Time reversed states in barrier tuneling Kanchan Meena P. Singha Deo S. N. Bose National Center for Basic Sciences Salt Lake Kolkata India 700106.

2025-05-06 0 0 680.62KB 13 页 10玖币

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Time reversed states in barrier tuneling

Kanchan Meena, P. Singha Deo

S. N. Bose National Center for Basic Sciences, Salt Lake, Kolkata, India 700106.

Abstract

Tunneling, though a physical reality, is shrouded in mystery. Wave packets cannot be con-

structed under the barrier and group velocity cannot be deﬁned. The tunneling particle can

be observed on either sides of the barrier but its properties under the barrier has never been

probed due to several problems related to quantum measurement. We show that there are

ways to bypass these problems in mesoscopic systems and one can even derive an expression

for the quantum mechanical current under the barrier. A general scheme is developed to derive

this expression for any arbitrary system. One can use mesoscopic phenomenon to subject the

expression to several theoretical and experimental cross checks. For demonstration we consider

an ideal 1D quantum ring with Aharonov-Bohm ﬂux Φ, connected to a reservoir. It gives clear

evidence that propagation occur under the barrier resulting in a current that can be measured

non-invasively and theoretically cross checked. Time reversed states play a role but there is

no evidence of violation of causality. The evanescent states are known to be largely stable and

robust against phase ﬂuctuations making them a possible candidate for device applications and

so formalizing current under barrier is important.

1 Introduction

Evanescent modes are one of the puzzling features of quantum mechanics wherein a particle with

energy less than the barrier height can tunnel through the barrier and this has no classical counter-

part. Although mathematical analogies can be found with electromagnetic wave-packets, physical

interpretations pose a serious challenge. It is not possible to draw any classical correspondence

because one cannot construct a wave-packet with such evanescent states and so one cannot deﬁne

propagation of the particle under the barrier. This has lead to study several aspects of states under

the barrier and one such aspect is tunneling time. Since without a wave-packet one cannot deﬁne a

group velocity and therefore addressing the problem lead to several puzzles [1]. In a situation where

a proper theory or formalism is missing making sense of experiments is diﬃcult [2]. One can address

the problem with the help of physical clocks like Larmor clock and Wigner delay time but that in-

variably appeals to semi-classical concepts like spin precision and stationary phase approximations

[3]. Feynman path approach has also been applied [4] and diﬀerent approaches give diﬀerent results.

Hartman eﬀect like phenomenon [5] show that a tunneling particle can come out of the barrier before

entering it which raises questions like if at all there is propagation under the barrier. We describe

here a situation wherein one can make a theoretical experiment to conﬁrm current due to evanescent

modes and hence conclude propagation under the barrier. That will also be a testing ground for the

validity of physical clock like the Larmor clock. Of course one way to validate Larmor clock is to

appeal to Berger’s circuit and analyticity [6]. There is no harm in making an alternate validation by

showing that current under the barrier as a measurable quantity can be established from analyticity

of scattering matrix elements even if one can not prove its existence from quantum mechanics alone.

For this we will use the set up described in the next section. For the complexity of physical inter-

pretations we limit ourselves to the simplest system of 1D rings. The sample in the form of a 1D

quantum ring is not a loss of generality. Such 1D rings can be achieved experimentally using lateral

conﬁnement and materials for which eﬀective mass approximation works [7]. Besides, as shown in

the book by S. Datta [8], one can make ﬁnite thickness rings that consist of many independent 1D

channels. A ﬁnite thickness lead and ring can have multiple channels but DOS per channel is just

the 1D DOS that reﬂects in conductance measurements [7].

arXiv:2210.04564v2 [cond-mat.mes-hall] 12 Oct 2022

2 The set up

We will ﬁrst provide a simple description of the mesoscopic set up and then discuss how this set up

helps us address the problems discussed in the previous section. The mesoscopic sample is taken

to be a ring pierced by a magnetic ﬂux Φ through the center of the ring such that there is no

magnetic ﬁeld on the electrons conﬁned to the ring. The ring is usually made up of gold or copper

or semiconductors that can accommodate an electron gas. We are interested in the equilibrium

response of this sample which in this case is response to a magnetic ﬁeld. There is a reservoir shown

Fig. 1 The set up we want to consider in this work is shown which is a typical mesoscopic grand

canonical system in equilibrium and described in detail in section 2.

as a 3D block which is a source of electrons at chemical potential µand temperature T, making it

a grand canonical system. The reservoir injects electrons to the sample or to the ring through a

lead shown as region A. The ring can thus exchange electrons with the reservoir through the lead.

This exchange does not result in a net current in the lead because electrons that go into the ring

also come out of it and escape to the classical reservoir which is very typical of a voltage probe.

The system therefore constitutes an equilibrium system where the reservoir also acts as a source of

decoherence according to the Landauer-Buttiker formalism.

The electrons in the lead and inside the ring are purely described by quantum mechanics. If the

ring is isolated then it will have some eigen functions and energy eigen states that can be obtained

from Schrodinger equation. But once connected to the reservoir, the states in the ring will be

aﬀected. They will no longer be eigen-states of the Hamiltonian but the eigen-states will acquire

some broadening due to life time related eﬀects as the state can now leak into the reservoir. The

states in the lead is unaﬀected by the ring because it is an ideal 1D system with a typical DOS given

by 2πm

h2k. A part of the ring (region C in Fig. 1) has a potential Uthat is shown as a thickened

line. Ucan be so adjusted that an electron can tunnel through this region C. This region C can be

extended to the entire ring wherein the entire ring can be made into a tunneling region. We know

the magnetic ﬂux can drive an equilibrium persistent current [6] in the ring and this current can

thus be a tunneling current in part of the ring or in the entire ring. This allows us to extend the

theoretical consistency of quantum currents to the tunneling regime. We have shown in Eq. 13 that

one has to follow a diﬀerent approach under the barrier using analytic continuity that does not lead

to a straightforward probabilistic interpretation of quantum mechanical currents. When the barrier

is located only in a part of the ring then the persistent current in the ring can ﬂow in a quantum

state that is partly in the propagating regime (for example in the region B in Fig. 1) and partly

in the evanescent regime (for example in the region C in Fig. 1). Consistency between them has

to conserve the current at the point of propagating to evanescent crossover at the junction between

regions B and C, there being no puzzle about the current in region B.

The primary advantage of theoretically studying such mesoscopic tunneling currents is that there

are alternate theoretical formalisms in mesoscopic regime to verify the existence of currents inside

the system under the barrier from propagating asymptotic states far from the barrier. That means

there are theoretical cross checks for propagation under the barrier without studying the states under

the barrier. Thus observations and calculation on current under the barrier can help us support or

eliminate ideas. In short there are asymptotic states for the current carrying states in the ring and

one can determine the current in the ring from these asymptotic states [9], that is again similar to

the theory of the Larmor clock [6]. Note that the derivation by Larmor clock [6] giving the LHS

of Eq. 18 does not need Schrodinger equation in particular. Now to calculate a scattering matrix

element we may need an equation of motion but the derivation of LHS of Eq. 18 is insensitive to

details like whether one is allowed to analytically continue the wave-function above the barrier to

below the barrier or one should have a separate equation of motion for evanescent states. Rather

one may interpret that since all that Eq. 16 and LHS of Eq. 18 cares is analyticity of scattering

matrix elements, it should be legitimate to analytically continue wave-functions inside the system

as long as it does not destroy the analyticity of scattering matrix elements. It has been claimed

[6] that the Larmor clock is more fundamental than Schrodinger equation as it gives a lot of new

measurable quantities like the hierarchy of DOS that Schrodinger equation does not. So if there are

doubts about what one gets from Schrodinger equation, we may expect to get conﬁrmation from

the theory of Larmor clocks. In the limit of a closed system it is also consistent with what one

gets from Schrodinger equation. One can make the entire ring to be in the evanescent regime and

the currents due to evanescent states has to be consistent with the asymptotic theory. A cartoon

observer is shown in region A or lead of Fig. 1 and another cartoon observer is shown inside the

ring. Observer A need not have any idea about the potential inside the ring. This observer does not

know if the tunneling region extend over the entire ring or only a part of the ring. This observer

only needs to know the inﬁnitesimal change of dU =without any knowledge of U. He can measure

the scattering phase shift (a theoretical measurement using analysis) in the wave-function in the

lead and from there infer the current inside the ring from the analyticity of the scattering matrix

elements. Observer A can vary incident energy and check analiticity of scattering matrix elements

from Cauchy-Riemann conditions or similar criterion. The observer B in the ring can extend or shrink

the tunneling region and also determine the tunneling current from the analytic continuation of the

internal wave-function which is an unsettled issue in quantum mechanics. If the two measurements

agree then the measurement by observer in the ring has to be correct.

In terms of practical measurements, there are problems associated with measuring tunneling

currents in quantum mechanics and whatever we know about quantum measurements through an

entanglement of sample states with the states of the detector do not apply for evanescent modes. One

can also not measure tunneling currents classically as that will require the detector to be placed under

the barrier and classical detectors can not work under the barrier. These problems can be avoided

in the above described mesoscopic set up. Because such currents can also cause magnetization

that can be observed in a practical experiment for further validation. This magnetization can be

measured without disturbing the state in the system and thus not invoking the unresolved issues

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TimereversedstatesinbarriertunelingKanchanMeena,P.SinghaDeoS.N.BoseNationalCenterforBasicSciences,SaltLake,Kolkata,India700106.AbstractTunneling,thoughaphysicalreality,isshroudedinmystery.Wavepacketscannotbecon-structedunderthebarrierandgroupvelocitycannotbedened.Thetunnelingparticlecanbeobservedon...

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Time reversed states in barrier tuneling Kanchan Meena P. Singha Deo S. N. Bose National Center for Basic Sciences Salt Lake Kolkata India 700106.

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