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NASA NTRS1993

Two photon annihilation operators and squeezed vacuum

Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than.

Roy, Anil K.↗

NASA NTRS1995

Coherent states for the relativistic harmonic oscillator

Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.

Aldaya, Victor↗

NASA NTRS1996

The Squeezing Operator and the Squeezing States of Superspace

In this paper ,the unitary squeezing operator of 'superspace' is introduced and by making this operator act on the supercoherent state, the squeezing supercoherent states are obtained, then come out the four orthonormalization eigenstates of the square of annihilation operator A of the supersymmetry harmonic oscillator, and their squeezing character is also studied.

Aiqun, Ma↗

NASA NTRS1992

Going through a quantum phase

Phase measurements on a single-mode radiation field are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator; its phase eigenkets generate the probability operator measure (POM) for maximum likelihood phase estimation. A commuting observables description for the SG-POM on a signal x apparatus state space is derived. It is analogous to the signal-band x image-band formulation for optical heterodyne detection. Because heterodyning realizes the annihilation operator POM, this analogy may help realize the SG-POM. The wave function representation associated with the SG POM is then used to prove the duality between the phase measurement and the number operator measurement, from which a number-phase uncertainty principle is obtained, via Fourier theory, without recourse to linearization. Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phase-measurement probability density function (PDF) for a single-mode field in an arbitrary quantum state. Finally, a two-mode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite average photon number. Application of this construct to interferometric precision measurements is briefly discussed.

Shapiro, Jeffrey H.↗

NASA NTRS1992

Coherent states and parasupersymmetric quantum mechanics

It is well known that Parafermi and Parabose statistics are natural extensions of the usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones. Due to this generalization, positive parameters appear: the so-called orders of paraquantization p (= 1, 2, 3, ...) and h sub 0 (= 1/2, 1, 3/2, ...), respectively, the first value leading in each case to the usual statistics. The superpostion of the parabosonic and parafermionic operators gives rise to parasupermultiplets for which mixed trilinear relations have already been studied leading to two (nonequivalent) sets: the relative Parabose and the relative Parafermi ones. For the specific values p = 1 = 2h sub 0, these sets reduce to the well known supersymmetry. Coherent states associated with this last model have been recently put in evidence through the annihilation operator point of view and the group theoretical approach or displacement operator context. We propose to realize the corresponding studies within the new context p = 2 = 2h sub 0, being then directly extended to any order of paraquantization.

Debergh, Nathalie↗

NASA NTRS1994

Berry phase in Heisenberg representation

We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.

Andreev, V. A.↗

DOE OSTI2021

Enhanced relaxed physical factorization preconditioner for coupled poromechanics

The relaxed physical factorization (RPF) preconditioner is a recent algorithm allowing for the efficient and robust solution to the block linear systems arising from the three-field displacement-velocity-pressure formulation of coupled poromechanics. For its application, however, it is necessary to invert blocks with the algebraic form C^ = (C + βFF T ), where C is a symmetric positive definite matrix, FF T a rank-deficient term, and β a real non-negative coefficient. The inversion of C^, performed in an inexact way, can become unstable for large values of β, as it usually occurs at some stages of a full poromechanical simulation. In this work, we propose a family of algebraic techniques to stabilize the inexact solve with C^. This strategy can prove useful in other problems as well where such an issue might arise, such as augmented Lagrangian preconditioning techniques for Navier-Stokes or incompressible elasticity. First, we introduce an iterative scheme obtained by a natural splitting of matrix C^. Second, we develop a technique based on the use of a proper projection operator annihilating the near-kernel modes of C^. Both approaches give rise to a novel class of preconditioners denoted as Enhanced RPF (ERPF). Furthermore, effectiveness and robustness of the proposed algorithms are demonstrated in both theoretical benchmarks and real-world large-size applications, outperforming the native RPF preconditioner.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

An efficient quantum circuit for block encoding a pairing Hamiltonian

We present an efficient quantum circuit for block encoding a pairing Hamiltonian often studied in nuclear physics. Our block encoding scheme does not require mapping the creation and annihilation operators to the Pauli operators and representing the Hamiltonian as a linear combination of unitaries. Instead, we show how to encode the Hamiltonian directly using controlled swap operations. We analyze the gate complexity of the block encoding circuit and show that it scales polynomially with respect to the number of qubits required to represent a quantum state associated with the pairing Hamiltonian. We also show how the block encoding circuit can be combined with the quantum singular value transformation to construct an efficient quantum circuit for approximating the density of states of a pairing Hamiltonian. The techniques presented can be extended to encode more general second-quantized Hamiltonians.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Clifford Circuit-Based Heuristic Optimization of Fermion-To-Qubit Mappings

Simulation of interacting Fermionic Hamiltonians is one of the most promising applications of quantum computers. However, the feasibility of analyzing Fermionic systems with a quantum computer hinges on the efficiency of Fermion-to-qubit mappings that encode nonlocal Fermionic degrees of freedom in local qubit degrees of freedom. While recent studies have highlighted the importance of designing Fermion-to-qubit mappings that are tailored to specific problem Hamiltonians, the methods proposed so far either are restricted to a narrow class of mappings or they use computationally expensive and unscalable brute-force search algorithms. Here, in this work, we address this challenge by designing a heuristic numerical optimization framework for Fermion-to-qubit mappings. To this end, we first translate the Fermion-to-qubit mapping problem to a Clifford circuit optimization problem and then use simulated annealing to optimize the average Pauli weight of the problem Hamiltonian. For all Fermionic Hamiltonians we have considered, the numerically optimized mappings outperform their conventional counterparts, including ternary-tree-based mappings that are known to be optimal for single creation and annihilation operators. We find that our optimized mappings yield between 15% and 40% improvements on the average Pauli weight when the simulation Hamiltonian has an intermediate level of complexity. Most remarkably, the optimized mappings improve the average Pauli weight for 6 × 6 nearest-neighbor hopping and Hubbard models by more than 40% and 20%, respectively. Surprisingly, we also find specific interaction Hamiltonians for which the optimized mapping outperforms any ternary-tree-based mapping. Our results establish heuristic numerical optimization as an effective method for obtaining mappings tailored for specific Fermionic Hamiltonian.

Hamiltonians↗

DOE OSTI2026

Violation of detailed balance in non-equilibrium magnons observed by inelastic neutron scattering

Traditional inelastic neutron scattering (INS) characterizes excitations—such as phonons and magnons—in condensed matter systems at thermodynamic equilibrium. However, the most intriguing and puzzling many-body effects in open quantum systems often emerge from dissipative dynamics that are inherently out of equilibrium. Here, we use a combination of laser pumping and INS to experimentally observe long-lived nonequilibrium magnons in a two-dimensional (2D) square-lattice Heisenberg antiferromagnet. These nonequilibrium magnons manifest themselves as a violation of detailed balance in the dynamic structure factor and reach steady states under periodic driving, analogous to nonequilibrium steady states in driven dissipative systems. Furthermore, we show that the violation of detailed balance reflects the quantum-mechanical nature of the underlying dynamical system, where out-of-time-ordered correlations of creation and annihilation operators do not satisfy commutation relations. The in operando INS technique developed here provides a new approach to studying nonequilibrium magnons in prototypical 2D quantum magnets and can be extended to other systems, including one-dimensional spin chains and topological many-body spin systems, where nonequilibrium effects are widespread and rich in discovery potential.

Hua, Chengyun [Oak Ridge National Laboratory (ORNL↗

DOE OSTI2021

Rotational bands beyond the Elliott model

Rotational bands are commonplace in the spectra of atomic nuclei. Inspired by early descriptions of these bands by quadrupole deformations of a liquid drop, Elliott constructed discrete nucleon representations of SU(3) from fermionic creation and annihilation operators. Ever since, Elliott's model has been foundational to descriptions of rotation in nuclei. Later work, however, suggested the symplectic extension Sp(3, R) provides a more unified picture. We decompose no-core shell-model nuclear wave functions into symmetry-defined subspaces for several beryllium isotopes, as well as 20 Ne, using the quadratic Casimirs of both Elliott's SU(3) and Sp(3, R). The band structure, delineated by strong B(E2) values, has a more consistent description in Sp(3, R) rather than SU(3). In particular, we confirm previous work finding in some nuclides strongly connected upper and lower bands with the same underlying symplectic structure.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1992

Quadratic squeezing: An overview

The amplitude of the electric field of a mode of the electromagnetic field is not a fixed quantity: there are always quantum mechanical fluctuations. The amplitude, having both a magnitude and a phase, is a complex number and is described by the mode annihilation operator a. It is also possible to characterize the amplitude by its real and imaginary parts which correspond to the Hermitian and anti-Hermitian parts of a, X sub 1 = 1/2(a(sup +) + a) and X sub 2 = i/2(a(sup +) - a), respectively. These operators do not commute and, as a result, obey the uncertainty relation (h = 1) delta X sub 1(delta X sub 2) greater than or = 1/4. From this relation we see that the amplitude fluctuates within an 'error box' in the complex plane whose area is at least 1/4. Coherent states, among them the vacuum state, are minimum uncertainty states with delta X sub 1 = delta X sub 2 = 1/2. A squeezed state, squeezed in the X sub 1 direction, has the property that delta X sub 1 is less than 1/2. A squeezed state need not be a minimum uncertainty state, but those that are can be obtained by applying the squeeze operator.

Hillery, M.↗

NASA NTRS1993

Which Q-analogue of the squeezed oscillator?

The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.

Solomon, Allan I.↗

NASA NTRS1995

The relativistic many body problem with an oscillator interaction

We start with the total energy E for a system of three scalar relativistic particles that, because of Einstein's relation, will have square roots of functions of the momenta. By taking powers of this relation, we finally get a fourth degree polynomial in E(exp 2), where the square roots have disappeared, and which we can convert into a type of Schroedinger equation. To be in the center of mass frame we pass to Jocobi momenta and then replace them by creation and annihilation operators. We thus get an equation in terms of the generators of a U(2) group, which, in principle, we can solve in an elementary way. Finally, we rewrite our equation in a Poincare invariant form.

Moshinsky, Marcos↗

NASA NTRS1995

Oscillator-like coherent states for the Jaynes-Cummings Model

A new way of diagonalizing the Jaynes-Cummings Hamiltonian is proposed, which allows the definition of annihilation operators and coherent states for this model. Mean values and dispersions over these states are computed and interpreted.

Berubelauziere, Y.↗

NASA NTRS1989

Absorbing boundary conditions for second-order hyperbolic equations

A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.

Jiang, Hong↗

NASA NTRS1990

Absorbing boundary conditions for second-order hyperbolic equations

Jiang, Hong↗

DOE OSTI2019

Nonlinear dynamics from linear quantum evolutions

Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ϕ{sub t} of transformations on a Hilbert space, H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schrödinger equation to a submanifold invariant under the flow ϕ{sub t}. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ϕ{sub t}. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗