Complete population inversion of Bose particles by an adiabatic cycle: an example of exotic quantum holonomy

A poster presentation at Quantum Thermodynamics: Thermalization and Fluctuations (201709-27/2017-09-30) @ Yukawa Institute for Theoretical Physics, Kyoto Univ., Japan

Transcript

1. Complete Population Inversion of Bose particles by an Adiabatic Cycle: an example of Exotic Quantum Holonomy Atushi Tanaka (Tokyo Metropolitan Univ.) [Collaboration with Tasku Cheon (Kochi Univ. Tech.)] Aim & Summary: Find an adiabatic closed path from the ground state to the inverted state of many-particle systems C ˜ C ground inverted The inverted state The populations of excited states are much larger. An ingredient for lasing. Made from pumping, usually. It will be shown that an adiabatic cycle induces the population inversion of weakly-interacting Bosons, which are initially in the ground state. Ref. AT and T. Cheon, New J. Phys. 18, 045023 (2016). Remark: adiabatic cycles in thermodynamics and mechanics The present example is quantum mechanical one. In Thermodynamics Quasi-static adiabatic cycles are always trivial. In Quantum Mechanics Adiabatic cycles may induce nontrivial changes, i.e., quantum holonomies. An adiabatic cycle C (= C1 +C2 +C3) for a 1-d box with N particles |ψ0 Prepare the system in an initial stationary state. C1 X1 Adiabatically insert a δ-wall at X1. C2 X1 X2 Adiabatically move the δ-wall from X1 to X2. C3 X2 Adiabatically remove the δ-wall at X2. |? The Hamiltonian returns to the initial one. N = 1: exotic quantum holonomy |1 →|2 for the adiabatic cycle C Parametric evolution of scaled eigenenergies ¯ k ∝ E/N. Parametric evolution of eigenfunctions Ref. S. Kasumie, M. Miyamoto and AT, PRA 93 (2016). N = 2: some crossings are broken due to interaction While |11 C → |22 is intact, the interaction destroys |33 C → |44 . This is because the crossing between |33 and |24 become avoided. On the other hand, the crossings in C2 is intact because the gaps are, at most, exponentially small as they involves tunneling. Selection rule V |2⊗N has nonzero overlap only with |2⊗N−2n1n2 , if V contains only two-body interactions. N > 2: the exotic holonomy |1⊗N C → |2⊗N survives This is thanks to the selection rule. N = 3 N = 4 Example (N = 3): |111 C → |222 is intact although |222 crosses with |113 . This is because 222|V |113 = 0 holds. Quantum Holonomy for Adiabatic Cycles (Conventional) Quantum Holonomy Adiabatic cycles may induce nontrivial geometric phase factors, which is identified with the holonomy of fiber bundles [Berry 1984]. Exotic Quantum Holonomy Adiabatic cycles may induce nontrivial change in eigenspaces, which can be associated with a fiber bundle with a discrete structure group (a covering space) [AT and T. Cheon, PLA 379 (2015)]. Outlook Analysis in the Gross-Pitaevskii equation (where the population inversion is related with dark solitons). cf. Karkuszewski, Sacha and Zakrzewski, A method for collective excitation of Bose-Einstein condensate, PRA 63, 061601(R)(2001). Fast driving of the adiabatic cycle cf. Mart´ ınez-Garaot, Palmero, Muga and Gu´ ery-Odelin, Fast driving between arbitrary states of a quantum particle by trap deformation, PRA 94, 063418 (2016). Application to light amplification/lasing (need to extend to charged particles). cf. TC and Shigehara, Fermion-Boson Duality of One-Dimensional Quantum Particles with Generalized Contact Interactions, PRL 82, 2536 (1999). Quantum Thermodynamics: Thermalization and Fluctuations @ YITP / P36