Quantum optical measurements with undetected photons through vacuum field indistinguishability

Quantum spectroscopy and imaging with undetected idler photons have been demonstrated by measuring one-photon interference between the corresponding entangled signal fields from two spontaneous parametric down conversion (SPDC) crystals. In this Report, we present a new quantum optical measurement scheme utilizing three SPDC crystals in a cascading arrangement; here, neither the detection of the idler photons which interact with materials of interest nor their conjugate signal photons which do not interact with the sample is required. The coherence of signal beams in a single photon W-type path-entangled state is induced and modulated by indistinguishabilities of the idler beams and crucially the quantum vacuum fields. As a result, the optical properties of materials or objects interacting with the idler beam from the first SPDC crystal can be measured by detecting second-order interference between the signal beams generated by the other two SPDC crystals further down the set-up. This gedankenexperiment illustrates the fundamental importance of vacuum fields in generating an optical tripartite entangled state and thus its crucial role in quantum optical measurements.


I. Fundamental Quantum Theory of Spontaneous Parametric Down Conversion (SPDC)
In this section, we present a brief description on the quantum theory of SPDC for the sake of completeness (see recent reviews, references S1 and S2, for detailed and complete theoretical descriptions). When an electric field ( , ) t Er propagates through a nonlinear and noncentrosymmetric medium, the i th component of the electric polarization of the optical medium is approximately given by (S1) where ( , ) j Et r is the j th component of the electric field vector and (1)  and (2)  are the first and second order susceptibility tensors, respectively. In Eq. (S1), all the higher-order contributions to the polarization are ignored. Throughout this Supplementary (S2) where D 0 ( ( , ) ( , )) tt   E r P r , B , and H are the displacement vector, the magnetic induction, and the magnetic field, respectively. Inserting the polarization in Eq. (S1) into Eq.
(S2), one can find the perturbation (nonlinear field-matter interaction) Hamiltonian, V  (S3) Here, ( , ) nl t Pr is the nonlinear component of the electric polarization. In the present work, we are interested in the nonlinear (second-order parametric down conversion) field-matter interaction in Eq. (S3) only.
3 To apply the quantization procedure, we expand the classical optical electric field, ( , ) ( , )  (S4) where is the field quantization volume, () F  is the transmission function for a filter placed in front of detector in a real experiment, and 2 ,0 ( , ) / (2 ( , )) n       k σ kk . Here, the linear refractive index of material is denoted as ( , ) n  k , k is the wave vector,  represents the two orthogonal components of the transverse plane wave, and ,  k is the mode amplitude. In Eq.
(S4), the interference filter that is fully characterized by the transmission function is taken into consideration to realistically describe experiments with frequency filters and detectors used to select specific frequency components of signal, idler, and detected fields. Quantization of the electric field is to replace the mode amplitude in Eq. (S4) with the corresponding photon annihilation operator so that the corresponding electric field amplitude becomes a field operator,  (S5) Inserting this electric field operator into Eq. (S3), one finds that the interaction Hamiltonian operator describing SPDC is given as   stands for Hermitian conjugate. The three fields involved in the SPDC process that is theoretically described by Eq. (S6) will be referred to as pump ( p ), signal ( s ), and idler (i ).
The quantum state at time t , when the initial state at time zero is assumed to be | (0)   , is then given by where the time-evolution operator is Typically, the pump field intensity is very weak and the field-matter interaction time is usually much shorter than the average time between down conversions. In this limit, the first two terms in the power series expansion of Eq. (S8) are sufficient enough to describe the quantum state at time t :  . In a real experiment, signal and/or idler beams are spatially filtered with dichroic mirrors and the corresponding detectors are placed far away from the SPDC NL crystal that can be considered as radiation source. In that case, the quantum state in Eq. (S10) can be simply written as In this limit, the electric field operator also takes a simple form (for js  and i ): Despite that a number of approximations were invoked to obtain both the quantum state (Eq. (S11) generated by a SPDC and the electric field operators (Eq. (S12)) of signal and idler photons, they properly describe most of the key features of temporal correlations between the quantum entangled signal and idler photons produced by SPDC. Often, such time correlation has been experimentally measured by detecting the coincidence counting rate that corresponds to the fourth-order (in the fields) correlation function or two-photon interference term: Often, since the pump beam is from a continuous wave laser, the pump spectrum is sufficiently narrow so that it can be approximately replaced by a Dirac delta function in frequency domain.
One can show that the coincidence counting rate is approximately proportional to the convolution of the Fourier transforms of the filter functions for signal and idler fields.
Furthermore, it depends on (i) the pump beam intensity, (ii) the efficiency of the detectors for 6 signal and idler fields, (iii) the magnitude of nonlinear susceptibility for SPDC, and so on. In the main text, the approximate results in Eqs. (S11) and (S12) are used to theoretically describe the quantum state, and one-photon (second-order) interferences of signal beams from triple-SPDC experimental setup are assumed to be detected. There, the idler field from the first SPDC NL crystal is allowed to interact with material of interest and various one-photon interferences (second-order correlation functions in the fields) among three signal fields are detected.

II. Cascading Configuration with Two SPDC Crystals
The experimental configuration shown in Figure S1, which corresponds to Figure 1(b) in the main text, is a schematic representation of Wang, Zou, and Mandel's experimental setup used in studying induced coherence without induced emission S4 . It involves two second-order NL crystals for SPDC processes. The two NL crystals are pumped by the same coherent laser. Each pumped crystal can convert a single pump photon into a pair of signal and idler photons, and the dichroic mirrors (shown in deep blue in Figure S1) are used to allow signal photons pass through them and to make idler photons reflected by them. A critical importance is the optical alignment that the idler beams from the two crystals are coherently superposed to make them indistinguishable. Furthermore, the idler beams are not detected, but the one-photon interference of the associated (conjugate) signal beams, 1 s and 2 s , is detected with D12 in Figure S1. To attenuate 1 i , Wang et al. S4 placed a 45  beam splitter (BS) between NL1 and NL2 (at the position of OS in our Figure S1), where the amplitude transmissivity and reflectivity from one side of the BS are T and R , respectively, whereas those from the other side are T  and R , respectively. They showed that the fringe visibility of one-photon interference between 1 s and 2 s detected at D12 is linearly proportional to the amplitude 7 transmissivity, because the degree of coherence between the conjugate idler beams is affected and reduced by the presence of the BS. Quite recently in 2014, Zeilinger and coworkers used the above cascading configuration with two down-converters ( Figure S1) to experimentally demonstrate quantum imaging technique S5 , where an imaging (phase) object instead of BS was placed between NL1 and NL2. In Wang et al.'s experiment with a 45  BS, if the transmission coefficient T depends on idler frequency, the spectrum of T with respect to the idler frequency would correspond to an elastic scattering spectrum of the beam splitter just in the case that the BS is a lossless medium. A very interesting quantum spectroscopy experiment employing the same idea with double-SPDC scheme was performed by Kalashnikov et al. S6 Since CO2 molecules in the gas phase absorbs infrared photons, the ro-vibrational spectrum of CO2 was obtained by measuring one-photon (second-order) interference of signal beams in visible frequency region. This demonstrates an IR spectroscopy with a visible photon detection instead of utilizing IR detectors.
Here, we wish to emphasize a critical difference of the experimental approach with two SPDC's from the more traditional coincidence count measurement method with a single SPDC, from the viewpoint of spectroscopic application. As shown by the Mandel group, signal and idler fields from a single SPDC are not coherent so that the corresponding second-order (in the fields) correlation function between signal and idler fields from the SPDC vanishes. Because the signal and idler fields are not coherent with each other, to investigate the signal-idler entanglement one cannot but measure coincidence counting rate or temporal correlation between signal and idler photons (Eq. (S13)). If a BS, object, or absorptive material is placed along the idler beam, the coincidence counting rate that requires detections of both signal and idler photons with two independent detectors will provide quantitative information on the property of the material, e.g., phase shift, transmission coefficient, or spatial shape (see Figure   1a). Thus, any quantum spectroscopy or imaging with just one SPDC requires detections of not 8 only signal photons but also idler photons. However, the double-SPDC scheme in Figure S1 ( Figure 1b in the main text) is drastically different from the single-SPDC quantum spectroscopy or imaging. Regardless of any real experimental conditions with either BS, object, or absorptive material placed between NL1 and NL2, it is absolutely necessary to detect 1 s field correlation with 2 s ; note that 1 s is SPDC-entangled with 1 i and that the idler beam 1 i directly interacts with the material of interest. The material placed between NL1 and NL2 modulates the degree of indistinguishability or coherence between 1 i and 2 i so that the onephoton interference between the conjugate entangled 1 s and 2 s fields carries direct information on the property of material located on the idler beam pathway. There are two critical differences between the single-SPDC spectroscopy (or imaging) and the double-SPDC technique that are to be emphasized. First, the single-SPDC technique requires fourth-order (in the fields) correlation or two-photon interference measurement, whereas the double-SPDC technique needs a second-order (in the fields) correlation or one-photon interference measurement.
Second, the single-SPDC technique detects not just signal photons not interacting with material but also idler photons directly interacting with material, whereas the double-SPDC technique does not require any detection of idler photons at all. Now, unlike the low-dimensional (in terms of experimentally controlled pairs of quantum entangled photons) quantum spectroscopy (or imaging) possibilities already explored before, we shall show that a triple-SPDC technique proposed here requires detections of neither idler photons that might interact with the material of interest nor their entangled (conjugate) signal photons at all. However, still one-photon interference between two signal fields generated from the other two NL crystals involved in the second and third SPDC's in a cascading arrangement ( Figure 1c in the main text) provides information on optical property of the material. Although this appears to be puzzling and quite non-intuitive, we shall show how the critical role of vacuum field correlating idler beams makes the triple-SPDC critically differ from the double-SPDC quantum spectroscopy.
Before we present theoretical results on our triple-SPDC quantum spectroscopy, for the sake of completeness and comparison with our new results with those of double-SPDC, we here summarize the basic aspects on one-photon interference of two signal fields ( 1 s and 2 s ) in the double-SPDC experiment shown in Figure S1 ( Figure 1b in the main text). Instead of considering an imaging application, we here will focus on the case that a lossless BS is placed between NL1 and NL2 for simplicity, even though the same principle applies to quantum imaging possibility. The BS can be any four-port device with two input and two output ports.
A typical BS with propagation directions at right angles or a partially reflecting film, dielectric slab, or planar material with light incident normally on both sides could be of use for BS (or OS) in Figure S1.
As emphasized earlier, the idler beam, 1 i , generated by NL1 transverses through NL2 and it is perfectly aligned with the idler beam, 2 i , generated by NL2. However, due to the presence of a BS between NL1 and NL2, the annihilation operator of 2 i can be related to that are the amplitude transmissivity from the upper side of the BS (OS in Figure S1) and the reflectivity from the other side. The phase gained by the idler mode due to beam propagation from NL1 to NL2 is denoted as NL1-NL2 ( , ) ii  k . In Eq. (S14), 0 a represents the vacuum field at the unused port of the BS (OS). From now on, it is assumed that the signal and idler beams have uniform linear polarization. Then, Eq. (S14) is simplified as The transmission and reflection coefficients should satisfy the following relations: in Eqs. (S14) and (S15) is mainly determined by not only the distances between the center of NL1 and the center of BS and between BS and NL2, which are denoted as NL1-BS d and BS-NL2 d , respectively, but also the thickness and refractive index of the BS used ( Figure S2).  i and 2 i beams given in Eq. (S15), one finds that Let us now consider the detection scheme in the experimental setup with two SPDC crystals in Figure S1 ( Figure 1b in the main text). The two signal beams from the two crystals, NL1 and NL2, are superposed by a 50:50 beam splitter and one of the outputs of the beam splitter is detected with D12. Then, the positive frequency part of the quantized signal field at the detector D12 can be expressed as a sum of two contributions associated with 1 s and 2 s , i.e., where  is the quantum efficiency of the signal detector D12. Hereafter, we shall assume that the quantum efficiencies of detectors used are the same and the conversion efficiencies of all the SPDC crystals are also the same.
Inserting Eqs. (S17) and (S19) into (S20) and carrying out a straightforward calculation, one can obtain the photon counting rate at D12 that is given by, apart from a constant proportionality constant, S4). As the transmission coefficient of the beam splitter (or optical sample) placed between NL1 and NL2 increases, the indistinguishability of two signal beams at D12 increases linearly so that the fringe visibility associated with interference between signal (not idler) beams becomes large. We shall compare these results for double-SPDC experiment summarized here with those for our triple-SPDC gedankenexperiment in the following section.

A. Distinguishable quantum vacuum fields
The cascading-type triple-SPDC experimental setup, which is a natural expansion of the double-SPDC setup in Figure S1, is shown in Figure S3. The critical difference from the double-SPDC scheme is just to add one more SPDC crystal, NL3, to the experimental setup, which is pumped by the same coherent laser. The generated idler beam, 3 i , from the NL3 is also assumed to be aligned with the other two idler beams, and the idler beam is not under detection. The signal beam 3 s is allowed to interfere with either 1 s or 2 s and the corresponding one-photon interferences are detected by D13 and D23. Since we are interested in utilizing the triple-SPDC setup for quantum spectroscopy or quantum measurement of dielectric (phase) properties of materials placed on the idler beam pathway right after the NL1, we put optical samples 1 and 2, denoted as OS1 and OS2 in Figure S3 between NL1 and NL2 and between 14 NL2 and NL3, respectively. We shall quantum mechanically treat them as quantum optical beam splitters. Here, it should be emphasized that the experimental configuration in Figure S3 is critically different from the triple-SPDC scheme (Figure 1c) considered in the main text.
Note that, since the OS1 and OS2 in Figure S3 are right-angle (45 • ) beam splitters, the quantum vacuum fields, 2 i  and 0 i , at the unused ports of OS1 and OS2, respectively, are completely uncorrelated, i.e., distinguishable. In fact, this ( Figure S3) is the experimental scheme with three SPDC crystals arranged in a cascading geometry that was considered by Ataman recently Here, it is assumed that the OS1 and OS2 are lossless materials for the sake of simplicity.
The amplitude transmissivities (reflectivities) of OS1 and OS2 for incident beams propagating from left to right are 11 () TR and 22 () TR , respectively, and those on the other sides are denoted as 11 () TR  and 22 () TR  , respectively. We shall only focus on whether the OS1 (an object that essentially played an important role in modulating the optical coherence between 1 s and 2 s beams in double-SPDC setup in Figure S1) affects the indistinguishability of 2 s and 3 s beams generated by two other down-converters or not.
We now need to consider the following relations of idler photon annihilation operators, Then, we can find that the quantum state becomes , 22  due to the complete distinguishability between the two quantum vacuum fields at the two unused ports of the beam splitters one cannot extract any spectral (or phase) information of the material of interest, OS1, through any one-photon (second-order) interference measurement of two signal beams, 1 s and 2 s , with D23.

B. Indistinguishable quantum vacuum fields: Optical resonator configuration
Now, we consider that the optical sample cell surfaces of OS1 and OS2 are normal to the incident idler beam ( Figure S4). In the case of double-SPDC with a single BS (or OS) in Figure S1 ( Figure 1b in the main text), the BS could be either a usual BS with beam propagation directions at right angles or a partially reflecting film or material with light incident normally on both sides. The results for the double-SPDC experiment do not depend on the angle of BS with respect to idler beam propagation direction at all. However, in the present case of the triple-SPDC experiment considered here ( Figure S4 or Figure 1c in the main text), the OS1 and OS2 are partially reflecting planar materials (or dielectric slabs) of which surfaces are normal to incident beams. This seemingly minor difference between the two configurations shown in Figures S3 and S4 has been found to be extremely important as shown below.
The quantum mechanical description of such partially reflecting film is still identical to that of conventional BS. The pair of OS1 and OS2 are therefore capable of forming a planar mirror resonator when they have non-zero reflection coefficients, which differs from the optical setup considered by Ataman recently S 7 , where he examined a special quantum imaging possibility of a similar but critically different triple-SPDC setup. He considered two right-angle (45 • ) beam splitters at the positions of OS1 and OS2 so that there are two uncorrelated vacuum (see Figure S3). Furthermore, the detection scheme considered in ref. S7 is completely different from ours depicted in Figure S4 ( Figure 1c in the main text).
Now, due to the existence of a resonator formed by OS1 and OS2 in Figure S4, the common vacuum mode is subject to the same planar materials, OS1 and OS2. Again, we shall particularly consider how the OS1 affects the indistinguishability of 2 s and 3 s beams. Note that neither 1 s nor 1 i beams that are generated by the so-called radiation source, NL1 crystal, arrives at the detector D23 at all, but the induced coherence among idler beams by means of aligning them in a collinear configuration and the indistinguishability of two quantum vacuum fields at the unused ports of OS1 and OS2 are shown to be critical in measuring optical properties of material (OS1) of interest placed after the spectroscopic radiation source NL1. In fact, this is quite counter-intuitive because neither the 1 i beams that could directly interact with material nor the 1 s beams that are correlated with 1 i beams via SPDC at NL1 are under the one-photon interference measurements with D23 at all.
Before we present the theoretical expression on the quantum state after all three SPDC processes, it would be necessary to discuss about the quantum description of idler state when all three idler beams are aligned like Figure S4. In Figure S5, the key components and definitions of distances are depicted. To obtain the idler field amplitudes at NL2 and NL3 that 18 are related to the idler field 1 i generated by NL1 crystal, let us first consider the amplitude of idler field 2 i at the position of NL2 that is given by where the phase factors are defined as . The phase factor  is that gained by the idler beam when it undergoes a single cycle of the resonator formed by OS1 and OS2. Due to the vacuum field on the right-hand side of OS2 in Figure S5, there appears the third term on the right-hand side of Eq. (S30).
Using these results for the annihilation operators of the three idler photons and the SPDC Hamiltonians describing the three SPDC processes, we obtain the quantum state of light after triple SPDC's that is where is a normalization factor, NL1-NL2 2 2 2 2 2 2 2 1 1 2 22 We note the normalization is required during the transformation from idler 2 ( 2 i ) to idler 1 ( 1 i ) and vacuum ( 0 i ).
We next consider the detection scheme in the experimental setup with three SPDC crystals in Figure S4. A pair of signal fields j s and k s from NL j and NL k are superposed by a properly placed beam splitter and its one-photon interference is detected by Djk, as shown in Figure S4.
where we assume that the quantum efficiency  of the three detectors are the same. Following the same arguments and carrying out a long but relatively straightforward calculation, we could obtain the photon counting rates at the detectors.
First of all, let us consider the photon counting rate at D12, which is found to be, apart from a constant proportionality constant,  Figure S1. A schematic layout of quantum spectroscopy or imaging experiment with two SPDC crystals.
Two nonlinear (NL) crystals, NL1 and NL2, are pumped by a common coherent laser. The generated signal and idler photons are separated by a dichroic mirror (DM). Idler beam, 1 i , from NL1 is allowed to interact with optical sample (OS) and is aligned collinearly with the idler 2, 2 i , from NL2. The idler beams are not under detection, but the one-photon interference, second order interference, between two signal beams, 1 s and 2 s , is measured with detector D12. Figure S2. The distances between NL1 and BS (or OS in Figure S1) and between BS (OS) and NL2. i , from NL1 is allowed to interact with 45  tilted Beam Splitter 1 (or Optical Sample 1, OS1) and is aligned collinearly with the idler 2, 2 i , from NL2. Similarly, the idler 2, 2 i , is allowed to interact with 45  tilted Beam Splitter 2 (OS2 in this figure) and is perfectly aligned in a way that the idler 3 cannot be distinguished from the other idler beams. The two quantum vacuum modes at the unused ports of OS1 and OS2 are denoted as 2 i  and 0 i , respectively. The idler beams are not detected, but the one-photon interferences, second-order interference, between any two signal beams,   Figure S6. a Visibility of interference fringe at D12, 12 V , with respect to amplitude transmissivity of OS1 at various 2 T from 0 to 1. b Visibility of interference fringe at D23, 23 V , with respect to amplitude transmissivity of OS1 at various 2 T from 0 to 1. We assume that the three NL crystals are pumped identically. a b 30 Figure S7. Visibility of interference fringe at D13, 13 V , with respect to the phase gained by a round trip of idler beam in the cavity with 2   , for various transmissivity( 1 T ) of OS1 for (a) 2 0.1 T  and (b) 2 1/ 2 T  . We note that the transmission becomes 1 on resonance for 21 TT  according to the cavity input-output relation. We assume that the three NL crystals are pumped identically. a b