Effect of interactions on the performance of ultracold quantum interferometers

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Facolta di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Scienze Fisiche Anno Accademico 2016/2017

Tesi di Laurea Magistrale

Effect of interactions on the

performance of ultracold quantum

interferometers

28 agosto 2017

Candidata:

Cosetta Baroni

Relatori:

Prof.ssa Maria Luisa Chiofalo

Dr. Andrea Trombettoni

Dr. Giacomo Gori

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δ 6= 0

. . . 14 2.7.2 Attractive interactions . . . 14

2.8 Spin Representation and Bloch sphere . . . 15

3 Interferometer Theory 17 3.1 Principles . . . 17

3.2 Squeezing and Sensitivity . . . 19

3.2.1 Defining squeezing . . . 19

3.2.2 Squeezing and entanglement. . . 19

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CONTENTS ii

II Effect of interactions 22

4 Two Mode Model Simulations 23

4.1 Energy spectrum . . . 23 4.2 Population Imbalance . . . 25 4.2.1 Repulsive interactions . . . 25 4.2.2 Attractive interactions . . . 25 4.2.3 Varying δ . . . 26 4.3 Dynamics . . . 27 5 Beam Splitter 28 5.1 Initial state Twin Fock . . . 28

5.1.1 Linear beam splitter . . . 29

5.1.2 Turning on the interactions . . . 30

5.1.3 Inter-well Interactions . . . 30

5.2 Initial state |N, 0i . . . 32

5.2.1 Linear beam splitter . . . 33

5.2.2 Turning on the interactions . . . 34

5.2.3 Inter-well interactions . . . 36

5.3 Initial state NOON . . . 38

5.4 Discussion . . . 38

6 Full interferometric protocol 39 6.1 Initial state Twin Fock . . . 40

6.1.1 Analytical results for U = 0 and N = 2 . . . 41

6.1.2 Behaviour for different holding times . . . 42

6.1.3 Projective measurement of Jx during the phase accumulation stage . . . . 52

6.1.4 Conclusions about initial state Twin Fock . . . 54

6.2 Initial state |N, 0i . . . 54

6.2.1 Analytical results for U = 0 and N = 2 . . . 55

6.2.3 Projective measurement of Jx during the phase accumulation stage . . . . 76

6.2.4 Conclusions about initial state |N, 0i . . . 79

6.3 Initial state NOON . . . 80

6.3.1 General considerations . . . 80

III Conclusions and future prospectives 83

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INTRODUCTION

T

he main issue we deal with in this thesis is to determine in what conditions interaction among particles can improve the sensitivity of an atom interferometer. For squeezed states, it is known that one can have such improvement. For instance, there are inter-ferometric protocols taking advantage of interactions in the splitting process, in order to reach squeezed states. However, in experimental set-ups, a certain amount of interaction is unavoidably always present and one can formulate the question: “In what conditions, interactions - when turned on during the whole interferometric process - can improve the sensitivity of an atom interferometer?” . We have found that, holding the particles for specific time intervals during the phase accumulation stage, it is possible to reach a bet-ter sensitivity (up to ∼ 50% in the cases analysed) with respect to the non inbet-teracting case. In this thesis we start by clarifying and contextualising the experiments with ultracold atoms in which interferometric tasks are performed. Historically, after the discovery of interference, physicists tried to use it as an useful tool to make measurements; one of the most famous interferometers was built in 1887 by Michelson and Morley to determine whether or not ether existed, an experiment that, indirectly, helped the birth of Einstein’s general relativity. In 1924 Louis de Broglie hypothesized that matter could be understood in terms of waves, as well as light could be thought as particles: this was the cornerstone for matter wave physics. With the prediction made in 1925 by Einstein and Bose of the existence of Bose-Einstein condensates and their realization in 1995 first by Cornell and Wieman at the NIST-JILA laboratory with87_{Rb gas [}₁_{] and few months later by Ketterle} at MIT with23_{Na atoms [}₂_{], a deep analogy with condensates and laser beams has been} pointed out. Actually the first interference pattern between two condensates was observed yet in the Ketterle’s experiment. Many experimental and theoretical works have been done in order to understand how matter waves behave and how they could be used for improved measurements. Indeed atoms beams have some features that make them a very different tool with respect to photons [3]: (i) they have short de Broglie wavelengths (∼ 10 pm for thermal atoms, up to ∼ 1 µm for ultracold ones) and very short coherence lengths (∼ 100 pm for thermal atoms, up to ∼ 10 µm for ultracold ones), (ii) they interact among them-selves, property that allows for non linearity, needed to reach particular useful states (as squeezed states) in order to get better measurements, without looking for some medium that can give the required non linearity for an optical interferometer (see, i.e. the Kerr effect), (iii) atoms can be trapped, giving rise to a new class of interferometer. Moreover, ultracold atoms show quantum effects that can be used for interferometric purpose [4]. These features make ultracold atom interferometers excellent tools for precision measure-ments of various physical quantity, such as accelerations and rotations [5], and to test basic quantum mechanics [6] and general relativity [7,8]. We will deal with atoms in a double well potential, with the possibility to tune the height of the barrier and the energy difference between the wells. This simple set-up gives rise to an interferometer,

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Introduction iv

where the splitting process is realized by letting the barrier high enough for the atoms in the two wells to be isolated, in order to accumulate a phase difference. After a phase difference is accumulated recombination is performed by lower the barrier or by letting the atom clouds fall and overlap during their expansion. This kind of double well po-tential interferometers are currently available and were realized in optical traps [9], and on atom chip [10], starting few years later the discovery of the Bose-Einstein condensation.

We want now to point out what place our findings have in the known literature. The main novel results concern the sensitivity of the interferometer, when it is fed with a |N,0i state. When population imbalance measurements are performed at the end of the whole interferometric sequence, an improved sensitivity with respect to the non interact-ing case is found for specific phase accumulation times (see Fig. 6.28). The possibility to obtain a better sensitivity when |N, 0i is used as input state was experimentally proved by the group of Treutlein [11] and by the one of Oberthaler [12]. The main difference is that in the cited works the splitting process is performed in absence of interactions and the im-provement in sensitivity is due only to the squeezing produced by the interactions, present during the phase accumulation time. In order to detect such improvement, an engineered measurement is needed since the final state is not squeezed along the desired direction. Our interferometric set-up is, instead, the simplest: is made up of a first beam splitter, a phase accumulation time and a second beam splitter identical to the first. With this scheme we are able to see improvements thanks to the onset of non linearities during the splitting process approaching the self trapping threshold, that lead to a larger δ-derivative (directly related to sensitivity, see Eq. 3.16). Another difference is that we demonstrate that this improvement is more evident the larger the value of the interaction energy is, while in the cited works an investigation at different interaction energies is absent. An-other novel finding is that not only we are able to find a better sensitivity with respect to the non interacting case, but also a better scaling with the particles number (∆δ = α/Nβ_, β = 0.5 for |N, 0i as initial state). At the self trapping threshold, by performing popula-tion imbalance measurements at the end of the interferometric sequence, we have found a β ∼0.7 (see Fig. 6.31), while by looking at the phase probability distribution before the last beam splitter β ∼ 1 (see Fig. 6.48), that is the Heisenberg limit.

Dealing with the Twin Fock as initial state, the novel result is that a value of the sensitivity close to the one for the non interacting case is found at specific phase accumulation times when population imbalance measurements are performed (see Fig. 6.6).

By looking, instead, at the phase probability distribution before the second beam splitter, an improvement is found for small values of the product UN (interaction energy and par-ticles number), while for larger values the sensitivity is greater than the one found in the non interacting case, but the scaling parameter β is the same (see Fig. 6.13). This last result is analogous to what found in [13], the main difference is that we use an optimized beam splitting process.

Other novel results concern the optimized splitting process for the initial state |N, 0i in the presence of intra-well interactions (see Fig. 5.11) and for both Twin Fock and N, 0i as initial states in the presence of inter-well interactions (paragraphs5.1.3and 5.2.3).

The analysis of the beam splitter when the interferometer is fed with the state |N, 0i and the characterization of the behaviour of the sensitivity as a function of the phase accumulation time with the tools itemized in the introduction of Chapter6, with the ex-ception of the analysis of the interferometer when interactions act only during different steps of the interferometric sequence, come from my personal ideas, while all the other investigations have been suggested by my supervisors’ suggestions, or inspired by helpful discussion with other physicists.

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Introduction v

I have performed all the simulations using my own codes, developed in C by starting from the Two Mode Hamiltonian .

This thesis is organized in three parts:

I Ultracold atoms and quantum interferometry, in which we briefly review the main physics behind atom interferometers.

In Chapter 1 we outline the physics of trapped interacting bosons and the Gross-Pitaevskii equation and we introduce the double well potential;

in Chapter2the two mode model, aimed at describing the double well potential and used in our simulations, is presented;

Chapter 3briefly sketches the main results of the theory of atom interferometry and squeezed states used in the following.

II Effects of interactions, in which the results of our simulations are presented.

Chapter 4 contains simulations of some know aspects of the physics of atoms in a double well, the theory of which has been presented in Chapter 2;

in Chapter5the realization of the beam splitter is analysed for the two input states we will use in the rest of this thesis (|N/2, N/2i and |N, 0i) and a comparison with the sensitivity reachable with a NOON state is presented;

Chapter 6 is the main chapter of this work. Here we study the behaviour of the interferometer sensitivity varying the holding time, the particles number and the interaction energy for different input states, looking for the possibility to reach a sensitivity better that the one for the non interacting case.

If not otherwise specified, when fits are reported, the reduced χ2 _{is always 0.93 ≤} χ2≤1.2.

III Conclusions and future prospectives, where we summarise the results of our work and propose further investigations.

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Part I

Ultracold atoms and quantum

interferometry

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CHAPTER

1 TRAPPED BOSONS

I

n this chapter we will briefly review some main results concerning trapped and inter-acting bosons [14].

We will start by outline the simplest case of non interacting trapped bosons in Section1.1; when there are interactions are the temperature is low enough, a suitable description for a system of bosons is given by the The Gross-Pitaevskii equation, which will be presented in Section1.2. This chapter closes with the depiction of an experimental realization of the double well potential and Josephson oscillations, Section1.3.

1.1 Non interacting bosons in harmonic trap

The starting point for studying trapped bosons is the non interacting case, for which the ground state wave function is a simple Gaussian. Most of the times one can safely approximate the confining potential by

Vtr(r) = m

2(ωx2x2+ ω2yy2+ ωz2z2). (1.1) If interactions are neglected the many body Hamiltonian is the sum of single-particle Hamiltonians with eigenvalues

nxnynz = nx+

1 2

~ωx+ ny+1₂~ωy+ nz+1₂~ωz. (1.2) The ground state of N non-interacting bosons is given by φ(r1, ..., rN) = Πiϕ0(ri), that is all the particles are in the lowest single-particle state (nx = ny = nz = 0), where, introducing the geometric average of the oscillator frequencies ωho = (ωxωyωz)1/3,

ϕ0(r) = _mω ho π~ 3/4 e−m2~(ωxx2+ωyy2+ωzz2). (1.3)

The density distribution is than given by n(r) = N|ϕ0|2, while the size of the cloud, independent on N, is given by the average width of the Gaussian (1.3)

aho = ~ mωho

, (1.4)

whose typical order is aho ∼1 µm.

1.2 Gross-Pitaevskii equation and effects of interactions

When considering N interacting bosons in a trapping potential, the many body Hamil-tonian is given, in second quantization, by

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CHAPTER 1. TRAPPED BOSONS 3

ˆ

H = ˆHkin+ ˆHpot+ ˆHint= =Z ˆΨ†_{(r, t)}−~2 2m4 ˆΨ(r, t)dr + Z ˆΨ†_{(r, t)V} tr(r) ˆΨ(r, t)dr+ 1 2 Z ˆΨ†_{(r, t) ˆΨ}†_{(r, t)U(r − r}0_{) ˆΨ(r, t) ˆΨ(r, t)drdr}0 _, (1.5)

where ˆΨ(r) ( ˆΨ†_{(r)) is the boson field operator that destroys (creates) a particle at} position r and U(r − r0_{) is the two-body inter-particle potential.}

A useful way to solve Eq. (1.5) is to adopt the Bogoliubov mean field approximation, whose key point consists in separating out the condensate contribution to the bosonic field operator.

On a general ground the field operator can be written as ˆΨ(r) =P

iΨi(r)ai, where Ψi(r) are single-particle wave functions and ai are the corresponding annihilation operators. In the Fock space the operators ai and a†_i are defined by the relations

a†_i|n₀, n1, ..., ni, ...i= √ ni+ 1|n0, n1, ..., ni+ 1, ...i; ai|n0, n1, ..., ni, ...i= √ ni|n0, n1, ..., ni−1, ...i, (1.6) and they obey the commutation rules

[ai, a † j] = δi,j , [ai, aj] = [a†i, a † j] = 0 . (1.7) The mean field prescription is given by

ˆΨ(r, t) = Φ(r, t) + ˆΨ0_{(r, t) ,} _(1.8) where ˆΨ0_{(r, t) describes the depletion of the condensate and Φ(r, t) = h ˆΨ(r, t)i is the} expec-tation value of the field operator, whose modulus fixes the condensate density (n0(r, t) = |Φ(r, t)|2_{) and, having a well-defined phase, allows for symmetry breaking} _.

Φ plays the role of an order parameter and assumes the name of wave function of the con-densate characterizing the off-diagonal long-range behaviour of the one-particle density matrix ρ1(r0, r, t) = h ˆΨ†(r0, t)Ψ(r, t)i. Indeed, even for a finite-sized system, where the concepts of broken symmetry and off-diagonal long-range order lose their meaning, the condensate wave function is still well defined, corresponding to the eigenfunction Φi with the largest eigenvalue Ni of the one-body density matrix, R dr0ρ1(r0r)Φi(r0) = NiΦi(r). At T = 0 the depletion of the condensate Ψ0 _{is negligible and an equation for the} con-densate wave function can be found using the Heisenberg equation of motion with the Hamiltonian 1.5 and replacing the operator Ψ with the classical field Φ. Dealing with a dilute and cold gas, the interaction term can be approximate considering that at low energy only two-body collisions are relevant, described solely by the s-wave scattering length, a:

U(r0− r) = gδ(r0− r) , (1.9) where the coupling constant is given by g =4π~2_a

m .

With these considerations one obtains a closed equation for the order parameter, the Gross-Pitaevskii equation (GP):

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CHAPTER 1. TRAPPED BOSONS 4 i~∂Φ(r, t) ∂t = − ~2 2m52Φ(r, t) + Vtr(r, t) + g|Φ(r, t)|2 Φ(r, t) . (1.10) Two parameters are needed to describe the dilution of the gas and the importance of interactions: the former is given by the number of atoms in a scattering volume |a|3_, namely, defining the average density of the gas as ¯n, the condition for the gas to be dilute is

¯n|a|3_1. _(1.11)

The latter can be estimate by comparing the interaction energy, Eint, for the ground state of the harmonic oscillator with the kinetic one due to the trap: Eint = gN ¯n, with ¯n ∝ N/a3

ho, so that Eint∝ N2|a|/a3ho; Ekin∝ N ~ωho ∝ N a−2ho; so that Eint

Ekin

∝ N |a| aho

, (1.12)

this parameter can be larger than 1, that is the interactions can be important, even if the gas is dilute.

1.3 Double well potential

Experimentally, a double well potential is obtained by superimposing an harmonic confinement and a 1D periodic potential: when the energy of the two minima of this new potential is significant lower than the energy of the other minima, this wells are much more populated than the others, and a double well is thus realized (see Fig. 1.1). The energy barrier and the distance between the wells are controlled by acting on the parameters of the optical lattice.

Figure 1.1: Experimental realization of a double well

by superimposing an optical lattice on an harmonic trap.

The trapping potential then reads V((r) = 1

2mω⊥2(x2+ y2) + VDW(z) , (1.13) where the double well potential along the z axis, VDW, is defined by

VDW(z) = 1

2mω2zz2+ V0cos2(kz) . (1.14)

The 1D periodical potential, V0cos2(kz), is created by an optical lattice obtained from two counter-propagating laser beams with k = 2π/λ, where λ = λlasersin(θ₂), λlaser being the wavelength of the laser beams and θ the angle between them. The spacing in the lattice is λ/2 and V0 is proportional to the laser power. ω⊥ and ωz are, respectively the axial and transverse frequencies of the harmonic confinement.

Typical experimental values are λ ∼ 1 − 10µm and ωz/2π ∼ 10 − 100 Hz; for λ ∼ 10 µm, the energy barrier is V0/h & 500 Hz.

A condensate in a double well represents a bosonic Josephson junction (BJJ), consist-ing of two superfluids coupled by tunnellconsist-ing through a potential barrier, very similar to a superconducting Josephson junction.

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CHAPTER 1. TRAPPED BOSONS 5

Figure 1.2: Josephson oscillations and self-trapping;

figure from [15].

The only relevant parameters, once the number of particles has been fixed, are the tunnelling coupling and the interaction energy between parti-cles in each well; varying them one can distinguish between two main be-haviours: oscillations of atoms be-tween the two wells and self-trapping, that is, when interactions are strong enough, all the particles remain in just one well during the system evolu-tion.

The first experiment realization of a single bosonic Josephson oscillation was carried out in 2005 by Oberthaler’s group [15] (see Fig. 1.2) and in an array of bosonic junctions in 2001 [16].

Dealing with fermionic superfluids across the BEC-BCS in a double well an

optical barrier only a few times wider than the correlation length of the system has been achieved at LENS in 2015 [17].

An useful tool to describe bosons in a double well potential is the two mode approxi-mation, which will be treated in the next chapter, consisting in neglecting all energy levels but the first two and rearranging them in such a way that the system can be described by two effective modes describing the two wells.

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CHAPTER

2 TWO MODE MODEL

A

s anticipated at the end of the previous chapter, we will now introduce the two mode (TM) approximation (Section2.1): thanks to it, the full many-body description of an interacting condensate in a double well potential is simplified to a two-sites Bose-Hubbard model with the following Hamiltonian:

ˆ H= −J(t)(â†ˆb + ˆb†â) + U(t) 2 (â †_â†_{ââ + ˆb}†ˆb†ˆbˆb) + δ(t) 2 (â †_{â − ˆb}†ˆb) , _(2.1)

where ˆa†_(ˆb†_{) creates a particle in the left (right) well and ˆa (ˆb) destroys it.}

J(t) is the tunnelling strength, U(t) the particle interaction in each well and δ(t) the en-ergy shift between the two wells and they all can vary with time.

For a review of the model and its applications to interferometry see [18].

Eq. 2.1 will be obtained form the full Hamiltonian in Section 2.2 and in Section 2.3 the system will be described in a classical fashion by making use of the number-phase repre-sentation.

In Sections2.5,2.6and 2.7some know results for the Hamiltonian2.1 will be presented. The system can be represented by a vector on the Bloch sphere, and this will be the sub-ject of the last section2.8.

The whole discussion will deal with a double well potential, but it also holds for an internal two level system.

In Chapter 4 some simulations carried out by making use of this model, in order to test the codes used in this thesis, will be presented.

2.1 Two Mode approximation

The TM approximation consists in describing the wave function of the atoms in a double well potential as restricted to a superposition of two static, localized spatial modes: left, φL(r), and right, φR(r). With this restriction and assuming indistinguishable particles, the dimension of the Hilbert space for the BJJ system is limited to N + 1, allowing exact calculations even for experimental atom numbers.

In a symmetric double well potential, this two modes can be built as a linear composition of the ground φg and the first excited φe states:

φL= φg_√+ φe 2 , φR= φg_√− φe 2 . (2.2)

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CHAPTER 2. TWO MODE MODEL 7

Figure 2.1: Double well potential: first four eigenstates (left) and the two localized modes as defined in

Eq.2.2(right). Figures obtained by simulating the Schrödinger equation with potential V (x) = Ax4_−Bx2_.

In a single-particle picture, the two lowest-energy solutions of the Schrödinger equation in the double well potential are good candidate for the TM approximation (see Fig. 2.1). In the presence of interactions, defining the spatial modes is less easy, being their shape depending on their occupation; nevertheless one may use the first two stationary solutions of the Gross-Pitaevskii equation, corresponding to a situation where all the particles are in the ground (first excited) state. This choice is found out to be a good one as long as the interactions do not strongly modify the spatial modes.

In any case, being φR/L not eigenstates of the system, an atom prepared in one of these two modes will oscillate between the two well and the frequency of this oscillation will be, for the non-interacting system, the Rabi frequency ΩR= Ee−E_~ g.

If the temperature or the energy scale associated with interactions is higher than the en-ergy spacing between the first and the second excited state, the model is no more valid. In an asymmetric double well there is no general way of defining the two modes; if the two lowest energy eigenstates are localized each in different well, a reasonable choice would be considering this states as candidates for the two modes.

2.2 Derivation from the full Hamiltonian

In this section we want to derive Eq. 2.1from a full many-body Hamiltonian, assuming that only two modes can be populated. This derivation can be found in [19–22] (among others).

The starting point is the full Hamiltonian: ˆ

H= ˆHkin+ ˆHpot+ ˆHint= =Z ˆΨ†_(r)−~2 2m4 ˆΨ(r)dr + Z ˆΨ†_{(r)V (r) ˆΨ(r) +}g 2 Z ˆΨ†_{(r) ˆΨ}†_{(r) ˆΨ(r) ˆΨ(r)dr ,} (2.3) where V (r) is the external potential, g = 4π~2_a

m , m the atomic mass and a the s-wave scattering length of the atoms, having assumed a delta-like contact interaction (U(r) = gδ(r)).

The field operator is then written as a superposition of left and right modes:

ˆΨ(r) = φL(r)ˆa + φR(r)ˆb , (2.4) without loss of generality φL/R can be considered real functions.

The creation and destruction operators obey the following commutation relations: [â, ˆb] = [â†_{, ˆ}_b†_{] = 0; [â, â}†_{] = [ˆb, ˆb}†_{] = 1 .} _(2.5)

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Inserting Eq. 2.4in Eq. 2.3and reordering the terms using the number operators ˆna= ˆa†ˆa, ˆnb = ˆb†ˆb one finds:

ˆ

H =E_L0ˆna+ ER0ˆnb+1₂I(4,0)ˆna(ˆna−1) + +1₂I(0,4)ˆnb(ˆnb−1) − J ˆa†ˆb + ˆb†ˆa

− I(3,1)â†ˆb − I(1,3)ˆb†â + I(3,1)ˆna â†ˆb + ˆb†â+ I(1,3)ˆnb â†ˆb + ˆb†â

+ 2I(2,2)_ˆn

aˆnb+1₂I(2,2) â†â†ˆbˆb + ˆb†ˆb†ââ.

(2.6)

Here E0

i is the sum of the mean kinetic and potential energies in the mode i:

E_i0= Z dr ~2 2m|∇φi|2+ V φ2i , (2.7) and I(i,j)_{= g}R φi_Lφj_Rdr.

Each term in the Hamiltonian2.6conserves the total atom number N = nL+ nR. - The first line in Eq. 2.6 describes the total energy of the left and right modes,

including interactions;

- The second corresponds to the tunnelling of one particles from one mode to the other, with the coupling strength given by

J = − Z dr ~2 2m5φL5φR+ φLV φR ; (2.8)

- The last two lines describe the interaction-induced transfers of atoms between the two modes.

In [23] these terms are retained in order to develop an improved TM model while in the standard TM model they are neglected.

To see why and when it is safe to neglect them one can compare the magnitude of the different terms allowing for particles transfers considering the one dimensional problem and approximating the double well potential by two harmonic potentials of frequency ω0 centred in ±x0 [22]. The left and right mode are defined as the two non-interacting Gaussian ground states centred in ±x0 so that the integrals in Eq. 2.6 describing the transfers can be performed and one finds

J = ~ω0 2 _x2 0 a2_ho−1 e−x20/a 2 ho , (2.9) I(1,3)= I(3,1) = √ g 2πaho e−(3/2)x20/a2ho _, (2.10) I(2,2)= √ g 2πaho e−2x20/a2ho , (2.11) where aho= p

~/(mω0) is the harmonic oscillator length.

The ratio between the tunnel coupling term and the dominant term responsible for interactions-induced transfers is J I(1,3) ∝ ~ω0aho g e x2 0/a2ho , (2.12)

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so that one can safely neglect the interactions-induced transfers terms as long as the ki-netic energy is greater that the interaction one, but this is nothing else that one of the condition of validity for the TM approximation.

With these assumptions, from the Hamiltonian2.6the two mode Bose-Hubbard Hamilto-nian is found:

ˆ

H= −J(ˆa†ˆb + ˆb†ˆa) +UL

2 (â†â†ââ) + UR 2 (ˆb†ˆb†ˆbˆb) + δ 2(â†â − ˆb†ˆb) , (2.13) where δ = E_L0 − E_R0 , UL,R= g Z φ4_L,Rdr . (2.14)

In our simulations we will always assume UL = UR, so that the Hamiltonian 2.13 coincides with the one in Eq. 2.1.

2.3 Number-Phase representation

For a condensate in a double well it is possible to define the relative phase φ = φL− φR and the fractional population imbalance z = na−nb

N .

1_{, the equations of motion for these}

quantities can be cast in a classical shape [24]:

˙z(t) = −q1 − z2(t) sin(φ(t)) , (2.15) ˙φ(t) = ∆E + U N 2J z(t) + z(t) p 1 − z2(t)cos(φ(t)) , (2.16) where ∆E ≡ _2Jδ +U1− U2 2J , (2.17)

and the time has been rescaled according to t2J/~ → t. In the following UL= UR so that ∆E = δ

2J.

These equations can be derived from the classical Hamiltonian H= 1 2 U N 2J z2+ ∆Ez − p 1 − z2cos φ , (2.18) that allows also to show that they are conjugate variables:

˙z = ∂H

∂φ; ˙φ = − ∂H

∂z . (2.19)

Here we sketch the derivation of the equations of motion (2.15), (2.15) by starting from the GP equation (1.10) [24]. The starting point is to use a trial solution

Φ(r, t) = ψR(t)φR(r) + ψL(t)φL(r) , (2.20) where

ψR,L= q

NR,L(t)eiθR,L(t) _, (2.21) 1_{Sometimes it will be helpful to use the simple population imbalance n = n}

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and φR,L(r) are the time-independent GP equation solutions for two independent conden-sate.

With the requirements of normalization and orthogonality between φR and φL and us-ing the parameters defined in Eqs. (2.7), (2.14), (2.8) one finds the equations for the amplitudes ψR,L: i~∂ψR ∂t = (E 0 R+ URNR)ψR− J ψL; (2.22) i~∂ψL ∂t = (E 0 L+ ULNL)ψL− J ψR. (2.23) By replacing Eq. (2.21) into Eqs. (2.22), (2.23) and remembering the definitions of z and φone finally recovers Eqs. (2.15), (2.16).

2.4 Self trapping regime

Since it will play a crucial role in our work, this section is dedicate to illustrate the self trapping regime.

When the interactions are strong enough compared to the tunnelling term, Eq.s (2.15) and (2.16) show a non linear behaviour and an exact solution for z(t) can be found in terms of elliptic functions [24]. If the initial population imbalance is non vanishing, the sinusoidal oscillations around z = 0 become anharmonic as the parameter Λ = U N

2J is increased (see Fig. 2.2: in c the loss of linearity is evident).

Figure 2.2: Solutions of Eq.s (2.15) and (2.16) with initial condition z(0) = 0.6 and φ(0) = 0 for different values of the parameter Λ: Λ = 1 (a), Λ = 8 (b), Λ = 9.99 (c), Λ = Λc= 10 (d) and Λ = 11 (e). Figure from [24].

From the Hamiltonian 2.18, one can see that the value z(t) = 0 is inaccessible at any time if

Λ

2z(0)2−

q

1 − z(0)2_{cos(φ(0)) > 1 ,} _(2.24) so that one can identify a critical initial population imbalance zc(0) at fixed Λ, and a critical Λc at fixed z(0): for z(0) > zc(0) or for Λ > Λc the particles are no more able to tunnel between the wells and z(t) starts to oscillate around a value different from

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zero. This phenomenon is called macroscopic quantum self trapping (see Fig. 2.2: d cor-responds to the critical value of Λ, after which the population imbalance is self trapped, e).

2.5 Energy spectrum

Considering a symmetric double well (δ = 0), the Hamiltonian (2.18) will depend only on the parameter γ = U

2J and three regimes can be identified (See Fig. 4.1,4.2): • Rabi2_{: γ} 1

N,

interactions are negligible and the tunnelling term is the dominant one; the energy spectrum is linear as an harmonic oscillator with levels separated by 2J;

• Fock: γ N,

interactions dominate and the spectrum has a quadratic form: pairwise quasi-degenerate states with opposite imbalance (|nL, nRi, |nR, nLi) (fragmented);

• Josephson: 1

N  γ N,

both interactions and tunnelling play their role and the spectrum is linear as well as quadratic;

For large and negative γ the ground state is a degenerate NOON state |N,0i+|0,N i_√ 2 [25].

2.6 Population imbalance

2.6.1 Repulsive interactions

When considering repulsive interactions, the ground state is always a symmetric state, with the same number of particles in each well and, therefore, the expectation value of the fractional population imbalance is z = 0.

For what concerns the variance of the population imbalance, two limiting cases can be identified:

U=0 : when there are no interactions the Hamiltonian is diagonal in the basis of the ground and first excited states:

ˆ

H= Egâ†gâg+ Eeâ†eâe, (2.25) and J = Ee−Eg

2 = ~Ω

R

2 .

At T = 0 all particles condense in the ground state which corresponds to |Ψi = √1 N!(ˆa † L+ ˆa † R) N_|_{0i =} _√1 2N N/2 X n=−N/2 N! (N 2 + n)!(N − N 2 − n)! |ni , (2.26)

that is an atomic coherent state. In the ground state n is distributed according to a binomial distribution: without interactions the ground state is a product state, each atom being in a superposition of right and left modes.

One can see that a coherent state saturates the Heisenberg uncertainty relation: it is a minimum uncertainty state, being ∆n2 _{= N/4, ∆φ}2_{= 1/N so that ∆n∆φ = 1/2.}

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J=0 : in this case, the interactions dominate and the Hamiltonian is diagonal in the basis of Fock states |nL, nRi3 _{with eigenvalues} U

2(n2L+ n2R− N). The minimum value for n2_L+ n2_Rwith the constrain nL+ nR= N is reached for nL= nR, so that the ground state is

|Ψi = |N/2, N/2i . (2.27)

In this state ∆n2 _{= 0 and the phase is uniformly distributed (random phase).} In the three regimes defined in section 2.5the variance behaves in the following way: • Fock regime: ∆n = _√1

2 J N

U 1;

• Rabi regime: the ground state is still close to a coherent state and one has ∆n ._√ N /2;

• Josephson regime: for low lying energy states one can see that ∆n2 _{= N/(4}√_{1 + Λ).} In general, for J 6= J → 0 one can find an analytical result [19]:

∆n = s N 2 2J 2J + NU 14 . (2.28)

For repulsive interactions the ground state is always number-squeezed, being ∆n < ∆ncoherent=

√ N

2 : number differences would be energetically costly (see Fig. 4.3). 2.6.2 Attractive interactions

For attractive interactions phase fluctuations are small and one can write the Hamil-tonian2.18 as [26,27] H = N U 2J z2+ 1 2 p 1 − z2_φ2₋p 1 − z2_. (2.29)

Making use of the z representation φ = −2i N

d

dz, an effective Hamiltonian can be written:

Hef f = − 2 N2 d dz p 1 − z2 d dz + W (z) , (2.30)

where, defining ˜γ = U/J,

W(z) = ˜γNz 2 2 −

p

1 − z2 _, _(2.31)

that, expanding around the value z = 0, becomes W (z) ' (˜γN +1)z2

2 +

z4

8: this potential is quadratic for ˜γN > −1, becomes quartic for ˜γN = −1 and double well shaped for ˜γN < −1 with two stable minima for z so that the majority of atoms is forced to localize randomly in one well or in the other (see Fig. 2.3and 2.4). The shape of the potential determines the shape of the ground state wave function: it is bell shaped, with width depending on 1/N, and centred at z = 0 for ˜γN > −1; double hump shaped and centred at ±z0 for ˜γN < −1 [27]. This structure suggests the existence of an order parameter, which can be identified with the population imbalance: the broken symmetry is characterized by a non-zero value of z, z(˜γ) =q1 − (˜γcN

˜

γN )2 for ˜γ ≤ ˜γc (second order phase transition) (see top panel of Fig. 4.4).

3_{Here we have assumed an even number of particles, otherwise the ground state reads (|(N − 1)/2, (N +} 1)/2i + |(N + 1)/2, (N − 1)/2i)/√2.

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γ N γcN

~

Figure 2.3: Ground state of system of bosons in a double well potential tuning inter-atomic interactions

to large negative values. Figure adapted from [26].

Figure 2.4: Effective potential (Eq. (2.31)) for different values of ˜γN ; see main text for details. Figure

from [26].

As already pointed out, for large and negative values of the interaction energy, the ground state is a NOON state |N,0i+|0,N i_√

2 and the variance of the population imbalance is given by ∆n = N/2 (∆z = 1) (see bottom panel Fig. 4.4).

2.6.3 Varying δ

An energy gap between the two wells (δ) provides a controlled symmetry breaking term, lifting the degeneracy between the z stable points [26]. One can see this effect introducing a term δz in the effective potential, whose minima represent stable and metastable points. For ˜γN > −1, z changes smoothly as a function of δ; as the interactions become more attractive, z depends more on δ and at the value ˜γN ≤ −1 one can see an abrupt change in z crossing δ = 0 (first order phase transition)(see Fig. 4.5).

2.7 Time-independent Hamiltonian

Some analytical results can be found in [28] and some numerical ones in [29]. From the equations of motion (2.15) and (2.16) for z and φ:

˙z = −p

1 − z2_{sinφ ,} (2.16)

˙φ = Λz + z √

1 − z2cosφ+ ∆E , (2.16) where we have introduced the parameter Λ = U N

2J , three principal regimes can be identified:

• Non-interacting: if the double well is symmetric and Λ → 0, the population imbalance performs Rabi oscillation with frequency ωR = 2J (see top left panel of Fig. 4.6);

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CHAPTER 2. TWO MODE MODEL 14 • Linear: for |z| 1 and |φ| 1, Eqs. (2.15,2.16) become

˙z ' −φ and

˙φ ' (Λ + 1)z ,

which represent small amplitude sinusoidal oscillations of z(t) with frequency ωL= √

2UNJ + 4J2.

To see the effect of a different potential energy in the two wells one can linearise Eqs.(2.15), (2.16) :

˙z ' −sinφ , ˙φ ' ∆E + (Λ + cosφ)z ,

so that, for large trap asymmetries, ∆E (Λ + cosφ)z, one has φ = φ(0) + ∆Et and z(t) oscillates with frequency ωac' δ/2;

• Non linear: when interactions are present, an exact solution for z(t) can be found in terms of elliptic functions [24] and a novel non linear effect appears: the self-trapping regime (see Section 2.4 and top right panel of Fig. 4.6).

For the self-trapping to occur the initial particles energy has to be just sufficient to reach the top of the potential (e.g φ = π, z = 0); given the Hamiltonian2.18this condition reads (for φ(0) = 0):

2.7.1

δ 6= 0

We want here to itemize the effect of a shift in the potential energy between the two wells in the different regimes:

• Extreme Rabi: z(t) oscillates around zero, as in the case δ = 0, but there is a different dependence on the initial state. (For example, the oscillation amplitude of z(t) is not zero for initial state |N/2, N/2i (see bottom panel of Fig. 4.6)).

The same behaviour is found for initial states |m, ni and |n, mi;

• Rabi-Josephson crossover: the z(t) oscillation amplitude depends on the initial state; |m, ni and |n, mi do not produce the same behaviour;

• Josephson and Fock: there are significant differences in the behaviour only if z small: if z is large enough, |m, ni and |n, mi produce same results, else, if z is close to zero, |m, ni and |n, mi produce different behaviours.

2.7.2 Attractive interactions

When the interactions are attractive, the dynamics is the same as for repulsive ones. Indeed, for the two-site Bose-Hubbard model, the reduced one-body density matrix can be written as [30]

ρ(1)(t; U) = ρLL(t : U) ρLR(t : U) ρ∗_LR(t : U) ρRR(t : U)

!

, (2.32)

Defining the unitary transformation ˆ R=

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CHAPTER 2. TWO MODE MODEL 15 and using the relation

ˆ

R ˆH(U) ˆR= − ˆH(−U) , (2.34) the reduced one-body density matrix for attractive interactions becomes

ρ(1)(t; −U) = ρLL(t : U) −ρ ∗ LR(t : U) −ρLR(t : U) ρRR(t : U) ! , (2.35)

so that the same eigenvalues are found for the matrices (2.32) and (2.35).

On a physical ground, considering a system of N bosons initially prepared in the left well, the Bose-Hubbard model underestimates the probability of finding bosons in the left well at time t for attractive interactions and overestimates it for repulsive ones.

2.8 Spin Representation and Bloch sphere

The internal state of an ensemble of N two level atoms can be described by a pseudo-spin ˆJ =PN

i=1ˆji, sum of the spins of all atoms.

In our two mode model the total angular momentum is N/2 and the projection m on the z-axis corresponds to states where, from an initial state with N/2 particles in the left and N/2 in the right wells, m atoms go from the right to the left well.

The link between the pseudo-spin operators and the creation/annihilation operators is given by the Schwinger boson representation:

ˆ Jx= 1₂(â†ˆb + ˆb†â) , ˆ Jy = 1 2i(â†ˆb − ˆb†â) , ˆ Jz= 1 2(â †_{â − ˆb}†ˆb) . (2.36)

Here ˆJx promotes an atom from the left to the right well (or vice versa) and can be consid-ered as the phase difference between the two modes; ˆJz is half the population imbalance. The operators2.36obey the commutation relations

[ ˆJx, ˆJy] = i ˆJz, [ ˆJy, ˆJz] = i ˆJx, [ ˆJz, ˆJx] = i ˆJy .

(2.37) This representation allows to map the TM wave function onto the Bloch sphere (see Fig.

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Figure 2.5: Bloch Sphere (Figure form [31]).

In this framework the Hamiltonian for the double well becomes H = −2J ˆJx+U

2Jˆz2+ δ ˆJz. (2.38) The tunnelling term rotates a state on the Bloch sphere around the x-axis, while the interaction term twists its components above and below the equator respectively to the right and to the left and the twist rate increases increasing the distance from the equator (see Fig. 2.6).

Figure 2.6: Time evolution of states on the Bloch sphere for (a) binomial, (b) number squeezed and (c)

phase squeezed state (Figure form [31]).

As we will see in section3.2, interactions can lead to squeezed states, that is states in which the variance along one axis is reduced at the cost of enhancing it along an orthogonal axis.

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CHAPTER

3 INTERFEROMETER THEORY

A

fter the introduction of the Two Mode model in Chapter 2, we will now outline the principles of a two mode interferometer (Section 3.1) and the concept of squeezing (Section3.2), useful in quantum metrology [32,33].

3.1 Principles

The main idea of an interferometer is shown if Fig. 3.1: here there are two beam splitter, the first separates the input beam in two paths, far from each other, in which some physical process allows for different phase accumulation; the last recombines them and from the readout it is possible to infer the phase accumulated difference.

Figure 3.1: Diagram of an optical (top) and an atomic (bottom) Mach-Zehnder interferometer. In the

bosonic double well interferometer the beam splitter is performed by increasing the height of barrier; a phase difference is accumulated thanks to the presence of an energy shift between the wells; by lowering the barrier again the second beam splitter is realized, allowing the recombination process; the accumulated phase can be estimated through population imbalance measurements.

For a linear interferometer (no interactions) one can see, by looking at the Hamiltonian of Eq. 2.38, the beam splitter as the operator e−iπ

2Jˆx (rotation of π

2 around an axis in the equatorial plane of a vector in the generalized Bloch sphere) and the phase shift as e−iφ ˆJz (rotation around the z axis), so that the total interferometer can be described by

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CHAPTER 3. INTERFEROMETER THEORY 18 the operator

ˆ

U = e−iπ2Jˆxe−iφ ˆJze−i π

2Jˆx = e−iφ ˆJye−iπ ˆJx , (3.1)

and the state evolves according to |Ψ(t)i = ˆU |Ψ(0)i.

Indeed one can make a connection between a linear interferometer with two input and two output ports and the group SU(2), connection that allows one to visualize the operations of beam splitter and phase accumulation as rotations in a 3D space [34]. The scattering matrix for the interferometer will be of the form

aout bout ! = U11 U12 U21 U22 ! ain bin ! , (3.2)

where the matrix U = U11 U12 U21 U22

!

has to be unitary for the creation/annihilation op-erators to satisfy the commutation relations2.5.

We are going now to describe the beam splitter and the phase shifter. Beam Splitter: consider a beam splitter with a scattering matrix

U = cos α

2 −isin α 2 −isinα₂ cosα₂

!

, (3.3)

so that ˆJ = [ ˆJx, ˆJy, ˆJz] transforms according to    ˆ Jx ˆ Jy ˆ Jz    out =    1 0 0 0 sin α − sin α 0 sin α cos α       ˆ Jx ˆ Jy ˆ Jz    in = eiα ˆJx    ˆ Jx ˆ Jy ˆ Jz    in e−iα ˆJx _, (3.4)

that is a rotation of angle α along the x-axis.

In the Schrödinger picture one has that the state vector, after the beam splitter is |outi= e−iα ˆJx_{|ini .} (3.5)

Phase shifter: let φa and φb be the phases accumulated by the two beams, the associated unitary matrix will be

U = e iφa 0 0 eiφb ! , (3.6) so that ˆJ transforms as    ˆ Jx ˆ Jy ˆ Jz    out =    cos(φa− φb) − sin(φa− φb) 0 sin(φa− φb) cos(φa− φb) 0 0 0 1       ˆ Jx ˆ Jy ˆ Jz    in = ei(φa−φb) ˆJz    ˆ Jx ˆ Jy ˆ Jz    in e−i(φa−φb) ˆJz _, (3.7) that is a rotation along the z-axis by the angle φa− φb: the relative phase shift between the two beams.

In the Schrödinger picture one has

|outi= e−i(φa−φb) ˆJz_{|ini .} (3.8)

The interaction term can be used to produce spin squeezed states in order to reduce the variance along one axis at the price of increasing it along one other.

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CHAPTER 3. INTERFEROMETER THEORY 19

3.2 Squeezing and Sensitivity

Since spin squeezed states can be used to overcome the shot noise limit in interferom-etry [35], in this section a brief reminder of the concept of squeezing and its connection to interferometry is presented.

3.2.1 Defining squeezing

The sensitivity for a linear interferometer can be understood thanks to classical argu-ments: if the input state is a coherent spin state with N atoms, the situation is equivalent to N individual measurements on a single particle. At the point of highest sensitivity for the interferometer, each particle has the same probability to be measured in each of the two states, so that the uncertainty for N particles is ∆n = √N /2. The maximal slope ∂n

∂φ = VN/2 is determined by the visibility, which is V ∼ 1 for a macroscopically populated coherent state [33]; the resulting minimal phase error is ∆φ ≡ ∆n/∂n

∂φ = 1/ √

N, known as standard quantum limit: this limit arises from the quantum noise due to measurements on a finite number of uncorrelated particles. If particles are correlated this classical limit can be overcame and the fundamental limit to take into account is the one coming from the Heisenberg uncertainty relation: the Heisenberg limit, where ∆φ = 1/N. The concept of squeezing is indeed aimed to reduce the variance in one direction, at the cost of enhancing it in the orthogonal ones.

Using the Schwinger representation (2.36) it is possible to define different criteria for a state to be squeezed.

The first one is to consider a state squeezed when the variance of one spin component is smaller than the shot-noise limit J/2 for a coherent spin state:

ξ_N2 ≡ 2

J(∆ ˆJ⊥,min)

2 _<_{1b ,} _(3.9)

this definition does not take into account the perpendicular spin direction; in real ex-periments, states that are not minimal uncertainty ones may be used: in this cases the variance in a direction that is not the squeezed one can be much larger that the one prescribed by the Heisenberg relation, bringing to a reduction of the effective mean spin length h ˆJ)i, which is required to be large for many standard metrological applications, being connected to the visibility by h ˆJ i = VN/2 [33]. In order to measure the useful squeezing, the following definition has been introduced:

ξR≡ √

2J∆ ˆJ⊥,min

h ˆJ i . (3.10)

When dealing with N particles, squeezing is connected to many-body entanglement, so that another criterion has been proposed:

ξ_S2 ≡ N(∆ ˆJ⊥,min) 2

h ˆJ i2 <1 . (3.11) 3.2.2 Squeezing and entanglement

A state of N particles in two modes is separable, and so not entangled, if it can be written as ˆρsep= X k pkˆρ (1) k ⊗ˆρ (2) k ⊗ · · · ⊗ˆρ (N ) k , (3.12)

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CHAPTER 3. INTERFEROMETER THEORY 20 where ˆρ(i)

k is the density matrix for the i-th particle and pk>0, P

kpk= 1.

If ˆρinp satisfies 3.11 the state is said to be entangled and spin squeezed along the ⊥ direction.

Another sufficient condition for entanglement can be introduced [36]: χ2 ≡ N

FQ[ˆρinp, ˆJ~n]

<1 , (3.13)

where ~n is an unitary vector and the quantum Fisher information is defined as FQ[ˆρinp, ˆJ~n] = 4(∆ ˆR)2, with ˆRan Hermitian operator, solution of { ˆR,ˆρinp}= i[ ˆJ~n,ˆρinp] (e.g., for a pure state ρinp = |ψinpihψinp|, ˆR= ˆJ~n).

It is possible to demonstrate that χ2 _{≤ ξ}2 _{so that there are states which are entangled,} χ2<1, but not spin squeezed, ξ2 ≥1.

Quantum interferometry aims to resolve a phase shift φ below the shot noise limit ∆φ = 1/√N. In general, the phase sensitivity is limited by the Quantum Cramer-Rao (QCR) bound, which depends only on the input state:

∆φQCR = q 1 FQ[ˆρinp, ˆJ~n]

= √χ

N , (3.14)

so that the condition 3.13 becomes also a necessary condition for measuring phase shift below the shot noise limit: χ < 1 is required in order to have states there are usefully entangled for sub-shot noise sensitivity.

It is possible, thanks to the non linear evolution naturally provided by inter-particles interactions, to create entanglement and squeezing starting from a coherent state [33,36]. Two main techniques are used: the adiabatic and diabatic approaches, both making use of the possibility to control the parameter characterizing the system, Λ = UN/(2J); at fixed particles number, both the tunnelling coupling J and the interactions strength U can be modified, for the last one it is possible to change the scattering length of the atoms via Fano-Feshbach resonances [37] or to control the wave functions overlap between the two states [11]. In the adiabatic approach, starting from a strong tunnelling coupling, Λ is changed slowly, driving the system deep into the Josephson regime; this technique has the advantages of generating squeezed state in a robust way but the problem of requiring long times [38]. The diabatic approach, instead, provides squeezing in shorter times by quenching the Hamiltonian parameters; starting from a coherent state with mean spin in Jxdirection, the tunnelling coupling J is suddenly switched off, letting the non linear term to dominate the evolution ˆU(t) = e−itU Jz2. In this way, since the rotation speed around

the Jz depends on Jz itself, non linearities cause a shearing of the uncertainty region on the Bloch sphere [39,40].

3.2.3 Squeezing and interferometry

To estimate the phase φ accumulated after the interferometric sequence, one measures an observable ˆO which has φ-dependent expectation values and variances. The error propagation formula gives

∆φ = ∆ Ô ∂h Ôi/∂φ = √ξ N , (3.15) where ∆ Ô= q h Ô2_{i − h ˆ}_Oi2.

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CHAPTER 3. INTERFEROMETER THEORY 21

The sensitivity can be improved both by a larger slope of the expectation value of the operator as a function of φ, or by a smaller variance. The slope can be enhanced by the use of cat-type entangled states, but unfortunately this states are very fragile against decoherence and, until now, have been realized only with few particles [41,42]. The variance of hOi can be decreased using spin squeezing, whose aim is to reduce the projection noise.

In this thesis we want to find the sensitivity with which the interferometer is able to measure a parameter δ, which couples via a Jz term in the Hamiltonian. During the phase accumulation stage, the spin rotates around the Jz axis with frequency δ, the second beam splitter will convert the accumulated phase, φ =R

δdt, into a measurable population imbalance and the sensitivity can be expressed as:

∆δ = ∆ ˆO ∂h ˆOi/∂δ . (3.16)

In the following we will use mainly two input state:

|N,0i : a coherent state, for which the sensitivity limit is the shot noise one, ∆δ = √1 N; |N/2, N/2i : a perfectly number squeezed state (in the sense of (3.9)), for which a sensitivity

near the the Heisenberg limit is predicted [43]. Being ∆z = 0, the phase is completely indeterminate and on the Bloch sphere this state is represented by an arbitrary thin line around the equator. A rotation of this state in the absence of interactions, does not lead to a population imbalance, so that other operators but z have to be used in order to make a measurement. Even if ˆJ_z2 gives some results, it has been demonstrated that the parity operator, Πb = eiπnb gives better sensitivity [44]. Such a measurement is realizable thanks to present day technologies, which allow for single atom detection [45].

Two operators will be used during this thesis: the fractional population imbalance z = na−nb

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Part II

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CHAPTER

4 TWO MODE MODEL SIMULATIONS

I

n this chapter we will present simulations carried out by numerically solving the TM model. This work has been developed with the twofold objective of studying the model and testing the codes against known available results.

For the characterization of the ground state a direct diagonalization of the Hamiltonian

2.1has been done in C using the LAPACK library.

The dynamics has been studied solving, with a 4th-order Runge-Kutta method, the fol-lowing equations: ˙cn(t) = −i _U 2(n(n − 1)) + (N − n)(N − n − 1) + δ 2(2n − N) cn − Jqn(N − n + 1)cn−1+ q (n + 1)(N − n)cn+1 , (4.1)

coming from the Schrödinger equation, i∂ψ(t)

∂t = Hψ(t), where H is the Hamiltonian 2.1 and ψ(t) =P

ncn(t)|ni, being {|ni = |n, N − ni}, with n = 0, ...., N, a Fock basis for the system.

All the simulations have been performed imposing δ 6= 0 only in the phase accumulation stage, in which J = 0, so that the code actually solves Eq. (4.1) in the splitting processes, while the coefficients during the accumulation stage evolve according to

cn(TH) = e−i U 2(2n(n−N )+N (N −1))+ δ 2(2n−N ) TH_c n(TBS) . (4.2)

4.1 Energy spectrum

Here the the simulations of the theoretical results presented in Section2.5are reported. As shown in Fig. 4.1, the energy spectrum for the different regimes has the following behaviour:

• it is linear, with levels separated by 2J in the Rabi regime, γ = U 2J

1

N (see upper left panel, green dots);

• it starts linear and it becomes quadratic with pairwise quasi-degenerate states in the Josephson regime, 1

Nγ N (upper left panel, blue dots);

• it is quadratic with pairwise quasi-degenerate states with opposite imbalance in the Fock regime, γ N (upper right panel);

• it starts quadratic, with pairwise degenerate state and than it becomes linear with levels separated by 2J for attractive energy such that 1

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CHAPTER 4. TWO MODE MODEL SIMULATIONS 24

• it is quadratic with pairwise degenerate states for large negative values of the inter-action energy, |γ| N lower right panel).

Figure 4.1: Energy spectrum for different values of γ. Rabi (green dots) and Josephson (blue dots) regimes

(upper left); Fock regime (upper right); attractive interaction with 1

N|γ| N (lower left); attractive interaction with |γ| N . N = 41 (lower right).

When the interactions are attractive, the ground state is always degenerate or quasi-degenerate, as illustrated in Fig. 4.2, where the first two energy levels are plotted against γN.

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4.2 Population Imbalance

In this section the numerical results related to the population imbalance are presented (see Section2.6).

4.2.1 Repulsive interactions

When the interactions among the particles are repulsive, the variance follows the equa-tion 2.28 and it shows a maximum in the Rabi regime, where the interactions are small, and a minimum in the Fock regime, for large values of the interaction energy (see Fig.

4.3).

Figure 4.3: Variance of population imbalance (n = na− nb) against positive values of the interaction energy.

4.2.2 Attractive interactions

In the case in which interactions are attractive, a second order phase transition is present, with the population imbalance playing the role of the order parameter, as shown in the upper panels of Fig. 4.4 (see paragraph 2.6.2 for details). The critical value ˜γcN tends to −1 for large values of the particles number (˜γ = U

J).

For large negative values of the interaction energy the ground state is a NOON state, with population imbalance variance ∆z = 1 (lower panels of Fig. 4.4).

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Figure 4.4: Absolute value of the fractional population imbalance varying negative ˜γ (top) for: N = 500

(left) and N = 1000 (right). Red dashed line is the theoretical curve z(˜γN ) =

q

1 − (γ˜cN ˜

γN)2. Variance of fractional population imbalance varying negative ˜γ (bottom) for: N = 500 (left) and N = 1000 (right).

4.2.3 Varying δ

By introducing an energy offset (δ) among the wells a first order phase transition is observed: for values of the interaction energy smaller than the critical threshold, ˜γN ≤ ˜γcN = −1, the population imbalance abruptly changes when it crosses the value δ = 0, as shown in Fig. 4.5(see paragraph2.6.3for more details).

Figure 4.5: Population imbalance ((na− nb)/N ) as a function of δ for different values of ˜γ. The critical value is ˜γN = −1. N = 200.

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4.3 Dynamics

In this section some simulations concerning the dynamics of the system (Section2.7) are reported. If the initial population imbalance is different from zero, in the non interacting case and for a symmetric double well, it oscillates around zero at the Rabi frequency ωR= 2J (upper left panel of Fig. 4.6). When interactions are sufficiently strong (see Eq.

2.24) a novel phenomenon appears: the particles are no more able to tunnel between the wells and are self trapped: the population imbalance oscillates around a value different from zero (upper right panel of Fig. 4.6). In the case, instead, of a perfectly balanced initial state (z = 0), if the interaction energy is not strictly equal to zero and if an asymmetry between the wells is present, the population imbalance oscillates around zero with a non vanishing amplitude (lower panel of Fig. 4.6).

Figure 4.6: Top: Initial state |N, 0i (z(0) = 1), N = 10: No interactions, Rabi oscillations for z(t) with

TR = 2π/ωR = 2π/2J ) (left). U = 1, self trapping regime (right). Bottom: Initial state |N/2, N/2i,

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CHAPTER

5 BEAM SPLITTER

B

efore describing the complete interferometric sequence in Chapter 6, here the first beam splitter will be analysed for the different initial states used in this thesis: a Twin Fock state, |N/2, N/2i (Section5.1) and a |N, 0i (Section5.2); the non interacting case will be considered, as well as the cases in which interactions among particles in the same well or between different wells are present. In order to make a comparison (Section 5.4), we will study the sensitivity reached when a NOON state, |N,0i+|0,N i_√

2 , is created after the beam splitter (Section 5.3).

The general way to realize a beam splitter for an atomic interferometer is to let the particles tunnelling by lowering the barrier, in the case of a double well interferometer, or by coupling the two states with resonant light, in the case of an internal two level system interferometer.

In characterizing the beam splitter we will define the sensitivity as the full width at half height (FWHH) of the narrowest peak in the phase probability distribution calculated as [46]:

P(φ, t) = N+ 1

2π hφ|ψout(t)ihψout(t)|φi = = _2π1 N X n=0 c∗_nei(N/2+n)φ 2 , (5.1) where |ψouti= N X n=0 cn(t)|ni , (5.2)

is the state after the first beam splitter. The coefficients cn(t) are given by cn(t) = hn|ψout(t)i, {|ni}with n = [0, N] being the N + 1 Fock basis vectors for the two mode model: |ni = |ni_a|N − nib. |φi are the normalized phase states |φi = _{N +1}1 PN

m=0eiφ(N/2+m)|mi.

5.1 Initial state Twin Fock

When the initial state is a Twin Fock (TF) state |N/2, N/2i, the optimal splitting time is de-fined as the time after which the main peak in the phase probability distribution is the narrowest (see Fig. 5.1). Sections 5.1.1and 5.1.2are a review of the results presented in [46], that we have reproduced in order to study the behaviour of the system and to find the optimal splitting times used in the complete interferometer with TF as input state (Note: In our code the Hamiltonian

ˆ

H = −J(â†ˆb +ˆb†â) +U₂(â†â†ââ + ˆb†ˆb†ˆbˆb) +δ₂(â†â − ˆb†ˆb) is used instead of the one in Eq (4) in the paper, so that Ec= 2U).

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CHAPTER 5. BEAM SPLITTER 29 5.1.1 Linear beam splitter

In the non interacting case, the 50/50 beam splitter is represented by the unitary transforma-tion |Ψ_outi= e−iπ2 ˆ a†ˆb+ˆb† ˆa 2 |Ψ_ini , (5.3)

and the optimal splitting time is given by TBS = π/(4J), corresponding to a π/2 Raman pulse (in the following we will refer to it as Tπ/2), as shown in Fig. 5.1, where relative number, |cn|2, and phase distributions are plotted after different times: the narrowest peak is obtained at t = Tπ/2.

Figure 5.1: Relative number (red, top) and phase probability (blue, bottom) distributions at different times.

From left to right: t=0, t=π/(8J), t=π/(4J); N = 40, U = 0 and J = 1. The optimal splitting time is reached at

t = π/(4J )

When interactions are not present the sensitivity is at the Heisenberg limit: ∆φ = α/Nβ _with β ∼ 1 (see Fig. 5.2, where the FWHH of the peak in the phase distribution is plotted against particle number).

Figure 5.2: FWHH of the phase probability distribution peak vs particles number when no interactions are present