gas. As already mentioned in Sec. 3.3, cooling fermions is more complicated than cooling bosons. The latter has been realized experimentally in 1999 [35], where experi-mentalists have reached temperature on the order of0.6 TF. Figure 3.4 shows the results
of another experiment [69], where experimentalists have reached temperatures on the or-der of0.25 TF and have imaged the composite gas. Figure 3.4 depicts the real density of
the atomic gas for different temperatures. The gas is a mixture of bosons and fermions. It is however possible to visualize separately the bosons (lithium 7, left) and the fermions (lithium 6, right). The bosonic gas is condensing in the fundamental mode of the trap. In the case of the fermionic gas, the situation is different. The gas is becoming degenerate atT = TF but due to the Pauli exclusion principle, the atomic cloud has a larger spatial
extension as compared to the bosonic sample.
3.5
Optical lattices and artificial gauge fields
Figure 3.4: First degenerate Fermi gas (lithium 6). The cooling mechanism involves a mixture of bosons and fermions (see Sec. 3.3). The image shows the density profile in real space of both species: bosons on the left and fermions on the right. At the beginning, the two atomic clouds have almost the same size. When the gas is cooled, the bosonic gas undergoes the phase transition to a BEC state with all the particles condensing in the fundamental state of the trap. In contrast, the fermionic gas is becoming quantum degenerate but due to the Pauli exclusion principle, the atomic cloud has a larger spatial extension as compared to the bosonic sample. Figure included by permission of R. Hulet [71].
atoms can be used to engineer artificial gauge fields. The latter is essential for the quan-tum simulation of topological phases.
Let us first describe an optical lattice in one dimension. The use of two counterpropa-gating laser beams with electric fieldE± = E0exp (±ikx) generates a standing wave with
wavelengthλ = 2π/k. The wavelength λ of the lasers is chosen such that they couple the two internal states|ei and |gi far off resonance (Ω ∆). As discussed in Sec. 3.2, in this regime, the dressed ground state undergoes an AC Stark shift of an energy as given by Eq. (3.3). In this case, the energy shift is proportional|E±|2and thus tocos2(kx), thereby
inherits the spatial dependence of the intensity profile: the coupling of the two internal states generates a periodic potential with spatial periodicityλ/2.
that due to the interference of the two laser beams, V0is
four times larger than Vtrapif the laser power and beam
parameters of the two interfering lasers are equal. Periodic potentials in two dimensions can be formed by overlapping two optical standing waves along differ-ent, usually orthogonal, directions. For orthogonal po-larization vectors of the two laser fields, no interference terms appear. The resulting optical potential in the cen-ter of the trap is then a simple sum of a purely sinusoidal potential in both directions.
In such a two-dimensional optical lattice potential, at-oms are confined to arrays of tightly confining one-dimensional tubes!see Fig.4"a#$. For typical experimen-tal parameters, the harmonic trapping frequencies along the tube are very weak "on the order of 10–200 Hz#, while in the radial direction the trapping frequencies can become as high as up to 100 kHz. For sufficiently deep lattice depths, atoms can move only axially along the tube. In this manner, it is possible to realize quantum wires with neutral atoms, which allows one to study strongly correlated gases in one dimension, as discussed in Sec. V. Arrays of such quantum wires have been real-ized"Greiner et al., 2001;Moritz et al., 2003;Kinoshita et al., 2004;Paredes et al., 2004;Tolra et al., 2004#.
For the creation of a three-dimensional lattice poten-tial, three orthogonal optical standing waves have to be overlapped. The simplest case of independent standing waves, with no cross interference between laser beams of different standing waves, can be realized by choosing orthogonal polarization vectors and by using slightly dif-ferent wavelengths for the three standing waves. The
resulting optical potential is then given by the sum of three standing waves. In the center of the trap, for dis-tances much smaller than the beam waist, the trapping potential can be approximated as the sum of a homoge-neous periodic lattice potential
Vp"x,y,z# = V0"sin2kx+ sin2ky+ sin2kz# "36#
and an additional external harmonic confinement due to the Gaussian laser beam profiles. In addition to this, a confinement due to the magnetic trapping is often used. For deep optical lattice potentials, the confinement on a single lattice site is approximately harmonic. Atoms are then tightly confined at a single lattice site, with trap-ping frequencies !0 of up to 100 kHz. The energy "!0
=2Er"V0/Er#1/2of 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 poten-tials. Typical values of Er are in the range of several
kilohertz for 87Rb.
Spin-dependent optical lattice potentials. For large de-tunings of the laser light forming the optical lattices compared to the fine-structure splitting of a typical alkali-metal atom, the resulting optical lattice potentials are almost the same for all magnetic sublevels in the ground-state manifold of the atom. However, for more near-resonant light fields, situations can be created in which different magnetic sublevels can be exposed to vastly different optical potentials "Jessen and Deutsch, 1996#. Such spin-dependent lattice potentials can, e.g., be created in a standing wave configuration formed by two counterpropagating laser beams with linear polar-ization vectors enclosing an angle#"Jessen and Deutsch, 1996;Brennen et al., 1999;Jaksch et al., 1999;Mandel et al., 2003a#. The resulting standing wave light field can be decomposed into a superposition of a $+- and a
$−-polarized standing wave laser field, giving rise to lat-tice potentials V+"x,##=V0cos2"kx+#/2# and V−"x,##
=V0cos2"kx−#/2#. By changing the polarization angle#,
one can control the relative separation between the two potentials %x="#/&#'x/2. When#is increased, both
po-tentials shift in opposite directions and overlap again when#=n&, with n an integer. Such a configuration has been used to coherently move atoms across lattices and realize quantum gates between them"Jaksch et al., 1999;
Mandel et al., 2003a,2003b#. Spin-dependent lattice po-tentials furthermore offer a convenient way to tune in-teractions between two atoms in different spin states. By shifting the spin-dependent lattices relative to each other, the overlap of the on-site spatial wave function can be tuned between zero and its maximum value, thus controlling the interspecies interaction strength within a restricted range. Recently, Sebby-Strabley et al. "2006#
have also demonstrated a novel spin-dependent lattice geometry, in which 2D arrays of double-well potentials could be realized. Such “superlattice” structures allow for versatile intrawell and interwell manipulation possi-bilities "Fölling et al., 2007; Lee et al., 2007; Sebby-Strabley et al., 2007#. A variety of lattice structures can be obtained by interfering laser beams under different
(a)
(b)
FIG. 4. "Color online# Optical lattices. "a# Two- and "b#
three-dimensional optical lattice potentials formed by superimposing two or three orthogonal standing waves. For a two-dimensional optical lattice, the atoms are confined to an array of tightly confining one-dimensional potential tubes, whereas in the three-dimensional case the optical lattice can be ap-proximated by a three-dimensional simple cubic array of tightly confining harmonic-oscillator potentials at each lattice site.
Figure 3.5: a. Optical lattice in 2D: two counterpropagating pairs of laser beams are used to generate two standing waves. This creates effectively an array of 1D tubes. b. By adding a pair of laser beams in the third direction, one creates an optical lattice in three dimen-sions. Figure reprinted with permission from Immanuel Bloch [29]. Copyright 2008 by the American Physical Society.
parameters of optical lattices can be controlled and tuned to a large extent: for example, the hopping, the onsite energy and the onsite interaction can be controlled by changing the lattice depth. Using this powerful tool in a seminal experiment, the transition be-tween the superfluid regime and the Mott insulating regime for the Hubbard model of a BEC loaded in an optical lattice has been observed [36].
We now briefly comment on the implementation of artificial gauge fields. Since atoms are neutral particles, various methods to engineer effective gauge fields have been pro-posed. The artificial gauge field should therefore be engineered. Here, we sketch the laser-tunneling-assisted method introduced by Jaksch and Zoller in Ref. [51]. More details on methods to implement light-induced gauge fields can be found in the review articles [30, 37].
The key idea is shown in Fig. 3.6. A state-dependent optical lattice is realized for atoms in the states|gi (white dots) and |ei (black dots): The optical potential is chosen such that only the hopping in they-direction is allowed. In the x-direction, the hopping is im-plemented between nearest neighbors with Raman transition from|gi to |ei. The Peierls phase can be engineered by choosing a spatially dependent Rabi frequency Ω1,2(r) =
Ωeiqy. For appropriate values of the parameters, this model corresponds to the Hofstadter
Figure 2. Optical lattice set-up. Open (closed) circles denote atoms in state |g⟩ (|e⟩). (a) Hopping in the y-direction is due to kinetic energy and described by the hopping matrix element Jy being the same for particles in states|e⟩ and
|g⟩. Along the x-direction hopping amplitudes are due to the additional lasers. (b) Trapping potential in the x-direction. Adjacent sites are set off by an energy ! because of the acceleration or a static inhomogeneous electric field. The laser "1
is resonant for transitions|g⟩ ↔ |e⟩ while "2is resonant for transitions|e⟩ ↔ |g⟩
due to the offset of the lattice sites. Because of the spatial dependence of "1,2
atoms hopping around one plaquette get phase shifts of 2πα = −ϕm+0+ϕm+1+0
where ϕm = mqλ/4π as indicated in (a).
for a number of studies related to the behaviour of charged particles in a 2D configuration subject to magnetic and electric fields and also to study strongly interacting and thus strongly correlated systems. Furthermore, it might be possible to extend this model to different geometries of optical lattices.
In this work we will concentrate on a possible set-up required to implement the effective magnetic field in an optical lattice. We will discuss in detail the laser set-up which leads to an effective magnetic flux through the optical lattice, and calculate the corresponding matrix elements in section2. We also show that it is possible to reach each point within the Hofstadter butterfly apart from a negligibly small region around α= 0 with the proposed set-up. In section3
we suggest one possibility of measuring some of the basic properties of the Hofstadter butterfly and discuss the limitations on the resolution for measuring the energy bands. We also give a brief account of the interaction effects. Finally we conclude with a short outlook on how the present set-up could be extended in section4. While the focus of the present work is the derivation of the single-particle terms in the Hubbard Hamiltonian mimicking a strong magnetic field, we see as one of the main motivations the extension to strongly correlated many-atom systems in strong (effective) magnetic fields.
2. Set-up and model
In this section we discuss the experimental set-up required to produce a Hofstadter butterfly for neutral atoms. We first present the optical lattice set-up, then introduce an additional acceleration or static electric field and finally describe in detail the additional lasers required for our purpose. Figure 3.6: Setup to engineer an artificial gauge field in an optical lattice, as suggested in
Ref. [51]: a. Two superimposed state-dependent optical lattices for atoms in the internal states|gi (white dots) and |ei (black dots). The optical potential is chosen such that only the hoppingJy in they-direction is allowed. b. The hopping in the x-direction is realized
with Raman transitions between the states|gi and |ei. A Peierls phase can be induced by working with spatially-dependent Rabi frequencies. Figure included by permission of D. Jaksch [51].
These are the basic techniques used in cold atom physics, and constituting the general framework of this thesis.
Topological Mott insulator
4.1
Introduction
In the preceding chapter, we discussed basic concepts of quantum simulation using cold atoms. Cold atoms constitute a very versatile platform that allow one to experimen-tally explore many-body quantum physics in a highly controllable setting. We also briefly described a basic mechanism to engineer gauge fields in optical lattice setups. There are two main avenues to realize topological insulating phases in cold atom systems:
i/ The original approach to simulate topological insulating (TI) systems with cold atoms in optical lattices stems from the idea of synthesizing gauge magnetic fields in an effective way [51, 76, 77, 78, 79], which has recently been demonstrated experimentally [38, 40, 41, 43, 44]. The reasoning behind these quantum engineering ideas is that background fixed classical gauge fields play a fundamental role in the theoretical construction of topological insulating systems.
Here, we are interested in the latter interacting scenario ii/ as a way to produce TI phases whose non-trivial topological properties arise purely as a consequence of the fermionic interactions, as opposed to the scenario i/ of gauge-field induced TIs. We will consider a model of interacting spinless fermions on a hexagonal lattice, which we spec-ify below, where this mechanism may result in the formation of a TMI phase, generated dynamically by repulsive Hubbard-type interactions and in the absence of any (synthetic) external gauge fields. Besides a theoretical analysis of the model, we will propose how this new physics could be observed in an experiment by implementing the interacting fermion model in a quantum simulator using cold Rydberg atoms in optical lattices. On the one hand, such quantum simulation is important for benchmarking theoretical pre-dictions with experimental observations, in particular as the problem of interacting lat-tice fermions in two or higher spatial dimensions is notoriously hard to solve. On the other hand, as we will show below, the flexibility to tune the parameters of the model of interest over a wide range of parameters in a quantum simulation offers the possibility to explore in the laboratory the physics in regimes that are not naturally realized or not readily accessible in known materials.
In the context of quantum simulation, we will see that as an alternative to engineering complex next-to-nearest neighbor hopping dynamics as required in the Haldane model [13], which supports a quantum anomalous Hall (QAH) phase that has so far not been ob-served in any material, introducing long-range interactions between the fermions might become an ally, as long as these turn out to favor the formation of a stable QAH phase without the need to implement synthetic gauge fields. We anticipate at this stage that the fact that both the Haldane model and the interacting model considered by Raghu et al. [45] and also by us [86] are defined on the same type of hexagonal lattice is rele-vant for the following reason: the next-to-nearest neighbor fermionic interactions, when treated within a mean-field theory (MFT) approach, give rise to an order parameter that resembles the single-particle next-to-nearest neighbor hopping in the Haldane model. This connection is the physical key mechanism underlying the dynamically created, i.e. interaction-induced TMI phase described above.
In this chapter, we
1. first review the properties of spinless fermions on a hexagonal lattice and the Haldane model;
2. then consider an interacting fermion model on the same lattice geometry. Here, we identify the relevant order parameters and determine the complete zero-temperature phase diagram within a MFT treatment, and compare our findings to related, previous work on this model [45, 81];
4. underpin the character of the topologically trivial and non-trivial phases by an explicit numerical calculation of Chern numbers [11, 59], which we compare with the behavior of local order parameters;
5. propose and analyze a realistic scheme for an analog quantum simulation of the model with cold fermionic Rydberg atoms in an optical lattice.
This chapter is based on the original work published in Ref. [86]. My contribution has been to perform the mean field analysis and the study of the phase diagram.
4.2
Spinless fermions on a hexagonal lattice and the Haldane
model
(m, n) (m + 1, n) (m, n + 1) a1 a2φ
ψ
χ
φ χψFigure 4.1: Hexagonal latticeΛ with sites j decomposed in two-site basis cells j = (m, n) connected by basis vectorsa1anda2. A fermion at theφ-type site (m, n) possesses three
nearest-neighbor sites (shaded black filled circles) and six next-to-nearest neighbor sites (shaded white open circles).
Before dealing with the full interacting Hamiltonian, we start first by reviewing the properties of the Hamiltonian of spinless fermions on a hexagonal lattice ˆHJ, in order
interaction terms in Sec. 4.3. Next, we review the Haldane model, which is related to the effective Hamiltonian which we will derive in Sec. 4.4.
The graphene Hamiltonian ˆHJ is a tight-binding Hamiltonian of spinless fermions on
the 2D honeycomb lattice:
ˆ
HJ =−J
X
hi,ji
c†icj, (4.1)
where the sum is performed over the nearest neighbors andc†i creates a spinless fermion on the sitei of the honeycomb lattice.
The bipartite nature of the hexagonal lattice makes it convenient to rewrite the Hamil-tonian ˆHJin terms of two-site basis cells [87] (see Fig. 4.1), labeled by an index pair(m, n),
each of which contains one site of theφ-sublattice (open white circles) and one of the ψ-sublattice (filled black circles):
ˆ HJ =−J X m,n (c†ψmncφmn+ c † ψmn+1cφmn+ c † ψm−1ncφmn) + h.c. . (4.2)
In addition to the tight-binding term, we consider a staggering potential, describing a chemical potential difference for fermions residing atφ- and ψ-sites, which reads
ˆ Hβ = β
X
m,n
(c†φmncφmn− c†ψmncψmn) . (4.3)
To determine the energy spectrum of the tight-binding Hamiltonian in combination with the staggering potential, we introduce Fourier-transformed fermionic operators,
cαmn = 1 √ NΛ X k∈BZ exp(ik rnm)cα(k), (4.4)
withα ∈ {φ, ψ}, to rewrite the Hamiltonian in momentum space as ˆ HJ + ˆHβ = X k ˆ Ψ†(k) β −JA∗ k −JAk −β ˆ Ψ(k). (4.5)
Here, ˆΨ†(k) = c†φ(k), cψ†(k), andrmn = na1 + ma2 are the real-space vectors of the
lattice, with a1 = (3/2, √ 3/2), a2= (−3/2, √ 3/2), (4.6)
denoting the basis vectors (for a unit length lattice spacing), which span the whole lattice in the two-site basis parametrization (see Fig. 4.1). The total number of two-site basis cells isNΛ, which is half of the total number of lattice sites,NΛ = N/2. The function
Ak= 1 + exp(ik · a1) + exp(−ik · a2) (4.7)
