Effective renormalized multi-body interactions of harmonically confined ultracold neutral bosons

We calculate the renormalized effective 2-, 3-, and 4-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming 2-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a defined at zero collision energy, which is necessary to obtain both the leading-order effective 4-body interaction and consistently include finite-range corrections for realistic 2-body interactions. The leading-order, effective 3- and 4-body interaction energies are U3 = -(0.85576...)(a/l)^2 + 2.7921(1)(a/l)^3 + O[(a/l)^4] and U4 = +(2.43317...)(a/l)^3 + O[(a\l)^4], where w and l are the harmonic oscillator frequency and length, respectively, and energies are in units of hbar*w. The one-standard deviation error 0.0001 for the third-order coefficient in U3 is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective 3- and 4-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range boson-boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.Comment: Updated introduction, correction of a few typos and sign error

Symmetries of Three Harmonically-Trapped Particles in One Dimension

We present a method for solving trapped few-body problems and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e. the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${\rm C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${\rm C}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.Comment: 27 pages, 5 figures, 6 tables, 37 references, 4 footnotes, 1 article; v2 has revised introduction and results sections as well as typos correcte

Frustrated quantum Ising spins simulated by spinless bosons in a tilted lattice: from a quantum liquid to antiferromagnetic order

We study spinless bosons in a decorated square lattice with a near-diagonal tilt. The resonant subspace of the tilted Mott insulator is described by an effective Hamiltonian of frustrated quantum Ising spins on a non-bipartite lattice. This generalizes an earlier proposal for the unfrustrated quantum Ising model in one dimension which was realized in a recent experiment on ultracold $^{87}$Rb atoms in an optical lattice. Very close to diagonal tilt, we find a quantum liquid state which is continuously connected to the paramagnet. Frustration can be reduced by increasing the tilt angle away from the diagonal, and the system undergoes a transition to an antiferromagnetically ordered state. Using quantum Monte Carlo simulations and exact diagonalization, we find that for realistic system sizes the antiferromagnetic order appears to be quasi-one-dimensional; however, in the thermodynamic limit the order is two-dimensional.Comment: 27 pages, 14 figure

From Rotating Atomic Rings to Quantum Hall States

Considerable efforts are currently devoted to the preparation of ultracold neutral atoms in the emblematic strongly correlated quantum Hall regime. The routes followed so far essentially rely on thermodynamics, i.e. imposing the proper Hamiltonian and cooling the system towards its ground state. In rapidly rotating 2D harmonic traps the role of the transverse magnetic field is played by the angular velocity. For particle numbers significantly larger than unity, the required angular momentum is very large and it can be obtained only for spinning frequencies extremely near to the deconfinement limit; consequently, the required control on experimental parameters turns out to be far too stringent. Here we propose to follow instead a dynamic path starting from the gas confined in a rotating ring. The large moment of inertia of the fluid facilitates the access to states with a large angular momentum, corresponding to a giant vortex. The initial ring-shaped trapping potential is then adiabatically transformed into a harmonic confinement, which brings the interacting atomic gas in the desired quantum Hall regime. We provide clear numerical evidence that for a relatively broad range of initial angular frequencies, the giant vortex state is adiabatically connected to the bosonic $\nu=1/2$ Laughlin state, and we discuss the scaling to many particles.Comment: 9 pages, 5 figure

A primer on quantum fluids

This book introduces the theoretical description and properties of quantum fluids. The focus is on gaseous atomic Bose-Einstein condensates and, to a minor extent, superfluid helium, but the underlying concepts are relevant to other forms of quantum fluids such as polariton and photonic condensates. The book is pitched at the level of advanced undergraduates and early postgraduate students, aiming to provide the reader with the knowledge and skills to develop their own research project on quantum fluids. Indeed, the content for this book grew from introductory notes provided to our own research students. It is assumed that the reader has prior knowledge of undergraduate mathematics and/or physics; otherwise, the concepts are introduced from scratch, often with references for directed further reading.Comment: 132 pages. Published as SpringerBriefs in Physics book. Typos corrected in this versio

Theory of ultracold Fermi gases

The physics of quantum degenerate Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions which play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, of the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to non-perturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare the theoretical predictions with the available experimental results.Comment: Revised and abridged version accepted for publication in Rev. Mod. Phys.: 63 pages, 36 figure

Quantum coherent phenomena in superconducting circuits and ultracold atoms

This thesis consists of theoretical studies of superconducting qubits, and trapped bosons and fermions at ultracold temperature. In superconducting qubits I analyze the resonant properties and decoherence behavior of dc SQUID phase qubits, in which one junction acts as a phase qubit and the rest of the device provides isolation from dissipation and noise in the bias lead. Typically qubit states in phase qubits are detected by tunneling it to the voltage state. I propose an alternate non-destructive readout mechanism which relies on the difference in the magnetic flux through the SQUID loop due to state of the qubit. I also study decoherence effects in a dc SQUID phase qubit caused by the isolation circuit. When the frequency of the qubit is at least two times larger than the resonance frequency of the isolation circuit, I find that the decoherence time of the qubit is two orders of magnitude larger than the typical ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the isolation circuit. This theory is extended to other similar superconducting quantum devices and has been applied to experiments from the group at the University of Maryland. I also demonstrate, theoretically, vacuum Rabi oscillations, analogous to circuit-QED, in superconducting qubits coupled to an environment with resonance. The result obtained gives an exact analytical expression for coherent oscillation of state between the system (the qubit) and the environment with resonance. Next I investigate ultracold atoms in harmonically confined optical lattices. They exhibit a `wedding cake structure' of alternating Mott shells with different number of bosons per site. In regions between the Mott shells, a superfluid phase emerges at low temperatures which at higher temperatures becomes a normal Bose liquid. Using finite-temperature quantum field theoretic techniques, I find analytically the properties of the superfluid, Bose liquid, and Mott insulating regions. This includes the finite temperature order parameter equation for the superfluid phase, excitation spectrum, Berezinskii-Kosterlitz-Thouless transition temperature and vortex-antivortex pair formation (in the two dimensional case), finite temperature compressibility and density - density correlation function. I also study interacting mixtures of ultracold bosonic and fermionic atoms in harmonically confined optical lattices. For a suitable choice of parameters I find emergence of superfluid and Fermi liquid (non-insulating) regions out of Bose-Mott and Fermi-band insulators, due to finite boson and fermion hopping. I also propose a possible experiment for the detection of superfluid and Fermi liquid shells through the use of Gauss-Laguerre and Gaussian beams followed by Bragg spectroscopy. Another area I explore is ultracold heteronuclear molecules such as KRb, RbCs and NaCs. I obtain the finite and zero-temperature phase diagram of bosons interacting via short range repulsive interactions and long-ranged isotropic dipolar interactions in two-dimensions. I build an analytical model for such systems that describes a first order quantum phase transition at zero temperature from a triangular crystalline phase (analogous to Wigner crystal phase of electrons) to superfluid phase. At finite temperature the crystalline phase melts, due to topological defects, to a hexatic phase where translational order is destroyed but hexagonal orientational order is preserved. Further temperature increase leads to the melting of the hexatic phase into a normal dipolar Bose liquid

Symmetries and Correlations in Strongly Interacting One-dimensional Quantum Gases

The main focus of this thesis is the theoretical study of strongly interacting quantum mixtures confined in one dimension and subjected to a harmonic external potential. Such strongly correlated systems can be realized and tested in ultracold atoms experiments. Their non-trivial permutational symmetry properties are investigated, as well as their interplay with correlations. Exploiting an exact solution at strong interactions, we extract general correlation properties encoded in the one-body density matrix and in the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In particular, we obtain substantial results about the short-range behavior, and therefore the high-momentum tails, which display typical $k^{-4}$ laws. The weights of these tails, denoted as Tan's contacts, are related to numerous thermodynamic properties of the systems such as the two-body correlations, the derivative of the energy with respect to the one-dimensional scattering length, or the static structure factor. We show that these universal Tan's contacts also allow to characterize the spatial symmetry of the systems, and therefore is a deep connection between correlations and symmetries. Besides, the exchange symmetry is extracted using a group theory method, namely the class-sum method, which comes originally from nuclear physics. Moreover, we show that these systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing to make our results as experimentally relevant as possible, we derive scaling laws for Tan's contact as a function of the interaction, temperature and transverse confinement. These laws display interesting effects related to strong correlations and dimensionality.Comment: PhD thesis, link to HAL version (original version, higher definition of images, bibtex file to cite this thesis...): https://tel.archives-ouvertes.fr/tel-0191714