Finite-size, boundary-condition, and detector-model corrections to fermion and spin counting statistics in the one-dimensional XY family

Braungardt, Sen(De), Sen, Glauber, and Lewenstein [Phys. Rev. A 78, 063613 (2008)] derived analytic fermion- and spin-counting statistics for the family of one-dimensional asymmetric XY models (equivalently, spinless fermions) realizable with ultracold atoms, using periodic boundary conditions and a phenomenological detector parameter $κ$. Two source-grounded limitations of that treatment were never quantified: (i) the counting distributions and their cumulant singularities were obtained under periodic boundary conditions, even though the Jordan-Wigner correspondence on which the fermion-spin mapping rests is exact only for open or infinite chains, so finite-size and boundary-condition corrections were not analyzed; and (ii) the detector physics was compressed into a single effective parameter $κ=1-\exp(-ζt)$, which the paper also adopts for quantum-nondemolition polarization-spectroscopy spin readout (citing Eckert et al. for the microscopic light-matter coupling scheme) without deriving $κ$ from the microscopic measurement parameters, so that quantitative relation was left unspecified. Neither limitation has been addressed in the later literature (Ivanov-Abanov 2013; Groha, Essler, and Calabrese 2018). The open problem is: derive the counting statistics for the open XY chain at finite $N$, quantify the finite-size and boundary-condition corrections to the published periodic-chain results (including cumulant criticality, even-odd oscillations, and imperfect detection), delimit the parameter range in which the periodic approximation is quantitatively reliable, and derive the effective counting parameter $κ$ from realistic quantum-nondemolition measurement parameters.

Background

Ultracold atoms in optical lattices can simulate strongly correlated one-dimensional (1D) quantum systems with an unusual degree of control, including over the detection: atom- and spin-counting experiments can in principle measure not only average densities but full probability distributions of particle number or magnetization. The statistics of such counts is called full counting statistics (FCS): the probability $p(m)$ of counting $m$ particles is encoded in a generating function $Q(λ)=\mathrm{Tr}(ρ:e^{-λI}:)$, where $ρ$ is the density operator, $I$ is the counting operator, and $:\ :$ denotes normal ordering (all creation operators placed to the left of all annihilation operators, with a sign change for each fermionic interchange). For fermions this theory was generalized by Cahill and Glauber using Grassmann variables (anticommuting variables suited to fermionic coherent states), which makes normally ordered fermion expectation values tractable.

The system studied in the source paper is a 1D gas of spinless fermions in an optical lattice, with Hamiltonian $H=-\frac{J}{2}\sum_{j=0}^{N-1}\left[c^\dagger_j c_{j+1}+γ c^\dagger_j c^\dagger_{j+1}+\text{h.c.}-2g c^\dagger_j c_j\right]+\frac{Ng}{2}$ (Eq. (1) of the source paper), where $J/2$ is the tunneling energy, $γ$ is the pairing anisotropy, and $g=h/J$ is the transverse field in reduced units. By the Jordan-Wigner transformation — an exact mapping between 1D spin-$\frac{1}{2}$ chains and spinless fermions in which each spin flip becomes a fermion operator multiplied by a string of its neighbors — this is equivalent to the asymmetric XY spin chain in a transverse field $h$, with $γ=0$ the XX chain, $γ=1$ the transverse-field Ising chain, and a quantum critical point at $g_c=1$ for $γ>0$ in the universality class of the 2D classical Ising model. Counting fermions is equivalent to measuring the total $z$-component of the spin, $\hat S_z=\hat N_f-N/2$, since each fermion corresponds to an 'up' spin; the source paper also treats spatially resolved counting of Fourier components of the spin density, e.g. counting every second site, relevant for antiferromagnetic correlations.

The counting operator used for total-number counting is $I=κ\sum_{j=0}^{N-1} c^\dagger_j c_j$, where $κ=1-\exp(-ζt)$ compresses the detector into a single effective parameter: $ζ$ is the detector efficiency and $t$ the exposure time, so $κ<1$ means each particle is missed independently with probability $1-κ$. The diagonalization proceeds by Fourier and Bogoliubov transformations — a momentum-space change of basis followed by a rotation mixing creation and annihilation operators that maps the quadratic Hamiltonian to free quasiparticles with dispersion $ε_k=2\sqrt{(\cos Φ_k-g)^2+γ^2\sin^2 Φ_k}$ (Eq. (10) of the source paper), $Φ_k=2πk/N$. With the ground state being the quasiparticle vacuum, the generating function factorizes over modes and yields closed-form counting distributions and recurrence relations for the mean and variance. The main physical results: the distributions are smooth at the transition but their cumulants (the coefficients of the series expansion of $\ln Q$, i.e. mean, variance, and higher irreducible moments) show critical behavior at $g_c=1$; the distributions are typically sub-Poissonian (variance below the mean, reflecting quantum correlations); an even-odd asymmetry (zero probability of odd counts, reflecting Cooper-like pairing of fermions) is visible only for $κ$ close to 1 and moderate $N$; and the critical signatures survive even at low efficiencies such as $κ=0.1$.

Two experimental routes motivate the counting program. One is direct atom counting, as in metastable-helium experiments. The other is quantum nondemolition (QND) polarization spectroscopy: a polarized off-resonant probe beam traversing the sample undergoes a quantum-Faraday rotation, described in the microscopic QNDPS coupling scheme of Eckert et al. — the scheme the source paper adopts for spin readout — by the interaction Hamiltonian $H=χ S_z J_z$, where $J_z$ is a Stokes operator of the light (the difference of right- and left-circular photon numbers), $S_z$ the total atomic spin, and $χ$ a coupling constant proportional to the optical density; a standing-wave version $H=χ S^{\mathrm{eff}}_z J_z$ with $S^{\mathrm{eff}}_z=\int dx\, s_z(x)\cos^2(k_Lx)$ provides spatial resolution. The theory of such QND light-matter interfaces was developed by Hammerer, Mølmer, Polzik, and Cirac, and its application to strongly correlated systems by Eckert et al., while Cherng and Demler had previously studied the noise and block-counting distributions of the transverse-field Ising chain. The source paper applies the same effective counting parameter $κ=1-\exp(-ζt)$ — including a spatially resolved QNDPS counting operator — to this readout scheme, but nowhere derives $κ$ from the microscopic light-matter measurement parameters, leaving that quantitative relation unspecified.

The remaining technical caveats of the 2008 analysis are precisely its two modeling approximations. First, the Jordan-Wigner transformation is exact for open chains and infinite chains, but the authors imposed periodic boundary conditions to enable the Fourier diagonalization, arguing only that this gives the correct leading behavior for large $N$; the finite-size and boundary-condition corrections to the counting distributions and cumulant singularities were not computed. Second, the detector is represented by the single phenomenological parameter $κ$, with no derivation from realistic measurement parameters. Later literature has independently computed related counting statistics — Ivanov and Abanov obtained the thermodynamic-limit FCS of the quantum XY chain as a phase diagnostic, and Groha, Essler, and Calabrese derived determinant representations for the transverse-magnetization FCS of a finite block of the transverse-field Ising chain for general Gaussian states (ground state, finite temperature, nonequilibrium steady states, and post-quench time evolution) — but both works consider perfectly efficient projective counting, and Groha et al. explicitly set aside the issue of boundary conditions. Neither line of work quantifies boundary or finite-size corrections to the 2008 predictions, nor connects them to a microscopic QND detector model.

Problem Statement

Extend the fermion- and spin-counting theory of Braungardt et al. (Phys. Rev. A 78, 063613 (2008)) for the one-dimensional asymmetric XY family (equivalently, the spinless-fermion Hamiltonian above) so that its predictions are quantitatively reliable for a realistic experiment on a finite, open chain with quantum-nondemolition readout. Concretely, two linked tasks, both grounded in approximations the source paper itself acknowledges:

(1) Boundary-condition and finite-size corrections. Derive the counting statistics — the generating function $Q(λ)$, the distributions $p(m)$, and the cumulants — for the open chain at finite $N$, where the Jordan-Wigner mapping underlying the fermion-spin correspondence is exact, and quantify the corrections to the periodic-boundary results of the source paper. The corrections to be quantified comprise: (i) the behavior of the cumulant singularities at the critical point $g_c=1$; (ii) the even-odd (parity) oscillations of the distributions; (iii) the spatially resolved (Fourier-component) counting variants treated in the source paper; (iv) the effect of imperfect detection with efficiency parameter $κ$; and (v) a delimitation of the region of parameter space (chain length $N$, anisotropy $γ$, transverse field $g$, detection efficiency $κ$) in which the periodic-boundary approximation used in the source is quantitatively accurate. Here 'quantitatively accurate' — and likewise the 'quantitatively reliable' and 'quantitatively testable' predictions the task asks for — means agreement between the open-chain and periodic-chain results to within a relative deviation of the order of a few percent, measured for each part on its corresponding observable: the cumulants, the even-odd parity weight of the distributions, and the spatially resolved counting moments, each also as a function of the detection efficiency $κ$.

(2) Microscopic detector mapping. Derive the quantitative relation between realistic quantum-nondemolition polarization-spectroscopy measurement parameters — the light-matter coupling $χ$ of the QND interaction Hamiltonians $H=χS_zJ_z$ and $H=χS^{\mathrm{eff}}_zJ_z$ (Eckert et al.), photon flux, interaction time, optical density, and decoherence or absorption channels — and the effective counting parameter $κ=1-\exp(-ζt)$ employed in the counting formulas. If the derived relation departs from the independent-miss (binomial) form of $κ$, that departure is part of the requested result for the same counting observables.

The two tasks are jointly tied to one objective — making the counting-statistics signatures of strongly correlated 1D phases and quantum criticality quantitatively testable — and they interact: the source paper shows that the even-odd oscillations are visible only for detection efficiency close to unity and moderate system size, so finite-size and detector effects must be controlled together in the same observable predictions.

References

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