Quantum field theoretical description of unstable behavior of a Bose-Einstein condensate with a highly quantized vortex in a harmonic potential

The Bogoliubov-de Gennes equations are used for a number of theoretical works to describe quantum and thermal fluctuations of trapped Bose-Einstein condensates. We consider the case in which the condensate has a highly quantized vortex. It is known that these equations have complex eigenvalues in this case. We give the complete set including a pair of complex modes whose eigenvalues are complex conjugates to each other. The expansion of the quantum fields which represent neutral atoms in terms of the complete set brings the operators associated with the complex modes, which are simply neither bosonic nor fermionic ones. The eigenstate of the Hamiltonian is given. Introducing the notion of the physical states, we discuss the instability of the condensates in the context of Kubo’s linear response theory.     

Quantum field theoretical analysis on unstable behavior of Bose-Einstein condensates in optical lattices

We study the dynamics of Bose-Einstein condensates flowing in optical lattices on the basis of quantum field theory. For such a system, a Bose-Einstein condensate shows an unstable behavior which is called the dynamical instability. The unstable system is characterized by the appearance of modes with complex eigenvalues. Expanding the field operator in terms of excitation modes including complex ones, we attempt to diagonalize the unperturbative Hamiltonian and to find its eigenstates. It turns out that although the unperturbed Hamiltonian is not diagonalizable in the conventional bosonic representation the appropriate choice of physical states leads to a consistent formulation. Then we analyze the dynamics of the system in the regime of the linear response theory. Its numerical results are consistent with those given by the discrete nonlinear Schrödinger equation.     

Analytical study of the splitting process of a multiply-quantized vortex in a Bose–Einstein condensate and collaboration of the zero and complex modes

We study the dynamics of a trapped Bose–Einstein condensate with a multiply-quantized vortex, and investigate the roles of the fluctuations in the dynamical evolution of the system. Using the perturbation theory of the external potential, and assuming the situation of the small coupling constant of self-interaction, we analytically solve the time-dependent Gross–Pitaevskii equation. We introduce the zero mode and its adjoint mode of the Bogoliubov–de Gennes equations. Those modes are known to be essential for the completeness condition. We confirm how the complex eigenvalues induce the vortex splitting. It is shown that the physical role of the adjoint zero mode is to ensure the conservation of the total condensate number. The contribution of the adjoint mode is exponentially enhanced in synchronism with the exponential growth of the complex mode, and is essential in the vortex splitting.     

Coherent states and quantum oscillations of BEC-BCS systems

We show by using the mean-field approximation that the states of composite Fermi-Bose superfluids created in cold-atom traps via a Feshbach resonance at zero temperature are generalized SU(2)⊗SU(1,1) coherent states. In response to a sudden change of the interaction between fermionic atoms and bosonic molecules, a Cooper pair can exhibit collapse and revival quantum behaviors for an initial generalized coherent state of molecules, and Rabi oscillation for a vacuum molecular state. Occurrence of the collapse and revival phenomenon is thus the manifestation of the formation of the Bose-Einstein condensate.     

Complete and orthonormal eigenfunctions of excitations to a one-dimensional dark soliton

We study linear excitations to a one-dimensional dark soliton described by a defocusing nonlinear Schödinger equation. By solving an eigenvalue problem for the excitations we obtain all eigenvalues and eigenfunctions and prove rigorously that these eigenfunctions are orthonormal and form a complete set. We then use the eigenfunctions to obtain the exact form of linear excitations for any given initial condition and to investigate the transverse stability of the dark soliton. The rigorous results reported in the present work can be applied to study the dynamics of dark solitons in various nonlinear optical media and Bose-Einstein condensates.     

Photo-induced dynamics of Bose-Einstein condensates in optical superlattice

The photo-induced dynamics of cold atoms in a one-dimensional optical superlattice is observed. Steady state distribution of the probability amplitudes and the site population in a one-dimensional optical superlattice is found. It is shown that this solution of the equations, which describes the temporal behavior of a Bose-Einstein condensate in a superlattice, is unstable at the sufficiently high level of boson density. The expression for the increment of modulational instability is obtained on the basis of the linear stability analysis. The numerical examples of non-stationary solutions for boson density in a superlattice for the general model are discussed as applied to both the attraction and repulsion potentials of boson interaction.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Influence of nonlinear quantum dissipation on the dynamical properties of the f-deformed Jaynes-Cummings model in the dispersive limit

We investigate the effect of a static electric field on photoionization of the He atom in the ground 1S and low-lying 2S and 2P excited states. The field-affected ionization potential and photoionization cross-section are determined from the complex eigenvalues of the time-dependent Schr?dinger equation solved by the complex rotation method in the Floquet ansatz. Accuracy of the method is enhanced by the use of the Hylleraas basis set. For the ground state of helium, we find that the total photoionization cross-section remains constant or decreases as a function of the DC field strength until this field reaches a certain critical value. For the low-lying excited states, effect of the static field is similar to the ordinary DC Stark effect.     

On quantum operations as quantum states

We formalize Jamiolkowski’s correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in Choi’s proof of the operator sum representation of Completely Positive-preserving linear maps; we go further and show that all of the important theorems concerning quantum operations can be derived directly from those concerning quantum states. As we do so the discussion first provides an elegant and original review of the main features of quantum operations. Next (in the second half of the paper) we find more results stemming from our formulation of the correspondence. Thus, we provide a factorizability condition for quantum operations, and give two novel Schmidt-type decompositions of bipartite pure states. By translating the composition law of quantum operations, we define a group structure upon the set of totally entangled states. The question whether the correspondence is merely mathematical or can be given a physical interpretation is addressed throughout the text: we provide formulae which suggest quantum states inherently define a quantum operation between two of their subsystems, and which turn out to have applications in quantum cryptography.     

Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise

We provide an efficient method for computing the maximum-likelihood mixed quantum state (with density matrix ρ) given a set of measurement outcomes in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d(4)) for the basis change plus O(d(3)) for finding ρ where d is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only O(d(3)) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.     

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A _k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived. Received: 15 May 1996 / Accepted: 25 November 1996     

Optimal approach to rotational state reconstruction of linear molecules

We present a simulation for the reconstruction of the rotational quantum state of linear molecules to retrieve the density matrix. An optimal approach in the sense of minimal error limit is proposed, in which a variable set of angular frequency is properly chosen and the least square inversion is then applied. This approach of reconstruction from time-dependent molecular-axis angular distribution is proved adaptable for various object states, which has a good numerical stability independent of the selected rotational space.     

Local Unitary Invariants for Multipartite States

We study the invariants of arbitrary dimensional multipartite quantum states under local unitary transformations. For multipartite pure states, we give a set of invariants in terms of singular values of coefficient matrices. For multipartite mixed states, we propose a set of invariants in terms of the trace of coefficient matrices. For full rank mixed states with non-degenerate eigenvalues, this set of invariants is also the set of the necessary and sufficient conditions for the local unitary equivalence of such two states.     

Self-consistency and search for collective effects in semiclassical pairing theory

A simple model, in which nuclei are represented as homogeneous spheres of symmetric nuclear matter, is used to study the effects of a self-consistent pairing interaction on the isoscalar nuclear response. Effects due to the finite size of nuclei are suitably taken into account. The semiclassical equations of motion derived in a previous paper for the time-dependent Hartree-Fock-Bogoliubov problem are solved in an improved (linear) approximation in which the pairing field is allowed to oscillate and to become complex. The new solutions are in good agreement with the old ones and also with the result of well-known quantum approaches. The role of the Pauli principle in eliminating one possible set of solutions is also discussed. The density response function is explicitly evaluated and it is shown that the energy-weighted sum rule is restored to its correct value by a part of the fluctuations of the imaginary pairing field. The remaining part of these imaginary fluctuations, together with the fluctuations of the real part, could give rise to collective excitations in the density response function. A detailed analysis of the monopole and quadrupole strength functions shows that there are practically no collective effects in these channels at low excitation energy.     

Dynamical behavior of supersymmetric Paul trap in semiunitary formulation of supersymmetric quantum mechanics

Using the semiunitary formulation of supersymmetric quantum mechanics quantum behavior of supersymmetric Paul trap is investigated, and correspondences of all observables in a complete set and their eigenvalues, wave functions and Schrödinger equations between the system and its superpartner are clarified.     

Control of indirect exciton population in an asymmetric quantum dot molecule

We analyze the problem of coherent population transfer to the indirect exciton state in an asymmetric double semiconductor quantum dot molecule that interacts with an external electromagnetic field. Using the controlled rotation method, we obtain analytical solutions of the time-dependent Schrödinger equation and determine closed-form conditions for the parameters of the applied field and the quantum system that lead to complete population transfer to the indirect exciton state, in the absence of decay effects. Then, by numerical solution of the relevant density matrix equations we study the influence of decay mechanisms to the efficiency of population transfer.     

Network coherence and imperfect superfluidity in the quantum Hall bilayer

Recently, superfluid-like properties have been observed in bilayer quantum Hall systems, in which the effective bosonic particles are understood to be electron-hole pairs. While experimental results are highly suggestive of superfluidity, the linear response of this system remains dissipative down to the lowest available temperatures. We demonstrate that these results may be understood in terms of a unique disorder-dominated state, in which the system organizes into a coherence network, with large incoherent regions separated by quasi-one-dimensional coherent strips with vortices and antivortices at their edges. We demonstrate that this novel state supports nearly dissipationless response at non-vanishing temperatures which can explain a number of puzzling experimental results.     

Antibunching Effect of Eigenstates of the Operator bQ－K in a Q-Deformed Non-harmonic Oscillator

We introduce a quantum antibunching effect. The quantum antibunching effect of the K eigenstates of the K th powere (K≥3) of the annihilation operator in the Q-deformed non-harmonic oscillator is investigated. The physical meaning of the K states are explored. The results show that there is the quantum antibunching effect in all of those states. All of them can be generated by a linear superposition of generalized coherent states produced by the time-dependent Q-deformation non-harmonic oscillator at different instants.     

Approximate solution of bound state problems through continued fractions

A method to solve ordinary linear differential equations through continued fractions is applied to several physical systems. In particular, results for the Schrödinger equation give a good accuracy for the eigenvalues of bound states in theS-wave Yukawa potential, and the lowest order approximations provide exact-values for the harmonic oscillator and Coulomb potential eigenvalues and eigenfunctions.