Distinguishability and "which pathway" information in multidimensional interferometric spectroscopy with a single entangled photon-pair.

Correlated photons inspire abundance of metrology-related platforms, which benefit from quantum (anti-) correlations and outperform their classical counterparts. While these mainly focus on entanglement, the role of photon exchange phase and degree of distinguishability has not been widely used in quantum applications. Using an interferometric setup, we theoretically show that, when a two-photon wave function is coupled to matter, it is encoded with “which pathway?” information even at low-degree of entanglement. An interferometric protocol, which enables phase-sensitive discrimination between microscopic interaction histories (pathways), is developed. We find that quantum light interferometry facilitates utterly different set of time delay variables, which are unbound by uncertainty to the inverse bandwidth of the wave packet. We illustrate our findings on an exciton model system and demonstrate how to probe intraband dephasing in the time domain without temporally resolved detection. The unusual scaling of multiphoton coincidence signals with the applied pump intensity is discussed.

INTRODUCTION

Interferometry introduces myriad of novel platforms to spectroscopy, aiming at revealing quantum information encoded in the wave function of multiple photons (–). One intriguing aspect of many-body quantum dynamics is the exchange statistics of indistinguishable particles, whereby the wave function acquires a phase upon the exchange of two particles. This affects their dynamics and is detectable via unique interference patterns in the correlations of two (or more) particles. Coincidence counting of photons (), current correlations of electrons (), and fractional charges (quantum Hall quasi-particles) (, ) are notable examples. In quantum electrodynamics, light-matter coupling can be represented as the sum of all possible interaction histories (pathways). These pathways differ by the temporal order of events; thus, multiphoton nonlinear processes are potentially imprinted with their relative phases. Although the exchange phase of photons is fixed and solely determined by their bosonic nature, it can be effectively manipulated using a well-established interferometric setup for a pair of entangled photons ().

Here, we develop an exchange phase cycling scheme that scans through different values of such a phase. We then demonstrate the capacity of the multiphoton wave function to encode and decipher matter information inaccessible otherwise. Matter information gain is physically manifested in a reduced number of light-matter configurations, the ability to switch between them, and a new set of time delay variables with unique characteristics. We study the multidimensional spectral information generated by coupling an entangled photon pair to matter, via combination of interferometers, as depicted in Fig. 1. Our scheme includes two interferometric stages: state preparation interferometer, followed by a “reading” setup for the quantum state encoded by light-matter information exchange. This results in several notable differences in comparison to the familiar semiclassical nonlinear optics. (i) While semiclassical techniques scan time delays between pulses, quantum interferometric setups introduce new type of time delay variables that are not conjugate to the wave packet bandwidth. (ii) Interferometric wave mixing (WM) of quantum light generates matter pathways unavailable classically due to the simultaneous detection of multiple photon propagation paths generated at different times (). (iii) It allows the separation of pathways into groups while delivering phase-sensitive read of each process, unmatched by classical light. (iv) Coincidence detection obeys unique scaling relations between the applied intensity I, the light-sample coupling, and the detected signal. This allows to avoid damaging disturbance to the sample and eliminates unwanted signal contributions (background). (v) Coincidence counting singles out the two-photon subspace from the total signal, therefore, restricting the number of possible microscopic pathways. Consequently, two-photon signals are sensitive to collective excitations (harmonic and anharmonic), in contrast to single-photon counting that are generated by matter anharmonicities (–).

Fig. 1.

Multidimensional interferometric spectroscopy setup.

(A) Entangled photons with continuously variable exchange phase θ described by Uθ interact with a sample. The pair is separated by a polarization beam splitter (PBS), combined into a Hong-Ou-Mandel (HOM) interferometer, and measured in coincidence following the detection protocol U. The HOM setup introduces a relative temporal shift to the photonic pathways, obtained by shifting the BS position . (B) The θ symmetrization denoted by Uθ is obtained using a modified Michelson interferometer setup. The exchange phase θ grants a varying control over the photon pair degree of distinguishability. (C) Two detection protocols: (I) denotes the total photon counting signal (generating a 1/0 event registry), and (II) denotes the frequency-resolved counting (generating frequency-dependent 1/0 list). M1, mirror 1; M2, mirror 2.

We propose an exchange phase cycling protocol that projects the information encoded in the multidimensional signals onto lower-dimension data, thereby revealing phase-dependent matter correlation functions. The interferometric time delays can probe the dephasing dynamics of the sample, without resorting to time-resolved detection. The latter is determined by the interferometer optical path difference and is not conjugate to the frequency measurement, paving the way to a high joint time-frequency resolution, beyond the Fourier limitation.

RESULTS

The setup

The interferometric spectroscopy setup depicted in Fig. 1 includes preparation and detection stages. Both play an essential role in acquisition of the nonlinear signal generated by the sample, through their control parameters. In the preparation process, a modified Michelson interferometer creates a photon pair with tunable degree of distinguishability, using the exchange phase engineering described below. At the detection stage, Hong-Ou-Mandel (HOM) interference is sensitive to the post-coupling degree of distinguishability. Symmetric, antisymmetric, and asymmetric optical pathways, each carrying valuable matter information corresponding to different light-matter coupling histories, enable temporal reconstruction. The control parameters available using this setup are summarized at the end of this section.

Preparation

Following (), a pump beam is directed into a modified Michelson interferometer using a dichroic mirror, as shown in Fig. 1B. The beam passes through a BBO (β-barium borate) crystal, which generates a pair of orthogonally (linearly) polarized entangled photons denoted as signal and idler. The pair is fully characterized by the joint spectral amplitude (JSA) ϕ(ω, ω). In our calculations, we have used a JSA given by ϕ(ω, ω) = A(ω + ω)φ(ω, ω), where is a symmetric pump Gaussian envelope with bandwidth σ, centered around the pump frequency ω (). The phase-matching factor breaks the frequency exchange symmetry. Here, are the signal and idler central frequencies, and , where L is the nonlinear crystal length and v is the inverse group velocity at the relevant central frequency . The JSA can exhibit strong exchange asymmetry, imprinting the horizontal and the vertical polarization quantum channels with distinct spectral signatures. On one arm, the pump is separated using a dichroic mirror (DM), and its optical path is adjusted equal to the signal and idler (piezo crystal PZT). On the other arm, the polarizations are swapped by passing twice through a plate. In the second passing through the BBO crystal, because of the exchanged polarizations, the spectral profile is flipped from ϕ(ω, ω) to ϕ(ω, ω) with a relative phase denoted as θ. The transmitted part of the beam from the DM is finally given by the θ-symmetrized amplitude, resulting in the two-photon wave functionwhere ∣vac⟩ is the noninteracting vacuum. Broadband pumping of a type II parametric down converter is known to generate photon pairs with strong spectral distinguishing information (see section S1) (, ). We shall show that the asymmetric part of the JSA to (ω, ω) exchange plays a substantial role in recovering the real part of the matter correlation function for microscopic matter processes that are symmetric to exchange. Figure 2A depicts the nonsymmetrized JSA in frequency domain. Figure 2 (B to D) then present , and π used in the cycling protocols presented below. The JSA was computed with ω = 4 eV for a 4-mm nonlinear crystal with T = 6.1 fs and T = 230 fs. For each value of θ, Fig. 2 (A to D) also presents the Schmidt decomposition in a bar plot of the respective panels. The latter can be interpreted as the probability for the arrival of the nth mode; it provides a measure for the dimensionality of the two-photon Hilbert space and the degree of entanglement (). Fewer participating modes conform to lower degree of entanglement, while many modes to high degree of entanglement (see Eq. 13 in Materials and Methods). For example, a single mode yields a separable pair, for which measurement of one does not affect the (pure) state of the other. Evidently, the values scanned in Fig. 2 show a relatively low number of participating modes, hence low degree of entanglement. Note that, for θ = π, Schmidt modes appear in pairs, as reported in (, ).

Fig. 2.

Joint spectral amplitude.

The JSA in Eq. 1b is presented for selection of θ values used on the exciton model system. (A) The nonsymmetrized amplitude two-photon amplitude in frequency domain, with Schmidt number κ = 1.01. Symmetrized amplitude using (B) with , (C) θ = 0 with κ0 = 1.58, and (D) antisymmetric θ = π with κπ = 2.87. Each bar plots signifies the Schmidt decomposition resulting from the corresponding amplitude.

Detection and coupling

The coincidence signal is obtained by a two-photon expectation value at the detection plane. Since a detection event involves an annihilation of a photon by the detector, we begin our analysis from the detection stage. We then describe the light-matter interaction stage using the joint photon-sample density operator. Interferometric setups can be described by a linear transformation of the field modes between the detection and interaction stages (). Two coincidence detection protocols are considered as depicted in Fig. 1. The corresponding coincidence observables and reflect two-photon populations in their respective (time frequency) domain. Both setups contain several independent control parameters that manipulate the excitation and detection mechanisms. Detection-related control parameters include the HOM relative delay T, and, when resolved , the detected frequencies (ω, ω). The excitation-related control parameters are the exchange phase θ, pump bandwidth σ, pump central frequency ω, and the pair central frequencies . The first three parameters (θ, σ, ω) can be scanned continuously in a single setup. The entangled pair central frequencies are additional control parameters, which require a special extended phase-matching preparation ().

Total coincidence signal –

The total coincidence count is described by a photon annihilation of two modes at the two detectors. When the signal is not resolved temporally, the annihilation time is integrated over and the corresponding observable is given by the operatorHere, E and E are electric field superoperators, corresponding to Hilbert space operators, that act from the right Eρ ≡ ρE and left Eρ ≡ Eρ of the density operator. The Hilbert space polarization-dependent field operator is given by , where is the σ-polarization vector, Ω is the quantization volume (c = 1), and are annihilation (creation) operators obeying the bosonic commutation relations . represents the photon detection as annihilation of two modes from the left and right of the density operator, projecting the two-photon subspace in the measurement. The coincidence signal is obtained by taking the expectation value of Eq. 2 in the interaction picturewhere Λ = {ω, σ, θ, T} represents the set of control parameters available in this measurement protocol. denotes the trace with respect to the initial state of the joint density operator ρ0 = ρ(t0), and 𝒯 is the time-ordering superoperator. The light-matter coupling is described by the interaction superoperator corresponding to the commutator of the Hilbert space interaction Hamiltonian and the density operator, Hint, −ρ ≡ [Hint, ρ]. We adopt the the multipolar interaction Hamiltonian in the rotating wave approximation (RWA) Hint = † · + H. c., where is the dipole-lowering operator. Within the RWA, emission (absorption) of a photon is associated with energy decrease (increase) of the sample. The total coincidence counting signal 𝒞 is obtained by integration over photon arrival times, unresolved due to the absence of temporal gating.

The frequency-resolved coincidence counting –

The frequency-resolved signal is obtained by a double annihilation of photons and is defined by the corresponding frequency-domain superoperators. These are related to time-domain counterparts using Fourier transform . Similarly, the observable is given byand the coincidence signal is obtained by the respective expectation valueHere, Λ = {ω, ω, ω, σ, θ, T,} are the corresponding control parameters.

When implementing Eqs. 3 and 5, it is crucial to note that the light-matter coupling takes place at different stages of the interferometer. Consequently, they are defined in different basis sets and require the linear transformation, as described below.

Interferometric photon basis-transformation

Because of the HOM interferometer, the optical modes involved in the light-matter coupling and the detected modes are given in different basis sets. The transformation (Jordan-Schwinger map) can be represented using an (2) rotation in the frequency domain (–), resulting in the input-output relationused in the derivation of the signal. Here, t and r are the transmission and reflection coefficients obeying ∣t∣2 + ∣r∣2 = 1, and T is the relative time delay. For the 50:50 beam-splitter (BS) considered here, . The field in vector notation is given by (, ω) = (E(, ω), E(, ω)). Under the HOM rotation, the detected field is given by . In the following, we express all field operators in the interaction domain basis ∣interaction ≡ (, ω), which requires the inverse rotation of the observable in Eq. 2 (see the Supplementary Materials for detailed derivation) ().

Application to an exciton model system

We consider the three-level model system depicted in Fig. 3A with energy manifolds (g, e, f) corresponding to the ground, singly, and doubly excited states, respectively. To resemble four WM (4-WM), we expand Eqs. 3 and 5 to fourth-order light-matter coupling Hint. In Fig. 3B, we show two groups of light-matter pathways: two-photon resonance (TPR) represented by diagram D1 and Raman processes (RP) (diagrams D2 and D3). The TPR pathways begin with a two-photon absorption event ∣{}⟩ → ∣{}⟩ → ∣{}⟩, while RPs solely represent ∣{}⟩ ↔ ∣{}⟩ transitions and do not involve the f manifold. The diagrams describe the light-matter interaction on a closed-time (Keldysh) contour in which the ket (bra) evolve forward (backward) in time (, ). The sample is taken to be initially in the ground-state ρμ(−∞) = ∣g1⟩⟨g1∣. Each inward (outward) arrow denotes interaction-induced excitation (de-excitation) of the sample, and the final state of the sample is stated at the top of each diagram. Note that reflection (interchanging the bra-ket arrows) of all D results in complex conjugation. We assume an initially uncorrelated field-matter density operator ρ(−∞) = ρφ(−∞) ⊗ ρμ(−∞), where ρφ(−∞) = ∣Ψθ⟩⟨Ψθ∣ is the two-photon initial pure state.

Fig. 3.

Exciton model system and coupling pathways.

(A) Three-level model system composed of ground ∣{}⟩, single ∣{}⟩, and doubly excited ∣{}⟩ manifolds. The red arrows pointing interchangeably up and down correspond to Raman pathways. The purple arrows arranged in two consecutive absorption followed by two emissions correspond to TPRs. (B) Diagrammatic representation of the contributing microscopic light-matter processes. Straight arrows represent an interaction corresponding to both fields. Curved arrows represent the detection (annihilation) process. Diagram D1 involves double excitations (reaching the f manifold) and referred to as TPR. D2 and D3 describe single excitations (e manifold) and denoted RP. The two distinct subgroups of processes are depicted in (A).

Below, we present the signals obtained using the two coincidence-counting detection protocols presented in Fig. 1, . The probability of observing each pathway (D) varies with the number of detected photons and is therefore sensitive to the final state of the sample, in contrast to signals obtained with a classical source. When single photons are detected, all processes contribute regardless to the number of generated photons (, ).

Total coincidence

The celebrated HOM dip is a minimum in the total coincidence count of photon pairs, obtained upon changing the degree of their distinguishability by varying their relative path delay T (). The signal attains its minimal value for T = 0 and vanishes altogether when the pair is completely indistinguishable. The detection protocol manifests a HOM interference of the pair, posterior to the interaction with the sample. Matter energy fluctuations shape the interference pattern of the applied field and then recorded as the signal.

The coincidence signal eq. S7 is derived using diagrams D1, D2, and D3 for general pulse parameters and the corresponding symmetrization procedure (initial state) in section S1. Simplified expressions are obtained for a narrowband pump pulse with frequency-degenerate entangled pair, fixing . This is possible by adopting phase-matching condition φ(ω, ω), which is maximal for identical central frequencies of the entangled pair. We assume a narrowband pump of bandwidth (full width at half maximum) , where λ is the central frequency of the pump. The central frequency is scanned in the 0.1- to 3-eV regime attainable by pulse duration τ ≈ 0.1 to 10 ps for a transform-limited Gaussian pulse. Combined with the degenerate phase-matching factor, the signal is maximal for . Under these conditions, it is possible to isolate the contribution of diagram D3 and probe the intraband dephasing in real time. This measurement requires no time-resolved detection and is exclusively based on the HOM relative delay, which is not conjugate to any frequency variable. The total coincidence signal (eq. S7) then reduces towhere we have used the fully symmetric initial state for the field (θ = 0), selectively isolating D3. Here, is the Green function of the sample and its Fourier transform , introducing a phenomenological dephasing rate γ (ħ = 1). It is convenient to interpret Eq. 7 using the density matrix: the sample optically pumped to an excited state population and then de-excited back to the ground manifold, where the observable 1 − iG(T) is measured (see eq. S9 for the sum-over-states expression). The time domain Green’s function is evaluated at the ∣{}⟩ manifold, revealing its temporal dynamics at time T. Scanning ω and performing the Fourier transform of the signal with respect to the pair {ω, T}, we obtain the two-dimensional (2D) signal depicted in Fig. 4A from Eq. 7. Note that T and ω are nonconjugate variables and can be jointly resolved to arbitrary accuracy. The spectral-temporal information in Fig. 4A can be reduced to a single dimension (time or frequency) in multiple ways. One way to achieve that is by tracing uniformly ω, keeping the HOM delay Tas depicted in Fig. 4B. The exponential decay envelope of the intraband dephasing is clearly visible along the variation of T. Two dephasing rates are illustrated γ = 1 meV and 2 meV, as well as the fast decay of γ. The temporal resolution of the intraband dephasing is determined solely by the HOM delay (T) step size. The signal peaks notably when is resonant with the interband transitions ω within the frequency scanning range. This can be rationalized by the sum-over-states expression of Eq. 14.

Fig. 4.

Coincidence counting of the exciton system.

(A) The total coincidence-counting 2D signal ( detection protocol) with respect to the HOM delay T and the pump central frequency obtained from Eq. 7. The signal is obtained by a narrowband frequency–degenerate photon pair, corresponding to the intraband dephasing rate γ = 1 meV (4 ps). (B) Illustration of Eq. 8: The total coincidence-counting signal as a function of the HOM delay for two chosen dephasing rates γ = 1 meV and 2γ and the respective exponential envelopes integrated over the pump frequency. The interband spectrum obtained from Eq. 9 is displayed for two interband transition rates in (C) γ = 10 meV and (D) γ = 1 meV.

To extract spectral information, it is convenient to look at cross sections of a given time delay. We find maximal contrast in the spectral features for almost vanishing delays T = 0+. We denote this cross sectionin Fig. 4 (C and D). Figure 4C depicts the spectrum obtained for γ = 1 meV (4 ps) and more rapid interband dephasing γ = 10 meV (0.4 ps) in logarithmic scale. Figure 4B presents the same for equal interband and intraband dephasing γ = γ = 1 meV (4 ps). The lineshapes are broadened because of the properties of the sample. We thus infer that rapid γ decay is a limiting factor in the spectral resolution of the interband transitions in this detection scheme.

Frequency-resolved coincidence

A natural extension of the total coincidence counting is provided by frequency-resolved photon detection (protocol in Fig. 1). This adds two control parameters to the above signal, resulting in the following set of control parameters Λ = {ω, ω, θ, T, ω, σ}. Expansion of Eq. 5 to fourth order in Hint introduces an additional phase factor exp {iη} associated with the HOM delayed path trajectories, where η = (ω − ω)T (see section S3 for detailed derivation). There are many ways to display the resulting high-dimensional signal. We use the shorthand notation 𝒞θ(η) ≡ 𝒞[Λ], where η can be fixed for any pair of frequencies (ω, ω) using the delay variable T. It is convenient to introduce the auxiliary functions corresponding to the real and imaginary parts of the respective pathway contributions. Here, i = {TPR, RP} and Ξi encapsulate all microscopic processes that contribute to each pathway (see section S3 for the full expressions). The choice (θ, η) = (0, π) giveswhere both groups of pathway are observed, as shown in Fig. 5A. In our calculations, we have used an entangled pair generated by a broadband pump with σ = 0.9 eV (≈1-fs pulse). The central frequency ω is scanned in the range of 2 to 5 eV, and all dephasing rates are identical γ = 5 meV. To quantify the degree of entanglement generated by the two-photon source, we compute the Schmidt number κθ ≡ [∑2(n∣θ)]−1 (see Materials and Methods). The Schmidt number for the above parameters is κ0 ≈ 2.7, using nonlinear crystal of length L = 0.4 mm. Figure 5A depicts all contributing pathways, where we observe distinct TPR features located along the diagonal lines in which ω + ω = ω such that ω = ω ω = ω. The RP pathways are observed along the lines corresponding to the transitions ω. Similar results are obtained with doubled bandwidth (σ) corresponding to κ0 ≈ 1.7. The coincidence signal vanishes for (θ, η) = (0,0) as expected for indistinguishable photons. For presentation purposes, all images in Fig. 5 are displayed by sweeping and summing the pump central frequency ω. In movies S1 and S2 (see Materials and Methods for more information), this protocol is shown to maintain the desired spectral information.

Fig. 5.

Frequency resolution of the exciton signal and exchange-phase cycling.

(A) 2D spectra obtained from Eq. 10 using ultrafast pump with σ = 0.9 meV and scanning ω in the range of 2 to 5 eV on logarithmic scale. The exchange-phase cycling protocol shows each of the components selectively. The imaginary (B) and real (C) part of the TPR pathway given by the cycling protocols in Eqs. 11a and 11b, respectively. The dashed diagonal lines follow the doubly excited transition ω + ω = ω. The imaginary (D) and real (E) part of the RP pathway following the cycling protocols in Eqs. 11c and 11d. The dashed lines denote transitions between the first excited and ground manifolds ω.

Exchange-phase cycling

Pathway selectivity can help in studying the molecular dynamics in greater detail. Frequency-resolved detection depicted in Fig. 1 achieves just that: isolating the real and imaginary part of various diagrams exclusively.

We propose exchange-phase cycling protocols, whereby several signals obtained with different control parameters are combined to selectively probe desired pathways. These exploit the Michelson and the HOM interferometers to manipulate the effective exchange phases before and after the interaction, respectively (θ, η). The Michelson interferometer (Fig. 1B) imprints any permutation of the initial photon pair with a relative phase factor. The HOM detection interferometer introduces path-related phase factor to the detected photons. Certain combinations of (θ, η) are useful for reconstructing the real and imaginary parts of the signals individually (see section S3 for derivation and final expressions). It is useful to examine the Fourier transform of the coincidence signal 𝒟θ = ∫ dT e−𝒞θ(η) (presenting 2D spectral map along the lines Ω = ω − ω). Combinations of 𝒞θ(η) and 𝒟θ allow the phase-sensitive reconstruction of the TPR and RP processes.

There are multiple choices of linearly dependent cycling protocols to achieve path selectivity. Here, we display one cycling protocol resulting in the 2D spectra presented in Fig. 5 (A to E). The information displayed in Fig. 5 (B-E) depict the cycling protocol below corresponding to the real ℛ and imaginary ℐ parts of the respective contribution

In the following calculations, we have used an entangled pair generated by a pump with varying bandwidth: from a narrowband σ = 1.8 meV corresponding to ≈1-ps pulse to broadband σ = 0.9 eV using a ≈1-fs pulse. The central frequency ω is scanned in the 2- to 5-eV regime, and all dephasing rates are identical γ = 5 meV. The real part of the response is strongly dependent on the degree of asymmetry of the initial state wave function. The asymmetric part of the JSA becomes more pronounced in the broadband regime and is negligible for a narrowband pulse (, ). The signals corresponding to the protocols in Eqs. 11a to 11d were calculated using an ultrafast pump with σ = 0.9 eV and are depicted in Fig. 5 (B to E), respectively. The Schmidt number κθ (Eq. 13) for the above parameters and a L = 0.4 mm varies between 1 and 5 (depending on θ). Figure 5 (B and C) presents the TPR pathways, where we observe dispersive features () located along the diagonal lines in which ω + ω = ω such that ω = ω and ω = ω. Note that the transition ω = ω = 2 eV is not resolved in ℐTPR, yet it appears in ℛTPR; this stems from the antisymmetric nature of the JSA Φπ(ω, ω), which vanishes when ω = ω. While ℐTPR is symmetric to the exchange, ℛTPR is antisymmetric and thus fully resolved. Nonetheless, different choices of cycling protocols may resolve this transition (see Fig. 5A and section S5 for such example). Figure 5 (D and E) depicts the RP pathway, scanning the single exciton manifold from which all transitions ω are visible. Last, one can verify that the information in Fig. 5A displays a combination of Fig. 5 (B and D), excluding the degenerate transitions in which ω = ω.

Comparison with ordinary WM

Nonlinear spectroscopic signals are usually described semiclassically using a sequence of temporally separated bright classical pulses (containing many photons) that trigger matter dynamics and generate a single photon (). This photon is modulated by m light-matter interactions, resulting in an m + 1 order correlation function. Nonlinear m + 1 WM is given by the expectation value of the integrated change of the electric field intensity (). The intensity is a single-photon quantity, related to the electric field operator ℐ = †. Quantum sources can achieve superior signal-to-noise ratio scaling with the number of photons. This superior scaling is referred to as the Heisenberg limit in quantum metrology (–), enabling reduced radiation exposure for comparable measurement certainty. Particularly, the offer to improve the resolution of fragile samples, limited by radiation dose constraints (). One way to benefit from the quantum properties of the electromagnetic field is via direct coupling of quantum light to the sample (e.g., entangled photons and squeezed states). Sample stimuli using quantum sources have shown to yield remarkable control over population dynamics and pathway selection (, , ). Alternatively, as done here, one can probe quantum effects of the emitted radiation directly via multiple photon counting [e.g., antibunching () and superesolved imaging (–)].

Multiple photon detection yields higher-order WM signals, denoted (n + m)-WM in analogy to the semiclassical (m + 1)-WM. It involves the application of m fields and the detection of n-generated photons that give rise to n + m order correlation function. While (m + 1)-WM depend on several pathways, (n + m)-WM contain fewer terms and may not be written in the form of amplitude square. In our setup, coincidence counting eliminates the single-photon diagram depicted in Fig. 6 denoted S, since only one photon is populated in the final state. This yields substantial change compared to single-photon (intensity) signal. Semiclassical nonlinear WM signals are generated from nonlinearities in matter (, ). Thus, collective excitations in the f manifold with energies ϵ = ϵ + ϵ vanish. This stems from the fact that the last light-matter interaction in D1 (Fig. 3) propagates forward in time (applied to the ket), while, with S in Fig. 6, backward in time (applied to the bra). Therefore, for small dephasing rates, the forward and backward resonances inherit opposite relative sign, verified using the Sokhotski-Plemelj theorem , where pp denotes the principle part. Because of the elimination of S, the coincidence signal here is sensitive to collective excitations in TPR processes.

Fig. 6.

TPR process missing in the coincidence signal.

(A) A single-photon TPR process labeled S, which involve double excitations (f manifold). The joint field matter is specified after each interaction event on the diagram. The highlighted (orange) line corresponds to the backward propagator responsible for the elimination of the (harmonic) collective excitations in single-photon nonlinear signals. (B) Schematic representation of the process.

(n + m)-WM also gives rise to different scaling with the pulse intensity. For example, double excitation signals induced by entangled pairs are known to scale linearly (rather than quadratically) with the pump intensity ∝Ip (–). This permits the study of doubly excited manifold with smaller probability of ground-state bleaching, thus potentially reduce sample damage. While single-photon detection events scale linearly with the pump intensity ∝Ip, the two-photon signal scales quadratically . This allows using lower intensities per desired detection gain, improving signal-to-noise ratio (–). This principle can be extended to n photon population detection in a straightforward manner.

DISCUSSION

The measurement of a specific number of photons using coincidence detection narrows down the observed dynamical pathways of the observed process. Pathways that the sample ends up in a certain final state are sorted out. In contrast, WM signals obtained with classical light probe a coherence and are not sensitive to the final state of the sample. Specifically, two-photon coincidence of a pair, which is weakly coupled to a sample, generates a unique (2 + 2) WM that probes a limited number of pathways. Similar to 4-WM, the signal considered here depends on a four-point dipole correlation function of matter. The difference is that the elaborate detection does not single out one field, as in semiclassical 4-WM signals, but two fields are detected. We can thus view the process as a generalized 4-WM. To avoid confusion, we simply refer to it as (2 + 2) WM, which specifies the number of applied and detected fields.

Our first detection scheme records the total photon coincidence count without any spectral and temporal resolutions. The two-natural control parameters of the setup depicted in Fig. 1 are the HOM delay T and the central frequency of the pair-generating pump ω. A remarkable effect occurs with the application of a degenerate photon pair [extended phase-matching condition (, )] with a symmetric narrowband JSA (θ = 0), such that both photons are sharply distributed around half the pump frequency . The TPR pathways are then eliminated, and only a single RP pathway (D3; Fig. 3) survives. The following dynamics is then observed: (i) The density matrix of the sample is excited and then de-excited back to the initial band ∣{}⟩. (ii) The Green function of the system at time T is measured. Since T is controlled with high precision, the intraband dephasing dynamics can be reconstructed in real time as shown in Fig. 4 (A to D). This allows to characterize the system’s coupling to its environment. Small T expansion can reveal moments of the sample Hamiltonian ⟨H⟩ and is given in section S2. This provides a compelling direction for future study aligned with the efforts invested in studying decoherence and energy leaks in quantum technologies.

Our second detection protocol involves frequency-resolve coincidence counting. By cycling values of T and the entangled pair exchange phase θ, we can separate the TPR and RP pathways. Pathway-selection protocols typically generate destructive interference to suppress certain populations () or induce distinct scaling of each process with the intensity (). Here, pathways are selected by projecting the high-dimensional signal to process-specific data, without reducing the event probability of other processes. Moreover, by combining signals with different θ and T(ω − ω) (phase cycling), we can obtain the phase of four-point matter correlation function, as shown in Fig. 5 (B to E). Temporal reconstruction is then possible using a simple Fourier transform. The cycling protocols related to the real part of the correlator require an asymmetric JSA with respect to exchange, inherited from the ultrafast pump. This suggests that obtaining temporal behavior of the correlators necessitates an ultrashort generating pump, which is compatible with temporal resolution, although this is a frequency domain measurement. An ultrashort pump generates a spontaneous photon pair that carries an identity-revealing spectral information (). The resulting distinguishability renders a “which pathway?” information available. The relation between the JSA asymmetry and the temporal resolution upper-bound warrants a further study, since the temporal and spectral control parameters are not conjugate quantities. These effects are more pronounced at low entanglement values and steered by the variable effective exchange phase.

In addition to the above merits, matter-induced field nonlinearities may also improve the frequency resolution (, ). Correlation-based detection techniques (e.g., coincidence) can reveal such electromagnetic field nonlinearities. These correlations have signatures in matter-induced photon-photon coupling (), free-electrons coupling (), and thus carry matter information that is imprinted in the postinteraction counting statistics. Characterization of the reading process that can access this information is one of the central goals of interferometric spectroscopy.

MATERIALS AND METHODS

Schmidt number calculation

The Schmidt number is a measure of photon entanglement that depends on the single photon density matrix. It is based on the Schmidt decomposition of the JSA ()Here, {ψ, ϕ} are an eigenfunctions of the single-photon reduced density matrix and p(n∣θ) are their weights. To obtain this representation, we solve p(n∣θ)ψ(ω) = ∫ dω′ K1(ω, ω′)ψ(ω′) and p(n∣θ)ϕ(ω) = ∫ dk′ K2(ω, ω′)ϕ(ω′). These kernels are found from the reductions , tracing the second frequency variable, and . They can be interpreted as single-photon correlation functions. The Schmidt decomposition computed by a diagnolization of the discretized single-photon reduced density matrix, following the procedure in (). Here, p(n∣θ) can be interpreted as the nth mode probability using θ-symmetrized JSA. It provides a measure for the effective two-photon Hilbert space. To obtain the Schmidt spectrum and characterize the degree of entanglement, we discretize the kernels and numerically solve the integral eigenvalue equations. We have used a 900 by 900 grid for the discretized kernel and calculated separately for each θ. The Schmidt number (also known as the participation ratio) is obtained byit is a measure of the number of relevant Schmidt modes.

Intraband dephasing signal computation

Equation 7 reveals the intraband dephasing in the time domain by scanning the HOM relative delay T. We express the correlation functions using the dipole-lowering operator V = ∑ μ ∣j⟩⟨i∣ and its complex conjugate, where (i, j) label energy eigenstates of the exciton system corresponding to ∣g⟩, ∣e⟩, and ∣f⟩. The expectation value in Eq. 7 requires Green’s function of the sample in frequency domain G(ω) = (ω − H + iγ)−1, where H is the Hamiltonian of the sample. With these definitions, we obtain the coincidence counting for our model systemused in the calculations presented in Fig. 4. The short-lived excited states of the first excited manifold serve as a prefactor to the relaxation process of the ground-state manifold. The calculation of Eq. 14 in Fig. 4 was carried out on a discretized grid by scanning 104 frequency and 103 points corresponding to ω and T, respectively.

2D spectra calculations

The results for the frequency-resolved coincidence section were obtained by direct implementation of eqs. S22 to S25 in section S4. The JSA was discretized within the shown interval in Fig. 5 in a 200 × 200 × 200 × 200 corresponding to ω, ω, ω, and T for the implementation of the numerical integration. The signal shown in Fig. 5 is obtained by integration over ω. Movies S1 and S2 demonstrate the spectra as a function of {ω, ω}, scanning ω as a function of time (V1) and also integrating as a function of time (V1). The desired dispersive features are expressed also by summing ω, justifying the 2D presentation in Fig. 5. An ultrashort pump induces large exchange asymmetry in addition to its broad frequency range coverage.