Low-energy relativistic effects and nonlocality in time-dependent tunneling

García–Calderón, Gastón; Rubio, Alberto; Villavicencio, Jorge

doi:10.1103/physreva.59.1758

Cited by 83 publications

(42 citation statements)

References 18 publications

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“…He observed a time-dependent oscillatory regime of the probability density near the semiclassical wavefront that he named diffraction in time, in analogy to the well known Fresnel optical diffraction. It is interesting to note the resemblance of the oscillatory pattern in figure 6, to the diffraction in time phenomenon observed in the free propagation case [11]. Moreover, in the low-energy regime (µ 0 /E ≫ 1) the solution (15) can be rewritten in a more concise form by using equation (12), namely,…”

Section: (Dotted Line)mentioning

confidence: 99%

“…To obtain the solution for x > 0 and t > 0, we shall proceed along the same lines as in our recent work [11]. We begin by Laplace transforming the equation (1) using the standard definition…”

Section: The Relativistic Shutter Problemmentioning

confidence: 99%

“…where µ 0 = (m 0 c/h), and the initial condition at t = 0 corresponds to a plane wave shutter [11] (see figure 1), given by,…”

Section: The Relativistic Shutter Problemmentioning

confidence: 99%

“…Over the years, several non-relativistic approaches based in cutoff wave initial conditions have been introduced in the literature in order to investigate the time-dependent features of wave evolution in evanescent media. Some of these theoretical models [1,2,3,4,5] were inspired in the pioneering work of Sommerfeld and Brillouin [6,7], while others [9,10,11,12] were based in the seminal work of Moshinsky [13,14], who a few decades ago started a fundamental discussion on the non-relativistic and relativistic transient effects. These models represent important steps towards the clarification of the dynamics in classical forbidden regions, and a renewed motivation to explore this problem has been recently stimulated by the issue of superluminal velocities in photon [15,16] and microwave [17,18] tunneling.…”

Section: Introductionmentioning

confidence: 99%

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Exact relativistic time evolution for a step potential barrier

Villavicencio

2000

J. Phys. A: Math. Gen.

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Abstract. We derive an exact analytic solution to a Klein-Gordon equation for a step potential barrier with cutoff plane wave initial conditions, in order to explore wave evolution in a classical forbidden region. We find that the relativistic solution rapidly evanesces within a depth 2xp inside the potential, where xp is the penetration length of the stationary solution. Beyond the characteristic distance 2xp, a Sommerfeld-type precursor travels along the potential at the speed of light, c. However, no spatial propagation of a main wavefront along the structure is observed. We also find a non-causal time evolution of the wavefront peak. The effect is only an apparent violation of Einstein causality. †

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Section: (Dotted Line)mentioning

confidence: 99%

Section: The Relativistic Shutter Problemmentioning

confidence: 99%

“…where µ 0 = (m 0 c/h), and the initial condition at t = 0 corresponds to a plane wave shutter [11] (see figure 1), given by,…”

Section: The Relativistic Shutter Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact relativistic time evolution for a step potential barrier

Villavicencio

2000

J. Phys. A: Math. Gen.

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“…An alternative configuration in which the beam tunnels through a δ-barrier was discussed in [4,23,24].…”

Section: Moshinsky Shuttermentioning

confidence: 99%

Exact propagators for atom–laser interactions

Campo¹,

Muga²

2006

J. Phys. A: Math. Gen.

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Abstract. A class of exact propagators describing the interaction of an N -level atom with a set of on-resonance δ-lasers is obtained by means of the Laplace transform method. State-selective mirrors are described in the limit of strong lasers. The ladder, V and Λ configurations for a three-level atom are discussed. For the two level case, the transient effects arising as result of the interaction between both a semi-infinite beam and a wavepacket with the on-resonance laser are examined. The spacetime propagator can be considered as one of the most important tools in quantum physics for it governs any dynamical process. However, the knowledge of propagators corresponding to non-quadratic Hamiltonians is severely restricted. In this line, the spacetime propagator for a δ-potential relevant to tunnelling problems has excited much attention [1,2,3,4,5]. Such interactions turn out to be particularly useful to gain physical insight in systems where only integrated quantities are to be considered. A thorough discussion of point interactions as solvable models using a functional approach can be found in [6], and a formalism to incorporate general pointinteractions and dealing with different boundary conditions has been developed by Grosche [7,8]. Even though the method is particularly suitable to calculate the energydependent Green function, a wide class of propagators was derived in such a fashion. The incorporation of time-dependent point-interactions has been possible through different approaches as Duru's method [9] or the use of integrals of motion [10]. However, most of the effort has been focused on the dynamics of structureless particles and to the knowledge of the authors no attention has been paid to problems involving internal levels. Such state of affairs contrasts dramatically with the current surge of activity in atom optics.In this paper we use the method of Laplace transform [4,5] to tackle particles with internal structure. In particular, we shall focus on exact propagators for atom-laser interactions, namely, those of an atom interacting with a set of δ-laser on-resonance with given interatomic transitions. The method is introduced in section 1 to obtain the exact propagator for a two-level atom. Details of the calculations relevant to the following sections are here provided. In section 2 the propagators for a ladder, V, and Λ configuration (see Fig. 1) of lasers interacting with a three-level atom are obtained. The

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Transient Effects in Time-Dependent Tunneling

García–Calderón

2002

phys. stat. sol. (b)

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An exact analytic solution of the time-dependent Schrö dinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.Introduction Quantum tunneling, that refers to the possibility that a particle traverses a classical forbidden region, constitutes one of the paradigms of quantum mechanics. In the energy domain, where the stationary Schrö dinger equation at a fixed energy E is solved, tunneling is a well understood process. A simple example is tunneling through a square barrier appearing in most textbooks. In the time domain, however, there are still aspects open to investigation. In fact, recent technological achievements as the possibility of constructing artificial quantum structures at nanometric scales [1] or the manipulation of individual atoms [2] have stimulated work on time-dependent tunneling both at applied and fundamental levels.In a recent paper García-Calderó n and Rubio [3] discussed the time-dependent solution of the Schrö dinger equation for tunneling through an arbitrary potential with the cutoff initial condition of a plane wave of momentum k and applied it to study the dynamics of tunneling in a double-barrier resonant tunneling system. More recently García-Calderó n and Villavicencio [4] studied the time dependence of the probability density at the barrier edge of a rectangular potential barrier as a function of time. At short times they found a time domain resonance that describes the early time response of the system to the initial condition. They also studied the properties of the time domain resonance peak t p as a function of the barrier width and the potential height and compared it with other tunneling time scales.In this work we extend our investigation of the behaviour of the time-dependent solution of the Schrö dinger equation using cutoff wave initial conditions to the internal and transmitted regions of a rectangular potential barrier. In particular we address the issue of the possible particle propagation along the classically forbidden internal region of the potential and analyze the propagation of the probability density along the corresponding transmitted region.

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Low-energy relativistic effects and nonlocality in time-dependent tunneling

Cited by 83 publications

References 18 publications

Exact relativistic time evolution for a step potential barrier

Exact relativistic time evolution for a step potential barrier

Exact propagators for atom–laser interactions

Transient Effects in Time-Dependent Tunneling

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