---
title: Quantum Imaging with Undetected Photons
url: https://www.emergentmind.com/topics/quantum-imaging-with-undetected-photons-qiup
type: topic
---

# Quantum Imaging with Undetected Photons

to=arxiv_search.search  体育彩票天天  天天中彩票谁  code: 
{"query":"all:\"quantum imaging with undetected photons\" OR ti:\"Quantum Imaging with Undetected Photons\"","max_results":10,"sort_by":"relevance"} 
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to=arxiv_search.search  天天中彩票无法  大发快三大小单双 code: 
{"query":"id:1401.4318 OR id:1504.00402 OR id:2010.07712 OR id:2101.02761 OR id:2106.11358 OR id:2106.06435 OR id:2203.06106 OR id:2208.03005 OR id:2306.01884 OR id:2501.05381","max_results":10,"sort_by":"relevance"} 

to=arxiv_search.search  大发彩票快三 平台总代理 code: 
{"query":"id:2408.02531 OR id:2406.05656 OR id:2409.18887 OR id:2505.21443 OR id:2507.06879 OR id:2507.20617","max_results":10,"sort_by":"relevance"} 

Quantum imaging with undetected photons (QIUP) is a nonlinear-interferometric imaging paradigm in which the photons that interrogate the object are not measured, while the image is reconstructed from partner photons that never interact with the object. In its canonical realization, two coherently pumped spontaneous parametric down-conversion (SPDC) sources are arranged so that the alternatives “pair created in source 1” and “pair created in source 2” are indistinguishable; the object is placed only in the undetected idler path, and its transmission or phase is transferred to the interference term of the detected signal beam [1401.4318][1504.00402].

## 1. Origins and induced-coherence basis

The experimental point of departure is the two-crystal induced-coherence arrangement of Lemos and co-workers, in which a 532 nm pump illuminates two identical ppKTP crystals, the signal photons are detected around 810 or 820 nm on an EMCCD, and the idler photons probe the object at 1550 or 1515 nm. In that demonstration, the method imaged an intensity mask, a silicon phase object opaque at 810 nm but transparent at 1550 nm, and a fused-silica phase object that was nearly invisible at 820 nm but visible when placed in the idler path [1401.4318]. The defining physical statement is that interference of the detected photons is controlled by whether the undetected partner photons preserve or destroy which-source information.

The first detailed theory established that the object information is present only in the interference term of the detected intensity and not in the individual intensities of the interfering beams. In the geometry analyzed there, the magnification is
\[
M=\frac{f_0 \bar\lambda_S}{f_I \bar\lambda_I},
\]
so it depends on both the average detected wavelength and the average illuminating wavelength [1504.00402]. This formulation also made explicit that QIUP is an instance of induced coherence without induced emission: the image is carried by first-order interference in the signal arm because the idler alternatives are rendered indistinguishable.

A persistent conceptual distinction from ghost imaging follows from that structure. In standard QIUP, object information is read out from single-photon interference in the detected beam, rather than from coincidence detection between two detected arms. Later near-field treatments emphasized that the method is closer in spirit to ghost imaging than to conventional imaging because image formation is governed by a two-photon correlation function, but still differs from standard ghost imaging in using single-photon interference without coincidence detection or post-selection [2106.11358].

## 2. Image-formation laws and geometrical regimes

In the far-field, momentum-correlation geometry, the biphoton state from one source is written as
\[
|\psi\rangle=\sum_{q_d,q_u} C(q_d,q_u)\,|q_d\rangle |q_u\rangle,
\]
with joint transverse-momentum probability
\[
P(q_d,q_u)=|C(q_d,q_u)|^2.
\]
For two weakly pumped crystals, the detected-camera intensity at position \(r_{q_d}\) takes the form
\[
I(r_{q_d})= \sum_{q_u} P(q_d,q_u) \left\{1+|T(r_{q_u})|\cos\!\big[\theta(r_{q_u})+\phi\big]\right\},
\]
so the image is a weighted average over the conditional momentum distribution of the undetected photon given the detected one. For a purely absorptive object, the visibility becomes
\[
V(r_{q_d})=\sum_{q_u} p(q_u|q_d)\,|T(r_{q_u})|,
\]
which makes the effective point-spread function the conditional distribution \(p(q_u|q_d)\) itself [2010.07712].

A complementary near-field formulation places both object and camera in image planes of the source. There the key quantity is the joint position probability density
\[
P(\boldsymbol{\rho}_{s},\boldsymbol{\rho}_{I}) \propto \left|\int d q_{s}\, d q_{I}\, C(q_{s}, q_{I})\, e^{i (q_{s}\cdot\boldsymbol{\rho}_{s} + q_{I}\cdot\boldsymbol{\rho}_{I})}\right|^{2},
\]
and the image for a purely absorptive object is extracted from two interferometer phases as
\[
G(\boldsymbol{\rho}_{c}) \propto \int d\boldsymbol{\rho}_{o}\, P\!\left(\frac{\boldsymbol{\rho}_{c}}{M_{S}}, \frac{\boldsymbol{\rho}_{o}}{M_{I}} \right) |T(\boldsymbol{\rho}_{o})|.
\]
In this regime QIUP becomes a linear convolution-like imaging system whose kernel is the twin-photon position-correlation function; for standard nondegenerate SPDC, the magnification is
\[
M=\frac{M_S}{M_I}.
\]
This is one of the sharpest distinctions from far-field QIUP, where wavelength ratios appear explicitly in the magnification law [2101.02761].

In the idealized limit of maximal position correlation,
\[
P(\boldsymbol{\rho}_s,\boldsymbol{\rho}_I)\propto \delta(\boldsymbol{\rho}_s-\eta \boldsymbol{\rho}_I),
\]
the near-field image reduces to point-to-point mapping with
\[
M=\eta \frac{M_S}{M_I},
\]
and for standard nondegenerate SPDC the paper states \(\eta=1\) [2101.02761]. This suggests that QIUP should be regarded less as a single fixed protocol than as a family of interferometric image-transfer schemes whose operative spatial variable is selected by geometry: momentum in the far field, position in the near field.

## 3. Spatial correlations, magnification, and resolution

The most developed far-field resolution theory showed that the decisive quantity is the transverse momentum correlation between detected and undetected photons. For a Gaussian pump, the conditional probability obeys
\[
p(q_u|q_d)\propto \exp\!\left(-|q_d+q_u|^2 w_p^2/2\right),
\]
so a larger pump waist \(w_p\) yields tighter momentum anti-correlation and better resolution. In that geometry the camera-plane blur width is
\[
\sigma=\frac{f_c\lambda_d}{\sqrt{2}\pi w_p},
\]
while the object-space blur is
\[
\frac{\sigma}{M}=\frac{f_u\lambda_u}{\sqrt{2}\pi w_p},
\qquad
M=\frac{f_c\lambda_d}{f_u\lambda_u}.
\]
The reported experiment found that weaker momentum correlation makes a 250 \(\mu\mathrm m\) test feature visibly blurrier, that all measured \(\sigma\) values differed by less than two standard deviations from theory, and that at fixed correlation strength a setup with \(\lambda_u=780\) nm resolved finer features than one with \(\lambda_u=1550\) nm even though the detected wavelength was slightly longer [2010.07712].

In the near-field, position-correlation regime, the principal resolution law is different. Under Gaussian pump and Gaussianized phase matching, the effective blur kernel has width
\[
\sigma_{\text{corr}} \sim \sqrt{L(\lambda_s+\lambda_I)},
\]
and the minimum resolvable distance derived from a dip-to-peak criterion \(\beta_{\max}=0.81\) is
\[
d_{\text{min}}\approx 0.53\, M_I \sqrt{L (\lambda_{I} + \lambda_{s})}.
\]
Here the detected and undetected wavelengths enter symmetrically through \(\lambda_s+\lambda_I\), and the correlation-imposed limit cannot be improved by conventional optical techniques in the detected arm [2106.11358].

A later experimental study of position-correlation QIUP varied crystal length \(L=2,5,10\) mm and pump waist \(p=50,142,214,308\ \mu\mathrm m\), and found that the visibility spread, rather than the amplitude-image spread, is the correct measure of image sharpness once finite-pump effects become important. In the large-pump limit, the magnification-adjusted visibility spread approaches
\[
\Delta_V \sim \sqrt{\frac{L(\lambda_d+\lambda_u)}{4\pi}},
\]
confirming the earlier crystal-length scaling, but strong pump focusing broadens \(\Delta_V\) and can drive the biphoton state toward a separability condition at which the visibility becomes spatially constant and object information is lost [2306.01884].

A significant controversy concerns which wavelength sets the ultimate far-field resolution. The focused far-field resolution theory above is explicit that, in its regime, the object-space resolution is governed by the undetected wavelength rather than the detected one [2010.07712]. A later beyond-paraxial treatment argued instead that, in the standard free-space far-field geometry, the ultimate diffraction-limited minimum resolvable distance is bounded by the longer wavelength of the pair,
\[
d_\mathrm{min}^{(\mathrm{limit})}\sim \frac{\max(\lambda_S,\lambda_I)}{2},
\]
more precisely with prefactors \(0.45\), \(0.40\), and \(0.37\) multiplying \(\max(\lambda_S,\lambda_I)\) in strongly nondegenerate and degenerate limits. In that account, the shorter-wavelength scaling found in earlier work is regime-specific rather than fundamental [2203.06106]. The coexistence of these results has become a central interpretive issue in the QIUP resolution literature.

## 4. Experimental regimes and application domains

QIUP has diversified from near-IR proof-of-principle interferometry into microscopy, phase imaging, terahertz sensing, holography, and tomography.

| Modality | Undetected / detected band | Reported capability |
|---|---|---|
| Mid-infrared microscopy [2106.06435] | \(3.74\,\mu\mathrm m\) / near \(800\,\mathrm{nm}\) | \(9\pm1\,\mu\mathrm m\) resolution; unstained mouse heart tissue |
| Single-shot phase quadrature [2208.03005] | \(\sim 1557\,\mathrm{nm}\) / \(\sim 808\,\mathrm{nm}\) | phase and visibility from one camera exposure |
| Terahertz QIUP [2408.02531] | \(1.5\,\mathrm{THz}\) / \(662.2\,\mathrm{nm}\) | amplitude and phase imaging; \(240\pm1\,\mu\mathrm m\) resolution |
| QUOPT tomography [2501.05381] | \(1550\,\mathrm{nm}\) / \(810\,\mathrm{nm}\) | 3D projection tomography; \(\sim 300\,\mu\mathrm m\) estimated resolution |
| Deep-learning QHUP [2409.18887] | QHUP signal detection near \(905\,\mathrm{nm}\) | one-shot hologram inversion; resolution improved from \(\sim 196\,\mu\mathrm m\) to \(116\,\mu\mathrm m\) |

Position-correlation microscopy provided the first experimental realization of near-field QIUP, with a Michelson-type nonlinear interferometer around a ppKTP crystal, \(3.74\,\mu\mathrm m\) idler probing, and measured field of view \(161\pm1\,\mu\mathrm m\) and resolution \(9\pm1\,\mu\mathrm m\) at \(M=4\). The same work reported ultralow mid-IR illumination of about 15 pW on the sample and demonstrated morphological imaging of mouse heart tissue on the cellular level [2106.06435].

Single-shot phase-quadrature QIUP replaced sequential phase stepping by a polarization-optics scheme that produces four simultaneous outputs with phase offsets \(0,\pi/2,\pi,3\pi/2\), allowing phase and visibility to be reconstructed from one camera frame. In calibration, the retrieved phase versus induced delay had \(r^2=0.9997\), with mean visibility \(67\%\) and absolute standard deviation \(1.5\%\); the method was demonstrated on a static phase mask opaque to the detected photons, drying isopropanol film, and stretched adhesive tape [2208.03005].

Terahertz QIUP extended the spectral separation to more than two orders of magnitude in frequency by probing the sample at 1.5 THz and detecting only visible photons at 662.2 nm on an uncooled sCMOS camera. The experiment reported spatial resolution \(240\pm1\,\mu\mathrm m\), field of view \(2.0\pm0.2\,\mathrm{mm}\), per-pixel interference visibility below \(0.15\%\), and a raw signal-to-noise ratio of at most 1:700 due to thermal and parasitic backgrounds; a Fourier-domain “quantum distillation” procedure isolated the desired interference term and enabled amplitude-sensitive, phase-sensitive, and spectrally selective imaging [2408.02531].

The tomographic extension QUOPT combined QIUP with optical projection tomography: a 1550 nm idler probed the sample, an 810 nm signal was recorded on a silicon camera, fringe visibility was extracted from phase scans via FFT, and 180 projections over 180° were reconstructed by inverse Radon transform into a three-dimensional infrared absorption volume. The demonstrated object was a twisted-wire figurine about 5 mm high, and the spatial resolution was estimated as \(\sim 300\,\mu\mathrm m\) [2501.05381].

Two further extensions altered the measurement architecture itself. Interaction-free single-pixel QIUP embedded an Elitzur–Vaidman-type Michelson interferometer in the idler arm and combined it with Hadamard-mask single-pixel imaging, using 1550 nm undetected idlers and 810 nm detected signals; in the binary-object regime, the relevant object information can be obtained from events in which the returning idler avoided the object arm [2212.06531]. A distinct four-photon protocol based on “phase-subtractive interference by path identity” used two independent sources and two undetected photons, yielding a coincidence interferogram that is independent of spatially uniform pump, signal, and idler phases while retaining spatially dependent object phases acquired by the undetected photons [2406.05656].

## 5. Information-theoretic reformulations and generalized tomography

QIUP has repeatedly been recast in explicitly quantum-informational language. One early reinterpretation described the original Lemos protocol as a form of ancilla-assisted process tomography in which the idler is the system passing through the unknown object channel \(\mathpzc{E}_O\), the signal is the ancilla, and the detected-photon measurement performs only a partial tomography of the object. In that model, the original setup measures only the combination \(T\cos\gamma\); a simple phase shifter before the final signal beamsplitter upgrades the protocol to full tomography of the two-parameter object model. The same analysis also argued that entanglement is not strictly necessary for image formation: for an extended Werner-state probe, the image visibility becomes \(\mathcal V=(1-\xi)T\), so sufficiently strong correlations can suffice even when the probe state is separable [1509.02031].

The induced-coherence paradigm has also been generalized from object imaging to tomography of the undetected subsystem itself. In “Quantum state tomography of undetected photons,” an unknown idler polarization qubit
\[
\rho_I = \begin{pmatrix} P_H & I\sqrt{P_H P_V}\,e^{-i\xi} \\ I\sqrt{P_H P_V}\,e^{i\xi} & P_V \end{pmatrix}
\]
was reconstructed from two signal-fringe measurements, with reported mean fidelities \(\bar F_{\mathrm{HWP}}=0.97\) and \(\bar F_{\mathrm{QWP}}=0.92\), and minimum fidelity \(F_{\min}=0.72\) [2211.10304]. This makes explicit that QIUP is not confined to spatial transmission maps; it is a broader ancilla-assisted measurement strategy for inaccessible quantum degrees of freedom.

A more recent conceptual development reframed QIUP in wave-particle-duality language. In the ideal case, the “imaging duality ellipse” is
\[
\frac{V^2}{T^2}+D^2=1,
\]
where \(V\) is visibility, \(D\) is path predictability, and \(T\) is the object transmittance. With source decoherence \(\gamma\) and idler misalignment \(\alpha\), the relation becomes
\[
\frac{V^2}{\gamma^2 T^2 \alpha^2}+D^2=1.
\]
Within that framework the object can be reconstructed from duality observables alone, and the ellipse form survives calibration against uniform imperfections [2505.21443].

Polarization-sensitive generalizations have begun to extend QIUP from scalar transmission to matrix-valued sample characterization. A theoretical Jones-matrix protocol uses selected idler polarization inputs, two special HWP settings, and signal-fringe visibilities and phase offsets to reconstruct a structured Jones matrix
\[
\hat{O}= \begin{pmatrix} \tau_H e^{i\phi_H} & \kappa e^{i\xi}\\ -\kappa e^{-i\xi} & \tau_V e^{i\phi_V} \end{pmatrix},
\]
thereby reducing the number of unknowns relative to a fully unconstrained complex \(2\times2\) fit [2507.20617]. By contrast, a separate theoretical study on polarization distinguishability concluded that if the two SPDC alternatives are polarization-distinguishable at generation, then post-generation optics and a Mach–Zehnder interferometer cannot restore the lost interference. That result sharpens a basic QIUP design rule: downstream optics reveal coherence already present between the generation alternatives; they do not create it after source-side distinguishability has destroyed it [2507.06879].

## 6. Limitations, controversies, and current directions

QIUP remains constrained by three interlocking resources: indistinguishability of the generation alternatives, usable transverse correlations in the biphoton state, and sufficiently stable interferometric readout. In practice, finite crystal size limits the pump waist and hence the achievable momentum correlation in far-field geometries; in one reported regime the diffraction-limited uncertainty from 1-inch optics was about \(5\,\mu\mathrm m\), whereas the uncertainty due to imperfect momentum correlation was about \(100\,\mu\mathrm m\) for \(w_p=200\,\mu\mathrm m\), making correlation rather than lens diffraction the dominant limit [2010.07712]. In the near field, the correlation-imposed blur cannot be undone by improving conventional optics in the detected arm, because the image has already been convolved with the twin-photon position kernel before detection [2106.11358].

Operational bottlenecks differ by modality. Single-shot phase-quadrature QIUP still required calibration of branch-dependent throughput, numerical spot registration, Gaussian filtering, and phase unwrapping for large phase excursions, with 500 ms integration time and stability only over several minutes [2208.03005]. THz QIUP achieved visible-only detection at extreme nondegeneracy, but with per-pixel visibility below \(0.15\%\), 1000 images averaged per delay step, and strong parasitic backgrounds imposed by room-temperature thermal photons and nonlinear side processes [2408.02531]. QUOPT tomography demonstrated 3D infrared reconstruction, but retained phase scanning and a proof-of-concept resolution of about \(300\,\mu\mathrm m\) [2501.05381]. Deep-learning-enhanced QHUP removed much of the multi-shot reconstruction burden, but explicitly acknowledged performance degradation when training and testing distributions are mismatched, especially between SLM-based virtual objects and real samples [2409.18887].

The most visible theoretical controversy remains the far-field wavelength question. One line of work, validated experimentally in a specific momentum-correlation regime, concludes that the camera-plane blur is set by the detected wavelength while the object-space resolution is governed by the undetected illumination wavelength [2010.07712]. Another, beyond-paraxial analysis argues that the ultimate free-space limit is instead set by the longer wavelength of the pair, regardless of which photon illuminates the object [2203.06106]. A plausible implication is that QIUP resolution must be discussed at two levels: regime-specific performance laws for a given geometry and source model, and the ultimate nonparaxial diffraction bound.

The current research trajectory is correspondingly bifurcated. One branch pushes spectral reach and modality: mid-IR microscopy, terahertz phase imaging, dynamic phase sensing, holography, and 3D tomography [2106.06435][2408.02531][2208.03005][2501.05381]. The other branch refines the formal structure of the protocol: ancilla-assisted tomography, duality-based reconstruction, polarization and Jones-matrix readout, and noise-resistant multi-photon interference [1509.02031][2505.21443][2507.20617][2406.05656]. At the device level, recurrent proposals include increasing the pump waist through larger transverse crystal dimensions, decreasing \(f_u\) in the undetected arm, using shorter undetected wavelengths, redesigning the geometry so that position rather than momentum correlations govern imaging, and extending nonlinear materials and interferometer architectures into still more difficult spectral bands, including explicit future concepts involving x-ray-scale illumination with optical detection [2010.07712].

Source: https://www.emergentmind.com/topics/quantum-imaging-with-undetected-photons-qiup