A 2011 Physical Review A paper by Braungardt, Rodriguez, Sen(De), Sen, Glauber, and Lewenstein showed that the full counting statistics of a strongly correlated one-dimensional fermionic gas in an optical lattice - equivalent, via the Jordan-Wigner transformation, to the anisotropic XY spin chain in a transverse field - carries sharp signatures of the zero-temperature quantum phase transition, of Cooper-like pair formation, and of thermalization, and that these survive as crossovers at low temperature. The analysis rests on four idealizations that the authors explicitly set aside: counting is performed in situ; the detector is much faster than the system-bath exchange; the bath is an abstract quasiparticle reservoir shown to be locally implementable only in limiting regimes; and the anisotropy is restricted to gamma > 0, excluding the isotropic XX chain. The problem is to remove these idealizations for the same model: compute the counting statistics after release and ballistic expansion, treat counting when detector exposure and bath-induced evolution occur on comparable time scales, construct a concrete reservoir implementing the quasiparticle coupling in intermediate regimes, and analyze the singular gamma = 0 limit.

Background

Full counting statistics (FCS) denotes the complete probability distribution p(m)p(m) of the number mm of particles registered by a detector, rather than only its mean and variance; the higher cumulants (standard combinations of the moments, of which the mean and variance are the first two) encode fine information about many-body correlations. The photo-counting theory developed in quantum optics extends to massive particles: for fermions absorbed by a detector with efficiency ε\varepsilon and exposure time τ\tau, the counting distribution of a lattice gas with NN sites reads

p(m)=⟨:(κ∑j=1Nc^j†c^j)me−κ∑j=1Nc^j†c^jm!:⟩p(m)=\left\langle :\left(\kappa\sum_{j=1}^{N}\hat c^\dagger_j\hat c_j\right)^{m}\frac{e^{-\kappa\sum_{j=1}^{N}\hat c^\dagger_j\hat c_j}}{m!}: \right\rangle

with κ=1−e−ετ\kappa=1-e^{-\varepsilon\tau}, where c^j\hat c_j annihilates a fermion on lattice site jj and the colons denote normal ordering (all creation operators placed to the left of annihilation operators).

The source paper applies this framework to a paradigmatic strongly correlated system: fermions in a one-dimensional optical lattice (a standing-wave light trap creating a periodic potential) described by

H^=−J∑j=1N(c^j†c^j+1+γ c^j†c^j+1†+h.c.−2g c^j†c^j+g),\hat H=-J\sum_{j=1}^{N}\left(\hat c^\dagger_j\hat c_{j+1}+\gamma\,\hat c^\dagger_j\hat c^\dagger_{j+1}+\text{h.c.}-2g\,\hat c^\dagger_j\hat c_j+g\right),

with tunnelling amplitude JJ, pairing of fermions on neighbouring sites controlled by the anisotropy γ\gamma, transverse-field (chemical-potential) parameter gg, and h.c. denoting the Hermitian conjugate of the preceding terms. Via the Jordan-Wigner transformation - an exact rewriting of a one-dimensional spin-1/2 chain in terms of fermionic operators - this Hamiltonian is equivalent to the anisotropic XY spin chain in a transverse field - a chain of spin-1/2 particles with nearest-neighbour exchange of the xx and yy spin components, their relative strength set by γ\gamma (γ=1\gamma=1 being the Ising chain with only xx-xx exchange), in a magnetic field along zz of strength set by gg (the transverse field) - realizable with cold fermionic atoms in an optical lattice. For γ≠0\gamma\neq 0 this model has a zero-temperature quantum phase transition at g=1g=1 between a magnetically ordered phase (g<1g<1) and a quantum paramagnet (g>1g>1); the pairing term produces Cooper-like fermion pairs on adjacent sites.

The paper's results are threefold. First, at zero temperature the mean and variance of p(m)p(m), and especially their derivatives with respect to gg, are non-analytic at the transition, and the pairing suppresses odd counts (p(m)=0p(m)=0 for odd mm at high detection efficiency). Second, at finite temperature the transition becomes a crossover whose visibility in the cumulants fades as kBTk_BT grows toward the single-particle energy scales, and thermal pair breaking makes odd counts probable above a temperature proportional to γ\gamma. Third, when the system is coupled to a heat bath modeled by a quantum master equation (an evolution equation for the density matrix ρ\rho) that exchanges the Bogoliubov quasiparticles d^k\hat d_k - the diagonal fermionic excitations of the paired Hamiltonian - with the bath at rate γ0\gamma_0, with thermal occupation NkdN^d_k of mode kk (loss jump d^k\hat d_k weighted by Nkd+1N^d_k+1, gain jump d^k†\hat d^\dagger_k weighted by NkdN^d_k), the system relaxes to the Boltzmann-Gibbs thermal equilibrium state, and the counting cumulants track this thermalization. Rewritten in the basis of the original lattice fermions, this master equation carries distance-dependent kernels FuF_u, FvF_v, FuvF_{uv} that decay rapidly with site separation, so that in limiting regimes it becomes a local exchange of particles and nearest-neighbour pairs - suggesting, in the authors' words, that it 'in principle ... could be realized using reservoir designs'.

The analysis nevertheless rests on four idealizations, each flagged in the paper. First, counting is assumed to be in situ, i.e. performed on atoms sitting in the lattice with single-site resolution, as in a quantum gas microscope (a fluorescence imaging device that detects individual atoms on individual lattice sites); yet many counting experiments release the cloud - for instance metastable atoms falling onto a microchannel plate (a particle detector registering single avalanches) - so ballistic expansion and its mode mixing precede detection, and the paper 'leav[es] the discussion of the interplay of atomic cloud expansion and atom counting to a separate publication'. Second, the detector is assumed much faster than the system-bath exchange, so each distribution is registered while the state is effectively frozen; the regime of comparable time scales is not analyzed. Third, the bath is an abstract quasiparticle reservoir; locality was demonstrated numerically and in limiting regimes only, and no concrete experimental reservoir was constructed for intermediate parameter regimes. Fourth, the treatment restricts the anisotropy to 0<γ≤10<\gamma\le 1: the paper states that for the isotropic XX chain (γ=0\gamma=0) 'the Jordan-Wigner transformation is ill defined and one cannot map it to the fermionic Hamiltonian', leaving counting statistics and thermalization in that limit outside its scope.

Problem Statement

Determine whether and how the counting-statistics signatures established for the strongly correlated fermionic system of the Hamiltonian above - the non-analytic behaviour of the counting cumulants at the g=1g=1 transition, the even-odd pair signature, and their evolution during bath-driven thermalization - survive when each idealization of the source analysis is removed. The four facets below serve one common objective: a counting-based characterization of this finite NN-site model that is valid under experimentally realistic detection, so that the predicted signatures can actually be read off measured distributions.

  1. Post-release counting. Compute the counting distribution p(m)p(m) when the fermions are released from the lattice and counted after ballistic expansion (including finite detector efficiency and an explicitly specified detector geometry), and establish which in-situ signatures of the transition, of the finite-temperature crossover, and of pair formation survive the expansion-induced mixing of the initially localized modes.

  2. Comparable time scales. Analyze the counting statistics when the detector exposure time is not negligible against the bath-induced evolution, so that the detection process and the quasiparticle-exchange dynamics must be treated concurrently - including the back-action of the absorbing detector on the evolving state - and determine what the recorded cumulants then reveal about the thermalization process and how these results reduce to the source paper's fast-detector limit.

  3. Reservoir realization. Construct a concrete reservoir design - a microscopic system-bath Hamiltonian or coupling architecture implementable with ultracold-atom or analogous techniques - from which the quasiparticle-exchange master equation, with its distance-dependent kernels FuF_u, FvF_v, FuvF_{uv}, follows in the intermediate parameter regimes where no local approximation was established, or establish that standard local-coupling reservoir engineering cannot implement it, in consistency with the source paper's results in the limiting regimes where locality was established.

  4. Isotropic limit. Treat the counting statistics and bath-driven thermalization at γ=0\gamma=0 (the isotropic XX chain), where the source paper's Jordan-Wigner-based route to the fermionic Hamiltonian breaks down, including the approach to this limit from γ>0\gamma>0 and its consistency with the free-fermion description of the XX chain.

References

  1. Sibylle Braungardt, Mirta Rodriguez, Aditi Sen(De), Ujjwal Sen, Roy J. Glauber, and Maciej Lewenstein, "Counting of fermions and spins in strongly correlated systems in and out of thermal equilibrium", Physical Review A 83, 013601 (2011). DOI: 10.1103/PhysRevA.83.013601; arXiv:1010.5099, https://arxiv.org/abs/1010.5099 (arXiv preprint titled "Fermion- and spin-counting in strongly correlated systems in and out of thermal equilibrium"); https://doi.org/10.1103/PhysRevA.83.013601
  2. Sibylle Braungardt, Mirta Rodriguez, Aditi Sen(De), Ujjwal Sen, and Maciej Lewenstein, "Atom counting in expanding ultracold clouds", Physical Review A 84, 043635 (2011). DOI: 10.1103/PhysRevA.84.043635; arXiv:1103.1868, https://arxiv.org/abs/1103.1868; https://doi.org/10.1103/PhysRevA.84.043635
  3. Sibylle Braungardt, Mirta Rodriguez, Roy J. Glauber, and Maciej Lewenstein, "Particle-counting statistics of time- and space-dependent fields", Physical Review A 85, 033818 (2012). DOI: 10.1103/PhysRevA.85.033818; arXiv:1110.4250, https://arxiv.org/abs/1110.4250; https://doi.org/10.1103/PhysRevA.85.033818
  4. Sibylle Braungardt, "Complex Systems for Quantum Technologies", PhD thesis, Universitat Politecnica de Catalunya (defended 12 July 2011). DOI: 10.5821/dissertation-2117-94467; https://upcommons.upc.edu/handle/2117/94467
  5. Izabella Lovas, Balazs Dora, Eugene Demler, and Gergely Zarand, "Full counting statistics of time-of-flight images", Physical Review A 95, 053621 (2017). DOI: 10.1103/PhysRevA.95.053621; arXiv:1612.02837; https://doi.org/10.1103/PhysRevA.95.053621
  6. Stephan Humeniuk and Hans Peter Buchler, "Full Counting Statistics for Interacting Fermions with Determinantal Quantum Monte Carlo Simulations", Physical Review Letters 119, 236401 (2017). DOI: 10.1103/PhysRevLett.119.236401; arXiv:1706.08951; https://doi.org/10.1103/PhysRevLett.119.236401
  7. Anton Mazurenko, Christie S. Chiu, Geoffrey Ji, Maxwell F. Parsons, Marton Kanasz-Nagy, Richard Schmidt, Fabian Grusdt, Eugene Demler, Daniel Greif, and Markus Greiner, "A cold-atom Fermi-Hubbard antiferromagnet", Nature 545, 462-466 (2017). DOI: 10.1038/nature22362; https://doi.org/10.1038/nature22362