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Quantum Imaging with Undetected Photons

I. INTRODUCTION

Information is essential to quantum mechanics. In particular, quantum interference occurs if and only if there exists no information that allows one to distinguish between the interfering states. It is not relevant if an observer chooses to notice this information or not. These are not just conceptual issues; they have direct practical consequences. In fact, interference of a photon can reveal information of another photon, which is not detected. As an example, here we apply this as a method of quantum imaging1.

In imaging, the ideal wavelength for illuminating the object normally depends on both the properties of the object to be imaged and the wavelength sensitivity of available detectors. This makes low-light imaging very difficult at wavelengths outside the range for which low-
light cameras are available. In order to circumvent this problem, one approach has used optical non-linearity to convert the light coming from the object into a shorter wavelength where efficient and low-noise detectors are available2,3. Another method is two-colour ghost
imaging4, in which the light field of one wavelength is used to illuminate an object and the image appears in the correlations between this and a light field in another wavelength, thus requiring coincident photon detection at both wavelengths5.

Our quantum imaging technique is based on a quantum interference phenomenon that was first shown in the early 1990's6. We begin with an explanation of this experiment, which is illustrated in Fig. 1. A pump laser beam divides at a 50:50 beam splitter (BS1) and coherently
illuminates two identical non-linear crystals, NL1 and NL2, where pairs of collinear photons called signal (yellow) and idler (red) can be created. The probability that a down-conversion occurs at each crystal is equal and very low so the chance that more than one pair of photons is produced at a given time can be neglected.

The idler photons created in NL1 are reflected at the dichroic mirror D1 into spatial mode d, and signal photons pass into spatial mode c. The idler then passes through the object that has a real transmittance coefficient and imparts a phase shift . We write this as | ⟩ | ⟩
| ⟩ | ⟩ √ | ⟩ | ⟩ where, for simplicity, we assume that all the idler photons that are not transmitted occupy a single state | ⟩ . After being reflected at dichroic mirror D2, the idler photons from NL1 are perfectly aligned with idler photons produced at NL2, | ⟩ | ⟩ . The quantum state at the grey dotted line in Fig.1 can be written as [( | ⟩ | ⟩ )| ⟩ √ | ⟩ | ⟩ ]. ( 1 )

The idlers are now reflected at dichroic mirror D3 and are not detected. The signal photon states | ⟩ and | ⟩ are combined at the 50:50 beam splitter BS2. The detection probabilities at the outputs, | ⟩ and | ⟩ are obtained by ignoring (tracing out) the idler modes, giving [ ] ( 2 )

This formula shows that fringes with visibility T can be seen at either output of BS2, even though the signal beams combined at BS2 have different sources. These fringes appear in the signal single photon counts; no coincidence detection is required. This is possible because the
idler photon that is reflected at the dichroic mirror D3 does not carry any information about the crystal where it was created, and therefore the two modes of the signal interfere when overlapped by BS2, i.e. now each signal photon came from both NL1 and NL2.

A very peculiar feature of this interferometer is that no photon that reaches the detectors can have gone through path (Fig. 1). Yet, in our experiment, it is precisely here that we put the object we want to image. The key to this experiment is that the signal source information
carried by the undetected idler photons depends on T. For instance, if T = 0, one could monitor the idlers reflected at D3. If an idler photon and a signal photon at | ⟩ or | ⟩ were