We investigate the classical and quantum metrology of performing parameter estimation with interacting trapped bosons, which we theoretically treat by a self-consistent many-body approach of the multiconfigurational Hartree type. Focusing on a tilted double-well geometry, we compare a self-consistently determined and monitored two-mode truncation, with dynamically changing orbitals, to the conventional two-mode approach of fixed orbitals, where only Fock space coefficients evolve in time. We demonstrate that, as a consequence, various metrological quantities associated to a concrete measurement such as the classical Fisher information and the maximum likelihood estimator are deeply affected by the orbitals' change during the quantum evolution. Self-consistency of the quantum many-body dynamics of interacting trapped ultracold gases thus fundamentally affects the attainable parameter estimation accuracy of a given metrological protocol.
We investigate the classical and quantum metrology of performing parameter estimation with interacting trapped bosons, which we theoretically treat by a self-consistent many-body approach of the multiconfigurational Hartree type. Focusing on a tilted double-well geometry, we compare a self-consistently determined and monitored two-mode truncation, with dynamically changing orbitals, to the conventional two-mode approach of fixed orbitals, where only Fock space coefficients evolve in time. We demonstrate that, as a consequence, various metrological quantities associated to a concrete measurement such as the classical Fisher information and the maximum likelihood estimator are deeply affected by the orbitals' change during the quantum evolution. Self-consistency of the quantum many-body dynamics of interacting trapped ultracold gases thus fundamentally affects the attainable parameter estimation accuracy of a given metrological protocol.
We theoretically investigate the possibility of performing high precision estimation of an externally imposed acceleration using scalar bosons in a single-well trap. We work at the level of a two-mode truncation, valid for weak to intermediate two-body interaction couplings. The splitting process into two modes is in our model entirely caused by the interaction between the constituent bosons and is hence neither due to an externally imposed double-well potential nor due to populating a spinor degree of freedom. The precision enhancement gained by using various initial quantum states using a two-mode bosonic system is well established. Here we therefore instead focus on the effect of the intrinsic dynamics on the precision, where, in a single well, the Hamiltonian assumes a form different from that of the typical double-well case. We demonstrate how interactions can significantly increase the quantum Fisher information maximized over initial states as well as the quantum Fisher information for a fragmented or a coherent state, the two many-body states that can commonly represent the ground state of our system.
In this work we develop a complete variational many-body theory for a system of N trapped bosons interacting via a general two-body potential. The many-body solution of this system is expanded over orthogonal many-body basis functions (configurations). In this theory both the many-body basis functions and the respective expansion coefficients are treated as variational parameters. The optimal variational parameters are obtained self-consistently by solving a coupled system of noneigenvalue---generally integro-differential---equations to get the one-particle functions and by diagonalizing the secular matrix problem to find the expansion coefficients. We call this theory multiconfigurational Hartree theory for bosons or MCHB(M), where M specifies explicitly the number of one-particle functions used to construct the configurations. General rules for evaluating the matrix elements of one- and two-particle operators are derived and applied to construct the secular Hamiltonian matrix. We discuss properties of the derived equations. We show that in the limiting cases of one configuration the theory boils down to the well-known Gross-Pitaevskii and the recently developed multi-orbital mean fields. The invariance of the complete solution with respect to unitary transformations of the one-particle functions is utilized to find the solution with the minimal number of contributing configurations. In the second part of our work we implement and apply the developed theory. It is demonstrated that for any practical computation where the configurational space is restricted, the description of trapped bosonic systems strongly depends on the choice of the many-body basis set used, i.e., self-consistency is of great relevance. As illustrative examples we consider bosonic systems trapped in one- and two-dimensional symmetric and asymmetric double well potentials. We demonstrate that self-consistency has great impact on the predicted physical properties of the ground and excited states and show that the lack of self-consistency may lead to physically wrong predictions. The convergence of the general MCHB(M) scheme with a growing number M is validated in a specific case of two bosons trapped in a symmetric double well.
In order to solve the many-boson Schr\"odinger equation we utilize the multiconfigurational time-dependent Hartree method for bosons (MCTDHB). To be able to attack larger systems and/or to propagate the solution for longer times, we implement a parallel version of the MCTDHB method, thereby realizing the recently proposed [Streltsov et al., Phys. Rev. A 81, 022124 (2010)] idea on how to construct efficiently the result of the action of the Hamiltonian on a bosonic state vector. As an illustrative example of its own interest, we study the real-space dynamics of repulsive bosonic systems made of N=12, 51, and 3003 bosons in triple-well periodic potentials. The ground state of this system is threefold fragmented. By suddenly strongly distorting the trap potential, the system performs complex many-body quantum dynamics. At long times it reveals a tendency to an oscillatory behavior around a threefold fragmented state. These oscillations are strongly suppressed and damped by quantum depletions. In spite of the richness of the observed dynamics, the three time-adaptive orbitals of MCTDHB(M=3) are capable of describing the many-boson quantum dynamics of the system for short and intermediate times. For longer times, however, more self-consistent time-adaptive orbitals are needed to correctly describe the nonequilibrium many-body physics. The convergence of the MCTDHB(M) method with the number M of self-consistent time-dependent orbitals used is demonstrated.
In this thesis, the behavior of ultracold bosonic systems in external potentials is investigated. Such systems are typically realized as Bose-Einstein condensates, where dilute atomic gases cooled to extremely low temperatures exhibit collective quantum behavior. Because of the large degree of control they allow over the external potential and over the interaction parameters of the many-body system, Bose-Einstein condensates constitute a testing ground for the dedicated study of coherence, correlations, and quantum phase transitions in interacting bosonic systems. This thesis presents an investigation of the many-body physics of ultracold bosons, focusing specifically on how the many-body properties of these ultracold bosonic systems change with changing external potential, how they manifest themselves in experimentally accessible observables, and how these properties can be analyzed using both numerical many-body methods and data-driven approaches. The first part of this thesis focuses on the theoretical description of ultracold bosonic systems in external trapping potentials. Using the multiconfigurational time-dependent Hartree method for bosons (MCTDH-X), numerically exact simulations were performed that allow the calculation of reduced density matrices, natural orbitals, and correlation functions. These quantities were used to analyze the coherence properties and fragmentation of trapped bosonic systems. In particular, double-well systems were investigated as model scenarios in which the gradual formation of a central barrier leads to a redistribution of natural occupations and a transition from a condensed to a fragmented many-body state. In this context, we demonstrate the usefulness of single-shot images, which represent projective measurements of the many-body wave function, for the experimental investigation of the many-body physics of fragmenting system. Single-shot images can be simulated numerically using MCTDH-X, but they are more commonly obtained through experimental measurements. The possibility of extracting many-body information in an experimental context, where said many-body information is not otherwise accessible, is discussed. Supervised machine learning models were trained on simulated single-shot data sets to investigate whether information about the underlying many-body state can be inferred directly from measurement outcomes. The results show that machine learning models are capable of learning patterns in the single-shot images that correlate with properties of the underlying many-body state and can therefore predict density-related quantities with good accuracy. The second part of this thesis focuses on a joint theoretical and experimental investigation of a system of ultracold bosons coupled to an optical cavity. In this system, the atoms interact with a dynamically generated optical potential produced by the cavity field. We use a new theoretical framework that combines a mean-field treatment of the cavity field with a many-body description of the atomic degrees of freedom to be able to capture the emergence of a superradiant self-organized phase and the subsequent transition to a self-organized Mott-insulating state. Our simulations show good agreement with the experimental results, and we discuss where the use of a beyond mean-field treatment in the simulations becomes essential in describing the physics of the different involved phases and quantum states. In summary, the results presented in this thesis demonstrate how numerical many-body simulations and machine learning methods can be used to analyze experimentally accessible observables and to gain insight into the many-body physics of ultracold bosonic systems.
We consider the estimation of an unknown parameter θ via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its θ sensitivity during the dynamics and/or in the steady state. We present bounds on the quantum Fisher information, and corresponding optimal interacting Hamiltonians, for two paradigmatic scenarios for encoding θ : (i) via unitary Hamiltonian dynamics (dynamical metrology), and (ii) in the Gibbs and diagonal ensembles (time-averaged dephased state), two ubiquitous steady states of many-body open dynamics. We then move to the specific problem of estimating the strength of a magnetic field via interacting spins and derive two-body interacting Hamiltonians that can approach the fundamental precision bounds. In this case, we additionally analyze the transient regime leading to the steady states and characterize trade-offs between equilibration times and measurement precision. Overall, our results provide a comprehensive picture of the potential of many-body control in quantum sensing.
In this thesis, the behavior of ultracold bosonic systems in external potentials is investigated. Such systems are typically realized as Bose-Einstein condensates, where dilute atomic gases cooled to extremely low temperatures exhibit collective quantum behavior. Because of the large degree of control they allow over the external potential and over the interaction parameters of the many-body system, Bose-Einstein condensates constitute a testing ground for the dedicated study of coherence, correlations, and quantum phase transitions in interacting bosonic systems. This thesis presents an investigation of the many-body physics of ultracold bosons, focusing specifically on how the many-body properties of these ultracold bosonic systems change with changing external potential, how they manifest themselves in experimentally accessible observables, and how these properties can be analyzed using both numerical many-body methods and data-driven approaches. The first part of this thesis focuses on the theoretical description of ultracold bosonic systems in external trapping potentials. Using the multiconfigurational time-dependent Hartree method for bosons (MCTDH-X), numerically exact simulations were performed that allow the calculation of reduced density matrices, natural orbitals, and correlation functions. These quantities were used to analyze the coherence properties and fragmentation of trapped bosonic systems. In particular, double-well systems were investigated as model scenarios in which the gradual formation of a central barrier leads to a redistribution of natural occupations and a transition from a condensed to a fragmented many-body state. In this context, we demonstrate the usefulness of single-shot images, which represent projective measurements of the many-body wave function, for the experimental investigation of the many-body physics of fragmenting system. Single-shot images can be simulated numerically using MCTDH-X, but they are more commonly obtained through experimental measurements. The possibility of extracting many-body information in an experimental context, where said many-body information is not otherwise accessible, is discussed. Supervised machine learning models were trained on simulated single-shot data sets to investigate whether information about the underlying many-body state can be inferred directly from measurement outcomes. The results show that machine learning models are capable of learning patterns in the single-shot images that correlate with properties of the underlying many-body state and can therefore predict density-related quantities with good accuracy. The second part of this thesis focuses on a joint theoretical and experimental investigation of a system of ultracold bosons coupled to an optical cavity. In this system, the atoms interact with a dynamically generated optical potential produced by the cavity field. We use a new theoretical framework that combines a mean-field treatment of the cavity field with a many-body description of the atomic degrees of freedom to be able to capture the emergence of a superradiant self-organized phase and the subsequent transition to a self-organized Mott-insulating state. Our simulations show good agreement with the experimental results, and we discuss where the use of a beyond mean-field treatment in the simulations becomes essential in describing the physics of the different involved phases and quantum states. In summary, the results presented in this thesis demonstrate how numerical many-body simulations and machine learning methods can be used to analyze experimentally accessible observables and to gain insight into the many-body physics of ultracold bosonic systems.