{
  "id": "1410.8701",
  "version": "v1",
  "published": "2014-10-31T10:56:18.000Z",
  "updated": "2014-10-31T10:56:18.000Z",
  "title": "Detection and Survival of a Quantum Particle on a Lattice",
  "authors": [
    "Shrabanti Dhar",
    "Subinay Dasgupta",
    "Abhishek Dhar",
    "Diptiman Sen"
  ],
  "comment": "13 pages, 6 figures",
  "categories": [
    "quant-ph"
  ],
  "abstract": "We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect it's arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals $\\tau$, and we consider the evolution of the wave function till the time a detection occurs. We study the probabilities of its first detection at some time and conversely the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, consisting of unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian. For some examples, of a particle moving on one and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with exact numerical results. A mean field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and non-trivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-31T10:56:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03.65.Ca",
      "03.65.Ta"
    ],
    "keywords": [
      "quantum particle",
      "non-hermitian hamiltonian",
      "survival probability",
      "regular time intervals",
      "mean field model"
    ],
    "publication": {
      "doi": "10.1103/PhysRevA.91.062115",
      "journal": "Physical Review A",
      "year": 2015,
      "month": "Jun",
      "volume": 91,
      "number": 6,
      "pages": "062115"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015PhRvA..91f2115D"
    }
  }
}