Journal of Computational Electronics (2021) 20:2232–2244 https://doi.org/10.1007/s10825-021-01798-1 Can Wigner distribution functions with collisions satisfy complete positivity and energy conservation? Matteo Villani1 · Xavier Oriols1 Received: 11 August 2021 / Accepted: 29 September 2021 / Published online: 23 November 2021 © The Author(s) 2021 Abstract To avoid the computational burden of many-body quantum simulation, the interaction of an electron with a photon (phonon) is typically accounted for by disregarding the explicit simulation of the photon (phonon) degree of freedom and just modeling its effect on the electron dynamics. For quantum models developed from the (reduced) density matrix or its Wigner–Weyl transformation, the modeling of collisions may violate complete positivity (precluding the typical probabilistic interpreta- tion). In this paper, we show that such quantum transport models can also strongly violate the energy conservation in the electron–photon (electron–phonon) interactions. After comparing collisions models to exact results for an electron interact- ing with a photon, we conclude that there is no fundamental restriction that prevents a collision model developed within the (reduced) density matrix or Wigner formalisms to satisfy simultaneously complete positivity and energy conservation. However, at the practical level, the development of such satisfactory collision model seems very complicated. Collision models with an explicit knowledge of the microscopic state ascribed to each electron seems recommendable (Bohmian conditional wavefunction), since they allow to model collisions of each electron individually in a controlled way satisfying both complete positivity and energy conservation. Keywords Matter–light interaction · Wigner Function · Complete positivity · Energy conservation · Bohmian conditional wavefunction Mathematics Subject Classification MSC 2020 81Q65 · MSC 2020 81U40 1 Introduction performance from the simulation of all particles, in a type of solution of the many-body Schrödinger equation, is directly Electron devices are quantum systems outside of thermo- impossible. This difficulty finds its origin in the so-called dynamic equilibrium with many interacting particles (elec- many body problem [1]. The typical strategy to reduce trons, atoms, photons, etc). In addition, one is not only such inaccessible computational burden is introducing an interested on the time-independent or steady state (DC) artificial division between the simulated particles, usually simulation of such devices, but also on their time-depend- named the open system, and the rest of non-simulated par- ent (AC, transients) performance, and even on their noise ticles, which are usually referred as the environment [1, 2]. properties. As a result, from a computational point of view, Such division, in turn, requires reintroducing the effect of an electron devices is one of the most difficult quantum sys- the non-simulated (environment) particles onto the simu- tems to be simulated. Any attempt to directly get the device lated ones through some new term in the equations of motion of simulated particles. This new element in the equation of motion is what is usually called the collision term. Typical * Matteo Villani matteo.villani@uab.es examples found in the literature following this strategy are Green function [3–5], the density matrix [6, 7], Wigner dis- Xavier Oriols xavier.oriols@uab.es tribution function [8–14], the master equation [15, 16], Kubo formalism [17], conditional wave functions [20–22], etc. 1 Department of Electronic Engineering, Universitat There are two main difficulties in these quantum transport Autònoma de Barcelona, Campus de la UAB, approaches dealing with collisions. The first difficulty is that 08193 Bellaterra, Barcelona, Spain 13 Vol:.(1234567890) Journal of Computational Electronics (2021) 20:2232–2244 2233 open quantum systems cannot be described by an (orthodox) uncontrolled change in the total energy, making the final pure state, but by a density matrix (or some transformation result of the collision process unphysical. Similar issues of it). Thus, we cannot assign a pure state to an electron have been underlined in the use of a Boltzmann superopera- and we have difficulties to identify which are the proper- tor [26, 27]. ties of each electron that are modified by collisions. The After this introduction, in Sect. 2, we will define the two second difficulty appears because the collisions are, at best, conditions, complete positivity and energy conservation, a reasonable approximation of the real interaction between for modeling collisions in a general density matrix formula- the simulated and non-simulated degrees of freedom (but tion of a quantum system. In Sect. 3, we will show an exact never an exact result). Both difficulties make the evaluation solution for the electron–photon interaction emphasizing of the physical soundness of a collision model for quantum the importance of energy conservation during this kind of transport a difficult task. A useful criteria for evaluating col- interaction. Then, we will perform a comparison between lisions models is checking about their complete positivity. approximate collision models that modify the energy of the The complete positivity means that the (reduced) density electron and approximate collision models that modify the matrix will not provide negative values of the probability momentum of the electron, respectively. In Sect. 4, we will presence along the device. The presence of such negative show the Wigner–Weyl representation of such energy and values is unphysical because then the typical probabilistic momentum collisions models for several cases showing the interpretation would be precluded [1, 2]. The Wigner dis- non-physical results originated by the collisions models that tribution function, as a Wigner–Weyl transformation of the exchange momentum in scenarios where energy and momen- density matrix [8, 23], does also suffer from such lack of tum do not commute. In Sect. 5, we define the two practical complete positivity. We clarify that we are not referring here requirements that are mandatory in a collision model imple- to the fact that the Wigner function is a quasi-probability in mented in the phase-space (Wigner) description to satisfy the phase space [24], but to the creation of negative prob- complete positivity and energy conservation. In Sect. 5, we ability of finding electrons at some positions in the physical also add a subsection about time reversibility and how it is space. For example, it is well-known that the typical use of a influenced by collisions. We conclude in Sect. 6 indicating Boltzmann-like superoperator in the description of the colli- that alternative collision models with an explicit knowledge sion term in the Wigner formalism can produce such regions of the state ascribed to each electron seems strongly recom- of negative probability presence [6, 7, 25]. mendable to avoid the previous two problems. In this work, we will focus on an additional criteria to verify the physical soundness of collision models. It is very reasonable to assume that most collisions between particles induce a well-defined change of the energy of the electron. 2 Problems in modeling collisions The physical justification of such assumption is that an hypothetical exact solution of the interaction of the particles The density matrix deals with mixed states which arise in involved in a collision, through a many-body Schrödinger quantum mechanics when the preparation of the system is equation, would show that the total energy (the ensemble not fully known, or when one wants to describe an open value of the Hamiltonian) is a constant of motion. However, system which is entangled to an environment [1]. Since in the density matrix (or some transformation of it), it is both conditions are typically observed in electron devices, not always possible to access to the energy of each parti- the density operator seems an adequate tool to study quan- cle. Without such information, it is not clear if a collision tum transport. The typical equation of motion for the den- model satisfies or not the mentioned energy conservation ̂ is given by the Liouville–Von Neumann sity operator 𝜌(t) requirements. A clear example of such difficulties appears equation in the formulation of quantum transport through the Wigner d𝜌(t) ̂ 1 ̂ 1 distribution function. In the Wigner function, the informa- = [H, 𝜌(t)] ̂ = [Ĥ 0 , 𝜌(t)] ̂ ̂ 𝜌(t)] + C[ ̂ . (1) tion of the quantum system is given by one position and dt i� i� one momentum degree of freedom. No information of the The first term in the right hand side of (1) provide the uni- energy of each individual particle is given by the Wigner tary evolution of the systems with H0 a single-particle Ham- formalism. One can assume that the kinetic energy of par- iltonian, while the collisions are introduced by the new term ticles can be obtained from the value of their momentum. given by the superoperator Ĉ acting on the density matrix However, in many practical scenarios, there is no one-to-one operator. relation between energy and momentum (the energy eigen- Due to a collision at time ts , the new density matrix states are not momentum eigenstates, and viceversa). Then, a 𝜌̂ (ts + 𝛥t) from (1) will be equal to the unitary den- s reasonable change in the momentum in the Wigner function sity matrix with a free evolution until time ts given by after a collision event can be translated into an unexpected/ 13 2234 Journal of Computational Electronics (2021) 20:2232–2244 ̂ s ) + 𝛥t 𝜌̂0 (ts ) = 𝜌(t ̂ s )], plus some perturbation due to [Ĥ 0 , 𝜌(t 2.1 The problem of complete positivity i� the collisions given by 𝛥𝜌(t ̂ s )]: ̂ 𝜌(t ̂ s ) = 𝛥t C[ For algorithms [29–31] that have access to the additional s 𝛥t ̂ ̂ 𝜌(t information in (3), the interaction of one electron with a 𝜌̂ (ts + 𝛥t) = 𝜌(t ̂ s) + ̂ s )] + 𝛥t C[ [H , 𝜌(t ̂ s )] i� 0 (2) photon at time ts, can be understood as an electron changing = 𝜌̂0 (ts ) + 𝛥𝜌(t ̂ s) . from its initial state �𝜓2 (ts )⟩ to its final state �𝜓2s (ts )⟩. Thus, the new scattered density matrix after the scattering is: Hereafter, when needed, we will use the superindex s to indicate elements that refer to the system after the scattering 1 1 event has taken place. Under the assumption that changes in 𝜌̂s (t) = 𝜌̂0 (t) − �𝜓2 (t)⟩⟨𝜓2 (t)� + �𝜓2s (t)⟩⟨𝜓2s (t)� . (4) M M the state of the system need some time to occur (an example of such time delay will be provided in the exact solution The last two terms in the above expression correspond to of the electron–photon in Sect. 3.1), a reasonable condition annihilating the old (description of the) electron �𝜓2 (t)⟩ and for the application of the model in (2) is a finite value of the creating a new (description of the) electron �𝜓2s (t)⟩. Such scattering time 𝛥t [26–28]. collision corresponds to the new collision term in (2) as: In most computational algorithms, it is not possible to 1 1 know which are the individual (pure) states that build the ̂ =− 𝛥𝜌(t) �𝜓2 (t)⟩⟨𝜓2 (t)� + �𝜓2s (t)⟩⟨𝜓2s (t)� . (5) M M density matrix. However, such (microscopic) knowledge can be obtained within the computational technique known The probability presence (also known as the charge den- as stochastic Schrödinger equation (for Markovian systems) sity) of the density matrix in (4) at any time t > ts can be or with the Bohmian theory (for either Markovian or non- computed as: Markovian systems) [29–31]. Within such techniques, we N � Mj can assume that the exact single-particle wave function ⟨Q(x, t)⟩ = ⟨x�𝜌(t)�x⟩ ̂ = �𝜓j (x, t)�2 𝜓j (x, t) = ⟨x�𝜓j (t)⟩ is known for each j-th electron, to build M j=1 (6) the density operator 𝜌(t) ̂ as: 1 1 − �𝜓2 (x, t)�2 + �𝜓2s (x, t)�2 . N M M 1 � ̂ = 𝜌(t) M �𝜓 (t)⟩⟨𝜓j (t)� , (3) Notice that �𝜓2 ⟩ and �𝜓2s ⟩ are totally different states because M j=1 j j the collision can change momentum, energy, spatial distri- where Mj is the number of states �𝜓j (t)⟩ that are present in bution of the probability, etc. The problem with collisions ∑N the system with M = j=1 Mj and N the maximum number for those algorithms that have no access to the information of possible types of states (to be able to deal with anni- in (3) is that the implementation of 𝛥𝜌(t) ̂ can correspond hilation and creation of electronic states, Mj can be incre- to subtracting a state that is not present in the initial den- mented or decremented by one at the scattering time, ts , but sity matrix 𝜌(t ̂ s ). Then, the term - M1 |𝜓2 (x, t)|2 can provide we do not write its time dependence explicitly to simplify negative probability presence ⟨Q(x, t)⟩ < 0 in some spatial the notation). regions or times, which is clearly unphysical. Certainly, the To better understand the problems of complete positivity computational algorithms with the additional knowledge of and energy conservation, in the next two subsections, we (3) could avoid such unphysical result by just not perform- will discuss the change of the density matrix due to a col- ing the scattering process from �𝜓2 (t)⟩ to �𝜓2s (t)⟩ in (2) when lision in (2) by distinguishing between the algorithms that there is no such initial state �𝜓2 (t)⟩. The problem is that most have access to the additional knowledge given by (3) and the of the computational algorithms dealing with quantum trans- ones that do not. The computation of collisions without such port do not have access to such microscopic information knowledge corresponds to most of the algorithms presented from (3) and they have to blindly apply a change on the den- in the literature dealing with quantum transport. We will see sity matrix given by 𝛥𝜌(t) ̂ in (2), without knowing which are that without the additional knowledge of (3) the collisions the implications of such change in terms of states. In other process implemented in (2) can lead to violations of the words, the study of quantum transport in terms of the (mac- complete positivity and energy conservation. Such violations roscopic) matrix 𝜌(x, x� ) = ⟨x� �𝜌(t)�x⟩ ̂ has the great computa- can easily be avoided using computational algorithms that tional advantage of not needing the (microscopic) knowledge have access to the additional information provided by (3). of the states building such matrix, but it has the drawback of losing control on the physical meaning of a change of density matrix by an amount equal to 𝛥𝜌(x, x� ) = ⟨x� �𝛥𝜌(t)�x⟩ ̂ after a collision. It is also well-known that the Boltzmann- like collision operator applied to the Wigner distribution function can violate such complete positivity [6, 7, 25]. 13 Journal of Computational Electronics (2021) 20:2232–2244 2235 2.2 The problem of energy conservation 3 Exact and approximate models for matter‑light interaction We discuss now an additional problem, not related with the probability presence, but with the conservation of We develop here an example of the role of the energy conser- energy. For those algorithms [29–31] that have access vation in the collision of an electron with a photon. First, we to the information in (3), the electron–photon colli- explain and exact electron–photon Schrödinger equation, and sion means that an electron with (ensemble) energy then we present two approximate collision models. The first ⟨E2 (ts )⟩ = ⟨𝜓2 (ts )�Ĥ 0 �𝜓2 (ts )⟩ at time ts changes its energy, approximate model for collisions will be based on changing due to interaction with a photon of energy E𝛾 , to the new the energy of the electron during the scattering process, while value ⟨E2s (ts )⟩ = ⟨𝜓2s (ts )�Ĥ 0 �𝜓2s (ts )⟩ . The requirement of the second one is based on changing the value of momentum. conservation of energy with such algorithms that have access to the microscopic states, ⟨E2s (ts )⟩ = ⟨E2 (ts )⟩ + E𝛾 , 3.1 Exact electron–photon interaction is trivially satisfied. The energy of the electron system at any time t > ts can The electromagnetic field in our quantum description is be computed as: assumed to be a monochromatic field inside a optical cavity of N length L𝛾 with the shape E(x, t) ∝ q cos(k𝛾 x − 𝜔t) with 𝜔 the � � � Mj angular frequency of the field and k𝛾 the wave vector giving a ⟨Es (t)⟩⟩ = Tr 𝜌̂s (t)H0 = ⟨E (t)⟩ M j speed of light c = 𝜔∕k𝛾 . The dependence of E(x, t) on x can be j=1 (7) 1 1 removed by assuming Lx << L𝛾 = 2𝜋 𝜔 , where Lx is the length − ⟨E2 (t)⟩ + ⟨E2s (t)⟩ , of the active region of the device, where electrons are simu- M M lated. The quantization of the electromagnetic field appears where ⟨Ej (t)⟩ = ⟨𝜓j (t)�Ĥ 0 �𝜓j (t)⟩. Again, for algorithms that because the amplitude q is not a fixed value, but a variable are not allowed to deal with the microscopic information degree of freedom. Then, the Hamiltonian of the electromag- of the state of each electron, before and after the collision, netic field in the q-representation H𝛾 can be written as [32]: the change of energy due to the collision done in (2) has to satisfy: Ĥ 𝛾 = − �2 𝜕 2 𝜔2 2 + q . (10) � � 2 𝜕q2 2 E𝛾 =⟨E2s (t)⟩ − ⟨E2 (t)⟩ = Tr 𝛥𝜌(t) ̂ H0 , (8) Inside the optical cavity, the vacuum state related to the with 𝛥𝜌(t) ̂ defined in (2). The problem is that it is not trivial absence of photons is �0⟩, corresponding to the ground state to ensure that expression (8) is satisfied, only by using the of an harmonic oscillator 𝜓0 (q) = ⟨q�0⟩ in (10). Identically, macroscopic information that we have 𝜌(x, x� ) = ⟨x� �𝜌(t)�x⟩ ̂ , the first state of the harmonic oscillator in (10) corresponds and without knowing the information of the states involved to the presence of one photon inside the optical cavity, is in such collision. described by 𝜓1 (q) = ⟨q�1⟩. For more details, see [22]. In summary, as we will see along the paper, the compu- The electron part of the system follows the well-known tational advantage of the density operator of encapsulat- electron Hamiltonian, in the x-representation: ing all the physical information into the density matrix 𝜌(x, x� ) = ⟨x� �𝜌(t)�x⟩ , avoiding to treat the heavy micro- ℏ2 𝜕 2 ̂ H0 = − + V(x) , (11) scopic description of the state of each electron, also pre- 2m 𝜕x2 sents some drawbacks when dealing with collisions. The where V(x) includes both internal and external scalar poten- representation of the density matrix in coordinate space tial and m is the electron effective mass. The Hamiltonian can be changed into the so-called Wigner distribution H0 is written assuming a 1D electron system. function fW (x, k, t) through the Wigner–Weyl transforma- Finally, the electron–photon wavefunction 𝛹 (x, q, t) will be tion [8, 23]: guided by the following Schrödinger-like equation of motion: x� x� 2𝜋 ∫ 1 ℏ2 𝜕 2 𝛹 (x, q, t) � fW (x, k, t) = dx� ⟨x + �𝜌(t)�x ̂ − )⟩ e−ikx . (9) iℏ 𝜕𝛹 (x, q, t) =− + V(x) 𝛹 (x, q, t) 2 2 𝜕t 2m 𝜕x2 Again, in general, the Wigner distribution function has no ℏ2 𝜕 2 𝛹 (x, q, t) 𝜔2 2 (12) − + q 𝛹 (x, q, t) information at all about the microscopic states. Then, the 2 𝜕q2 2 collision operator can provoke unphysical violations of the +𝛼 � xq 𝛹 (x, q, t) , conservation of energy, as we will see in following sections. 13 2236 Journal of Computational Electronics (2021) 20:2232–2244 where the last term in (12) is the interaction term between At time ts the electron undergoes a collision with a pho- the photon and electron in the typical dipole approximation ton. It is worth mentioning that the use of a finite time for the [32]. The parameter 𝛼 ′ indicates the interaction strength. implementation of the scattering event is key to maintain con- Since we are only interested in a dynamic process involv- tinuity of the (conditional) wave function in space and time, ing only two possible photons states, 𝜓0 (q) for zero photons but to simplify the discussion we will assume an instantaneous and 𝜓1 (q) for one photon, the whole wave function 𝛹 (x, q, t) scattering process here. Later, in the numerical results, we will can be decomposed as: explicitly consider the finite duration of the scattering process. The process of collision 𝜓(x, ts ) → 𝜓 s (x, ts ) can be 𝛹 (x, q, t) = 𝜓A (x, t) 𝜓0 (q) + 𝜓B (x, t) 𝜓1 (q) , (13) understood as a transition between the initial and final with states 𝜓(x, ts ) and 𝜓 s (x, ts ) , respectively, which satisfy ⟨Es (ts )⟩ = ⟨E(ts )⟩ + E𝛾, with E𝛾 the energy of a photon. Within the energy representation, the wave packet can be decomposed ∫ (14) ∗ 𝜓A∕B (x, t) = dq 𝜓0∕1 (q) 𝛹 (x, q, t) . into a superposition of Hamiltonian eigenstates 𝜙E (x) of the electron with Hamiltonian Ĥ 0 in (11) as: The wave functions 𝜓A (x, t) and 𝜓B (x, t) describe how the total wave function is projected in 𝜓0 (q) and 𝜓1 (q) , ∫ (17) 𝜓(x, ts ) = dE a(E, ts ) 𝜙E (x) , respectively. with a(E, ts ) = ∫ dx 𝜓(x, ts ) 𝜙∗E (x) . The ensemble energy The equation of motion of 𝜓A (x, t) and 𝜓B (x, t) can be obtained by introducing the definition (13) into (12) and using the orthogonality of 𝜓0 (q) and 𝜓1 (q) as follows ⟨Es (ts )⟩ after the scattering has to increase of an amount E𝛾 , so the new state after the collision can be written as: 𝜕𝜓A (x, t) 2 ℏ2 𝜕 𝜓A (x, t) iℏ =− ∫ 𝜕t 2m 𝜕x2 (15) 𝜓 s (x, ts ) = dE a(E − E𝛾 , ts ) 𝜙E (x) ( ) 1 (18) + V(x) + ℏ𝜔 𝜓A (x, t) + 𝛼x 𝜓B (x, t) ∫ 2 = dE as (E, ts ) 𝜙E (x) , and 2 where we have defined as (E, ts ) = a(E − E𝛾 , ts ) . This 𝜕𝜓 (x, t) ℏ2 𝜕 𝜓B (x, t) transition corresponds to absorption of energy by the iℏ B =− 𝜕t 2m 𝜕x2 ( ) (16) electron. Emission can be identically modeled by using 3 ⟨Es (ts )⟩ = ⟨E(ts )⟩ − E𝛾 . In few words, the collision model + V(x) + ℏ𝜔 𝜓B (x, t) + 𝛼x 𝜓A (x, t) , 2 performs a change in the wave packet ensemble energy of where we have defined 𝛼 = 𝛼 � ∫ 𝜓0 (q) q 𝜓1 (q) dq and we have an amount ± E𝛾. used ∫ 𝜓0 (q) q 𝜓0 (q) dq = ∫ 𝜓1 (q) q 𝜓1 (q) dq = 0. In this way the closed electron–photon system is now 3.3 Modeling electron–photon collisions described with an exact model acting only on two coupled as momentum exchange electron wavefunctions. The present model can be extended to an open system through the use of the so-called Bohmian It is straightforward to see that the process 𝜓(x, ts ) → 𝜓 s (x, ts ) conditional wavefunction for electron wavefunctions 𝜓A (x, t) satisfying a controlled change of the momentum can be done and 𝜓B (x, t). For an overview on Bohmian mechanics, we using 𝜓 s (x, ts ) = eip𝛾 x∕ℏ 𝜓(x, ts ), which ensures that the state recall [18, 19], and for a definition and properties of the 𝜓 s (x, ts ) has a increase of momentum by an amount p𝛾 with Bohmian conditional wavefunction see [20–22]. respect to 𝜓(x, ts ). In fact, it can be easily demonstrated that such change on the momentum of 𝜓 s (x, t) can be produced from the Schrödinger equation of the form: 3.2 Modeling electron–photon collisions as energy ( )2 𝜕𝜓(x, t) 1 𝜕 exchange iℏ = −iℏ + p𝛾 𝛩ts 𝜓(x, t) 𝜕t 2m 𝜕x (19) We consider now an electron defined by a single-particle + V(x) 𝜓(x, t) , state 𝜓(x, t). As indicated above, for an open system like an where 𝛩ts is a function equal to zero for t < ts and 1 for electron device, such state can be understood as a Bohmian t > ts . Notice that the probability presence of the scattered conditional wavefunction [20–22] where only the degree of freedom of the electron is considered 𝜓(x, t) ≡ 𝜓(x, q(t), t) wave packet satisfies |𝜓 s (x, ts )|2 = |𝜓(x, ts )|2 because only a global phase eip𝛾 x∕ℏ is added. For a deeper explanation of the while the degree of freedom of the photon q is fixed to some derivation of (19) we recall [33]. particular (Bohmian) value q(t). See [22] for more details. 13 Journal of Computational Electronics (2021) 20:2232–2244 2237 In an analogous way as the previous section, before the will also plot the Wigner distribution through the Wigner- scattering, the state can be written as a superposition of Weyl transform defined from Eq. (9). In particular, we will momentum eigenstates 𝜙p (x) (which is also a basis of the consider scenarios where electrons are suffering emission electron in the x space) as: and absorption of photons, while travelling in free space or impinging on a resonant tunneling diode. Such structure is ∫ 𝜓(x, t) = dp b(p, t) 𝜙p (x) , (20) composed by two barriers with thickness 2nm, height 0.3 eV and a distance between the two barriers of 16 nm. The with b(p, t) = ∫ dx 𝜓(x, t) 𝜙∗p (x). The ensemble momentum electron has effective mass m = 0.041 m0 , where m0 is the ⟨p(t)⟩ is increased of an amount p𝛾 , so in the momentum mass of the electron at rest. The resonant energies of the representation the state after the collision is: double barrier structure are 0.023 eV for first resonant state and 0.096 eV for the second resonant state. ∫ 𝜓 s (x, ts ) = dp b(p − p𝛾 , ts ) 𝜙p (x) 4.1 Exact evolution of the Wigner function (21) ∫ = dp bs (p, ts ) 𝜙p (x) In this subsection we will show electron wave functions as =e ip𝛾 x∕ℏ 𝜓(x, ts ) , defined in (13). In particular, we start by considering 𝜓A (x, 0) as a Gaussian wave packet in the left contact outside the where we have defined bs (p, t) = b(p − p𝛾 , t). Notice that: barrier with an energy equal to the second resonant energy, while 𝜓B (x, 0) = 0, meaning that initially there is no photon ∫ b(p − p𝛾 , t) = dx 𝜓(x, t) 𝜙∗p−p (x) in the structure. Since both electron wavefunctions have a coupled evolution in (15) and (16), step by step, the wave 𝛾 (22) ∫ =eip𝛾 x∕ℏ dx 𝜓(x, t) 𝜙∗p (x) . function 𝜓B (x, t) grows and 𝜓A (x, t) decreases indicating that a photon is being created. The Wigner function of the whole process is computed as the sum of the Wigner func- Unfortunately, as it will be demonstrated in the following tion linked to 𝜓A (x, t) (with (9)) plus the Wigner function numerical results, a general model of scattering valid for linked to 𝜓B (x, t) (again with (9)). Notice that, from (13), we get ∫ dq |𝛹 (x, q, t)|2 = |𝜓A (x, t)|2 + |𝜓B (x, t)|2 because 𝜓0 (q) scenarios with the presence of potential barriers requires dealing with the energy conservation, and not with momen- and 𝜓1 (q) are orthogonal. The Wigner function is plotted in tum conservation. Fig. 1 from a to c in three different times corresponding to For the sake of clarity, the relationship of the approxi- the electron impinging upon a double barrier structure with mated models in Sects. 3.2 and 3.3 with the exact model of an (initial) energy equal to the second resonant level of the Sect. 3.1 is here described. In the exact model, for each elec- double barrier. Inside the well the electron is interacting with tron, both wavefunctions 𝜓A (x, t) and 𝜓B (x, t) are computed the electromagnetic field and Rabi oscillations of the elec- through (15) and (16). This ensures the possibility of a con- tron between first and second resonant levels are observed. tinuous transition from 𝜓A (x, t) (linked with the zero photon) This can be seen from the wavefunction that, showing one to 𝜓B (x, t) (one photon), and viceversa. In the approximate maximum in the probability distribution inside the quantum models, we just simulate one wave function 𝜓(x, t) for each well, is occupying the first level in Fig. 1b and e. On the electron that takes into account the perturbation of the state other hand, from Fig. 1c to f, two maxima appear inside due to scattering. In particular, in the case of a emission of the well, so that the second level of the double barrier is a photon, the initial state is 𝜓(x, 0) = 𝜓B (x, 0) (one photon), occupied. and we wanted the final state at time ts provided by approxi- mate model to be as similar as possible to 𝜓 s (x, ts ) ≈ 𝜓A (x, ts ) 4.2 Approximate evolution of the Wigner function (zero photon). For photon absorption, we have as the initial in free space state 𝜓(x, 0) = 𝜓A (x, 0) (zero photon), and the final state 𝜓 s (x, ts ) ≈ 𝜓B (x, ts ) (one photon). In this subsection we show the evolution of the Wigner func- tion defined from a single electron evolving in free space and undergoing scattering through the (approximate) collision 4 Numerical results models explained in the Sects. 3.2 and 3.3. The goal of this section is to prove the equivalence of two previous collisions In this section, we will compare the exact collision approach models (energy exchange and momentum exchange) in the developed in the previous section, with the two collisions case of an electron system in free space. In free space, the models mentioned there. Apart from plotting the evolution momentum of a photon is much smaller than the momentum of the involved states before and after the collisions, we of an electron and the conservation of momentum (not only 13 2238 Journal of Computational Electronics (2021) 20:2232–2244 Fig. 1  Wigner function of the total electron wavefunction interact- emitting a photon, so that the electron energy is ⟨E(t)⟩ = 0.023 eV . In ing with a double barrier structure (green lines), at different times of d–f are shown projections along the energy (top), momentum (mid- its evolution: a at the beginning of the evolution, b when first enter- dle) and position (bottom) axis of the Wigner transform, respectively, ing inside the double barrier structure with ⟨E(t)⟩ = 0.096 eV , c after of (a), (b), and (c) of energy) has to be satisfied. Thus, in this Sect. 4.2, from a momentum of ⟨p(t)⟩ = 1.573 ⋅ 108 m −1. The final wave a physical point of view, we consider the interaction of an packet given by the collision models in Sects. 3.2 or 3.3 electron with a phonon (instead of a photon). has ensemble value of the energy ⟨Es (t)⟩ = 0.096 eV, cor- In (2), is has been shown [26–28] that the time of scat- responding to a momentum of ⟨ps (t)⟩ = 3.214 ⋅ 108 m−1. This tering 𝛥t has to be finite. In the present work, the transition process 𝜓(x, ts ) → 𝜓 s (x, ts ) mimics, for example, an electron shown in (18) for a scattering of energy Es , is divided into absorbing a phonon. The Wigner function is computed from 40 steps of energy change 𝛥Es = Es ∕40. Between each step, (9) for the states 𝜓(x, ts ) → 𝜓 s (x, ts ). the system is evolved for a time of around 6 fs. As a result, In Fig. 3, the electron is injected with initial ensemble the total scattering time for the approximate model will be energy ⟨E(t)⟩ = 0.096 eV, or momentum ⟨p(t)⟩ = 3.214 ⋅ 108 240 fs. An analogous implementation is done for (21). In m−1. This electron also undergoes scattering in a finite time Fig. 2, the electron is injected with an ensemble energy of approximately 0.1 ps and emits a phonon reaching an equal to the first resonant level, ⟨E(t)⟩ = 0.023 eV, meaning Fig. 2  Wigner function for photon absorption in free space: a before conditions. In c and d are shown projections along the energy (top), scattering b scattered with the energy exchange and the momentum momentum (middle) and position (bottom) axis of the Wigner func- exchange model, which is equivalent to the former in free space tions, respectively, of (a) and (b) 13 Journal of Computational Electronics (2021) 20:2232–2244 2239 ensemble energy of the wave packet equal to ⟨Es (t)⟩ = 0.023 of the well. However this is not observed in Fig. 4c and f, eV, corresponding to a momentum of ⟨ps (t)⟩ = 1.573 ⋅ 108 where a momentum exchange model is used. Now, inside m −1. This process corresponds to an emission of phonon the well, the maximum of probability after the scattering and the transition 𝜓(x, ts ) → 𝜓 s (x, ts ) is computed from the belongs still to the first resonant level (only one maximum collision models in Sects. 3.2 or 3.3 and the Wigner function is observed). This is an unphysical result whose reason is linked to such states from (9). that there is no one-to-one correspondence between energy We conclude that the evolution of the Wigner function and momentum. Thus, despite having control on the change of an electrons scattered in free space behaves equivalently of the momentum, we do not have control on the change of with both models, the one of Sect. 3.2 with energy conser- the energy. In the case of photon emission, shown in Fig. 5, vation and the one of Sect. 3.3 with momentum conserva- the same physical transition is observed from Fig. 5a to b tion. The physical reason of such agreement is because, in (and on their respective projections in Fig. 5d and e, where free space, there is a one-to-one correspondence between the two maxima of probability inside the quantum well are the eigenvalues of momentum and energy operators (since transformed into just one maxima later), while the transition they commute). between levels is not observed from Fig. 5a to c (where the number of maxima remains equal to two). This is another 4.3 Approximate evolution of the Wigner function proof of the difficulties to use a momentum exchange model with potential barriers in case of an arbitrary potential. In other words, in typi- cal device scenarios for nanoscale devices, the energy and In this subsection, we show a comparison between the momentum operators do not commute so that the eigenstates energy exchange and momentum exchange scattering mod- of the momentum are not eigenstates of the energy and vice- els, respectively, explained in Sects. 3.2 and 3.3, for an elec- versa. Then, the change in momentum can have an arbitrary tron wavefunction interacting with the double barrier struc- translation into change of energy. The dramatic consequence ture described at the beginning of Sect. 4. The momentum of is that the conservation of energy during the electron–pho- an electron inside a quantum well tends to be very small (the ton collision is not guaranteed by ensuring conservation of wave function tends to become real) and the conservation of momentum. momentum is not a requirement because of the translation symmetry is broken by the barriers. Thus, in this Sect. 4.3, we do consider the interaction of a photon with an electron also impinging on a double barrier structure. 5 Does Wigner function satisfy complete In Fig. 4, the electron is first injected at the first reso- positivity and energy conservation? nant level of the double barrier structure and undergoes a photon absorption, thus occupying, after the scattering, the The typical way of reducing the computational burden second level of the double barrier structure. Energetically by including the role of photons and phonons in quantum the initial an final energy values are the same as in Fig. 2. transport is through a collision model. Such collision model This transition is clear in Fig. 4b and e, where the energy introduces some new term in the equation of motion of the exchange model is used. In fact, while in Fig. 4d there is simulated degrees of freedom. For example, in the Wigner a single maximum of the probability presence in the bot- distribution formalism, we can visualize the effect of the col- tom image, there are two maxima in Fig. 4e. This is a proof lision as a change from the initial (before collision) Wigner of a transition from the first to the second resonant state function fW (x, k, t) to the final (after collision) Wigner Fig. 3  Wigner function for photon emission in free space: a before conditions. In c and d are shown projections along the energy (top), scattering b scattered with the energy exchange and the momentum momentum (middle) and position (bottom) axis of the Wigner func- exchange models, which is equivalent to the former in free space tions, respectively, of (a) and (b) 13 2240 Journal of Computational Electronics (2021) 20:2232–2244 Fig. 4  Comparison of Wigner functions undergoing photon absorp- tom) axis of the Wigner transform, respectively, of (a), (b), and (c). tion in a double barrier structure (green lines), at different times: a Notice the change for the transition from the first resonant level to the before scattering b scattered with the energy exchange model and c second from (d) to (e), while such transition is not present from (d) with the momentum exchange model. In d—f are shown projections to (f) along the energy (top image), momentum (middle) and position (bot- Fig. 5  Comparison of Wigner functions undergoing photon emission energy (top), momentum (middle) and position (bottom) axis of the in a double barrier structure (green lines), at different times: a before Wigner transform, respectively, of (a–c). Notice the change for the scattering b scattered with the energy exchange model and c with the transition from the second resonant level to the first from (d) to (e), momentum exchange model. In d–f are shown projections along the while such transition is not present from (d) to (f) 13 Journal of Computational Electronics (2021) 20:2232–2244 2241 function fWs (x, k, t). But, is there any mandatory requirement wavefunctions. The algorithm explained below will not be that such change of the Wigner function has to satisfy? Or is easily implementable in a general Wigner transport formal- any possible change physically acceptable? ism, but it will clearly confirm that it is possible to develop By construction, the Wigner function without (external) a successful transition fW (x, k, ts ) → fWs (x, k, ts ) due to the collisions do satisfy complete positivity in the sense that collision of an electron with a photon (phonon), while satis- ∫ dk fW (x, k, ts ) > 0 for all positions x. The Wigner function, fying conditions (23) and (25). The algorithm deals with a after being modified by the collision operator, does also have single electron (as everywhere else in this paper), and it has to satisfy the same complete positivity condition: the following three steps: ∫ ̂ s )�x⟩⟨x�) = Tr(𝜌(t dk fWs (x, k, ts ) > 0 ∀ x . (23) • First step The Wigner function before the scattering fW (x, k, t) is translated into a wave function 𝜓(x, ts ) in Notice that for a discrete Wigner function (used in the com- the following way: putational algorithms), for a grid with Nx position points and 1 Nk wave vector points, the condition (23) imposes Nx equa- 𝜓(x, ts ) =𝜌(x, 0, ts ) 𝜓 ∗ (0, ts ) tions, each equation involving Nk elements of the estimated ( ) ∫∫ 1 � discrete fWs (x, k, ts ). = dk dx� 𝜌 y, y� , ts eik(x−x ) (26) 2𝜋𝜓 ∗ (0, ts ) On the other hand, during the collision we can assume a ( ) 𝜓 (0, ts ) ∫ 1 x controlled change of the energy of the system. To verify such = ∗ dk fW , k, ts eikx , energy conservation, the expectation value of the electron 2 energy within the Wigner approach is defined as follows: where we have used that 𝜌(x, 0) = 𝜓 ∗ (0, t)𝜓(x, t) for a pure state, and y = (x + x� )∕2 and y� = (x − x� )∕2 . The ∫∫ ⟨E(t)⟩ = Tr(𝜌̂Ĥ0 ) = dk dx fW (x, k, t) h0 (x, k) , (24) complex number 𝜓 ∗ (0, ts ) is irrelevant here since it can be understood as a normalization constant [34]. where h0 (x, k) = ∫ dx� ⟨x + x2 �Ĥ0 �x − x2 ⟩e−ikx with Ĥ0 the � � • Second step Once we have 𝛹 (x, ts ) we can apply the col- � Hamiltonian without collisions defined in (11). lision algorithm explained in Sect. 3.2 to get the transi- The initial energy of the Wigner function is tion 𝛹 (x, ts ) → 𝛹 s (x, ts ) that we know satisfy complete ⟨E(ts )⟩ = ∫ ∫ dk dx fW (x, k, ts )h0 (x, k), then after the absorp- positivity and energy conservation. tion (or emission) of a photon with energy E𝛾 , the final • Third step Once we get the pure state after the collision Wigner function has to satisfy: 𝛹 s (x, ts ), we use expression (9) to compute the Wigner distribution function after the collision: fWs (x, k, ts ). ̂ ⟨Es (ts )⟩ =⟨E(ts )⟩ ± E𝛾 = Tr(𝜌̂s H) (25) Indeed, an example of such Wigner functions, from ∫∫ = dk dx fWs (x, k, ts )h0 (x, k) . fW (x, k, ts ) before collision to fWs (x, k, ts ) after collision, is already plotted from Fig. 4a to b, respectively, for photon The condition (25) implies one additional equation involv- absorption. Identically, another example is provided from ing Nx × Nk elements of the new Wigner function. Together, Fig. 5a to b for photon emission. In Fig. 6 it is shown how conditions (23) and (25) do only define Nx + 1 elements of such algorithm (based on the electron–photon collision the total Nx × Nk elements of the Wigner function after the model of Sect. 3.2) reproduce the exact electron–phonon scattering fWs (x, k, ts ). Notice that the Wigner distribution interaction computed from Sect. 3.1. The evolution of the function has Nx × Nk elements so that these two conditions ensemble value of the energy for simulations in Fig. 1 do not totally determine which is the new Wigner function (black dashed line in Fig. 6), Fig. 4a and 4b (blue line in after scattering. In other words, there are many possible Fig. 6), Fig. 5a and 5b (red line in Fig. 6). The ensemble Wigner functions that can satisfy (23) and (25), but there value ⟨E(t)⟩ is computed using expression (24) at different are also many Wigner function that are invalid because they times. To better approximate the exact result the change of do not satisfy (23) and (25). energy E𝛾 is not done in a single time step of the simulation, We want to emphasize again that there is no fundamental but in a time interval identical to the time it takes the exact limitation for developing a successful collisions model from solution to produce such energy transition (which is related the Wigner function formalism. In other words, it is possible to the frequency of the Rabi oscillations [32]). to find fW (x, k, ts ) and fWs (x, k, ts ), before and after the scatter- The excellent agreement in Fig. 6 was possible because ing respectively, that satisfy complete positivity and energy the evolution of such system is treated on the level of the conservation. To underline this point, we explain an algo- Bohmian conditional wavefunction. Without such knowl- rithm that uses the collision model developed in Sect. 3.2 for edge, the task of satisfying conditions (23) and (25) 13 2242 Journal of Computational Electronics (2021) 20:2232–2244 emission or absorption, when described by (12) because it is perfectly possible to go back in time in this exact model, and the process of emission and absorption are time reversible at the microscopic level. In the approximations done in Sects. 3.2 and 3.3, we elim- inate the explicit simulation of the photon degree of freedom and introduce the effect of the photon into the electron as a collision. The electron alone, without the photon, is an open (non-Markovian) system. But, if we know exactly when the collision with the electron takes place, then we can still con- sider that approximations done in Sects. 3.2 and 3.3 as time reversible (as it was Eq. (12) itself) because it is possible to know how to go back in time, and the process of emission and absorption are time reversible at the microscopic level. The conclusions above for time reversibility of the equa- tion of motion of one electron interacting with one pho- Fig. 6  Comparison between the expectation value of energy of the ton can be straightforwardly generalized to the equation of system undergoing collision modeled from the exact model (black motion of thousands of electrons interacting with thousands dashed line) and with the energy exchange algorithm, for photon of photons, as far as we only look at the microscopic infor- emission (red line) or photon absorption (blue line). These models are implemented from the Wigner distribution function of a pure state mation. However, for such many-body system, if instead of looking at the microscopic degrees of freedom, we are interested in a macroscopic equation of motion for the aver- becomes very complicated. In a real electron device, more age energy translated from the electrons to the photons and than one electron have to be simultaneously simulated and viceversa, the conclusion can be different. If we are able then expression (26) cannot be applied because the density to simulate a many-particle Schrödinger equation of such matrix, and its Wigner transform, will hide the knowledge system, including the electron device and the surroundings about every single-particle state. We conclude that without in a scenario outside of thermodynamic equilibrium, we will the knowledge of the state it seems quite complicated to realize that a photon emitted from the electron device will model a reasonable collision model that satisfies energy con- hardly be able to return to the electron device again because servation. For example, there is no guarantee at all that the the surroundings have larger extension, in space, than the direct implementation of the Boltzmann collision operator electron device. Thus, inside the electron device there will in the Wigner distribution can satisfy the above conditions. be more emissions than absorption, in average. In fact, the expected net dissipation of electron energy (Joule effect) will take place. We conclude that the equation of motion 5.1 Are collisions a source of time irreversibility ? for the macroscopic average energy in the electrons inside the device is time irreversible. Notice that the time irrevers- At this point, although not directly related with the goal of ibility of the (macroscopic) equation of motion of the aver- this paper, it is interesting to discuss if collisions in electron age energy of electrons inside the device is fully compatible devices are a source of time irreversibility or not. While with the time reversibility of the (microscopic) equation of typical microscopic laws are time reversible, an arrow of motion of electrons and photons everywhere. time appears in macroscopic phenomena. For example, we In conclusion, are collisions a source of time irrevers- know that electrons lose energy, in average, when traversing ibility? Not in a microscopic description, but yes in a macro- the device (Joule effect). Therefore, we can know if we are scopic one. Microscopically, the collisions provide an equa- looking forward or backward in time by looking at the mac- tion of motion for the dynamics of electrons and photons that roscopic heating or cooling of the device. But, why micro- is time reversible. Macroscopically, the collisions inside the scopic time-reversible laws become time irreversible at the device ensure that the dissipation of the average energy of macroscopic level? What is the role of collisions? the electrons is time irreversible and satisfy the Joule effect. We have shown in Sect. 4.1 that electron–photon interac- We have learned a practical lesson to implement a col- tion can be studied in an exact way as a solution of the many- lision model here. Since the exact solution of thousands of body Schrödinger equation in a closed system including the electrons interacting with thousands of photons is not pos- degrees of freedom of one electron and one photon. Thus, sible, the practical implementation of the scattering model we can conclude that interaction of a single electron with a with the approximations done in Sect. 4.2 has to satisfy the single photon are a time-reversible phenomena, either for microscopic time reversibility of the equation of motion and, 13 Journal of Computational Electronics (2021) 20:2232–2244 2243 simultaneously, the time irreversibility of the macroscopic Author contributions MV, XO involved in conceptualization; MV, XO average energy. This is typically achieved by randomly participated in methodology; MV, XO involved in formal analysis and investigation; MV, XO involved in writing—original draft prepara- selecting the scattering rates with a distributions that sat- tion; MV, XO participated in writing—review and editing; XO involved isfies the macroscopic requirement: higher emission than in funding acquisition; XO participated in resources; XO involved in absorption rates. supervision. All authors read and approved the final manuscript. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. This project has received funding from the European 6 Conclusions Union’s Horizon 2020 research and innovation programme under the Maria Skłodowska-Curie Grant Agreement No. 765426 (TeraApps). This research was also funded by Spain’s Ministerio de Ciencia, Inno- In this paper we have shown that a physically reasonable vación y Universidades under Grant No. RTI2018-097876-B-C21 collision model, within the Wigner description, has to (MCIU/AEI/FEDER, UE), the ”Generalitat de Catalunya” and FEDER satisfy the following two mandatory requirements: Com- for the project 001-P-001644 (QUANTUMCAT). plete positivity and energy conservation. After presenting the two conditions and studying its implementation for an Availability of data and material The datasets generated during and/or analyzed during the current study are available from the corresponding electron inside a double barrier interacting with a photon, author on reasonable request. we compare exact results to a collision model based on energy exchange in Sect. 3.2 and to a collision model based Declarations on momentum exchange in Sect. 3.3. We conclude that, unfortunately, it is very complicated to develop a practical Conflicts of interest The authors have no conflicts of interest to declare collision model in the Wigner formalism that satisfies both that are relevant to the content of this article. The funders had no role conditions. in the design of the study; in the collection, analysis, or interpretation of data; in the writing of the manuscript, or in the decision to publish The good conclusion is that there is no fundamental the results. reason that disqualifies, a priori, the possibility of imple- menting collisions in the Wigner distribution framework. Open Access This article is licensed under a Creative Commons Attri- The practical requirement in (23) and in (25) provides less bution 4.0 International License, which permits use, sharing, adapta- restrictions than the degrees of freedom (number of ele- tion, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, ments) of a (discretized) Wigner distribution function. In provide a link to the Creative Commons licence, and indicate if changes fact, for a pure state, we have shown an algorithm that mod- were made. The images or other third party material in this article are els electron–phonon collisions in the Wigner function for- included in the article's Creative Commons licence, unless indicated malism with an exact approach. The problem for its gener- otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not alization to realistic quantum transport is that the mentioned permitted by statutory regulation or exceeds the permitted use, you will algorithm needs the wave function information of each elec- need to obtain permission directly from the copyright holder. To view a tron inside the device, however such additional information copy of this licence, visit http://​creat​iveco​mmons.​org/​licen​ses/​by/4.​0/. is not available in the typical Wigner function algorithms. Finally, we want to mention that most of the difficulties of the Wigner distribution function to satisfy energy conser- vation in collisions, are inherited form the difficulties of its References father description, the density matrix, to tackle the properties of individual particles. At the end of the day, these prob- 1. Breuer, H.P., Petruccione, F.: Theory of Open Quantum Sys- tems, pp. 5–23. 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