Geometric Phase For Mixed States Research Articles

Mixed state behavior of Hermitian and non-Hermitian topological models with extended couplings

Geometric phase is an important tool to define the topology of the Hermitian and non-Hermitian systems. Besides, the range of coupling plays an important role in realizing higher topological indices and transition among them. With a motivation to understand the geometric phases for mixed states, we discuss finite temperature analysis of Hermitian and non-Hermitian topological models with extended range of couplings. To understand the geometric phases for the mixed states, we use Uhlmann phase and discuss the merit-limitation with respect extended range couplings. We extend the finite temperature analysis to non-Hermitian models and define topological invariant for different ranges of coupling. We include the non-Hermitian skin effect, and provide the derivation of topological invariant in the generalized Brillouin zone and their mixed state behavior also. We also adopt mixed geometric phases through interferometric approach, and discuss the geometric phases of extended-range (Hermitian and non-Hermitian) models at finite temperature.

Mixed state geometric phase for neutrino oscillations

The geometric picture of neutrino oscillations offers a unique way to study the quantum mechanics of this phenomenon. In this picture, the propagation of a neutrino beam is described by a density matrix evolving in a state space with non-trivial geometry. We derive explicit expressions of the mixed state geometric phase which arise during such an evolution for both two and three flavor neutrino oscillations. We show that, in the case of two flavor neutrino oscillations, the geometric phase is independent of the Majorana phase and it can be used as a measure of coherence of the neutrino beam.

Geometry along evolution of mixed quantum states

The metric underlying the mixed state geometric phase in unitary and nonunitary evolution [Phys. Rev. Lett. {\bf 85}, 2845 (2000); Phys. Rev. Lett. {\bf 93}, 080405 (2004)] is delineated. An explicit form for the line element is derived and shown to be related to an averaged energy dispersion in the case of unitary evolution. The line element is measurable in interferometry involving nearby internal states. Explicit geodesics are found in the single qubit case. It is shown how the Bures line element can be obtained by extending our approach to arbitrary decompositions of density operators. The proposed metric is applied to a generic magnetic system in a thermal state.

Geometric phases for mixed states of the Kitaev chain.

The Berry phase has found applications in building topological order parameters for certain condensed matter systems. The question whether some geometric phase for mixed states can serve the same purpose has been raised, and proposals are on the table. We analyse the intricate behaviour of Uhlmann's geometric phase in the Kitaev chain at finite temperature, and then argue that it captures quite different physics from that intended. We also analyse the behaviour of a geometric phase introduced in the context of interferometry. For the Kitaev chain, this phase closely mirrors that of the Berry phase, and we argue that it merits further investigation.

Reference Section Method for Mixed State Geometric Phase in Nonunitary Evolution* *Supported in part by the Natural Science Foundation of China under Grant Nos. 10821504, 11075194, 11135003, 11275246, and 11475238, and by the National Basic Research Program of China (973 Program) under Grant No. 2010CB833000 (TL)

We propose the reference section method to obtain a geometric phase for the mixed states in nonunitary evolution. The reference connection on density operator space is defined. That the parallel transport method and reference section method are essentially two aspects of in phase concept is pointed out.

Spectral weighted geometric phase for mixed quantum states

Geometric phase has found a broad spectrum of applications in both classical and quantum physics. In this work we discuss a geometric phase for mixed quantum states based on traces of spectral weighted holonomies. Our approach applies to general unitary evolutions of both nondegenerate and degenerate mixed states, and it generalizes the standard definition of geometric phase for mixed states, which is based on quantum interferometry. We provide an explicit formula for the geometric phase that can be easily implemented for computations in quantum physics, and we discuss higher order analogs of the geometric phase that might be defined at points where the ordinary geometric phase is undefined.

Geometric Phase for High-Temperature Master Equation

Geometric phase of mixed state is investigated for three-level system obeying a high-temperature master equation. The results show the Berry phase of mixed phase is strongly dependent on the initial condition. For the different initial angle, the turning region is different. In addition, the decrease of Berry phase is most slow around the coupling strength α=5 with an increasing of evolving time.

Geometric phase and spinorial representation of mixed state

Abstract A novel approach to geometric phase of mixed state is proposed by using the normalized spinorial representation in connecting the density matrix with mixed state vector. We find that though the spinor involves N separate U ( 1 ) phases correspondingly to any N-level decomposition of the density matrix, both geometric phase and density matrix of mixed state are holonomy ⨂ k = 1 N U ( 1 ) gauge invariants. This noncyclic invariant is conceptually useful in analyzing geometric phase and CP violation of open system. Under a quasicyclic case, the geometric phase depends only on the symplectic area spanned in a given closed evolving curve with the classical probabilities relating to the Bloch radius, in which quantifies mixed degree of open system, in the Bloch sphere structure.

A geometric phase for superconducting qubits under the decoherence effect

We propose a relaxation rate or dissipative cavity-based parameters that can be used as indicators of the stationary limit of a mixed state geometric phase. We perform our considerations for the system of a superconducting qubit in an open transmission line or interacting with a dissipative cavity. This system is very useful for performing an effective quantum computation by exhibiting the long collapse time of the geometric phase. It is shown that the geometric phase in the stationary limit does not depend on interaction time if the decay time exceeds an upper bound.

Operational geometric phase for mixed quantum states

The geometric phase has found a broad spectrum of applications in both classical and quantum physics, such as condensed matter and quantum computation. In this paper, we introduce an operational geometric phase for mixed quantum states, based on spectral weighted traces of holonomies, and we prove that it generalizes the standard definition of the geometric phase for mixed states, which is based on quantum interferometry. We also introduce higher order geometric phases, and prove that under a fairly weak, generically satisfied, requirement, there is always a well-defined geometric phase of some order. Our approach applies to general unitary evolutions of both non-degenerate and degenerate mixed states. Moreover, since we provide an explicit formula for the geometric phase that can be easily implemented, it is particularly well suited for computations in quantum physics.

C*-geometric phase for mixed states: entanglement, decoherence and the spin system

We study a kind of geometric phase for entangled quantum systems, and particularly a spin driven by a magnetic field and entangled with another spin. The new kind of geometric phase is based on an analogy between open quantum systems and dissipative quantum systems which uses a C*-module structure. We show that the system presents, from the viewpoint of their geometric phases, two behaviours. The first one is identical to the behaviour of an isolated spin driven by a magnetic field, as the problem originally treated by Berry. The second one is specific to the decoherence process. The gauge structures induced by these geometric phases are then similar to a magnetic monopole gauge structure for the first case, and can be viewed as a kind of instanton gauge structure for the second case. We study the role of these geometric phases in the evolution of a mixed state, particularly by focusing on the evolution of the density matrix coherence. We investigate also the relation between the geometric phase of the mixed state of one of the entangled systems and the geometric phase of the bipartite system.

Non-Abelian quantum holonomy of hydrogenlike atoms

We study the Uhlmann holonomy [Rep. Math. Phys. 24, 229 (1986)] of quantum states for hydrogenlike atoms where the intrinsic spin and orbital angular momentum are coupled by the spin-orbit interaction and are subject to a slowly varying magnetic field. We show that the holonomy for the orbital angular momentum and spin subsystems is non-Abelian while the holonomy of the whole system is Abelian. Quantum entanglement in the states of the whole system is crucially related to the non-Abelian gauge structure of the subsystems. We analyze the phase of the Wilson loop variable associated with the Uhlmann holonomy and find a relation between the phase of the whole system and corresponding marginal phases. Based on the results for the model system, we provide evidence that the phase of the Wilson loop variable and the mixed-state geometric phase [E. Sj\"oqvist et al., Phys. Rev. Lett. 85, 2845 (2000).] are generally inequivalent.

Geometric phase of mixed states for three-level open systems

Geometric phase of mixed state for three-level open system is defined by establishing in connecting density matrix with nonunit vector ray in a three-dimensional complex Hilbert space. Because the geometric phase depends only on the smooth curve on this space, it is formulated entirely in terms of geometric structures. Under the limiting of pure state, our approach is in agreement with the Berry phase, Pantcharatnam phase, and Aharonov and Anandan phase. We find that, furthermore, the Berry phase of mixed state correlated to population inversions of three-level open system.

Geometric Phase and Sidebands

Geometric phase of mixed state is expanded for a two-level atom interacting with a quantized field mode, where master equation is a modified Bloch equation with constant terms that are used to explain emergence of sidebands in the spectrum of the fluorescent light. The results show that geometric phase transition is observed for medium initial angle when the Rabi frequency associated with the driving field becomes comparable to the spectral width of the atom. We find that the geometric phase transition depends on population inversion and Bloch radius, which is helpful to understand mechanism of sideband.

Fidelity between partial states as a signature of quantum phase transitions

We introduce a partial state fidelity approach to quantum phase transitions. We consider a superconducting lattice with a magnetic impurity inserted at its center, and look at the fidelity between partial (either one-site or two-site) quantum states. In the vicinity of the point of the quantum phase transition, we observe a sudden drop of the fidelity between two one-site partial states corresponding either to the impurity location or its close vicinity. This enables us to identify the on-site magnetization as the order parameter for the phase transition studied. In the case of two-site states, the fidelity reveals the transition point as long as one of the two electron sites is located at the impurity, while the other lies elsewhere in the lattice. We also determine the Uhlmann mixed state geometric phase, recently introduced in the study of the structural change of the system state eigenvectors in the vicinity of the lines of thermal phase transitions, and find it to be trivial, both for one- and two-site partial states, except when an electron site is at the impurity. This means that the system partial state eigenvectors do not contribute significantly to the enhanced state distinguishability around the point of this quantum phase transition. Finally, we use the fidelity to analyze the total amount of correlations contained within a composite system, showing that, even for the smallest one-site states, it features an abrupt quantitative change in the vicinity of the point of the quantum phase transition.

Macroscopic distinguishability between quantum states defining different phases of matter: Fidelity and the Uhlmann geometric phase

We study the fidelity approach to quantum phase transitions (QPTs) and apply it to general thermal phase transitions (PTs). We analyze two particular cases: The Stoner-Hubbard itinerant electron model of magnetism and the BCS theory of superconductivity. In both cases we show that the sudden drop of the mixed state fidelity marks the line of the phase transition. We conduct a detailed analysis of the general case of systems given by mutually commuting Hamiltonians, where the nonanalyticity of the fidelity is directly related to the nonanalyticity of the relevant response functions (susceptibility and heat capacity), for the case of symmetry-breaking transitions. Further, on the case of BCS theory of superconductivity, given by mutually noncommuting Hamiltonians, we analyze the structure of the system's eigenvectors in the vicinity of the line of the phase transition showing that their sudden change is quantified by the emergence of a generically nontrivial Uhlmann mixed state geometric phase.

Geometric phase for mixed states: Formulation based on a connection on a vector bundle over a Grassmann manifold

A geometrical description of open quantum systems is presented. This is based on a connection on a vector bundle over a Grassmann manifold, as well as the Anandan geometric phase for pure states [J. Anandan, Phys. Lett. A 133, 171 (1988)]. The geometric phase proposed here also naturally includes the geometric phase in adiabatic open quantum systems proposed by Sarandy and Lidar [Phys. Rev. A 73, 062101 (2006)]. The present geometrical description can be applied to all the quantum systems described by time-local master equations for density operators.

Geometric phases for mixed states and decoherence

The gauge invariance of geometric phases for mixed states is analyzed by using the hidden local gauge symmetry which arises from the arbitrariness of the choice of the basis set defining the coordinates in the functional space. This approach gives a reformulation of the past results of adiabatic, non-adiabatic and mixed state geometric phases. The geometric phases are identified uniquely as the holonomy associated with the hidden local gauge symmetry which is an exact symmetry of the Schrödinger equation. The purification and its inverse in the description of de-coherent mixed states are consistent with the hidden local gauge symmetry. A salient feature of the present formulation is that the total phase and visibility in the mixed state, which are directly observable in the interference experiment, are manifestly gauge invariant.

Non-Cyclic Geometric Phases in Mixed State Neutron Polarimetry

We have measured the non-cyclic geometric phase acquired during a unitary SU(2) evolution of a neutron spinor wave function for the mixed state case. The data is in good agreement with theory, verifying the predicted values for the mixed state geometric phase.

Geometric Phases for Mixed States in Trapped Ions

The generalization of geometric phase from the pure states tothe mixed states may have potential applications in constructing geometricquantum gates. We here investigate the mixed state geometric phases andvisibilities of the trapped ion system in both non-degenerate and degeneratecases. In the proposed quantum system, the geometric phases are determinedby the evolution time, the initial states of trapped ions, and the initialstates of photons. Moreover, special periods are gained under which thegeometric phases do not change with the initial states changing of photonparts in both non-degenerate and degenerate cases. The high detectionefficiency in the ion trap system implies that the mixed stategeometric phases proposed here can be easily tested.