We consider a one-dimensional gas of cold atoms with strong contact interactions and construct an effective spin-chain Hamiltonian for a two-component system. The resulting Heisenberg spin model can be engineered by manipulating the shape of the external confining potential of the atomic gas. We find that bosonic atoms offer more flexibility for independently tuning the parameters of the spin Hamiltonian through interatomic (intraspecies) interaction, which is absent for fermions due to the Pauli exclusion principle. Our formalism can have important implications for control and manipulation of the dynamics of few- and many-body quantum systems; as an illustrative example relevant to quantum computation and communication, we consider state transfer in the simplest nontrivial system of four particles representing exchange-coupled qubits.
We consider a one-dimensional gas of cold atoms with strong contact interactions and construct an effective spin-chain Hamiltonian for a two-component system. The resulting Heisenberg spin model can be engineered by manipulating the shape of the external confining potential of the atomic gas. We find that bosonic atoms offer more flexibility for independently tuning the parameters of the spin Hamiltonian through interatomic (intraspecies) interaction, which is absent for fermions due to the Pauli exclusion principle. Our formalism can have important implications for control and manipulation of the dynamics of few- and many-body quantum systems; as an illustrative example relevant to quantum computation and communication, we consider state transfer in the simplest nontrivial system of four particles representing exchange-coupled qubits.
Interactions between quantum particles can give rise to behaviors greater than the sum of their parts. The low-dimensional cold atomic framework provides a unique platform with exquisite control and enhanced correlations to develop a deeper understanding of such quantum phenomena, with promise for emerging technologies. In this thesis, I study novel quantum many-body physics in spinor gases trapped in optical lattices and develop efficient theoretical tools for such investigations. My work illustrates the central role of spin and charge degrees of freedom for comprehending and manipulating intricate quantum behaviors. With a broad goal of understanding the emergent physics in strongly correlated many-body systems, in Chapter 2, I investigate the two-component coupled Bose gas. An unprecedented phase emerges from the coupling of two components in the nearest-neighbor sites via a resonant two-photon Raman transition. Furthermore, coupling provides a novel tuning knob for the controlled manipulation of spin phases. Probing for behaviors away from the Mott state, the limitations of existing theoretical approaches are exposed. To address this, in Chapter 3, I develop a powerful theoretical framework for the unrestricted study of strongly correlated systems, valid even for large spin systems. This generalized effective spin-chain (GESC) formalism has numerous advantages: elegant mathematical form, computationally efficient parameterization, generalized for arbitrary spin and statistics, and universality: not limited to lattice systems. Importantly, leveraging a spin–charge separated form, it provides a unique vantage into system behaviors originating from quantum fluctuations. Motivated by the immense potential of anyons in cold atomic setups, in Chapter 4, I investigate the combined effect of fractional statistics and the presence of multiple components on their ground state and dynamical properties. This reveals the potential of statistical-phase-induced transition between different system phenomena. The GESC formalism is extended to consider anyons and used to obtain insights into the origin of exchange-phase-dependent and independent behavior. I demonstrate the versatility of the GESC formalism, in Chapter 5, to Bose–Fermi mixtures, beyond one-dimensional systems, and to consider long-range interactions. In doing so, avenues are provided for leveraging the immense utility of this formalism in exploring the novel physics such systems have to offer.
We present a general form of the effective spin-chain model for strongly interacting atomic gases with an arbitrary spin in one-dimensional (1D) traps. In particular, for high-spin systems the atoms can collide in multiple scattering channels, and we find that the resulting form of spin-chain model generically follows the same structure as that of the interaction potentials. This is a unified form working for any spin, statistics (Bose or Fermi), and confinement potentials. We adopt the spin-chain model to reveal both the ferromagnetic (FM) and antiferromagnetic (AFM) magnetic orders for strongly interacting spin-1 bosons in 1D traps. We further show that by adding the spin-orbit coupling, the FM and AFM orders can be gradually destroyed and eventually the ground state exhibits universal spin structure and contacts that are independent of the strength of spin-orbit coupling.
In this paper, we provide a theoretical analysis of strongly interacting quantum systems confined by a time-dependent external potential in one spatial dimension. We show that such systems can be used to simulate spin chains described by Heisenberg Hamiltonians in which the exchange coupling constants can be manipulated by time-dependent driving of the shape of the external confinement. As illustrative examples, we consider a harmonic trapping potential with a variable frequency and an infinite square well potential with a time-dependent barrier in the middle.
Quantum many-body physics has been studied for many decades. A lot of intriguing phenomena have been observed and theories have been developed. Yet many problems remain unsolved, which is largely due to a lack of general efficient classical computation method. One-dimensional (1D) systems have also drawn much attention over the past few decades. One reason is that many unique strongly correlated quantum phenomena only appear in low dimensions. Another reason is that many exact results can be obtained in 1D for cross benchmarking, such as Bethe Ansatz, Bosonization, Bose-Fermi mapping, etc. In addition, many numerical methods, such as Matrix-Product-State based and Monte Carlo methods, work most efficiently in 1D. In recent years, 1D quantum gases have been realized in many cold atom labs, providing experimental motivation for their studies. One of the most mysterious assumptions about a system of identical quantum particles is that the wavefunction must be symmetric (for bosons) or anti-symmetric (for fermions). This means that no two fermions can occupy the same state, while multiple occupancy is allowed for bosons. Bosons and fermions are therefore generally very different. But in 1D this distinction could become rather subtle, provided that the bosonic multiple occupancy is suppressed, which can happen if strong repulsion exists between them. However, when considering spin degrees of freedom, things become more complicated. In this work, we develop a generalized Bose-Fermi mapping theory, under which, the charge degrees of freedom is mapped to a spinless fermions, while the spin degrees of freedom to an effective spin chain. This mapping works for arbitrary spin, arbitrary trapping potentials, arbitrary spin-dependent interaction potential, and for either spinor bosons or fermions. In the strong interaction limit, the wavefunction of the system is represented by the strong coupling ansatz wavefunction (SCAW), which takes the form of a direct product of spinless fermions wavefunction and spin wavefunction. Using this mapping technique, we study the dynamics and collective modes of the system. Furthermore, we develop a very efficient method to calculate the one-body density matrix, from which we can calculate the momentum distribution of the system.
We consider a one-dimensional trapped gas of strongly interacting few spin-1 atoms which can be described by an effective spin chain Hamiltonian. Away from the SU(3) integrable point, where the energy spectrum is highly degenerate, the rules of ordering and crossing of the energy levels and the symmetry of the eigenstates in the regime of large but finite repulsion have been elucidated. We study the spin-mixing dynamics which is shown to be very sensitive to the ratio between the two channel interactions g0/g2 and the effective spin chain transfers the quantum states more perfectly than the Heisenberg bilinear-biquadratic spin chain.