A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at
T=0. Taking into account the density fluctuations at the outset we get the effective action where the density $\ensuremath{\rho}$ is the conjugated momentum to the phase $\ensuremath{\theta}$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field $\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.$ The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.
Although quantum mechanics is linear, there are nevertheless quantum systems with multiple interacting particles in which the effective evolution of a single particle is governed by a nonlinear equation. This includes Bose-Einstein condensates, which are governed by the Gross-Pitaevskii equation, which is a cubic nonlinear Schrodinger equation with a term proportional to /[psi]/ ²[psi] scaling of Grover's algorithm. Since the Gross- Pitaevskii equation effectively approximates the multi- particle Schrodinger equation, for which Grover's algorithm is optimal, our result leads to a quantum information-theoretic bound on the number of particles needed for this approximation to hold, asymptotically. The Gross-Pitaevskii equation is not the only nonlinearity of the form f(/[psi]/²)[psi] that arises in effective equations for the evolution of real quantum physical systems, however : The cubic-quintic nonlinear Schrodinger equation describes light propagation in nonlinear Kerr media with defocusing corrections, and the logarithmic nonlinear Schrodinger equation describes Bose liquids under certain conditions. Analysis of computation with such systems yields some surprising results; for example, when time-measurement precision is included in the resource accounting, searching a database'' when there is a single correct answer may be easier than searching when there are multiple correct answers. These results lead to quantum information-theoretic bounds on the physical resources required for these effective nonlinear theories to hold, asymptotically. Furthermore, strongly regular graphs, which have no global symmetry, are sufficiently complete for quantum search on them to asymptotically behave like unstructured search. Certain sufficiently complete graphs retain the improved runtime and resource scalings for some nonlinearities, so our scheme for nonlinear, analog quantum computation retains its benefits even when some structure is introduced
Consider a system of
N bosons in three dimensions interacting via a repulsive short range pair potential
N2V(N(xi−xj)), where $\bx=(x_1, >..., x_N)$ denotes the positions of the particles. Let
HN denote the Hamiltonian of the system and let
ψN,t be the solution to the Schrodinger equation. Suppose that the initial data
ψN,0 satisfies the energy condition
≤CN>. and that the one-particle density matrix converges to a projection as
N→∞. Then, we prove that the
k-particle density matrices of
ψN,t factorize in the limit
N→∞. Moreover, the one particle orbital wave function solves the time-dependent Gross-Pitaevskii equation, a cubic non-linear Schrodinger equation with the coupling constant proportional to the scattering length of the potential
V. In \cite{ESY}, we proved the same statement under the condition that the interaction potential
V is sufficiently small; in the present work we extend this result to large potentials.
Dynamics of the repulsive Bose–Einstein condensate (BEC) in a double-well trap is explored within the 3D time-dependent Gross–Pitaevskii equation. The model avoids numerous common approximations (two-mode treatment, time-space factorization, fixed values of the chemical potential and barrier penetrability, etc) and thus may provide a realistic description of BEC dynamics, including both weak-coupling (sub-barrier) and strong-coupling (above-barrier) regimes and their crossover. The strong coupling regime is achieved by increasing the number N of BEC atoms and thus the chemical potential. The evolution with N of Josephson oscillations (JO) and macroscopic quantum self-trapping (MQST) is examined and the crucial impact of the BEC interaction is demonstrated. At weak coupling, the calculations well reproduce the JO/MQST experimental data. At strong coupling with a significant overlap of the left and right BECs, we observe a remarkable persistence of the Josephson-like dynamics: the JO and MQST converge to a high-frequency JO-like mode where both population imbalance and phase difference oscillate around the zero averages.
We study the ensemble of pseudo-spin 1/2 ultracold bosons, performing Lévy flights, confined in a parabolic potential. The (pseudo-) spin-orbit coupling (SOC) is additionally imposed on these particles. We consider the structure and dynamics of macroscopic pseudospin qubits based on Bose-Einstein condensates, obtained from the above "fractional" bosons. Under "fractional" we understand the substitution of the ordinary second derivative (kinetic energy term) in the Gross-Pitaevskii equation by a so-called fractional Laplacian, characterized by the Lévy index μ. We show that the joint action of interparticle interaction, SOC, and Zeeman splitting in a synthetic magnetic field makes the dynamics of corresponding qubit highly nontrivial with evident chaotic features at both strong interactions and Lévy indices μ→1 when the Lévy trajectories of bosons with long jumps dominated over those derived from ordinary Gaussian distribution, corresponding to μ=2. Using analytical and numerical arguments, we discuss the possibilities to control the above qubit using the synergy of SOC, interaction strength, and "fractionality," characterized by the Lévy index μ.