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Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

Abstract

A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method (Colangeli et al. in J Stat Mech Theor Exp 6:P06006, 2015). This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs. Analytical and numerical results prove that the stationary density profile of the particle system overlaps with the quantum mass density profile of the stationary Schrödinger equation, when the parameters of the two models are suitably matched. The equivalence between the quantum model and a stochastic particle system would mainly be fruitful in a disordered setup. Indeed, we also show, here, that this connection, analytically proven to hold for periodic barriers, holds even when the width of the barriers and the distance between barriers are randomly chosen.

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Acknowledgements

MC and ENMC acknowledge financial support from FFABR 2017. LR acknowledges that the present research has been partially supported by MIUR grant Dipartimenti di Eccellenza 2018–2022.

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Appendix A: The \(\gamma ,L\)-continuum Limit of the Hopping Probabilities

Appendix A: The \(\gamma ,L\)-continuum Limit of the Hopping Probabilities

In this appendix we consider the Kronig–Penney model introduced in the Sect. 2 in the case \(D=0\) (see the comment below (6)). As mentioned below the Eq. (4), see also [10], the \(\gamma ,L\)-continuum limit is realized by keeping fixed all the parameters of the Kronig–Penney model but the number of barriers N, which tends to infinity. Recalling the matrix \(\mathbf {M}\) defined in (15), following [10], we let

$$\begin{aligned} \varPhi = \mathfrak {R}{(M_{11})} = \cos (k\delta ) \cosh (z\gamma \delta ) + \frac{z^2 - k^2}{2 k z} \sin (k\delta ) \sinh (z\gamma \delta ). \end{aligned}$$

Denoting the eigenvalues of \(\mathbf {M}\) by \(\mu _1\) and \(\mu _2=\mu _1^{-1}\), one finds

$$\begin{aligned} \mu _1= \varPhi - \sqrt{\varPhi ^2- 1} \quad \text {and} \quad \mu _2=\varPhi + \sqrt{\varPhi ^2- 1} \end{aligned}$$

which can be real or complex-valued, depending on the value of \(\varPhi \). It is important to remark that \(\mu _1\ne \mu _2\), indeed, by expanding \(\varPhi \) in Taylor series with respect to \(\delta \) in a neighborhood of \(\delta =0\), one has

$$\begin{aligned} \varPhi (\delta ) = 1 -\frac{1}{2}L^2(E-E_0)\delta ^2 +\frac{1}{24}L^4\Big [E^2-2EE_0+E_0^2\frac{\gamma (2+\gamma )}{(1+\gamma )^2} \Big ]\delta ^4 +o(\delta ^4), \end{aligned}$$

which proves that, for \(\delta \) small, \(\varPhi (\delta )<1\) (resp. \(\varPhi (\delta )>1\)) for \(E>E_0\) (resp. \(E<E_0\)). Moreover, for \(E=E_0\) the above expansion becomes \(\varPhi (\delta )=1-L^4E_0^2\delta ^4/[24(1+\gamma )^2]+o(\delta ^4)\) which proves that, for \(\delta \) small, \(\varPhi (\delta )<1\) even for \(E=E_0\).

In the sequel we shall often use the nth power \(\mathbf {M}^n\) of \(\mathbf {M}\) for \(n=1,\dots ,N\). By slightly abusing the notation, we shall denote its elements by \(M^N_{ij}\). One can use the Eq. (14) to express the coefficients \(C_{n+1}\) and \(D_{n+1}\) in terms of the boundary condition C (recall that we assumed \(D_{N+1}=D=0\)). One first writes (14) for \(n=N\) and finds \(D_1=C M^N_{21}e^{-i2k\delta }/M^N_{11}\), then, using again (14) for a general n, one gets

$$\begin{aligned} C_{n+1} = C\frac{e^{- i k \ell _n}}{M^N_{11}} (M^N_{11}M^n_{22}-M^N_{21}M^n_{12}) \;\;\text { and }\;\; D_{n+1} = C\frac{e^{i k (\ell _n-2 \delta )}}{M^N_{11}} (M^N_{21}M^n_{11}-M^N_{11}M^n_{21}) \end{aligned}$$
(A.38)

which for \(n=N\) reproduce \(C_{N+1}\) and D, while for \(n=0\) they yield C and \(D_{1}\).

Recalling the definition of hopping probabilities (33), we wish to express in terms of the boundary condition C the two quantities \(|D_{n}|^2+S_{n}\) and \(|C_{n+1}|^2+S_{n+1}\), respectively the average hopping rates to the left and to the right from the nth site of the ZRP model. Using (A.38) we find

$$\begin{aligned} \frac{|D_{n+1}|^2+S_{n+1}}{|C|^2}= \frac{ |M^N_{21}M^n_{11}-M^N_{11}M^n_{21}|^2 + \mathfrak {R}{[(M^N_{11}M^n_{22}-M^N_{21}M^n_{12}) (M^N_{12}M^n_{22}-M^N_{22}M^n_{12}) e^{ik\delta } ]} }{|M^N_{11}|^2} \end{aligned}$$

for \(n=0,\dots ,N-1\) and

$$\begin{aligned} \frac{|C_{n+1}|^2+S_{n+1}}{|C|^2} = \frac{ |M^N_{11}M^n_{22}-M^N_{21}M^n_{12}|^2 + \mathfrak {R}{[ (M^N_{11}M^n_{22}-M^N_{21}M^n_{12}) (M^N_{12}M^n_{22}-M^N_{22}M^n_{12}) e^{ik\delta }]} }{|M^N_{11}|^2} \end{aligned}$$

for \(n=1,\dots ,N\).

To compute the N large limit of the quantities above express the nth power of the matrix \(\mathbf {M}\) as in [10, Eq. (3.15)]:

$$\begin{aligned} \mathbf {M}^n = \frac{\mu _1^n-\mu _2^n}{\mu _1-\mu _2}\mathbf {M} - \frac{\mu _2\mu _1^n-\mu _1\mu _2^n}{\mu _1-\mu _2}\mathbf {I} \,\,, \end{aligned}$$
(A.39)

where \(\mathbf {I}\) is the identity matrix.

Recalling the definition (33) of right hopping probability \(p_n\), we let \(x=n/N\in (0,1]\) and, in the limit \(N\rightarrow \infty \), we find

$$\begin{aligned} p(x)= \left\{ \begin{array}{ll} \frac{1}{2}+\frac{E_0-E}{E_0(1+\cosh \left( 2 L\sqrt{E_0-E} (1-x)\right) )-2 E} &{}E<E_0\\ \frac{1}{2}+\frac{1}{2+2 E_0 L^2 (1-x)^2} &{} E=E_0\\ \frac{1}{2}+\frac{E-E_0}{2 E -E_0(1+\cos \left( 2 L\sqrt{E-E_0} (1-x)\right) )} &{}E>E_0\\ \end{array} \right. \end{aligned}$$
(A.40)

One readily notes that \(p(x) \in [0,1]\), as illustrated in Fig. 5 (see, also, the comment at the end of the paragraph below (34)).

Finally, the \(\gamma ,L\)-continuum limit of the stationary current (31), takes the form:

$$\begin{aligned} \overline{J}=\lim _{N\rightarrow \infty }\left( |C_{n+1}|^2-(|D_{n+1}|^2 \right) = \left\{ \begin{array}{ll} \left| C\right| ^2\frac{8 E (E-E_0)}{8 E (E-E_0)+E_0^2\left( 1-\cosh \left( 2 L\sqrt{E_0-E} \right) \right) }&{} E<E_0\\ \left| C\right| ^2\frac{4}{4+E_0 L^2} &{} E=E_0 \\ \left| C\right| ^2\frac{8 E (E-E_0)}{8 E (E-E_0)+E_0^2\left( 1-\cos \left( 2 L\sqrt{E-E_0} \right) \right) } &{} E>E_0\\ \end{array}\right. \end{aligned}$$
(A.41)

For \(C=1\), the expression (A.41) yields the asymptotic value of the transmission coefficient \(\overline{S}\) obtained in [10, Eq. (4.3)].

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Cirillo, E.N.M., Colangeli, M. & Rondoni, L. Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice. J Stat Phys 176, 692–709 (2019). https://doi.org/10.1007/s10955-019-02317-0

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Keywords

  • Kronig–Penney model
  • Zero range process
  • Transfer matrix
  • Stationary current
  • Non-equilibrium steady states