Quantum FTIR: Undetected Photon Spectroscopy

• Quantum Fourier Transform Infrared Spectroscopy is a quantum-optical method that uses SPDC-generated correlated photons to probe mid-IR samples through Fourier analysis of interferograms.
• It circumvents mid-IR detection challenges by leveraging nonlinear interferometric techniques and induced coherence to transfer spectral information to silicon-based visible or near-IR detectors.
• Experimental implementations, ranging from low-gain to high-gain setups, demonstrate high photon-efficiency and low sample power operation, though trade-offs exist with SNR and bandwidth compared to conventional FTIR.

Quantum Fourier Transform Infrared Spectroscopy (QFTIR) denotes a class of Fourier-transform mid-infrared spectroscopic methods in which a sample is probed by mid-IR photons generated in spontaneous parametric down-conversion (SPDC), while detection is performed only on correlated visible or near-IR photons. In the demonstrated implementations, spectral information is recovered from a scanned interferogram by Fourier transformation, so QFTIR is a quantum-optical or nonlinear-optical realization of FTIR based on induced coherence and sensing with undetected photons, rather than the quantum Fourier transform algorithm of quantum computing (Lindner et al., 2019). The method has developed from visible-light-detected mid-IR transmission spectroscopy in the $3.2$–$3.9~\mu\mathrm{m}$ range (Lindner et al., 2019) to fingerprint-region complex spectroscopy at $8$–$10.5~\mu\mathrm{m}$ (Mukai et al., 2021), high-gain SU(1,1) implementations in the $3~\mu\mathrm{m}$ region (Hashimoto et al., 2024), open-path detection of volatile organic compounds in ambient air (Neves et al., 2024), and explicit benchmarking against commercial FTIR instrumentation (Gattinger et al., 17 Mar 2025).

1. Conceptual basis and scope

QFTIR addresses a standard limitation of mid-IR spectroscopy: the spectral region is chemically valuable because fundamental vibrational bands of many molecules lie there, but direct mid-IR sources and detectors are comparatively poor, often noisy, and frequently require thermoelectric or cryogenic cooling (Lindner et al., 2019). The central idea is therefore to interrogate the sample with the idler photon of a non-degenerate SPDC pair in the mid-IR while reading out the correlated signal photon in the visible or near-IR, where silicon-based detection is mature, low-noise, and can operate at room temperature (Gattinger et al., 17 Mar 2025).

Operationally, QFTIR is the quantum-optical analogue of classical FTIR. A nonlinear interferometer replaces the classical Michelson source-and-beamsplitter arrangement, but the measured quantity remains an interferogram as a function of optical delay, and the spectrum is reconstructed numerically by Fourier transformation. Several papers describe this as FTIR with undetected photons, or as sensing with undetected photons, because the sample-interacting mid-IR field is never directly detected (Hashimoto et al., 2024).

The term does not designate a single instrument architecture. Reported implementations include Michelson-type nonlinear interferometers based on a single crystal traversed twice (Lindner et al., 2019), folded two-pass interferometers based on induced coherence without induced emission (Gattinger et al., 17 Mar 2025), low-gain fingerprint-region systems using AgGaS$_2$ (Mukai et al., 2021), high-gain SU(1,1) interferometers using KTP-family crystals (Hashimoto et al., 2024), and open-path gas sensors using long-arm nonlinear Michelson geometries (Neves et al., 2024). What unifies them is the FTIR-style delay scan and Fourier reconstruction combined with nonlinear interferometric transfer of mid-IR sample information into a shorter-wavelength detection channel.

2. Nonlinear interferometry, induced coherence, and spectral encoding

The physical principle is interference between two indistinguishable SPDC alternatives. A coherent pump illuminates a nonlinear crystal twice, or equivalently two coherently related SPDC stages. If the alternatives “pair generated in first pass” and “pair generated in second pass” are mode-matched and indistinguishable, the detected signal field shows interference whose visibility and phase depend on what happened to the undetected idler between the two generation events (Lindner et al., 2019).

A compact expression used for the benchmarked QFTIR implementation is

$$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$

where $R_s$ is the normalized detected signal photon rate, $|T|$ is the absolute value of the sample transmission in the idler arm, $\Delta \phi$ is the phase shift between pump, signal, and idler, and $3.9~\mu\mathrm{m}$0 is the sample-induced phase delay (Gattinger et al., 17 Mar 2025). In this form, absorption in the undetected mid-IR arm reduces visibility, while sample dispersion shifts the interference phase. A closely related expression used in an earlier visible-light-detected implementation is

$3.9~\mu\mathrm{m}$1

with the phase-matching envelope $3.9~\mu\mathrm{m}$2, transmission factor $3.9~\mu\mathrm{m}$3, and relative phase $3.9~\mu\mathrm{m}$4 (Lindner et al., 2019).

All implementations rely on non-degenerate SPDC and energy conservation,

$3.9~\mu\mathrm{m}$5

with $3.9~\mu\mathrm{m}$6, $3.9~\mu\mathrm{m}$7, and $3.9~\mu\mathrm{m}$8 the pump, signal, and idler angular frequencies (Lindner et al., 2019). Phase matching is provided either by birefringent crystals such as AgGaS$3.9~\mu\mathrm{m}$9 in the fingerprint region (Mukai et al., 2021) or by quasi-phase-matched media such as PPLN, ppKTP, and apKTP in the $8$0 region (Lindner et al., 2019). In several systems, the sample is placed in the idler arm of a Michelson-type interferometer, so it affects only the probing mid-IR photon, but its effect is read out on the correlated visible or near-IR signal (Neves et al., 2024).

Architecturally, the scanned arm need not always be the idler arm. In one benchmarked implementation, the scanned path contains both pump and signal, and the authors state that scanning $8$1 is quantitatively equivalent to scanning the idler-arm path length (Gattinger et al., 17 Mar 2025). This suggests that QFTIR is better characterized by which relative phase is scanned than by a unique geometric layout.

3. Fourier-transform acquisition and reconstruction

The defining measurement step is the acquisition of a delay-dependent interferogram and its Fourier transformation into a mid-IR spectrum. In the 2019 visible-light implementation, the idler-arm mirror is scanned, a 2D visible interference pattern is recorded at each delay step, and a DFT based on FFT is applied to the interferogram of each camera pixel individually; partial spectra are then summed to obtain the total spectrum (Lindner et al., 2019). Pixel-wise reconstruction is essential there because different pixels exhibit different phase offsets and effective path delays. The reported acquisition parameters were an image size of $8$2 pixels, $8$3 integration time per frame, $8$4 displacement step, $8$5 total mirror displacement, $8$6 total optical path difference, $8$7 frames, and $8$8 total acquisition time (Lindner et al., 2019).

In the fingerprint-region low-gain implementation, the measured quantity is the visible-photon count rate $8$9 as a function of optical path-length difference $10.5~\mu\mathrm{m}$0, referred to as a quantum interferogram. Its Fourier transform,

$10.5~\mu\mathrm{m}$1

yields the idler spectral response, and normalization by a reference measurement gives

$10.5~\mu\mathrm{m}$2

where $10.5~\mu\mathrm{m}$3 is the complex transmittance coefficient of the sample in the idler arm and $10.5~\mu\mathrm{m}$4 is a phenomenological dephasing parameter (Mukai et al., 2021). Because the sample is traversed twice in that Michelson geometry, the squared transmittance appears naturally.

The 2025 benchmarking study uses a single-pixel silicon photodetector rather than a camera. Its processing chain is explicitly: record interferograms, use a HeNe reference to track scanning, linearize and resample the interferograms, apply apodization (Blackman Harris), apply zero filling, perform Fourier transformation, and reconstruct single-channel spectra using standard FT algorithms (Gattinger et al., 17 Mar 2025). For absorbance measurements, transmission through air serves as the background. In the open-path VOC system, the finite scan window is modeled by an instrumental response

$10.5~\mu\mathrm{m}$5

and absorbance is retrieved from reference and sample spectra with additional differential absorption spectroscopy processing (Neves et al., 2024).

Despite these instrumental differences, the FTIR logic is shared. The maximum optical path difference sets the spectral resolution, finite scan range produces window-function artifacts, and reference/sample normalization removes the source spectrum and instrument response. What changes from one implementation to another is the detection modality—camera, APD, or silicon photodetector—and the degree of post-processing required for phase linearization, per-pixel dephasing compensation, or baseline suppression.

4. Experimental realizations and spectral coverage

The first clear FTIR-with-visible-light implementation combined a Michelson-type nonlinear interferometer, non-degenerate SPDC, and broadband non-collinear PPLN emission. It achieved continuous spectra with a spectral bandwidth of more than $10.5~\mu\mathrm{m}$6 and approximately $10.5~\mu\mathrm{m}$7 resolution in the $10.5~\mu\mathrm{m}$8–$10.5~\mu\mathrm{m}$9 region, and demonstrated a polypropylene transmission spectrum with an absorption band around $3~\mu\mathrm{m}$0 in good agreement with a commercial Bruker Vertex 80 FTIR reference (Lindner et al., 2019).

A major extension was the demonstration of QFTIR in the fingerprint region using an AGS crystal pumped at $3~\mu\mathrm{m}$1. By tuning the source, visible signal wavelengths from $3~\mu\mathrm{m}$2 to $3~\mu\mathrm{m}$3 were mapped to idler wavelengths from $3~\mu\mathrm{m}$4 to $3~\mu\mathrm{m}$5, and complex spectra were reconstructed in the $3~\mu\mathrm{m}$6–$3~\mu\mathrm{m}$7 range. The reported demonstrations were a $3~\mu\mathrm{m}$8 silicon wafer around $3~\mu\mathrm{m}$9 and a polytetrafluoroethylene sheet showing clear absorption due to symmetric and asymmetric stretching modes of C–F bonds near $_2$0 and $_2$1 (Mukai et al., 2021).

High-gain QFTIR replaced the low-gain induced-coherence regime with a high-parametric-gain SU(1,1) interferometer. Using pulsed $_2$2 pumping and KTP-family crystals, this approach demonstrated polymer spectroscopy in the $_2$3 region and reported three major advantages: a high photon number at the interferometer output, a considerably lower photon number at the sample, and improved interference contrast (Hashimoto et al., 2024). Spectral broadening was pursued by both temperature gradient in ppKTP and aperiodic poling in apKTP. The broadest reported FTIR bandwidth in that work was a $_2$4 dB width of $_2$5 and FWHM $_2$6 (Hashimoto et al., 2024).

Open-path QFTIR extended the method from sealed-cell reference-gas spectroscopy to ambient-air sensing. A nonlinear Michelson interferometer with $_2$7-long arms, a $_2$8 PPLN crystal at $_2$9, and a near-$$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$0 signal/$$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$1 idler pair measured a broad $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$2 mid-IR band in the $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$3–$$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$4 region (Neves et al., 2024). The system identified acetone, methanol, and ethanol vapors in ambient air, including mixed-vapor experiments and overnight time evolution during evaporation. Reported average-concentration detection limits were $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$5 for acetone, $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$6 for methanol, and $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$7 for ethanol over the $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$8 path (Neves et al., 2024).

The most systematic evaluation to date benchmarks a ppKTP-based QFTIR spectrometer operating from around $$R_{s} = \frac{1 + |T| \cos(\Delta \phi + \gamma)}{2},$$9 to $R_s$0, with detection around $R_s$1, against commercial FTIR systems (Gattinger et al., 17 Mar 2025). Under matched resolution and integration conditions, it reproduces the absorbance spectra of $R_s$2 PET and $R_s$3 PP films qualitatively well, including thin-film interference in PET and polymer bands in the $R_s$4–$R_s$5 window (Gattinger et al., 17 Mar 2025).

5. Performance, advantages, and present limitations

The primary practical attraction of QFTIR is detector substitution. The sample is interrogated in the mid-IR, but detection occurs on silicon-compatible visible or near-IR hardware, avoiding direct use of noisy, costly, and frequently cooled mid-IR detectors (Gattinger et al., 17 Mar 2025). Several implementations also emphasize operation at extremely low mid-IR power. The benchmarked ppKTP system estimates a total mid-IR probing power of $R_s$6, or about $R_s$7 effective after accounting for signal coupling efficiency, while still recovering vibrational spectra over the $R_s$8–$R_s$9 region (Gattinger et al., 17 Mar 2025). The high-gain SU(1,1) implementation reports $|T|$0 detected signal power at the output for a ppKTP case with an estimated idler power at the sample of only $|T|$1 (Hashimoto et al., 2024).

Benchmarking against commercial FTIR reveals a characteristic trade-off. In absolute signal-to-noise ratio, the commercial FTIR remains superior: for the Bruker LUMOS comparison, the reported single-channel single-shot SNR is $|T|$2 for FTIR versus $|T|$3 for QFTIR, and with $|T|$4 averages $|T|$5 versus $|T|$6 (Gattinger et al., 17 Mar 2025). Yet the power-normalized figure of merit favors QFTIR strongly: $|T|$7 for QFTIR versus $|T|$8 for the benchmarked commercial FTIR, leading the authors to conclude that for the same optical power the two methods would differ by $|T|$9–$\Delta \phi$0 orders of magnitude in SNR (Gattinger et al., 17 Mar 2025). A different metric, reported for the fingerprint-region AGS system, states that the signal-to-noise ratio per unit spectral width and unit probe light intensity outperforms conventional FTIR by a factor of $\Delta \phi$1 (Mukai et al., 2021). These are not identical benchmarks, but both point toward photon-efficiency as a defining strength of QFTIR.

The limitations are equally explicit. Present systems often have narrower bandwidth than classical FTIR, because coverage is set by the SPDC gain profile, crystal transparency, phase matching, and collection geometry (Gattinger et al., 17 Mar 2025). Absolute SNR is still typically $\Delta \phi$2–$\Delta \phi$3 orders of magnitude lower than in state-of-the-art conventional instruments (Gattinger et al., 17 Mar 2025). Long-term stability is sensitive to drift, especially when single-mode coupling is used; in one Allan-Werle analysis, stable averaging in a single-reference mode was possible only up to about $\Delta \phi$4 measurements for QFTIR, compared with $\Delta \phi$5 for the commercial FTIR, while in consecutive referencing mode the QFTIR global minimum occurred at $\Delta \phi$6 measurements (Gattinger et al., 17 Mar 2025).

Scan time and alignment burden remain substantial. The 2019 camera-based system required $\Delta \phi$7 for a scan with $\Delta \phi$8 frames (Lindner et al., 2019). The open-path VOC system required averaging over $\Delta \phi$9 scans for the reported vapor measurements (Neves et al., 2024). The low-gain fingerprint implementation reported only $3.9~\mu\mathrm{m}$00 interferometric visibility and identified improved mode matching as a major route to higher SNR (Mukai et al., 2021). High-gain operation alleviates some flux limitations but introduces its own engineering issues, including bandwidth narrowing, sidebands, gray tracking, and relative intensity noise (Hashimoto et al., 2024).

6. Terminology, interpretive boundaries, and related approaches

The expression “Quantum Fourier Transform Infrared Spectroscopy” is frequently misunderstood. In the QFTIR literature, “quantum” refers to quantum-optical resources—SPDC-generated correlated or entangled photons, induced coherence without induced emission, and nonlinear interferometry—and “Fourier transform” refers to classical FTIR-style interferogram inversion (Lindner et al., 2019). It does not refer to the quantum Fourier transform algorithm used in quantum information processing. A separate line of work on quantum computing does use a Split Operator–Quantum Fourier Transform propagation scheme for time-dependent simulation of infrared spectra, but there the QFT is an internal basis-change primitive in a grid-based vibrational dynamics algorithm rather than the sensing mechanism of a laboratory spectrometer (Feng et al., 22 Mar 2026).

A second boundary concerns related quantum infrared spectroscopies that are not, strictly speaking, FTIR. Broadband induced-coherence spectroscopy in the fingerprint mid-IR region has been demonstrated by reading out spectral interference fringes directly in the near-IR, without scanning a delay and without Fourier inverting a time-domain interferogram (Paterova et al., 2022). That method is closely related in application space and physical principle, but it is spectral-domain quantum interferometry rather than QFTIR in the strict instrumental sense.

There is also an interpretive question about how “quantum” the observed interference is. Several papers explicitly frame the experiments in terms of induced coherence, entangled or correlated photon pairs, and low-gain SPDC, and one fingerprint-region implementation estimates an average number of generated pairs per biphoton correlation time of $3.9~\mu\mathrm{m}$01, far below unity (Mukai et al., 2021). At the same time, one overview notes that aspects of the interferometric behavior are also compatible with a classical nonlinear-interferometry description in the low-gain regime (Lindner et al., 2019). The most conservative formulation is therefore that QFTIR is quantum-optical in origin and operational principle, while some of its observable interference phenomenology admits classical wave descriptions of nonlinear interferometers.

Taken together, the literature defines QFTIR as a technically specific subset of undetected-photon spectroscopy: a delay-scanned, Fourier-reconstructed, nonlinear-interferometric method for retrieving mid-IR absorption and, in some implementations, phase information through visible or near-IR detection. Its demonstrated scope now includes the $3.9~\mu\mathrm{m}$02 vibrational region, the $3.9~\mu\mathrm{m}$03–$3.9~\mu\mathrm{m}$04 fingerprint region, polymer spectroscopy, ambient-air VOC sensing, and detailed performance benchmarking against commercial FTIR systems (Gattinger et al., 17 Mar 2025). The present record is therefore not one of wholesale replacement of conventional FTIR, but of a distinct spectroscopic architecture whose strongest regime is low-flux, detector-limited, or deployment-constrained mid-IR sensing.