Geometric phase in entangled bipartite systems

The geometric phase (GP) for bipartite systems in transverse external magnetic fields is investigated in this paper. Two different situations have been studied. We first consider two non-interacting particles. The results show that because of entanglement, the geometric phase is very different from that of the non-entangled case. When the initial state is a Werner state, the geometric phase is, in general, zero and moreover the singularity of the geometric phase may appear with a proper evolution time. We next study the geometric phase when intra-couplings appear and choose Werner states as the initial states to entail this discussion. The results show that unlike our first case, the absolute value of the GP is not zero, and attains its maximum when the rescaled coupling constant J is less than 1. The effect of inhomogeneity of the magnetic field is also discussed.     

Unconventional geometric phase gate and multiqubit entanglement for hot ions with a frequency-modulated field

We present an alternative scheme for implementing the unconventional geometric two-qubit phase gate and prepar-ing multiqubit entanglement by using a frequency-modulated laser field to simultaneously illuminate all ions.Selecting the index of modulation yields selective mechanisms for coupling and decoupling between the internal and the external states of the ions.By the selective mechanisms,we obtain the unconventional geometric two-qubit phase gate,multipar-ticle Greenberger-Horne-Zeilinger states and highly entangled cluster states.Our scheme is insensitive to the thermal motion of the ions.     

Quantum geometric phase between orthogonal states

We show that the geometric phase between any two states, including orthogonal states, can be extracted and measured using the notion of projective measurement, and we show that a topological number can be extracted in the geometric phase change in an infinitesimal loop near an orthogonal state. Also, the Pancharatnam phase change during the passage through an orthogonal state is shown to be either pi or zero (mod 2pi). All the off-diagonal geometric phases can be obtained from the projective geometric phase calculated with our generalized connection.     

Geometric phase and Pancharatnam phase induced by light wave polarization

We use the quantum kinematic approach to revisit geometric phases associated with polarizing processes of a monochromatic light wave. We give the expressions of geometric phases for any, unitary or non-unitary, cyclic or non-cyclic transformations of the light wave state. Contrarily to the usually considered case of absorbing polarizers, we found that a light wave passing through a polarizer may acquire in general a nonzero geometric phase. This geometric phase exists despite the fact that initial and final polarization states are in phase according to the Pancharatnam criterion and cannot be measured using interferometric superposition. Consequently, there is a difference between the Pancharatnam phase and the complete geometric phase acquired by a light wave passing through a polarizer. We illustrate our work with the particular example of total reflection based polarizers.     

Geometric phase gate with trapped ions in thermal motion by adiabatic passage

We propose a scheme for implementing two-qubit geometric phase gate via the adiabatic evolution for trapped ions in thermal motion, leveraging on the stimulated Raman adiabatic passage with the geometric phase mechanism. Evolution along a dark state makes our scheme not only immune from decoherence due to spontaneous emission from excited states, but also rid off the dynamical phase. Furthermore, due to the opposite detuning of the driving lasers, the vibrational states of the trapped ions are only virtually excited during the operations, so our scheme is also insensitive to the occupation number of the vibrational mode.     

A new class of adiabatic cyclic states and geometric phases for non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field.     

Off-diagonal geometric phase for mixed states

We extend the off-diagonal geometric phase [Phys. Rev. Lett. 85, 3067 (2000)]] to mixed quantal states. The nodal structure of this phase in the qubit (two-level) case is compared with that of the diagonal mixed state geometric phase [Phys. Rev. Lett. 85, 2845 (2000)]]. Extension to higher dimensional Hilbert spaces is delineated. A physical scenario for the off-diagonal mixed state geometric phase in polarization-entangled two-photon interferometry is proposed.     

Unconventional geometric quantum phase gates with two SQUIDs in a cavity

We present a potential scheme to implement two-qubit quantum phase gates through an unconventional geometric phase shift with two four-level SQUIDs in a cavity. The SQUID qubits undergo no transitions during the gate operation, while the cavity mode is displaced along a circle in the phase space, acquiring a geometric phase depending conditionally upon the SQUIDs’ states. Under certain conditions, the SQUID qubits are disentangled with the cavity mode and the SQUIDs’ states remain in their ground states during the gate operation, thus the gate is insensitive to both the SQUIDs’ “spontaneous emission” and the cavity decay.     

Geometric phase contribution to quantum nonequilibrium many-body dynamics

We study the influence of geometry of quantum systems underlying space of states on its quantum many-body dynamics. We observe an interplay between dynamical and topological ingredients of quantum nonequilibrium dynamics revealed by the geometrical structure of the quantum space of states. As a primary example we use the anisotropic XY ring in a transverse magnetic field with an additional time-dependent flux. In particular, if the flux insertion is slow, nonadiabatic transitions in the dynamics are dominated by the dynamical phase. In the opposite limit geometric phase strongly affects transition probabilities. This interplay can lead to a nonequilibrium phase transition between these two regimes. We also analyze the effect of geometric phase on defect generation during crossing a quantum-critical point.     

Non-Adiabatic Geometric Phase in a Dispersive Interaction System

We investigate the geometric phase and dynamic phase of a two-level fermionic system with dispersive interaction, driven by a quantized bosonic field which is simultaneously subjected to parametric amplification. It is found that the geometric phase is induced by a counterpart of the Stark shift. This effect is due to distinct shifts in the field frequency induced by interaction between different states （｜e〉 and ｜g〉 ） and cavity field, and a simple geometric interpretation of this phenomenon is given, which is helpful to understand the natural origin of the geometric phase.     

Quantum Phase and Nonlocal Correlations for a Three-Level System Interacting with Laser Light in a Nonlinear Kerr Medium Under Decoherence

In this paper, we study the time evolution of the geometric phase and nonlocal correlations for a three-level atom interacting with the quantum field emerged in a nonlinear Kerr medium. We discuss the dependence of the physical quantifiers on the phase damping effect. We examine the effects of the initial state and different system parameters on the evolution of the nonlocal correlation and geometric phase with and without the phase damping effect. Furthermore, we explore the link between the geometric phase and the nonlocal correlation during the time evolution. Finally, we show that the model proposed will be very useful to avoid the phase damping effect by a proper choice of the physical parameters in the field for both cases of the initial pure and mixed states of the three-level atom.     

Geometric Phases for Mixed States in Trapped Ions

The generalization of geometric phase from the pure states to the mixed states may have potential applications in constructing geometric quantum gates. We here investigate the mixed state geometric phases and visibilities of the trapped ion system in both non-degenerate and degenerate cases. In the proposed quantum system, the geometric phases are determined by the evolution time, the initial states of trapped ions, and the initial states of photons. Moreover,special periods are gained under which the geometric phases do not change with the initial states changing of photon parts in both non-degenerate and degenerate cases. The high detection efficiency in the ion trap system implies that the mixed state geometric phases proposed here can be easily tested.     

Geometric Phases for Mixed States in Trapped Ions

The generalization of geometric phase from the pure states to the mixed states may have potential applications in constructing geometric quantum gates. We here investigate the mixed state geometric phases and visibilities of the trapped ion system in both non-degenerate and degenerate cases. In the proposed quantum system, the geometric phases are determined by the evolution time, the initial states of trapped ions, and the initial states of photons. Moreover, special periods are gained under which the geometric phases do not change with the initial states changing of photon parts in both non-degenerate and degenerate cases. The high detection efficiency in the ion trap system implies that the mixed state geometric phases proposed here can be easily tested.     

Geometric phases and coherent states

A cyclic evolution of a pure quantum state is characterized by a closed curve γ in the projective Hilbert space , equipped with the Fubini-Study geometry. It is known that the geometric phase for this evolution is given by the integral of the symplectic form of the Fubini-Study geometry over an arbitrary surface spanning γ. This result extends to an infinite-dimensional Hilbert space for a bosonic quantum field. We prove that is bounded above by the infimum area over all surfaces spanning γ, and that the bound is attained if γ can be spanned by a holomorphic curve. Using an earlier result concerning the intrinsic Euclidean geometry of the coherent state submanifold , we derive an expression for the geometric phase for a cyclic evolution amongst coherent states. We indicate how the intensity of a classical configuration can be inferred from the winding number of the exponential geometric phase about the origin in the complex plane. In the case of photon states we present group theoretic and 2-component spinor representations of . We derive an expression for in the case of a sequence of measurements such that the resulting states are coherent at each step, in terms of a sequence of projection operators. The situation in relation to some earlier experiments of Pancharatnam and Tomita–Chiao is explained.     

Interacting spin pairs in rotational magnetic fields and geometric phase

In virtue of the quantum invariant theory, we obtain the rigorous solution of the isotropic bipartite system in rotational magnetic fields, based on which the general expression of the noncyclic geometric phase is worked out and the entanglement dependence of the noncyclic geometric phase in this model is investigated. We show that the influence of the coupling on noncyclic geometric phase depends on the initial condition of the system. We also show that when the magnetic fields are stationary, there is a more general class of states existed of which the noncyclic geometric phase could be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parameterized by the evolving Schmidt coefficients.     

Preparation of Macroscopic Quantum-Interference States for a Collection of Trapped Ions Via a Single Geometric Operation

We describe a scheme for the generation of macroscopic quantum-interference states for a collection of trapped ions by a single geometric phase operation. In the scheme the vibrational mode is displaced along a circle with the radius proportional to the number of ions in a certain ground electronic state. For a given interaction time, the vibrational mode returns to the original state, and the ionic system acquires a geometric phase proportional to the area of the circle, evolving from a coherent state to a superposition of two coherent states. The ions undergo no electronic transitions during the operation. Taking advantage of the inherent fault-tolerant feature of the geometric operation, our scheme is robust against decoherence.     

Reference Section Method for Mixed State Geometric Phase in Nonunitary Evolution

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.     

Reference Section Method for Mixed State Geometric Phase in Nonunitary Evolution

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.     

Floquet states and geometrical phase of a multi-level system in cyclic evolution

We study the electronic states of a mesoscopic system whose Hamiltonian has a complicated static multi-level energy structure and undergoes periodic evolution in time. By using the Floquet theory, we derive the quasienergies, the Floquet states, and the geometrical phase. It is shown numerically that the geometrical phase is strongly dependent on the evolution circuits in the parameter space and on the evolution frequency of the varying Hamiltonian. In some cases the nonadiabatic geometric phases can exhibit chaotic behavior. We also show a trend of phase compensation in pairs of states which could restore the phase coherence if the pairing occurs.