Sub-Geometric Phases in Density Matrices

Wang, Zheng-Chuan

doi:10.1038/s41598-019-49770-1

Cited by 3 publications

(3 citation statements)

References 18 publications

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“…A recent understanding of the origins of the geometric phases led to a conceptual advancement of nonadiabatic motion by introducing sub-geometric phases, thus unifying the adiabatic and nonadiabatic evolution of the Hamiltonian. 32 The sub-geometric phase expression has a form equivalent to an adiabatic Berry phase,

Here, the geometric phase G is the sum of the geometric phases of two possible initial conditions of the spinors

and

for the solution of the Schrödinger equation [Eq. (1) ].…”

Section: Theorymentioning

confidence: 99%

“…We then introduce the following definitions of the wave components:

Following the work of Wang et al 32 we define the sub-geometric phases as

Here,

( i , j = 1, 2) is the definition of the sub-geometric phase component of the spinor solution of Schrödinger equations. Notably, the sum of sub-geometric phase components for the initial condition of spinors

and

is opposite in sign, namely,

.…”

Section: Theorymentioning

confidence: 99%

“…For the convenience of the reader, we define total dynamic phases D 1 ( t ) and D 2 ( t ) for the initial conditions of the spinors

and

, respectively, as follows:

with sub-dynamic phases given by

Here, H ( τ ) is defined to be a rank 2 matrix operator. 32 To completely define sub-eigenfunctions, we operate in terms of a modified Schrödinger equation,

where i corresponds to the initial conditions of the solutions of the Schrödinger equation and refers to the identity of the eigenfunction. Next, we use index j = 1, 2 to define the order for definition in the spinor column vector as follows:

or, for completeness,

…”

Section: Theorymentioning

confidence: 99%

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On the geometric phases during radio frequency pulses with sine and cosine amplitude and frequency modulation

Sorce¹,

Michaeli

2023

AIP Advances

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In this work, we describe the formation of geometric phases during nonadiabatic frequency swept (FS) radio frequency (RF) pulses with sine amplitude modulation and cosine frequency modulation functions. The geometric phases during the FS pulse were analyzed using a Schrödinger equation formalism, and the unified analytical expression for the geometric phase was derived. We present the solutions for sub-geometric phase components incorporated in spinor wavefunctions for the RF Hamiltonian of spin ½ nuclei. We demonstrate that the geometric phases during sine/cosine RF pulses are opposite in signs for different initial conditions of the spinor and that geometric phases can accumulate in correspondence to different magnetization trajectories. The derived formalism could be extended for the evaluation of the geometric phases during a wide class of amplitude- and frequency-modulated pulses used in MRI and in high-resolution NMR.

show abstract

Here, the geometric phase G is the sum of the geometric phases of two possible initial conditions of the spinors

and

for the solution of the Schrödinger equation [Eq. (1) ].…”

Section: Theorymentioning

confidence: 99%

“…We then introduce the following definitions of the wave components:

Following the work of Wang et al 32 we define the sub-geometric phases as

Here,

and

is opposite in sign, namely,

.…”

Section: Theorymentioning

confidence: 99%

“…For the convenience of the reader, we define total dynamic phases D 1 ( t ) and D 2 ( t ) for the initial conditions of the spinors

and

, respectively, as follows:

with sub-dynamic phases given by

Here, H ( τ ) is defined to be a rank 2 matrix operator. 32 To completely define sub-eigenfunctions, we operate in terms of a modified Schrödinger equation,

or, for completeness,

…”

Section: Theorymentioning

confidence: 99%

See 1 more Smart Citation