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arXiv:1104.3743 · Modern Physics Letters B

GAUGE THEORY AND TWO LEVEL SYSTEMS

Angelo Bruno, Antonio Capolupo, Subhash C. Kak, Gianluca Raimondo, Giuseppe Vitiello

Published Jul 14, 2011• 15 citations
arXivHTMLPDF

Abstract

We consider the time evolution of a two level system (a two level atom or a qubit) and show that it is characterized by a local (in time) gauge invariant evolution equation. The covariant derivative operator is constructed and related to the free energy. We show that the gauge invariant characterization of the time evolution of the two level system is analogous to the birefringence phenomenon in optics. The relation with Berry-like and Anandan–Aharonov phase is pointed out. Finally, we discuss entropy, environment effects and the distance in projective Hilbert space between two level states in their evolution.

DOI
10.1142/s021798491102698x

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We consider the time evolution of a two level system (a two level atom or a qubit) and show that it is characterized by a local (in time) gauge invariant evolution equation. The covariant derivative operator is constructed and related to the free energy. We show that the gauge invariant characterization of the time evolution of the two level system is analogous to the birefringence phenomenon in optics. The relation with Berry-like and Anandan–Aharonov phase is pointed out. Finally, we discuss entropy, environment effects and the distance in projective Hilbert space between two level states in their evolution.
Neither geometric phases nor differences in geometric phases are generally invariant under time-dependent unitary transformations (unlike differences in total phases), in particular under local gauge transformations and Galilei transformations. (This was pointed out originally by Aharonov and Anandan, and in the case of Galilei transformations has recently been shown explicitly by Sjoeqvist, Brown and Carlsen.) In this paper, I introduce a phase, related to the standard geometric phase, for which phase differences are both gauge- and Galilei-invariant, and, indeed, invariant under transformations to linearly accelerated coordinate systems. I discuss in what sense this phase can also be viewed as geometric, what its relation is to earlier proposals for making geometric phases invariant under gauge or Galilei transformations, and what is its classical analogue. I finally apply this invariant phase to Berry's derivation of the Aharonov-Bohm effect.
The concept of Geometric Phase in Quantum Mechanics is generally formulated en- tirely in terms of geometric structure of complex Hilbert Space. This tutorial article gives a general mathematical overview of the Geometric phase introduced by Berry with the conditions over the system’s evolution embedded through adiabaticity and cyclicity, which were subsequently relaxed sequentially by Aharonov-Anandan, Samuel-Bhandari, and later by Mukunda and Simon by using the idea of Bargmann Invariants. The arti- cle presents a thorough and illustrative overview regarding the mathematical derivation behind the upliftment of the conditions, which results in the generalized definition of the Geometric phase for quantum systems. In addition to that, the article also gives an overall idea about the general analytical aspects of finding the Geometric phase for three level open quantum systems undergoing decoherence and dephasing due to interaction with the surroundings in the weak system reservoir coupling limit described by quantum master equations.
The generic two-level model with time-dependent matrix elements becomes soluble in the limit that its diagonal and off-diagonal terms vary along a flat ellipse, encircling the diabolical singular point in the parameter space. The time evolution of the state vector is explicitly obtained, and the condition for its evolution to form a closed circuit in the projective Hilbert space of rays is given as a result of destructive interference at level crossing. The Aharonov-Anandan geometrical phase is shown to be related to the Stokes phase for the Landau-Zener model, which is a natural extension of Berry's phase to nonadiabatic evolutions.
The time-evolution operator is in general gauge dependent. Its gauge transformation property follows from the gauge transformation of the wave function and ensures gauge-invariant matrix elements. The same transformation property is shown here to follow from the formal solution of the Schr\"odinger equation for the time-evolution operator, which is a time-ordered exponential of the time integral of the Hamiltonian. The gauge transformation property of the time-evolution operator in the interaction picture is also obtained. The perturbation expansion of the time-evolution operator in one gauge can be transformed to give the perturbation expansion for the time-evolution operator in another gauge. The A\ensuremath{\cdot}p versus E\ensuremath{\cdot}r controversy in the electric dipole approximation is resolved by specifying the correct initial and final states.