Traffic models and traffic-jam transition in quantum (N+1)-level systems Andrea Nava, Domenico Giuliano, Alessandro Papa, and Marco Rossi Dipartimento di Fisica, Universit`adella Calabria, Arcavacata di Rende I-87036, Cosenza, Italy INFN - Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy (Dated: April 14, 2021) We propose a model to implement and simulate different traffic-flow conditions in terms of quan- tum graphs hosting an (N+1)-level dot at each site, which allows us to keep track of the type and of the destination of each vehicle. By implementing proper Lindbladian local dissipators, we derive the master equations that describe the traffic flow in our system. To show the versatility and the reliability of our technique, we employ it to model different types of traffic flow (the symmetric three-way roundabout and the three-road intersection). Eventually, we successfully compare our predictions with results from classical models. I. INTRODUCTION one implements the analogy with quantum mechanical systems by introducing an effective Hamiltonian H^ , as well as vehicles creation and annihilation operators, sim- In the last decades the traffic behavior and the as- ilar to creation and annihilation operators in quantum sociated traffic-jam transition have been widely studied mechanics. Yet, the expectation value of an observable within the master equation formalism for stochastic ex- O^ for a system in the state j i has not to be com- clusion processes of many-body systems, using models puted as h j O^ j i, but with respect to the \sum-vector" such as, for instance, the asymmetric exclusion process P (ASEP) as well as the total asymmetric exclusion process hsj = hnj with jni being a basis of the Fock space, i.e., ^ (TASEP)1{4. All these models rely on the possibility to it is defined as hsj O j i. simulate the dynamical behavior of the system at a coarse Despite its potentially wide applicability, the formal- graining scale with vehicles living on a lattice with many- ism based on the effective Hamiltonian H^ is not easy to body interactions, rather than considering a continuum implement, since it is not easy to compute the operator of possible states (representing vehicles position and ve- expectation values as described above. For this reason locity). Therefore, the evolution of the system is not in the last years a fully quantum formulation in terms expressed in terms of deterministic equations of motion, of a Markovian Lindblad master equation description for but by means of a set of stochastic rules that describe all density matrix has been introduced14,15, which allows for the different paths that a system can follow starting from expressing the stochastic classical rules in terms of Lind- a given initial state5,6. In fact, despite the traffic flow is a blad jump operators that ensure (vehicle) particle conser- classic problem and, therefore, it might seem appropriate vation exclusion processes by allowing, at the same time, to describe it by resorting to appropriate hydrodynamic for an easy implementation of open boundary conditions. models, the finite dimensions of vehicles introduce a nat- Furthermore, the approach allows to treat at the same ural spatial \quantization" of space. Indeed, this concept level classical incoherent evolution and pure quantum co- has been widely used in cellular-automaton descriptions herent evolution opening the possibility to use quantum of the traffic-flow problem, where both space and time are dot systems as experimental devices to simulate classical discrete5{7 (incidentally, it is worth stressing how a sim- traffic behavior16,17. ilar \quantization" of both space and time, followed by Remarkably, while a huge amount of work has been a systematic implementation of cellular-automaton ap- already done for traffic description in one-dimensional proach, has proved to be pretty effective in, e.g., the systems18{23, in the last years a wide interest is arising 8,9 analysis of the infection spread in real space ). around the possibility to describe different kinds of inter- In Ref.[10], the stochastic rules have been encoded sections24{26, searching for optimization procedures27{30. in the transition probabilities for elementary evolutions Along this line of research, in this paper we analyze from one state to another. Moreover, they have been graphs with multiple in/out sites in terms of multilevel shown to be equivalent to the consolidated cellular au- quantum dots. In particular, we are interested in the in- tomaton approach of Biham and Nagatani7,11,12. Of par- tersection between three possible routes where the exit arXiv:2104.06289v1 [cond-mat.stat-mech] 13 Apr 2021 ticular relevance is the fact that the rules can be con- direction of each vehicle is not stochastically chosen dur- verted into a set of linear operators acting on the vec- ing the hopping process, but it is rather an internal prop- tors of a proper Fock space, taking advantage of the for- erty of the vehicle. mal analogy between classical stochastic processes and Specifically, we discuss in detail three-road junctions, quantum mechanical formalism. The Fock space descrip- which we model in terms of a quantum system on a graph, tion has been developed in terms of both Bosonic and realized by means of multilevel quantum dots coupled to Fermionic operators on periodic and open systems (see, external reservoirs. Within our formalism, we use quan- for instance, Ref. [13] for a review on the subject). tum hopping operators between quantum dots to repre- Within the Fock space formulation of the problem, sent the flux of vehicles in the junction. At variance, 2 to account for vehicles entering the junction from out- the other N states describe different vehicle types and side/exiting from the junction, we introduce hopping op- destinations. Assuming m different vehicle types and n erators from/to external reservoirs to/from the internal different destinations, we have N = m × n 'occupied' dots. Making use of quantum Lindblad master equation states, which we label as jji = j(v − 1) n + di with inte- formalism we recover results in agreement with classical gers v 2 f1; : : : ; mg and d 2 f1; : : : ; ng. Within our nota- experiments. In doing so, we develop and discuss a three- tion, the levels jji, with j = 1; :::; n, represent a vehicle of dot model, which encodes the general information about type 1 directed towards one of the n possible destinations, the intersection of three routes, and eventually generalize the levels jji, with j = n + 1;:::; 2n represent a vehicle it to a six-dot model. This allows us to implement differ- of type 2 directed towards one of the n possible destina- ent priority rules such as a roundabout, a right-hand pri- tions, and so on. On each site we introduce the operators: ority junction, and an intersection between a major and σj;0 = jji h0j (σ0;j = j0i hjj), that create (destroy) a given 0 a minor road. Remarkably, despite the \minimal" setup vehicle-destination combination, and σj0;j = jj i hjj, with we employ, our model exhibits all the relevant features of (v −1)n+1 ≤ j; j0 ≤ vn, that describe a vehicle of type v a classical traffic-flow diagram, such as the existence of a changing its destination. These operators satisfy the con- N critical density of vehicles beyond which the traffic-jam y P 7,11,12 ditions σi;jσk,λ = δj;kσi,λ, σi;j = σj;i and σj;j = 1. phase transition sets in . j=0 Moving backwards along the correspondence with As a consequence the dot can only be empty or occu- quantum lattice systems, our model can be re- pied by a single vehicle/destination combination as it is garded as a \minimal", pertinently adapted, version impossible to create more than a vehicle on the same site. 31,32 of the Y-junction of one-dimensional fermionic We can now build up the Hilbert space for a complex 33{35 and/or bosonic quantum systems, including spin network of L multi-level dots as the tensor product of the 36{39 chains , as well as junctions hosting the remarkable Hilbert spaces of each dot. It is more convenient, as we 40{42 \topological" realization of Kondo effect , which have will show later, to let N be site dependent. Therefore, we been largely addressed in the recent literature on corre- L Q lated quantum systems as simplest examples of devices have, in general, (N` +1) basis vectors in the enlarged exhibiting nontrivial phases/quantum phase transitions `=1 space, jj1; : : : ; jLi. They are defined as the Kronecker in their phase diagram. tensor product of the basis vectors of each dot The paper is organized as follows: jj1; : : : ; jLi = jj1i ⊗ ::: ⊗ jjLi : (1) • In Section II we present our multi-directional model, in which we represent vehicles as hardcore (`) In the same way, we label as σi;j the creation, annihi- bosons on a graph. Consistently with our identifi- lation and conversion operators acting on a given dot `: cation, we define the creation and annihilation op- these are realized as the tensor product of (L−1) identity erators for vehicles and write the Markovian Lind- matrices of dimension N` × N` and the σi;j operator at blad equation governing the time evolution. the site `, that is In Section III we present a minimal three-dot model • σ(`) = ⊗ ::: ⊗ σ ⊗ ::: ⊗ : (2) that allows us to address the main features of the i;j I i;j I three-route intersection. For example, the state j1; 0; 2i represents a graph of three • In Section IV we extend the analysis of Section III dots where we have an empty road on site two, while by introducing a six-dot model that allows us to sites one and three are filled by two different vehicles implement different priority rules.