The Theory Of The Family Of Garrison-Wright’s Phases And Its Application

Quantal geometric phase is one of the most important discoveries in physics in the past thirty years. In 1984, Berry in his insightful and beautiful work Quantal Phase Factors Accompanying Adiabatic Changes formally proposed the concept of quantum geometric phase, and found that there exists an additional phase merely related to evolution path besides the known dynamical phase after a quantum state evolved adiabatically and cyclically, and thus the additional phase, which is also called Berry’s phase, merely relies on the circuit on a parameter manifold and thence has geometric meaning. Substantially, Berry’s phase is a real-valued abelian phase of a non-degenerate eigenstate of the Hamiltonian operator of the quantum system where the eigenstate is chosen as a cyclic state controlled by parameters. Driven by the Berry’s work, quantal geometric phase hidden and ignored in probability amplitude began to be attracted much attention, and the family of which has been constantly growing. However, the main path of its development was given by the following works:In 1984 Wilczek and Zee introduced the concept of quantum non-Abelian geometric phase; In 1987 Aharonov and Anandan proposed the concept of quantum nonadiabatic geometric phase; In 1988 Samuel and Bhandari presented the concept of quantum noncyclic geometric phase; In 2000 Manini and Pistolesi addressed the concept of off-diagonal geometric phase; These concepts were based on quantum pure state, while more practical quantum geometric phase of quantum mixed states was firstly introduced by Uhlmann in 1986; A more physical concept of quantum geometric phase of quantum mixed states under unitary evolution was proposed by Sjoqvist et al. in 2000, while a more general concept of quantum ge-ometric phase of quantum mixed states under nonunitary evolution was given by Tong et al. in 2004; Quantal geometric phases has also triggered directly a broad and deep discussion on how to use geometric and topological methods in branches of physics including classical and quantum mechanics, classical and quantum op-tics, atomic and molecular physics, particle and nuclear physics, condensed matter physics, gravitation and cosmology, and so forth. And moreover, quantal geometric phase has also played a very important role in practical applications. For example, quantum geometric phase plays a key role in the field of geometric quantum com-putation because it owns a certain geometric invariance which can resist to certain types of external noise; The singularity of quantum geometric phase in quantum critical points can be used to characterize quantum phase transition; Berry’s curva-ture can be used to fabricate artificial electromagnetic field which can act on neutral atoms as real electromagnetic field acts on charged particles. Quantum simulators based on that mechanism can be used to mimic physical experiments or conjectures which can not be easily verified in reality.One of the axioms in conventional quantum mechanics requires that observ-ables are represented by self-adjoint or Hermitian operators. As an attempt to gen-eralize conventional quantum mechanics, non-Hermitian quantum mechanics aban-dons the necessity of Hermiticity of operators as a major contribution to quantum mechanics. The formulation of non-Hermitian quantum mechanics tolerates physi-cists to enlarge the set of possible and even non-Hermitian Hamiltonian to describe novel physical phenomena effectively and provide powerful numerical and analytical methods. A well-known non-Hermitian quantum model was in 1987 proposed by Lamb et al. when solving the problem that the matter-field interaction was sensitive to the choice of gauge as decaying states were used in two-level systems. And later soon, Garrison and Wright in 1988 used this model to introduced a new concept of quantal geometric phase, complex-valued quantal geometric phase, and also showed how to calculate it for two special case, adiabatic and nonadiabatic cyclic cases. As one of the investigations during doctoral program, the author of the dissertation generalized Garrison-Wright’s quantum geometric phase into a more general case and established a general theory for the family of Garrison-Wright’s phases, which not only admits Garrison and Wright’s work but also introduces the concepts of noncyclic and off-diagonal Garrison-Wright’s phases, respectively. And Moreover, a scheme for a phase-shift gate in quantum computations was proposed by combining the theory with the sophisticated single molecule spectroscopy techniques.The dissertation is organized as follows.The first chapter is an introduction to show the development of theoretical and applied physics before and after the discovery of Berry’ phase as a watershed from the viewpoint of history, and especially the work of Berry etal. Finally, unsolved issues in quantal geometric phase are given for subsequent chapters.The second Chapter is to introduce and establish a unified geometrical theory for the concept of cyclic Garrison-Wright’s phase by giving the concept of gauge-invariant distance. Finally, the famous Bethe-Lamb model is used to show how to apply the theory to calculate the cyclic Garrison-Wright’s phases in adiabatic and nonadiabatic cases.The third chapter is to generalize and unify the family of Garrison-Wright’s phases using axiomatic approach. At first, the fixed rule for the dynamical system in Garrison-Wright’s sense is postulated with an axiom, and some necessary pre-liminaries are also listed. Secondly, the state manifold is structured from the fixed rule to establish a framework for subsequent discussions. Thirdly, the interference formula in Garrison-Wright’s sense are generalized for introducing the generalized Bargmann invariant and giving the topological explanations on it. Finally, after in-troducing two new classes of Garrison-Wright’s phase in noncyclic and biorthogonal cases, a new formulation is used to unify the family of Garrison-Wright’s phases。The fourth chapter is to theoretically implement the open-loop control of Garrison-Wright’s phase in a driven single molecule model in the sophisticated field of single molecule spectroscopy, where the theory of the family of Garrison-Wright’s phases is applied combining with u(1) (?) su(1,1) algebra.The fifth chapter is to summarize the dissertation and prospect the orientation of future projects.An overview over the main results of this dissertation is given below.1 A geometrical theory of cyclic Garrison-Wright’s phasesThe concept of gauge-invariant distance is introduced by the following theorem,Theorem §2.2.1 If and only if the functional(?)([ψ1], [ψ2]) is stationary with respect to the phase a, namely, δ(?)/δα= 0, the functional(?) is gauge-invariant. Here and [ψ1,2] denote the two equivalence classes of the states|ψ1> and |ψ2), respectively.Definition §2.2.2 The functional(?)([ψ1], [ψ2]) as is stationary with respect to the phase is called the gauge-invariant distance between the two rays [ψ1] and [ψ2] .Furthermore, a geometrical theorem on cyclic Garrison-Wright’s phases is given below by the above definition of gauge-invariant distance, which can describe Gar-rison and Wright’s work on adiabatic and non-adiabatic complex geometric phases by a unified approach.Theorem §2.2.3 Assume that a quantum dissipative system describing by a hamil-tonian H(t) with finite non-degenerate instantaneous eigenenergies is under cyclic evolution, then the cyclic Garrison-Wright’s phase must satisfy the following inte-gral, Here denotes the infinitesimal length of the circuit Cclosed: t ∈ [0, T]â†'|ψc(t)>, d(?)= AEdt/h denotes the infinitesimal gauge-invariant dis-tance between two neighboring rays, is the energy variance,|ψ(t)>～|ψc(t)> is the solution of non-hermitian Schrodinger equation.Corollary §2.2.4 Assume that a quantum dissipative system describing by a hamil-tonian H(R(t)) with finite non-degenerate instantaneous eigenenergies is under cyclic evolution on a certain non-degenerate eigenstate|n(R(t))>, then the adia-batic Garrison-Wright’s phase must satisfy the following integral Here Cclosed:t ∈ [0, T]â†' R(t) is a circuit on parameter manifold, s.t. R(0)= R(T).2 The general theory of the family of Garrison-Wright’s phasesThe concept of gauge-invariant interference intensity is introduced by the fol-lowing theorem,Theorem §3.3.1 If and only if the functional (?)2([ψ1].[ψ2] is stationary with respect to θ, namely, δ(?)2/δθ = 0, the functional (?)2 is gauge invariant. HereDefinition §3.3.2 |ψ1,|ψ2 ∈V are the two states in Garrison-Wright’s sense. Then the functional (?)2([ψ1].[ψ2] as is stationary with respect to the phase B is called the gauge-invariant interference intensity between two states |ψ1＞ and |ψ2＞.Gauge-invariant interference intensity can generalize the concept of Pancharat-nam’s "in-phase" such that any two non-local states can compare the generalized Pancharatnam’s phase difference between them. A theorem for establishing a re-lationship between the local generalized Pancharatnam connection 1-form and the global Pancharatnam phase is given by the following lemma.Lemma §3.3.6 The geodesic linking any two state |ψ1＞1,|ψ2＞ ∈ V- {0} (or |ψ1＞, |ψ2＞ ∈ V - {0} ) must satisfy the following two geodesic equations with gauge invariance, Here c = is a constant, |ψ(s)＞ ∈ V - {0} and |ψ(s)＞ ∈{0} denote the geodesics linking |ψ1＞ with |ψ2＞ and linking |ψ1＞ and |ψ2＞, respectively.Theorem §3.3.7 Let |ψ1＞, |ψ2＞ ∈ V - {0} in Garrison-Wright’s sense connected by a geodesic (?)1,2 ∈ V - {0}. Then the generalized Pancharatnam phase BG2 between the two states is determined by the following equation, Here .AGP is the local generalized Pancharatnam connection 1-form.The above theorem both gives the following generalized Bargmann invariants, and implies the following theorem to unify and expand the family of Garrison-Wright’s phases.Theorem §3.4.1 Assume that|ψ(t)> is the evolving state of the quantum system in Garrison-Wright’s sense and that the initial state |ψ(0)> is not biorthogonal to the final state |ψ(T)＞. Then the Garrison-Wright’s phase between the two state is determined by The above theorem not only admits the work of Garrison and Wright’s but also introduce the concept of non-cyclic and off-diagonal Garrison-Wright’s phases.Corollary §3.5.1 Adiabatic and non-adiabatic Garrison-Wright’s phases are the two special cases of the Theorem §3.4.1.Corollary §3.5.2 There is a well-defined off-diagonal Garrison-Wright’s phase be-tween biorthogonal states of the multi-level quantum system in Garrison-Wright’s phases, which is independent of the choice of intermediate states.