Session Y13: Advances and Applications of Numerical Methods in Ultracold Atomic Physics

Numerical Methods for Cold Atoms

I will discuss merits and practical applications of some numerical methods for simulations of equilibrium finite-temperature properties of cold atomic systems. For bosonic systems without frustration, the methods of choice are the Path-Integral Monte Carlo and Stochastic Series Expansion (with worm-type updates). They allow one to produce accurate results that can be compared to experiments `'as is'' not only for standard thermodynamic quantities but also for time-of-flight images and linear response functions. As far as simulations of frustrated systems (spins or bosons with spin-orbit coupling) or interacting fermions at low, but finite, temperature are concerned, I will discuss the status of the Diagrammatic Monte Carlo method and the possibilities it offers for the dual-fermion approach.   

Deterministic alternatives to the full configuration interaction quantum Monte Carlo method for strongly correlated systems

The development of exponential scaling methods has seen great progress in tackling larger systems than previously thought possible. One such technique, full configuration interaction quantum Monte Carlo, allows exact diagonalization through stochastically sampling of determinants. The method derives its utility from the information in the matrix elements of the Hamiltonian, together with a stochastic projected wave function, which are used to explore the important parts of Hilbert space. However, a stochastic representation of the wave function is not required to search Hilbert space efficiently and new deterministic approaches have recently been shown to efficiently find the important parts of determinant space. We shall discuss the technique of Adaptive Sampling Configuration Interaction (ASCI) and the related heat-bath Configuration Interaction approach for ground state and excited state simulations. We will present several applications for strongly correlated Hamiltonians.   

Real-space renormalization group methods and the prospect of observing conformal Calabrese-Cardy scaling

Over the past few decades, real-space renormalization group methods, implementable numerically, have contributed greatly to understanding the phase structure of lattice models in both condensed matter physics and lattice gauge theory. Using two of these methods, the tensor renormalization group and the density matrix renormalization group, we consider the possibility of experimentally observing the conformal Calabrese-Cardy scaling, and measuring the conformal charge in the superfluid phase of the Bose-Hubbard model in one spatial dimension. We suggest using existing experimental methods to measure the quantum purity, however our approach is unique in that the ground state of our proposed experimental set-up is adiabatically prepared at relatively small $J/U$ and at \emph{half-filling}.   

Matrix Product States in the continuum: Integrability and beyond

Building on recent ideas for the formulation of matrix product states in the continuum (cMPS), we have proposed a type of cMPS ansatz that can approximate ground states of interacting spin-1/2 fermions with spin-imbalance in 1D [1]. That system is integrable (Gaudin-Yang) and we showed that cMPS recovers all the expected features of the exact solution and is in good quantitative agreement with it. We now extend that effort to describe a more general system, having both spin- and mass-imbalance. Due to the mass-imbalance, the integrability of the Gaudin-Yang Hamiltonian is lost but we show how the cMPS ansatz continues to perform well in capturing the physics of the system. We also examine the question of different boundary conditions and exploit integrable cases to assess what is the most natural cMPS formulation for studying the thermodynamic limit of this type of problems [2].\\ \vskip 1mm \noindent [1] S. S. Chung and C. J. Bolech, Phys. Rev. B 91, 121108(R) (2015)\\ \noindent [2] Z. Mei and C. J. Bolech, arXiv:1609.08045   

Symmetric minimally entangled typical thermal states, grand-canonical ensembles, and the influence of the collapse bases

Based on DMRG, strongly correlated quantum many-body systems at finite temperatures can be simulated by sampling over a certain class of pure matrix product states (MPS) called minimally entangled typical thermal states (METTS). Here, we show how symmetries of the system can be exploited to considerably reduce computation costs in the METTS algorithm. While this is straightforward for the canonical ensemble, we introduce a modification of the algorithm to efficiently simulate the grand-canonical ensemble under utilization of symmetries. In addition, we construct novel symmetry-conserving collapse bases for the transitions in the Markov chain of METTS that improve the speed of convergence of the algorithm by reducing autocorrelations.   

Studies of Single Component Fermi Gas near a $p$-wave Resonance with LOCV Method

We study a single component Fermi gas near a $p$-wave resonance with the Lowest Order Constrained Variation (LOCV) method. With a many-body trial wave function, we calculate the ground state energy as well as contacts of p-wave resonant Fermi gas. The general behavior of the contacts are in accordance with experimental findings. We also calculate the compressibility of the system and identify a possible region of instability close to the p-wave resonance.   

Stochastic methods for driven-dissipative quantum optics

Driven-dissipative non-linear quantum circuits may display dynamics that is difficult to accurately capture using a master equation approach. To keep pace with rapid experimental progress and increasing complexity in superconducting circuit quantum electrodynamics systems, an effective computational approach is needed that is not limited by Hilbert space truncation. We present a stochastic differential equation method to calculate dynamical observables which is based on an exact phase-space representation of the quantum density matrix. This formalism is an extension of our previous work [1] and we present results for the full quantum dynamics of driven cavity-qubit systems for which the semiclassical dynamics does not settle to a steady-state. In this parameter regime, the quantum system under consideration is most sensitive to noise. This stochastic approach elucidates the role of quantum noise in the system dynamics and allows for an intuitive understanding of how it modifies the dynamics of observables. \newline [1] S. Mandt \emph{et al.}, New J. Phys. \textbf{17}, 053018 (2015)   

Ab initio pairing gap, spectral and response functions of two-dimensional ultracold fermionic gases

We study dynamical correlations of two-dimensional (2D) Fermi atomic gases at zero temperature: pairing gap, spectral function, as well as density and spin dynamical structure factors. Building on recent exact numerical calculations of static ground-state properties of the 2D Fermi gas \footnote{H. Shi, S. Chiesa, and S. Zhang, {\em{Phys. Rev. A}} {\bf{92}}, 033603 (2015)} and methodological developments for the computation of dynamical correlation functions \footnote{E. Vitali, H. Shi, M. Qin, and S. Zhang, {\em{Phys. Rev. B}} {\bf{94}}, 085140 (2016)}, we carry out exact numerical calculations using the auxiliary-field quantum Monte Carlo (AFQMC) method which, for the attractive Fermi gas, is free of the fermion sign problem. The correlation functions are analyzed via state-of-art analytic continuation methodologies in order to infer real time dynamical properties. Dynamical correlation functions provide probes of the excitations and allow potentially direct comparisons with spectroscopy and scattering experiments.   

Designing mixtures of ultracold atoms ​to boost $T_{c}$

​Mixtures of light and heavy atoms, with the light atom a fermion (and the heavy atom having either statistics), have a density wave ordering pattern that is analogous to the antiferromagnetic phase transition in the Hubbard model. ​For a single species mixture, the $T_{c}$ is about half the Hubbard model $T_{c}$ for the same interaction strength. As the degeneracy of the heavy species increases, the $T_{c}$ is boosted by entropic effects that scale approximately linearly in the number of species. Hence, increasing the species to $3$ or $4$ states for the heavy particle will be enough to raise $T_{c}$ above temperatures already reached for the ​three-dimensional Hubbard model on an optical lattice. \footnote{A. Hart, P. M. Duarte,T. L. Yang, X. Liu, T. Paiva, E. Khatami, R. T. Scalettar, N. Trivedi, D .A. Huse & R. G. Hulet, Nature 519, 211–214 (2015)}. We discuss a number of different possible mixtures which might achieve this enhanced $T_{c}$ ​and ​​allow ultracold atoms on optical lattices to study ordered quantum phases in the low-temperature regime.   

Full counting statistics with determinantal quantum Monte Carlo

Within the framework of determinantal quantum Monte Carlo a method is presented for computing the probability distribution of the total particle number and magnetization on a subregion of a system of interacting fermions. Such full counting statistics can be obtained from repeated projective measurements in cold atoms experiments with single-site and single-atom resolution. Applied to the attractive Hubbard model, the full counting statistics reveals the size of a preformed pair or Cooper pair as a function of interaction strength.   

Mott insulating states and quantum phase transitions of correlated SU($2N$) Dirac fermions

We investigate the competing orders in the half-filled SU($2N$) Hubbard model on a honeycomb lattice, which can be accurately realized in optical lattices with ultracold large-spin alkaline-earth fermions. Employing large-scale projector determinant quantum Monte Carlo simulations, we have explored quantum phase transitions from the gapless Dirac semimetals to the gapped Mott insulating phases in the SU(4) and SU(6) cases. Both of these Mott insulating states are found to be columnar valence bond solid (cVBS) and to be absent of the antiferromagnetic Neel ordering and the loop current ordering. Inside the cVBS phases, the dimer ordering is enhanced by increasing fermion components and behaves nonmonotonically as the interaction strength increases. Although the transitions generally should be of first order due to a cubic invariance possessed by the cVBS order, the coupling to gapless Dirac fermions can soften the transitions to second order through a nonanalytic term in the free energy. Our simulations provide guidance for the experimental exploration of new states with alkaline-earth fermions.   

Purifications for canonical ensembles -- matrix product state approximation and entanglement

Matrix product purifications (MPPs) are a very efficient tool for the simulation of strongly correlated quantum many-body systems at finite temperatures. It is straightforward to compute an MPP of a grand-canonical ensemble. In this talk, we present methods for the efficient computation of MPPs of canonical ensembles under utilization of symmetries. Furthermore, we discuss their entanglement properties and a scheme for the efficient evaluation of global quantum number distributions.\\[0.5em] Reference: T.\ Barthel, Phys.\ Rev.\ B \textbf{94}, 115157 (2016).   

The Volkov Basis: Improved Simulation of Time Evolution for Systems in Intense Laser Fields

By employing a basis comprised of time-dependent Volkov states, one is afforded improved accuracy for the simulation of electron dynamics in intense laser fields as compared to propagation using other popular bases such as that of plane waves or the real-space grid representation. As a result, larger time-steps are allowed leading to faster computational run times and the ability to describe longer time-scale phenomena. One-dimensional, one-electron time-dependent simulations of both finite and periodic systems under laser fields using the Volkov basis are presented and compared to existing methods. The basis is also incorporated in three-dimensional time-dependent density functional theory calculations.