Fig. 2(a) shows the geometry of the link and Fig. 2(b) the signal chain. The TX array is prepared in a single-excitation state \( \rho_{T}\) and radiates through its own collective coupling; the photon crosses the free-space channel, described by the dyadic Green tensor between TX and RX dots; the RX array absorbs it through its own collective coupling and is left in the spatial state \( \rho_{R}\) . The RX emission is collected element by element and detected by heterodyne receivers, giving snapshots \( \boldsymbol{x}[k]\) , which feed two separate branches: DOA estimation, returning \( (\hat{\theta},\hat{\phi})\) , and state reconstruction, returning \( \hat{\rho}_{R}\) .
Quantum operators evolve according to the Schr
specialChar{34}odinger equation, so the physics time convention \( e^{-i\omega t}\) is used throughout, and an outgoing spherical wave carries \( e^{+ik_{0}R}\) . With this convention, the array vector is defined as

where \( \boldsymbol{r}_{n}\) is the position of element \( n\) , \( k_{0}=2\pi/\lambda_{0}\) is the free-space wavenumber, \( \theta\) and \( \phi\) are the elevation (from \( +z\) ) and azimuth angles, and \( \hat{\boldsymbol{s}}\) is the corresponding unit vector. The same vector describes two situations: emission toward \( \hat{\boldsymbol{s}}\) , and reception of a plane wave arriving from a source that lies in the direction \( \hat{\boldsymbol{s}}\) , whose wave vector is therefore \( -k_{0}\hat{\boldsymbol{s}}\) . A single vector serving both roles is the array form of reciprocity. A steering state with the opposite phase sign would point toward \( -\hat{\boldsymbol{s}}_{0}\) , which for a planar array appears as a \( 180^{\circ}\) azimuth error; all equations and the code use (3) as written.

Each dot is treated as a two-level emitter with ground state \( |g\rangle\) , excited state \( |e\rangle\) (the \( \hat{\boldsymbol{x}}\) -polarized exciton \( |X\rangle\) in the inset of Fig. 2(a)) and transition angular frequency \( \omega_{0}=2\pi c/\lambda_{0}\) , where \( c\) is the speed of light and \( \lambda_{0}=1550\,\mathrm{n}\mathrm{m}\) . This gives \( f_{0}=\omega_{0}/2\pi=193.41\,\mathrm{T}\mathrm{Hz}\) and a photon energy \( \hbar\omega_{0}=0.7999\,\mathrm{eV}\) . No transition energy is derived from a confinement model. The material parameters are those measured on droplet-epitaxy InAs/InP dots [19, 20] and are listed with their origin in Table 2.
All decay rates in this paper are angular-frequency quantities in \(\mathrm{s}^{-1}\). The radiative decay rate \( \Gamma\) is the decay rate of the excited-state population, \( \Gamma=1/T_{1}\) , where \( T_{1}\) is the radiative lifetime. A single dot then emits a Lorentzian spectrum whose full width at half maximum (FWHM) is \( \Gamma\) in angular frequency, i.e., \( \Gamma/2\pi\) in hertz. The pure-dephasing rate \( \gamma_{\phi}\) is defined through the dissipator of Section 3, so that the optical coherence of one dot decays at \( T_{2}^{-1}=\Gamma/2+\gamma_{\phi}\) and its emission FWHM becomes \( \Gamma+2\gamma_{\phi}=2/T_{2}\) . The Fourier limit is \( T_{2}=2T_{1}\) . The measured Fourier-limited coherence time \( 2T_{1}=2.45\,\mathrm{n}\mathrm{s}\) [20] gives \( T_{1}=1.225\,\mathrm{n}\mathrm{s}\) , \( \Gamma=8.163\times10^{8}\,\mathrm{s}^{-1}\) (\( \Gamma/2\pi=130\,\mathrm{M}\mathrm{Hz}\) ) and \( \hbar\Gamma=0.537\,\mu\mathrm{eV}\) . The non-resonant coherence time \( T_{2}=447\,\mathrm{p}\mathrm{s}\) gives \( \gamma_{\phi}=1/T_{2}-\Gamma/2=2.24\Gamma\) , and the resonant bound \( T_{2}\geq1.5T_{1}\) gives \( \gamma_{\phi}\leq(1/1.5-1/2)\Gamma=0.167\Gamma\) . Two further linewidths are kept separate from \( \Gamma\) : \( \Gamma_{T}\) , the radiative rate of the TX dots, varied in Section 8 to change the photon bandwidth (\( \Gamma_{T}=\Gamma\) meaning identical TX and RX dots), and \( \gamma\) , the FWHM of the transmitted photon spectrum, fixed by the collective TX state and proportional to \( \Gamma_{T}\) (further details in Section 5).

The orientation of the dipoles plays a critical role in quantum dot arrays. The neutral exciton of a self-assembled dot has two orthogonal in-plane linear dipoles, split by the fine-structure splitting (FSS), measured as \(22.77\,\mu\mathrm{eV}\) [20], which is \( 42\hbar\Gamma\) . A polarization-selective resonant drive therefore addresses one linear dipole, and we take the unit dipole vector \( \hat{\boldsymbol{p}}=\hat{\boldsymbol{x}}\) for every dot. An out-of-plane dipole, sometimes assumed for convenience, is not the natural orientation of such dots. Another critical factor is spectral tuning. As the dots differ in transition energy, so collective effects require tuning to a common frequency, as demonstrated with local strain [15] and electrical control [16]. The nominal model assumes tuned, identical dots; residual spectral disorder is a real limitation and is quantified in Section 9.
If in a quantum transmit-receive propagation link, the host medium is homogeneous free space, \( n_{h}\) becomes \( 1\) . The vacuum-equivalent dipole moment reproducing the measured rate through \( \Gamma=\omega_{0}^{3}\mu_{eg}^{2}/(3\pi\varepsilon_{0}\hbar c^{3})\) is \( \mu_{eg}=3.284\times10^{-28}\,\mathrm{C}\,\mathrm{m}\) . Every array result depends on \( \Gamma\) and \( k_{0}\) only through normalized rates and the products \( k_{0}r\) . Both arrays are \( 4\times4\) uniform planar arrays (\( N=N_{T}=N_{R}=16\) ) with element spacing \( d=\lambda_{0}/2=775\,\mathrm{n}\mathrm{m}\) in \( x\) and \( y\) , i.e., a \( 2.325\,\mu\mathrm{m}\times2.325\,\mu\mathrm{m}\) aperture. The default TX–RX distance is \( R=200\lambda_{0}=310\,\mu\mathrm{m}\) , and the default source direction seen from the RX is \( (\theta_{s},\phi_{s})=(40^{\circ},30^{\circ})\) . The distance is a numerical electromagnetic link length chosen to lie well inside the far field of both apertures.

The free-space dyadic Green tensor for the time convention \( e^{-i\omega t}\) is

where \( \boldsymbol{R}=\boldsymbol{r}-\boldsymbol{r}'\) , \( R=|\boldsymbol{R}|\) , \( \hat{\boldsymbol{R}}=\boldsymbol{R}/R\) , \( \boldsymbol{I}\) is the \( 3\times3\) identity and \( \hat{\boldsymbol{R}}\hat{\boldsymbol{R}}\) the dyadic product, normalized so that \( \nabla\times\nabla\times\overline{\overline{\boldsymbol{G}}}-k_{0}^{2}\overline{\overline{\boldsymbol{G}}}=\boldsymbol{I}\,\delta(\boldsymbol{R})\) . Eliminating the field in the Born–Markov approximation [1, 2] couples two dots \( m\neq n\) through [3]

where \( \mu_{0}\) is the vacuum permeability and \( J_{mn}\) , \( \Gamma_{mn}\) (both in \(\mathrm{s}^{-1}\)) are the coherent (frequency-shift) and dissipative (cross-decay) couplings: \( J_{mn}=-(3\pi\Gamma/k_{0})\,\hat{\boldsymbol{p}}\cdot\operatorname{Re}\overline{\overline{\boldsymbol{G}}}\cdot\hat{\boldsymbol{p}}\) and \( \Gamma_{mn}=(6\pi\Gamma/k_{0})\,\hat{\boldsymbol{p}}\cdot\operatorname{Im}\overline{\overline{\boldsymbol{G}}}\cdot\hat{\boldsymbol{p}}\) . Since \( \operatorname{Im}\overline{\overline{\boldsymbol{G}}}(\boldsymbol{r},\boldsymbol{r})=(k_{0}/6\pi)\boldsymbol{I}\) , the same formula gives \( \Gamma_{nn}=\Gamma\) , while the divergent \( \operatorname{Re}\overline{\overline{\boldsymbol{G}}}(\boldsymbol{r},\boldsymbol{r})\) is a Lamb shift already contained in \( \omega_{0}\) ; unequal radiative rates replace \( \Gamma\) by \( \sqrt{\Gamma_{m}\Gamma_{n}}\) . The couplings are collected in the non-Hermitian matrix

where \( \boldsymbol{\Delta}=\operatorname{diag}(\delta_{n})\) contains the detunings \( \delta_{n}\) of the dots from \( \omega_{0}\) , \( \boldsymbol{J}=[J_{mn}]\) has a zero diagonal, and \( \boldsymbol{\Gamma}=[\Gamma_{mn}]\) has \( \Gamma\) on its diagonal. Throughout the paper the lightface \( \Gamma\) is the scalar single-dot decay rate, \( \Gamma_{mn}\) its pairwise collective counterpart, and the boldface \( \boldsymbol{\Gamma}\) the matrix of those entries; the same convention distinguishes \( \Gamma_{T}\) (scalar, TX dots) from \( \boldsymbol{\Gamma}_{TT}\) and \( \boldsymbol{\Gamma}_{RR}\) (matrix blocks). The eigenvalues \( \lambda_{q}\) of \( \boldsymbol{M}\) give the collective frequency shifts, \( \operatorname{Re}\lambda_{q}\) , and half the collective decay rates, \( -\operatorname{Im}\lambda_{q}\) ; Fig. 3 shows the couplings for the arrays used here (further details in Section 8-B).

Pan/Zoom

Without a drive the density matrix \( \rho\) stays in the span of the collective ground state \( |G\rangle\) and the single-excitation states \( |e_{n}\rangle\) (dot \( n\) excited, all others in \( |g\rangle\) ). Its excited-state block \( \boldsymbol{C}(t)\) , with elements \( C_{mn}=\langle e_{m}|\rho|e_{n}\rangle\) , obeys

where \( (\cdot)^{H}\) is the conjugate transpose and \( \operatorname{diag}\boldsymbol{C}\) keeps only the diagonal of \( \boldsymbol{C}\) . The last term is the pure-dephasing dissipator \( \mathcal{D}[\rho]=(\gamma_{\phi}/2)\sum_{n}(\hat{\sigma}^{z}_{n}\rho\hat{\sigma}^{z}_{n}-\rho)\) restricted to one excitation. For one dot it damps the optical coherence \( \langle e|\rho|g\rangle\) at \( \gamma_{\phi}\) , giving \( T_{2}^{-1}=\Gamma/2+\gamma_{\phi}\) with radiative decay, the relation used in Table 2; between two dots each \( \hat{\sigma}^{z}\) factor contributes \( \gamma_{\phi}\) , hence \( 2\gamma_{\phi}\) in (7). For \( \gamma_{\phi}=0\) the solution is \( \boldsymbol{C}(t)=e^{-i\boldsymbol{M} t}\boldsymbol{C}(0)e^{i\boldsymbol{M}^{H}t}\) . With the eigen-decomposition \( \boldsymbol{M}=\boldsymbol{V}\boldsymbol{\Lambda}\boldsymbol{V}^{-1}\) , \( \boldsymbol{\Lambda}=\operatorname{diag}(\lambda_{k})\) , the time integral that governs every energy-type quantity below has the closed form {

where the bracket denotes the matrix of the \( (k,l)\) -th element. The denominator never vanishes, because every collective mode decays (\( \operatorname{Im}\lambda_{k}<0\) ). For \( \gamma_{\phi}>0\) , (7) is a linear system \( \tfrac{d}{dt}\operatorname{vec}\boldsymbol{C}=\mathcal{L}\operatorname{vec}\boldsymbol{C}\) , and \( \operatorname{vec}\boldsymbol{X}=-\mathcal{L}^{-1}\operatorname{vec}\boldsymbol{C}(0)\) , where \( \operatorname{vec}\) stacks columns and \( \mathcal{L}\) is the Liouvillian matrix of (7).

Far from the array, the positive-frequency field radiated toward \( \hat{\boldsymbol{s}}\) is proportional to \( (\boldsymbol{I}-\hat{\boldsymbol{s}}\hat{\boldsymbol{s}})\cdot\hat{\boldsymbol{p}}\sum_{n}a_{n}(\hat{\boldsymbol{s}})\hat{\sigma}_{n}^{-}\) , where \( (\boldsymbol{I}-\hat{\boldsymbol{s}}\hat{\boldsymbol{s}})\) projects onto the plane transverse to \( \hat{\boldsymbol{s}}\) . The first-order (intensity) pattern is therefore

where \( \mathrm{EF}(\hat{\boldsymbol{s}})=1-(\hat{\boldsymbol{p}}\cdot\hat{\boldsymbol{s}})^{2}\) is the element factor of a \( \hat{\boldsymbol{p}}\) -directed dipole and \( [\boldsymbol{R}_{Q}]_{mn}=\langle\hat{\sigma}^{+}_{m}\hat{\sigma}^{-}_{n}\rangle\) . In the single-excitation manifold \( \hat{\sigma}^{+}_{m}\hat{\sigma}^{-}_{n}=|e_{m}\rangle\langle e_{n}|\) , so \( \langle\hat{\sigma}^{+}_{m}\hat{\sigma}^{-}_{n}\rangle=\langle e_{n}|\rho|e_{m}\rangle=C_{nm}\) ; that is, \( \boldsymbol{R}_{Q}=\boldsymbol{C}^{T}\) . The transpose is not an error: it reflects the two index orders used for the same matrix, operator order for \( \boldsymbol{R}_{Q}\) and ket–bra order for \( \boldsymbol{C}\) . The element factor is not a free modeling choice. Integrating (9) over the sphere and using the plane-wave representation \( \operatorname{Im}\overline{\overline{\boldsymbol{G}}}(\boldsymbol{R})=(k_{0}/16\pi^{2})\oint(\boldsymbol{I}-\hat{\boldsymbol{s}}\hat{\boldsymbol{s}})e^{ik_{0}\hat{\boldsymbol{s}}\cdot\boldsymbol{R}}d\Omega\) gives the sum rule

where \( d\Omega\) is the element of solid angle and the right-hand side is the collective decay matrix \( \boldsymbol{\Gamma}=[\Gamma_{mn}]\) divided by the scalar single-dot rate \( \Gamma\) . Hence \( (3\Gamma/8\pi)\oint G^{(1)}d\Omega=\operatorname{Tr}(\boldsymbol{\Gamma}\boldsymbol{R}_{Q})=-\tfrac{d}{dt}\operatorname{Tr}\boldsymbol{C}\) for \( \gamma_{\phi}=0\) : the photon flux through a large sphere equals the loss of excited population only if the element factor belongs to the dipole that enters \( \boldsymbol{\Gamma}\) , which isotropic elements with dipole coupling do not.
Two patterns are reported in the next. The prepared-state pattern from \( \boldsymbol{R}_{Q}(0)\) , and the emitted-photon pattern from \( \boldsymbol{X}^{T}\) of (8), the angular distribution of the complete photon. They differ only because the collective modes in the prepared state decay at different rates. The directivity is \( D_{0}=4\pi\max_{\hat{\boldsymbol{s}}}G^{(1)}/\oint G^{(1)}d\Omega\) ; we write \( D_{0}\) for the prepared state and \( D_{0}^{\mathrm{int}}\) for the emitted photon.

Equation (9) factorizes the pattern: \( \boldsymbol{a}(\hat{\boldsymbol{s}})\) and the element factor are fixed by the positions and the dipole orientation, \( \boldsymbol{R}_{Q}\) by the state, whose diagonal \( [\boldsymbol{R}_{Q}]_{nn}\) is the excitation probability of dot \( n\) and whose off-diagonal entries carry the mutual coherence of two dots and their relative optical phase. Reconfiguring the array means changing \( \boldsymbol{R}_{Q}\) while aperture, spacing, and elements stay untouched: such an array is state-programmable rather than feed-programmable.

The factorization also fixes what state control cannot do. For a planar array in \( z=0\) , \( \hat{\boldsymbol{s}}\cdot\boldsymbol{r}_{n}\) depends on \( \theta\) only through \( \sin\theta\) , and for an in-plane dipole \( \mathrm{EF}\) depends on \( \theta\) the same way, so both factors of (9) are invariant under \( \theta\rightarrow180^{\circ}-\theta\) :

numerically, the largest relative asymmetry over the steering states of Section 8-C is \( 1.3\times10^{-15}\) . Every steered beam therefore carries a mirror lobe at the same azimuth in the lower half space that no choice of \( \boldsymbol{R}_{Q}\) removes, so single-hemisphere operation needs a ground plane or a directive element, as for a classical planar array, and this symmetry – not an estimator heuristic – is why the DOA search of Section 7 is restricted to the upper half space. Asymmetry in azimuth, where the array has extent, remains state-controlled and is measured by \( \eta_{A}\) (Section 8-C); it is a pattern property only, since reciprocity constrains the transmit–receive relation and not the spatial symmetry of a pattern.

Following [8] and [9], the timed Dicke state

with commanded direction \( \hat{\boldsymbol{s}}_{0}\) and \( \boldsymbol{C}(0)=\boldsymbol{c}\boldsymbol{c}^{H}\) , gives \( G^{(1)}(\hat{\boldsymbol{s}},0)=\mathrm{EF}(\hat{\boldsymbol{s}})|\sum_{n}c_{n}a_{n}(\hat{\boldsymbol{s}})|^{2}\) , exactly the pattern of a classical phased array of \( \hat{\boldsymbol{p}}\) -directed dipoles with weights \( w_{n}=c_{n}\) . The equivalence holds for every pure single-excitation state and settles what can be claimed: first-order radiation of a state-controlled array is steered, shaped and limited as a classical array is. Two consequences follow. The array factor peaks at \( \hat{\boldsymbol{s}}_{0}\) but its product with \( \mathrm{EF}\) does not, so for \( \hat{\boldsymbol{p}}=\hat{\boldsymbol{x}}\) the peak is displaced toward broadside and the \( yz\) plane; we call this great-circle offset the element-factor pointing displacement \( \Delta\Omega\) , a property of the element rather than an error of the state. And (12) is not an eigenmode of \( \boldsymbol{M}\) : it decays at \( \Gamma_{\mathrm{st}}=\boldsymbol{c}^{H}\boldsymbol{\Gamma}\boldsymbol{c}\) and is shifted by \( \boldsymbol{c}^{H}\boldsymbol{J}\boldsymbol{c}\) only at \( t=0\) , so the emitted-photon pattern departs from the prepared one. Figs. 4–6 illustrate both points numerically (Section 8-C).

Pan/Zoom

Pan/Zoom

Pan/Zoom

A classical phased array cannot reproduce the photon statistics. With \( \hat{A}(\hat{\boldsymbol{s}})=\sum_{n}a_{n}(\hat{\boldsymbol{s}})\hat{\sigma}^{-}_{n}\) and the shorthand \( \hat{A}_{i}=\hat{A}(\hat{\boldsymbol{s}}_{i})\) , \( \mathrm{EF}_{i}=\mathrm{EF}(\hat{\boldsymbol{s}}_{i})\) ,

which vanishes identically on the single-excitation manifold, two lowering operators annihilating any state with one excitation. Real sources contain a small two-excitation admixture, modelled as \( \rho=(1-p_{2})|\psi_{1}\rangle\langle\psi_{1}|+p_{2}|\psi_{2}\rangle\langle\psi_{2}|\) with two-excitation probability \( p_{2}\) and \( |\psi_{2}\rangle\propto\sum_{m<n}c_{m}c_{n}|e_{m}e_{n}\rangle\) ; in the commanded direction this gives \( G^{(1)}\propto(1-p_{2})N+2p_{2}(N-1)\) and \( G^{(2)}\propto2p_{2}N(N-1)\) , so \( g^{(2)}=G^{(2)}/(G^{(1)})^{2}\) is

The coherent-state reference has \( g^{(2)}=1\) and thermal light \( g^{(2)}=2\) ; Table 3 lists (14) computed from the full operator algebra, which agrees with the closed form to all printed digits. These statistics identify the source as a single-photon emitter and justify the one-photon measurement model of Section 6, but, as Section 5 shows, they are not the limiting factor for the receiver. Dephasing, by contrast, converts part of the coherent steered emission into incoherent element-like emission (Fig. 7).

Pan/Zoom

The RX array lies in the plane \( z=0\) , centred at the origin, and the TX array is a parallel copy centred at \( R\hat{\boldsymbol{s}}_{s}\) , where \( \hat{\boldsymbol{s}}_{s}\) is the source direction seen from the RX. The positions are \( \boldsymbol{r}^{T}_{n}\) (\( n=1,\dots,N_{T}\) ) and \( \boldsymbol{r}^{R}_{m}\) (\( m=1,\dots,N_{R}\) ). Stacking the TX amplitudes above the RX amplitudes and applying (5) to all \( N_{T}+N_{R}=32\) dots gives

for every pair of distinct dots, with the diagonal entries \( \delta_{n}-i\Gamma/2\) of (6) on the diagonals of \( \boldsymbol{M}_{TT}\) and \( \boldsymbol{M}_{RR}\) . The first index of a block refers to the dot that is acted on, the second to the dot that acts; hence \( \boldsymbol{M}_{RT}\) carries excitation from TX to RX and \( \boldsymbol{M}_{TR}=\boldsymbol{M}_{RT}^{T}\) by reciprocity. We write \( \boldsymbol{G}_{XY}=[\hat{\boldsymbol{p}}\cdot\overline{\overline{\boldsymbol{G}}}(\boldsymbol{r}^{X}_{m},\boldsymbol{r}^{Y}_{n})\cdot\hat{\boldsymbol{p}}]\) for the corresponding Green blocks. \( \boldsymbol{G}_{TT}\) shapes the emitted photon (Section 3), \( \boldsymbol{G}_{RT}\) is the propagation channel, and \( \boldsymbol{G}_{RR}\) distorts the receive manifold (Section 5). The TX is prepared in the steering state that points back at the receiver. Since \( \hat{\boldsymbol{s}}_{s}\) is defined from the RX toward the TX, the photon must propagate toward \( -\hat{\boldsymbol{s}}_{s}\) , so the state phase of (12) is taken with \( \hat{\boldsymbol{s}}_{0}=-\hat{\boldsymbol{s}}_{s}\) and the sign of the exponent is opposite to that in (12): \( c_{n}=e^{-ik_{0}\hat{\boldsymbol{s}}_{s}\cdot\boldsymbol{r}^{T}_{n}}/\sqrt{N_{T}}\) . The RX starts in its ground state, and

where \( \boldsymbol{X}_{RR}\) is the lower-right \( N_{R}\times N_{R}\) block of (8) evaluated for the full \( 32\times32\) matrix \( \boldsymbol{M}\) , and \( \boldsymbol{\Gamma}_{RR}\) is the RX block of \( \boldsymbol{\Gamma}\) . Because all excitation that enters the RX is eventually radiated by it, \( P_{\mathrm{abs}}\) is the probability that the photon is absorbed by the RX array. Unlike a semiclassical receiver model, this formulation contains no expectation value of the field operator, so it applies to a Fock state [30]; it conserves probability photon by photon, including RX-to-TX back-action through \( \boldsymbol{M}_{TR}\) ; and the full spatial near-field terms of the dyadic Green tensor are retained in every TX–RX pair.
To understand what the above Markov elimination retains and neglects, it keeps the full spatial dependence of \( \overline{\overline{\boldsymbol{G}}}\) between every pair of dots, including the retardation phase \( e^{ik_{0}R}\) and all \( 1/R\) , \( 1/R^{2}\) and \( 1/R^{3}\) terms, but neglects the propagation delay \( \tau=R/c\) in the envelopes, replacing \( c_{n}(t-\tau)\) by \( c_{n}(t)\) , and evaluates \( \overline{\overline{\boldsymbol{G}}}\) at \( \omega_{0}\) rather than across the photon band. The neglected delay and the single-frequency evaluation both give errors of relative order \( \Gamma\tau=\Gamma R/c\) , so the model requires \( \Gamma\tau=\Gamma R/c\ll1\) ; this quantity is \( 8.4\times10^{-4}\) at the default \( R=310\,\mu\mathrm{m}\) and \( 4.2\times10^{-3}\) at the largest distance studied, \( 1000\lambda_{0}\) . Once \( \Gamma\tau\) approaches \( 0.1\) (\( R\) of the order of centimetres) the delayed amplitudes must be kept and the link treated as a cascaded open system [38, 39], the RX–TX back-action then arriving after the round-trip delay \( 2\tau\) .

Pan/Zoom

The field that reaches the RX dots at frequency \( \omega\) is represented by the vector \( \boldsymbol{h}(\omega)=\boldsymbol{M}_{RT}\,\boldsymbol{c}_{T}(\omega)\) , in \(\mathrm{s}^{-1}\), where \( \boldsymbol{c}_{T}(\omega)\) is the Fourier transform of the TX amplitudes defined in Section 5. For \( R\) well beyond the Fraunhofer distance of the apertures, \( |\boldsymbol{r}^{R}_{m}-\boldsymbol{r}^{T}_{n}|\approx R-\hat{\boldsymbol{s}}_{s}\cdot\boldsymbol{r}^{R}_{m}+\hat{\boldsymbol{s}}_{s}\cdot(\boldsymbol{r}^{T}_{n}-R\hat{\boldsymbol{s}}_{s})\) , and (4) and (15) provide

where the scalar \( \kappa\) contains the path loss, the element factors and the TX array factor. For the steering state the sum equals \( \sqrt{N_{T}}\) times a unit-modulus phase. The far-field channel is therefore rank one, its only angular content being (3), and the departure from (17.a) is measured by the plane-wave error

which scales as \( (L_{\mathrm{ap}}^{2}/\lambda_{0}R)^{2}\propto R^{-2}\) for an aperture of largest dimension \( L_{\mathrm{ap}}\) ; here \( L_{\mathrm{ap}}=\sqrt{2}\times2.325\,\mu\mathrm{m}\) and the Fraunhofer distance is \(14\,\mu\mathrm{m}\), or \( 9\lambda_{0}\) . Since \( |\kappa|^{2}\propto R^{-2}\) , \( P_{\mathrm{abs}}\) also decays as \( R^{-2}\) ; this is the free-space Friis law applied to one photon. Fig. 8 confirms both of these \( R^{-2}\) scalings numerically.

For a pure state with \( \gamma_{\phi}=0\) , (7) is equivalent to the amplitude equation \( i\dot{\boldsymbol{c}}=\boldsymbol{M}\boldsymbol{c}\) with \( \boldsymbol{c}=[\boldsymbol{c}_{T};\boldsymbol{c}_{R}]\) and \( \boldsymbol{C}=\boldsymbol{c}\boldsymbol{c}^{H}\) . We define the transform \( \boldsymbol{c}(\omega)=\int_{0}^{\infty}\boldsymbol{c}(t)e^{i\omega t}dt\) , with \( \omega\) measured from \( \omega_{0}\) . With \( \boldsymbol{c}_{R}(0)=\boldsymbol{0}\) the RX block of the amplitude equation gives

where \( \boldsymbol{I}\) is now the \( N_{R}\times N_{R}\) identity. Equation (19), written with the full excitation vector \( \boldsymbol{h}(\omega)=\boldsymbol{M}_{RT}\boldsymbol{c}_{T}(\omega)\) , is exact within the present model; it is not restricted to a plane-wave excitation. In the far-field regime, where the TX-to-RX field factorizes as \( \boldsymbol{h}(\omega)\approx\kappa(\omega)\boldsymbol{a}(\hat{\boldsymbol{s}})\) by (17.a), the response reduces to the coupling-aware form \( \boldsymbol{c}_{R}(\omega)\propto\kappa(\omega)(\omega\boldsymbol{I}-\boldsymbol{M}_{RR})^{-1}\boldsymbol{a}(\hat{\boldsymbol{s}})\) , which inherits the far-field approximation and should not be read as exact in the general near-field case. The TX amplitudes follow from \( \boldsymbol{c}_{T}(\omega)=i(\omega\boldsymbol{I}-\boldsymbol{M}_{TT})^{-1}\boldsymbol{c}_{T}(0)\) up to the back-action term \( \boldsymbol{M}_{TR}\boldsymbol{c}_{R}\) , of relative order \( (k_{0}R)^{-1}\) , which the numerics retain in any case. The usual semiclassical receiver is recovered as follows. A weak coherent field of frequency \( \omega\) , described by the Rabi-frequency vector \( \boldsymbol{\Omega}\) at the RX dots, produces in linear response the steady amplitudes \( \boldsymbol{\beta}(\omega)=\tfrac{1}{2}(\omega\boldsymbol{I}-\boldsymbol{M}_{RR})^{-1}\boldsymbol{\Omega}\) , i.e., \( \boldsymbol{\beta}=-\tfrac{1}{2}\boldsymbol{M}_{R}^{-1}\boldsymbol{\Omega}\) with \( \boldsymbol{M}_{R}=\boldsymbol{M}_{RR}-\omega\boldsymbol{I}\) .

Proposition 1 answers the question raised in Section 1. In the far-field regime the coupling-aware manifold \( \boldsymbol{M}_{R}^{-1}\boldsymbol{a}\) remains valid under single-photon excitation, but only frequency by frequency. The receiver therefore responds to the spectrum of the photon, and that spectrum is set by the transmitter.

By Parseval’s theorem the time-integrated RX excitation is {

} We define the receiver state as the unit-trace matrix

If each RX dot radiates at the same rate into its own collection channel (Section 6), the photon leaving the RX is in the state \( |\psi_{\mathrm{out}}\rangle\propto\int dt\sum_{n}c_{R,n}(t)|1_{n}(t)\rangle\) , with \( |1_{n}(t)\rangle\) one photon in channel \( n\) at time \( t\) ; tracing out the time (equivalently frequency) degree of freedom leaves \( \int\boldsymbol{c}_{R}\boldsymbol{c}_{R}^{H}dt/\operatorname{Tr}(\cdot)=\rho_{R}\) . Thus \( \rho_{R}\) is the spatial state of the absorbed and re-emitted photon, conditioned on absorption, and the symbol is used only in this sense.
Although the joint TX–field–RX state is pure, \( \rho_{R}\) is mixed whenever \( (\omega\boldsymbol{I}-\boldsymbol{M}_{RR})^{-1}\boldsymbol{h}(\omega)\) turns in direction across the band occupied by \( |\boldsymbol{h}(\omega)|^{2}\) : spatial and spectral degrees of freedom are then entangled, and tracing out one leaves the other mixed. Since the collective RX modes are split and broadened on the scale of \( \Gamma\) (Section 8) and a lifetime-limited photon has the same bandwidth, a photon from an emitter identical to the receiver dots is not narrowband with respect to the collective receiver modes in the geometry studied here. The purity \( \operatorname{Tr}\rho_{R}^{2}\) measures the effect. Fig. 9 shows \( \rho_{R}\) for the default link and its purity as a function of the photon linewidth.

Pan/Zoom

In the far field \( \boldsymbol{h}(\omega)\approx\kappa(\omega)\boldsymbol{a}(\hat{\boldsymbol{s}})\) . The spectral weight \( |\kappa(\omega)|^{2}\) follows from \( \boldsymbol{c}_{T}(\omega)\) and in general contains contributions from several collective TX modes, since the steering state (12) is not an eigenvector of \( \boldsymbol{M}_{TT}\) . When the transmitted state is dominated by a single collective resonance, that weight can be approximated by a Lorentzian with centre \( \omega_{c}=\operatorname{Re}(\boldsymbol{c}^{H}\boldsymbol{M}_{TT}\boldsymbol{c})\) and FWHM \( \gamma=-2\operatorname{Im}(\boldsymbol{c}^{H}\boldsymbol{M}_{TT}\boldsymbol{c})=\boldsymbol{c}^{H}\boldsymbol{\Gamma}_{TT}\boldsymbol{c}\) , i.e., the collective shift and decay rate of the prepared state. The centre frequency and the width are both proportional to \( \Gamma_{T}\) , and both can be obtained by spectroscopy of the source. We adopt this single-resonance approximation only to obtain a closed-form manifold; three objects must therefore be kept apart: the exact spectral weight \( |\kappa(\omega)|^{2}\) obtained numerically from the TX model, its Lorentzian approximation, and the reduced spectrally matched manifold that follows. Substituting the Lorentzian into (20) gives the spectrally matched (SM) model

with the unit-area Lorentzian \( S(\omega)=(\gamma/2\pi)/[(\omega-\omega_{c})^{2}+\gamma^{2}/4]\) and \( \boldsymbol{a}=\boldsymbol{a}(\hat{\boldsymbol{s}})\) ; \( \operatorname{Tr}(\cdot)\) denotes the trace of the numerator. The integral converges because the integrand decays as \( |\omega|^{-4}\) . After the shift \( \omega\rightarrow\omega+\omega_{c}\) , let \( \boldsymbol{M}_{RR}-\omega_{c}\boldsymbol{I}=\boldsymbol{V}\boldsymbol{\Lambda}\boldsymbol{V}^{-1}\) with eigenvalues \( \lambda_{k}\) and let \( \boldsymbol{u}=\boldsymbol{V}^{-1}\boldsymbol{a}\) . Closing the contour in the upper half plane encloses the poles \( \omega=i\gamma/2\) of \( S\) and \( \omega=\lambda_{l}^{*}\) of the conjugate resolvent (all \( \lambda_{l}\) lie in the lower half plane), and the residues give

Evaluating (23) costs one eigen-decomposition of \( \boldsymbol{M}_{RR}\) plus \( O(N^{2})\) operations per look direction. For \( \gamma\rightarrow0\) , \( \boldsymbol{R}_{\mathrm{SM}}\) collapses to the rank-one narrowband model \( \boldsymbol{b}\boldsymbol{b}^{H}/\|\boldsymbol{b}\|^{2}\) with

which we call the coupling-aware (CA) manifold; it is the linear-response manifold of a semiclassically driven coupled array, here including the centre-frequency offset \( \omega_{c}\) . To summarize the status of the three expressions: (20)–(21) define the spectrum-integrated receiver state within the TX–Green-function–RX model and use no spectral or far-field assumption beyond the Markov elimination; (22)–(23) are a reduced spectrally matched representation obtained after the far-field factorization (17.a) and the single-resonance spectral approximation; and (24) is the further narrowband limit of that reduction. Whether the Lorentzian step is quantitatively harmless for the arrays studied here is checked against the numerically computed spectrum in Section 8-A. Pure dephasing of the source broadens the photon to \( \gamma+2\gamma_{\phi}\) , and (23) accepts this linewidth directly. The spatially matched (SM) model is a statement about linear electrodynamics and the photon spectrum; a classical receiver facing a broadband classical field could use the same correction.

Each RX dot is coupled, with equal efficiency and negligible crosstalk, to its own single-mode collection channel (a local waveguide or grating coupler), so the channel amplitude is proportional to \( \hat{\sigma}^{-}_{n}\) ; the \( N\) outputs are mixed with a common local oscillator on balanced heterodyne detectors and integrated over a window long compared with \( T_{1}\) , which traces out time as in (21). One repetition of the link produces one snapshot \( \boldsymbol{x}[k]\in\mathbb{C}^{N}\) , \( k=1,\dots,K\) , whose element \( x_{n}[k]\) is the complex heterodyne output of channel \( n\) , scaled so that vacuum gives \( E|x_{n}|^{2}=1\) ; with probability \( P_{d}\) it contains one photon in the spatial state \( \rho_{R}\) , otherwise only vacuum.
It is to note here that \( P_{\mathrm{abs}}\) and \( P_{d}\) are different quantities. \( P_{\mathrm{abs}}\) is the absorption probability of the bare link, \( 1.0\times10^{-4}\) at \( R=200\lambda_{0}\) (Section 8). \( P_{d}\) is the probability that a retained snapshot contains a detected photon, that is, a probability conditioned on the gating or heralding rule that selects which repetitions are kept. Writing \( P_{\mathrm{g}}\) for the fraction of repetitions retained and \( \eta_{\mathrm{c}},\eta_{\mathrm{d}}\) for the collection and detection efficiencies, \( P_{d}=\min\{1, P_{\mathrm{abs}}\eta_{\mathrm{c}}\eta_{\mathrm{d}}/P_{\mathrm{g}}\}\) , which satisfies \( 0\leq P_{d}\leq1\) because the retained fraction cannot be smaller than the fraction of repetitions that actually produce a detection. The two limits used below are therefore \( P_{d}\ll1\) , a loosely gated receiver in which most snapshots are vacuum, and \( P_{d}=1\) , a perfectly heralded receiver in which every retained snapshot carries a photon; neither is the bare-link absorption probability. The Monte Carlo results are conditioned on the retained gated or heralded snapshot ensemble rather than on the total number of transmitted photons.
Heterodyne outcomes are distributed according to the Husimi \( Q\) function of the detected field [40, 41]. For the mixture of vacuum and a one-photon state with spatial density matrix \( \rho_{R}\) the density of one snapshot is

with covariance and sample estimate

The identity in \( \boldsymbol{R}_{x}\) is the vacuum (shot) noise that heterodyne detection adds to every channel, one unit of noise per mode; it plays the part of receiver noise. Detector excess noise of variance \( \sigma_{e}^{2}\) would add \( \sigma_{e}^{2}\boldsymbol{I}\) . The density (25) is not necessarily Gaussian, and this matters for the bound in Section 6-C. Equation (26) follows from \( E[\boldsymbol{w}\boldsymbol{w}^{H}|\boldsymbol{u}^{H}\boldsymbol{w}|^{2}]=\boldsymbol{I}+\boldsymbol{u}\boldsymbol{u}^{H}\) for \( \boldsymbol{w}\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{I})\) : the photon adds one unit of variance in its own mode. Snapshots are drawn exactly from (25), a vacuum snapshot being \( \boldsymbol{w}\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{I})\) and a photon snapshot \( \boldsymbol{x}=\boldsymbol{w}+(z-\boldsymbol{u}^{H}\boldsymbol{w})\boldsymbol{u}\) , where the mode \( \boldsymbol{u}\) is drawn from the eigen-decomposition of \( \rho_{R}\) with probabilities equal to its eigenvalues, \( |z|^{2}\sim\mathrm{Gamma}(2,1)\) and \( \arg z\) is uniform.

The DOA branch returns a direction; the second branch of Fig. 2 returns an estimate \( \hat{\rho}_{R}\) of the true state \( \rho_{R}\) . From (26), \( \hat{\boldsymbol{R}}_{x}-\boldsymbol{I}\) is an unbiased estimate of \( P_{d}\rho_{R}\) . Its trace estimates \( P_{d}\) , so dividing by the trace removes the unknown \( P_{d}\) . Sampling noise leaves the normalized matrix Hermitian but generally with some negative eigenvalues. We keep its eigenvectors and replace its eigenvalues by their Euclidean projection onto the probability simplex (non-negative, summing to one). For additive Gaussian noise this projection is the maximum-likelihood physical state [42]. The quality of \( \hat{\rho}_{R}\) is measured by the Uhlmann fidelity [43]

in the squared convention, so that \( 0\leq F\leq1\) and \( F=1\) only for \( \hat{\rho}_{R}=\rho_{R}\) . The same definition is used for every fidelity in the paper, including the fidelity of \( \rho_{R}\) to the FS, CA and SM models.

For \( K\) independent snapshots and the parameter vector \( \boldsymbol{\vartheta}=(\theta,\phi)\) with \( \partial_{i}=\partial/\partial\vartheta_{i}\) , the Fisher information matrix of the snapshot density (25), exact for that density and not a quantum Fisher information over all possible measurements, is

which we evaluate by Monte Carlo with \( 2\times10^{5}\) samples and central-difference derivatives of \( \rho_{R}(\hat{\boldsymbol{s}})\) . For comparison, a zero-mean complex Gaussian vector with the same covariance \( \boldsymbol{R}_{x}\) has \( [\boldsymbol{F}_{G}]_{ij}=K\operatorname{Tr}(\boldsymbol{R}_{x}^{-1}\partial_{i}\boldsymbol{R}_{x}\boldsymbol{R}_{x}^{-1}\partial_{j}\boldsymbol{R}_{x})\) . We call \( \boldsymbol{F}^{-1}\) the exact CRB and \( \boldsymbol{F}_{G}^{-1}\) the Gaussian CRB. Under the regularity conditions of the Fisher-information formulation, the exact CRB bounds unbiased estimators based on the full snapshot distribution (25); the Gaussian CRB is the corresponding asymptotic bound for estimators that use only \( \hat{\boldsymbol{R}}_{x}\) and treat the data as Gaussian. Both are classical Fisher-information bounds for the specified heterodyne measurement and are not quantum Fisher information bounds optimized over all measurements. Both are reported as great-circle standard deviations, \( \mathrm{CRB}=\sqrt{[\boldsymbol{F}^{-1}]_{\theta\theta}+\sin^{2}\theta\,[\boldsymbol{F}^{-1}]_{\phi\phi}}\) , so a ratio of two such values is a ratio of standard deviations and the variance ratio is its square.

Three receive manifolds are compared, each normalized to unit norm. FS is the free-space plane-wave vector \( \boldsymbol{a}(\hat{\boldsymbol{s}})\) of (3), which ignores \( \boldsymbol{G}_{RR}\) ; CA is the narrowband vector \( \boldsymbol{b}(\hat{\boldsymbol{s}})\) of (24), exact for a monochromatic field by Proposition 1; SM: the matrix \( \boldsymbol{R}_{\mathrm{SM}}(\hat{\boldsymbol{s}})\) of (23) is the full spectrum-weighted spatial state predicted for direction \( \hat{\boldsymbol{s}}\) , and for the subspace estimators below, which require a vector-valued manifold, we use its principal eigenvector \( \boldsymbol{e}_{1}(\hat{\boldsymbol{s}})\) ; this is the step that adds the centre frequency and linewidth of the photon to the manifold. All three are tabulated on a grid; CA and SM need \( \boldsymbol{M}_{RR}\) , from the geometry through (5) or from calibration. FS and CA are the manifolds a classical receiver would use with and without mutual-coupling compensation [27]; SM is the manifold that the single-photon model of Section 5 actually predicts.

With \( \boldsymbol{g}(\hat{\boldsymbol{s}})\) denoting any of the three manifolds, we use the Bartlett spectrum \( P_{\mathrm{B}}=\boldsymbol{g}^{H}\hat{\boldsymbol{R}}_{x}\boldsymbol{g}\) , the MVDR spectrum \( P_{\mathrm{MV}}=(\boldsymbol{g}^{H}\hat{\boldsymbol{R}}_{x}^{-1}\boldsymbol{g})^{-1}\) [34] and the MUSIC spectrum \( P_{\mathrm{MU}}=(\boldsymbol{g}^{H}\hat{\boldsymbol{E}}_{n}\hat{\boldsymbol{E}}_{n}^{H}\boldsymbol{g})^{-1}\) [33]. Here \( \hat{\boldsymbol{E}}_{n}\) holds the \( N-Q\) eigenvectors of \( \hat{\boldsymbol{R}}_{x}\) with the smallest eigenvalues, and \( Q\) is the number of sources. The spectra are evaluated on a \( 2^{\circ}\times2^{\circ}\) grid and the \( Q\) largest well-separated maxima refined by a Nelder–Mead search of the continuous spectrum. The search is restricted to the upper half space because of the mirror symmetry (11): a planar array of in-plane dipoles cannot tell \( \theta\) from \( 180^{\circ}-\theta\) . Fig. 10 shows MUSIC spectra with each of the three manifolds.

Pan/Zoom

ESPRIT [35] requires the manifold to be invariant under a translation of a subarray. The FS vector of a uniform planar array has this property. The CA vector \( \boldsymbol{M}_{R}^{-1}\boldsymbol{a}\) does not, because edge and interior dots sit in different coupling environments, so ESPRIT applied directly to coupled data is biased. For a narrowband field the invariance is restored exactly by decoupling: the signal eigenvector is multiplied by \( (\boldsymbol{M}_{RR}-\omega_{c}\boldsymbol{I})\) before the least-squares shift equations in \( x\) and \( y\) are formed [27]. For a photon of finite bandwidth the signal subspace is not spanned by a single CA vector, so decoupling is exact only as \( \gamma\rightarrow0\) . The decoupled 2-D ESPRIT is included to measure that residual. Beamspace processing is not pursued, since with \( N=16\) the element-space eigen-decomposition is already cheap.

The SM receiver needs three one-time steps – diagonalizing \( \boldsymbol{M}_{RR}\) at cost \( O(N^{3})\) , measuring \( \omega_{c}\) and \( \gamma\) once per source type, and storing the principal eigenvector of \( \boldsymbol{R}_{\mathrm{SM}}\) on the search grid – and one per-data step, forming \( \hat{\boldsymbol{R}}_{x}\) at cost \( O(KN^{2})\) and searching the stored table. Only the last depends on the data. Once the spectrally matched manifold has been precomputed and tabulated, the per-snapshot computational structure is therefore comparable to conventional subspace DOA estimation with a stored calibration table [27, 36]; the total cost is larger by the offline eigen-decomposition and the grid tabulation, which are performed once per array and source type.

Relative agreement is \( 2.3\times10^{-15}\) between (8) and the Kronecker-form Liouvillian solution, \( 2.1\times10^{-12}\) between the dephasing solver at \( \gamma_{\phi}\rightarrow0\) and the undephased result, and \( 1.8\times10^{-10}\) between (23) and quadrature of (22); the radiated photon number equals one to within \( 10^{-6}\) and the operator-algebra \( g^{(2)}\) reproduces (14). The last of these figures verifies the closed-form integration of (23); the Lorentzian approximation itself is checked separately. Computing the spectral weight directly from the TX model, \( |\boldsymbol{h}(\omega)|^{2}\) with \( \boldsymbol{h}(\omega)=\boldsymbol{M}_{RT}\boldsymbol{c}_{T}(\omega)\) , and normalizing it to unit area gives a line of FWHM \( 0.975\,\Gamma\) against \( \gamma=0.891\,\Gamma\) for the single-resonance Lorentzian built from the same state; the two normalized weights differ by \( 5.4\%\) in normalized RMS error and overlap by \( (\int\sqrt{S_{\mathrm{num}}(\omega)S(\omega)}\,d\omega)^{2}=0.9994\) . Replacing the Lorentzian in (22) by the numerical weight and repeating the default-link MUSIC estimate of Section 8-F gives \( 0.015^{\circ}\) , against \( 0.031^{\circ}\) for the closed-form SM manifold and \( 1.813^{\circ}\) for CA; the quadrature is converged to \( \pm0.001^{\circ}\) for grids of \( 1200\) points or more over \( \pm60\,\Gamma\) . Roughly half of the small SM residual is therefore attributable to the single-resonance approximation, and none of the two orders of magnitude that separate SM from CA. The two analytic limits of the model, an exact CA manifold for \( \Gamma_{T}\rightarrow0\) and identical manifolds for an uncoupled receiver, are recovered as well (Section 8-F).

Fig. 3(a) shows the pair couplings (5) for separations along and across the dipole axis; at \( d=\lambda_{0}/2\) (\( k_{0}r=\pi\) ) both \( |\Gamma_{mn}|/\Gamma\) and \( |J_{mn}|/\Gamma\) are a sizeable fraction of one, the collinear pair being dominated by near-field terms. The collective decay rates [Fig. 3(b)] span \( 0.154\Gamma\) to \( 2.426\Gamma\) at \( d=\lambda_{0}/2\) and the shifts \( -0.611\Gamma\) to \( 0.371\Gamma\) . In DOA terms [Fig. 3(c)], the CA–FS mismatch \( \epsilon_{m}=1-|\boldsymbol{a}^{H}\boldsymbol{b}|^{2}/(\|\boldsymbol{a}\|^{2}\|\boldsymbol{b}\|^{2})\) is \( 0.109\) at \( d=\lambda_{0}/2\) , where FS-MUSIC is biased by \( 3.09^{\circ}\) ; the bias peaks at \( 16.2^{\circ}\) near \( 0.30\lambda_{0}\) (\( \epsilon_{m}=0.668\) ) and falls below \( 1^{\circ}\) only for \( d\geq0.57\lambda_{0}\) . Coupling cannot be ignored at half-wavelength spacing.

A four-element array along \( z\) with \( \hat{\boldsymbol{x}}\) dipoles reproduces the classical equivalence of Section 3 (Fig. 4): the \( W\) state (\( c_{n}=1/\sqrt{N}\) ) and the uniformly fed classical array have identical patterns, with \( D_{0}=8.74\)  dBi, half-power beamwidth (HPBW) \( 26.32^{\circ}\) and side-lobe level (SLL) \( -11.30\)  dB, while the state steered to endfire at \( d=0.3\lambda_{0}\) gives \( D_{0}=7.75\)  dBi, HPBW \( 103.3^{\circ}\) , front-to-back ratio \( 12.04\)  dB and pattern asymmetry \( \eta_{A}=\int|G^{(1)}(\theta)-G^{(1)}(\pi-\theta)|d\theta/\int G^{(1)}d\theta=1.806\) ; at \( d=0.5\lambda_{0}\) the same phase law gives the familiar symmetric bidirectional pattern. None of this is quantum.
To demonstrate the planar steering, Fig. 5 shows prepared-state patterns on the \( (u,v)\) disc; Fig. 6 and Table 4 summarize the scan at \( \phi_{0}=45^{\circ}\) . Broadside gives \( D_{0}=13.92\)  dBi and HPBW \( 25.4^{\circ}\) ; scanning to \( \theta_{0}=60^{\circ}\) lowers \( D_{0}\) to \( 11.79\)  dBi, widens the beam to \( 52.9^{\circ}\) and grows \( \Delta\Omega\) from \( 0.27^{\circ}\) at \( 5^{\circ}\) to \( 5.34^{\circ}\) [Fig. 6(b),(c)], the peak moving toward broadside and toward \( \phi=90^{\circ}\) where the \( \hat{\boldsymbol{x}}\) dipole radiates most strongly, as Section 3 predicts. The emitted-photon directivity stays within \( 0.1\)  dB of \( D_{0}\) up to \( 45^{\circ}\) and \( 0.39\)  dB behind at \( 60^{\circ}\) , and the steering states, not being eigenmodes, decay at \( 0.93\Gamma\) to \( 1.59\Gamma\) .

Table 3 gives \( g^{(2)}\) in the commanded direction. A pure single-excitation state is perfectly antibunched, and a \( 10\%\) two-excitation admixture raises \( g^{(2)}\) only to \( 0.136\) (\( N=4\) ) and \( 0.159\) (\( N=16\) ), far below the coherent-state reference. This is the principal nonclassical distinction considered for the transmitter in this model, and the results below show that direction finding does not use it. Under pure dephasing of every dot (Fig. 7) TX, \( D_{0}^{\mathrm{int}}\) of the \( (30^{\circ},45^{\circ})\) state falls by \( 1.13\)  dB (from \( 13.17\) to \( 12.04\)  dBi) at the resonant bound \( \gamma_{\phi}=0.167\Gamma\) and by \( 6.20\)  dB (to \( 6.97\)  dBi) at the non-resonant \( 2.24\Gamma\) , while the coherence ratio \( \eta_{\mathrm{coh}}\) – the \( \ell_{1}\) norm of the off-diagonal part of the normalized \( \boldsymbol{X}\) , relative to its undephased value – drops to \( 0.715\) and \( 0.181\) . The beam direction hardly moves (\( 0.2^{\circ}\) over the whole sweep) while the HPBW grows from \( 30.2^{\circ}\) to \( 36.0^{\circ}\) : dephasing costs directional contrast, not pointing, because it suppresses the off-diagonal entries of \( \boldsymbol{R}_{Q}\) without biasing their phases. Reading the sweep as a specification, a \( 0.5\)  dB directivity budget is met at \( \gamma_{\phi}=0.05\Gamma\) (\( 0.38\)  dB, \( \eta_{\mathrm{coh}}=0.884\) ) but not at \( 0.1\Gamma\) (\( 0.72\)  dB, \( \eta_{\mathrm{coh}}=0.801\) ), so the transmitter must be driven resonantly and kept near the transform limit; this is a budget derived from a chosen criterion, not a physical boundary.

Fig. 8 varies the communication link distance from \( 3\lambda_{0}\) to \( 1000\lambda_{0}\) ; this studies the channel and the manifold, not a practical long-distance 1550-nm link. The plane-wave error (18) follows the \( R^{-2}\) reference line from \( 1.5\times10^{-1}\) to \( 1.8\times10^{-6}\) (\( 4.5\times10^{-5}\) at \( R=200\lambda_{0}\) ), and the absorption probability the same law, from \( 0.22\) to \( 4.0\times10^{-6}\) (\( 1.01\times10^{-4}\) at \( 200\lambda_{0}\) ). The DOA errors [Fig. 8(b)] show two regimes. For \( R\leq10\lambda_{0}\) the incident field is not a plane wave and no plane-wave-based manifold is adequate (SM-MUSIC errs by \( 3.72^{\circ}\) at \( 3\lambda_{0}\) ). Beyond \( 20\lambda_{0}\) the RX coupling alone sets the error: FS levels off near \( 1.67^{\circ}\) and CA near \( 1.81^{\circ}\) , while SM stays between \( 0.014^{\circ}\) and \( 0.042^{\circ}\) , about half of that residual coming from the single-Lorentzian description of the TX spectrum (Section 8-A). The RX purity is \( 0.949\) for every \( R\geq20\lambda_{0}\) , so the mixedness is set by the RX coupling and the photon spectrum, not the channel.

Fig. 9 shows the receiver state \( \rho_{R}\) of (21) for the default link. The magnitudes \( |[\rho_{R}]_{mn}|\) are dominated by the plane-wave phase gradient, modulated at the edges by the coupling. The eigenvalues of \( \rho_{R}\) are \( 0.974\) , \( 0.022\) and \( 0.004\) , followed by smaller ones, and its purity is \( 0.949\) . The fidelity (27) of \( \rho_{R}\) is \( 0.918\) to the FS model \( \boldsymbol{a}\boldsymbol{a}^{H}/N\) , \( 0.955\) to the narrowband CA model, and \( 0.99997\) to the SM model of (23). Fig. 9(c) contains the central result of the paper. The radiative rate \( \Gamma_{T}\) of the TX dots, and with it the photon linewidth, is varied while the RX is left unchanged; \( \Gamma_{T}=\Gamma\) is the lifetime-limited photon of an identical emitter and \( \Gamma_{T}\rightarrow0\) the narrowband limit of Proposition 1. As \( \Gamma_{T}/\Gamma\) rises from \( 0.01\) to \( 2\) the purity falls from \( 0.998\) to \( 0.943\) and the CA error grows from \( 0.038^{\circ}\) to \( 2.05^{\circ}\) , while the FS error falls from \( 3.05^{\circ}\) to \( 1.43^{\circ}\) because averaging over a wider band washes out the frequency-selective coupling; the curves cross near \( \Gamma_{T}\approx0.8\Gamma\) , leaving the lifetime-limited photon on the wrong side for CA, and SM stays below \( 0.036^{\circ}\) throughout. Every photon in this sweep is a single photon, so photon statistics cannot cause the effect: the narrowband manifold fails because of the photon bandwidth.

Fig. 10 shows MUSIC spectra in the \( \phi=30^{\circ}\) plane and Table 5 the errors of all estimators. For a given manifold the three spectral estimators agree – FS \( 1.64^{\circ}\) –\( 1.67^{\circ}\) , CA \( 1.80^{\circ}\) –\( 1.81^{\circ}\) , SM \( 0.031^{\circ}\) – so the manifold matters far more than the estimator. The Bartlett and MVDR spectra vary by only a few tenths of a decibel over the cut, since at \( P_{d}=0.1\) the signal term is one tenth of the vacuum noise, yet their peaks are placed correctly with the SM manifold. Direct ESPRIT errs by \( 1.95^{\circ}\) and decoupled ESPRIT by \( 0.58^{\circ}\) , the residual left by the finite photon bandwidth in the shift-invariance structure (Section 7-C).
The four-system baseline in the lower half of Table 5 separates the ingredients, the narrowband system using \( \Gamma_{T}=10^{-3}\Gamma\) , which by Proposition 1 responds as to a weak monochromatic drive. With an uncoupled RX all manifolds are exact to within the \( 0.009^{\circ}\) residual of the finite link distance; with a coupled RX, FS is biased by \( 3.09^{\circ}\) while CA is exact (\( 0.005^{\circ}\) ); a single photon on an uncoupled RX is again unbiased (\( 0.014^{\circ}\) ). Only a broadband single photon on a coupled RX breaks the CA manifold: collective electrodynamics acting on a finite-bandwidth wavepacket, not photon statistics.

Pan/Zoom

Pan/Zoom

Pan/Zoom

Fig. 11 gives Monte Carlo results from 100 independent sets of \( K\) snapshots drawn from (25), conditioned on gated or heralded repetitions (Section 6); the error bars are the approximate \( 95\%\) confidence interval of an RMSE from 100 trials, \( \pm1.96/\sqrt{200}\approx\pm14\%\) . With \( K=10^{4}\) [Fig. 11(a)] all estimators show the usual threshold: below \( P_{d}\approx0.05\) the signal term \( P_{d}\rho_{R}\) is buried in the vacuum noise and the CRB is not informative. Above it the three SM estimators track the bounds, with RMSEs of \( 2.25^{\circ}\) –\( 2.34^{\circ}\) against an exact CRB of \( 2.04^{\circ}\) at \( P_{d}=0.1\) and \( 0.80^{\circ}\) against \( 0.71^{\circ}\) at \( P_{d}=0.3\) , while FS-MUSIC levels off at its \( 1.67^{\circ}\) bias and decoupled ESPRIT approaches its \( 0.58^{\circ}\) residual (\( 0.79^{\circ}\) at \( P_{d}=1\) ).
At \( P_{d}=1\) the SM estimators reach \( 0.284^{\circ}\) , almost on the Gaussian CRB (\( 0.276^{\circ}\) ) but \( 41\%\) above the exact CRB (\( 0.202^{\circ}\) ), which is \( 27\%\) lower in standard deviation, a factor \( 1.86\) (\( 2.7\)  dB) in variance. The heterodyne data of a nearly pure one-photon state are strongly non-Gaussian, so part of their Fisher information is not carried by the covariance and only a maximum-likelihood estimator built on (25) could recover this margin. At \( P_{d}=0.3\) [Fig. 11(b)] the SM-MUSIC error decreases as \( K^{-1/2}\) , from \( 2.63^{\circ}\) at \( K=10^{3}\) to \( 0.228^{\circ}\) at \( K=10^{5}\) , which is within \( 1.5\%\) of the exact CRB (\( 0.224^{\circ}\) ); FS-MUSIC stalls at \( 1.69^{\circ}\) . Reconstructing the state [Fig. 11(c)] is far harder than finding the direction, since all \( N^{2}-1=255\) real parameters of \( \rho_{R}\) must be resolved rather than two angles: the fidelity rises from \( 0.850\) to \( 0.980\) between \( K=10^{3}\) and \( 10^{5}\) at \( P_{d}=1\) , reaches \( 0.869\) at \( P_{d}=0.1\) , and stays below \( 0.24\) at \( P_{d}=0.01\) . The DOA branch needs one to two orders of magnitude fewer detected photons than the state branch.

Fig. 12 gives the probability of resolving two mutually incoherent single-photon sources at \( (40^{\circ}\mp\Delta\theta/2,30^{\circ})\) from \( K=10^{4}\) heralded snapshots (\( P_{d}=1\) , photons shared equally), a trial counting as resolved when both estimates lie within \( \Delta\theta/2\) of distinct true directions. SM-MUSIC resolves in \( 95\) of the \( 100\) trials at \( \Delta\theta=20^{\circ}\) and in \( 100\%\) of the trials at \( 25^{\circ}\) and above, whereas FS-MUSIC reaches \( 0.24\) at \( 20^{\circ}\) and \( 0.55\) at \( 25^{\circ}\) , its manifold bias pulling the peaks together, and needs \( 30^{\circ}\) . MVDR-SM and Bartlett-SM are beamwidth-limited (HPBW \( \approx30^{\circ}\) near \( \theta=40^{\circ}\) ) and resolve only at \( 40^{\circ}\) . No method resolves \( \Delta\theta\leq12^{\circ}\) : the second eigenvalue of the two-source \( \rho_{R}\) is then \( 0.094\) , against a noise-eigenvalue spread of about \( \pm2\sqrt{N/K}=\pm0.08\) . These values are specific to this array, criterion and snapshot budget.