Production and High Resolution Imaging of a quantum degenerate Ytterbium atomic gas

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Universit`

a degli studi di Pisa

DIPARTIMENTO DI FISICA ’E. FERMI’

Tesi di Laurea Magistrale in Fisica

Production and High-Resolution Imaging of a

quantum degenerate Ytterbium atomic gas

Candidato

Elisa Soave

Relatori

Fabrice Gerbier

Donatella Ciampini

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Introduction

Associated to a particle of mass m moving at temperature T there exists the so-called thermal (or De Broglie) wavelength λDB “

b

2π¯h2

mkBT, that is the position uncertainty associated to the thermal

momen-tum of the particle. At high temperature, this quantity is large enough that the system can be treated according to classical mechanics. As the temperature decreases, eventually λDB becomes comparable

with the average distance among particles. The wave nature of the particles can no longer be neglected, and phenomena related to the quantum nature of the system start to manifest themselves. In particular, if a system of bosons attains its critical temperature Tc, condensation takes place, as predicted for the

first time by A. Einstein in 1925. Particles of an ideal gas cooled below the critical temperature start to occupy all the same quantum state, i.e., the less energetic one. What is extremely important, is the fact that, in this condition, the condensed particles start to behave as a whole. From a mathematical point of view, this has the consequence that instead of speaking of N separated wavefunctions, we can describe the system as a unique, macroscopic wavefunction. These systems have thus the peculiarity of behaving according to quantum mechanics and at the same time being experimentally reproducible, controllable and observable: that is, these systems allow for an experimental investigation of the quantum properties of matter, providing a benchmark for a huge number of theories that have been developed so far. Despite the astonishing power of such kind of systems, the experimental realization of the first Bose Ein-stein Condensate (BEC) required almost a century because of the difficulties arising in the achievement of the low temperature that is required for entering the quantum regime. This has been the major goal of atomic, molecular and optical physics experimentalists for many years. It was clear that light has a key role in answering that question: light can be used not only for investigating atoms (spectroscopy), but it can be used also for acting on them, manipulating and controlling them. Nowadays, almost all the techniques currently employed in cold atoms experiments are based on the interactions between light and atoms.

At the dawn of cold atoms physics, the achievement of the degenerate regime was the primary goal of the majority of the experiments on atomic physics. Nowadays, on the contrary, the condensate has began mostly a tool, to be manipulated at will and used for getting a deeper insight into a wide range of physics’ fields. In particular, thanks to the high accuracy with which cold atomic systems can be controlled, cold atoms physics has begun the backing for solid state physics, since it permits to simulate condensed matter system, and, more in general many body physics, and to investigate limit regimes (for instance, high magnetic fields) that are hard to obtain in standard laboratories acting directly on the sample under quest. This interplay among condensed matter physics and cold atoms physics has the effect of pushing both these disciplines further. Optical lattice have been soon implemented, permitting to obtain artificial crystal of light.

The fact that atoms are neutral are one of the factor that limit the variety of systems that can be simulated: the neutral nature of atoms in principle prevents any direct implementation of magnetic systems. This problem has been circumvented simulating not the system itself but rather its Hamiltonian.

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In other words we look for a system whose Hamiltonian has the same mathematical form as the one of the system we aim to analyze.

Gauge potentials’ simulations One of the first attempt of magnetic field simulation with cold atoms takes advantage of the analogy among the Lorentz force and the Coriolis one. The idea is thus to confine atoms in a trapping potential that is rotating at the angular frequency Ω “ Ωˆz. In the rotating frame solidal with the particle, the Hamiltonian has the same form as the Hamiltonian for a particle of mass mand charge q freely moving on a magnetic field B “ 2mΩ{q ([1]).

A second way to simulate the magnetic Hamiltonian exploits the adiabatic theorem. The implementation of this method requires a two-levels (g, e) atom, the transition g Ñ e driven by a light field. It is important that this field is not homogeneous. Indeed, atom-light interaction results also in a momentum transfer: if this transfer follows adiabatically the transition, the atom stays resonant with the light field, even if its external energy is varying. It can be shown that the resulting Hamiltonian is formally the same as the one for a magnetic system ([2],[3],[4]). This "laser dressing" technique mimics the so-called Aharanov-Bohm (AB) phase. The Aharanov-Bohm phase is one of the benchmark of magnetism, and is proportional to the integral of the vector potential along the path followed by the particle. This phase appears naturally to coincide with the Berry phase encountered in the context of the adiabatic theorem. The analog of the AB phase for particles in a lattice is the so-called Peierls phase, a phase factor that is added to the tight binding tunneling amplitude of the lattice Hamiltonian and that accounts for the presence of the Gauge potential [5]. Magnetic field simulation in optical lattice aim to implement directly this complex amplitude tunneling, instead of the whole magnetic field. Since the Peierls factor is more directly related to the vector potential than to the magnetic field itself, we rather refers to Gauge potential simulation instead of magnetic field simulation.

Also in this case, the simulation requires a two-levels (g, e) atom: the artificial Peierls phase arises from the light induced coupling between the two states (we refer to this technique as laser assisted tunneling; see for instance [2],[3],[4]). Nowadays, there are lots of experiment conduced in this sense: we mention the one in Münich or the one at NIST where g and e are two different Zeeman substates, coupled by a two-photon Raman field ([6],[7]). In this thesis I’ll report on a third method, that, on the contrary, exploits a one-photon optical transition: this is possible thanks to the presence of a metastable state in the spectrum of the alkaline-earth atoms, that plays the role of the state |ey . Since the excited state is a metastable one, this line is extremely narrow (clock transition): this is challenging from the point of view of the experimental implementation.

Optical lattice Implementation of artificial Gauge fields in optical lattices is of considerable impor-tance for several reasons. From a purely experimental point of view, it permits to overcome some of the limitations that can be encountered implementing directly the whole magnetic Hamiltonian1 _([4]).

Second, the introduction of a periodic potential in the Hamiltonian opens an alternative way for study-ing magnetic phenomena on crystals: indeed, the presence of a magnetic field change enormously the energetic band structure of a periodic lattice, introducing subbands separated by finite gaps. This turns in important consequences for the transport properties of the system (e.g. integer quantum hall effect; see for instance [6]).

The third reason that makes optical lattice so interesting in the simulation of magnetic fields is the fact that optical lattices add the possibility to include the effect of correlations. In many-body physics a system of stongly-correlated particles confined in a periodic potential can be described through the Hubbard Hamiltonian. Mainly two parameters characterizes the Hubbard Hamiltonian: one is accoun-ting for the interactions among particles in the same situ (U) and one is accounaccoun-ting for the tunneling

1_{For instance the strength of the effective magnetic field is limited from the centrifugal force in the case of rotating gas,}

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amplitude among neighboring sites (J). Optical lattices give an high degree of freedom on the choice of those two parameters, providing thus the possibility to investigate several regimes, depending on the ratio U{J. U and J can be tuned changing the depth of the potential, i.e., mainly acting on the power of the laser beam. Interactions among particles can be tuned thanks to Feshbach resonances[8]. Content of this thesis This thesis aims to describe the work I’ve done within the Bose Einstein Condensate group of the Laboratoire Kastler Brossel, at Collège de France. For eights month I’ve been part of the Yb group, led by Fabrice Gerbier and Jérôme Beugnon.

The main goal of my work has been the design and the test of an high resolution microscope objective, characterized by a long working distance and great optical properties both at λ “ 399 nm and λ “ 760nm.

Imaging systems with enhanced resolving power are fundamental tools for probing and manipulating systems where single-atom precision is required [9]. Such an high resolution is hardly achievable with standard imaging systems, generally made of a first collimating lens that collects the light coming from atoms, and a second one that focuses the image on a camera. The resolution of these systems is enormously reduced by aberrations (deviations of light from focus). On the contrary, imaging systems that make use of a microscope objective have a resolution that is really close to the diffraction limit, that is, the limit imposed by the wave nature of light. Microscope objective are designed for reducing aberrations as much as possible. In order to accomplish their own experiment’s requirements, atomists prefer to design and assemble their own microscope objective. This task is made easier by several optical design softwares, that are able to evaluate and optimize optical systems.

The timing of my internship has been such that when I arrived at Collège de France the vacuum cham-ber was completely disassembled and the main set up was undergoing some changes and improvements. Therefore, I had the great fortune to take part of the reconstruction and optimization of the whole cooling system, as well as the measurements required for the characterization of the atomic cloud after each step. This offered me the possibilities to have an insight on the standard techniques used in atoms cooling and to experience first hand technical stuff, from optics to coils, from standard measurement techniques to optimization procedures.

In this manuscript, I aim to report not only on the main results perceived during my stage, but also on the experiment as a whole, including some interesting results achieved before I joined the group. The manuscript is divided into four chapter:

• Chapter I I introduce a few theoretical elements on how a Gauge potential affects a system of charged particles in a bidimensional lattice. In particular, the Hamiltonian describing these systems (Harper Hamiltonian) is derived. Starting from this Hamiltonian, I present the technique used by the Yb group for simulating these systems, based on a state dependent lattice. This is possible thanks to the peculiar level structure of 174

Yb, that presents two low energy metastable states.

• Chapter II Here the experimental set up is described. Each section is dedicated to one of the typical steps that lead a bosonic system to the quantum regime. Starting from the Zeeman slower, we explain the physical principles beneath each technique, and than presents how this technique is practically implemented in our experiment. For each steps, a few of the main considerations that have dictated the choice of the experimental parameters (beams’ waist, lasers’ power, magnetic fields’ intensity, etc...) are reported, together with a characterization of the intermediate state of the system after optimization.

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• Chapter III This chapter is dedicated to the description of the condensate that we obtained, together with a brief report on the spectroscopy on the metastable state, performed within the previous set up. The final part of this chapter is devoted to describe the first attempt for the construction of a repumper for the metastable state (this was a second side project that I was supposed to develop. The laser breakage prevented me to go further).

• Chapter IV This last chapter is completely dedicated to the imaging systems. A few elements on both geometrical and physical optics are reported, mostly for introducing the notation used in the following. I then describe the preliminary test that I effectuated on the standard imaging system used for imaging atoms at intermediate steps or where less accuracy is required. Finally the microscope objective that I designed is introduced, together with the evaluations and tests that I performed on it.

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Chapter 1

Simulation of Gauge potential in

optical lattices

The purpose of this first chapter is to introduce the basic concepts beneath the experiment I took part in. The final goal of the experiment is to simulate the presence of a magnetic field in a single two-dimensional layer of ultracold atoms in an optical lattice. This problem is of huge interest in the frame of condensed matter physics. Indeed, the combined effect of a magnetic field and a periodic potential gives rise to a wide variety of phenomena, in particular related to the transport properties of the system. The argument is particularly subtle to handle, because it mixes together the quantum nature of the particles, the magnetic field and the periodicity of the lattice.

In the first part of this chapter we will derive the Hamiltonian of a charged particle (eg, an electron) moving under the combined effect of a magnetic field and a periodic potential (Harper Hamiltonian): its diagonalization will highlight a fractal band structure that depends on the magnetic flux across a single lattice cell. This band structure is the starting point for the study of lots of other effects, related to the transport properties of the system, such as the integer and fractal quantum hall effect (see for instance [2], [6], [10]).

In the second part we will present the model we use to simulate this kind of systems in a bidimensional 3D optical lattice. There will be given a few elements about neutral atoms in optical lattices, and then presented how we can practically implement the Harper Hamiltonian.

1.1 Charged particles in a magnetic field

A classical charged particle, of charge q moving at velocity v in the presence of a magnetic field B experiences a force FL“ qv ˆ B(Lorentz force). In the absence of other forces, the particle moves in a

cyclotronic orbit lying in the plane perpendicular to the direction of the field.

From Maxwell equations it follows that the magnetic field B can be derived from a vector potential A such that B “ ∇ ˆ A. Since the dynamics of the particle is fully described in terms of fields, we could expect the vector potential not to have any physical meaning. In 1959 Bohm and Aharanov pointed out that contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish[11]. These effects arise because of the gauge invariance of quantum wavefunctions [12].

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1.1.1 Charged particle freely moving in a magnetic field: the

Aharanov-Bohm effect

The electric E and magnetic B fields are related to the scalar and vector potentials φ and A through the relations:

E “ ´∇φ ´BA Bt,

B “ ∇ ˆ A. (1.1)

Given a scalar function χ, the fields are left invariant under the transformation: A ÝÑ A ` ∇χ,

φ ÝÑ φ ´Bχ_Bt (1.2)

We refer to this as a gauge transformation.

Let’s consider a particle, of mass m and charge ´e, moving in the absence of electromagnetic fields according to the Hamiltonian H “ p2_{{2m ` V prq. The Hamiltonian of the same particle, now in the}

presence of the fields, is easily obtained through the so-called minimal coupling: this method consists in replacing V and p in the "old" Hamiltonian by:

V ÝÑ V ´ eφ,

p ÝÑ p ` eA. (1.3)

In the frame of classical physics, under the transformation (1.2) the Hamiltonian acquires additional terms that depend on the chosen χ. Nevertheless the physics of the particle doesn’t change because its dynamics is fully described in terms of the fields E and B, that are gauge invariants.

In quantum mechanics things are slightly different, and the gauge transformation gives rise to the so called Aharanov-Bohm effect, as shown in the following. Let’s call H1 the Hamiltonian of a charged

particle in a electromagnetic field, and H2the Hamiltonian of the same system, after the gauge

transfor-mation (1.2), obtained through the minimal coupling technique. The wavefunctions of the two systems are such that Hiψi“ Eψi. Since the physics behind the two Hamiltonians must be the same, it can be

shown [12] that ψ1has to change according to1:

ψ1prq ÝÑ ψ2prq “ e´i

eχprq ¯

h ψ₁_prq, (1.4)

where we have made explicit the dependence on space.

The appearance of a phase factor should not be surprising (wavefunctions in quantum mechanics are defined up to a global phase): what is worthy to note is the fact that here we are dealing with a local phase instead of a global one, and thus we can not get rid of this phase with a simple rotation of the system.

As a consequence of this fact, a particle moving from a point r0 to a point r ‰ r0 along a path γ,

accumulates a phase2 ϕ “ ´e ¯ h ż γ Aprq dr. (1.5)

The phase ϕ is said to be geometrical , since depends on the path followed by the particle and not on its dynamics: we thus see that even if the dynamics of the particle is left unchanged by (1.2), the wavefunction does depend on the scalar function χprq. Gauge invariance is restored if the path γ is

1_{Hamiltonian defined in different points are operators acting on different Hilbert spaces. In order to compare}

wave-functions in different space points, we have to take into account the variation of χ between the points: what we do, is transporting the function along the path followed by the particle. This transportation gives rise to the geometrical phase (1.5)

2_{The arising of this phase factor can be demonstrated in several ways: with a semiclassical argument, in the frame of}

WKB approximation ([13]); observing the evolution of the wavefunction of a particle moving around a solenoid ([2]); in the frame of the adiabatic theorem ([14]).

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Fig. 1.1: Particle moving from r0 to r in a

plane perpendicular to a magnetic field. Its wavefunction acquire a phase that depends on the geometry of the path γ, and that is proportional to the path integral of the vector potential Aalong γ.

closed. Indeed, from Stoke theorem follows: ϕ “ ´e ¯ h ¿ γ Aprq dr “ ´e ¯ h ż D Bprq dS “ 2πΦpBq Φ0 (1.6)

where ΦpBq is the flux of the magnetic field across the area D delimited by γ, and where we have introduced the elementary flux Φ0“ h{e.

The appearance of this phase factor that depends on the magnetic flux, is widely known as Aharanov-Bohm effect, and it is one of the most important manifestations of magnetism in quantum systems. I want to underline the fact that this geometrical phase is proportional to the charge of the particle: more generally, magnetic behavior is a property of charged matter3_{, which makes simulations with neutral}

atoms challenging, as we will discuss in the following paragraphs.

1.1.2 Harper Hamiltonian in solid state physics

Bloch theorem We are interested on finding the eigenvalues and eigenvectors of the Schrödinger equation: Hψprq “ ˆ ´¯h 2 ∇2 2m ` V prq ˙ ψprq “ Eψprq (1.7)

where V prq is a periodic potential of period a, i.e., V prq “ V pr`aq. In the following we will restrict to the one dimensional case, mainly following the reasoning in [10]. In the absence of the periodic potential, the eigenstates of (1.7) are plane waves4 _p

kpxq “ p1{

?

Lqe´ikx _{where k “}

b

2mE{¯h2. We rewrite the periodic potential as V pxq “ řnV phnqe

ihnx, where h

n “ 2πn{a, and introduce the subspace Sk “

tpkpxq, pk˘h1pxq, pk˘h2pxq, ...pk˘hnpxqu: this subspace is closed under the application of the Hamiltonian

in (1.7). In other words, we can reduce the analysis of the problem to the subspaces Sk with ´π{a ď

k ď π{a. For all the wavevectors k1

“ k ` 2πn{a with n integer the problem is the same as for the wavevector k. The region ´π{a ď k ď π{a is called first Brillouin zone or simply Brillouin zone (BZ). The effect of the potential is to lift the degeneracy between the edge states of the BZ: the spectrum of the system results in well separated energy bands, referred to as Enpkqwith k varying in the BZ.

From what explained above results that for each k in the BZ, the eigenstate of (1.7) has to be found as a linear combination of functions in Sk:

ψkpxq “ 1 ? L ÿ n cneipk`hnqx“ 1 ? Le ikxÿ n cneihnx“ 1 ? Lukpxqe ikx _(1.8)

where ukpxqis a function with the same periodicity as the lattice potential. This is the content of the

Bloch theorem: in the presence of a periodic potential the eigenstates of the system have to be as in (1.8), that is, a plane waves modulated by a second periodic function, with the same periodicity as the potential.

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Tight binding model In the previous chapter we have derived the eigenstates of the system starting from plane waves, that is, delocalized functions: the reasoning is well fitted, for instance, in the case of metals, where electrons move almost freely in the crystal. We consider here the opposite case, where electrons are supposed to be localized in the lattice sites.

If we suppose infinite potential depth, the system is N times degenerate (with N the number of lattice sites), with energy E0, and the eigenstates are the electronic functions φpx ´ naq “ φnpxq, centered in

the (n-th) lattice site. The degeneracy is lifted as soon as we consider tunneling among different sites: in the tight binding (TB) model, we assume that particles are so localized that only tunneling between neighboring sites is allowed. The Hamiltonian of the system will be a tridiagonal matrix, whose elements are given by:

xφn|H|φny “ E0 xφn|H|φn˘1y “ ´J (1.9)

Because of the translational invariance of the system, both E0and J do not depends on n. Translational

invariance also implies that J is real.

The eigenstates of the system are given by a linear combination of the atomic functions. This combination has to be in agreement with the Bloch theorem. These requirements are satisfied by the Bloch functions

ψkpxq “ 1 N ÿ n eikanφpx ´ anq, (1.10)

where we have exploited the fact that the periodic potential doesn’t mix different ks in the BZ (the Hamiltonian is a block diagonal matrix with respect to k).

Evaluation of the matrix elements (1.9) within each block gives the energetic spectrum of the system:

Epkq “ E0` 2J cospkaq. (1.11)

Wannier functions It is useful for the following to introduce the Wannier functions wpx´anq. These functions form a basis of orthonormal, localized functions and are obtained from the linear combination of Bloch functions ψkpxq: wpx ´ anq “ ?1 N ÿ k e´ikanψkpxq (1.12)

where, in turn, the Bloch functions can be obtained starting from another set of localized functions φpx´ anq(in general, the electronic functions that diagonalize the Hamiltonian in the absence of tunneling). It is important to underline the fact that, unlike the Block functions, the Wannier functions a priori do not diagonalize the Hamiltonian of the system.

Wannier functions can be defined for any kind of periodic system, independently on the validity of the tight binding approximation[15]: obviously, the more the potential is deep, the more the Wannier functions are similar to the atomic ones.

Magnetic field in a 2D periodic system We now extend our analysis to 2D systems, and introduce a magnetic field B. We take B uniform and constant, propagating along ˆz, whereas the 2D system is assumed to lay in the ˆxy plane. We introduce a basis of Wannier functions t|n, my “ wpx ´ naxqwpy ´

mayqu; axand ay are the sizes of a lattice cell. We will work in the Landau gauge

A “ B ¨ ˝ ´y 0 0 ˛ ‚, Bzzˆz (1.13)

In the TB regime the Hamiltonian in the absence of magnetic field is:

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Fig. 1.2: Particle moving in a plane lattice: tunneling can occur only among neighboring sites. Moving parallel to the vector potential, the tunneling amplitude acquire a complex phase

Extending to the 2D case what we said so far, the system energy band is Epkx, kyq “ E0´2J rcospkxaxq`

cospkyayqs, with a band width of 8J.

When we introduce the vector potential (1.13), the system is no longer periodic along the ˆy direction, and thus in principle the Bloch theorem and the band structure of the spectrum are no longer valid. Nevertheless, under the assumption that the magnetic field is weak enough so that we can neglect contri-butions from different bands[5], it can be shown that the eigenstates of the system are one-dimensional Bloch functions, calculated as in (1.8), along the direction where the periodicity is conserved.

To account for the presence of the magnetic field, the matrix elements in (1.14) have to be modified according to the so-called Peierls substitution J ÝÑ J exppiφpn, m Ñ n1_{, m}1qq, with

φpn, m Ñ n1, m1q “ ´e ¯ h n1,m1 ĳ n,m Apx, yq dx dy, (1.15)

where the integral has to be performed along the path. In the given gauge, this phase will be zero if we are moving along ˆy (n, m ÝÑ n, m ˘ 1). On the contrary, if we move parallel to the vector potential it will be: φ “ ´e ¯ h pn˘1,mq ĳ pn,mq Apx, yq dx dy “ eB ¯ h xy| pn˘1,mq pn,mq “ ˘ eB ¯ h axaym “ 2πΦpBq{Φ0m “ 2παm (1.16) where we α is the ratio ΦpBq{Φ0. We have thus retrieved the analog of the Aharanov-Bohm phase for

a particle moving in a lattice. Indeed, if we move around the unit cell (closed path), the total phase acquired by the wavefunction is exactly 2πΦpBq{Φ0: the dependence on the index lattice m has dropped,

and we are left with the same expression as (1.6). The Hamiltonian (1.14) becomes:

H “ ÿ m,n E0|n, myxn, m| ` ÿ m,n ´

J eiφpn`1,mÑn,mq|n ` 1, myxn, m| ` J |n, m ` 1yxn, m| ` h.c. ¯

. (1.17)

This is known as Harper Hamiltonian, and is exactly the Hamiltonian we want to implement in a system of cold atoms confined in a bidimensional lattice. It’s important to underline the fact that the assumptions of validity for the Peierls substitution are not really important in the context of quantum simulation, because we aim to implement directly the Hamiltonian with the complex tunneling amplitude.

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Fig. 1.3: Hofstadter butterfly: spectrum of a particle moving in a 2D lattice under the influence of a magnetic field, as a function of the magnetic flux across the lattice cell. In the absence of magnetic field, the sy-stem is periodic along the two directions and the periodic potential individuates an energy band of width 8J. In the presen-ce of a magnetic field, the system loses the periodicity along one direction, say ˆy. If Φ{Φ0“ p{p1, with p and p1integer, the

sy-stem recovers a new periodicity along ˆy, that causes the energy band to split in-to p1 _{subbands. Only rational values of α}

with denominator less than 50 are shown. Image taken from [17].

magnetic field breaks the periodicity along ˆy, but not along ˆx. Thus we can still look for eigenstates of the form: ψkxpnaq “ ÿ n,m eikxnax_{wpx ´ na} xqwpy ´ mayq. (1.18)

Performing the calculations, we obtain that the Hamiltonian is a block diagonal matrix respect to kx.

Each block is of the form: Hpkxq “ ¨ ˚ ˚ ˝ E0` 2J β1 J 0 0 ... J E0` 2J β2 J 0 ... 0 J E0` 2J β3 J ... ... ... ... ... ... ˛ ‹ ‹ ‚ (1.19)

where βm “ cospkxax´ 2παmq. The energy bands arise from the diagonalization of the tridiagonal

matrix (1.19): the analysis can be limited to the cases where 0 ď α ď 1.

It is interesting to observe what happens when α assumes rational values [16]: setting α “ p{p1_{, with p}

and p1 _{integers, we recover a new periodicity along ˆy, with period p}1_a

y. Indeed:

φpn, m ` jp1

Ñ n ˘ 1, m ` jp1q “ 2παpm ` jp1

q “ 2παm ` 2παjp1 (1.20)

“ 2παm ` 2πpj “ φpn, m Ñ n ˘ 1, mq ` 2πjp.

Independently on the nature of α, kxis still varying within the first BZ, ´π{axď kxď π{ax. If α “ p{p1,

ky is varying within the reduced zone ´π{pp1ayq ď ky ď π{pp1ayq: this new periodicity individuates p1

energy bands (separated by well defined energy gaps) inside the original energy band of with 8J, typical of the 2D lattice in the absence of magnetic field. The subband structure varies enormously for small variation of α, as we can see in fig. 1.3, where the allowed energies are plotted as a function of α.

In the systems that we have described above, we can individuate two natural lengths scales: the cell size a (for simplicity, we will assume that ax “ ay“ a), and the cyclotron radius (or magnetic length)

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lmag “

a ¯

h{peBq. From eq.(1.16) it follows that α can be expressed in terms of these lengths rather than the magnetic flux. Indeed we have

2πα “ 2π Φ Φ0 “eB ¯ h a 2 “ ˆ a lmag ˙2 . (1.21)

For α ! 1, the cyclotron radius is much bigger than the lattice size: thus magnetic phenomena are not modified by the presence of the lattice potential. We can treat the system as a continuous one, regardless of the discretization of the space. For α Ñ 0, the energy levels are well discretized: we actually recover the Landau levels, that in turns arise when we consider a free electron gas in a magnetic field. On the opposite, for α » 1, both magnetism and periodicity of the crystal have to be taken into account in the description of the system.

1.2 Implementation of the Harper Hamiltonian with cold atoms

1.2.1 Optical lattices

Before presenting the scheme that we will use to implement the Harper Hamiltonian, I want to recall here a few elements of the physics relative to a bosonic system in an optical lattice. A brief on the experimental realization of optical lattices is reported in Appendix D : for the moment it is sufficient to know that this potential is of the form V0sin2pklxq, where kl is the wavevector of the laser used for the

lattice. The period of the lattice is half the wavelength of the laser: a “ λl{2. In the following we will

limit the discussion to the 1D case.

Single particle in an optical lattice The Hamiltonian of a particle in an optical lattice is given by:

H “ p 2 2m´ V0 2 cosp2klxq ` V0 2 . (1.22)

Since we are dealing with a periodic potential, from section 1.1.2 it follows that the eigenstates of the system will be Bloch functions of quasi-momentum k; the energy spectrum will be divided in bands, and the analysis of the problem can be confined to the first BZ, that is, for ´π{a ď k ď π{a. Two energy scales arise from the problem: the potential depth V0 and the recoil energy Er“

¯ h2_k2

l

2m , that coincides

with the maximum kinetic energy achievable by a particle in the first band. Depending on the ratio between these two values, the system will be in the delocalized regime (V0ă Er), or in the tight binding

regime (V0ą Er). This latter is the one we are interested in.

Since we are dealing with a system in the TB regime, it is natural to introduce a basis of Wannier functions wn “ wpx ´ anq. In the particular case of a well depth V0" Er, it is logical to assume that the

wavefunction is strongly confined at the well center, and we can thus develop the potential for x » an.

The Hamiltonian assumes the form of an harmonic oscillator of frequency ω “ 2?V0Er{¯h. The Wannier

function wn for the ground state will be approximated with a Gaussian centered in an and of width

aw[18]: wn» exp ˆ ´px ´ anq 2 2aw ˙ , aw» pV0Erq´1{4. (1.23)

The width of the gaussian decrease with the strength of the periodic potential, and the wavefunctions become more and more localized (fig. 1.4).

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Fig. 1.4: Wannier functions relative to two neighboring sites, in the gaussian approximation, plotted for different values of V0{Er. The dotted curves are the same function, displaced of a lattice site. With the increase of this

ratio, the width of the gaussian diminishes, together with the overlap between neighboring functions.

System of bosons in an optical potential: the Bose Hubbard Hamiltonian We now consider several atoms moving in the same optical lattice.

Within the Bose Hubbard (BH) model, the Hamiltonian of the system is obtained in the frame of the second quantization, starting from a many-body Hamiltonian where interactions among particles are described by a contact pseudopotential. Two important approximations are done: 1) the lattice potential depth is such that the TB model holds; as a consequence, we can express the atomic wavefunctions in terms of Wannier functions; we can neglect tunneling except among first neighboring sites; 2) the energy of the particles is low compared to the separation among two bands. Thus we can retain only the Wannier functions of the first band.

Under these assumptions, the Bose Hubbard Hamiltonian reads: H_BH“ E0 ÿ n ˆ nn` J ÿ xn,n1_y ˆ a: nˆan1` ˜U ÿ n ˆ nnpˆnn´ 1q (1.24) where ˆn “ řiˆa :

iˆai is the number operator, E0“ş wnprq

´ ´¯h_2m2∇2 ` Vlatprq ¯ wnprq dr, J “ ş wnprq ´ ´¯h_2m2∇2 ` Vlatprq ¯ wn1prq dr, ˜U “ 1 24π¯h 2_a s{m ş

|wnprq|4dr, with as the scattering length.

The factor ˆnnpˆnn´ 1q follows from the fact that we are considering binary interactions, and that an

isolated particle does not interact with itself. It can be shown ([18]) that J{Er» V03{4expp´

a V0{Erq.

The dynamics of such a system is strongly related to the ratio U{J. This topic is widely treated in the literature ([19], [1]): we will only report a few qualitative considerations. If U ! J (superfluid regime), the tunneling is dominant with respect to the interactions: the BEC is thus delocalized in the optical lattice. It can be described as a superposition of Fock states (i.e., states with a well defined number of particles per site). In the superfluid regime coherence is preserved between the wells. The Mott insulator regime occurs for J ! U: in this case, particles are well localized in the lattice sites. The occupation number per well is well defined, whereas there is no phase correlation between particles.

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1.2.2 State dependent lattice and laser assisted tunneling

Equation (1.22) can be easily extended to the 2D case. With a suitable choice both of the trapping and lattice potential, the Hamiltonian takes additional terms that are formally identical to the first ones, but are a function of the new variable y. Consequently, the single site wavefunctions and the annihilation and creation operators will be respectively wn,m“ wpx ´ axnqwpy ´ mayqand a:n,m, an,m.

Despite this substitution, the Hamiltonian (1.24) is still far from having the same structure as the Harper Hamiltonian that we want to simulate. Indeed, the marker indicating the presence of a magnetic field is the complex value of the tunneling amplitude, obtained in (1.17) through the Peierls substitution. On the contrary, the tight binding approximation alone implies only that J is real.

How can we get a complex value for the tunneling factor J in the Bose Hubbard Hamiltonian? As pointed out by Jaksch and Zoller in [20], the only parameter that we can experimentally modify is the periodic potential Vlatpx, yq. Acting only on Vlat is not sufficient, since J stays real. Moreover if we

modify J letting unaltered U, we could induce a variation in the ratio U{J, and consequently in the kind of system we are dealing with (superfluid/Mott insulator phase transition).

The solution they propose is to use two different atomic configurations |gy and |ey , and a state dependent optical lattice, as sketched in fig. 1.6: tunneling between sites occupied by atoms in the same state occurs as for the single species case, with a real amplitude; on the contrary, tunneling between sites occupied by different species requires an internal state transition, which involves the coupling with a light potential VAL. Thus in this second case, the tunneling amplitude depends both on the overlap of the neighboring

wavefunctions and on the off diagonal matrix elements of VAL. We will show briefly that atoms-light

interaction leads to a complex phase factor in the tunneling amplitude, that plays the role of the Peierls phase in condensed matter: in this sense, laser assisted tunneling is said to be the key point in gauge fields simulations.

In our experiment, we exploit the presence of the metastable state3_P

0among Ytterbium lines(see section

3.3). The metastable state will play the role of the excited state. The state dependent lattice is obtained thanks to the fact that there exist, for Ytterbium, particular wavelengths such that the polarizability of the |gy and |ey states are the same (magic wavelength λmagic) or opposite (antimagic wavelength λantim),

as we can see from picture1.5. When the two states have the same polarizability, they are both trapped in the minima of the potential. On the contrary, if they have opposite polarizability, one is trapped in the minima and the other in the maxima. Referring to figure 1.6, along the ˆy direction atoms are trapped in a 1D lattice at λmagic: each particle, no matter which its internal state is, occupies a site

m, where m indicates the position of the potential minimum (the lattice size is ay“ λmagic{2). Along

the ˆx direction the lattice is at λantim: in the minima (n, n ˘ 1, ...) there are trapped particles in |gy ,

whereas in the maxima (n ˘ 1{2, ...) there are trapped particles in |ey . This results in two sublattices, displaced along ˆx of a quantity ax{2 “ λantim{4. It’s important to note that in this configuration the

two wavefunctions wn,mand wn˘1{2,m are not orthogonal, since they belong to different sublattices, and

their overlap can be substantial.

The Bose Hubbard model can be reformulated so that the presence of two different atomic species is taken into account. We introduce the annihilation operator bn1_,m1 for particles in the excited state,

with n1_{“ n ` 1{2} (a

n,mis the annihilation operator for particles in the ground state), and the number

operators ˆnpgq_“ř iˆa : iˆai and ˆnpeq“ ř i1ˆb :

i1ˆbi1 5. According to this, eq.(1.24) reads:

H_BH“ÿ i εpeq_i â:_iâi` ÿ i1 εpgq_i1 ˆb : i1ˆbi1` Jpxqp ÿ xi,i1_y ˆ a:_iˆbi1` h.c.q ` Jpyqp ÿ xi,i1_y pˆb:_iˆbi1` â:_iâi1q ` h.c.q (1.25)

5_{The couple of index m, n has been replaced by i: this notation is kept as far as it’s not necessary to explicit along}

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Fig. 1.5: Polarizability of the Yb states1

S0(dashed blue line) and3P0(red line), used respectively as the |gy and

the |ey of our system. Polarizability is plotted as a function of the wavelegth. We can individuate three points where the two species have equal polarizability (λmagic=760 nm) and opposite ones (λp1qantim “ 617nm and

λp2q_antim“ 1116nm)[4].

Fig. 1.6: Lattice configuration (on the right). Sites are spaced of λmagic/2 along ˆy and of λantim/2 along ˆx. The

lattice at λmagicactually individuates two sublattices, displaced of λantim/4, one for each particle in the internal

state. The insights show how particles in different internal states are trapped. The lattice at λmagicis such that

all the particles are trapped in the minima of the potential, whereas in the other one particles are trapped in the minima or in the maxima depending on whether they are in the ground or excited state.

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Fig. 1.7: Tunneling along the two directions: for the λantimlattice (right), the two different states are coupled

thanks to a Rabi impulse. This gives rise to a complex phase in front of the tunneling amplitude J. Along the λmagicdirection (left), tunneling amplitude is real

We have written separately the local energies εpαq

i of the ground and excited states, because they depend

on the internal state α of the atoms (through the polarizability) and thus could be in principle different6_.

Tunneling along the ˆy direction occurs exactly in the same manner as before, since it involves particles in the same internal state (xα|βy “ δα,β). Problems arise if we consider the tunneling along ˆx: indeed, in the

TB regime the tunneling n, m ÝÑ n ` 1, m, that is, between neighboring sites of the same sublattice, is inhibited because of the weak overlap between the wavefunctions. Tunneling between neighboring sites, that is, n, m ÝÑ n ` 1{2, m is forbidden because, even if the two spatial wavefunctions overlap, the two internal states are orthogonal. This problem is circumvented introducing a light field that couples the two internal states (fig.1.7): this can be obtained using a running wave laser beam with wavevector ~kR.

The interaction is described, in the rotating wave approximation, by the potential VAL:

V_AL“ ˆ ∆ hΩ¯ R 2 e´ikR r ¯ hΩR 2 e ikRr ₀ ˙ (1.26) where ∆ is the detuning of the laser field and ΩRis the Rabi frequency (real) characterizing the coupling

strength. The addition of this term does not modify the Hamiltonian (1.25), except for the term in Jpxq

(the effect of the detuning can be included in the single site energies εi). A direct evaluation leads to

J_˘pxq“ť dx dy wnpx ´ naxqwmpy ´ mayqV_AL˘wnpx ´`n `1₂˘ axqwmpy ´ mayq (1.27)

“ ¯hΩR

2 e

˘ikymay_I

xIy

where Ix,y depends on the overlap between the Wannier functions respectively along x, y. The sign

depends on the transition we are considering: e Ñ g (´), g Ñ e (`).

Substituting Jpxqin (1.25), we are left with the Hamiltonian we were looking for. The sign appearing in

the phase of Jpxq depends on the atom transition, but it is completely independent on whether we are

moving in the positive or negative direction; in terms of operator this means that the Hamiltonian of the system is not hermitian. Therefore, in order to perfectly simulate the model presented in section 1.1.2, we have to fix the sign of the phase in the proper way. The exponent kymay plays the role of the phase

2παm, found in eq. (1.16). It’s worthy to note that changing the angle of incidence of the coupling laser is equivalent to change the value α in the Harper model: we are thus able to experimentally explore all the α values included in the interval7_[0,2λ

R{λmagic].

6_{We have also neglected interactions among particles: these would have carried three additional terms ˜}_U

eeř_ipa:_iq2a2 i`

˜

Uggři1pb:_i1q2b2i1` ˜Ugeř_xi,i1_ya:_iaib:_i1bi1. The terms in ˜Uαα describe the on-site interactions. The term in ˜Ueg describes

interactions among particles of neighboring sites: this term in principle can not be neglected, because it depends on the overlap between neighboring wavefunctions, that, as already pointed out, is not zero. As explained in Chapter III, interactions are responsible for a widening of the transition line g Ñ e (together with other effects).

7_{Actually, we can also vary a}

yadding an angle θ between the beams that form the lattice, as explained in Appendix D.

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1.2.3 Rectification of the phase

We have seen above that the sign of the phase is positive or negative depending on whether the tran-sition is from g to e or viceversa. Since the internal states are alternated along ˆx, this implies that while passing from a cell to the neighboring one the sign of the phase flips from `kymay to ´kymay.

Carrying on the analogy with the Harper model, this means that the magnetic flux alternates its sign along the lattice (staggered magnetic flux, picture 1.8). It is therefore necessary to artificially induce the right phase sign: we refer to the phase fixing as rectification.

Several methods have been proposed in the literature. The basic idea is to introduce an energy shift between g and e, in such a way that the transitions g Ñ e and e Ñ g are resonants with two different frequencies. Once achieved this situation, the two impulses can be made propagating in opposite direc-tions, so that the proper signs in front of the phase appears naturally.

In our case, the flux rectification is obtained using a so-called superlattice[3]: a second lattice, with period asl “ 2ax is superimposed to the antimagic lattice. This introduces a shift in the resonance

energies. In principle, four different frequencies are needed. Nevertheless, choosing properly the relative phase between the two lattices allows for the use of only three frequencies. Referring to picture 1.9, the phase between the two waves is such that the zero of the superlattice potential occur at a site where a paricle in the state g is trapped, at an energy Eg´ V1, where V1 is the potential depth felt by the

particle, and where we set to zero the potential energy felt by a particle in the state e (in the absence of the superlattice. We are neglecting all the other terms concurring to the energy of the particles). In the presence of the superpotential, particles are shifted versus higher or lower energy, depending on their internal state (at λantimthe polarizabilities of the two states have opposite sign). The shift V2 is, in

absolute value, the same for the two particles. Starting from a particle in the nth site, the new energies are:

En“ Eg´ V1 (1.28)

En`1{2“ Ee´ V2

En`1“ Eg´ V1` V2

E_n`3{2“ Ee

The new frequencies are therefore:

ω1“ ωgÑe“ ¯hω0` V1´ V2 (1.29)

ω2“ ωeÑg“ ´¯hω0` 2V1´ V2

ω3“ ωgÑe“ ¯hω0` V1´ V2

ω4“ ωeÑg“ ´¯hω0` ´V1

The same beam can be used for inducing the transition g ÝÑ e. The two opposite transitions are induced by two other beams, propagating in opposite direction with respect to the first one. We thus obtain the proper sign for the tunneling phase factor. The Hamiltonian is now Hermitian. The phase rectification based on a superimposed super-lattice is peculiar of our setup, and has been proposed by F.Gerbier and J.Dalibard ([3]). The proposal of Jaksch and Zoller ([21]) is based on the addition of a linear potential V pnq “ nη along the ˆx direction: with a proper choice of the gradient of this potential, it can be shown the two following transition e Ñ g and g Ñ e occurs at two different frequency: the transition can be addressed with two different laser, propagating in opposite directions. In the choice of the laser powers, it is important to avoid excitation to higher Bloch bands (the energy shift between neighboring sites has to be small compared with the depth of the optical lattice).

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Fig. 1.8: Alternation of the phase sign in the tun-neling amplitude along the ˆx direction. This is the analog of an alternated magnetic flux across the lattice site in solid state physics.

Fig. 1.9: Potentials felt by the particles in the two internal states, in the presence of a superlattice of amplitude V2. Blue arrows indicate the

transi-tions g ÝÑ e (1,3) and e ÝÑ g (1,4): the phase between the lattice and the superlattice can be chosen in such a way that ω1“ ω3. In the

pictu-re, the amplitude of the pseudopotential has been exaggerated on purpose.

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Chapter 2

Experimental setup

"I am betting on nature to hide Bose condensation from us. The last 15 years she’s been doing a great job".

Steve Chu, one year before the achievement of the first gaseous condensate.

Fig. 2.1: Piece of Yb

This chapter is devoted to the description of the way the Bose Einstein Condensate of Ytterbium is obtained. First, a sample of solid Ytterbium is heated at 500°C: even if we are far below the sublimation temperature, at 500°C the vapour pressure of Yb is sufficiently low to allow for the formation of a gaseous cloud. The atomic jet exits out from the oven, passes through a collimating tube, and gets into the Zeeman slower, through which it thus joins the first vacuum chamber: at this point, the temperature of the atoms is a few Kelvins. Here atoms are trapped in the magneto optical trap (MOT1_{) and cooled down close to the Doppler limit. An optical trap is}

subsequently switched on. From the MOT chamber, atoms are transferred to the so-called science chamber, where a second optical trap crosses the first one, forming a crossed dipole trap. After evaporation, atoms get a temperature below 1 µK, and the condensation begins.

In the following, I aim to describe each experimental step leading to the BEC, both from a theoretical and a practical point of view. At the beginning of my internship, in order to get rid of a bug in the vacuum system, the experiment was forced to be disassembled: I therefore had the big fortune to take part, later, in the rebuilding phase. For this reason I will report here also on measurements, optimizations and technical details of the assembly, that have been a substantial part of my internship.

2.1 Ytterbium

Ytterbium (Z=70) is the 14th_{element of the Lanthanide series, and its electronic configuration is similar}

to the one of alkaline earth metals, with two electrons in the outer s-shell: [Xe]4f14_6s2_{. In nature there}

are seven different stable isotopes: of those ones, five are bosons, with I=0 (168_Yb,170_Yb,172_Yb,174_Yb, 176_{Yb), whereas the two left are fermions, with I=1/2 (}171_{Yb) and I=5/2 (}173_Yb).

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Yb melts at 1097°C, and boils at 1469°C [Alcock, 1984]. Nevertheless, at 500°C it has a vapor pressure of 0.73 mTorr: those conditions of temperature and pressure are sufficient for getting a vapor of Yb. The energy structure of Yb is pretty rich. The fact that Yb has two electrons in the outer shell, implies that it presents both singlet and triplet states2_{. The availability of metastable states makes this}

che-mical species one of the favorite candidate for investigating a wide variety of phenomena, such as clock transitions, two level systems physics, parity violation, etc...

The bosonic isotope that we use is the most abundant one3₍174_{Yb). Because of the zero nuclear spin, its}

energy structure is slightly less complicated that the one of the fermionic isotopes (no hyperfine structu-re).

Fig. 2.2: Energetic levels of Yb.

The ground state of Yb (and in general of atoms with two electrons in the outer s-shell) is a singlet state,1_S

0. The vanishing electronic spin implies

that only optical method can be used for trapping atoms. According to the selection rules, atoms from the ground state can be optically excited to the singlet state1_P

1(∆S=0, ∆L=1, ∆J=1): this

transition (λ “ 399 nm, blue) has a large natural width (Γ399 “ 2π ˆ 29MHz), and thus is well

fitted for the Zeeman slower, where a big amount of momentum has to be transferred away from atoms. The 1_S

0 ÝÑ 1P1 is an open transition:

atoms naturally decay to the states3_D

1and3D2,

and then to the 3_P

J states: the J=0 and J=2

are metastable states, and thus atoms fallen here are in principle lost. There are experiments (for instance [22]) where this transition is used also for the MOT: in this case a repumper recycles atoms to1_S

0.

In our case, the transition we do exploit for the MOT is the green one: 1_S

0ÝÑ3P1(λ=556 nm). This

transition is made possible because of the LS coupling (Russel Saunder regime): L and S are no longer good quantum number, and J has to be considered. Under the effect of a vector potential (∆J “ ˘1), the transition 1_S

0ÝÑ3P1is thus allowed. As will be explained in section 2.4.1, spontaneous emission

imposes an inferior limit to the temperature achievable during the cooling mechanism (Doppler limit): the green transition has a natural width of Γ556 “ 2π ˆ 182kHz, that translates in a low Doppler

temperature (4 µK). This is particularly important since the ground state has J=0, and thus it would be impossible to perform Sisyphe subcooling4_.

The transitions1_S

0ÝÑ3P0,2 are optically forbidden (∆J “ 0, 2). Nevertheless, we can circumvent the

problem weakly perturbing the system: the additional Hamiltonian term has to be choosen in such a way that the transition will be partially permitted. We will return on this in chapter III.

2_{In the following, we will use the Russel-Saunders notation:} 2S`1_L

J, where J=L+S and L is indicated with letters,

following the usual convention s,p,d,f,... for increasing values of L.

3_{Its abundance is of 32.02%; it is followed by}172_{Yb (21.68%),}173_{Yb (16.10%),}171_{Yb (14.09%),}176_{Yb (12.99%);}170_Yb

and168_{Yb are the less abundant: respectively 2.98% and 0.12%.}

4_{Cooling technique that allows for temperature lower than the Doppler limit. It requires the ground state to be}

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Fig. 2.3: Experimental setup. Atoms travel from the Oven to the MOT chamber, through the Zeeman slower, and then from the MOT chamber to the Science chamber thanks to a small tube 20cm long. A typical experimental cycle (to obtain the condensate) last few seconds.

2.2 Vacuum system

The experimental set up is mainly constituted by the oven, the MOT chamber and the science chamber, each one connected to the other. In particular, atoms pass from the oven to the MOT chamber traveling through the Zeeman slower. The idea is to spatially separate where the BEC is prepared (oven, Zeeman slower, MOT chamber) from where its manipulation takes place (the science chamber). The same reasoning has been followed for the the optical setups, as far as it was possible.

It’s of fundamental importance that all the experiment is performed at conditions of almost zero pressure. Indeed, collisions with residual particles heat the cloud of Yb with consequent losses in the number of atoms that can be condensed. The characteristic time between two collisions is inversely proportional to the pressure of the residual gas itself (for simplicity we will assume N2 even if actually it’s not the only

one): 1{τ “ σYb´N2vN2nN2. For a thermal gas we can perform the two substitution vN2“

a

3KBT {mN2

and nN2 “ P {pKBT q. At room temperature KbT » 1{40eV; the cross section for Yb and N2is » 10

´7_m2_.

Thus we get τP » 10´8s¨mbar. Thus, if we want a characteristic time of e.g. 10 s, we need a pressure

of about 10´9_{mbar ([4]).}

Such low pressures are obtained thanks to a system of vacuum pumps connected to the setup at different points. Aside the oven, two ionic pumps are present, whereas a third one is placed in the junction between the Zeeman slower and the oven itself. In the MOT chamber the vacuum is assured by a big ionic pump and a getter one5_{; same happens at the Science chamber, where two smaller pumps are used (a getter}

and a ionic one).

The optical access to the two chambers is provided through viewports of different diameters. For what concerns the MOT chamber, six viewports (two horizontal CF100 and four diagonal CF63, 45° tilted respect to the optical table), give access to the MOT beam; additional other eight CF16 ports are

5_{Ionic pumps basis their working principle on the ionization of particles: in this sense, they are called "active" pumps.}

Getter pumps are made of absorptive material: particules encountering this material are absorbed and therefore "leave" the chamber: those ones are in some sense "passive".

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present6_{: among these, one is used for the Zeeman beam and one for an harm of the dipole trap. The}

MOT and the science chambers are connected by a tube of diameter 16 mm, long about 20 cm. The science chamber, where the investigation on gauge potential is carried on, should has an optical access as good as possible, so that atoms can be manipulated easily by optical means. The viewport are all CF40: two along the vertical axis, and eight on the equatorial plane.

2.3 Zeeman slower

Atoms exit from the oven at almost 500°C, that is, a temperature at which the vapor pressure of Yb is low enough to allow for part of the atoms entering the vapor state. Hereafter, atoms can be treated as a thermal gas which obeys the laws of classical thermodynamics. In particular, velocities are distributed according to the Maxwell Boltzmann:

fM Bpvq “ ˆ 2m KBπT ˙3{2 e´2KB Tmv2 (2.1)

2.3.1 Principle of the Zeeman slower

The first step for laser cooling exploits the radiative pressure exerted on a moving atom by a plane wave propagating in the opposite direction with respect to the atom.

Consider fig. 2.4(a): an atom of velocity v and energy E is traveling in the positive direction, whereas a photon (of energy ¯hωl) propagates in the opposite direction with momentum ´¯hk, k “ ωl{cpositive. If

the photon is resonant with the atom, it excites the atom. Because of the conservation of energy and mo-mentum, the atom results with a final energy E1_{“ E `¯}_hω

land a velocity, in modulus, v1“ v ´¯hk{m: its

velocity is diminished. When the atom de-excites fig. 2.4(b), it re-emits a photon of the same energy ¯hωl.

It’s velocity will be vf “ v1`¯hk1cos θ{m, where k1is the wavevector of the "new" photon: it is unchanged

in modulus (k1

“ k “ ωl{c), but it is directed along a random direction (individuated by the angle θ).

Averaging in several cycles, this randomness ends up in a decrease of the total velocity (momentum) of the atom.

Fig. 2.4: Excitation (a) and disexcitation (b) of the atom. The photon is emitted in a random direction

The extension of this picture to an ensemble of coheren-tly propagating photons (i.e., the laser beam) and an ensemble of moving atoms (i.e., the atomic jet, who-se velocity distribution has a certain width and cen-tral value that, according to statistical mechanics, de-pends on the temperature) leads to the concept of ra-diative pressure[23]: the light field exercises on atoms a radiative force given by:

F prq “ ¯hkΓ 2

sprq

1 ` sprq (2.2)

where s is the saturation parameter7

s “ _δ2Ω_`Γ2{22_{4 “

I{Isat

1`p2δ{Γq2.

6_{CF stands for ConFlat: the fused silica window is already mounted in the flange. The various pieces are assembled}

to the main chamber. Vacuum is preserved thanks to a knife-edge mechanism: when assembles, between the pieces is placed a copper gasket. When the screws are clamped, the knife-edge deforms the copper and assures an airtight seal. The number indicates the dimensions of the viewport (optically effective diameter 99,68, 36, 16 mm, respectively for CF100, 63, 40, 16).

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To maximize the deceleration, the photon has to be resonant with the transition. This is far from trivial, since the atom is moving at a given velocity v and thus we have to take into account the Doppler effect. Ergo we see that if the photon was resonant at a given time, it will not at a later one, when the atom velocity is decreased.

The condition for the resonance is obtained for ωl “ ωeg ´ kv, with v changing along the path. To

face the problem, a magnetic field is introduced: because of the Zeeman effect, the atom levels undergo an energy shift ∆EpBq, that depends both on their magnetic moment and on the field strength. The detuning in the presence of the magnetic field reads:

ωl“ ωeg´ kv `

∆EpBq ¯

h (2.3)

Accurately choosing the profile of the magnetic field, we can thus circumvent the problem arising from the Doppler effect and so exploit the maximum of the force eq. (2.2) along the whole distance traveled by the atoms.

2.3.2 Magnetic field profile

In our experiment we exploit the blue transition1_S

0ÝÑ 1P1: this choice is dictated by the large natural

width of the transition, that for small detuning translates into a big force. In this case, the ground state has zero total angular momentum, while for the excited one J “ 1: thus only the state1_P

1 undergoes

the Zeeman energy shift:

∆EpBqz“ ´µB “ gjmjµBB “ ˜µB (2.4)

where gj is the Landé factor for the state1P1 (» 1), and where we have assumed B propagating along

the ˆz direction. In the following, we will work in terms of modulus, since we are considering only the ˆz direction.

In order to determine the expression for the magnetic field profile, let us introduce the quantity amax“

´¯hkΓ_2m, the maximal deceleration. This maximum is obtained for saturated intensity (eq.(2.2) with s Ñ 8): we can handwavingly figure this upper bound for the deceleration thinking to the fact that high intensities exert strong forces, but at the mean time induce not negligible stimulated emission, since they augment the number of atoms that are in the excited state (and therefore no longer disposable for excitation)8_.

Rewriting the force of radiation as F “ mamax, we can derive the equation of motion of the particle. In

particular, the dependence of the velocity on the distance z is given by: vpzq “ v0 d 1 ´ 2amaxz v2 0 (2.5) We can substitute this value in the expression for the detuning eq. (2.3): taking then into account the Zeeman shift, and forcing the detuning to be zero (resonant condition), we obtain the expression for the magnetic field profile as a function of z:

Bpzq “ ´¯h ˜ µδ0´ ¯ h ˜ µkv0 d 1 ´2amaxz v2 0 (2.6) where δ0 “ ωl´ ωge is the detuning for the atom at rest, in absence of magnetic field. δ0 is chosen in

order to maximize the efficiency of the cooling and the number of atoms that can be trapped in the

8_{In simulation and calculation, the value a}

maxis substituted by ηamax, with η ă 1. The factor η permits to account

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MOT (next chapter).

In our case, we have fixed the detuning at δ399“ ´2π ˆ 400MHz, that is, about 15 times Γ399. With this

choice, the laser is not resonant with atoms at the center of the MOT chamber (if it does, the efficiency of the MOT loading would have been reduced).

The length of the Zeeman slower is dictated by the initial and final velocities of the atoms: in the calculation ([4]) is taken v0 “

a

3KBT {m » 300 m/s for the first value and the capture velocity of

the MOT for the second one (that in turn has been fixed9 _{at a value below 10 m/s). Those two values,}

together with the acceleration, fix the length of the Zeeman slower: Lzs“

v20´ vf2

2amax (2.7)

In our case, Lzs=23 cm. This has the shape of a truncated cone. It is made of a tube with coils

Fig. 2.5: Zeeman slower cut (the MOT chamber is on the left pf the picture). The big coil at the end of the Zeeman allows to get rid of residual field to whom atoms in the MOT would be very sensitive. The bias (constant) field is produced by the inner coil, whereas the gradient is obtained varying the number of wire turns along the coil.

wrapped around it: the number of turns varies along the tube, providing the desired gradient for the field. A second series of coils, with a constant number of turns, produces the bias field (the constant term in (2.2)). At the level of the MOT chamber, a third coil compensates undesired residual gradients. The actual values for the fields (both of the Zeeman and for the compensation) have been optimized observing the number of atoms captured in the MOT after a fixed loading time.

2.3.3 Laser beam profile

During the deceleration, the atomic jet spreads along the transversal direction. In order to maximize the number of atoms captured in the MOT, it is important to limit the transversal spread. That is, it is important that atoms are not faster than a certain velocity, dictated by the MOT beams configuration (this point will be clarified in the next section). In turns, this maximal transversal velocity imposes an upper limit in the divergence of the atomic jet.

In order to limit the beam divergence, at the exit of the oven is placed a collimating tube, 10 cm long and with a diameter of 2 mm: doing so, only atoms with a ratio vt{vz ă r{l » 10´2 can travel across 9_{The capture velocity for the MOT is fixed by the waists of the trap beams, that in turn define the area of capture.}

(27)

Fig. 2.6: Optical setup used for the λ “ 399nm beam. The presence of part of the MOT optics forced this kind of configura-tion. The first waveplate together with the cube permit to control the power of the beam; the two others permit the control on the polarization of the beam. Actually the last mirror is inclined also respect to the normal at the optical table, sending light to an additional series of two mirrors: this set up bring the beam at the same height as the viewport (hereafter, the optical axis is parallel to the table).

the whole tube, where vt is the transversal velocity and vz is the longitudinal one. The axial velocity

distribution can be derived from the Maxwellian eq.2.1, after integration over the radial velocity. The limits of integration have to be chosen considering that only atoms with a velocity vrď vzr{lcan traverse

the tube. For small divergence, the distribution is given by: f pvzq “ π r2 l2 ˆ 2m KBπT ˙3{2 v2_ze´ mv2_z 2KB T (2.8)

In addition to the natural divergence of the atomic jet, other effects come into play, modifying the transversal spread of the atomic jet: in particular, I mention gravity (even if its effect is negligible) and spontaneous emission.

The real divergence of the atomic jet is almost impossible to be determined exactly. An approximative idea is nevertheless necessary in order to properly fix the waist of the blue beam. In a first attempt, the beam had been prepared with a wide waist, so that it results almost collimated when incising in atoms. Nevertheless, measurements show that a smaller waist (thus a tiny divergence in the beam profile) allows for a higher number of captured atoms. The divergence of the beam is set in such a way that its waist is about 1 mm at the entrance of the collimating tube and 5 mm at the entrance of the Zeeman (MOT side): this latter fix the divergence of the beam to θ » 6 mrad, comparable with the atomic beam divergence (according to a rough estimation).

Optical set up The beam coming from the fiber is collimated with a waist of 660 µm: the waist is first magnified to 5 mm and than focused thanks to a collimating lens (focal length f “ 1000 mm) placed 50 cm before the viewport. Actual waists of the beam in such configuration have been measured aside the Zeeman slower: distances have been optimized in order not to be clipped by 1 inch diameter optics. For the 1_S

0 ÝÑ 1P1 transition (λ “ 399 nm), the saturation intensity is Isat “ 60mW/cm2. The

intensity of the beam is tuned thanks to a waveplate and a polarizing cube placed at the exit of the fiber collimator. The polarization is fixed by a system of two waveplates.

Before describing the next step, I want to underline the fact that, in their path between the oven and the MOT chamber, atoms are not simply decelerated. In fact, what happens it is a bit more sophisticated: the effect of the Zeeman slower is to select a class of velocity (the mechanism is efficient only on atoms slower than a certain velocity), compress the density in the momentum space (the final

Production and High Resolution Imaging of a quantum degenerate Ytterbium atomic gas

Universit`

a degli studi di Pisa

DIPARTIMENTO DI FISICA ’E. FERMI’

Production and High-Resolution Imaging of a

quantum degenerate Ytterbium atomic gas

Candidato

Elisa Soave

Relatori

Fabrice Gerbier

Donatella Ciampini

Table of contents

Introduction

Chapter 1

Simulation of Gauge potential in

optical lattices

1.1

Charged particles in a magnetic field

1.1.1

Charged particle freely moving in a magnetic field: the

Aharanov-Bohm effect

1.1.2

Harper Hamiltonian in solid state physics

1.2

Implementation of the Harper Hamiltonian with cold atoms

1.2.1

Optical lattices

1.2.2

State dependent lattice and laser assisted tunneling

1.2.3

Rectification of the phase

Chapter 2

Experimental setup

2.1

Ytterbium

2.2

Vacuum system

2.3

Zeeman slower

2.3.1

Principle of the Zeeman slower

2.3.2

Magnetic field profile

2.3.3

Laser beam profile