Counting statistics in multiple path geometries and the fluctua- tions of the integrated current in a quantum stirring device

The amount Q of particles that are transported via a path of motion is characterized by its expectation value 〈Q〉 and by its variance Var(Q). We analyze what happens if a particle has two optional paths available to get from one site to another site, and in particular what is Var(Q) for the current which is induced in a quantum stirring device. It turns out that coherent splitting and the stirring effect are intimately related and cannot be understood within the framework of the prevailing probabilistic theory. A new light is shed on the operation of a quantum pump in a closed circuit, and on the feasibility of observing either “fractional” or “mega” charges. The possibility to induce DC currents by periodic (AC) modulation of the potential is familiar from the context of electronic devices. If an open geometry is concerned, it is known as “quantum pumping” [1–6], while for closed geometry [7,8] we use the term “quantum stirring” [9,10]. The simplest model that captures the physics of quantum stirring is a 3 site Bose-Hubbard Hamiltonian [11–13]. While in a parallel study [14] we explore the role of interactions in the stirring process, in the present work we would like to explore a new aspect of the problem that has to do with counting statistics. For simplicity we neglect the interactions and accordingly the problem reduces to “one particle physics”. It is well established [15–24] that counting statistics in the context of shot noise studies provides information on the fluctuations of the occupation and on the random probabilistic nature of the the quantum-mechanical transmission/reflection process. In fact these two effects combine together. The prototype example is barrier crossing. If the average channel occupation is f and the transmission probability is p, then the emerging number of particles Q is proportional to fp, while the variance (perparticle) is Var(Q) = (1− fp)fp. Furthermore the results which are derived using classical methods (Master equation; Boltzmann-Langevin) are consistent with the quantum mechanical calculation (Scattering approach; Green function techniques), and the quantum-classical crossover is related to the statistics of the transmission coefficients as reflected by the Fano factor. The purpose of this Letter is to argue that the above common wisdom does not apply in the context of quantum stirring. In order to demonstrate this point we analyze the prototype 3-site system of Fig. 1. We measure the current that flows through a section in the c1 bond, and define the splitting ratio as λ = c1/(c1 + c2). If the passage probability from left-to-right is p we do not get Var(Q) = (1−λp)λp as implied by the naive probabilistic considerations, but rather Var(Q) = λ2(1−p)p. Then we turn to analyze a full pumping cycle that consists of two sequential Landau-Zener adiabatic passages. During the first half of the cycle λ = λ while during the second half of the cycle λ = λ . If λ 6= λ then it follows that the net integrated current is 〈Q〉 6 = 0, and we ask what is the variance Var(Q). There are some qualitative observations that are associated with our results and we would like to enumerate them in advance: (1) Coherent splitting of a wavepacket does not generate a noisy current in the system; (2) The “fractional charge” of a partial wavepacket is determined by the splitting ratio, and is physically meaningful. (3) The splitting ratio can be greater than unity or negative. This has the interpretation of having an induced circulating current in the system. (4) The splitting ratio concept allows an easier better understanding of quantum stirring, in comparison with the complicated Kubo formalism [7]. (5) The fluctuations in Q reflect the non-adiabaticity of the driving cycle. (6) The fluctuations of the integrated current grow with time as t and not as √ t.