arXiv subjects

Lincoln D. Carr

Publications and source records attributed to Lincoln D. Carr.

At least 55 records · Page 3Linked to original sources

arXiv2018

Open source Matrix Product States: Exact diagonalization and other entanglement-accurate methods revisited in quantum systems

Tensor network methods as presented in our open source Matrix Product States software have opened up the possibility to study many-body quantum physics in one and quasi-one-dimensional systems in an easily accessible package similar to density functional theory codes but for strongly correlated dynamics. Here, we address methods which allow one to capture the full entanglement without truncation of the Hilbert space. Such methods are suitable for validation of and comparisons to tensor network algorithms, but especially useful in the case of new kinds of quantum states with high entanglement violating the truncation in tensor networks. Quantum cellular automata are one example for such a system, characterized by tunable complexity, entanglement, and a large spread over the Hilbert space. Beyond the evolution of pure states as a closed system, we adapt the techniques for open quantum systems simulated via the Lindblad master equation. We present three algorithms for solving closed-system many-body time evolution without truncation of the Hilbert space. Exact diagonalization methods have the advantage that they not only keep the full entanglement but also require no approximations to the propagator. Seeking the limits of a maximal number of qubits on a single core, we use Trotter decompositions or Krylov approximation to the exponential of the Hamiltonian. All three methods are also implemented for open systems represented via the Lindblad master equation built from local channels. We show their convergence parameters and focus on efficient schemes for their implementations including Abelian symmetries, e.g., U(1) symmetry used for number conservation in the Bose-Hubbard model or discrete Z2 symmetries in the quantum Ising model. We present the thermalization timescale in the long-range quantum Ising model as a key example of how exact diagonalization contributes to novel physics.

cond-mat.quant-gas↗

arXiv2018

Correlation Between Student Collaboration Network Centrality and Academic Performance

We compute nodal centrality measures on the collaboration networks of students enrolled in three upper-division physics courses, usually taken sequentially, at the Colorado School of Mines. These are complex networks in which links between students indicate assistance with homework. The courses included in the study are intermediate Classical Mechanics, introductory Quantum Mechanics, and intermediate Electromagnetism. By correlating these nodal centrality measures with students' scores on homework and exams, we find four centrality measures that correlate significantly with students' homework scores in all three courses: in-strength, out-strength, closeness centrality, and harmonic centrality. These correlations suggest that students who not only collaborate often, but also collaborate significantly with many different people tend to achieve higher grades. Centrality measures between simultaneous collaboration networks (analytical vs. numerical homework collaboration) composed of the same students also correlate with each other, suggesting that students' collaboration strategies remain relatively stable when presented with homework assignments targeting different skills. Additionally, we correlate centrality measures between collaboration networks from different courses and find that the four centrality measures with the strongest relationship to students' homework scores are also the most stable measures across networks involving different courses. Correlations of centrality measures with exam scores were generally smaller than the correlations with homework scores, though this finding varied across courses.

physics.ed-ph↗

arXiv2018

Internal Oscillations of a Dark-Bright Soliton in a Harmonic Potential

We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multicomponent Bose-Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas-Fermi profile. The wavefunction ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with psuedo-spectral Runge-Kutta methods. We find, analytically and numerically, for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.

cond-mat.quant-gas↗

arXiv2018

For high-precision bosonic Josephson junctions, many-body effects matter

Typical treatments of superconducting or superfluid Josephson junctions rely on mean-field or two-mode models; we explore many-body dynamics of an isolated, ultracold, Bose-gas long Josephson junction using time-evolving block decimation simulations. We demonstrate that with increasing repulsive interaction strength, localized dynamics emerge that influence macroscopic condensate behavior and can lead to formation of solitons that directly oppose the symmetry of the junction. Initial state population and phase yield insight into dynamic tunneling regimes of a quasi one-dimensional double well potential, from Josephson oscillations to macroscopic self-trapping. Population imbalance simulations reveal substantial deviation of many-body dynamics from mean-field Gross-Pitaevskii predictions, particularly as the barrier height and interaction strength increase. In addition, the sudden approximation supports localized particle-hole formation after a diabatic quench, and correlation measures unveil a new dynamic regime: the Fock flashlight.

cond-mat.quant-gas↗

arXiv2018

Quantum phase transition modulation in an atomtronic Mott switch

Mott insulators provide stable quantum states and long coherence times to due to small number fluctuations, making them good candidates for quantum memory and atomic circuits. We propose a proof-of-principle for a 1D Mott switch using an ultracold Bose gas and optical lattice. With time-evolving block decimation simulations -- efficient matrix product state methods -- we design a means for transient parameter characterization via a local excitation for ease of engineering into more complex atomtronics. We perform the switch operation by tuning the intensity of the optical lattice, and thus the interaction strength through a conductance transition due to the confined modifications of the "wedding cake" Mott structure. We demonstrate the time-dependence of Fock state transmission and fidelity of the excitation as a means of tuning up the device in a double well and as a measure of noise performance. Two-point correlations via the $g^{(2)}$ measure provide additional information regarding superfluid fragments on the Mott insulating background due to the confinement of the potential.

cond-mat.quant-gas↗

arXiv2018

Macroscopic quantum escape of Bose-Einstein condensates: Analysis of experimentally realizable quasi-one-dimensional traps

The variational-JWKB method is used to determine experimentally accessible macroscopic quantum tunneling regimes of quasi-bound Bose-Einstein condensates in two quasi one-dimensional trap configurations. The potentials can be created by magnetic and optical traps, a symmetric trap from two offset Gaussian barriers and a tilt trap from a linear gradient and Gaussian barrier. Scaling laws in barrier parameters, ranging from inverse polynomial to square root times exponential, are calculated and used to elucidate different dynamical regimes, such as when classical oscillations dominate tunneling rates in the symmetric trap. The symmetric trap is found to be versatile, with tunneling times at and below one second, able to hold $10^{3}$ to $10^{4}$ atoms, and realizable for atoms ranging from rubidium to lithium, with unadjusted scattering lengths. The tilt trap produces sub-second tunneling times, is able to hold a few hundred atoms of lighter elements like lithium, and requiring the use of Feshbach resonance to reduce scattering lengths. To explore a large parameter space, an extended Gaussian variational ansatz is used, which can approximate large traps with Thomas-Fermi profiles. Nonlinear interactions in the Gross-Pitaevskii equation are shown to produce additional effective mean-field barriers, affecting scaling laws.

cond-mat.quant-gas↗

arXiv2018

A complex network description of thermal quantum states in the Ising spin chain

We use network analysis to describe and characterize an archetypal quantum system - an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Renyi mutual information, concurrence, and negativity. We analytically calculate the spin-spin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases.

cond-mat.stat-mech↗

arXiv2018

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

Ultra-cold quantum turbulence is expected to decay through a cascade of Kelvin waves. These helical excitations couple vorticity to the quantum fluid causing long wavelength phonon fluctuations in a Bose-Einstein condensate. This interaction is hypothesized to be the route to relaxation for turbulent tangles in quantum hydrodynamics. The local induction approximation is the lowest order approximation to the Biot-Savart velocity field induced by a vortex line and, because of its integrability, is thought to prohibit energy transfer by Kelvin waves. Using the Biot-Savart description, we derive a generalization to the local induction approximation which predicts that regions of large curvature can reconfigure themselves as Kelvin wave packets. While this generalization preserves the arclength metric, a quantity conserved under the Eulerian flow of vortex lines, it also introduces a non-Hamiltonian structure on the geometric properties of the vortex line. It is this non-Hamiltonian evolution of curvature and torsion which provides a resolution to the missing Kelvin wave motion. In this work, we derive corrections to the local induction approximation in powers of curvature and state them for utilization in vortex filament methods. Using the Hasimoto transformation, we arrive at a nonlinear integro-differential equation which reduces to a modified nonlinear Schrödinger type evolution of the curvature and torsion on the vortex line. We show that this modification seeks to disperse localized curvature profiles. At the same time, the non-Hamiltonian break in integrability bolsters the deforming curvature profile and simulations show that this dynamic results in Kelvin wave propagation along the dispersive vortex medium.

cond-mat.quant-gas↗

arXiv2018

Exact results for a fractional derivative of elementary functions

We present exact analytical results for the Caputo fractional derivative of a wide class of elementary functions, including trigonometric and inverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian, quartic Gaussian, and Lorentzian functions. These results are especially important for multi-scale physical systems, such as porous materials, disordered media, and turbulent fluids, in which transport is described by fractional partial differential equations. The exact results for the Caputo fractional derivative are obtained from a single generalized Euler's integral transform of the generalized hyper-geometric function with a power-law argument. We present a proof of the generalized Euler's integral transform and directly apply it to the exact evaluation of the Caputo fractional derivative of a broad spectrum of functions, provided that these functions can be expressed in terms of a generalized hyper-geometric function with a power-law argument. We determine that the Caputo fractional derivative of elementary functions is given by the generalized hyper-geometric function. Moreover, we show that in the most general case the final result cannot be reduced to elementary functions, in contrast to both the Liouville-Caputo and Fourier fractional derivatives. However, we establish that in the infinite limit of the argument of elementary functions, all three definitions of a fractional derivative - the Caputo, Liouville-Caputo, and Fourier- converge to the same result given by the elementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourier fractional derivatives.

math-ph↗

arXiv2018

Expansion of fractional derivatives in terms of an integer derivative series: physical and numerical applications

We use the displacement operator to derive an infinite series of integer order derivatives for the Grünwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We demonstrate that all three definitions of a fractional derivative lead to the same infinite series of integer order derivatives. We find that functions normally represented by Taylor series with a finite radius of convergence have a corresponding integer derivative expansion with an infinite radius of convergence. Specifically, we demonstrate robust convergence of the integer derivative series for the hyperbolic secant (tangent) function, characterized by a finite radius of convergence of the Taylor series $R=π/2$, which describes bright (dark) soliton propagation in non-linear media. We also show that for a plane wave, which has a Taylor series with an infinite radius of convergence, as the number of terms in the integer derivative expansion increases, the truncation error decreases. Finally, we illustrate the utility of the truncated integer derivative series by solving two linear fractional differential equations, where the fractional derivative is replaced by an integer derivative series up to the second order derivative. We find that our numerical results closely approximate the exact solutions given by the Mittag-Leffler and Fox-Wright functions. Thus, we demonstrate that the truncated expansion is a powerful method for solving linear fractional differential equations, such as the fractional Schrödinger equation.

math.NA↗

arXiv2017

Non-Hamiltonian Dynamics of Quantized Vortices in Bose-Einstein Condensates

The dynamics of quantized vortices in weakly interacting superfluids are often modeled by a nonlinear Schrödinger equation. In contrast, we show that quantized vortices in fact obey a non-Hamiltonian evolution equation, which enhances dispersion along the vortex line while introducing a gain mechanism. This allows the vortex medium to support a helical shock front propagating ahead of a dissipative soliton. This dynamic relaxes localized curvature events into Kelvin wave packets. Consequently, a beyond local induction model provides a pathway for decay in low-temperature quantum turbulence.

cond-mat.quant-gas↗

arXiv2017

Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks

We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of EEG/fMRI measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, $Z_2$, mean field superfluid/Mott insulator, and a BKT crossover.

cond-mat.stat-mech↗

arXiv2017

Open source Matrix Product States: Opening ways to simulate entangled many-body quantum systems in one dimension

Numerical simulations are a powerful tool to study quantum systems beyond exactly solvable systems lacking an analytic expression. For one-dimensional entangled quantum systems, tensor network methods, amongst them Matrix Product States (MPSs), have attracted interest from different fields of quantum physics ranging from solid state systems to quantum simulators and quantum computing. Our open source MPS code provides the community with a toolset to analyze the statics and dynamics of one-dimensional quantum systems. Here, we present our open source library, Open Source Matrix Product States (OSMPS), of MPS methods implemented in Python and Fortran2003. The library includes tools for ground state calculation and excited states via the variational ansatz. We also support ground states for infinite systems with translational invariance. Dynamics are simulated with different algorithms, including three algorithms with support for long-range interactions. Convenient features include built-in support for fermionic systems and number conservation with rotational $\mathcal{U}(1)$ and discrete $\mathbb{Z}_2$ symmetries for finite systems, as well as data parallelism with MPI. We explain the principles and techniques used in this library along with examples of how to efficiently use the general interfaces to analyze the Ising and Bose-Hubbard models. This description includes the preparation of simulations as well as dispatching and post-processing of them.

cond-mat.quant-gas↗

arXiv2017

Critical Phenomena and Kibble-Zurek Scaling in the Long-Range Quantum Ising Chain

We investigate an extension of the quantum Ising model in one spatial dimension including long-range $1 / r^α$ interactions in its statics and dynamics with possible applications from heteronuclear polar molecules in optical lattices to trapped ions described by two-state spin systems. We introduce the statics of the system via both numerical techniques with finite size and infinite size matrix product states and a theoretical approaches using a truncated Jordan-Wigner transformation for the ferromagnetic and antiferromagnetic case and show that finite size effects have a crucial role shifting the quantum critical point of the external field by fifteen percent between thirty-two and around five-hundred spins. We numerically study the Kibble-Zurek hypothesis in the long-range quantum Ising model with Matrix Product States. A linear quench of the external field through the quantum critical point yields a power-law scaling of the defect density as a function of the total quench time. For example, the increase of the defect density is slower for longer-range models and the critical exponent changes by twenty-five per cent. Our study emphasizes the importance of such long-range interactions in statics and dynamics that could point to similar phenomena in a different setup of dynamical systems or for other models.

cond-mat.quant-gas↗

arXiv2017

Macroscopic Quantum Tunneling Escape of Bose-Einstein Condensates

Recent experiments on macroscopic quantum tunneling reveal a non-exponential decay of the number of atoms trapped in a quasibound state behind a potential barrier. Through both experiment and theory, we demonstrate this non-exponential decay results from interactions between atoms. Quantum tunneling of tens of thousands of 87 Rb atoms in a Bose-Einstein condensate is modeled by a modified Jeffreys-Wentzel-Kramers-Brillouin model, taking into account the effective time-dependent barrier induced by the mean-field. Three-dimensional Gross-Pitaevskii simulations corroborate a mean-field result when compared with experiments. However, with one-dimensional modeling using time-evolving block decimation, we present an effective renormalized mean-field theory that suggests many-body dynamics for which a bare mean-field theory may not apply.

cond-mat.quant-gas↗

arXiv2017

Absence of Landau damping in driven three-component Bose-Einstein condensate in optical lattices

We explore the quantum many-body physics of a three-component Bose-Einstein condensate (BEC) in an optical lattices driven by laser fields in $V$ and $Λ$ configurations. We obtain exact analytical expressions for the energy spectrum and amplitudes of elementary excitations, and discover symmetries among them. We demonstrate that the applied laser fields induce a gap in the otherwise gapless Bogoliubov spectrum. We find that Landau damping of the collective modes above the energy of the gap is carried by laser-induced roton modes and is considerably suppressed compared to the phonon-mediated damping endemic to undriven scalar BECs.

cond-mat.quant-gas↗

arXiv2017

Engineering and Manipulating Exciton Wave Packets

When a semiconductor absorbs light, the resulting electron-hole superposition amounts to a uncontrolled quantum ripple that eventually degenerates into diffusion. If the conformation of these excitonic superpositions could be engineered, though, they would constitute a new means of transporting information and energy. We show that properly designed laser pulses can be used to create such excitonic wave packets. They can be formed with a prescribed speed, direction and spectral make-up that allows them to be selectively passed, rejected or even dissociated using superlattices. Their coherence also provides a handle for manipulation using active, external controls. Energy and information can be conveniently processed and subsequently removed at a distant site by reversing the original procedure to produce a stimulated emission. The ability to create, manage and remove structured excitons comprises the foundation for opto-excitonic circuits with application to a wide range of quantum information, energy and light-flow technologies. The paradigm is demonstrated using both Tight-Binding and Time-Domain Density Functional Theory simulations.

cond-mat.mes-hall↗

arXiv2017

Damping-free collective oscillations of a driven two-component Bose gas in optical lattices

We explore quantum many-body physics of a driven Bose-Einstein condensate in optical lattices. The laser field induces a gap in the generalized Bogoliubov spectrum proportional to the effective Rabi frequency. The lowest lying modes in a driven condensate are characterized by zero group velocity and non-zero current. Thus, the laser field induces roton modes, which carry interaction in a driven condensate. We show that collective excitations below the energy of the laser-induced gap remain undamped, while above the gap they are characterized by a significantly suppressed Landau damping rate.

cond-mat.quant-gas↗