Zitterbewegung (ZB, the trembling motion) of free relativistic electrons in a vacuum in the presence of an external magnetic field is calculated. It is shown that the motion of an electron wave packet has intraband frequency components, corresponding to the classical cyclotron motion, and several interband frequency components corresponding to the Zitterbewegung. For a two-dimensional situation, the presence of a magnetic field makes the ZB motion stationary, i.e. not decaying in time. We show how to simulate the ZB in a magnetic field using trapped ions and laser excitations in the spirit of recently observed proof-of-principle ZB simulation by Gerritsma {\it et al.} Nature {\bf 463}, 68 (2010). It is demonstrated that, for the parameters of the Dirac equation simulated by the above experiment, the effect of a magnetic field on the Zitterbewegung is considerable.
We review recent research on Zitterbewegung (ZB, trembling motion) of electrons in semiconductors. A brief history of the subject is presented, the trembling motion in semirelativistic and spin systems is considered and its main features are emphasized. Zitterbewegung of charge carriers in monolayer and bilayer graphene as well as in carbon nanotubes is elaborated in some detail. We describe effects of an external magnetic field on ZB using monolayer graphene as an example. Nature of electron ZB in crystalline solids is explained. We also review various simulations of the trembling motion in a…
We present a canonical formalism facilitating investigations of the dynamical Casimir effect by means of a response theory approach. We consider a massless scalar field confined inside of an arbitaray domain $G(t)$, which undergoes small displacements for a certain period of time. Under rather general conditions a formula for the number of created particles per mode is derived. The pertubative approach reveals the occurance of two generic processes contributing to the particle production: the squeezing of the vacuum by changing the shape and an acceleration effect due to motion af the boundari…
The physical model (PhsMdl) of a Schrodinger nonrelativistic quantized electron (ShEl) is built by means of a transition of the quadratic differential particle equation of Hamilton-Jacoby into the quadratic differential wave equation of Schrodinger in this work, which interprets the physical reason of its quantum (wave and stochastic) behaviour by explanation of the physical reason, which forces the classical Lorentz electron (LrEl) to participate in Furthian quantized stochastic oscillation motion, which turn it into quantum ShEl. It is performed that this transition is realized by my conside…
The Klein-Gordon equation is used to calculate the Zitterbewegung (ZB, trembling motion) of spin-zero particles in absence of fields and in the presence of an external magnetic field. Both Hamiltonian and wave formalisms are employed to describe ZB and their results are compared. It is demonstrated that, if one uses wave packets to represent particles, the ZB motion has a decaying behavior. It is also shown that the trembling motion is caused by an interference of two sub-packets composed of positive and negative energy states which propagate with different velocities. In the presence of a mag…
Zitterbewegung (ZB) is a phenomenon in relativistic quantum systems where the electron wave packet exhibits a trembling or oscillating behavior during its motion, caused by its interaction or coupling with the negative energy state. To directly observe ZB in electronic systems is difficult, due to the challenges associated with the atomic scale wavelength of the electron. Photonic systems offer an alternative paradigm. We exploit the concept of pseudo parity-time (pseudo PT ) symmetry to study ZB in non-Hermitian quantum systems implemented as an experimentally feasible optical waveguide array…
Zitterbewegung (ZB), the trembling of free relativistic electrons in a vacuum could be simulated by a single trapped ion. We focus on the variations of ZB under different parity conditions and find no ZB in the case of odd or even parity. ZB occurs only for admixture of the odd and even parity states. We also show the similar role played by the parity operator for the trapped ion in Fock-state representation and the space inversion operator for a realistic relativistic electron. Although the ZB effect is invisible in a relativistic electron, preparation of the trapped ion in different parity s…
Using Dirac equation together with the Wigner distribution function,the trembling motion,known as Zitterbewegung effect,of moving electrons in quasi-one-dimensional relativistic quantum plasma is theoretically investigated.The relativistic Wigner matrix is used to calculate the mean values of the position and velocity operators for a Dirac gas of electrons.It is found that the oscillatory behavior of measurable quantities could be associated with the Zitterbewegung effect which manifests itself as a damping interference pattern stemming from mixing the positive and negative dispersion modes of…
We study the effect of an in-plane magnetic field on the zitterbewegung (ZB) of electrons in a semiconductor quantum well (QW) and in a quantum dot (QD) with the Rashba and Dresselhaus spin-orbit interactions. We obtain a general expression of the time-evolution of the position vector and current of the electron in a semiconductor quantum well. The amplitude of the oscillatory motion is directly related to the Berry connection in momentum space. We find that in presence of the magnetic field the ZB in a quantum well does not vanish when the strengths of the Rashba and Dresselhaus spin-orbit in…
We study classical and quantum dynamics of a kicked relativistic particle confined in a one dimensional box. It is found that in classical case for chaotic motion the average kinetic energy grows in time, while for mixed regime the growth is suppressed. However, in case of regular motion energy fluctuates around certain value. Quantum dynamics is treated by solving the time-dependent Dirac equation for delta-kicking potential, whose exact solution is obtained for single kicking period. In quantum case, depending on the values of the kicking parameters the average kinetic energy can be quasi pe…
We derive a general and simple expression for the time-dependence of the position operator of a multi-band Hamiltonian with arbitrary matrix elements depending only on the momentum of the quasi-particle. Our result shows that in such systems the Zitterbewegung like term related to a trembling motion of the quasi-particle, always appears in the position operator. Moreover, the Zitterbewegung is, in general, a multi-frequency oscillatory motion of the quasi-particle. We derive a few different expressions for the amplitude of the oscillatory motion including that related to the Berry connection m…
Using a minimal-coupling-scheme we investigate the quantum Brownian motion of a particle in an anisotropic-dissipative-medium under the influence of an arbitrary potential in both relativistic and non-relativistic regimes. A general quantum Langevin equation is derived and explicit expressions for quantum-noise and dynamical variables of the system are obtained. The equations of motion for the canonical variables are solved explicitly and an expression for the radiation-reaction of a charged particle in the presence of a dissipative-medium is obtained. Some examples are given to elucidate the …
A general formalism for obtaining the Lagrangian and Hamiltonian for a one dimensional dissipative system is developed. The formalism is illustrated by applying it to the case of a relativistic particle with linear dissipation. The relativistic wave equation is solved for a free particle with linear dissipation.
Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.
The quantum mechanical motion of a relativistic particle in a non-continuous spacetime is investigated. The spacetime model is a dense, rationale subset of two-dimensional Minkowski spacetime. Solutions of the Dirac equation are calculated using a generalized version of Feynman's checkerboard model. They turn out to be closely related to the continuum propagator.
Around 1930, both Gregory Breit and Erwin Schroedinger showed that the eigenvalues of the velocity of a particle described by wavepacket solutions to the Dirac equation are simply $\pm$c, the speed of light. This led Schroedinger to coin the term Zitterbewegung, which is German for "trembling motion", where all particles of matter (fermions) zig-zag back-and-forth at only the speed of light. The result is that any finite speed less than $c$, including the state of rest, only makes sense as a long-term average that can be thought of as a drift velocity. In this paper, we seriously consider this…
Relativistic free-motion time-of-arrival theory for massive spin-1/2 particles is systematically developed. Contrary to the nonrelativistic time-of-arrival operator studied thoroughly in previous literatures, the relativistic time-of-arrival operator possesses self-adjoint extensions because of the particle-antiparticle symmetry. The nonrelativistic limit of our theory is in agreement with the nonrelativistic time-of-arrival theory. By comparing the time-of-arrival operator with the Hamiltonian operator of a free Dirac particle, one can show a duality between position space and momentum space.
We show that the operational definition of non-classicality of a quantum state depends on the motion of the observer. We use the relativistic Unruh-DeWitt detector model to witness nonclassicality of the probed field state. It turns out that the witness based on the properties of the P-representation of the quantum state depends on the trajectory of the detector. Inertial and noninertial motion of the device have qualitatively different impact on the performance of the witness.
One-electron 3+1 and 2+1 Dirac equations are used to calculate the motion of a relativistic electron in a vacuum in the presence of an external magnetic field. First, calculations are carried on an operator level and exact analytical results are obtained for the electron trajectories which contain both intraband frequency components, identified as the cyclotron motion, as well as interband frequency components, identified as the trembling motion (Zitterbewegung, ZB). Next, time-dependent Heisenberg operators are used for the same problem to compute average values of electron position and veloc…
The complete set of operators commuting with the Dirac Hamiltonian and exact analytic solution of the Dirac equation for the two-dimensional Coulomb potential is presented. Beyond the eigenvalue $μ$ of the operator $j_{z}$, two quantum numbers $η$ and $κ$ are introduced as eigenvalues of hermitian operators $P=βσ_{z}'$ and $K=β(σ_{z}'l_{z}+1/2)$, respectively. The classification of states according to the full set of constants of motion without referring to the non-relativistic limit is proposed. The linear Paschen-Back effect is analyzed using exact field-free wave-functions as a zero-order a…
The purpose of the present note is to propose, in the framework of relativistic quantum mechanics, a new Poincare-invariant equation for two particles with masses m_1, m_2 and spin s_1=s_2=1/2. It is a first-order linear differential equation for the eight-component wave function. With the help of this equation the description of the motion of two-particle systems is reduced to the description of one-particle systems in the (1+6)-dimensional Minkowski space which can be in two spin states (s=0 or s=1).
The physical model (PhsMdl) of the relativistic quantized Dirac's electron (DrEl) is proposed. The DrEl is regarded as a point-like (PntLk) elementary electric charge (ElmElcChrg), taking simultaneously part in following four disconnected different motions: a/ in Einstein's random trembling harmonic shudders as a result of momentum recoils (impulse kicks), forcing the DrEl's PntLk ElmElcChrg at its continuous emission and absorption of high energy stochastic virtual photons (StchVrtPhtns) by its PntLk ElmElcChrg ; b/ in Schrodinger's fermion vortical harmonic oscillations of DrEl's fine spread…
A Wigner-Klein-Kramers equation is proposed, which merges relativistic, quantum and thermo dynamics. The relativistic effect on quantum Brownian motion is studied via the Breit-Fermi Hamiltonian applied into a dissipative Madelung hydrodynamics. A new thermo-quantum Smoluchowski equation is derived, which accounts for the relativistic correction of the Bohm quantum potential.
Based on the mathematics of noncommutative geometry, we model a 'classical' Dirac fermion propagating in a curved spacetime. We demonstrate that the inherent causal structure of the model encodes the possibility of Zitterbewegung - the 'trembling motion' of the fermion. We recover the well-known frequency of Zitterbewegung as the highest possible speed of change in the fermion's 'internal space'. Furthermore, we show that the latter does not change in the presence of an external electromagnetic field and derive its explicit analogue when the mass parameter is promoted to a Higgs-like field. We…
We study the dynamics of an Airy wavepacket moving in a one-dimensional lattice potential. In contrast to the usual case of propagation in a continuum, for which such a wavepacket experiences a uniform acceleration, the lattice bounds its velocity, and so the acceleration cannot continue indefinitely. Instead, we show that the wavepacket's motion is described by relativistic equations of motion, which surprisingly, arise naturally from evolution under the standard non-relativistic Schrödinger equation. The presence of the lattice potential allows the wavepacket's motion to be controlled by mea…
A non-Grassmanian path integral representation is given for the solution of the Klein-Gordon and the Dirac equations. The trajectories of the path integral are rendered differentiable by the relativistic corrections. The nonrelativistic limit is briefly discussed from the point of view of the renormalization group.
We show that the relativistic motion of a quantum system can be used to generate quantum gates. The nonuniform acceleration of a cavity is used to generate well-known two-mode quantum gates in continuous variables. Observable amounts of entanglement between the cavity modes are produced through resonances which appear by repeating periodically any trajectory.
Relativistic Brownian motion can be inexpensively demonstrated on a graphene chip. The interplay of stochastic and relativistic dynamics, governing the transport of charge carrier in graphene, induces noise-controlled effects such as (i) a stochastic effective mass, detectable as a suppression of the particle mobility with increasing the temperature; (ii) a transverse ratchet effect, measurable as a net current orthogonal to an ac drive on an asymmetric substrate, and (iii) a chaotic stochastic resonance. Such properties can be of practical applications in the emerging graphene technology.
In this paper, we introduce a deterministic approach of quantum mechanics for particles with spin 1 2 moving in one dimension. We present a Lagrangian of a spinning particle ($s ={1 \over 2} $), and deduce the expression of the conjugate momentum related to the velocity of the particle.
The quantum channel between two particle detectors provides a prototype framework for the study of wireless quantum communication via relativistic quantum fields. In this article we calculate the classical channel capacity between two Unruh-DeWitt detectors arising from couplings within the perturbative regime. To this end, we identify the detector states which achieve maximal signal strength. We use these results to investigate the impact of relativistic effects on signaling between detectors in inertial and uniformly accelerated motion which communicate via a massless field in Minkowski spac…
We analyse particle creation and mode mixing for a quantum field in an accelerated cavity, assuming small accelerations but allowing arbitrary velocities, travel times and travel distances, and in particular including the regime of relativistic velocities. As an application, we identify a desktop experimental scenario where the mode mixing resonance frequency in linear sinusoidal motion or in uniform circular motion is significantly below the particle creation resonance frequencies of the Dynamical Casimir Effect, and arguably at the threshold of current technology. The mode mixing acts as a b…
The work reported in arXiv:1311.5619v1 proposes to produce continuous-variable cluster states through relativistic motion of cavities. This proposal does not produce the states claimed by the authors. The states actually produced are in general not known to be useful for measurement-based quantum computation.
We show that particle detectors, such as 2-level atoms, in non-inertial motion (or in gravitational fields) could be used to build quantum gates for the processing of quantum information. Concretely, we show that through suitably chosen non-inertial trajectories of the detectors the interaction Hamiltonian's time dependence can be modulated to yield arbitrary rotations in the Bloch sphere due to relativistic quantum effects.
We show how the dynamical modulation of the qubit-field coupling strength in a circuit quantum electrodynamics architecture mimics the motion of the qubit at relativistic speeds. This allows us to propose a realistic experiment to detect microwave photons coming from simulated acceleration radiation. Moreover, by combining this technique with the dynamical Casimir physics, we enhance the toolbox for studying relativistic phenomena in quantum field theory with superconducting circuits.
We show that the relativistic signatures on the transition probability of atoms moving through optical cavities are very sensitive to their spatial trajectory. This allows for the use of internal atomic degrees of freedom to measure small time-dependent perturbations in the proper acceleration of an atomic probe, or in the relative alignment of a beam of atoms and a cavity.
We show how to use relativistic motion and local phase shifts to generate continuous variable Gaussian cluster states within cavity modes. Our results can be demonstrated experimentally using superconducting circuits where tuneable boundary conditions correspond to mirrors moving with velocities close to the speed of light. In particular, we propose the generation of a quadripartite square cluster state as a first example that can be readily implemented in the laboratory. Since cluster states are universal resources for universal one-way quantum computation, our results pave the way for relati…
We follow recent work and study the relativistic d-brane system in (d+1,1) dimensions and its connection with a Galileo invariant system in (d,1) dimensions. In particular, we solve d-brane systems in (2,1), (3,1) and (4,1) dimensions and show that their solutions solve the corresponding Galileo invariant systems in (1,1), (2,1), and (3,1) dimensions. The results are extended to higher dimensions.
A vacuum medium model is advanced. The motion of a relativistic particle in relation to its interaction with the medium is discussed. It is predicted that elementary excitations of the vacuum, called "inertons," should exist. The equations of the particle path in Euclidean space are derived. The motion is marked by the relations basic for quantum mechanics: $E=hν$ and $Mv = h/ λ$ (here, $λ$ is the amplitude of spatial oscillations of the particle along the trajectory, i.e., the interval at which the velocity of the particle is periodically altered from $v$ to 0 and then from 0 to $v$; $ν$ is t…
We study the influence of three laser beams on the center of mass motion of cold atoms with internal energy levels in a tripod configuration. We show that similar to electrons in graphene the atomic motion can be equivalent to the dynamics of ultra-relativistic two-component Dirac fermions. We propose and analyze an experimental setup for observing such a quasi-relativistic motion of ultracold atoms. We demonstrate that the atoms can experience negative refraction and focussing by Veselago-type lenses. We also show how the chiral nature of the atomic motion manifests itself as an oscillation o…
We solve the problem of formulating Brownian motion in a relativistically covariant framework in 3+1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of d'Alembert form. Treating the spacelike and timelike fluctuations separately in order to maintain the covariance property, we show that it is essential to take into account the analytic continuation of "unphysical" fluctuations. We discuss the notion of locality in this framework and possible implications for entangled states.
We give an argument that a broad class of geometric models of spinning relativistic particles with Casimir mass and spin being separately fixed parameters, have indeterminate worldline (while other spinning particles have definite worldline). This paradox suggests that for a consistent description of spinning particles something more general than a worldline concept should be used. As a particular case, we study at the Lagrangian level the Cauchy problem for a spinor particle and then, at the constrained Hamiltonian level, we generalize our result to other particles.
