PaperlayerView exact revision on arXiv

Source-linked AI summary

A Theory of Transmit-Receive-Manifold of Single-Photon Quantum-Dot Antenna Arrays

Angeru Pavan Kumar, Sandip Ghosal

arXiv:2610.37542v1eess.SP

TL;DR

A coupled QD receiver can acquire a spatial state that is not represented by the conventional plane-wave manifold, particularly for a finite-bandwidth single photon. This paper models transmission, propagation, and reception together and compares receive manifolds for direction finding. In the default-link comparison, the spectrally matched model has the highest reported fidelity to the receiver state.

  • Problem

    Prior work on QD antenna arrays treated them as radiators, leaving open what spatial state a second mutually coupled array acquires from a received single photon and which manifold should guide direction finding.

    Source: §I · p. 1

  • Method

    The paper models the single-photon transmitter, Green-function channel, and coupled receiver together, then compares free-space, narrowband coupling-aware, and spectrally matched receive manifolds.

    Source: §I · p. 3, §VII · p. 10, A. Manifolds · p. 10

  • Results

    At the default link, receiver-state fidelity is 0.99997 to the SM model, 0.955 to the narrowband CA model, and 0.918 to the FS model.

    Source: E. Receiver state · p. 12

  • Takeaways & Limitations

    Finite bandwidth interacting with frequency-selective collective coupling makes receiver-state mixedness, rather than photon statistics, the reason a narrowband manifold can fail.

    Source: §XII · p. 15

    The closed-form SM model uses a single-resonance Lorentzian approximation, and receiver dephasing is represented only through the photon linewidth.

    Source: §XI · p. 15

Abstract

from arXiv · show

A quantum-dot array that receives a single photon is a collectively coupled antenna, and its receive manifold is not always the plane-wave manifold. So, this work aims to develop a complete transmit-channel-receive manifold model for quantum dot (QD) arrays. For the sake of validation, it considers a reference target pattern which can be obtained using classical array excitation.

I. INTRODUCTION

The paper extends quantum-dot antenna work from state-controlled radiation to a single-photon transmit–channel–receive link, asking what spatial state a coupled receiver acquires and which manifold direction finding should use. It models the link as a collectively coupled system and routes heterodyne snapshots to direction-finding and state-reconstruction branches.

I. INTRODUCTION

  • The central question is what spatial state a mutually coupled QD receiver acquires from a directional single photon, and which receive manifold its direction finder should use.Earlier work treated QD arrays as radiators; this paper shifts the problem to reception.Source: §I · p. 1
  • A lifetime-limited photon has finite bandwidth, so the coupled receiver maps its spectral components to distinct spatial responses and tracing over frequency can leave a mixed spatial state.A carrier-frequency coupling-aware manifold is not generally adequate when the photon bandwidth spans receiver-mode splitting.Source: §I · p. 2
  • The model treats transmitter radiation, dyadic Green-function propagation, and collective reception as one single-photon link, extending prior work on state-controlled directional emission.The work identifies quantum-excitation, electrodynamic-consistency, and finite-bandwidth modeling gaps.Source: §I · pp. 1–2

A. Architecture and conventions

  • The link’s signal flow is ρT → GRT → ρR → x[k], with heterodyne snapshots feeding separate direction-finding and state-reconstruction branches.The arrays and link form a theoretical architecture for numerical investigation; the QD parameters are measured values drawn from prior literature.Source: Figure 2 · p. 3, §I · p. 3
  • The array vector a(ŝ) represents both emission toward ŝ and reception of a plane wave arriving from ŝ, with its phase sign fixed by the paper’s convention.Its elements are e^(−ik0 ŝ·rn), where rn is element position and k0 is the free-space wavenumber.Source: A. Architecture and conventions · p. 3

B. Effective C-band quantum-dot emitter

The C-band model uses measured emitter parameters and tuned, identical in-plane dipoles in two 4 × 4 arrays. The dyadic Green tensor supplies their coherent and dissipative couplings, which determine the collective dynamics.

B. Effective C-band quantum-dot emitter

  • The model uses InAs/InP dots at 1550 nm, with measured coherence and decay parameters rather than transition energies derived from a confinement model.The radiative rate is Γ = 1/T1; pure dephasing broadens the single-dot emission linewidth to Γ + 2γϕ.Source: B. Effective C-band quantum-dot emitter · pp. 3–4
  • The nominal arrays are tuned, identical two-level emitters with in-plane x-polarized dipoles, arranged as 4 × 4 planar arrays at d = λ0/2.The default model uses R = 200λ0 and source direction (40°, 30°); spectral disorder is treated as a limitation.Source: B. Effective C-band quantum-dot emitter · p. 4

A. Collective dynamics from the dyadic Green tensor

  • The dyadic Green tensor determines pairwise coherent frequency shifts Jmn and dissipative cross-decays Γmn, which are collected with dot detunings into the non-Hermitian coupling matrix M.The real and imaginary parts of the Green tensor provide the respective couplings; eigenvalues of M encode collective frequency shifts and half-decay rates.Source: A. Collective dynamics from the dyadic… · p. 4
  • Without a drive, the state remains in the ground and single-excitation subspaces, and its excited-state block evolves under the collective dynamics with a pure-dephasing term.For zero pure dephasing, the block has the stated matrix-exponential solution; for nonzero dephasing, the evolution is a linear system in vec C.Source: A. Collective dynamics from the dyadic… · pp. 4–5

B. Far-field pattern and the element factor

The far-field pattern combines the dipole element factor with the first-order spatial coherence of the emitter array. Its angular integral is tied to the collective decay matrix, so the element factor must match the dipole used in that coupling.

B. Far-field pattern and the element factor

  • The far-field field is transverse-projected, and its first-order intensity pattern uses the dipole element factor together with the array’s spatial coherence matrix.The element factor is EF(ŝ) = 1 − (p̂·ŝ)^2, and RQ = CT in the paper’s index convention.Source: B. Far-field pattern and the element… · p. 5
  • Integrating the first-order pattern over solid angle gives the collective decay matrix normalized by the single-dot rate.The element factor is not an independent modeling choice: it must correspond to the dipole entering Γ.Source: B. Far-field pattern and the element… · p. 5
  • The prepared-state and emitted-photon patterns differ because collective modes in the prepared state decay at different rates.With the matching dipole element factor, photon flux through a large sphere equals excited-population loss for γϕ = 0.Source: B. Far-field pattern and the element… · p. 5

C. State–geometry factorization and planar mirror symmetry

The pattern factorizes into fixed geometry and element terms and a state-dependent term, so changing the state reconfigures the array without changing its hardware. For planar arrays, that control cannot remove the mirrored lobe; first-order steering remains classically equivalent.

C. State–geometry factorization and planar mirror symmetry

  • State reconfiguration changes RQ while aperture, spacing, and elements remain fixed; the geometry and element factors stay fixed.The diagonal entries of RQ give dot excitation probabilities, while its off-diagonal entries encode mutual coherence and relative optical phase.Source: C. State–geometry factorization and… · p. 5
  • For a planar array with in-plane dipoles, every steered beam has an equal mirror lobe in the lower half space that no RQ choice removes.Single-hemisphere operation therefore requires a ground plane or directive element, and the symmetry motivates restricting the DOA search to the upper half space.Source: C. State–geometry factorization and… · p. 5

D. Steering by state preparation and classical equivalence

  • Every pure single-excitation state has the same first-order pattern as a classical phased array with weights wn = cn.Thus first-order radiation can be steered, shaped, and limited as in a classical array.Source: D. Steering by state preparation and… · p. 5
  • In the four-element example, the W-state and classical patterns coincide, while the phased state at d = 0.3λ0 radiates toward endfire.For the prepared steering state, the element factor displaces the pattern peak toward broadside and the yz plane; this is an element-factor pointing displacement, not a state error.Source: Figure 4 · p. 6, D. Steering by state preparation and… · p. 6

E. Second-order coherence as the quantum discriminator

The paper uses second-order coherence to distinguish single-photon emission from classical array excitation, while modeling transmitter, channel, and receiver together. The model retains full spatial Green-function terms but assumes negligible envelope propagation delay over the link.

E. Second-order coherence as the quantum discriminator

  • A pure single-excitation state has vanishing second-order coherence, while a small two-excitation admixture raises g(2); these statistics identify the source as a single-photon emitter.The coherent-state and thermal-light references have g(2) values of 1 and 2, respectively; dephasing converts part of coherent steered emission into incoherent element-like emission.Source: E. Second-order coherence as the… · p. 6

A. Block formulation of transmitter, channel and receiver

  • The transmitter is prepared with phase cn = e−ik0 ŝs·rTn/√NT so its photon propagates toward the receiver along −ŝs.The sign is reversed because ŝs points from the receiver toward the transmitter.Source: A. Block formulation of transmitter,… · p. 6
  • The coupled TX–RX model conserves probability photon by photon, includes RX-to-TX back-action, and retains near-field Green-tensor terms for every TX–RX pair.It applies to a Fock state without a field-operator expectation value; the stated Markov treatment neglects envelope delay while retaining spatial dependence, retardation phase, and 1/R, 1/R², and 1/R³ terms.Source: A. Block formulation of transmitter,… · p. 6
  • The Markov model requires ΓR/c ≪ 1: this parameter is 8.4 × 10−4 at 310 µm and 4.2 × 10−3 at 1000λ0.When ΓR/c approaches 0.1 at distances of order centimetres, delayed amplitudes and a cascaded open-system treatment are needed.Source: A. Block formulation of transmitter,… · p. 7

B. Near- and far-field forms of the channel

The channel maps the transmitter spectrum and amplitudes to the field at the receiver, with a rank-one far-field form beyond the Fraunhofer distance. The paper reports R−2 scaling for both plane-wave error and receiver absorption probability.

B. Near- and far-field forms of the channel

  • Beyond the Fraunhofer distance, the far-field channel is rank one, with angular content given only by the plane-wave manifold.The scalar κ contains path loss, element factors, and the transmitter array factor.Source: B. Near- and far-field forms of the… · p. 7
  • The plane-wave channel error scales as (Lap/λ0R)², or R−2, for an aperture of largest dimension Lap.The supplied figure describes numerical electromagnetic link lengths, not demonstrated communication ranges.Source: B. Near- and far-field forms of the… · p. 8, Figure 8 · p. 8
  • Receiver absorption probability Pabs also decays as R−2, consistent with the one-photon free-space Friis law; Fig. 8 numerically confirms both R−2 scalings.The field reaching the receiver is represented by h(ω) = MRT cT(ω).Source: B. Near- and far-field forms of the… · pp. 7–8

V. COLLECTIVELY COUPLED RX

The coupled receiver responds to a single photon frequency by frequency like a classical receiver driven by the same field, but its response depends on the photon spectrum. Tracing out frequency yields a spatial receiver state that can be mixed when the response direction varies across the occupied band.

A. Single-photon response of the receiver

  • At each frequency, the single-photon receiver amplitude is parallel to the linear-response vector of a weak coherent field radiated by the same transmitter.The equivalence concerns spatial response, not identity of the physical states or their spectral weights and higher-order correlations.Source: A. Single-photon response of the… · p. 8
  • The coupling-aware manifold remains valid for single-photon excitation frequency by frequency, while a frequency-independent manifold is exact only in the narrowband limit.In the far-field plane-wave model, the frequency-specific response is (ωI − MRR)^−1a.Source: A. Single-photon response of the… · p. 8

B. Receiver state and its mixedness

  • The receiver state ρR is the spatial state of the absorbed and re-emitted photon conditioned on absorption, after tracing out time or frequency.The state is mixed when the receiver response changes direction across the photon’s occupied spectral band, entangling spatial and spectral degrees of freedom.Source: B. Receiver state and its mixedness · pp. 8–9

C. Spectrally matched receive model

The spectrally matched model integrates the receiver response over the photon spectrum, using a Lorentzian approximation to obtain a reduced representation. Its narrowband limit is the coupling-aware manifold.

C. Spectrally matched receive model

  • The spectrally matched model uses a single-resonance Lorentzian approximation to the transmitter’s spectral weight, with centre frequency ωc and linewidth γ.The approximation is used to obtain a closed-form manifold; the paper distinguishes it from the numerically computed exact spectral weight.Source: C. Spectrally matched receive model · p. 9
  • The reduced spectrally matched receiver state follows by integrating the frequency-dependent response with the Lorentzian spectral weight.The derivation evaluates the integral using the eigenvalues of the shifted receiver matrix and the transformed look direction.Source: C. Spectrally matched receive model · p. 9
  • As γ approaches zero, the spectrally matched state collapses to the rank-one coupling-aware manifold; pure dephasing broadens the photon linewidth to γ + 2γϕ.The exact spectrum-integrated state uses no spectral or far-field assumption beyond Markov elimination, whereas the reduced model assumes far-field factorization and a single-resonance spectrum.Source: C. Spectrally matched receive model · p. 9

VI. READOUT, MEASUREMENT MODEL AND STATE RECONSTRUCTION

The readout model treats each retained repetition as a heterodyne snapshot drawn from vacuum or a one-photon receiver state. It reconstructs that state from sample covariance and evaluates direction-estimation bounds for both the exact snapshot distribution and a Gaussian approximation.

A. Heterodyne snapshots

  • Each heterodyne snapshot is an N-channel output vector from one retained repetition, containing a photon with probability Pd and otherwise vacuum.Pd is conditional on the gating or heralding rule and is distinct from the bare-link absorption probability Pabs.Source: A. Heterodyne snapshots · p. 9
  • The receiver-state purity is reported as a function of transmitter radiative rate, which sets photon linewidth; the lifetime-limited case is ΓT = Γ and the narrowband limit is ΓT → 0.The figure caption identifies the plotted receiver-state components as matrix magnitudes, eigenvalues, MUSIC error, and purity.Source: A. Heterodyne snapshots · p. 10, Figure 9 · p. 10

B. State reconstruction

  • The unbiased covariance estimate ˆRx − I estimates PdρR, and trace normalization removes the unknown Pd before the estimated state is made physical by simplex projection of its eigenvalues.For additive Gaussian noise, the projection is the maximum-likelihood physical state.Source: B. State reconstruction · p. 10

C. Exact and Gaussian Cramér–Rao bounds

  • The exact and Gaussian Cramér–Rao bounds are classical Fisher-information bounds for the specified heterodyne measurement, not quantum bounds optimized over measurements.The exact bound uses the full snapshot distribution; the Gaussian bound assumes estimators use only the covariance and treat data as Gaussian.Source: C. Exact and Gaussian Cram´er–Rao bounds · p. 10

VII. DIRECTION FINDING WITH THE COUPLED QUANTUM-DOT RECEIVER

Direction finding matches the measured covariance to directional receiver-state models derived from the coupled quantum-dot link. The methods compare free-space, coupling-aware, and spectrally matched manifolds, while accounting for the limits of shift invariance in ESPRIT.

VII. DIRECTION FINDING WITH THE COUPLED QUANTUM-DOT RECEIVER

  • The direction-finding problem matches modeled receiver states to measured covariance, so the manifold is mismatched when it differs from the arriving photon’s state.The data covariance is Rx = I + PdρR(ŝ), where the signal subspace is the receiver state produced by the Green-function link.Source: §VII · p. 10

A. Manifolds

  • The three unit-norm manifolds are free-space (FS), narrowband coupling-aware (CA), and spectrally matched (SM), with SM represented by the principal eigenvector of its predicted state for subspace estimators.FS ignores receiver coupling, CA is exact for a monochromatic field, and SM includes the photon centre frequency and linewidth.Source: A. Manifolds · pp. 10–11

B. Estimators

  • Bartlett, MVDR, and MUSIC spectra are evaluated on a 2° × 2° grid, with candidate maxima refined continuously and the search restricted to the upper half space.The restriction follows planar-array mirror symmetry; Fig. 10 shows normalized MUSIC spectra for the three manifolds at Pd = 0.1.Source: B. Estimators · p. 11, Figure 10 · p. 11

C. Shift invariance and ESPRIT

  • Direct ESPRIT is biased for coupled data because the CA manifold lacks translation invariance, while decoupling restores invariance exactly only for a narrowband field.Finite-bandwidth photons leave residual error; the paper includes decoupled 2-D ESPRIT to measure it.Source: C. Shift invariance and ESPRIT · p. 11

D. Receiver procedure and complexity

The receiver procedure uses a precomputed spectrally matched manifold, while numerical checks validate the model and show how coupling and dephasing shape array performance.

D. Receiver procedure and complexity

  • Once the manifold is tabulated, each data step forms the covariance estimate at O(KN^2) and searches the table; one-time work includes an O(N^3) eigendecomposition.The total cost also includes measuring source parameters and grid tabulation, performed once per array and source type.Source: D. Receiver procedure and complexity · p. 11

A. Numerical verification

  • The numerical checks recover the analytic limits, and replacing the Lorentzian approximation lowers SM-MUSIC error to 0.015° versus 0.031° for closed-form SM and 1.813° for CA.The numerical spectral weight differs from the Lorentzian by 5.4% normalized RMS error; the small SM residual is partly attributable to the single-resonance approximation, not the SM–CA gap.Source: A. Numerical verification · p. 11

B. Mutual coupling and manifold distortion

  • At d = 0.5λ0, FS-MUSIC has 3.09° bias, while the bias peaks at 16.2° near 0.30λ0; it falls below 1° only for d ≥ 0.57λ0.At half-wavelength spacing the coupling cannot be ignored.Source: B. Mutual coupling and manifold… · p. 12

C. Transmitter: validation, steering and photon statistics

  • The four-element W state and uniformly fed classical array share the same pattern, while planar steering to 60° lowers directivity from 13.92 to 11.79 dBi and widens HPBW from 25.4° to 52.9°.The emitted-photon directivity stays within 0.1 dB of the prepared-state value through 45° and is 0.39 dB lower at 60°.Source: C. Transmitter: validation, steering… · p. 12, Table IV · p. 12
  • A 10% two-excitation admixture gives g(2) = 0.136 for N = 4 and 0.159 for N = 16; dephasing reduces directional contrast while barely shifting pointing.A 0.5 dB directivity-loss budget is met at γϕ = 0.05Γ (0.38 dB) but not at 0.1Γ (0.72 dB); this is a chosen criterion, not a physical boundary.Source: C. Transmitter: validation, steering… · p. 12

D. TX-RX communication channel

Finite photon bandwidth interacting with receiver coupling produces a mixed receive state that defeats the narrowband coupling-aware manifold. The spectrally matched manifold substantially reduces model-mismatch error, and its estimators approach the corresponding bounds as snapshots accumulate.

D. TX-RX communication channel

  • Beyond 20λ0, SM error stays between 0.014° and 0.042°, while FS and CA level off near 1.67° and 1.81°; at R ≤ 10λ0, plane-wave manifolds are inadequate.For R ≥ 20λ0, receiver-state purity is 0.949, with mixedness set by receiver coupling and photon spectrum rather than the channel.Source: D. TX-RX communication channel · p. 12

E. Receiver state

  • For the default link, SM-MUSIC errs by 0.031°, versus 1.80°–1.81° for CA; the receive-state sweep shows CA error rising to 2.05° as photon linewidth increases, while SM stays below 0.036°.The receiver-state purity is 0.949, and the curves cross near ΓT ≈ 0.8Γ; the photon remains a single photon throughout, so the mismatch is attributed to bandwidth rather than photon statistics.Source: E. Receiver state · p. 12, F. DOA spectra and the four-system… · p. 12

F. DOA spectra and the four-system baseline

  • For the default link, the three spectral estimators give errors of 1.64°–1.67° with FS, 1.80°–1.81° with CA, and 0.031° with SM; the baseline isolates bandwidth as the cause of CA mismatch on a coupled receiver.On a coupled receiver, FS errs by 3.09° while CA is exact to 0.005° for the narrowband source; with an uncoupled receiver, all tested manifolds are unbiased within the finite-link residual.Source: F. DOA spectra and the four-system… · pp. 12–13, Table V · p. 13

G. Statistical performance, Cramér–Rao bounds and reconstruction

  • At Pd = 0.3, SM-MUSIC error falls from 2.63° at K = 10^3 to 0.228° at K = 10^5, within 1.5% of the exact CRB (0.224°); FS-MUSIC stalls at 1.69°.At Pd = 1, state fidelity rises from 0.850 to 0.980 between K = 10^3 and 10^5, while DOA estimation needs one to two orders of magnitude fewer detected photons than state reconstruction.Source: G. Statistical performance,… · p. 13

H. Two-source resolution

  • SM-MUSIC resolves sources separated by 20° in 95 of 100 trials and at 25° or more in all trials, while FS-MUSIC needs 30° and SM beamformers need 40°.No method resolves separations ≤12° in this array, criterion, and snapshot budget.Source: H. Two-source resolution · p. 13

IX. ROBUSTNESS

Robustness depends on receiver conditions and calibration: unmodelled dephasing, position disorder, and spectral disorder can degrade the nominal spectrally matched manifold. The paper distinguishes classical-equivalent first-order steering from single-photon measurement statistics and states modeling and validation boundaries.

IX. ROBUSTNESS

  • With RX position disorder, nominal-SM error reaches 0.77° at σr = 0.02λ0 and 1.97° at 0.05λ0; spectral disorder is more demanding, raising error to 1.07° already at σδ = 0.1Γ.At σδ = Γ, SM and FS errors are similar (6.3° and 6.1°); calibrating positions and the coupling matrix restores low errors, including 0.048° at σδ = 0.2Γ.Source: §IX · pp. 13–14

X. CLASSICAL–QUANTUM COMPARISON AND LIMITS

  • First-order patterns are reproduced by a classical phased array, while the SM correction removes manifold mismatch rather than creating quantum-enhanced resolution.The non-Gaussian one-photon heterodyne statistics can place the exact Fisher-information bound up to 27% below the covariance-only Gaussian bound; exploiting that margin requires likelihood-based estimation.Source: §X · p. 14

XI. DISCUSSION AND LIMITATIONS

  • The reported results are numerical only, and the model assumes identical frequency-tuned dots; receiver dephasing is represented through photon linewidth rather than a full Liouvillian treatment.The SM manifold uses a single-resonance Lorentzian approximation, which can be replaced by spectrum-based quadrature when its residual bias matters.Source: §XI · pp. 14–15

XII. CONCLUSION

  • Finite photon bandwidth combined with frequency-selective collective receiver coupling creates a mixed spatial state that invalidates the narrowband manifold; SM reduces the default-link error to 0.031° from CA’s 1.81°.The conclusion calls for resonant excitation and in-situ receiver-coupling calibration; realistic position and spectral disorder are tolerated with both.Source: §XII · p. 15
Loading 2610.37542v1…