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Theory of Two Distinguishable Atoms in an Optical Lattice

Otim Odong

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

We study the Bose-Hubbard model of two atoms in a one-dimensional optical lattice, with onsite atom-atom interaction and periodic boundary conditions. Though this model has been solved analytically, the majority of the analytical solutions do not seem to include stationary states. These are needed to analyze the spectroscopy of a dimer of two atoms in an optical lattice. We develop a unique analytical method and remedy this problem in the case of two identical bosons in a fixed one-dimensional optical lattice. Our results compare well with the existing experimental results. Then, we extend this method to two distinguishable atoms and find that, among other things, a dimer of two distinguishable atoms can be formed in an optical lattice. We highlight the differences in these results from those in the case of two identical bosons. These include stationary states that do not depend on the atom-atom interactions and the ability of the dimer to dissociate into a continuum of states with odd symmetry. Lastly, we investigate the two atoms in the lattice when the lattice is allowed to rotate. The main outcome of this is that the atoms in the lattice can be controlled through adjusting the rotation speed of the lattice. ^

Related papers

Heterodimer of two distinguishable atoms in a one-dimensional optical lattice

Otim Odong, Jerome Sanders, Juha Javanainen

Within the Bose-Hubbard model, we theoretically determine the stationary states of two distinguishable atoms in a one-dimensional optical lattice and compare with the case of two identical bosons. A heterodimer has odd-parity dissociated states that do not depend on the interactions between the atoms, and the lattice momenta of the two atomic species may have different averages even for a bound state of the dimer. We discuss methods to detect the dimer. The different distributions of the quasimomenta of the two species may be observed in suitable time-of-flight experiments. Also, an asymmetry in the lineshape as a function of the modulation frequency may reveal the presence of the odd-parity dissociated states when a heterodimer is dissociated by modulating the depth of the optical lattice.

arXiv:1105.6324•Physical Review A•Sep 13, 2011•1 citation

Dimer of Two Bosons in a One-dimensional Optical Lattice

Jerome Sanders, Otim Odong, Juha Javanainen

We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.

Bulletin of the American Physical Society•May 28, 2010•5 citations

Dimer of two bosons in a one-dimensional optical lattice

Juha Javanainen, Otim Odong, Jerome Sanders

We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.

arXiv:1004.5118•Physical Review A•Apr 13, 2010•35 citations

Two-Channel Theory for Two Bosons in an Optical Lattice

Jerome Sanders

We theoretically and numerically study the Bose-Hubbard model for two bosons in a one-dimensional optical lattice. We begin with a single-channel model describing two interacting atoms, known as a lattice dimer, and then examine the more nuanced two-channel model. The two-channel model accounts for the presence of an association resonance that facilitates the conversion of an atom pair into a molecule and adds a separate molecular degree of freedom. Lattice dimers in the single-channel case have been extensively investigated, but we resolve the conundrum between these dimers and molecules by employing a two-channel model. The stationary states for the time-independent Schrödinger equation are determined and these states reveal some unusual features, including the existence of the celebrated repulsively bound state. We examine the similarities and differences between the single- and two-channel models and find a peculiar qualitative difference between the two. Whereas the usual atoms-only theory yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Possible methods for experimentally detecting both the dimers and the simultaneous bound states are discussed and we identify several systems for observing this novel aspect of molecular physics. In addition to our theoretical work, we also present the results and verification of our quantum Monte Carlo simulations of the two-channel model. ^

•Jan 1, 2011•0 citations

Bound states of two bosons in an optical lattice near an association resonance

Jerome Sanders, Otim Odong, Juha Javanainen +1 more

We model two bosons in an optical lattice near a Feshbach or photoassociation resonance, focusing on the Bose-Hubbard model in one dimension. Whereas the usual atoms-only theory with a tunable scattering length yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, for a sufficiently small direct background interaction between the atoms a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Such unusual molecular physics may be observable for an atom-molecule coupling strength comparable to the width of the dissociation continuum of the lattice dimer, which is the case, for instance, with narrow Feshbach resonances in Na, $^{87}\mathrm{Rb}$, and $^{133}\mathrm{Cs}$ or low-intensity photoassociation in $^{174}\mathrm{Yb}$.

arXiv:1011.0599•Physical Review A•Mar 21, 2011•13 citations

EXOTIC FEW-BODY BOUND STATES IN A LATTICE

D. PETROSYAN, M. VALIENTE

Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above the two-particle scattering continuum, respectively. Adding then the third particle -- "monomer" -- we find that, apart from the simple strongly-bound "trimer" corresponding to all three particles occupying the same lattice site, there are two peculiar families of weakly-bound trimers with energies below and above the monomer-dimer scattering continuum, the corresponding binding mechanism being an effective particle exchange interaction.

arXiv:1002.2778•Jun 1, 2010•2 citations
We study the Bose-Hubbard model of two atoms in a one-dimensional optical lattice, with onsite atom-atom interaction and periodic boundary conditions. Though this model has been solved analytically, the majority of the analytical solutions do not seem to include stationary states. These are needed to analyze the spectroscopy of a dimer of two atoms in an optical lattice. We develop a unique analytical method and remedy this problem in the case of two identical bosons in a fixed one-dimensional optical lattice. Our results compare well with the existing experimental results. Then, we extend this method to two distinguishable atoms and find that, among other things, a dimer of two distinguishable atoms can be formed in an optical lattice. We highlight the differences in these results from those in the case of two identical bosons. These include stationary states that do not depend on the atom-atom interactions and the ability of the dimer to dissociate into a continuum of states with odd symmetry. Lastly, we investigate the two atoms in the lattice when the lattice is allowed to rotate. The main outcome of this is that the atoms in the lattice can be controlled through adjusting the rotation speed of the lattice. ^
Within the Bose-Hubbard model, we theoretically determine the stationary states of two distinguishable atoms in a one-dimensional optical lattice and compare with the case of two identical bosons. A heterodimer has odd-parity dissociated states that do not depend on the interactions between the atoms, and the lattice momenta of the two atomic species may have different averages even for a bound state of the dimer. We discuss methods to detect the dimer. The different distributions of the quasimomenta of the two species may be observed in suitable time-of-flight experiments. Also, an asymmetry in the lineshape as a function of the modulation frequency may reveal the presence of the odd-parity dissociated states when a heterodimer is dissociated by modulating the depth of the optical lattice.
We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.
We investigate theoretically the stationary states of two bosons in a one-dimensional optical lattice within the Bose-Hubbard model. Starting from a finite lattice with periodic boundary conditions, we effect a partial separation of the center-of-mass and relative motions of the two-atom lattice dimer in the lattice momentum representation, and carefully analyze the eigenstates of the relative motion. In the limit when the lattice becomes infinitely long, we find closed-form analytic expressions for both the bound state and the dissociated states of the lattice dimer. We outline the corresponding analysis in the position representation. The results are used to discuss three ways to detect the dimer: by measuring the momentum distribution of the atoms, by finding the size of the molecule with measurements of atom number correlations at two lattice sites, and by dissociating a bound state of the lattice dimer with a modulation of the lattice depth.
We theoretically and numerically study the Bose-Hubbard model for two bosons in a one-dimensional optical lattice. We begin with a single-channel model describing two interacting atoms, known as a lattice dimer, and then examine the more nuanced two-channel model. The two-channel model accounts for the presence of an association resonance that facilitates the conversion of an atom pair into a molecule and adds a separate molecular degree of freedom. Lattice dimers in the single-channel case have been extensively investigated, but we resolve the conundrum between these dimers and molecules by employing a two-channel model. The stationary states for the time-independent Schrödinger equation are determined and these states reveal some unusual features, including the existence of the celebrated repulsively bound state. We examine the similarities and differences between the single- and two-channel models and find a peculiar qualitative difference between the two. Whereas the usual atoms-only theory yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Possible methods for experimentally detecting both the dimers and the simultaneous bound states are discussed and we identify several systems for observing this novel aspect of molecular physics. In addition to our theoretical work, we also present the results and verification of our quantum Monte Carlo simulations of the two-channel model. ^
We model two bosons in an optical lattice near a Feshbach or photoassociation resonance, focusing on the Bose-Hubbard model in one dimension. Whereas the usual atoms-only theory with a tunable scattering length yields one bound state for a molecular dimer for either an attractive or repulsive atom-atom interaction, for a sufficiently small direct background interaction between the atoms a two-channel atom-molecule theory may give two bound states that represent attractively and repulsively bound dimers occurring simultaneously. Such unusual molecular physics may be observable for an atom-molecule coupling strength comparable to the width of the dissociation continuum of the lattice dimer, which is the case, for instance, with narrow Feshbach resonances in Na, 87Rb^{87}\mathrm{Rb}87Rb, and 133Cs^{133}\mathrm{Cs}133Cs or low-intensity photoassociation in 174Yb^{174}\mathrm{Yb}174Yb.
Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above the two-particle scattering continuum, respectively. Adding then the third particle -- "monomer" -- we find that, apart from the simple strongly-bound "trimer" corresponding to all three particles occupying the same lattice site, there are two peculiar families of weakly-bound trimers with energies below and above the monomer-dimer scattering continuum, the corresponding binding mechanism being an effective particle exchange interaction.