Quantum imaging with undetected photons (QIUP) is an emerging technique that enables mid-infrared microscopy using only visible-light detection, exploiting nonlinear interferometry with entangled photon pairs generated by spontaneous parametric down-conversion. As QIUP approaches application in biological tissue imaging, a central open question is how quantum correlations between signal and idler photons degrade when the idler traverses a structured, lossy, dispersive medium. We propose a trapped-ion SU(1,1) interferometer with engineered decoherence on one motional mode as a controllable testbed for channel-level models of QIUP in structured media. The trapped-ion system reproduces not the microscopic physics of molecular absorption, but the quantum-channel structure experienced by the idler mode, providing controlled validation for theoretical models that can subsequently inform QIUP system design. The key literature bridging the trapped-ion, quantum-optics, and open-quantum-systems communities is mapped, the formal analogy at the level of quantum channel estimation is developed, and the experimental feasibility within current trapped-ion technology is assessed. This document is an open invitation to collaborate.
This document is a cross-disciplinary concept note intended to connect four communities: trapped-ion quantum control, nonlinear quantum optics, open quantum systems, and mid-infrared biological imaging. Its purpose is not to present completed results, but to formulate a concrete experimental and theoretical programme that appears feasible with current technology.
The central claim is modest and specific: a trapped-ion SU(1,1) interferometer with engineered decoherence on one motional mode can serve as a controlled simulator of the channel-level dynamics relevant to nonlinear interferometric sensing in structured environments. The note does not claim a microscopic simulation of biological tissue or a complete model of multimode QIUP imaging.
Readers from different backgrounds may wish to enter the note at different points. Sections 1 and 2 provide the conceptual overview and formal mapping. Sections 4 and 5 develop the experimental and theoretical programme. Section 6 explains the biomedical motivation and the limits of translation. Appendix A offers tutorial remarks for readers approaching from a neighbouring field.
Quantum imaging with undetected photons exploits the phenomenon of induced coherence without induced emission, first described by Zou, Wang and Mandel (1991) and demonstrated for imaging by Lemos et al. (2014). The protocol uses a nonlinear crystal (or pair of crystals) pumped to generate entangled photon pairs via spontaneous parametric down-conversion (SPDC). One photon of each pair — the idler — illuminates the sample. The other — the signal — is detected. By arranging the nonlinear interaction such that the which-crystal information for each photon pair is erased, the signal photons exhibit interference fringes that encode the absorption and phase properties of the sample, despite the idler photons never being detected.
The technique has been extended to the mid-infrared (mid-IR) spectral region, where molecular vibrational modes provide chemically specific contrast for biological imaging. Kviatkovsky et al. (2020) demonstrated wide-field QIUP microscopy covering 3.4–4.3 µm with 35 µm resolution using a standard silicon CMOS camera for visible-light detection. Subsequent work has achieved hyperspectral capability via Fourier-transform methods (Placke et al., 2023) and scanning implementations that overcome spatial-correlation constraints (León-Torres et al., 2025). The elimination of cryogenically cooled mid-IR detectors and the ultralow irradiation intensities achievable at the single-photon level make QIUP attractive for photodamage-sensitive biological samples.
In all existing QIUP implementations, the sample is modelled as a classical transfer function: a spatially varying complex transmission t(r) = |t(r)| exp[iφ(r)] acting on the idler field mode. This treatment is adequate for thin polymer films and fixed tissue sections where the interaction is well approximated by wavelength-independent attenuation.
It is well understood that loss and decoherence degrade the induced coherence underlying QIUP, and beam-splitter channel models have been applied to quantify this degradation. However, the treatment of the sample as a general quantum channel — with spectral structure, memory, and non-trivial thermal properties — has received comparatively little attention. Biological tissue in the mid-IR fingerprint region (6–11 µm) presents precisely such a regime: water absorption is intense, molecular vibrational modes are spectrally structured and thermally populated at 300 K, and scattering, dispersion, and spatial inhomogeneity on cellular length scales all contribute.
A controlled platform for studying how structured decoherence channels affect nonlinear interferometric sensitivity would provide quantitative input for QIUP system design and help delineate the regime boundary between quantum advantage and classical superiority.
Trapped-ion systems offer an exceptionally well-controlled platform for quantum simulation and sensing. The motional degrees of freedom of a confined ion (quantised harmonic oscillator modes) can be coherently manipulated using laser-driven or electric-field-driven interactions, and their quantum state can be read out with near-unit efficiency via the ion's internal (spin) degree of freedom.
Critically, both the SU(2) algebra of optical beamsplitters and the SU(1,1) algebra of parametric amplifiers — the mathematical structures underlying nonlinear interferometry — have been experimentally realised in trapped-ion motional modes:
Furthermore, engineered decoherence — controllable coupling of the motional state to an artificial reservoir — was demonstrated by Myatt et al. (2000) at NIST, producing tuneable amplitude damping and phase damping channels.
The standard QIUP implementation uses a Michelson-type nonlinear interferometer formed by double-passing a periodically poled crystal (e.g. ppKTP). The Hamiltonian for the SPDC interaction in the undepleted-pump approximation is the two-mode squeezing Hamiltonian
HSPDC = iℏξ ( âs† âi† − âs âi ),where âs and âi are the annihilation operators for the signal and idler modes, and ξ is the parametric gain proportional to the pump amplitude and the nonlinear susceptibility χ(2). This generates entangled signal–idler photon pairs from the vacuum.
Between the two passes, the idler mode interacts with the sample. In the standard treatment, this is modelled as a unitary phase shift and amplitude attenuation: âi → t âi + r v̂, where t is the transmission amplitude, r the corresponding loss amplitude, and v̂ a vacuum mode representing the loss channel. The SU(1,1) group structure of the interferometer has been extensively analysed (Yurke et al., 1986; Chekhova and Ou, 2016; Ou and Li, 2020).
In the trapped-ion implementation of Metzner et al. (2024), two motional modes of a single 25Mg+ ion — labelled a (high-frequency) and b (low-frequency) — serve as the two bosonic modes. Parametric driving of the trap electrodes at the sum frequency ωa + ωb implements the two-mode squeezing Hamiltonian
HTMS = iℏg ( ↠b̂† − â b̂ ),where g is the coupling rate set by the parametric drive amplitude. This is algebraically identical to HSPDC under the identification â ↔ âs and b̂ ↔ âi.
The interferometric protocol is:
The interference fringe in the spin readout probability as a function of φ encodes the phase accumulated by the "idler" mode b — which is never directly measured.
The conceptual innovation proposed here is to replace step 3 — the coherent phase shift — with an engineered decoherence channel ε acting on mode b while leaving mode a unperturbed. This channel plays the role of the "sample" in QIUP.
Scope of the analogy. The trapped-ion experiment does not reproduce the microscopic physics of mid-infrared molecular absorption. Rather, it reproduces the quantum-channel structure experienced by the idler mode: the mapping from the channel's parameters (loss rate, spectral density, memory time, temperature) to the interferometric observables (visibility, phase shift, photon statistics of mode a). A large part of the QIUP sensing performance can be organised at this channel level, even though platform-specific details (spatial multimodeness, pump coherence, spectral correlations, detector model) still matter for quantitative translation to the optical case.
Native decoherence. The trapped-ion motional modes experience native anomalous heating and background dephasing. The total channel is therefore the composition εtotal = εnative ∘ εengineered. Separating the engineered contribution from the native background is an essential part of the experimental protocol, achievable by independent characterisation of εnative prior to applying the engineered channel.
| QIUP optical system | Trapped-ion analogue | Status |
|---|---|---|
| SPDC crystal (two-mode squeezing, SU(1,1)) | Parametric drive on trap electrodes | Demonstrated (Metzner et al., 2024) |
| Signal photon (visible, detected) | Motional mode a (read out via spin) | Demonstrated |
| Idler photon (mid-IR, undetected) | Motional mode b (not directly measured) | Demonstrated |
| Sample (absorption, phase, dispersion) | Engineered decoherence channel ε on mode b | Proposed |
| Native background | Anomalous heating, background dephasing | Characterised; must be subtracted |
| Interferometer closure | Second parametric drive pulse | Demonstrated |
| Visibility readout | Spin fluorescence measurement | Demonstrated |
| Structured environment | Coupling to auxiliary mode or shaped noise | Markovian demonstrated (Myatt, 2000); structured baths require further engineering |
The natural language for comparing performance across platforms is quantum channel estimation. The experiment amounts to preparing a bipartite probe state ρab (via two-mode squeezing), applying ida ⊗ εθ (engineered channel on mode b only), and measuring the output state to estimate the channel parameter θ. The channel process matrix χε (or equivalently the Choi state) provides a complete description of ε. The quantum Fisher information framework (Giovannetti et al., 2006; Demkowicz-Dobrzański et al., 2012; Escher et al., 2011; Fujiwara, 2001) yields fundamental precision bounds and determines whether entangled probes outperform classical probes for a given channel class.
Single-mode limitation. The trapped-ion system simulates single-mode bosonic channels. Biological tissue presents a multimode environment. The single-mode results can be extended via theoretical modelling, but direct multimode simulation would require multi-ion crystals — a significant but not insurmountable experimental extension.
A systematic literature search suggests that the following connections have not been exploited:
This work proposes to bridge these gaps by combining existing experimental capabilities in a new configuration, guided by quantum channel estimation theory.
Two-mode SU(1,1) interferometer. Demonstrated by Metzner et al. (2024) using 25Mg+ with parametric drives from oscillating electrode potentials.
Engineered decoherence. Myatt et al. (2000) demonstrated controllable amplitude damping and phase damping on trapped-ion motional states. The decoherence can be applied selectively to one motional mode by frequency-targeting.
Structured spectral density. Beyond Markovian damping, structured environments can be engineered by coupling mode b to an auxiliary motional mode (discrete-mode environment) or by applying noise with a controlled spectral shape to trap electrodes (continuous-mode environment). The Myatt (2000) demonstrations focused on Markovian reservoirs; engineering structured non-Markovian spectral densities represents an additional technical challenge. The classes realisable in the near term are likely limited to: (a) Markovian amplitude and phase damping with variable rate; (b) discrete-mode environments via auxiliary motional modes; (c) band-limited noise of moderate spectral complexity.
The primary observable is the spin-state probability P(↑ | φ) as a function of the interferometer phase φ. The interference fringe visibility is
V = (Pmax − Pmin) / (Pmax + Pmin),and the fringe phase shift Δφ is the displacement relative to the zero-decoherence reference. The key experimental programme is the measurement of V(θ) and Δφ(θ) as functions of the channel parameters θ.
Visibility data provide a lower-bound route to the classical Fisher information associated with the chosen measurement scheme; full optimisation requires the complete output statistics, not visibility alone. Full channel tomography — reconstructing χε from multiple measurement bases — is achievable via standard motional-state tomography. Crucially, the channel can be independently characterised without the interferometric protocol, providing a closed-loop validation.
No new major hardware is required. Conservative estimate for first results (Markovian channels): 12–18 months. Extension to structured non-Markovian channels would require additional development.
The interferometric protocol prepares an entangled probe state ρab, applies the channel ida ⊗ εθ, and measures the output to estimate θ. The quantum Cramér–Rao bound provides the fundamental precision limit:
Δθ ≥ 1 / √(N · F(θ)),where N is the number of repetitions and F(θ) is the quantum Fisher information. The central question: does the entangled SU(1,1) probe yield larger F(θ) than the best classical probe, and for which channel classes?
In the Markovian (Lindblad) limit, the reduced dynamics of mode b coupled to a bath produce exponential visibility decay:
V(τ) = V0 exp(−κτ/2),the direct analogue of Beer–Lambert absorption. For a structured environment, the dynamics acquire a memory kernel, and the visibility may exhibit non-exponential decay, oscillations, or revivals — signatures of the environmental spectral density. Process-tensor methods (Pollock et al., 2018) provide a numerically exact non-perturbative treatment.
The relevant dimensionless parameters governing translation between platforms are: the ratio of environmental correlation time to interaction time (ωcτ), the loss parameter (κτ), and the thermal parameter (n̄). The mapping can be made exact for matched channel families and probe-measurement models, expressed through these common dimensionless control parameters. Translation to the multimode optical case requires additional theoretical modelling beyond the single-mode framework validated by the trapped-ion data.
The benchmark for label-free mid-IR biological imaging is mid-infrared optoacoustic microscopy (MiROM), demonstrated by Pleitez et al. (2020), achieving ~5 µm resolution for lipid, protein, and carbohydrate imaging in living cells. The emerging quantum-enhanced programme (Kviatkovsky et al., 2020; Placke et al., 2023; León-Torres et al., 2025) aims to replace the classical source-and-detector chain with SPDC and visible detection.
The trapped-ion experiment proposed here does not itself perform biological imaging. It provides a controlled testbed for theoretical models of how structured decoherence channels affect nonlinear interferometric sensitivity. These models, once validated where every parameter is independently known, can then inform QIUP system design — with appropriate caution regarding the translation from single-mode to multimode channels.
This concept note is published as an open-science contribution. It is intended to seed collaboration across four communities:
The concept is offered freely. Collaborative arrangements and authorship will be discussed openly with any contributing partners. The goal is to advance the science.
Contact: Ulrich Warring — ulrich.warring@physik.uni-freiburg.de
Repository: to be announced
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This appendix provides brief orientation for readers approaching from one of the four fields connected by this concept note. It is not a comprehensive tutorial; references are provided for further reading.
Quantum imaging with undetected photons exploits a counterintuitive quantum-optical effect: information about a sample can be obtained by detecting a photon that never interacted with that sample. The technique uses SPDC in a nonlinear crystal to generate pairs of photons — signal and idler — that are quantum-correlated. The idler illuminates the sample; the signal is detected. If the nonlinear interaction is arranged so that the origin of each pair is fundamentally indistinguishable, the signal photons exhibit interference fringes whose visibility and phase encode the sample's absorption and refractive index at the idler wavelength.
The practical appeal: the idler can be in the mid-infrared (3–11 µm), where molecules have chemically specific vibrational fingerprints, while the signal is in the visible, where cheap silicon cameras are available. For a comprehensive introduction see Chekhova and Ou (2016) and Ou and Li (2020).
A conventional Mach–Zehnder interferometer uses beamsplitters (SU(2) transformations) to split and recombine two modes. An SU(1,1) interferometer replaces the beamsplitters with parametric amplifiers — devices that create or annihilate pairs of excitations simultaneously (hyperbolic rotations, or two-mode squeezing). The QIUP nonlinear interferometer is an SU(1,1) interferometer: the SPDC crystal generates correlated signal–idler pairs on each pass, and interference arises from the indistinguishability of pair-creation events. SU(1,1) interferometers can achieve phase sensitivities below the standard quantum limit, as demonstrated in both optics (Yurke et al., 1986) and trapped-ion motional modes (Metzner et al., 2024).
A quantum channel is the most general physically allowed transformation of a quantum state — encompassing unitary evolution, decoherence, loss, and noise. Formally, it is a completely positive, trace-preserving (CPTP) linear map. In this note, the "sample" in QIUP is modelled as a quantum channel acting on the idler mode. In the simplest case it is a beamsplitter channel (loss + vacuum noise). For complex media it may exhibit memory (non-Markovianity) and spectral structure. The process matrix provides a complete mathematical description; the goal of quantum channel estimation is to infer its parameters from measurements on the output state.
A single laser-cooled ion in an electromagnetic trap provides two types of quantum degrees of freedom: internal electronic states (effective spin-1/2) and quantised motional modes (harmonic oscillators at MHz frequencies). These can be coherently coupled, and the spin can be measured with near-unit efficiency. The motional modes can be prepared in the ground state, subjected to displacement, squeezing, and two-mode squeezing, and read out via the spin. Critically, they can also be coupled to engineered reservoirs — controllable noise sources producing specific decoherence dynamics (Myatt et al., 2000). The combination of coherent control and engineered decoherence makes trapped ions uniquely suited for studying how environmental interactions degrade quantum correlations and interferometric sensitivity.
| Version | Date | Changes |
|---|---|---|
| 0.1.0 | Mar 2026 | Initial concept note |
| 0.2.0 | Mar 2026 | Critical physics review: channel-level scoping, process-matrix framing, visibility definition, claim calibration |
| 0.2.1 | Mar 2026 | Final calibration: native decoherence, single-mode limitation, realisable channel classes, Fisher-information qualification, platform-translation narrowing |
| 0.3.0 | Mar 2026 | Added "About this note" and Appendix A (tutorial background); no changes to core scientific content |