The continuous-counting formalism of Braungardt, Rodríguez, Glauber and Lewenstein (Phys. Rev. A 85, 033818 (2012)) gives the full counting distribution of a time- and space-dependent quantum field while including the detector's back-action on the field, reducing the problem to a nonunitary single-particle propagation e−i(t−t0)(H^0−iϵℏΩ^)/ℏe^{-i(t-t_{0})(\hat{H}_{0}-i\epsilon\hbar\hat{\Omega})/\hbar} in the imaginary potential created by the detector. It is demonstrated exactly only for a one-dimensional freely expanding Bose-Einstein condensate with a point-like absorber, with uncontrolled Born-type approximations as the only alternative. The open problem is to make this description usable for experimentally realistic detection: exact or certified-accuracy computation of the counting distribution for general detector geometries, higher-dimensional and nontrivial dynamics, provable error bounds for the perturbative schemes including their long-time behavior, and extension of the formalism beyond linear fields and purely absorptive detector models.

Background

Counting statistics — the full probability distribution p(m,τ)p(m,\tau) of registering mm particles (or photons) at a detector during an opening time τ\tau — is a standard tool for characterizing quantum states of light and, more recently, of ultracold atomic gases. The foundational description is Glauber's quantum counting formalism (1963), which obtains p(m,τ)p(m,\tau) from time-ordered, normally ordered correlation functions of the field operators; normal ordering (rearranging all creation operators to the left of annihilation operators) encodes that detected particles are annihilated at the detector. This formalism treats the detection intervals as statistically independent: the field is assumed to evolve as if no detector were present, so the back-action of the detector — the absorption of particles out of the field — is neglected. For long detection times this leads to unphysical predictions, such as negative counting probabilities or mean counted particle numbers exceeding the total number of particles present.

The continuous-measurement alternative of Srinivas and Davies (1981) models counting by a one-count operator and a no-count operator that propagate the joint detector–field state; it keeps the back-action and avoids the unphysical limits, but its closed counting formula holds only for a single field mode. Chmara (1987) generalized the continuous-counting formula to multimode fields, but no practical computation of the resulting time-dependent intensities was reported. For massive particles — ultracold atoms detected by absorption or ionization on, e.g., microchannel plates (a standard single-particle detector in which an incident particle triggers an electron avalanche in narrow microscopic channels) — the same group as the source paper had earlier used counting statistics of expanding clouds as a probe of the initial trapped state, still within the standard approach without back-action (Atom counting in expanding ultracold clouds, 2011).

The source paper (Braungardt, Rodríguez, Glauber, Lewenstein, 2012) closes this gap for linear (noninteracting) many-body fields. Starting from a Lindblad-type master equation (the standard Markovian evolution equation for the density matrix of a system weakly coupled to its environment) for the field density matrix, in which quantum jumps (particle absorptions) occur inside the detector volume Ω(r)\Omega(\mathbf{r}) — a function describing the detector's spatial configuration, with detection efficiency ϵ\epsilon — the authors derive the counting formula

p(m,τ)=⟨T:(I(τ))mm!e−I(τ):⟩,p(m,\tau)=\left\langle T:\frac{(I(\tau))^{m}}{m!}e^{-I(\tau)}:\right\rangle,

formally identical to the quantum counting formalism, but with the intensity I(τ)=ϵ∫0τdt∫dr Ω(r) Ψ~†(r,t)Ψ~(r,t)I(\tau)=\epsilon\int_{0}^{\tau}dt\int d\mathbf{r}\,\Omega(\mathbf{r})\,\tilde{\Psi}^{\dagger}(\mathbf{r},t)\tilde{\Psi}(\mathbf{r},t) built from modified field operators

Ψ~(r,t)=∫dr′ G(r,r′,t,t0) Ψ(r′,t0),G(r,r′,t,t0)=⟨r∣e−i(t−t0)(H^0−iϵℏΩ^)/ℏ∣r′⟩,\tilde{\Psi}(\mathbf{r},t)=\int d\mathbf{r}'\,G(\mathbf{r},\mathbf{r}',t,t_{0})\,\Psi(\mathbf{r}',t_{0}),\qquad G(\mathbf{r},\mathbf{r}',t,t_{0})=\langle\mathbf{r}|e^{-i(t-t_{0})(\hat{H}_{0}-i\epsilon\hbar\hat{\Omega})/\hbar}|\mathbf{r}'\rangle,

where H^0\hat{H}_{0} is the field Hamiltonian without the detector and Ω^\hat{\Omega} is multiplication by Ω(r)\Omega(\mathbf{r}). The field thus propagates nonunitarily in an imaginary (absorptive) potential −iϵℏΩ-i\epsilon\hbar\Omega created by the detector. The derivation relies on the commutation relations of linear fields, so the closed formula holds for linear many-body systems; the worked example is a condensate of NN atoms occupying one single-particle state.

The paper demonstrates the formalism on a one-dimensional freely expanding Bose-Einstein condensate (a cloud of ultracold bosons all in one quantum state) with a point-like absorber Ω(z)=δ(z−z0)\Omega(z)=\delta(z-z_{0}) and a Lorentzian initial wave function. There the modified propagator is obtained exactly by solving a Lippmann–Schwinger integral equation (an iterative rewriting of the propagator in terms of the free propagator and the absorber) via Laplace transformation, following Kleber's quantum-source method (1994); the result is expressed through the Moshinsky function, a known closed-form solution of the time-dependent free Schrödinger equation behind a shutter (the time-domain analogue of a diffraction integral). The exact counting distribution is then compared numerically with (i) the standard quantum counting formalism, (ii) the second-order Born approximation (the propagator expanded to first order in ϵ\epsilon), and (iii) the exponentiated Born approximation (that expansion resummed into an exponential). The comparison shows the standard formalism becoming unphysical at long opening times, the second-order Born approximation diverging there, and the exponentiated Born approximation remaining bounded but approaching an asymptotic intensity exceeding the exact bound — a purely numerical observation with no accompanying error analysis.

The paper's own text states that the formula allows, in principle, calculation of the counting distribution for arbitrary detector geometries, but that solving the modified-field evolution is, in its words, a highly non-trivial task in general. The derivation is restricted to linear systems and to a purely absorptive detector model, and the exact solution exists only for the one-dimensional point-absorber case. These acknowledged limitations delimit the open problem.

Problem Statement

Starting point. For a time- and space-dependent quantum field detected by a continuous absorptive detector, the counting distribution of Braungardt, Rodríguez, Glauber and Lewenstein (2012) is

p(m,τ)=⟨T:(I(τ))mm!e−I(τ):⟩,I(τ)=ϵ∫0τdt∫dr Ω(r) Ψ~†(r,t)Ψ~(r,t),p(m,\tau)=\left\langle T:\frac{(I(\tau))^{m}}{m!}e^{-I(\tau)}:\right\rangle,\qquad I(\tau)=\epsilon\int_{0}^{\tau}dt\int d\mathbf{r}\,\Omega(\mathbf{r})\,\tilde{\Psi}^{\dagger}(\mathbf{r},t)\tilde{\Psi}(\mathbf{r},t),

with modified field operators determined by the nonunitary propagator

Ψ~(r,t)=∫dr′ ⟨r∣e−i(t−t0)(H^0−iϵℏΩ^)/ℏ∣r′⟩ Ψ(r′,t0).\tilde{\Psi}(\mathbf{r},t)=\int d\mathbf{r}'\,\langle\mathbf{r}|e^{-i(t-t_{0})(\hat{H}_{0}-i\epsilon\hbar\hat{\Omega})/\hbar}|\mathbf{r}'\rangle\,\Psi(\mathbf{r}',t_{0}).

Here p(m,τ)p(m,\tau) is the probability of counting mm particles during the detector opening time τ\tau, and TT and the surrounding colons denote time ordering and normal ordering (rearrangement of all creation operators to the left of all annihilation operators) of the field operators. In the exponent, ϵ\epsilon is the detector efficiency, Ω(r)\Omega(\mathbf{r}) the detector's spatial profile, Ω^\hat{\Omega} the operator that multiplies by Ω(r)\Omega(\mathbf{r}), and H^0\hat{H}_{0} the field Hamiltonian without the detector. This has been evaluated exactly only for H^0\hat{H}_{0} the one-dimensional free-particle Hamiltonian and Ω\Omega a point-like (delta-function) absorber, with a Lorentzian initial condensate wave function.

Question. Extend the certified computation of this counting distribution to experimentally realistic situations beyond that demonstration case. For an explicitly specified system class (Hamiltonian, spatial dimension, detector geometry, detector model, initial state), establish results of one or more of the following kinds, with the common objective that the counting distribution p(m,τ)p(m,\tau) or its generating function becomes available either exactly or with a provable error bound, in regimes — long opening times, strong absorption — where the standard quantum counting formalism yields unphysical results:

  1. Exact solutions: a closed-form, or provably convergent algorithmic, computation of the modified propagator GG and hence of the counting distribution, for detector geometries beyond point-like absorbers (finite apertures, extended detector surfaces) and/or dynamics beyond one-dimensional free expansion (higher spatial dimensions, nontrivial H^0\hat{H}_{0}).
  2. Certified approximations: explicit error bounds and convergence criteria for perturbative schemes for the absorptive (nonunitary) propagation — the Born series and its exponentiated resummation — including the long-opening-time regime, where the second-order truncation is known to diverge and the exponentiated Born approximation approaches an asymptotic intensity that exceeds the exact bound.
  3. Formalism extensions: a derivation of the counting distribution when either model assumption of the original derivation is relaxed: (a) interacting (nonlinear) many-body fields, for which the linear-field commutation relations underlying the single-particle reduction no longer hold; (b) detector models beyond a purely absorptive imaginary potential with constant efficiency, for example including internal detector degrees of freedom, nonlinear detector response, or finite detection bandwidth.

A result of this kind is certified when the counting distribution satisfies the consistency requirements — normalization ∑mp(m,τ)=1\sum_{m}p(m,\tau)=1, non-negativity 0≤p(m,τ)≤10\leq p(m,\tau)\leq 1, a mean counted number bounded by the total particle number, reduction to the solved one-dimensional point-absorber case, and recovery of the quantum counting formula at short opening times — and when it is exhibited on at least one physically motivated worked example lying beyond the solved demonstration case.

References

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