Preparation of atomic Fock states by trap reduction

Abstract

We describe the preparation of atom-number states with strongly interacting bosons in one dimension, or spin-polarized fermions. The procedure is based on a combination of weakening and squeezing of the trapping potential. For the resulting state, the full atom number distribution is obtained. Starting with an unknown number of particles $N_i$, we optimize the sudden change in the trapping potential which leads to the Fock state of $N_f$ particles in the final trap. Non-zero temperature effects as well as different smooth trapping potentials are analyzed. A simple criterion is provided to ensure the robust preparation of the Fock state for physically realistic traps.Comment: 5 pages, 5 figure

Microscopic theory of a phase transition in a critical region: Bose–Einstein condensation in an interacting gas

“…It is immediate to generalize them to any trapping potentials and boundary conditions. They open a way to solve the long-standing problem of the BEC and other phase transitions [1][2][3][4][5][6][7][8][9][10][11][12], including a restricted canonical ensemble problem [2], and describe numerous modern laboratory and numerical experiments on the critical phenomena in BEC of the mesoscopic systems [22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”

“…It becomes possible due to the newly developed methods of (a) the nonpolynomial averages and contraction superoperators [15,16], (b) the partial difference (recurrence) equations [17][18][19] (a discrete analog of the partial differential equations) for superoperators, and (c) a characteristic function and cumulant analysis for a joint distribution of the noncommutative observables. They allow us to take into account (I) the constraints in a many-body Hilbert space, which are the integrals of motion prescribed by a broken symmetry in virtue of a Noether's theorem, and constraintcutoff mechanism, responsible for the very existence of a phase transition and its nonanalytical features, [4,20,21] (II) an insufficiency of a grand-canonical-ensemble approximation, which is incorrect in the critical region [2,8] because of averaging over the systems with different numbers of particles, both below and above the critical point, i.e., over the condensed and noncondensed systems at the same time, that implies an error on the order of 100% for any critical function, (III) a necessity to solve the problem for a finite system with a mesoscopic (i.e., large, but finite) number of particles N in order to calculate correctly an anomalously large contribution of the lowest energy levels to the critical fluctuations and to avoid the infrared divergences of the standard thermodynamic-limit approach [5][6][7][8][9][10][11] as well as to resolve a fine structure of the λ-point, (IV) a fact that in the critical region the Dyson-type closed equations do not exist for true Green's functions, but do exist for the partial 1-and 2-contraction superoperators, which reproduce themselves under a contraction.The problem of the critical region and mesoscopic effects is directly related to numerous modern experiments and numerical studies on the BEC of a trapped gas (including BEC on a chip), where N ∼ 10 2 − 10 7 , (see, for example, [22][23][24][25][26][27][28][29][30][31][32][33]) and superfluidit...…”

Adiabaticity in a time-dependent trap: The passage near a continuum threshold

“…External manipulation of Hamiltonians with both discrete and continuum spectra routinely occur in applications such as metrology and quantum information processing. The presence of a continuum plays an important role in atom lasers [1,2], in the preparation of atomic pulses with a known velocity distribution [3], or in the production of few-body number states [4][5][6][7][8]. Quite often a continuum is responsible for undesirable loss of trapped particles, as it happens in transport of trapped ions, or in trapped ion atomic clocks.…”

Atomic Fock states by gradual trap reduction: From sudden to adiabatic limits

“…The aim of such a preparation is to create a quantum state with the mean number of atoms n equal to the chosen M and its mean variance as small as possible. Different proposals based on a time-dependent modulation of the confining potential have been put forward both for optical traps [1][2][3][4][5] and optical lattices [6,7].…”

“…This technique is referred as to atom culling [1]. For ultracold gases confined in tight waveguides, it works optimally in the strongly interacting regime for bosonic samples [1][2][3][4], that is, in the Tonks-Girardeau gas, where the repulsive interactions lead to an effective Pauli exclusion principle [9]. Alternatively, Fock-state preparation is optimized as well with a spin-polarized noninteracting Fermi gas [3,5].…”