{
  "annotation_id": "64e7f72d-6c79-4102-a241-999aa97c6ab5",
  "date_created": "2026-03-02T18:02:23.737000Z",
  "date_modified": "2026-03-02T18:02:23.737000Z",
  "file_hash": "48527bf371798c2635543b38b84a9e197d9765f337bd5cdc796d38bbbfbe666f",
  "private": false,
  "record": {
    "abstract": "In this paper we consider the minimum time population transfer problem for\nthe $z$-component of the spin of a (spin 1/2) particle driven by a magnetic\nfield, controlled along the x axis, with bounded amplitude. On the Bloch sphere\n(i.e. after a suitable Hopf projection), this problem can be attacked with\ntechniques of optimal syntheses on 2-D manifolds. Let $(-E,E)$ be the two\nenergy levels, and $|\\Omega(t)|\\leq M$ the bound on the field amplitude. For\neach couple of values $E$ and $M$, we determine the time optimal synthesis\nstarting from the level $-E$ and we provide the explicit expression of the time\noptimal trajectories steering the state one to the state two, in terms of a\nparameter that can be computed solving numerically a suitable equation. For\n$M/E\u003c\u003c1$, every time optimal trajectory is bang-bang and in particular the\ncorresponding control is periodic with frequency of the order of the resonance\nfrequency $\\omega_R=2E$. On the other side, for $M/E\u003e1$, the time optimal\ntrajectory steering the state one to the state two is bang-bang with exactly\none switching. Fixed $E$ we also prove that for $M\\to\\infty$ the time needed to\nreach the state two tends to zero. In the case $M/E\u003e1$ there are time optimal\ntrajectories containing a singular arc. Finally we compare these results with\nsome known results of Khaneja, Brockett and Glaser and with those obtained by\ncontrolling the magnetic field both on the $x$ and $y$ directions (or with one\nexternal field, but in the rotating wave approximation). As byproduct we prove\nthat the qualitative shape of the time optimal synthesis presents different\npatterns, that cyclically alternate as $M/E\\to0$, giving a partial proof of a\nconjecture formulated in a previous paper.",
    "arxiv_id": "quant-ph/0512074",
    "authors": [
      "Ugo Boscain",
      "Paolo Mason"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1063/1.2203236",
    "title": "Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field",
    "url": "https://arxiv.org/abs/quant-ph/0512074"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "7e8072cc-4675-495a-8311-2576454cdb64",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}