Transport in quantum multi–barrier systems as random walks on a lattice
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 . 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.
Dipartimento di Scienze di Base e Applicate per l’Ingegneria,
Sapienza Università di Roma, via A. Scarpa 16, I–00161, Roma, Italy.
Dipartimento di Ingegneria e Scienze dell’Informazione e
Università degli Studi dell’Aquila, via Vetoio, 67100 L’Aquila, Italy.
Dipartimento di Scienze Matematiche,
Dipartimento di Eccellenza 2018–2022,
Politecnico di Torino, corso Duca degli Abruzzi 24, I–10129, Torino, Italy.
INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy.
Pacs: 64.60.Bd, 68.03.g, 64.75.g
Keywords: Kronig–Penney model; Zero Range Process; Transfer Matrix; Stationary current; Non–equilibrium steady states.
A variation of the Anderson model  for disordered solids has been recently introduced and investigated in order to describe systems that are finely structured, but not macroscopic, [12, 11, 25]. In particular, for 1D chains of potential barriers, the large limit was performed in a way that prevents the application of standard techniques, such as Furstenberg type theorems . Numerically, it has been observed that such systems do not lead to localization, as indeed natural for systems whose length remains finite and fixed, when the number of barriers grows. Unlike the tight–binding model introduced by Anderson, the one of Refs. [12, 11, 25] enjoys a purely off–diagonal disorder that affects the tunneling couplings among the wells, but not the energies of the bound states within the wells. In Refs. [12, 11], it has been numerically shown that the limit for the model there treated leads to the conclusion that a large deviation principle holds for the fluctuations of the transmission coefficient, with a proper scaling for the rate function. In Ref. , the transmission coefficient has been investigated for physically relevant numbers of potential barriers, referring to the scales of nano–structured sensors. In particular, the effect of compression has been numerically investigated, for different compression rules, finding that the rate of convergence to the behavior of the Kronig–Penney model [18, 10] strongly depends on the disorder degree; that compression induces a decrease of the transmission coefficient; and that a moderate number of barriers together with strong disorder imply high sensitivity to compression, which can be used for pressure sensors.
In search for a more detailed treatment of the model of Refs. [12, 11, 25], the present paper investigates the possibility of adopting an effective description, that replaces the original one with models that behave in an equivalent fashion, while being amenable to analytical treatment. The first case that ought to be investigated, in this respect, is the regular one, corresponding to the finite Kronig–Penney model. We show that the wave function time independent profile of the Kronig–Penney model can be associated with the stationary particle profile of independent random walkers with suitable jumping rates. For convenience, we deal with this model, using the language of the Zero Range Process (ZRP) [15, 19, 20, 4, 5, 6].
The paper is organized as follows. In Section 2, we illustrate our model, which consists of a 1D sequence of regularly placed conducting and insulating regions, and we solve the time independent Schrödinger equation for the density of particles, at the center of each conducting region, as a function, in particular, of and of the energy of the particles . We observe that the transport of particles is approximately ballistic when the energy is sufficiently higher than a reference energy .
Our main result is reported in Section 3, where we summarize the fundamental properties of the Zero Range Process, and we show how the parameters of the quantum and of the stochastic process can be tuned so that their profiles coincide. In Section 4, we test numerically the validity of our method comparing the Monte Carlo results with the exact expressions. We show that our technique also applies to the case of random environments, for which no general ergodic–like results seem to have been so far developed. Conclusions are drawn in Section 5.
2 The quantum multi–barrier model
and the position of the left side of the –th barrier is given by:
For the potential barrier, we take for , , and for the other positions . The zero potential regions can be interpreted as electrically conducting regions and the potential barriers as made of a given insulating material. As in Refs. [12, 11, 25, 10], the total length , and the fraction of insulating to conducting material are fixed, while the number of barriers can be varied to represent more or less finely structured media. Therefore, the width of the conducting regions is given by
Like , most of the quantities introduced in this paper depend on , but for sake of notation simplicity we omit to explicitly note it where there is no risk of confusion.
In Ref. , it was found that a single barrier of height
leads to the same asymptotic transmission coefficient of the Kronig–Penney model in the –continuum limit, namely, when all the parameters of the model are kept fixed but the number of barriers which tends to infinity. The potential , which was there called zero point energy in Ref. , will be used also here to discriminate various transport regimes.
We now write the stationary Schrödinger equation in units such that the constant ( the mass of the electron) equals 1, denoting by a prime the space derivatives:
As in Ref. , the solution corresponding to positive energy, , can be written as
where , , and . The boundary conditions provide the real constants and , below simply denoted by and respectively. The remaining coefficients and , that respectively represent the amplitudes of the waves entering and being reflected from the left boundary of the –th barrier, are then computed in terms of and , imposing the continuity of both the wave function and its first derivative.
As in , we introduce the notation
and we observe that
where denotes the identity. Moreover, we set
and we remark that for any
where denotes matrix transposition.
With such notation, the continuity of the wave function and its first derivative at the points and can be written as
Furthermore, commutes with , hence one can write:
for , where
Iterating (2.12) it is possible to relate and to and , as follows:
for . Then, using (2.13), the entries of the matrix are found to be:
where denotes complex conjugation, and the number is either real or purely imaginary. It is simple to verify that .
We now compute the square modulus of the wave function at the points :
for . For , this quantity is the square modulus of the wave function at the center of the conducting regions . Then, each potential barrier can be associated with the mass density:
The quantity is due to the interference within a conducting region of waves coming from two consecutive barriers. The resulting profile is plotted in Fig. 2.2 as a function of , while the interference term is plotted in Fig. 2.3 as a function of the energy . Unless otherwise stated, we consider models with a real positive constant and with , i.e., cases with particles input only at the left boundary of the system of interest. As one may have expected, very large values of compared to , imply that the profile is only little affected by the material inhomogeneity. In particular, it is approximately uniform in space, with approximatively constant, indicating that the transport of particles is essentially ballistic, since particles move almost exclusively in the left-to-right direction.
In our units the quantum current is defined by [22, equation (3.84)]. Between the consecutive barriers and , it takes the value which in a stationarity state is uniform in . Because also the plane wave is constant along the multi–barrier system, hence the wave vector is, the uniformity of the current is expressed by:
so that is the quantum current. Using (2.14) with one can express in terms of and the current can be written as:
Finally, taking , it is natural to define the transmission coefficient as:
which is proportional to the current.
3 Boundary driven heterogeneous Zero Range Process
In this section, we show how the parameters of the quantum model illustrated in Section 2 can be tuned so that the stationary profile coincides with the stationary density profile of a Zero Range Process (ZRP) [15, 19, 20, 4, 5, 6] on a 1D lattice. The ZRP considered hereafter consists of independent random walkers that jump to nearest neighboring sites with site–dependent hopping probabilities. The lattice is coupled, at its horizontal boundaries, to two external (infinite) particle reservoirs characterized by fixed injection rates, cf. Fig. 2.4.
More precisely, let us denote by a lattice with sites, and let the corresponding finite state or configuration space be denoted by , which consists of the configurations , where the non–negative integer represents the number of particles at site . Given such that for some , we let be the configuration obtained by moving a particle from the site to the site . We understand and to be the configurations obtained by removing a particle, respectively, from the sites and . Similarly, and denote the configurations obtained by adding a particle to the sites and , respectively. The intensity function is chosen to be , for , so that the ZRP is equivalent to a superposition of independent random walkers.
Let , with denote the hopping probabilities on , and let be the injection rates from the left and the right reservoirs, respectively. The ZRP is then defined as the continuous time Markov jump process , , with rates
for particles injection at the boundaries,
for bulk left displacements, and
The generator of this dynamics can be written as
for any real function on . This means that particles hop on the lattice to the neighboring sites to the left and to the right with rates, respectively, and . The system is “open” in the sense that a particle hopping from the site or can leave the system via, respectively, a left or a right jump, with rates and . Finally, particles are injected in the system at the left and right boundaries with rates, respectively, and .
The stationary measure for this process  is the product measure on the space
which, recalling that for any , can be rewritten as
The main quantities of interest, in our study, are the stationary occupation number profiles
where denotes the mean computed with respect to the measure , and the stationary current
for . The stationary current represents the difference between the average number of particles crossing a bond between two adjacent sites on the lattice from the left to the right and the corresponding number hopping in the opposite direction. From (3.28) it easily follows that the stationary current does not depend on the site , as required for stationary states, therefore we shall simply write .
The equivalence of the stationary profiles of the stochastic ZRP and of the quantum Kronig–Penney model is now obtained tuning the parameters as follows. First, introduce
with defined by Eq. (2.19). Then, in order to match the density profiles of the quantum process with those of the ZRP, when the parameters of the quantum process are given, we take
as hopping probabilities, and
as the left and right injection rates of the ZRP. The idea underlying this identification is that , namely the square modulus of the wave function evaluated at the center of the conducting region on the right of the –th barrier (i.e. at ) yields two contributions: one, corresponding to , associated with the average rate of the right jump of a particle from the –th site and another, given by , associated with the average rate of the left jump from the –th site.
It is important to note that, the rules (3.33) and (3.34) give non–negative hopping probabilities that sum to 1 when and for all . As explained in the Appendix A, this is the case in the –continuum limit  with , namely, for large enough when all the other parameters are kept fixed.
The values of the hopping probabilities for several different values of the energy are plotted in Fig. 3.5. Note that in the various panels the thick black dashed lines indicate the behavior of the probability in the –continuum limit considered in the Appendix A, see Eq. (A.40).
The top left panel of this figure shows that, for , such probabilities are practically constant and smaller than in most of the space. As the energy grows, the hopping probabilities develop oscillations till they settle about the value 1.
for . Using the conservation of the quantum current, Eq. (2.20), in Eqs. (3.35) and (3.36), and by comparing with Eq. (3.29), one thus obtains for , and the equivalence between the quantum current and the current (3.31) of the stochastic model is established. Moreover, since Eq. (2.20) yields , one realizes that equations (2.17) and (3.30) imply for every , which is the equivalence of the quantum mass profile and the stationary occupation profile of the stochastic particle system.
We conclude this section recalling that, as also noted at the end of Section 2, the quantum current (2.20) reduces to the transmission coefficient in (2.22), when and . Thus, the recipe given in Section 3. allows us to interpret the transport properties of the quantum model in terms of the stationary current of a boundary driven random walk on a 1D lattice .
4 Discussion of profiles in stationary states
In this section we compare the exact expression (2.17) of and with the profile obtained from Monte Carlo simulations of the ZRP model. We also compare the current of the stochastic process, obtained from Monte Carlo simulations, with the of (2.20), that is the quantum current divided by . To do that, we take the parameters values prescribed by equations (3.33) and (3.34). We further take , , , , , and , which yields , cf. (2.4). Because , given an energy , must be correspondingly large, cf. comment below (3.34) and the Appendix A.
The numerical simulations are performed as follows: the process starts with zero particles, but the injection of particles at the left boundary quickly makes the lattice populated. After a sufficiently large time, the particles distribution reaches a stationary state. At stationarity, the occupation number profile is measured with two different methods, that are mathematically equivalent, but numerically independent: i) the number of particles at each site is averaged collecting its values at time intervals larger than the decorrelation time, that is of the order of the number of sites; ii) the total number of particles jumping from a generic site to the left is computed and, at the end of the simulation, is divided by the total time, that is given by the sum of exponentially distributed time intervals between two consecutive jumps, and by the probability to perform the left jump. Using (3.31) also this ratio should yield . The accord between these two independent calculations is one test of the accuracy of our numerical simulations.
We shall now distinguish the regimes with energy , and cases with energy , as suggested in [10, Remark 4.4]. In our study .
Case , monotonic decay: The top left panel of Fig. 2.3 shows that the is constant in most of the system positions, and smaller than . At the right end of the system a sudden increase takes place. The occupation number profile is, instead, monotonically decreasing, see Fig. 4.6, and convex.
Case , transition behavior: The profile in this case is shown in the top right panel of Fig. 2.3, and it turns out to be monotonically increasing. The occupation number profile, plotted in Fig. 4.7, is monotonically decreasing, but the total mass on the lattice is larger with respect to the previous case. This can be ascribed to the fact that in that case the injection rate from the left reservoir has increased with respect to the case with .
Case , oscillatory behavior: The probability profile is shown in the bottom left panel of Fig. 2.3 and it is an oscillatory function. As a consequence, the occupation number profile shown in in Fig. 4.8 is also an oscillating function.
Case , ballistic behavior: The transition probability profile is shown in the bottom panel of Fig. 2.3 and it is very close to . Therefore, the stochastic model is made of particles undergoing a strongly asymmetric random walk on the lattice. Fig. 4.9 shows the occupation number profile which is, indeed, almost uniform.
Finally, we investigate numerically the equivalence between a disordered version of the Kronig–Penney model introduced in  and a boundary driven ZRP with random hopping probabilities, cf. e.g. [17, 16]. The quantum disordered model is constructed by drawing, first, the random variables from a uniform distribution, and then by rescaling them, in order to fulfill the two constraints
with fixed. Remarkably, the correspondence method developed in Section 3 still holds here, the only difference being the definition of the multi–barrier matrix appearing in (2.14). In the disordered case, indeed, this matrix is given by the product of single random matrices M defined in Eq.(2.13), that clearly differ from the expression (A.39). The results of the MC simulations are reported in Fig. 4.10, which shows a striking equivalence between the quantum and the stochastic processes here investigated. Note that the black dashed line displayed in the various panels of Fig. 4.10 corresponds to the exact quantum profile in Eq. (2.17), obtained by using the transfer matrix formalism discussed in Section 2, whereas the filled and empty symbols denote the results of the MC simulations based on the two alternative methods outlined before.
The correspondence highlighted in Fig. 4.10 can be used to investigate, or to interpret, some of the peculiar properties of the random Kronig–Penney model  with the techniques developed in the study of the ZRP. In particular, the disordered version of the quantum multi–barrier model, with the random variables scaled as in Eq.(4.37), can be studied as an instance of a random walk in a random environment (with site randomness), for which a rich theory has been developed [23, 24, 26, 27]. The opposite is of course also possible: casting the particle dynamics in terms of our random Kronig–Penney model may offer novel insights into some interesting phenomena, e.g. the Sinai’s localization , observed in random walks in random environments.
In this paper we have shown that the equivalence between the quantum multi–barrier finite Kronig–Penney model and the ZRP is achieved when the hopping probabilities of the stochastic process are properly tuned with the the parameters of the quantum system. In particular, we have evidenced that, for the finite Kronig–Penney model with unitary input only from the left boundary, the transmission coefficient corresponds to the stationary current in a boundary driven heterogeneous ZRP , realized by independent particles performing a random walk on a lattice with site–dependent hopping probabilities.
If and are taken as prescribed by equation (3.33) and and as by equation (3.34), the stationary profile and the current of the quantum model are correctly recovered from the ZRP, for any value of the energy .
To see how the choice of parameters is essential, let us drop the dependence of and on and , in (3.33). In this case, one may still recover the correct value of the stationary current but not the stationary profile for an arbitrary value of the energy.
It is reasonable to expect that our correspondence method produces the equivalence of the two models considered in this paper also in other situations. After all, everything in the ZRP depends on the hopping probabilities and on the boundary conditions, like they depend on the boundary conditions and on the potential barriers in the quantum model. This may allow us to treat rigorously the large limit of mesoscopic disordered systems numerically investigated in Refs. [12, 11, 25], for which no general ergodic–like results seem to have been so far developed.
We have also shown that a random walk in a random environment correctly reproduces the transport properties of the quantum disordered model. Thanks to this equivalence, one concludes for the stochastic process that the stationary profiles may vary from monotonically decreasing to oscillating because of the variation of the energy of an incoming plane wave, of the associated quantum multi–barrier model. When this energy exceeds a critical value, the stationary states turn from monotonically decaying to oscillating.
On the basis of the studied equivalence between quantum multi–barrier models and stochastic particle systems, there are various open questions that demand further investigation. We have in mind, in particular, the onset of uphill currents for the considered open quantum models, since analogous currents for interacting particle (or spin) systems on a lattice have been recently investigated and reported in the literature [7, 8, 4, 9, 1].
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.
Appendix A The –continuum limit of the hopping probabilities
In this appendix we consider the Kronig–Penney model introduced in the Section 2 in the case (see the comment below (2.6)). As mentioned below the equation (2.4), see also , the –continuum limit is realized by keeping fixed all the parameters of the Kronig–Penney model but the number of barriers , which tends to infinity. Recalling the matrix defined in (2), following , we let
Denoting the eigenvalues of by and , one finds
which can be real or complex–valued, depending on the value of . It is important to remark that , indeed, by expanding in Taylor series with respect to in a neighborood of , one has
which proves that, for small, (resp. ) for (resp. ). Moreover, for the above expansion becomes which proves that, for small, even for .
In the sequel we shall often use the –th power of for . By slightly abusing the notation, we shall denote its elements by . One can use the equation (2.14) to express the coefficients and in terms of the boundary condition (recall that we assumed ). One first writes (2.14) for and finds , then, using again (2.14) for a general , one gets
which for reproduce and , while for they yield and .
Recalling the definition of hopping probabilities (3.33), we wish to express in terms of the boundary condition the two quantities and , respectively the average hopping rates to the left and to the right from the –th site of the ZRP model. Using (A.38) we find
To compute the large limit of the quantities above express the –th power of the matrix as in [10, Eq. (3.15)]:
where is the identity matrix.
Recalling the definition (3.33) of right hopping probability , we let and, in the limit , we find
One readily notes that , as illustrated in Fig. 3.5.
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