The effect of inter-subsystem couplings on the Berry phase of a composite system as well as that of its subsystem is investigated in this paper. We analyze two coupled spin-$\frac 1 2 $ particles with one driven by a quantized field as an example, the pure state geometric phase of the composite system as well as the mixed state geometric phase for the subsystem is calculated and discussed.
The Berry phase in a composite system with only one subsystem being driven has been studied in this Letter. We choose two spin-$\frac 1 2 $ systems with spin-spin couplings as the composite system, one of the subsystems is driven by a time-dependent magnetic field. We show how the Berry phases depend on the coupling between the two subsystems, and what is the relation between these Berry phases of the whole system and those of the subsystems.
Berry phase, which had been discovered for more than two decades, provides us a very deep insight on the geometric structure of quantum mechanics. Its classical counterpart--Hannay's angle is defined if closed curves of action variables return to the same curves in phase space after a time evolution. In this paper, we study the Berry phase and Hannay's angle in a quantum-classical hybrid system under the Born-Oppenheimer approximation. By quantum-classical hybrid system, we denote a composite system consists of a quantum subsystem and a classical subsystem. The effects of subsystem-subsystem c…
Due to the potential application in quantum information process, geometric phase of interacting system arouse many interests. Some physicists concentrate on the system in pure classical envi- ronment, while others study the system in pure quantized environment. So a natural question is asked: how about an interacting system in composite environments made up of both classical and quantized field. In this letter, we analyze a quantum system composed of two interacting spins, of which one is in classical magnetic field and the other is in quantized field. First, classical magnetic field driven Be…
We present a general theoretical framework for the exact treatment of a hybrid system that is composed of a quantum subsystem and a classical subsystem. When the quantum subsystem is dynamically fast and the classical subsystem is slow, a vector potential is generated with a simple canonical transformation. This vector potential, on one hand, gives rise to the familiar Berry phase in the fast quantum dynamics; on the other hand, it yields a Lorentz-like force in the slow classical dynamics. In this way, the pure phase (Berry phase) of a wavefunction is linked to a physical force.
A quantized fermion can be represented by a scalar particle encircling a magnetic flux line. It has the spinor structure which can be constructed from quantum gates and qubits. We have studied here the role of Berry phase in removing dynamical phase during one qubit rotation of a quantized fermion. The entanglement of two qubit inserting spin-echo to one of them results the change of Berry phase that can be considered as a measure of entanglement. Some effort is given to study the effect of noise on the Berry phase of spinor and their entangled states.
We show in this comment that the results obtained in a recent work by Yi et al. [Phys. Rev. Lett. 92, 150406 (2004)] are quantitatively not correct and the proposed subsystem Berry phase is not well-defined.
The Berry phase of an anisotropic spin system that is adiabatically rotated along a closed circuit C is investigated. It is shown that the Berry phase consists of two contributions: (i) a geometric contribution which can be interpreted as the flux through C of a non-quantized Dirac monopole, and (ii) a topological contribution which can be interpreted as the flux through C of a Dirac string carrying a non-quantized flux, i.e., a spin analogue of the Aharonov-Bohm effect. Various experimental consequences of this novel effect are discussed.
In this reply, we show that the adiabatic theorem would break down in the weak coupling limit, and the definition for the subsystem geometric phase is well defined.
The Berry phase (BP) in a quantized light field demonstrated more than a decade ago (Phys. Rev. Lett. 89, 220404) has attracted considerable attentions, since it plays an important role in the cavity quantum electrodynamics. However, it is argued in a recent paper ( Phys. Rev. Lett. 108, 033601) that such a BP is just due to the rotating wave approximation (RWA) and the relevant BP should vanish beyond this approximation. Based on a consistent analysis we conclude in this letter that the BP in a generic Rabi model actually exists, no matter whether the RWA is applied. The existence of BP is al…
The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided c…
We calculate the Berry phase of a spin-1/2 particle in a magnetic field considering the quantum nature of the field. The phase reduces to the standard Berry phase in the semiclassical limit and eigenstate of the particle acquires a phase in the vacuum. We also show how to generate a vacuum induced Berry phase considering two quantized modes of the field which has a interesting physical interpretation.
The experimental realisation of the basic constituents of quantum information processing devices, namely fault-tolerant quantum logic gates, requires conditional quantum dynamics, in which one subsystem undergoes a coherent evolution that depends on the quantum state of another subsystem. In particular, the subsystem may acquire a conditional phase shift. Here we consider a novel scenario in which this phase is of geometric rather than dynamical origin. As the conditional geometric (Berry) phase depends only on the geometry of the path executed it is resilient to certain types of errors, and o…
Starting from the Schrödinger-equation of a composite system, we derive unified dynamics of a classical harmonic system coupled to an arbitrary quantized system. The classical subsystem is described by random phase-space coordinates entangled with the quantized variables of the complementary subsystem. Our semiclassical equation is {\it true} in a sense that its predictions are identical to those of the fully quantized composite dynamics. This exact method applies to a broad class of theories, including e.g. the relativistic quantum-electrodynamics and the electron-fonon dynamics.
Berry phases and quantum fidelities for interacting spins have attracted considerable attention, in particular in relation to entanglement properties of spin systems and quantum phase transitions. These efforts mainly focus either on spin pairs or the thermodynamic infinite spin limit, while studies of the multipartite case of a finite number of spins are rare. Here, we analyze Berry phases and quantum fidelities of the energetic ground state of a Lipkin-Meshkov-Glick (LMG) model consisting of three spin-1/2 particles (qubits). We find explicit expressions for the Berry phase and fidelity susc…
he Jaynes-Cummings model (JCM) is an very important model for describing interaction between quantized electromagnetic fields and atoms in cavity quantum electrodynamics (QED). This model is generalized in many different direction since it predicts many novel quantum effects that can be verified by modern physics experimental technologies. In this paper, the Berry phase and entropy of the ground state for arbitrary photon number $n$ of a two-atom Jaynes-Cummings model with Kerr like medium are investigated. It is found that there are some correspondence between their images, especially the exi…
For a quantum system subject to external parameters, the Berry phase is an intra-level property, which is gauge invariant module $2π$ for a closed loop in the parameter space and generally is non-quantized. In contrast, we define a inter-band character $Θ$ for a closed loop, which is gauge invariant and quantized as integer values. It is a quantum mechanical analogy of the Euler character based on the Gauss-Bonnet theorem for a manifold with a boundary. The role of the Gaussian curvature is mimicked by the difference between the Berry curvatures of the two levels, and the counterpart of the ge…
The Berry phase acquired by an electromagnetic field undergoing an adiabatic and cyclic evolution in phase space is a purely quantum-mechanical effect of the field. However, this phase is usually accompanied by a dynamical contribution and cannot be manifested in any light-beam interference experiment because it is independent of the field state. We here show that such a phase can be produced using an atom coupled to a quantized field and driven by a slowly changing classical field, and it is manifested in atomic Ramsey interference oscillations. We also show how this effect may be applied to …
We present both the gauge theoretic description and the numerical calculations of the Berry phases with the real eigenstates, involving one with a many-body system as a background and the other with no such background. We demonstrate that for the former the sign of the Berry phase factor for a spin $\f{1}{2}$ particle (hole) coupled to a slow subsystem (phonon) depends on both the strength of electron correlations and the characteristics of the closed paths, unlike the cases for the latter.
We investigate the quantization of adiabatic charge transport in the insulating ground state of finite systems. Topological charge pumps are used in experiments as an indicator of topological order. In the thermodynamic limit the transport can be related to a topological Berry phase and is thus strictly quantized. This is no longer true for finite systems. We derive finite-size corrections to the transport both for non-interacting and interacting systems and relate them to analytic properties of the single- and many-body Berry curvature. We find that they depend on the details of experimental …
We investigate quantum phase transitions, quantum criticality, and Berry phase for the ground state of an ensemble of non-interacting two-level atoms embedded in a non-linear optical medium, coupled to a single-mode quantized electromagnetic field. The optical medium is pumped externally through a classical electric field, so that there is a degenerate parametric amplification effect, which strongly modifies the field dynamics without affecting the atomic sector. Through a semiclassical description the different phases of this extended Dicke model are described. The quantum phase transition is…
We show that bosonic fields may present anyonic behavior when interacting with a fermion in a Jaynes-Cummings-like model. The proposal is accomplished via the interaction of a two-level system with two quantized modes of a harmonic oscillator; under suitable conditions, the system acquires a fractional geometric phase. A crucial role is played by the entanglement of the system eigenstates, which provides a two-dimensional confinement in the effective evolution of the system, leading to the anyonic behavior. For a particular choice of parameters, we show that it is possible to transmute the sta…
Geometric phases are well known in classical electromagnetism and quantum mechanics since the early works of Pantcharatnam and Berry. Their origin relies on the geometric nature of state spaces and has been studied in many different systems such as spins, polarized light and atomic physics. Recent works have explored their application in interferometry and quantum computation. Earlier works suggest how to observe these phases in single quantum systems adiabatically driven by external classical devices or sources, where, by classical, we mean any system whose state does not change considerably …
The level crossing problem is neatly formulated by the second quantized formulation, which exhibits a hidden local gauge symmetry. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval $T$. The topological proof of the Longuet-Higgins' phase-change rule, for example, thus fails in the practical Born-Oppenheimer approximation where $T$ is identified …
With reference to the vacuum induced Berry phase (VIBP) obtained in the interaction of a spin-1/2 particle with quantized irradiation field under rotating-wave approximation (RWA), we present completely different treatment for the VIBP by a fully quantum mechanical treatment beyond the RWA, which gives a new definition of the VIBP and indicates the validity of RWA to be relevant not only to the energy conservation and comparison of characteristic parameters, but also to more subtle physics, such as the geometric property of the state evolution. Our result is of conceptual importance and also o…
By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval $T$. The topological interpretation of Berry's phase such as the topological proof of phase-change rule thus fails in the practical Bor…
We show how the entropy operators for two subsystems may be calculated. In the case of the atom-field interaction we obtain the associated Wigner function for the entropy operator for the quantized field.
We examine Berry phase pertaining to purely quadrupolar state ($\langle ψ| \vec{S} | ψ\rangle = 0$) of a spin-$1$ system. Using the Majorana stellar representation of these states, we provide a visualization for the topological (zero or $π$) nature of such quadrupolar Berry phase. We demonstrates that the $π$ Berry phase of quadrupolar state is induced by the Majorana stars collectively tracing out a closed path (a great circle) by exchanging their respective positions on the Bloch sphere. We also analyse the problem from the perspective of dynamics where a state from the quadrupolar subspace …
The geometric (Berry) phase of a two-level system in a dissipative environment is analyzed by using the second-quantized formulation, which provides a unified and gauge-invariant treatment of adiabatic and nonadiabatic phases and is thus applicable to a quantitative analysis of transitional regions away from ideal adiabaticity. In view of the recent experimental observation of the Berry phase in a superconducting qubit, we illustrate our formulation for a concrete adiabatic case in the Ohmic dissipation. The correction to the total phase together with the geometry-dependent dephasing time is g…
The dynamics of an atomic few-level system can depend on the phase of driving fields coupled to the atom if certain conditions are satisfied. This is of particular interest to control interference effects, which can alter the system properties considerably. In this article, we discuss the mechanisms of such phase control and interference effects in an atomic three-level system in $Λ$ configuration, where the upper state spontaneously decays into the two lower states. The lower states are coupled by a driving field, which we treat as quantized. This allows for an interpretation on the single ph…
We argue that the entanglement Chern number proposed recently is invariant under the adiabatic deformation of a gapped many-body groundstate into a {\it disentangled/purified} one, which implies a partition of the Chern number into subsystems (disentangled Chern number). We generalize the idea to another topological number, the Z$_2$ Berry phase for a system with particle-hole symmetry, and apply it to a groundstate in a weak topological phase where the Chern number vanishes but the groundstate nevertheless has edge states. This entanglement Berry phase is especially useful for characterizing …
We investigate some aspects of the dynamics and entanglement of bipartite quantum system (atom-quantized field), coupled to a third ``external" subsystem (quantized field). We make use of the Raman coupled model; a three-level atom in a lambda configuration interacting with two modes of the quantized cavity field. We consider the far off resonance limit, which allows the derivation of an effective Hamiltonian of a two-level atom coupled to the fields. We also make a comparison with the situation in which one of the modes is treated classically rather than prepared in a quantum field (coherent …
Quasi-periodically driven quantum systems are predicted to exhibit quantized topological properties, in analogy with the quantized transport properties of topological insulators. We use a single nitrogen-vacancy center in diamond to experimentally study a synthetic quantum Hall effect with a two-tone drive. We measure the evolution of trajectories of two quantum states, initially prepared at nearby points in synthetic phase space. We detect the synthetic Hall effect through the predicted overlap oscillations at a quantized fundamental frequency proportional to the Chern number, which character…
Quantized Chern-Simons Axion Coupling in Anomalous Floquet Systems
Université Libre de Bruxelles（布鲁塞尔自由大学）；International Solvay Institutes（国际索尔维研究所）；Collège de France（法兰西公学院）；CNRS（法国国家科学研究中心）；ENS-Université PSL（巴黎高等师范学院-巴黎文理研究大学）；Sorbonne Université（索邦大学）；Centro Atómico Bariloche（巴里洛切原子能中心）；Instituto Balseiro（巴尔塞罗研究所）；Comisión Nacional de Energía Atomica (CNEA)（国家原子能委员会）；Universidad Nacional de Cuyo (UNCUYO)（门多萨国立大学）；Instituto de Nanociencia y Nanotecnología (INN-Bariloche)（纳米科学与技术研究所）；Consejo Nacional de Investigaciones Científicas y Tecnicas (CONICET)（国家科学研究委员会）
We have studied here the geometrodynamics of relativistic electron vortex beams from the perspective of the geometric phase associated with the scalar electron encircling the vortex line. It is pointed out that the electron vortex beam carrying orbital angular momentum is a natural consequence of the skyrmion model of a fermion. This follows from the quantization procedure of a fermion in the framework of Nelson's stochastic mechanics when a direction vector (vortex line) is introduced to depict the spin degrees of freedom. In this formalism a fermion is depicted as a scalar particle encirclin…
Physics arising from two-dimensional~(2D) Dirac cones has been a topic of great theoretical and experimental interest to studies of gapless topological phases and to simulations of relativistic systems. Such $2$D Dirac cones are often characterized by a $π$ Berry phase and are destroyed by a perturbative mass term. By considering mean-field nonlinearity in a minimal two-band Chern insulator model, we obtain a novel type of Dirac cones that are robust to local perturbations without symmetry restrictions. Due to a different pseudo-spin texture, the Berry phase of the Dirac cone is no longer quan…
