The equation of state of the two-dimensional Bose gas 

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Phase transitions with ultracold two-dimensional Bose gases

The aim of this first chapter is to introduce the theoretical basis upon which our experiments are built. The intention is to give an overview of the features of the two-dimensional Bose gas, without aiming for exhaustivity. A more com-plete picture can be found in recent reviews, such as [21, 74]. Already in 1934, Peierls noticed that the properties of system are strongly af-fected by its dimensionality [56]. In particular, Bose–Einstein condensation (BEC) occurs in the infinite ideal tri-dimensional Bose gas, but only takes place at zero temperature in the bi-dimensional case. Indeed, this is a particular ex-ample of a more general theorem: as noted by Mermin and Wagner [57] and Hohenberg [58], spontaneous symmetry breaking and long-range order are im-possible at non-zero temperature in 1D and 2D systems with short-range inter-actions and a continuous Hamiltonian symmetry. However, there exists a dif-ferent phase transition from a normal state to a superfluid state for an infinite interacting 2D Bose gas: the Berezinskii–Kosterlitz–Thouless (BKT) transition [60, 61].
In any experimental realization of the 2D Bose gas, the two transitions are rel-evant: the finite size of the system restores the BEC transition at a non-zero temperature. In the first section, we will review the properties of the ideal Bose gas, both in an infinite system, and in a trapped geometry. We will then provide a description of the interactions in two dimensions [75–77], and provide a mean field description of the system. Finally, we will introduce the BKT transition, and present numerical results directly relevant for our experiments [78, 79].

The ideal gas

In this section, we describe the way non-interacting bosons can arrange them-selves among the available energy levels in a two-dimensional geometry. This means that energy levels can only be accessed in a single plane: the particles are considered to be confined to a single quantum state along the remaining direc-tion, which we choose to be z. This is achieved as long as the energy necessary to reach the first excited state along z is large compared to the temperature T.

The infinite uniform two-dimensional Bose gas

The usual argument for the presence of a Bose–Einstein condensate (BEC) in a non-interacting system is the saturation of the excited single-particle states at non-zero temperature, following Einstein’s standard argument [4, 5]. For a given temperature, if such a maximal occupation of the excited states Nexc(T) exists, all particles beyond this critical number must accumulate in the single-particle ground state, leading to a macroscopically populated quantum state. It is important to note that this is a sufficient condition for the presence of a BEC, but not a necessary one.
A generalization of this phenomenon was proposed by Penrose and Onsager [80], which associates Bose–Einstein condensation with the existence of a macro-scopic eigenvalue in the one-body correlation function g1(r). Finite size effects, or the presence of interactions can lead to a BEC even in the absence of satu-rated excited states (see for example [59]).
We place ourselves in the grand canonical ensemble, with chemical potential µ and temperature T. In a non-interacting case, the number of particles in the excited states is given by:
Nexc = ∑ (1.1)
i=1 eβ (Ei −µ) − 1
∞ D(ǫ) (1.2)
Nexc =
dǫ.
0 eβ(ǫ−µ) − 1
Eq. 1.1 describes a system with discrete energy levels Ei and becomes Eq. 1.2 in the case of an infinite system. In this equation, D(ǫ) is the density of states and β = 1/kB T. For the the non-interacting Bose gas, µ is necessarily inferior to the ground state energy, and the excited levels are maximally populated when µ reaches the ground state energy. For the homogeneous 2D Bose gas, the density of states is uniform: D(ǫ) = mS/(2πh¯ 2) where S is the surface of the system, and m is the mass of the particles. In this case, the occupation of the excited states simplifies to:
N = − mSkBT ln 1 − eµ/kBT . (1.3)
exc 2πh¯2
Taking the thermodynamic limit N → ∞, S → ∞ with N/S = n, we obtain the result in terms of phase-space density:
2 µ/kBT , (1.4)
D ≡ n λT = − ln 1 − e
where λT = 2πh¯2/(mkB T) is the thermal wavelength. Contrarily to the three dimensional case, the phase-space density can become arbitrarily large: this in-dicates that there is no saturation of the excited levels in two dimensions, and no BEC for the infinite 2D ideal Bose gas. Eq. 1.4 does not include the ground state, but since the system never undergoes condensation, the population of the ground state is always negligible. Therefore, it describes properly the whole system, and it is the first equation of state that we can write: it connects directly the phase space density to chemical potential and temperature. This result is in accordance with the Mermin-Wagner theorem: since long-range order is for-bidden in our system at non-zero temperature, no eigenstate can be macroscop-ically populated.
The absence of long-range order can be seen more directly by looking at the one-body correlation function:
g(1)(R) ≡ Ψ∗(R)Ψ(0)
= 1 ∞ ei K R d2k, (1.5)
(2π)2 0 eβ(ǫK −µ) − 1
where K is the characteristic wave-vector of the plane wave of energy ǫK = h¯2k 2/(2m). This quantity always vanishes at r → ∞, again in accordance with the Mermin-Wagner theorem. For a non-degenerate gas (D ≪ 1), the one-body correlations decays as a gaussian, and the correlation length is the thermal wavelength. Even though there is no phase transition, in the degenerate regime (D ≫ 1), the correlation function decays exponentially for large distances:
g(1)(r) ≈ e−r/lC ,
with the one body correlation length lC = λT
Even though no condensation occurs in an infinite system, it is known that the finite size of a system can significantly affect its properties. In particular,we saw that the one-body correlation length lC grows exponentially with the phase-space density in the degenerate regime (Eq. 1.6). For a homogeneous system in a finite box of characteristic size L, there exists a non-zero tempera-ture T such that the correlations span the entire system. In this case, there is a significant phase correlation between any two points, and the system undergoes Bose-Einstein condensation. This occurs when lC ≈ L, or equivalently when the phase space density reaches the critical value:
Dc ≈ ln 4π L2 . (1.7)
However, this criterion is not a quantitative definition for the critical atom num-ber. This can be derived more accurately for a given geometry by calculating the maximal population of the excited states following Eq. 1.1. This shall be done in the following in three different geometries.
Square box In a square box, the eigenfunctions are of the form: ψ(x, y)i,j = 2 sin i π x sin j π y (1.8)
L L L with i and j strictly positive integers. The eigenenergies are Ei,j = π2 h¯2/(2 m L2) (i2 + j2 − 2), taking the energy of the ground state to be 0. In this case, the max-imal population of the excited levels is given by:
Nc = ∑ 1 ≡ fsq L2 . (1.9)
e β Ei,j − 1 λ2
i>1,j>1 T
Circular box In a circular box of radius L, the eigenfunctions are of the form: ψ(r, θ)s,n = √ 1 J|n|(js,|n|r/L) ei n θ (1.10)
where n is an integer, Jn is the Bessel function of order n of the first kind, and js,n is its s-th zero. The eigenenergies are therefore Es,n = h¯2/(2 m L2)(js2,n − j0,02), taking the energy of the ground state to be 0. In this case, the maximal population of the excited levels is given by:
Nc = ∑ ≡ fcirc . (1.11)
e β Es,n − 1 2
s,n λT
The functions fsq and fcirc can be evaluated numerically, and compared to the functional form introduced in Eq. 1.7. We choose the following fitting function:
Nc,fit = ξ S ln η4 π S (1.12)
λT2 λT2
where S is the surface of the box, and ξ and η are the fitting parameters. The results are shown in Fig. 1.1, along with the prediction from Eq. 1.7. The numer-ical results are well described by the prediction from Eq. 1.12, with the shape of the box influencing mainly the coefficient η.
Harmonic trap Experimentally, the most relevant geometry for the Bose gas is the harmonically trapped system. In this case, owing to the different density of states, the gas always undergoes a phase transition at a non-zero temperature, even in the thermodynamic limit. 1 We can assume without loss of generalit that the trap is isotropic, with a trapping frequency ω. The density of states for a two-dimensional harmonic trap is D(ǫ) = ǫ/(h¯ω)2, and Eq. 1.2 yields the following for the maximal population of the excited states:
c,harm π2 kB T 2 (1.13)
6 h¯ ω
It is also possible to derive the equation of state for the trapped 2D Bose gas, by using the local density approximation (LDA). If the trapping potential is varying smoothly enough, it can be considered as constant over a region in which the particles are at the thermodynamic equilibrium. In this case, parti-cles in this region will be described by the equation of state for a homogeneous system, with a local potential µ(r) = µ0 − V(r), where µ0 is the chemical po-tential at the center of the trap, and V(r) the trapping potential. In the case of a harmonic trap, this substitution in Eq. 1.4 gives the density distribution for the excited states: n(r)λT2 = − ln 1 − eµ0/kB T− mω2 r2 . (1.14) 2kB T
The number of atoms occupying excited states in a harmonic trap can be ob-tained by integrating the previous equation: N(µ0, T) = Li2 eµ0/kB T kB T 2 (1.15) where Li2 is the dilogarithm function. In particular, when µ0 → 0, we re-cover the critical atom number from Eq. 1.13. Note that Eq. 1.14 indicates that the density in the center of the trap diverges when µ0 → 0, which is in fact due to the integration: performing the calculation with Eq. 1.1 gives a finite value. This divergence in the semi-classical limit resolves an apparent paradox: in Eq. 1.13 when taking the thermodynamic limit N → ∞, ω → 0 with n = N m ω2/(2π kBT) kept constant, the critical temperature remains ap-parently finite. However, in this case, the semi-classical limit applies, and the density n has to diverge, indicating the critical temperature must be zero. Contrarily to the uniform case, Eq. 1.15 does not always give the equation of state for the whole system: when µ0 → 0, the atoms in excess of Nexc,harm will occupy the ground state, and are not accounted for in Eq. 1.14.

The interacting two-dimensional Bose gas

Up to this point, we did not take into account the existence of a third dimen-sion, and assumed the problem was strictly two-dimensional. However, the experimental realization of such a system relies on imposing a confining poten-tial, which will freeze the excitations along one dimension. Let us suppose that the confining potential is harmonic: U(z) = mωz2 z 2/2. When the temperature is low enough (typically h¯ωz ≫ k B T), the system occupies a single quantum state along z and its wave function can be factorized: 1 (r, θ)e− z2
Ψ(r, θ, z) = ψ 2 lz . (1.16)
(2D)
(π lz2) /4
In this case, the dynamics of the system is contained in ψ(2D)(r, θ), which con-stitutes a realization of the 2D Bose gas. The third direction does not have a direct effect on the dynamics of the 2D Bose gas, though it introduces a charac-teristic thickness of the system: the spread of the ground state of the harmonic oscillator lz = √h¯/mωz. As we will see, the third dimension still has an influ-ence on the collisional properties of the particles.

Interactions in two dimensions: the quasi 2D regime

Since experimental realizations of the 2D Bose gas with cold atoms are carried out at low enough densities, we can restrain the interactions to binary collisions, which are well characterized by a contact potential in three dimensions [81]: V(Ri − Rj) = g3D δ(3D)(Ri − Rj) = 4πh¯ 2 asδ(3D)(Ri − Rj) (1.17) where as is the scattering length (for 87Rb , as = 5.1 nm). This describes a three dimensional collision process, and cannot be directly transposed to the 2D case [75]. In order to determine the collisional properties, we need to distinguish two different regimes, depending on the thickness of the system lz:
– the true 2D regime, where the motion of the atoms is strictly confined to the xy plane. This regime is reached when lz ≪ Re, where Re is the effective range of the interaction potential. In this case, the scattering amplitude is energy-dependent, and cannot be characterized by a constant scattering amplitude.
– the quasi 2D regime, where the microscopic motion of the atoms remains three-dimensional, which corresponds to lz ≫ Re. This condition is typi-cally realized in our experiment, where lz ≈ 180 nm and Re ∼ as = 5.1 nm  (see for example [82] for a derivation of Re). In this regime, the interactions can be expressed by a contact potential as in Eq. 1.17, and a general expres-sion for the two-dimensional coupling constant was given by Petrov et al. [76, 77]
√ h¯2 1
8π where q2 = 2mµ/¯h2.(1.18)
g = lz/as − ln(π q2 lz2)/√
For our experimental parameters, the logarithmic contribution is negligi-ble, and the 2D scattering amplitude remains constant, as in three dimen-sions. In the case of a harmonic confinement along the z direction, it is given by:
h¯2 g˜ where g˜ = √ as
g = 8π
m lz
In this limit, the interaction energy of the 2D Bose gas is:
g h¯ 2g˜
E = n2(R) d2r = n2(R) d2r
int 2 2m
Note that the same result can be obtained by a naive integration of the three-dimensional interaction energy, assuming the gas is described by Eq. 1.16. However, the condition lz ≫ as remains hidden in this procedure.
One can also estimate the strength of the interactions by comparing the interaction energy to the kinetic energy in a uniform gas of N particles. The interaction energy can be estimated by neglecting the density fluctuations ( n2 = n 2):
h¯2 N2
Eint ≈ g˜ (1.21)
2m S
Using the density of states in 2D D(ǫ) = mS/(2πh¯2), the kinetic energy can be estimated :
EN πh¯ 2 N 2 where N = EN
E = ǫD(ǫ)dǫ = D(ǫ)dǫ (1.22)
kin 0 m S 0
The strongly interacting regime is defined as Eint = Ekin, which corre-sponds to g˜ = 2π. This result does not depend on the density, as opposed to the 3D case, where the strength of interactions depends explicitly on the density through the parameter n3D a3s. According to Eq. 1.19, the strongly interacting limit also corresponds to as ≈ lz. Consequently, the quasi 2D regime is only an appropriate description for the weakly interacting 2D Bose gas. Experimentally, values of g˜ for cold atoms range from 0.01 to 3 [64, 65, 83–85], and can reach 1 in the more strongly interacting helium films [86].

Scale invariance in 2D

One important consequence of Eq. 1.19 is the scale invariance of the equation of state of the 2D Bose gas. To properly characterize the 2D Bose gas, we aim to establish an equation of state (EoS), for example for the density. The general form of such an EoS is: F(n, µ, T, g˜) = 0 (1.23) where F is the function to be determined. Using dimensional analysis, Eq. 1.23 must take the form: G(D, µ/kBT, g˜) = 0 or equivalently D = f (µ/kBT, g˜). (1.24)
Not only is g˜ dimensionless, it is also independent of the density in the weakly interacting regime. Therefore, contrarily to the three dimensional case, the in-teractions do not introduce a characteristic energy scale. In this respect, the EoS for the 2D Bose gas is said to be scale invariant: multiplying chemical potential and temperature by the same amount will not change the phase space density of the system. Note that this is only an approximate result, which breaks down if the logarithm in Eq. 1.18 is large enough.
In particular, the EoS for the ideal gas (Eq. 1.4) has the correct functional form: the phase-space density only depends on µ/kBT. More generally, all dimension-less thermodynamic quantities characterizing the homogeneous gas can only depend on g˜ and µ/kBT.

The mean-field Hartree–Fock approximation

In the presence of interactions, the EoS is established by the mean-field Hartree– Fock method. In this approach, the interactions are taken into account by replac-ing µ by µ − 2gn in the equation of state of the ideal gas [87, 88]:
This self-consistent equation for the phase space density can be solved numeri-cally for D. Remarkably, in the presence of interactions, the chemical potential is unconstrained, and can take any value. In particular, the singularity at µ = 0 for the ideal gas vanishes. In the case of the harmonically trapped Bose gas, using the LDA gives us:
µ − mω2 r2−g˜D(r)/π
D(r) = − ln 1 − e kB T 2kB T (1.26)
Since µ is not constrained to negative values anymore, the number of atoms derived from this equation can be made arbitrarily large. The condensation phenomenon which occurred for the ideal gas disappears in the presence of interactions in a harmonic trap.
However, the underlying assumption is that the density fluctuations are im-portant, so that n2 = 2 n 2. A Bogoliubov analysis shows that when the phase-space density reaches D ≫ 2π/g˜, density fluctuations decrease, until the regime where n2 = n 2 is reached [74]. In the zero temperature limit, the gas is described by the Thomas-Fermi approximation : µ = g n. From this ob-servation, the Hartree–Fock approximation is only a good description for small D .

The Berezinskii-Kosterlitz-Thouless (BKT) transition

Though it appears Bose–Einstein condensation does not occur in the pres-ence of interactions, there exists a phase transition at low temperatures to a superfluid state. This transition is unusual, since it cannot break any contin-uous symmetry, in accordance with the Mermin–Wagner–Hohenberg theorem. For this reason, there is no long range order in the superfluid state. It is often classified as a transition of infinite order, since most thermodynamics quantities vary smoothly at the transition point: the only exception is the superfluid den-sity, which jumps from 0 in the normal phase to 4/λ2T in the superfluid phase at the transition point.
The microscopic theory of the transition was developed by Berezinskii [60] and Kosterlitz and Thouless [61]. It is associated with the existence of vortices in the gas. These are points around which the phase winds by a multiple of 2π. At the center of vortices, the density drops to zero, over a typical length scale given by the healing length: ξ = 1/√gn˜. We can restrict our description to the single-charged vortices with phase winding of ±2π, which are energetically stable. Above the critical temperature, the vortices form a disordered gas, and each vortex modifies the phase of the gas significantly, preventing the appearance of a superfluid state. However, below the transition temperature, formation of pairs of vortices of opposite circulation is energetically favorable. The total cir-culation of the phase around such pairs is 0, which indicates that the pair only perturbs locally the gas, thus allowing the existence of a superfluid state.
Even though the value of the superfluid density at the transition point is uni-versal, the BKT theory does not tell us when this transition occurs. In general, computing the transition parameters as a function of the total density n and the interaction strength is a difficult problem. However, in the weakly inter-acting limit, classical field Monte-Carlo calculations performed by Prokof’ev, Ruebenacker and Svistunov [78] give a value for the phase space density and the chemical potential at the transition point:
Dc = n 2πh¯2 = ln ξ D (1.27)
m kB TBKT g˜
µc = kBT g˜ ln ξ µ (1.28)
with ξ D = 380(3) and ξ µ = 13.2(4). In our experiments, g˜ ≈ 0.1, so we expect DC≈8.
In a following article, Prokof’ev and Svistunov [79] numerically calculated the equation of state around the critical point for an infinite system. This allows us to interpolate between the two known limits:
– for D ≪ DC, the Hartree–Fock analysis remains valid
– for D ≫ DC, the gas is well described by the Thomas–Fermi approximation The resulting contributions to the equation of state for the phase space density are shown in Fig. 1.3. From these three predictions, we can create a composite equation of state, which we will later use to model our data.
Since this equation of state is calculated for a homogeneous system, one needs to use the LDA to describe a weakly interacting trapped Bose gas. The validity of this procedure was addressed by quantum Monte–Carlo calculations per-formed by Holzmann and Krauth [89] and Holzmann, Chevallier and Krauth [90], for harmonically trapped systems and for atom numbers similar to our experimental observations.

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Producing and imaging two–dimensional Bose gases

Since this experimental apparatus was built for a large part by previous PhD students, a detailed description of it can be found in [66–68]. Therefore, in the first half of this chapter, we provide a simple description of the experimental sequence. We will detail further both the most recent modifications, and the experimental steps specific to the preparation of a two-dimensional Bose gas. In the second half, we will describe in detail our image acquisition process, both in its theoretical and experimental aspects.

Experimental setup

Experimental sequence

Our experimental setup consists of a vacuum system composed of two cham-bers: a steel “MOT chamber” and a glass “science cell”, linked by a differential pumping stage. A scheme of the vacuum chamber with the magnetic coils is presented in Fig. 2.1. Preparation and measurement of a BEC requires a com-plex sequence of different computer-controlled events, with a precise timing. To this end, we installed a program created at MIT by Aviv Keshet: the Cicero Word generator [91]. Using this software, a series of different phases are de-fined, and described below.
We load a magneto-optical trap from the background 87Rb gas: the loading rate is typically 5 • 109 atoms/s. Eventually, the cloud contains ≈ 7 • 109 atoms. The cloud is then compressed during a cMOT phase [92], the atoms are optically pumped into the internal state | F = 2, mF = 2 and trans-ferred to a quadrupole trap with magnetic gradient bz′ = 140 G/cm. At the end of this sequence, the quadrupole trap contains ≈ 5 • 109 atoms, at 320 µK.

Magnetic transport (6 s)

We transport the atoms from the MOT chamber to the science cell, using a series of 9 pairs of coils, and load a quadrupole trap with bz′ = 90 G/cm. The transport is nearly adiabatic: heating mainly comes from the collisions with the background gas in the first stages of the transport. In the most recent version of the experiment, we cannot readily estimate the number of atoms in the science cell: in situ imaging fails because the magnetic field is not homogeneous over the cloud, and time-of-flight imaging is strongly perturbed by eddy currents. Measurements on a previous version of the experiment indicated a transport efficiency of ≈ 50 %. Alternatively, the two-way transport can provide a lower bound for the one-way transport efficiency.

Radio-frequency evaporation in the quadrupole trap (16 s)

The quadrupole trap gradient is increased to bz′ = 140 G/cm in order to increase the collision rate. We apply a radio-frequency evaporation ramp (on the transition between different Zeeman substates) from 30 MHz to 2 MHz, bringing the atom number down to 2.5 • 107 and the temperature down to 25 µK.

Optical evaporation in the hybrid trap (9 s)

The hybrid trap (see 2.1.2) is loaded in 1 s. We then lower the dipole trap power from 4.5 W to ≈ 85 mW in 8 s to obtain a BEC: the end point of the ramp can be adjusted to choose the final temperature of the atomic cloud.

Transfer into the Hermite–Gauss trap and final evaporation (4 s)

The Hermite-Gauss beam (see 2.1.3) is ramped up in 1 s, splitting the cloud in three parts as shown in Fig. 2.5a. The atoms outside of the Hermite– Gauss beam are removed, leaving us with a single two-dimensional plane (see Fig. 2.5b). We then perform another stage of evaporative cooling to achieve degeneracy in our samples.

A new setup: the hybrid trap

The first stage of evaporative cooling takes place in a quadrupole trap, which is the easiest magnetic trap to build. It is also well suited for radio-frequency evaporation: indeed, as opposed to the optical evaporation, lowering the trap depth does not weaken the confinement. However, a quadrupole trap has a ma-jor drawback: since the magnetic field cancels at its center, atoms crossing this region can be lost through Majorana spin-flips. The loss rate is inversely propor-tional to the temperature: hot clouds can be efficiently trapped and evaporated, but Majorana losses put an upper bound on the phase space density obtained in this trap [93]. For realistic experimental parameters, it is not possible to create a condensate in these conditions.
On the first version of our experimental setup, this problem was avoided by applying a time-orbiting bias field, to realize a TOP trap [1]. In this setup, the effective magnetic field seen by the atoms does not cancel at the center of the trap, thus suppressing the spin-flip induced losses. However, this approach led to weak confinement frequencies: 2π × (32, 32, 92) Hz in our experiment. Consequently, the collision rate could not be made very large, thus evaporative cooling in this trap was quite slow: our optimal condensation sequence took 80 s to complete. In order to shorten this duration, we chose instead to circum-vent Majorana losses by adding an attractive optical trap to the quadrupole, as demonstrated in [94] and [95].
The idea is superpose an attractive dipole trap of depth UH with the quadru-pole trap. When the cloud is sufficiently cold (kB T ≤ UH) but before Majorana losses start to be significant, the gradient of magnetic field is adiabatically low-ered, which simultaneously cools down the atoms, and transfers them inside the dipole trap. To prevent Majorana losses both during and after the transfer, the center of the dipole trap is offset from the center of the quadrupole. The remaining magnetic field gradient provides a harmonic confinement along the direction of propagation of the dipole trap. The total potential is then given by:
(y − y0)2 + (z − z0)2
V (r) = µ b′ x2 + y2 + z2 − UH exp − 2 + m g z
B z 4 w2
Experimentally, we choose the following parameters:
– The dipole trap is generated with a 1560 nm laser beam, propagating along the x axis (see Fig. 2.2). It is focused on the atoms with a waist of w = 50 µm, and the available laser power is 4.5 W. From these parameters, we can expect a trap depth UH = 52 µK. Its position (y0, z0) relative to the center of the quadrupole trap (y = 0, z = 0) is adjusted by a mirror mount with micrometric screws, and is chosen to maximize the number of atoms transferred in the dipole trap. The optimum is found for z0 = −90 µm, with a transfer efficiency of 20 %. While the precise value of y0 does not influence strongly the loading of the dipole trap, it significantly affects the properties of the trap for low depths, and we thus chose y0 = 0. Along the propagation direction, the Rayleigh range of the dipole trap is too large (ZR = 5 mm) for it to affect significantly the position of the cloud: it is therefore imposed by the magnetic field.
– The magnetic gradient is initially bz′ = 140 G/cm and is lowered in 1 s to bz′ = 12.5 G/cm, which corresponds to a gradient slightly lower than grav-ity for atoms in the | F = 2, mF = 2 state. For the optimal value of z0 presented above, this creates a field of 110 mG at the location of the atoms: this is not sufficient to fully prevent Majorana losses. After the decompres-sion of the quadrupole, the zero of magnetic field is therefore moved to z1 = −150 µm to increase the field at the location of the atoms, and thus increase the lifetime in the trap.
– In the final configuration, a linearization of the potential in Eq. 2.1 around its minimum yields the following trap frequencies:
ω y = ω z = 4UH = 2π × 421 Hz (2.2)
ωx = µB bz = 2π × 18 Hz (2.3)
4 m z1
and the bias field at the location of the atoms is
B = |bz′ z1| = 188 mG. (2.4)

Preparing two-dimensional Bose gases and reaching degeneracy

The Hermite–Gauss beam As we saw in Sec. 1.1, the experimental realiza-tion of a two-dimensional Bose gas requires a strong confinement along one direction of space, which is chosen to be the vertical z direction. The tempera-ture of the cloud is typically 100 nK: this requires a confining harmonic potential of characteristic angular frequency ωz > kB/¯h 100 nK = 2π × 2 kHz. This har-monic trapping is realized by a 532 nm laser beam (blue detuned with respect to the atomic resonance). To achieve a suitable geometry for the optical trap, we shine a collimated laser beam on a phase plate. This phase plate imprints a phase of π on the upper half of the beam (z > 0) with respect to the other half (z < 0). The beam then propagates, and is focused on the atoms by a converging lens (see Fig. 2.2 for the geometrical arrangement).
At the focus of the lens, the electric field is the Fourier transform of the electric field after the phase plate in the paraxial approximation. The expected intensity profile can then be calculated: the two halves of the beam interfere destruc-tively, and the intensity distribution presents a minimum in the center (Fig. 2.3). In the nodal plane, the intensity distribution around the minimum can be approximated by:
4 z 2 − 2 y2
2 (2.5)
I(y, z ≪ wz) ≈ I0 π w2z e
where wz is the waist of the beam in the z direction. Therefore, the smaller wz, the tighter the confinement. However, we want the trap to have a significant ex-tension in the xy plane, in order to have a uniform confinement over the whole cloud. Along the propagation direction, the extension of the trap is given by the Rayleigh range of the laser (ZR = 140 µm for wz = 5 µm). Along the remaining direction, the extension of the trap is given by the waist wy, so the aspect ratio of the beam must be properly chosen. These constraints can be summed up as follows:
– wz must be as small as possible to achieve a tight confinement
– wz is limited by the numerical aperture of the focusing lens. In our setup, this means wz ≥ 5 µm
– wy must be larger than the size of the cloud. For a typical sample, this means wy ≫ 20 µm
– on the other hand, for a fixed laser power, increasing wy decreases the in-tensity, and thus the confinement, so wy cannot be too large.
To satisfy all these conditions, we choose wy = 150 µm and wz = 5 µm. The trapping potential is given by U = α I = mωz2 z2/2, with α = 6.5 • 10−8 µK m2/W. For a laser power of 1W, we expect a confinement of ωz = 2π × 4.2 kHz. This corresponds to a harmonic oscillator length scale lz = 165 nm, and a coupling constant g˜ = 0.15.
It must be noted that the previous result assumes a perfect optical setup, and is only true at the focus of the Hermite–Gauss beam. In practice, the harmonic oscillator frequency is closer to ωz = 2π × 2 kHz. Indeed, when loading the Hermite–Gauss trap, a balance must be found between achieving a strong con-finement and loading a large number of atoms. As we saw, the confinement increases with decreasing wz. However, the number of atoms loaded into the Hermite–Gauss beam is given by the overlap between the trap and the atomic distribution. In particular, it is directly proportional to the distance between the two barriers, which is in turn proportional to wz. Consequently, the number of loaded atoms increases with wz. Of course, if wz is too large, the height of the barriers will not be sufficient to keep the atoms in the nodal plane of the beam. For this reason, since we cannot dynamically control wz, a compromise must then be found between a high atom number and a high harmonic oscillator frequency. Experimentally, the optimum was found when the atoms are trapped between one and two Rayleigh ranges away from the focus, with the precise value depending on the quality of the optical alignment.

Table of contents :

Introduction 
1. Phase transitions with ultracold two-dimensional Bose gases 
1.1. The ideal gas
1.1.1. The infinite uniform two-dimensional Bose gas
1.1.2. Bose-Einstein condensation in a finite system
1.2. The interacting two-dimensional Bose gas
1.2.1. Interactions in two dimensions: the quasi 2D regime
1.2.2. Scale invariance in 2D
1.2.3. The mean-field Hartree–Fock approximation
1.2.4. The Berezinskii-Kosterlitz-Thouless (BKT) transition
2. Producing and imaging two–dimensional Bose gases 
2.1. Experimental setup
2.1.1. Experimental sequence
2.1.2. A new setup: the hybrid trap
2.1.3. Preparing two-dimensional Bose gases and reaching degeneracy
2.2. Imaging two-dimensional Bose gases and processing the data
2.2.1. Using absorption images to determine the density
2.2.2. Imaging setup
2.2.3. An algorithm for image processing: the Principal Component Analysis (PCA)
3. The equation of state of the two-dimensional Bose gas 
3.1. Exploring the thermodynamics of a two-dimensional Bose gas
3.1.1. Experimental preparation of 2D samples
3.1.2. Thermodynamic analysis
3.1.3. Measuring the interaction energy
3.2. A fit-free equation of state: compressibility, density and pressure
3.2.1. Choosing the correct dimensionless variables
3.2.2. Characterizing the trapping potential
3.2.3. Measuring the equation of state with the global method
3.2.4. Thermometry on single images
3.2.5. Excited levels in the transverse direction
4. Superfluidity in two dimensions 
4.1. A brief theoretical overview
4.1.1. The three-dimensional case
4.1.2. The two–dimensional case
4.2. Superfluid character of a two-dimensional Bose gas
4.2.1. Experimental scheme
4.2.2. Observation of a critical velocity
4.2.3. Comparison with theory
4.3. Closing remarks
5. Fluctuations of the two-dimensional Bose gas
5.1. Experimental procedure
5.2. Local density fluctuations
5.2.1. Characterizing the density minima
5.2.2. Quantifying the distribution of minima
5.2.3. Qualitative interpretation
5.3. Density correlation function
5.3.1. Correlation in real space
5.3.2. Correlations in reciprocal space
5.4. Concluding remarks
6. The uniform two-dimensional Bose gas
6.1. A brief theoretical analysis
6.1.1. Ideal gas in a stadium potential
6.1.2. Interacting gas in a stadium potential
6.1.3. Experimental perspectives
6.2. Experimental realization: preliminary studies
6.2.1. Creating a box-like potential: a holographic method
6.2.2. Creating a box-like potential: by forming the image of a mask
7. Single atom imaging scheme 
7.1. Working principles
7.1.1. Optical molasses
7.1.2. The pinning lattice
7.1.2.1. Influence of the hyperfine structure of the excited states
7.1.2.2. Influence of the hyperfine structure of the ground state
7.1.2.3. Heating by spontaneous emission of photons
7.1.2.4. “Gray molasses” effect from the lattice
7.1.3. Choosing the parameters
7.2. Characterizing our implementation
7.2.1. Experimental setup
7.2.2. Preliminary results
Concluding remarks 
Summary
Perspectives
On the two dimensional Bose gas
On strongly correlated states
Appendix A. Contribution of the excited states to the EoS
Appendix B. Conversion of phase fluctuations into density fluctuations
B.1. Density distribution, in real and reciprocal space
B.2. Case of a small perturbation
B.3. Interpretation in terms of Talbot effect
Appendix C. Diagonalization of the vectorial light shift in a lattice
Appendix D. Collection efficiency of the imaging system
Bibliography

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