Noncyclic Geometric Phase Research Articles

State‐Independent Geometric Quantum Gates via Nonadiabatic and Noncyclic Evolution

AbstractGeometric phases are robust to local noises and the nonadiabatic ones can reduce the evolution time, thus nonadiabatic geometric gates have strong robustness and can approach high fidelity. However, the advantage of geometric phase has not been fully explored in previous investigations. Here,a scheme is proposed for universal quantum gates with pure nonadiabatic and noncyclic geometric phases from smooth evolution paths. In the scheme, only geometric phase can be accumulated in a fast way, and thus it not only fully utilizes the local noise resistant property of geometric phase but also reduces the difficulty in experimental realization. Numerical results show that the implemented geometric gates have stronger robustness than dynamical gates and the geometric scheme with cyclic path. Furthermore, it proposes to construct universal quantum gate on superconducting circuits, with the fidelities of single‐qubit gate and nontrivial two‐qubit gate can achieve 99.97% and 99.87%, respectively. Therefore, these high‐fidelity quantum gates are promising for large‐scale fault‐tolerant quantum computation.

Dynamically tunable broadband output coupling of optical oscillators based on non-cyclic geometric phase mirror

We present a uniquely versatile and efficient mirror system capable of real-time fine-tuning in reflection and transmission properties across a broad wavelength range and at a high optical power. Leveraging the principles of the non-cyclic geometric phase (GP) acquired by the clockwise and counterclockwise beams of the Sagnac interferometer satisfying the anti-resonant condition on propagation through the quarter-wave plate, half-wave plate, and quarter-wave plate combination having fast axes oriented at 45° (fixed), θ (variable), and −45° (fixed) with respect to the vertical, respectively, our mirror system offers dynamic transmission control across 0–100% without the need for realignment. Notably, the GP-based mirror (GP-mirror) preserves the polarization state of the reflected beam, making it ideal for polarization-sensitive applications. The wavelength insensitivity of the GP enables seamless operation of the mirror across a wide wavelength range. As a proof-of-principle, we use the GP-mirror as the output coupler of a continuous-wave, green-pumped, doubly resonant optical parametric oscillator (DRO) based on a 30-mm-long MgO:sPPLT crystal and obtain stable operation at high powers over a wide wavelength tuning range. For a pump power of 5 W, the DRO provides an output power of 2.45 W at an extraction efficiency as high as 49% when operated at optimum output coupling. The DRO shows a maximum pump depletion of 89% and delivers an optimum output power across a tuning range ≥90 nm. The demonstrated concept offers a promising approach for advancing the capabilities and control of coherent optical sources tunable across different spectral regions and in all time scales from continuous-wave to ultrafast femtosecond domain.

Comment on: “Geometric phase of neutrinos: Differences between Dirac and Majorana neutrinos” [Phys. Lett. B 780 (2018) 216

In [5] (2018), Capolupo et al. calculate the noncyclic geometric phases associated with neutrino flavor transitions in the two-neutrino scenario, and claim that the results depend on the choice of representations of neutrino mixing matrix thus can be used to distinguish between Dirac and Majorana neutrinos. In this paper, we show that their calculation is flawed due to incorrect application of the definition of noncyclic geometric phase and the omission of one term in Wolfenstein effective Hamiltonian. Their results are neither gauge invariant nor lepton field rephasing invariant. We present an alternative calculation, which is solely in order to demonstrate that the Majorana CP-violating phase enters the geometric phase essentially by lepton field rephasing transformation. We point out that the seemingly nontrivial dependence of geometric phase on Majorana CP-violating phase presented in Capolupo et al. (2018) [5] is unphysical and thus unmeasurable.

An experimental test of the geodesic rule proposition for the noncyclic geometric phase.

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. In noncyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental demonstration of this geodesic rule proposition in different systems is of great interest, especially due to the potential use in quantum technology. Here, we report a previously unshown experimental confirmation of the geodesic rule for a noncyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and π jumps. We show the connection between our results and the Pancharatnam phase. Last, we point out that the geodesic rule may be applied to obtain the red shift in general relativity, enabling a new quantum tool to measure gravity.

Cosmological birefringence and the geometric phase of photons

Regarding axion electrodynamics in the background flat FRW universe, we show that cosmological birefringence arises from an adiabatic noncyclic geometric phase that appears in the quantum state of photons because of their interaction with the axion field. We also show that the axion electrodynamics is equivalent to standard electrodynamics in time-dependent bi-isotropic magnetoelectric Tellegen media, which serves as an analogue system that can simulate cosmological birefringence.

Dynamical quantum phase transitions in non-Hermitian lattices

In closed quantum systems, a dynamical phase transition is identified by nonanalytic behaviors of the return probability as a function of time. In this work, we study the nonunitary dynamics following quenches across exceptional points in a non-Hermitian lattice realized by optical resonators. Dynamical quantum phase transitions with topological signatures are found when an isolated exceptional point is crossed during the quench. A topological winding number defined by a real, noncyclic geometric phase is introduced, whose value features quantized jumps at critical times of these phase transitions and remains constant elsewhere, mimicking the plateau transitions in quantum Hall effects. This work provides a simple framework to study dynamical and topological responses in non-Hermitian systems.

Geometric phase of neutrinos: Differences between Dirac and Majorana neutrinos

We analyze the non-cyclic geometric phase for neutrinos. We find that the geometric phase and the total phase associated to the mixing phenomenon provide a theoretical tool to distinguish between Dirac and Majorana neutrinos. Our results hold for neutrinos propagating in vacuum and through the matter. We feed the values of the experimental parameters in our formulas in order to make contact with experiments. Although it remains an open question how the geometric phase of neutrinos could be detected, our theoretical results may open new scenarios in the investigation of the neutrino nature.

Noncyclic geometric phases and helicity transitions for neutrino oscillations in a magnetic field

We show that neutrino spin and spin-flavor transitions involve nonvanishing geometric phases. The geometric character of neutrino spin rotation is explored by studying the neutrino spin trajectory in the projective Hilbert space representation and its relation to the geometric phase. Analytical expressions are derived for noncyclic geometric phases. Several calculations are performed for different cases of rotating and nonrotating magnetic fields in the context of solar neutrinos and neutrinos produced inside neutron stars. Also the effects of adiabaticity, critical magnetic fields and cross boundary effects in the case of neutrinos emanating out of neutron stars are examined.

Noncyclic geometric quantum computation and preservation of entanglement for a two-qubit Ising model

After presenting an exact analytical solution of time-dependent Schrodinger equation, we study the dynamics of entanglement for a two-qubit Ising model. One of the spin qubits is driven by a static magnetic field applied in the direction of the Ising interaction, while the other is coupled with a rotating magnetic field. We also investigate how the entanglement can be controlled by changing the external parameters. Because of the important role of maximally entangled Bell states in quantum communication, we focus on the generalized Bell states as the initial states of the system. It is found that the entanglement evolution is independent of the initial Bell states. Moreover, we can preserve the initial maximal entanglement by adjusting the angular frequency of the rotating field or controlling the exchange coupling between spin qubits. Besides, our calculation shows that the entanglement dynamics is unaffected by the static magnetic field imposed in the direction of the Ising interaction. This is an interesting result, because, as we shall show below, this driving field can be used to control and manipulate the noncyclic geometric phase without affecting the system entanglement. Besides, the nonadiabatic and noncyclic geometric phase for evolved states of the present system are calculated and described in detail. In order to identify the unusable states for quantum communication, completely deviated from the initial maximally entangled states, we also study the fidelity between the initial Bell state and the evolved state of the system. Interestingly, we find that these unusable states can be detected by geometric quantum computation.

Unification of the family of Garrison-Wright's phases.

Inspired by Garrison and Wight's seminal work on complex-valued geometric phases, we generalize the concept of Pancharatnam's “in-phase” in interferometry and further develop a theoretical framework for unification of the abelian geometric phases for a biorthogonal quantum system modeled by a parameterized or time-dependent nonhermitian hamiltonian with a finite and nondegenerate instantaneous spectrum, that is, the family of Garrison-Wright's phases, which will no longer be confined in the adiabatic and nonadiabatic cyclic cases. Besides, we employ a typical example, Bethe-Lamb model, to illustrate how to apply our theory to obtain an explicit result for the Garrison-Wright's noncyclic geometric phase, and also to present its potential applications in quantum computation and information.

Noncyclic geometric phase in counting statistics and its role as an excess contribution

We propose an application of fiber bundles to counting statistics. The framework of the fiber bundles gives a splitting of a cumulant generating function for a current in a stochastic process, i.e., contributions from the dynamical phase and the geometric phase. We will show that the introduced noncyclic geometric phase is related to kinds of excess contributions, which have been investigated a lot in nonequilibrium physics. Using a specific nonequilibrium model, the characteristics of the noncyclic geometric phase are discussed; in particular, we reveal differences between a geometric contribution for the entropy production and the ‘excess entropy production’ which has been used to discuss the second law of steady-state thermodynamics.

The nonmodular topological phase and phase singularities

Generalizing an earlier definition of the noncyclic geometric phase [R. Bhandari, Phys. Lett. A 157 (1991) 221], a nonmodular topological phase is defined with reference to a generic time-dependent two-slit interference experiment involving particles with N internal states in which the internal state of both the beams undergoes unitary evolution. A simple proof of the shorter geodesic rule for closure of the open path is presented and several useful new insights into the behavior of the dynamical and geometrical components of the phase shift presented. An effective Hamiltonian interpretation of the observable phase shifts is also presented.

Entanglement dynamics via geometric phases in quantum spin chains

We introduce a connection between entanglement induced by interaction and geometric phases acquired by a composite quantum spin system. We begin by analyzing the evaluation of cyclic (Aharonov-Anandan) and non-cyclic (Mukunda-Simon) geometric phases for general spin chains evolving in the presence of time-independent magnetic fields. Then, by considering Heisenberg chains, we show that the interaction geometric phase, namely, the total geometric phase with subtraction of free spin contributions, is directly related to the global (Meyer-Wallach) entanglement exhibited by an initially separable state during its evolution in Hilbert space. This is analytically shown for N=2 spins and numerically illustrated for larger chains. This relationship promotes the interaction geometric phase to an indicator of global entanglement in the system, which may constitute a useful tool for quantum tasks based on entanglement as a resource to their performance.

Geometric phases for nonlinear coherent and squeezed states

The geometric phases for standard coherent states which are widely used in quantum optics have attracted considerable attention. Nevertheless, few physicists consider the counterparts of nonlinear coherent states, which are useful in the description of the motion of a trapped ion. In this paper, the non-unitary and non-cyclic geometric phases for two nonlinear coherent and one squeezed states are formulated, respectively. Moreover, some of their common properties are discussed, such as gauge invariance, non-locality and nonlinear effects. The nonlinear functions have dramatic impacts on the evolution of the corresponding geometric phases. They speed the evolution up or down. So this property may have an application in controlling or measuring geometric phase. For the squeezed case, when the squeezed parameter r → ∞, the limiting value of the geometric phase is also determined by a nonlinear function at a given time and angular velocity. In addition, the geometric phases for standard coherent and squeezed states are obtained under a particular condition. When the time evolution undergoes a period, their corresponding cyclic geometric phases are achieved as well. And the distinction between the geometric phases of the two coherent states may be regarded as a geometric criterion.

Non-cyclic geometric phase of nuclear quadrupole resonance signals of powdered samples

The non-cyclic geometric phase of 14 N and 35 Cl NQR signals induced by the character of trajectory of nuclear magnetization motion upon pulse r.f. excitation of powdered samples is studied. Analytical expressions for the geometric phases of NQR signals of the nuclei of spins I=1 and 3/2 upon nuclear magnetization rotation induced by means of r.f. pulses with frequency detuned from the resonance and for any impulse duration for a separate crystallite are obtained. It is shown that the geometric phase recorded for the signal from a powdered sample at Δ ω=0 can be different from zero and can oscillate upon changes in duration of the r.f. excitation pulse. An alternative variant of the nutation experiment aimed at obtaining the asymmetry parameter η from locations of frequency singularities in the nutation phase spectrum for nuclei of spin I=3/2 in powder substances is proposed.

Adiabatically steered open quantum systems: Master equation and optimal phase

We introduce an alternative way to derive the generalized form of the master equation recently presented by J. P. Pekola et al. [Phys. Rev. Lett. 105, 030401 (2010)] for an adiabatically steered two-level quantum system interacting with a Markovian environment. The original derivation employed the effective Hamiltonian in the adiabatic basis with the standard interaction picture approach but without the usual secular approximation. Our approach is based on utilizing a master equation for a nonsteered system in the first superadiabatic basis. It is potentially efficient in obtaining higher-order equations. Furthermore, we show how to select the phases of the adiabatic eigenstates to minimize the local adiabatic parameter and how this selection leads to states which are invariant under a local gauge change. We also discuss the effects of the adiabatic noncyclic geometric phase on the master equation.

Geometric Phase for Two Entangled Spin-1/2 Particles in a Magnetic Field

In this paper the geometric phases of two entangled spin-1/2 particles in the presence and absence of spin-spin interaction are calculated. We also discuss the geometric phases when only one of the two particles is affected by the external magnetic field. Our results show that the geometric phase in this case is not equal to that of a single particle under the same evolution condition because of the effect of entanglement. We further study the entanglement dependence of the noncyclic geometric phases in the interacting and noninteracting spins under a time-independent uniform magnetic field. A general entanglement-dependence geometric phase is formulated.

Homotrinuclear Spin Cluster with Orbital Degeneracy in a Magnetic Field: Algebraic Dynamic Studies of the Geometric Phase January 25, 2008

Based on the homotrinuclear spin cluster having SU(2)⊗SU(2) symmetry with twofold orbital degeneracy τ = 1/2) and the SU(2) algebraic structures of both ŝ and τˆ subspaces in the external magnetic field, we calculate exactly the non-adiabatic energy levels and the cyclic and non-cyclic non-adiabatic geometric phase of the homotrinuclear spin cluster by making use of the method of algebraic dynamics. The solution will show that the Berry phase is much influenced by the parameters N =γs/γτ (γs and γτ are the magnetic momentums of ŝ and τ̂ subspaces, respectively) in addition to ω/Ω in a rotating magnetic field. The change of the Berry phase in the basis state of the system is demonstrated from the changing diagram.

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.

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.