Quadratic time dependent Hamiltonians and separation of variables

Anzaldo-Meneses, A.

doi:10.1016/j.aop.2017.04.002

Cited by 3 publications

(5 citation statements)

References 46 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where Q(t) is a time-dependent linear combination of the generators analogous to H and Q is a (m + n ) × (m + n ) matrix. Having a solution M(t), the transformation U can be obtained, as explained in [3], using standard techniques of matrix algebra, obtaining first Q and from this Q. Taking advantage of the fact that the matrices M are block diagonal, they can be written as M = Diag(M o , M e ) with M o = exp( Qo /i ) and M e = exp( Qe /i ), and Q = Diag( Qo , Qe ).…”

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

“…Alternatively, the corresponding orthogonal, respectively symplectic, algebra can be decomposed into subalgebras, as mentioned above, as g l = x l + y l + z l , l = e, o, with g o = so(m ) and g e = sp(n ), and the subalgebras can be associated with the decomposition M = LDU , where L is a lower-diagonal, D is a diagonal and U is an upper-diagonal matrix. This last observation leads to several equivalent BCH-like representations of U , which are tantamount to a separation of variables method rooted in Inönü-Wigner contractions, as explained in detail in [3]. For periodic Hamiltonians with H(t) = H(t + T ), the binary form Q(T ) gives an equivalent Floquet Hamiltonian.…”

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

“…where the matrix elements are functions of time determined by first-order linear differential equations (3). Correspondingly decompose sp(2) in y = {p 2 /2}, x = {x 2 /2},and z 0 = {(px + xp)/2} and decompose the unitary transformation as U = U (3) U (2) U (1) , with U (1) = exp( Q1 x 2 /2i ), U (2) = exp( Q2 (px + xp)/2i ), U (3) = exp( Q3 p 2 /2i ). The exponentiation of the decomposition matrices of M e leads to Q3 = −c/a, Q2 = log(a), Q1 = b/a.…”

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

“…Some time later, for physical applications in nuclear and in condensed matter physics, Moshinsky et al and Manko and coworkers [6] have done similar calculations for the finite-dimensional cases with commuting variables. Some of these results were extended in [3], for commuting variables, to problems with time-dependent coefficients and the time evolution operators were additionally obtained. In this work, time-dependent quadratic Hamiltonians in both variables are deduced using the mentioned standard approach of supersymmetric quantum mechanics and a method is given to obtain their time evolution operators, giving therefore a unified presentation for quadratic finite-dimensional problems and extending many results to time-dependent problems.…”

mentioning

confidence: 99%

“…In a recent work [3] the time evolution operator has been explicitely obtained for quadratic time-dependent Hamiltonians in commuting variables. The method is rooted in the Heisenberg picture and the equations of motion, which are essentially given by the adjoint action of the underlying symplectic Lie algebra.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Supercanonical transformations and time-dependent Hamiltonians

Anzaldo-Meneses

2021

EPL

Self Cite

View full text Add to dashboard Cite

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp . The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

show abstract

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

Section: Orthosymplectic Canonical Transformations -mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Supercanonical transformations and time-dependent Hamiltonians

Anzaldo-Meneses

2021

EPL

Self Cite

View full text Add to dashboard Cite

show abstract

Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fields

Büyükaşık

Çayiç

2022

Journal of Mathematical Physics

View full text Add to dashboard Cite

The time-dependent Schrödinger equation describing a generalized two-dimensional quantum parametric oscillator in the presence of time-variable external fields is solved using the evolution operator method. For this, the evolution operator is found as a product of exponential operators through the Wei–Norman Lie algebraic approach. Then, the propagator and time-evolution of eigenstates and coherent states are derived explicitly in terms of solutions to the corresponding system of coupled classical equations of motion. In addition, using the evolution operator formalism, we construct linear and quadratic quantum dynamical invariants that provide connection of the present results with those obtained via the Malkin–Man’ko–Trifonov and the Lewis–Riesenfeld approaches. Finally, as an exactly solvable model, we introduce a Cauchy–Euler type quantum oscillator with increasing mass and decreasing frequency in time-dependent magnetic and electric fields. Based on the explicit results for the uncertainties and expectations, squeezing properties of the wave packets and their trajectories in the two-dimensional configuration space are discussed according to the influence of the time-variable parameters and external fields.

show abstract

Supercanonical coordinates for orthosymplectic evolution operators

Anzaldo-Meneses

2022

Journal of Mathematical Physics

View full text Add to dashboard Cite

A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum [Formula: see text], of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra [Formula: see text] elements times [Formula: see text] elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of bosons and, separately, of fermions with coefficients given in terms of the same coefficients of Π. Intertwining is invariant under transformations of Π, which leave certain binary forms of the coefficients of Π in the Hamiltonian unchanged. Alternatively, the coefficients can be interpreted as simultaneous time-dependent (super-) control parameters for both spin-statistics. Time-dependent inhomogeneous linear supercanonical transformations of wave vectors leave invariant the Heisenberg superalgebra [Formula: see text] and belong to the semidirect product Osp( m′/ n′) ⋉ N e( n′ + 1) of the orthosymplectic supergroup with the even Heisenberg group. The unitary time evolution operator is constructed using the adjoint map in canonical coordinates determined by the supercanonical transformation. The method is a generalization of an Inönu–Wigner contraction procedure and a Wei–Norman method for superalgebras with a selection of subalgebras associated with the root space decomposition of the Lie superalgebra. Analogously, this is a separation of variables method for quantum mechanical problems in systems with bosons and fermions. The standard Floquet theory leads to new results concerning stability for locally periodic coefficients. The lowest dimensional cases are explicitly computed. The intertwining of boson and fermions systems and the Hamiltonians considered here are of interest in quantum control theory for systems including fermions and bosons, in quantum optics, and quantum computation.

show abstract

Quadratic time dependent Hamiltonians and separation of variables

Cited by 3 publications

References 46 publications

Supercanonical transformations and time-dependent Hamiltonians

Supercanonical transformations and time-dependent Hamiltonians

Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fields

Supercanonical coordinates for orthosymplectic evolution operators

Contact Info

Product

Resources

About