Coupling undetected sensing modes by quantum erasure Open Access

In 1991, Zou, Wang, and Mandel demonstrated the capability of sensing changes in the phase and the transmission of a beam of light that need not be detected.¹ They achieved this by interference of two distinct photon-pair sources, which were pumped coherently from a single laser beam. Each nonlinear crystal in their setup produced a pair of outputs via spontaneous parametric down-conversion (SPDC) whose shorter/longer wavelengths are commonly known as the signal/idler beams. The idler output of the first crystal was directed through the second crystal, while the two signal beams were combined at a beam splitter—making generation in the first or second crystal indistinguishable from one another—meaning it was impossible to determine which crystal a particular photon had come from. The resulting absence of “which-way” information along with the common idler path causes “induced coherence” and thus interference on the signal beam that is observed when one of the path lengths is scanned. An example of this setup is shown in Fig. 1(a). In this scenario, the interference fringes are recorded in the signal beam, which is often in the more readily detected visible spectral region (the idler output—commonly in the infrared—is discarded). The interference fringes have a phase that is dependent on the total phase between the two crystals, including that of the idler path, and an amplitude that is dependent on the photon indistinguishability between the two crystal outputs. The latter is sensitive to how well the output beams of the two crystals have been overlapped optically, and it can be modulated by changing the transmission of the idler path between the crystals.² 

In 2014, Lemos et al. demonstrated how to use this effect to create images. Combining the induced coherence effect with the anti-momentum photon pair correlations gave position information.³ This ushered in a field of research, so-called “Imaging with Undetected Photons” (IUP). There are a variety of experimental configurations, but in each case, the photon pair correlations are harnessed to transfer image information from wavelengths where detectors may be expensive, slow, bulky, or noisy, to wavelengths that are much easier to detect (within the detection range of silicon, for example).

As there is some ambiguity in the terminology in the literature, here we define some specific configurations. Figure 1(a) shows IC-IUP, where IC denotes induced coherence, as in the original work by Wang, Zou, and Mandel.^1,4 Figure 1(b) shows an alternative common configuration, where both signal and idler photons from the first crystal are directed through the second. We label this NI-IUP, where NI denotes nonlinear interference. We recognize that these naming conventions are not ideal (induced coherence can be described as a nonlinear interference and vice versa); their use in this article is to provide a distinction only between the experimental arrangements without introducing further terms into the field.

NI-IUP is most commonly performed in a folded arrangement where the pump, signal, and idler are reflected back through the same crystal,^5–20 as opposed to unfolded versions that require two crystals. It should be noted that the main reason this is done is to make it easier to overlap the beams accurately and achieve the indistinguishability required for interference. This folded configuration has resulted in some compact and robust practical implementations.²¹ Recently, both the works by Kim et al.²² and León-Torres et al.²³ showed how hybrid nonlinear interferometers could be constructed, where the idler paths were folded back through the crystal, but the signal paths were mixed on an external beam splitter without passing back through the crystal. These interferometers displayed advantages for undetected imaging (such as increased data acquisition rates), although at the expense of a larger experimental footprint.

In Figs. 1(c) and 1(d), we show how a single folded style setup can implement both IC-IUP and NI-IUP sensing modes (respectively) simply by rotating a quarter wave-plate (QWP) inside the signal arm of the interferometer.²⁴ We note that these configurations are usually operated in the low-gain (or quantum) regime, where only a single photon pair is in the system at any given time. Thus, there is negligible interaction between the photon pairs generated in the first pass of the crystal with the other optical fields present for the second pass.⁴ 

While there have been individual attempts to theoretically characterize the IC-IUP and NI-IUP arrangements of this experiment,^25–27 to our knowledge, the two have not been compared directly, either experimentally or theoretically with a single theory or experiment capable of encompassing both. However, using the setup shown in Figs. 1(c) and 1(d), we are able to compare and even interfere these two IUP sensing modes using quantum erasure. Quantum erasure is a term that describes the act of removing which-way (“welcher-weg”), distinguishing information from a system in a way that generates interference effects where none were previously present.²⁸ In our setup, the pump beam passes the crystal twice, creating photon pair generation events from each pass. The signal beam from the first crystal pass traverses a QWP twice, rotating its polarization in a way that can introduce distinguishability relative to the second pass. A θ₁ = 0° polarization rotation results in signal photons from the first pass being aligned (and thus indistinguishable) with those generated in the second pass, defining an NI-IUP system. A θ₁ = 90° polarization rotation gives perfect distinguishability between the two passes and stops the interference completely. Inserting a half wave plate (HWP) at the output rotates both signal polarizations by θ₂ = 45°, which after a polarizing beam splitter, effectively restores indistinguishability and, therefore, interference.^29–32 This now defines an IC-IUP configuration. We can also use this arrangement to investigate partial distinguishability by setting the QWP and HWP angles such that both effects are seen simultaneously; i.e., we are able to use both IC-IUP and NI-IUP sensing modes, either independently or when coupled.

The visibility, $V$⁠, can be maximized by balancing the signal beam intensities that reach the detector from the two crystal passes.

Using Eq. (1), we can now model our experiment for arbitrary idler transmission; different internal QWP and external HWP angles to compare NI-IUP, IC-IUP, and combined setups; and uneven gain parameters for the two passes of the nonlinear crystal.

Equations (4) and (5) highlight that the behavior of the interferometer can be mediated by the phase difference of the horizontal and vertical components of the first signal; at θ₁ = 0° or 90°, Eqs. (4) and (5) are equivalent (since there is no mixing of NI and IC sensing modes—simply detection of one or the other). However, at θ₁ = 45° (full mixing), $V=0$ when $ϕHs=0$⁠, and $V=22/3$ when $ϕHs=π$⁠.

Figure 2 shows a schematic of the experimental optical setup designed to test the model described above. A 1064 nm wavelength continuous wave (CW) pump laser beam is set in power and polarization via an automated half-wave plate (HWP) and polarizing beam splitter (PBS). A second HWP and PBS create the two pump beam paths for nonlinear processes A and B as described in the theory [the polarization of pump beam B is rotated to match A via a quarter-wave plate (QWP) and reflection from another PBS].

The two pump beams enter the crystal where signal and idler photons are generated through SPDC at wavelengths of 1550 nm and 3.4 μm, respectively. The temperature-controlled periodically poled lithium niobate (PPLN) crystal, supplied by Covesion, has a poling period of Λ = 30.5 μm and is anti-reflection coated to R < 1.5% @ 1064 nm, to R < 1% @1400–1800 nm, and to R ∼ 6%–3% @ 2600–4800 nm, on both input/output facets. The pump beams are focused through two off-axis parabolic mirrors, which also serve to collect and nominally collimate the signal and idler beams. Signal and idler photons from process A (the first pass) are separated from the other beams via a pair of dichroic mirrors and reflected back into the crystal along with the pump for process B (the second pass).

A QWP in the signal arm allows for a polarization rotation to switch between the IC-IUP and NI-IUP operating modes. Signal and idler path lengths are matched macroscopically via a computer-controlled stepper-motor translation stage, while sub-wavelength phase shifts are implemented through a closed-loop piezoelectric translation stage, also in the idler path. The reflected signal beams from process A and from process B are separated from the pump with another short-pass dichroic mirror.

A 5 mm long beta barium borate (BBO) crystal is used to minimize the relative path lengths of the two signal polarizations, $ϕHs$ and $ϕVs$⁠, while the phase difference $ϕHs−ϕVs$ is adjusted by BBO’s tilt angle. An additional HWP and PBS before detection enables the two orthogonal components of the signal beam to be mixed. The signal is then filtered through a 2100 nm longpass filter (Layertec), a 12 nm wide 1550 nm bandpass filter (Thorlabs), and a 1400 nm longpass filter (Thorlabs) before being coupled into a single-mode fiber (SMF-28). The signal beam is detected with a superconducting nanowire single-photon detector (SNSPD, IDQuantique, $∼80$% detection efficiency at 1550 nm), with count rates monitored by a pulse counting module (ID900) with a 50 ms acquisition time.

The upper color plot in Fig. 3 shows experimental data (raw signal single-photon count rate), where the difference in length between the signal and idler paths has been changed by small (sub-wavelength) shifts using the piezo-stage (plotted on the x-axis), as well as larger (mm) shifts using the stepper motor (plotted on the y-axis). Here, the system has been put into a hybrid state, part way between the IC-IUP and NI-IUP operating modes, by setting the internal signal QWP to 30°. This introduces partial distinguishability between photon pairs from the first and second passes of the crystal that is subsequently erased by the external output HWP and PBS.

Two regions of interference, which occur at different macroscopic path length changes, can be clearly identified; they correspond to the NI-IUP (centered at ∼0.25 mm) and the IC-IUP (centered at ∼0.6 mm) modes of operation. As discussed above, the change in the refractive index experienced by the orthogonal components of the signal beam through the PPLN is compensated for at different idler delay positions. The apparent changes in the “angle” of the interference plots at the two macroscopic positions are artefacts caused by stepper motor non-linearities that cause aliasing effects in the data on the y axis, where the step size is larger than the wavelength (see supplementary material 1). The period of oscillation on the x axis remains constant at the wavelength of the idler (the field being scanned). The lower plot in Fig. 3 shows the behavior of the system as predicted by Eq. (1) for realistic input parameters (T = 0.25, L_coh = 0.2 mm). The visibility of the interference [as calculated by Eq. (2) from the experimental x-axis data shown in the upper plot in Fig. 3] as a function of the macroscopic path length shift (the y axis shown in the upper plot of Fig. 3) is shown in Fig. 4. Three regions of interest are highlighted, showing IC-IUP, NI-IUP, and mixed behavior. We note that the maximum observed visibility for the IC-IUP and NI-IUP cases is different, which we attribute to polarization-dependent background levels in the experiment. In supplementary material 1, we show background-free density plots of varying QWP and HWP angles for the IC-IUP and NI-IUP cases.

The upper plot in Fig. 5 shows the visibilities of the interference fringes—extracted from fits to experimental data—at three different idler mirror delays, which correspond to the NI-IUP, IC-IUP modes, and part way in between. The lower plot shows the visibilities calculated from the corresponding theoretical model. These visibilities have been measured and plotted as a function of the angle of the output HWP (θ₂), and they clearly demonstrate differences in how the interference is balanced in the NI-IUP and IC-IUP operating modes. Tuning the HWP can be used to balance the interference in the latter but not the former.

In the NI-IUP case, the only way to achieve a balance of counts from either crystal (and thereby maximize the fringe visibility) is to adjust the gain of one relative to the other, something that is difficult to do in a folded setup where the pump may be reflected back through the crystal. The IC-IUP mode offers the possibility of changing the beam splitter ratio away from the 50:50 setting (as used by Lemos et al.); in our setup, we can conveniently achieve the same effect simply by altering the HWP angle.

A further advantage of the IC-IUP setup over NI-IUP versions is the fact that the signal fields are mixed on a beam splitter and thus has access to two output ports, which could be measured simultaneously. This gives access to a constant total photon number output regardless of phase or amplitude. In principle (although not explored further here), this could have dramatic implications for low sample transmissivity applications.

Of additional interest is the region in between IC-IUP and NI-IUP modes, where the two modes of operation overlap with each other. These data show a fringe visibility minimum at an HWP angle of $∼30°$ and a maximum at $∼65°$ due to addition of the two modes of operation. The positions of the minimum and maximum visibility are dependent on the phase difference between the IC-IUP and the NI-IUP interference fringes. By tilting the BBO crystal, it is possible to introduce a relative phase shift between the horizontal and vertical components of the signal photons $(ϕHs−ϕVs)$⁠, changing the interplay between the two interference effects and adjusting the positions of the maximum and minimum (see supplementary material 1).

We have built a novel interferometer that can showcase both IC-IUP and NI-IUP sensing modes. This work furthers our understanding of the differences between the operation of these systems. In the low-gain regime (as shown here), the differences would at first seem minimal, since they operate on the same fundamental principle. However, on closer inspection, one will see that the IC-IUP version has additional functionality given by the mixing of signal fields at the final beam splitter. The HWP and PBS allow continuous control of the balance between the two signal fields, enabling losses internal to the interferometer (either loss of signal from the first crystal, or loss of pump between crystals leading to lower second signal) to be compensated. Furthermore, the dual output of the interferometer could be utilized for a greater signal to noise ratio in amplitude and phase measurements. Thus, this system gives access to a wide range of advantages of IC-IUP but in the compact single crystal design associated with folded NI-IUP setups. Our theoretical model includes all features of this design, including the limited coherence lengths of the signal and idler fields, giving a complete description of the system that fits well with the experimental data. This model (and indeed the experimental system shown here) is only valid for the low-gain regime, where there is no stimulation of the second PDC by the first.

In the high-gain regime, we would expect differences in the IC-IUP and NI-IUP to become significant, as in the NI-IUP version both signal and idler fields generated in the first crystal stimulate PDC in the second crystal. In the IC-IUP version, only the idler stimulates, leading to higher gains needed to reach the same levels of stimulation as in NI-IUP. In an idler seeded interferometer (where a beam matching the expected idler mode is injected into the interferometer), we may speculate that the idler seeding would dominate any signal stimulation, and thus, the two modalities would again behave in a similar fashion.

In the case where the two modalities are present simultaneously, we have also shown that there is a third linear interference term that is dependent on the phase difference between the horizontal and vertical signal components. We anticipate that the combination of these three interference terms can be exploited for enhanced sensing platforms in the future.

Supplementary material 1 contains the full theoretical derivation of our model, added information on the behavior of the interferometer due to balancing, phase shifts between the interferometer modes, and aliasing effects in the experimental data.

We acknowledge funding from the UK National Quantum Hub for Imaging (QUANTIC, Grant No. EP/T00097X/1), an EPSRC DTP, and the Royal Society (Grant No. UF160475).

The authors have no conflicts to disclose.

Nathan R. Gemmell: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yue Ma: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Emma Pearce: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Jefferson Flórez: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Olaf Czerwinski: Conceptualization (equal); Investigation (equal); Software (equal). M. S. Kim: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal). Rupert F. Oulton: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Alex S. Clark: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Chris C. Phillips: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.