scholarly journals Multi-time correlations in the positive-P, Q, and doubled phase-space representations

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 455
Author(s):  
Piotr Deuar
Keyword(s):  
Phase Space ◽  
Simple Algorithm ◽  
Stochastic Simulations ◽  
Time Dependent ◽  
Derivative Free ◽  
Time Correlations ◽  
Large Systems ◽  
Free Operator ◽  
Heisenberg Operators ◽  
Conversion Rules

A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

scholarly journals Time-dependent relaxed magnetohydrodynamics: Inclusion of cross helicity constraint using phase-space action

2020 ◽  
Vol 27 (6) ◽  
pp. 062504 ◽  
Author(s):  
R. L. Dewar ◽  
J. W. Burby ◽  
Z. S. Qu ◽  
N. Sato ◽  
M. J. Hole
Keyword(s):  
Phase Space ◽  
Time Dependent ◽  
Space Action

CHAOTIC SCATTERING IN TIME-DEPENDENT HAMILTONIAN SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 667-679 ◽  
Author(s):  
YING-CHENG LAI ◽  
CELSO GREBOGI
Keyword(s):  
Phase Space ◽  
Measure Zero ◽  
Invariant Set ◽  
Time Dependent ◽  
Invariant Sets ◽  
Physical Origin ◽  
Chaotic Scattering ◽  
Periodic Oscillations ◽  
Surface Of Section ◽  
Oscillating Potential

We consider the classical scattering of particles in a one-degree-of-freedom, time-dependent Hamiltonian system. We demonstrate that chaotic scattering can be induced by periodic oscillations in the position of the potential. We study the invariant sets on a surface of section for different amplitudes of the oscillating potential. It is found that for small amplitudes, the phase space consists of nonescaping KAM islands and an escaping set. The escaping set is made up of a nonhyperbolic set that gives rise to chaotic scattering and remains of KAM islands. For large amplitudes, the phase space contains a Lebesgue measure zero invariant set that gives rise to chaotic scattering. In this regime, we also discuss the physical origin of the Cantor set responsible for the chaotic scattering and calculate its fractal dimension.

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
Author(s):  
S.S. MIZRAHI ◽  
M.H.Y. MOUSSA ◽  
B. BASEIA
Keyword(s):  
Phase Space ◽  
Unitary Transformation ◽  
Uniform Magnetic Field ◽  
Evolution Operator ◽  
Single Mode ◽  
Time Dependent ◽  
Special Cases ◽  
Complete Set ◽  
Hamiltonian Evolution ◽  
General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

scholarly journals Time-dependent quantum correlations in phase space

2017 ◽  
Vol 95 (6) ◽  
Author(s):  
F. Krumm ◽  
W. Vogel ◽  
J. Sperling
Keyword(s):  
Phase Space ◽  
Quantum Correlations ◽  
Time Dependent

Extracting Mode Shapes for Beams Through a Passing Auxiliary Mass

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Yao Zhang ◽  
Hai-Sheng Zhao ◽  
Seng-Tjhen Lie
Keyword(s):  
Experimental Validation ◽  
Simple Algorithm ◽  
Coupled System ◽  
Time Dependent ◽  
New Technique ◽  
Mode Shapes ◽  
Lumped Mass ◽  
Ridge Detection ◽  
Instantaneous Frequencies ◽  
Auxiliary Mass

This paper shows an approach to evaluate mode shapes for beams through using a passing auxiliary mass. The coupled system of an auxiliary mass passing over a beam is time-dependent, and the corresponding instantaneous frequencies (IFs) are equivalent to the mode shapes. Hence, reconstruction of the mode shapes is easy to be achieved through estimating the IFs. A simple algorithm based on ridge detection is proposed to reconstruct the mode shapes. This method is effective if the beam is light or the lumped mass is heavy. It is convenient since it requires an accelerometer mounted on the passing auxiliary mass rather than a serious of sensors mounted on the structure itself. It is also more practical because it is usually difficult to install external exciter. A lab-scale experimental validation shows that the new technique is capable of identifying the first three mode shapes accurately.

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