arXiv subjects
Publications and source records attributed to M. D. Girardeau.
Using the exact $N$-particle ground state wave function for a one-dimensional gas of hard-core bosons in a harmonic trap we develop an algorithm to compute the reduced single-particle density matrix and corresponding momentum distribution. Accurate numerical results are presented for up to N=8 particles, and the momentum distributions are compared to a recent analytic approximation.
The ground state of a one-dimensional (1D) quantum gas of dipoles oriented perpendicular to the longitudinal axis, with a strong 1/x^3 repulsive potential, is studied at low 1D densities $n$. Near contact the dependence of the many-body wave function on the separation x_{jl} of two particles reduces to a two-body wave function Ψ_{rel}(x_{jl}). Immediately after a sudden rotation of the dipoles so that they are parallel to the longitudinal axis, this wave function will still be that of the repulsive potential, but since the potential is now that of the attractive potential, it will not be stationary. It is shown that as nd^2 -> 0 the rate of change of this wave function approaches zero. It follows that for small values of nd^2, this state is metastable and is an analog of the super Tonks-Girardeau state of bosons with a strong zero-range attraction. The dipolar system is equivalent to a spinor Fermi gas with spin $z$ components σ_{\uparrow}=\perp (perpendicular to the longitudinal axis) and σ_{\downarrow}=|| (parallel to the longitudinal axis). A Fermi-Fermi mapping from spinor to spinless Fermi gas followed by the standard 1960 Fermi-Bose mapping reduces the Fermi system to a Bose gas. Potential experiments realizing the sudden spin rotation with ultracold dipolar gases are discussed, and a few salient properties of these states are accurately evaluated by a Monte Carlo method.
A harmonically trapped ultracold 1D spin-1 Bose gas with strongly repulsive or attractive 1D even-wave interactions induced by a 3D Feshbach resonance is studied. The exact ground state, a hybrid of Tonks-Girardeau (TG) and ideal Fermi gases, is constructed in the TG limit of infinite even-wave repulsion by a spinor Fermi-Bose mapping to a spinless ideal Fermi gas. It is then shown that in the limit of infinite even-wave attraction this same state remains an exact many-body eigenstate, now highly excited relative to the collapsed generalized McGuire cluster ground state, showing that the hybrid TG state is completely stable against collapse to this cluster ground state under a sudden switch from infinite repulsion to infinite attraction. It is shown to be the TG limit of a hybrid super Tonks-Girardeau (STG) state which is metastable under a sudden switch from finite but very strong repulsion to finite but very strong attraction. It should be possible to create it experimentally by a sudden switch from strongly repulsive to strongly attractive interaction, as in the recent Innsbruck experiment on a spin-polarized bosonic STG gas. In the case of strong attraction there should also exist another STG state of much lower energy, consisting of strongly bound dimers, a bosonic analog of a recently predicted STG gas which is an ultracold gas of strongly bound bosonic dimers of fermionic atoms, but it is shown that this STG state cannot be created by such a switch from strong repulsion to strong attraction.
The ground state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground state wave function which permits to solve the Fermi sign problem in the following respects (i) the nodal surface is known, permitting exact calculations (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system, thus allowing simulation of a large number of fermions. Due to the mapping the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground state energy has been calculated in homogeneous and trapped geometries by using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the large-density ideal Fermi gas/Tonks-Girardeau limit, where low lying excitations are phonons. Exact density profiles are confronted to the ones calculated within the local density approximation which predicts a change from a semicircular to inverted parabolic shape of the density profile as the value of the charge is increased.
Motivated by previous suggestions that three-body hard-core interactions in lower-dimensional ultracold Bose gases might provide a way for creation of non-Abelian anyons, the exact ground state of a harmonically trapped 1D Bose gas with three-body hard-core interactions is constructed by duality mapping, starting from an $N$-particle ideal gas of mixed symmetry with three-body nodes, which has double occupation of the lowest harmonic oscillator orbital and single occupation of the next $N-2$ orbitals. It has some similarity to the ground state of a Tonks-Girardeau gas, but is more complicated. It is proved that in 1D any system of $N\ge 3$ bosons with three-body hard-core interactions also has two-body soft-core interactions of generalized Lieb-Liniger delta function form, as a consequence of the topology of the configuration space of $N$ particles in 1D, i.e., wave functions with \emph{only} three-body hard core zeroes are topologically impossible. This is in contrast with the case of 2D, where pure three-body hard-core interactions do exist, and are closely related to the fractional quantized Hall effect. The exact ground state is compared with a previously-proposed Pfaffian-like approximate ground state, which satisfies the three-body hard-core constraint but is not an exact energy eigenstate. Both the exact ground state and the Pfaffian-like approximation imply two-body soft-core interactions as well as three-body hard-core interactions, in accord with the general topological proof.
A harmonically trapped ultracold 1D spinor Fermi gas with a strongly attractive 1D even-wave interaction induced by a 3D Feshbach resonance is studied. It is shown that it has two different super Tonks-Girardeau (sTG) energy eigenstates which are metastable against collapse in spite of the strong attraction, due to their close connection with 1D hard sphere Bose gases which are highly excited gas-like states. One of these sTG states is a hybrid between an sTG gas with strong $(\uparrow\downarrow$ attractions and an ideal Fermi gas with no $(\uparrow\uparrow)$ or $(\downarrow\downarrow)$ interactions, the sTG component being an exact analog of the recently observed sTG state of a 1D ultracold Bose gas. It should be possible to create it experimentally by a sudden switch of the $(\uparrow\downarrow)$ interaction from strongly repulsive to strongly attractive, as in the recent Innsbruck experiment on the bosonic sTG gas. The other is a trapped analog of a recently predicted sTG state which is an ultracold gas of strongly bound $(\uparrow\downarrow)$ fermion dimers which behave as bosons with a strongly attractive boson-boson interaction leading to sTG behavior. It is proved that the probability of a transition from the ground state for strongly repulsive interaction to this dimer state under a sudden switch from strongly repulsive to strongly attractive interaction is $\ll 1$, contrary to a previous suggestion.
Recent theoretical and experimental results demonstrate a close connection between the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gas with hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ of the sTG gas, a highly excited gas-like state with nodes only at interparticle separations $|x_{j\ell}|=x_{node}\approx a_{1D}$. It is shown herein that when the coupling constant $g_B$ in the Lieb-Liniger interaction $g_Bδ(x_{j\ell})$ is negative and $|x_{12}|\ge x_{node}$, the sTG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the sTG and HSB wave functions is that the sTG wave function allows a small penetration into the region $|x_{12}| 2$. The sTG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ge 2$, $x_{node}$ is given accurately by a simple parabola. The metastability of the sTG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.
A model of two Calogero-Sutherland Bose gases A and B with strong odd-wave AB attractions induced by a p-wave AB Feshbach resonance is studied. The ground state wave function is found analytically by a Bose-Bose duality mapping, which permits one to accurately determine static physical properties by a Monte Carlo method. The condensation of particles or particle pairs (molecules) is tested by analyzing the presence of the off-diagonal long-range order in one- or two-body density matrices. The p-wave symmetry of AB interaction makes possible quasi-condensation of type A particles at the Fermi momentum of the B component. The zero-temperature phase diagram is drawn in terms of densities and interaction strengths.
In their Comment [1] Giraud and Combescot point out that the contribution to the impurity-boson distribution function $ρ_{bi}(x-y)$ of a term we dropped is not negligible, rather than being negligible in the thermodynamic limit as we had conjectured. We now agree with them, but nevertheless our results for $ρ_{bi}$ are highly accurate for large impurity-boson mass ratio $m_i/m$ and remain qualitatively correct for all values of $m_i/m$ and all values of the boson-impurity coupling constant.
A model of two 1D ideal Bose gases A and B with strong AB attractions induced by a p-wave AB Feshbach resonance is studied. The model is solved exactly by a Bose-Bose duality mapping, and it is shown that there is no A-component or B-component Bose-Einstein condensation and no AB-pair off-diagonal long-range order (ODLRO), but both AA-pair and BB-pair ODLRO. After generalization by adding even-wave AA and BB repulsion and reducing the strength of the odd-wave AB attraction by Feshbach resonance detuning, a quantum phase transition occurs between a phase with AB contact nodes and one with no such nodes.
The low-lying eigenstates of a one-dimensional (1D) system of many impenetrable point bosons and one moving impurity particle with repulsive zero-range impurity-boson interaction are found for all values of the impurity-boson mass ratio and coupling constant. The moving entity is a polaron-like composite object consisting of the impurity clothed by a co-moving gray soliton. The special case with impurity-boson interaction of point hard-core form and impurity-boson mass ratio $m_i/m$ unity is first solved exactly as a special case of a previous Fermi-Bose (FB) mapping treatment of soluble 1D Bose-Fermi mixture problems. Then a more general treatment is given using second quantization for the bosons and the second-quantized form of the FB mapping, eliminating the impurity degrees of freedom by a Lee-Low-Pines canonical transformation. This yields the exact solution for arbitrary $m_i/m$ and impurity-boson interaction strength.
In this paper we elucidate the physics underlying the fact that both bright and dark solitary waves can arise in a one-dimensional spin-polarized gas of fermionic atoms with attractive three-dimensional p-wave interactions in a hard-wall trap. This is possible since the one-dimensional fermion system can be mapped to a system of bosons described by the Lieb-Liniger model with either repulsive or attractive delta-function interactions which can support solitary waves.
The major finding of this paper is that a one-dimensional spin-polarized gas comprised of an even number of fermionic atoms interacting via attractive p-wave interactions and confined to a mesoscopic ring has a degenerate pair of ground states that are oppositely rotating. In any realization the gas will thus spontaneously rotate one way or the other in spite of the fact that there is no external rotation or bias fields. Our goal is to show that this counter-intuitive finding is a natural consequence of the combined effects of quantum statistics, ring topology, and exchange interactions.
A generalized Fermi-Bose mapping method is used to determine the exact ground states of several models of mixtures of strongly interacting ultracold gases in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). We detail the case of a Bose-Fermi mixture with TG boson-boson (BB) and boson-fermion (BF) interactions. Exact results are given for density profiles in a harmonic trap, single-particle density matrices, momentum distributions, and density-density correlations. Since the ground state is highly degenerate, we analyze the splitting of the ground manifold for large but finite BB and BF repulsions.
A generalized Fermi-Bose mapping method is used to determine the exact ground states of six models of strongly interacting ultracold gases of two-level atoms in tight waveguides, which are generalizations of the Tonks-Girardeau (TG) gas (1D Bose gas with point hard cores) and fermionic Tonks-Girardeau (FTG) gas (1D spin-aligned Fermi gas with infinitely strong zero-range attractions). Three of these models exhibit a quantum phase transition in the presence of an external magnetic field, associated with a cooperative ground state rearrangement wherein Fermi energy is traded for internal excitation energy. After investigation of these models in the absence of an electromagnetic field, one is generalized to include resonant interactions with a single photon mode, leading to a possible thermal phase transition associated with Dicke superradiance.
The two and three-body correlation functions of the ground state of an optically trapped ultracold spin-1/2 Fermi gas (SFG) in a tight waveguide (1D regime) are calculated in the plane of even and odd-wave coupling constants, assuming a 1D attractive zero-range odd-wave interaction induced by a 3D p-wave Feshbach resonance, as well as the usual repulsive zero-range even-wave interaction stemming from 3D s-wave scattering. The calculations are based on the exact mapping from the SFG to a ``Lieb-Liniger-Heisenberg'' model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, and indicate that the SFG should be stable against three-body recombination in a large region of the coupling constant plane encompassing parts of both the ferromagnetic and antiferromagnetic phases. However, the limiting case of the fermionic Tonks-Girardeau gas (FTG), a spin-aligned 1D Fermi gas with infinitely attractive p-wave interactions, is unstable in this sense. Effects due to the dipolar interaction and a Zeeman term due to a resonance-generating magnetic field do not lead to shrinkage of the region of stability of the SFG.
The ground and excited states of a one-dimensional (1D) spin-1/2 Fermi gas (SFG) with both attractive zero-range odd-wave interactions and repulsive zero-range even-wave interactions are mapped exactly to a 1D Lieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, such that the complete SFG and LLH energy spectra are identical. The ground state in the ferromagnetic phase is given exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in the antiferromagnetic phase by a variational method combining Bethe ansatz solutions of the LL and 1D Heisenberg models. There are excitation branches corresponding to LL type I and II phonons and spin waves, the latter behaving quadratically for small wave number in the ferromagnetic phase and linearly in the antiferromagnetic phase.
The Fermi-Bose mapping method for one-dimensional (1D) Bose and Fermi gases with zero-range interactions is generalized to an anyon-fermion mapping and applied to exact solution of several models of ultracold gases with anyonic exchange symmetry in tight waveguides: anyonic Calogero-Sutherland model, anyons with point hard core interaction ("anyonic TG gas"), and spin-aligned anyon gas with infinite zero-range odd-wave attractions ("anyonic FTG gas"). It is proved that for even $N\ge 4$ there are states of the anyonic FTG gas on a ring, with anyonic phase slips which are odd integral multiples of $π/(N-1)$, of energy lower than that of the corresponding fermionic ground state. A generalization to a spinor Fermi gas state with anyonic symmetry under purely spatial exchange enables energy lowering by the same mechanism.