Analysis of Berry's phase by the evolution operator method

Cheng, C M; Fung, P. C. W.

doi:10.1088/0305-4470/22/17/015

Cited by 17 publications

(5 citation statements)

References 30 publications

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“…Our aim is to employ equations (5) to form a relation between the phase factor in a wave function and its amplitude-modulus. If a wave-function amplitude ψ(t) is written in the form:…”

Section: Theorymentioning

confidence: 99%

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Englman

Yahalom

Baer³

2000

Eur. Phys. J. D

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We calculate the open path phase in a two state model with a slowly (nearly adiabatically) varying time-periodic Hamiltonian and trace its continuous development during a period. We show that the topological (Berry) phase attains π or 2π depending on whether there is or is not a degeneracy in the part of the parameter space enclosed by the trajectory. Oscillations are found in the phase. As adiabaticity is approached, these become both more frequent and less pronounced and the phase jump becomes increasingly more steep. Integral relations between the phase and the amplitude modulus (having the form of Kramers-Kronig relations, but in the time domain) are used as an alternative way to calculate open path phases. These relations attest to the observable nature of the open path phase.

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“…Our aim is to employ equations (5) to form a relation between the phase factor in a wave function and its amplitude-modulus. If a wave-function amplitude ψ(t) is written in the form:…”

Section: Theorymentioning

confidence: 99%

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Englman

Yahalom

Baer³

2000

Eur. Phys. J. D

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“…We shall apply the theorem to the amplitudes in a wave function (t), which is expanded in some (time independent) set in the form (t) = i i (t) i> (13) Then (t) in (1) may stand for any of the amplitudes i (t) in the expansion. Before we make applications to "cyclic" or to "adiabatically periodic" wave functions, of the type discussed in [1][2][3][4][5][6], we note that the physical i (t)'s are not strictly cyclic, since when continuity of the phase is imposed on the wave function, then the phase undergoes changes between periods ( generally in a monotonic manner). We can correct for these phase changes, which include the dynamic phase [2], by multiplying the true wave-function with a phase factor exp[i (t)] (with (t) real) which then leaves us with a properly periodic function.…”

Section: Formalismmentioning

confidence: 99%

“…If we now expand log ( /c 0 ) as (infinite) cos and sine-series log ( /c 0 ) = n A n cos(nt)+ i n B n sin(nt) (5) since the m's in (4) are positive. We obtain by Abel's theorem [8] that A n = B n (all real, since c m are such).…”

Section: Formalismmentioning

confidence: 99%

“…In textbook expositions of Quantum Mechanics the squared moduli of the wave function components , which serve as measures of the state occupation probability, play a central role. In the last 15 years increasing emphasis is put on another quantity, namely on the phases in the wave-function, and in particular on the Berry-phase part which reflects the geometry or topology in which the physical system exists [1][2][3][4][5][6] and whose practical manifestation is in some interference experiments [7]. By considering cyclic wave functions we derive theoretically and demonstrate computationally connections between phases and amplitude moduli.…”

Section: Introductionmentioning

confidence: 99%

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Phase-modulus relations in cyclic wave functions

Englman¹,

Yahalom

Baer³

1999

Physics Letters A

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We derive reciprocal integral relations between phases and amplitude moduli for a class of wave functions that are cyclically varying in time. The relations imply that changes of a certain kind (e.g. not arising from the dynamic phase) obligate changes in the other. Numerical results indicate the approximate validity of the relationships for arbitrarily (non-cyclically) varying states in the adiabatic (slowly changing) limit.

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“…Following Cheng et al [14] we first expand the time-dependent Hamiltonian H(t) of a two-level system in the identity operator I, the raising and lowering operators σ ± , and Pauli spin matrix σ 3…”

Section: Resultsmentioning

confidence: 99%

The quantum mechanical geometric phase of a particle in a resonant vibrating cavity

Yuen¹,

Fung²,

Cheng³

et al. 2003

J. Phys. A: Math. Gen.

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We study the general-setting quantum geometric phase acquired by a particle in a vibrating cavity. Solving the two-level theory with the rotating-wave approximation and the SU(2) method, we obtain analytic formulae that give excellent descriptions of the geometric phase, energy, and wavefunction of the resonating system. In particular, we observe a sudden π-jump in the geometric phase when the system is in resonance. We found similar behaviors in the geometric phase of a spin-1/2 particle in a rotating magnetic field, for which we developed a geometrical model to help visualize its evolution.

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Analysis of Berry's phase by the evolution operator method

Cited by 17 publications

References 30 publications

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

The open path phase for degenerate and non-degenerate systems and its relation to the wave-function modulus

Phase-modulus relations in cyclic wave functions

The quantum mechanical geometric phase of a particle in a resonant vibrating cavity

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