Study On The Noisy Discrete-time Quantum Walk

As a generalization of classical random walk to the field of quantum mechanics,quantum walk displays many characteristics different from those of classical random walk because of quantum coherence.On the one hand,quantum walk could realize the unitary evolution of quantum superposition states,which corresponds to the parallel computing in quantum algorithm.Therefore,the quantum algorithm based on quantum walk displays exponential acceleration compared with the classical algorithm.On the other hand,quantum walk could simulate the dynamic behaviors of many physical systems such as Anderson localization and topology,which provides a broad platform for the simulation of novel physical phenomena.According to the characteristics of evolution time of the walker,quantum walk can be categorized into two types:continuous-time quantum walk and discrete-time quantum walk.The latter is widely concerned by researchers because of its simple and diversified physical implementations and adjustable parameters.The walker of discrete-time quantum walk has spin(coin)and position degrees of freedoms.The unitary evolution protocol is generally implemented by combining conditional shift operators and coin operators,which act on the position of a walker and its associated coin,respectively.A realistic physical system is always inevitably affected by the environment or noise.A large number of studies have explored the dynamic characteristics of discrete-time quantum walk with spatial or temporal non-uniform noise.However,there is no temporal or spatial correlation in noise.In this paper,we explored the dynamics of discrete-time quantum walk with time-correlated noise.The noise is modeled as a unitary stochastic coin-type opera-tor and it is generated by a sample path of the Ornstein-Uhlenbeck process.Continuously applying the stochastic coin-type operator and the unitary evolution operator to the wave function,the dynamic of the particle in a single noise realization is obtained.Finally,the exact dynamic of the noisy quantum walk is obtained by averaging the dynamic over all pos-sible noise realizations.Based on the first-order approximation of Baker-Campbell-Hausdorff formula,the master equation of noisy quantum walk is derived.Similarity is defined to mea-sure the consistency between the particle's dynamics described by the master equation and the exact dynamic.If the similarity is greater than 0.8,we say the master equation can well describe the noisy quantum walk.The results show that with the increase of evolution time,the similarity will gradually decrease from the maximum value 1.If the time that the similarity decreases from 1 to 0.8 is called the valid time of the master equation,it can be found that the longer the noise correlation time is,the smaller the amplitude is,the longer the effective time of the main equation will be.Because the master equation only holds in the valid time,we also numerically simulate the dynamics of the noisy quantum walk.It is found that the noise always makes the dynamics of the walker transit from quantum walk to classical random walk,which is reflected by the space distribution,standard deviation of position,interference and other physical quantities.The unitary evolution protocol of discrete-time quantum walk couples particle's coordi-nate and spin degree of freedoms.In this sense,the random walk of particles simulates the influence of spin orbit coupling on particle dynamics.More precisely,the unitary quantum-walk protocol simulates continuous evolution under an effective spin-orbit Hamiltonian,which presents the spin-orbit coupling effect.Since the unitary protocol is translation invariant,the effective Hamiltonian exhibits Bloch energy band,and each energy level has a certain spin polarization.When the walker's momentum traverses the first Brillouin region,the spin in the energy band gradually changes back to the initial state,the band structure of such spin-orbit-coupled Hamiltonians can feature quantized topological invariants.The spin-orbit coupling effect enables the discrete-time quantum walk could simulate the topological non-trivial system,which is also the fundamental reason for the mental surface state in topological materials.Due to the spin orbit coupling effect in topological insulator materials,the spin and momentum of electron in the upper(lower)band are locked.Now we study the spin po-larization of the surface-state electron coupled to a single-mode quantum field.Firstly,we derive the operator expressions for quantized?-polarized light,?-polarized light and circular-polarized light.Taking the spin polarization of the surface-state electron in Bi₂Se₃coupled to?-polarized light as an example,beyond the rotating wave approximation,the approximate eigenstates are obtained.The expected value of pseudo-spin(Pauli operators)on the eigen-state could be derived.Then,multiplying the expected value of pseudo-spin with a constant g-factor,we can get the spin polarization of the electron coupled to a single-mode,quantized external field.The electron's spin polarization in the other two cases can be calculated by the same method.Besides,we also studied the spin polarization of the electron in Sm B₆coupled to quantized external field.These results show that interacting with the quantized single-mode field,the orientation of the electron's spin polarization remain unchanged but the magnitude of spin will be smaller compared with the case without external field.Adjusting the photon incident angle to the sample surface,the spin polarization of the electron will rotate synchronously with the change of azimuth.