Experiments on trapped quantum gases can probe challenging regimes of quantum many-body dynamics, where strong interactions or nonequilibrium states prevent exact solutions. Here we present a different kind of exact result, which applies even in the absence of actual solutions: a class of space-time mappings of different experiments onto each other. Since our result is an identity relating second-quantized field operators in the Heisenberg picture of quantum mechanics, it is extremely general; it applies to arbitrary measurements on any mixtures of Bose or Fermi gases, in arbitrary initial states. It represents a strong prediction of quantum field theory which can be tested in current laboratories, and whose practical applications include perfect simulation of interesting experiments with other experiments which may be easier to perform.
Experiments on trapped quantum gases can probe challenging regimes of quantum many-body dynamics, where strong interactions or nonequilibrium states prevent exact solutions. Here we present a different kind of exact result, which applies even in the absence of actual solutions: a class of space-time mappings of different experiments onto each other. Since our result is an identity relating second-quantized field operators in the Heisenberg picture of quantum mechanics, it is extremely general; it applies to arbitrary measurements on any mixtures of Bose or Fermi gases, in arbitrary initial states. It represents a strong prediction of quantum field theory which can be tested in current laboratories, and whose practical applications include perfect simulation of interesting experiments with other experiments which may be easier to perform.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. In particular I will present large-size calculations of the frequency- and momentum-resolved spectral function of strongly repulsive lattice bosons and a symmetry-breaking exact solution for a Bose-Bose mixture. Our approach directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. This Review collects the predictions coming from a family of exact solutions which allows us to obtain the many-body wavefunction of strongly correlated quantum fluids confined by a tight waveguide and subjected to any form of longitudinal confinement. It directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved. The exact solution applies to bosons, fermions, and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. The Review illustrates the idea of the solution, presents the key theoretical achievements, and the main experiments on strongly correlated one-dimensional quantum gases.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
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.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures, and in particular our new exact solution for the case of symmetric-breaking mixtures.The solutions allow one to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach, as well as to describe finite-temperature conditions, spin, and magnetization effects. I will present some uses of these solutions to obtain the exact quantum dynamics, both for the small oscillation regime and for the strongly out-of-equilibrium spin dynamics, where I will propose a protocol to observe symmetry oscillations in binary bosonic mixtures.