Exploration of properties and phases of correlated bosons in optical potentials

The work in this thesis entitled "Exploration of the properties and phases of correlated bosons in optical potentials" was carried out by me under the supervision of Prof.

ACKNOWLEDGMENTS

I would like to thank all the inspiring people who have motivated me in one way or another in my life. Last but not least, I want to express my heartfelt gratitude to my family for their role in my life.

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A BSTRACT

L IST OF F IGURES

The time in the x-axis is measured in units of the tunneling frequency, J and is true for all plots. U (non-interaction), right-angled triangles with solid line (red) indicate αvirV=0.5U and diamonds with dashed line (blue) indicateαvirV=2U. it is in units of inverse of the energy scale,². a) Presence of Rabi oscillations for κ=0.

C ONTENTS

47 3 Study of tunnel dynamics and phases using BHM 49 3.1 Tunnel dynamics of correlated bosons in a double well potential. 89 4 Exploration of phases and dynamics using the MCTDHB approach 91 4.1 Managing the correlations of ultracold bosons in triple wells.

I NTRODUCTION

Bose-Einstein condensation: Developments so far

At the time, it was thought that interactions present in the superfluids would change the physics of Bose-Einstein condensation. In the beginning, spin-polarized hydrogen, due to its light mass, was the prime candidate for realizing Bose-Einstein condensation.

BOSE-EINSTEIN CONDENSATION: DEVELOPMENTS SO FAR

Noninteracting bosonic gas
Interacting ultracold bosonic gas

Weakly coupled bosonic gas
Strongly correlated bosonic gas

These fluctuations cause a depletion of the condensed state, and this depletion is of the order of (na3s)1/2, which is usually very small (about 1% in experiments). However, there is a problem with this approach, which is a sharp reduction in condensate lifetime due to three-body losses [44] and increases with increasing scattering density and length.

Optical lattice potentials

OPTICAL LATTICE POTENTIALS

Dipole potentials
AC Stark effect
Different geometries of optical lattices
Atomic interactions via Feshbach Resonance

The scattering rate in terms of resolution and trap depth is given as,. Finally, different optical gratings can be achieved by tuning the orientation and signs of the diffraction parameter of the laser beams [51].

THEORETICAL METHODS FOR ULTRACOLD BOSONIC GAS

Theoretical methods for ultracold bosonic gas

However, the detailed physics involved in a quantum many-body system is known to a limited extent, or at best approximately known.

Simulation techniques for ultracold bosonic gas

INTRIGUING FEATURES OF ULTRACOLD BOSONS

Intriguing features of ultracold bosons

Figure 1.4(b), included for comparison, shows the dynamics of the occupancy density expectation values of the two wells. The first is the superfluid state in which bosons are completely delocalized over the entire lattice due to overlapping of the system's wave functions.

MOTIVATION AND OVERVIEW OF THE THESIS

Motivation and Overview of the Thesis

It is observed that a chirp modulation actually restores order to the tunnel dynamics of the system. We further study the effect of time-dependent measuring fields on the energy spectrum of the system.

F ORMALISMS USED IN THE STUDY

Second Quantization

Fock ground state for the complete orthonormal set of states of one particle (orbital) {φi}; where i=1, ..,∞, expressed as,. The number of state occupations in relation to the total number of particles, N for a system with a constant number of bosons is read as,.

BOSE-HUBBARD MODEL (BHM)

Bose-Hubbard Model (BHM)

Exact Dynamics
Dynamics using EQMs for SU(2) generators
Single-Site Mean Field Technique (MFT)

The dynamics of the system is obtained by calculating the Heisenberg equation of motion (EOM) as,. Here the Bloch vector passes through the poles of the Bloch sphere, as shown in Figure 2.1(a).

MULTI-CONFIGURATIONAL TIME-DEPENDENT HARTREE APPROACH FOR BOSONS (MCTDHB)

Multi-configurational Time-Dependent Hartree Approach for Bosons (MCTDHB)

The many-boson Hamiltonian
Time-dependent variational principle
The many-boson ansatz (The MCTDHB wave function)
Reduced Density Matrices (RDM)
p-th order coherence
The building equations of MCTDHB
Condensation and Fragmentation

The measurement of the p-positions is correlated when g(p)>1 and is anti-correlated if g(p)<1. To derive the equations of motion of MCTDHB, the Lagrangian formulation of the time-dependent variational principle is used [123]. In the next step, stationarity of the action with respect to its arguments, {φj(r,t)} and {Cn(t)} is required.

S TUDY OF TUNNELING DYNAMICS AND PHASES EMPLOYING BHM

Tunneling dynamics of correlated bosons in a double well potential

Introduction

In a simplified way, the tunneling dynamics of correlated particles in the presence of confining potential can be investigated by considering a single particle or a few particles in a double-well potential. The time evolution of the Fock space for a simple two-site Bose Hubbard model (BHM) (without the offsite density exchange term) with two bosons has been investigated and the tunneling probabilities calculated as a function of time [117, 146] . While a detailed investigation of the tunneling dynamics in strong and weak coupling regimes and the sensitivity of the time-evolved state to a variety of initial configurations was lacking.

TUNNELING DYNAMICS OF CORRELATED BOSONS IN A DOUBLE WELL POTENTIAL

The BHM and EOMs for a few bosons
Physical Observables and Results

Admixture of states
Time averaged dynamics - extrapolation to large N

Conclusions

Similarly, (010) indicates one in each well, while (001) implies both in the left well with the right well empty. Similarly, the case corresponding to (N−1) bosons in the right well and one in the left can be expressed as, . In the limitN→ ∞, α becomes small 0.1−0.2 (the extrapolated value), which again emphasizes the onset of the trapping effects as the time-averaged probability for the particles to spend in one of the wells (right here) becomes small.

DYNAMICS OF INTERACTING BOSONS IN A DOUBLE WELL POTENTIAL

Dynamics of interacting bosons in a double well potential

Introduction
Equations of motion for the SU (2) generators
Results

Harmonic modulation

Conclusions

Without any interaction (κ=0), the phase space projection shows a usual elliptic trajectory (Rabi) that represents the periodic behavior of the system (Fig. 3.11(a)). In Figure 3.12 we show the temporal evolution of the population imbalance,Jz(t) in this case. At large values of the interaction strength, the system makes a transition from a regular to a chaotic state.

Ultracold gases in presence of time-dependent synthetic field

Introduction

To confirm the re-entry of periodic behavior, we also study the phase space projection of the population imbalance Jz(t). In this work, we have investigated the tunneling dynamics of a system of bosons in a double-well potential in the presence of a harmonic interaction potential. In particular, in the case of the harmonic interaction potential, different paths to chaos emerge depending on whether the amplitude or the phase of the interaction term is tuned.

ULTRACOLD GASES IN PRESENCE OF TIME-DEPENDENT SYNTHETIC FIELD

Model
Results
Conclusions

The MI-SF phase transition is changing as a function of time-dependent magnetic flux in Table 3.1. It is also observed that the inclusion of time in the synthetic field destroys the periodicity of the network, but with the proper tuning of the time dependence, the periodicity can be restored again. Therefore, the periodicity of the one-dimensional superfluid (SF) order parameter strongly depends on the nature of the time-dependent gauge field.

E XPLORATION OF PHASES AND DYNAMICS USING

MCTDHB APPROACH

Management of the Correlations of Ultracold Bosons in Triple Wells

Introduction

Thus, the inclusion of tilt in the system broadens the spectrum of controllable parameters and enriches the emerging physics. Here, a reduced density matrix (RDM) of many-body states is used to quantify the correlations. Henceforth, the phenomenon of correlation and fragmentation in the many-body system is investigated as a function of the interaction strength and the inclination of the triple well.

MANAGEMENT OF THE CORRELATIONS OF ULTRACOLD BOSONS IN TRIPLE WELLS

Hamiltonian

Quantities of interest

Results

Convergence of results with respect to the number of orbitals
Natural occupations
First-order correlation
Inter-well correlation
Natural orbitals: variation as a function of the tilt

Conclusions

We begin our investigation by plotting the one-body densityρ(x) in Fig.4.1(b) as a function of the tilt α. Henceforth, we chose the values of the interaction strength (λ=6) and barrier height (V0=180) such that the ground state is triply fragmented in the absence of tilt (α=0). For large values of the tiltα, the natural orbitals are mainly localized in the central and the right wells [Fig.4.7(d)].

Dynamics of interacting bosons in a quantum seesaw potential

Introduction

A natural extension of our work and in light of recent technical developments [214] would concern bosons with an internal structure and/or embedded in an optical cavity. Furthermore, we note that, due to the long decoherence time, the many-body state of ultracold atoms can provide a way to store correlations and entanglement that arise in the processing of quantum information [215]. For this purpose, protocols are needed to monitor and quantify correlations in the many-body state of ultracold atoms, such as the ones we have outlined in this work [216].

DYNAMICS OF INTERACTING BOSONS IN A QUANTUM SEESAW POTENTIAL

Results

First-order correlation
Population imbalance and Power spectral density

Conclusions

Let us now explore the dynamics with the inclusion of the time-driven tilt in the potential. Similar features for the time evolution of|g(1)(x,x0;t)|2 are observed with the inclusion of the driving force,α. For small amplitude, i.e. α=1, the observed white noise dominates the dynamics of the bosons [see Fig.4.14(c)].

Simulation of rotating condensates confined in a 2D harmonic trap

Introduction

With the addition of the periodic drive, the dynamics of the transition from a depleted to a fully fragmented state becomes faster. To delve deeper into particle dynamics, we also investigated the time evolution of the population imbalance, together with the power spectral density, PSD. Moving on from PSD, it is observed that for finite interaction strength without any actuation, the PSD exhibits the presence of white noise in the particle dynamics trajectory.

SIMULATION OF ROTATING CONDENSATES CONFINED IN A 2D HARMONIC TRAP

Magnetic translational group and magnetic Fourier transform

Results

One-body density and natural orbitals

Conclusions

It is also important to mention that in the absence of the synthetic field (ωωω=0), we. It is observed that the one-body density,ρ shows the maximum, which is centered at the origin of the harmonic trap Fig.4.17(a). As time progresses, for t=33, it is observed that ρ is fragmented into two maxima around the center of the harmonic trap [Fig.4.17(f)].

I MPORTANCE OF THE MCTDHB FORMALISM : A STUDY FOR CONTACT AND NON - CONTACT INTERACTIONS

Introduction
Variation of natural orbitals with interaction strength
VARIATION OF NATURAL ORBITALS WITH INTERACTION STRENGTH
Long range interaction
LONG RANGE INTERACTION

Natural Occupations
First-order correlation function

Conclusions

Fig. (5.3) shows the behavior of the natural occupations, nNi as a function of the slope parameter, α for fixed barrier height, V0 and the interaction force, λ, in the case of long-range interactions as given by equation (5.1). However, for a larger interaction strength, λ=20, the pitches are not affected by the presence of tilt in the trapping potential. However, the second natural occupation, n2/N remains constant regardless of increasing α (unlike the contact interaction case, see Fig.4.4(c) in Chapter 4).

C ONCLUSIONS

In section 4.2, we have explored the tunneling dynamics of the quantum seesaw, where a system of correlated bosons is confined in a temporally driven tilted double-well potential. With the increase of the interaction strength, for a certain value of the interaction, it is observed that the system goes from being a depleted condensate to a threefold fragmented state with time. Further increase in interaction and the periodic drive, it is observed that the initially frozen dynamics of the population imbalance depicts some features with rapid oscillations.

E QUATION OF MOTION FOR TRIPLE WELL : EFFECT OF INITIAL CONDITIONS

The proper well population, PR(t) of three bosons confined to a triple well potential is defined as[117],. A.5) The different initial conditions, pi of the bosons in the triple well potential are represented by the different possible combinations of pi as given in table (A.1). Fig.(A.1) depict the time evolution of the proper well population, PR(t) in the weak coupling regime for both attractive and repulsive interactions with an initial configuration in which the system is prepared considered as. Here, we assess the generality of our results in Chapter 4 for a trap potential of the form given as , .

B IBLIOGRAPHY

V ITA