Quantum imaging and metrology with undetected photons: tutorial

Gabriela Barreto Lemos; Mayukh Lahiri; Sven Ramelow; Sven Ramelow; Radek Lapkiewicz; William N. Plick; William N. Plick

doi:10.1364/JOSAB.456778

1. INTRODUCTION

One thousand years ago, in 1021, Ḥasan Ibn al-Haytham (a.k.a. Alhazen) completed the Book of Optics, in which he laid the foundations of modern optics and detailed the apparatus that later Kepler dubbed camera obscura (pinhole camera). In the intervening 10 centuries, imaging technology has progressed immeasurably—and up to the present day, developing ever-better imaging and sensing devices remains an extremely active field of research. Evolving approaches to imaging technology have enabled new abilities such as extreme sensitivity and resolutions, imaging at non-visible wavelengths, and many others too diverse to fully enumerate.

One of the very latest areas of research is a suite of new technologies enabled by the development of quantum theory, which contains the most accurate description of optical fields. Devices that take advantage of the co-called “quantum nature of light” include new imaging schemes that may beat the classical limits of sensitivity [1–4], spatial resolution [5–10], and phase metrology [11–13]. Quantum optics has also enabled the emergence of imaging schemes where the light that interacts with the sample is not captured by the pixelated detector/camera, e.g., interaction free imaging [14], ghost imaging [15–21], and quantum imaging with undetected photons (QIUP) [22–24].

With ever more sophisticated camera and photon-source technologies emerging in recent years, quantum imaging and quantum-inspired imaging techniques have become even more promising avenues for research and development—and will likely be key components of the 21st century quantum revolution, alongside quantum computing, quantum communication, and quantum metrology.

Here we focus on QIUP and its spin-offs. In these experiments, coherence is induced between light produced in two twin-photon sources placed within a nonlinear interferometer. Quantum imaging with nonlinear interferometers was introduced in [22], and applications of this method to bio imaging, spectroscopy, optical coherence tomography (OCT), and moving images were first proposed in [25,26]. The most important advantage of this kind of technique is that one can obtain information about an object probed by a light beam of one wavelength by detecting only a separate light field at a different wavelength. The light field that illuminates the sample is not detected at all. This is especially useful when the illumination wavelength is one for which detectors are not available or unsatisfactory, and in the case of delicate samples that require low intensity illumination.

How to read this tutorial. In this tutorial, we will prepare you both theoretically and experimentally to investigate a few methods of quantum imaging and metrology with undetected photons, i.e., quantum-imaging-based technologies using induced coherence without induced emission (ICWIE) within a nonlinear interferometer in the low-gain regime. We will assume basic knowledge of the quantum optics formalism and familiarity with spontaneous parametric downconversion (SPDC).

In Section 2, we give an extended introduction to basic interferometric devices and the effect of ICWIE, including the basic state-vector description and conceptual overview. In Section 3, we present an introduction to one of the main applications of interferometry: phase metrology, and show how ICWIE may be exploited in this field, yielding a number of advantages and technological prospects. In Section 4, we provide the theoretical description for a multi-mode nonlinear interferometer and its application to phase and absorption imaging. In Section 5, we describe OCT, holography, and spectral imaging with undetected photons. Sections 6 and 7 are aimed at giving experimental guidelines to researchers who are building a Zou–Wang–Mandel interferometer (ZWMI) or a SU(1,1) interferometer in the low-gain regime. In Section 8, we give an outlook of some interesting research directions one could explore. Appendix A is dedicated to a basic overview of the main quantum optics states and operators, for those who have not encountered quantum optics formalism before or would like a recap.

This tutorial does not have to be read in a linear fashion, but if this is your first encounter with imaging and metrology using ICWIE, we suggest you start by reading Section 2. Those who would like to revisit key points of basic quantum optics theory might want to read Appendix A at that point, before proceeding on. Basic theoretical background on quantum optics and quantization of electromagnetic fields is required for understanding Sections 3 and 4. Good accounts of these topics can be found in standard textbooks on quantum optics (e.g., [27,28]). Also, readers with different interests might jump to different sections. In particular, readers interested in the application of ICWIE to phase metrology can learn about it in Section 3, which can be skipped by those not interested in phase metrology. Researchers aiming at understanding the rigorous theory behind imaging with undetected photons should refer to Section 4, which can be skipped by experimentalists who are not planning on doing detailed calculations. On the other hand, researchers who are not planning to build a setup in the laboratory might want to skip Sections 6 and 7. Please note that background knowledge in elementary Fourier optics and multi-mode theory of SPDC will be useful for a thorough understanding of imaging, particularly of the theory developed in Section 4. An excellent account of Fourier optics can be found in [29], and a multi-mode description of the theory of SPDC can be found in [30].

Fig. 1. (a) Mach–Zehnder interferometer. A light field is split at the 50:50 beam splitter, ${{\rm{BS}}_1}$, and is recombined at ${{\rm{BS}}_2}$. A phase shifter is placed in path $A$ and an object with complex field transmittance $T$ placed in path $B$. Interference is analyzed at the detector. (b) Two-Photon Interferometer. A pump is split into paths ${P_1}$ and ${P_2}$ at a 50:50 beam splitter (${{\rm{BS}}_1}$) and illuminates two nonlinear sources, ${Q_1}$ and ${Q_2}$, producing correlated photon pairs. When a pair is produced in source ${Q_1}$ (${Q_2}$), the so-called signal photon is emitted into path ${S_1}$ (${S_2}$) and so-called idler photon is emitted into path ${I_1}$ (${I_2}$). Signal paths ${S_1}$ and ${S_2}$ are combined at ${{\rm{BS}}_2}$, and idler paths ${I_1}$ and ${I_2}$ are combined at ${{\rm{BS}}_3}$. No interference is observed in the signal intensity at the detector because which-way information is in principle obtainable. One can observe interference only using post-selection, i.e., by detecting idlers at one output of ${{\rm{BS}}_3}$ in coincidence with signals at one output of ${{\rm{BS}}_2}$.

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2. INDUCED COHERENCE WITHOUT INDUCED EMISSION

Richard Feynman considered quantum interference the biggest mystery in quantum mechanics [31]. Quantum interference is a phenomenon observed at a detector if and only if it is impossible to associate each detected quantum (e.g., photon, electron, atom, molecule) to a particular path, between the two or more paths connecting that detection apparatus to the quantum source [31,32]. Moreover, the fringe visibility in a two-way interferometer gives an upper bound on the available which-way (welcher-weg) information [33]. The quintessential example is the double slit experiment, where no interference is observed if the alternative paths between the source and the detector are distinguishable and interference is observed if those paths are indistinguishable [31,34]. Other important examples in optics are the Michelson, Mach–Zehnder, and Sagnac interferometers [35–37].

When setting up any interferometer in the laboratory, to observe good interference visibility, one aligns the beams incoming to the detector and adjusts the path lengths of the interferometer so that they differ by not more than the coherence length of the quanta. In the language of quantum information, the alignment and the length adjustment of the interferometer paths amount to ensuring indistinguishability between quanta arriving at the detector, thus enabling interference [32].

A. Mach–Zehnder Interferometer

Let us consider what happens to light in a Mach–Zehnder interferometer (MZI), illustrated in Fig. 1. The incoming light field is split at the first beam splitter (BS) ${{\rm{BS}}_1}$ and recombined at a second BS, ${{\rm{BS}}_2}$. For interference to be seen, the optical path lengths of paths $A$ and $B$ between the two BSs must be equal to within the coherence length of the light field, such that each photon is described as being in a superposition, $({|1{\rangle _A} + {e^{- i{\phi _A}}}|1{\rangle _B}})/\sqrt 2$, where $|1{\rangle _x}$ denotes a photon in path $x$. The phase ${\phi _A}$ is associated with relative optical delays between the two modes of the interferometer and can be adjusted by slightly shifting a mirror or BS or by inserting a slab of a transparent material, such as silica. The count rate at a detector placed after ${{\rm{BS}}_2}$ in path $A$ (or $B$) is given by

(1)$${{\cal R}_{A/B}} = \frac{{\left({1 + |T{|^2}} \right) \pm 2|T|\cos ({\phi _A} - \gamma)}}{4},$$

where $T = |T|{e^{{i\gamma}}}$, with $0 \le |T| \le 1$ and $0 \le \gamma \lt 2\pi$, is the complex field transmittance of an object placed in path $B$. The interference visibility obtained by scanning ${\phi _A}$ is defined as

(2)$${\cal V} = \frac{{{{\cal R}_{{\max}}} - {{\cal R}_{{\min}}}}}{{{{\cal R}_{{\max}}} + {{\cal R}_{{\min}}}}}.$$

Assuming equal optical path lengths, the interference visibility of the MZI is thus

(3)$${{\cal V}_{{MZ}}} = \frac{{2|T|}}{{1 + |T{|^2}}}.$$

In the language of quantum information, the reduction of visibility for $|T| \lt 1$ is due to the path distinguishability introduced by the object. One application of the MZI is interaction free measurements [38,39], which can be used for imaging [14]. An object is placed in one arm of a MZI where photons are sent one at a time. The object affects the interference pattern at the output and, in a fraction of the experimental runs, the presence of the object can be deduced without the photon having interacted with it.

B. Two-Photon Interferometer

Now let us consider an interferometer that uses two identical sources of photon pairs, e.g., nonlinear crystals that can generate photon pairs, commonly referred to as signal and idler, through SPDC. The sources are prepared such that the biphoton fields emerging from them are mutually coherent. These crystals are weakly pumped by mutually coherent laser beams, for example, generated by splitting a laser beam into two as shown in Fig. 1(b). Let us denote the two sources as ${Q_1}$ and ${Q_2}$, and the emitted beams will be referred to as signal beams (${S_1}$ and ${S_2}$) and idler beams (${I_1}$ and ${I_2}$). Beams ${S_1}$ and ${S_2}$ are combined at a BS, the outputs of which are sent to detectors.

A few simple calculations can help us understand what is going on. In a first approach to this problem, it is instructive to write down the photon states in the device as relatively simple state vectors. In this picture, the action of a nonlinear source is simply to add one photon each to the appropriate modes. For example, in reference to Fig. 1(b), the action of source ${Q_j}$, with $j = \{1,2\}$, is modeled as taking the vacuum input in modes ${S_j}$ and ${I_j}$ and transforming them as $|0{\rangle _{{S_j}}}|0{\rangle _I} \to |1{\rangle _{{S_j}}}|1{\rangle _{{I_j}}}$, where $|1{\rangle _{{S_j}}},\;|1{\rangle _{{I_j}}}$ represent a photon occupying mode ${S_j},\;{I_j}$, respectively. Given this assumption, it is also very unlikely that both nonlinear sources, ${Q_1}$ and ${Q_2}$, fire in sync. Please note that here we keep the kets as single-photon number states so as to maintain the same notation as the rest of the paper, but since first quantization does not use the Fock basis (all modes are assumed to have only one photon) we could equally well omit the occupation number—that is, $|1{\rangle _I} = |I\rangle$, for example.

Let us first consider that $T = 1$ in Fig. 1(b), which is equivalent to removing that object/sample. Assuming both sources are identical and their emissions are coherent, the state just before the BS ${{\rm BS}_2}$ will be the superposition

(4)$$\frac{{|1{\rangle _{{S_1}}}|1{\rangle _{{I_j}}} + {e^{- i\phi}}|1{\rangle _I}|1{\rangle _{{S_2}}}}}{{\sqrt 2}}.$$

Now let us consider the more general case. We will see in Section 4 that the object/sample in the idler path is modeled as a BS with complex field transmittance function $T = |T|{e^{{i\gamma}}}$, with $0 \le |T| \le 1$ and $0 \le \gamma \lt \pi /2$. In this picture, the action of the object in Fig. 1(b) is to transform the state produced in source ${Q_1}$ as

(5)$$|1{\rangle _{{S_1}}}|1{\rangle _{{I_1}}} \to |1{\rangle _{{S_1}}}\left({T|1{\rangle _{{I_1}}} + i\sqrt {1 - |T{|^2}} |1{\rangle _0}} \right),$$

where $|1{\rangle _0}$ represents a photon absorbed or scattered by the object. Hence, when we apply this to the interferometer we are considering, Eq. (4) is replaced by

(6)$$\frac{{\left({T|1{\rangle _{{I_1}}} + i\sqrt {1 - |T{|^2}} |1{\rangle _0}} \right)|1{\rangle _{{S_1}}} + {e^{- i\phi}}|1{\rangle _{{I_2}}}|1{\rangle _{{S_2}}}}}{{\sqrt 2}}.$$

The action of a BS is described in Appendix A [Eq. (A21)]. Essentially, ${{\rm BS}_2}$ combines the signal modes as $|1{\rangle _{{S_1}}} \to ({|1{\rangle _{{S_1}}} - \imath |1{\rangle _{{S_2}}}})/\sqrt 2$ and $|1{\rangle _{{S_2}}} \to ({|1{\rangle _{{S_2}}} - \imath |1{\rangle _{{S_1}}}})/\sqrt 2$. Similarly, ${{\rm BS}_3}$ combines the idler modes, such that the two-photon state after ${{\rm{BS}}_2}$ and ${{\rm{BS}}_3}$ in Fig. 1(b) is

(7)$$\begin{split}\!\!\!&\left({\frac{{{f_ -}|1{\rangle _{{I_1}}} + i{f_ +}|1{\rangle _{{I_2}}} + i\sqrt {2(1 - |T{|^2})} |1{\rangle _0}}}{{2\sqrt 2}}} \right)|1{\rangle _{{S_1}}} \\&\quad+ \left({\frac{{i{f_ +}|1{\rangle _{{I_1}}} - {f_ -}|1{\rangle _{{I_2}}} - \sqrt {2(1 - |T{|^2})} |1{\rangle _0}}}{{2\sqrt 2}}} \right)|1{\rangle _{{S_2}}},\!\end{split}$$

where ${f_ \pm} \equiv ({T{e^{{i\phi}}} \pm 1})$.

If the idler photons are not detected, mathematically we perform a partial trace over the idler modes and obtain a constant ($= 1/2$) signal photon counting rate at either output of ${{\rm{BS}}_2}$. In other words, interference is not directly observed in intensity measurements at the signal detector without any post-selection (coincidence detection). An intuitive explanation for this is that which-source information is retrievable in principle; for example, one could (hypothetically) add a fourth BS combining the two idler outputs of ${{\rm BS}_3}$, which could reveal which idler came from which source. In that case, effectively, spatial modes ${I_1}$ and ${I_2}$ would be inputs to a MZI [Fig. 1(a)] defined by ${{\rm BS}_3}$ and (hypothetical) ${{\rm BS}_4}$, such that the output of ${{\rm BS}_4}$ could be set to be the original ${I_1}$ and ${I_2}$ fields. The astonishing thing is that even if one does not actually perform that measurement, by leaving the idlers undetected, one leaves the possibility, in principle, of using these idlers to extract which-source information of the signal photons. That mere possibility, even if it is not actually realized, is enough to inhibit signal intensity modulation due to interference.

Fig. 2. Three architectures of the Zou–Wang–Mandel interferometer. (a), (b) The idler paths produced in sources ${Q_1}$ and ${Q_2}$ are aligned, and a 50:50 beam splitter, ${{\rm{BS}}_2}$, combines signal paths ${S_1}$ and ${S_2}$; which-path information is erased, and single-photon interference can be observed in the detector, even though idler photons are not detected at all. The field transmittance function, $T$, of an object illuminated by the idler field $I$ can be observed in the interference pattern of the signal beams at the detector, even though signal photons have not interacted with that object. The phase $\phi$ can be tuned by adjusting signal, idler, or pump optical path lengths. (b), (c) The signal and idler are emitted in the same direction as the pump (collinear emission), and if they are all at distinct frequencies, dichroic mirrors can be used to separate them. In (b), the dichroic mirror ${{\rm{DM}}_1}$ reflects idler photons and transmits signal wavelength, whereas ${{\rm{DM}}_2}$ transmits the pump and reflects idler photons. In (c), a single crystal is pumped from both sides. ${{\rm{DM}}_1}$ reflects the pump and transmits both signal and idler, whereas ${{\rm{DM}}_2}$ reflects idler and transmits the signal. Finally, ${{\rm{DM}}_3}$ reflects the signal while transmitting the idler and pump. Notice that in this architecture, undetected light goes twice through the same sample ($T^\prime = {T^2}$).

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Note that if one detects one output of ${{\rm{BS}}_2}$ in coincidence with an output of ${{\rm{BS}}_3}$, i.e., by using post-selection, it is possible to observe interference [40,41], and the visibility is ${\cal V} = 2|T|/(|T{|^2} + 1)$.

C. Zou–Wang–Mandel Interferometer

In the MZI, as well as the Michelson interferometer and the Sagnac, classical wave models, including classical electromagnetism, can describe the interference visibility due to (mis)alignment, path length difference, and/or an object placed in a path of the interferometer. Seeking to unravel the connection between quantum indistinguishability and interference, in 1991, Zou, Wang, and Mandel, with essential insight from Jeff Ou, created an interferometer that cannot be described by classical wave optics models [42,43]. The difference between this interferometer and the two-photon interferometer described above is that now both sources emit idler beams into the same spatial mode, $I$. Considering that the idler photons are not detected at all, do you expect that interference fringes can be observed in the detected signal outputs? Why or why not?

Let us take a look at the state of a photon pair just before ${{\rm{BS}}_2}$ in Fig. 2:

(8)$$\frac{{\left({T|1{\rangle _I} + i\sqrt {1 - |T{|^2}} |1{\rangle _0}} \right)|1{\rangle _{{S_1}}} + {e^{- i\phi}}|1{\rangle _I}|1{\rangle _{{S_2}}}}}{{\sqrt 2}},$$

where $T = |T|{e^{{i\gamma}}}$ is the complex field transmittance of the object/sample placed in the idler path, $0 \le \phi \lt 2\pi$ is an adjustable interferometric phase, and $|1{\rangle _0}$ represents a photon absorbed or scattered by the object, i.e., mode 0 corresponds to the “loss mode.” Note that mode $I$ does not acquire a second photon, as the single photon in that mode is assumed to have come from either the second or first crystal. At this point, the origin of this photon could be determined by seeing which of modes ${S_1}$ and ${S_2}$ contains a photon.

The final BS ${{\rm{BS}}_2}$ combines the signal fields, after which the state of the twin photons can be written as

(9)$$\begin{split}\!\!\!|{\psi _f}\rangle & = \frac{1}{2}\left({({T + i{e^{- i\phi}}} )|1{\rangle _I} + i\sqrt {1 - |T{|^2}} |1{\rangle _0}} \right)|1{\rangle _{{S_1}}}\\&\quad + \frac{1}{2}\left({({{e^{- i\phi}} + iT} )|1{\rangle _I} + \sqrt {1 - |T{|^2}} |1{\rangle _0}} \right)|1{\rangle _{{S_2}}}.\!\end{split}$$

By tracing out the idler mode $I$ and the loss mode 0, we can obtain the count rate at a detector placed at either output of ${{\rm{BS}}_2}$:

(10)$${{\cal R}_{{S_1}({S_2})}} = \frac{{1 \pm |T|\cos (\phi + \gamma)}}{2},$$

where—strikingly—an interference pattern modulated by the object $T$ can now be observed in the signal photon counting rate, despite the fact that neither ${S_1}$ nor ${S_2}$ interacted with that object. Coherence is thus “induced” between the two modes as the result of aligning as precisely as possible the shared idler mode. We explain in Section 7 how to do this alignment in the laboratory. Idler mode $I$, which contains no phase information, is typically discarded.

In the ZWMI that we have just described, the interference visibility is directly proportional to the absolute value of the field transmission coefficient:

(11)$${{\cal V}_{{\rm{ZWM}}}} \propto |T|.$$

This relationship holds even if the intensities of the signal beams are not equal. In fact, any photon loss in the idler arm results in a reduction of the total visibility, as it introduces partial path distinguishability (welcher-weg information).

Let us compare this with the nonlinear effect of loss ($|T| \lt 1$) in an arm of a MZI [Eq. (3)]. The linear relation between interference visibility and loss in the undetected idler photon path between the two sources is a distinguishing feature of the ZWMI. It is shown in [42,44] that this characterizes the non-classicality of induced coherence without induced (stimulated) emission. This is a very important point: stimulated emission at the second source ${Q_2}$ due to the input idler field $I$ is not necessary for induced coherence (interference), a fact that highlights the non-classicality of the phenomenon [45]. In cases where stimulated emission is not negligible in ${Q_2}$, the interference visibility is a nonlinear function of the field transmittance $|T|$ [44,46]. This regime can be achieved using very high gain sources (e.g., using very high pump power), or by seeding ${Q_1}$ and ${Q_2}$ via mode $I$ with a coherent state (laser beam) with the idler beam wavelength [40,47].

Fig. 3. Three architectures for the SU(1,1) interferometer. (a) Both signal and idler paths $S$ and $I$ from the first nonlinear source (${Q_1}$) are aligned with signal and idler paths originating in the second nonlinear source (${Q_2}$). (b), (c) The laser pumps a crystal and is reflected back through the same crystal. The undetected light traverses twice through the same sample ($T^\prime = {T^2}$). In (c), the pump, signal, and idler leave the crystal collinear with each other, and if they are all at different frequencies, they can be separated using three dichroic mirrors: ${{\rm{DM}}_1}$ transmits the pump but reflects signal and idler photons; ${{\rm{DM}}_2}$ transmits the pump and idler photons, but reflects signal photons; ${{\rm{DM}}_3}$ transmits the pump and signal photons, but reflects idler photons. In all three architectures, single-photon interference is seen in signal and idler outputs without post-selection (coincidence detection). The field transmittance function of an object placed in either path $S$ or $I$ can be seen in the interference pattern appearing in a camera placed in either output path.

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The ZWMI was generalized to many spatial modes using spatially correlated photon pairs to produced QIUP [22,23]. Quantum interference and spatial correlations between signal and idler photons [30] together produce in the detected signal photons images of an object placed in the idler beam [23,24,48,49]. Both the real and imaginary parts of the object’s spatially varying field transmittance function can be observed. The real part $|T|$ is encoded in the interference visibility. The imaginary part $\gamma$ appears as an interferometric phase.

D. SU(1,1) Interferometer

Let us now consider that both signal and idler photons from source ${Q_1}$ are fed into source ${Q_2}$ [50,51], as shown in Fig. 3. Here we will refer to this interferometer as an “SU(1,1) interferometer,” also known as a “nonlinear Mach–Zehnder” [50,52–54]. This experiment can be thought of as a nonlinear adaptation of the MZI, which has two BSs, whereas the SU(1,1) has instead two nonlinear media, ${Q_1}$ and ${Q_2}$. The object with transmission function $T = |T|{e^{{i\gamma}}}$ is placed in the idler mode between the crystals. At the output modes, the two-photon state can be written as

(12)$$|\psi \rangle = \frac{{(|T|{e^{\gamma + \phi}} + 1)|1{\rangle _S}|1{\rangle _I} + \sqrt {1 - |T{|^2}} |1{\rangle _S}|1{\rangle _0}}}{2}.$$

The (singles) counting rate at detectors placed on either output path is therefore

(13)$${{\cal R}_{S/I}} = \frac{{1 + |T|\cos (\phi + \gamma)}}{2},$$

giving the interference visibility ${{\cal V}_Y} = |T|$, just as in the case of the ZWMI [Eq. (11)]. As the interferometer path of the signal, idler, or pump is adjusted, both signal and idler count rates oscillate, a clear manifestation of interference. The very unique feature of this particular interferometer is that the single photon output ports are in phase with each other, though out of phase with the laser output port. That means that if maximum (minimum) counts are observed in output mode $S$, maximum (minimum) counts are simultaneously observed in output mode $I$. This leads to the curious phenomenon of frustrated downconversion, analyzed in [51]. If one introduces which-path information in the signal or idler paths between the crystal, for example, by unaligning the modes, interference is reduced or even lost in both signal and idler modes, showing complementarity between welcher-weg (which-path) information and interference visibility.

The experiments described above and illustrated in Figs. 2 and 3 can be viewed as “quantum eraser” experiments [55,56]. In the ZWMI (Fig. 2), after the crystals but before the final BS, no interference would appear in either of the detection modes since the path itself marks which crystal experienced the photon-generating downconversion; however, after the final BS, this information is erased (a photon in either mode could have come from either crystal), and thus interference appears. In the SU(1,1) interferometer (Fig. 3) which-source information is erased at the second crystal. Also note that, unlike interferometers such as Mach–Zehnder, Michelson, and Sagnac, where only a single phase shift is possible, in the nonlinear interferometers we discuss here, one can independently change the phases of the pump, of the signal field, and of the idler fields, and usually these have different frequencies.

Notice that in the setup in Fig. 4(a), the signal and idler go through the object, and in Fig. 4(a), all three fields, signal, idler, and pump, go through the imaged object (labeled each time with $T$). In that case, the equations in the theory sections must be adapted accordingly. In addition, we show in this tutorial the imaging due to light transmitted through an object, but it is trivial to adapt the theory to the case of a reflective object, which is the case of the OCT setup [Fig. 13(b)], described in Section 5.B.

Fig. 4. These are two alternative versions of the SU(1,1) interferometer. (a) The same crystal acts as sources ${Q_1}$ and ${Q_2}$, as pump, signal, and idler are reflected back into that crystal. the dichroic mirror (DM) reflects only the signal field, and long pass (LP) reflects only the pump. (b) Setup for collinear non-degenerate downconversion, where both signal and idler pass through the sample. Dichroic mirror ${{\rm DM}_1}$ separates the signal $S$ from the other fields, and dichroic mirror ${{\rm DM}_2}$ separates the idler $I$ from the pump.

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It turns out that, in addition to the ZWMI and the SU(1,1) configurations, there is a whole range of interferometric architectures that involve two or more nonlinear sources [50,52]. These “nonlinear interferometers” have proved to be interesting and useful for imaging, spectroscopy, metrology, and other applications [54,57,58,60–66]. In this tutorial, we will explain how to build and model some of them.

In Section 3, we will introduce the theoretical underpinnings of phase metrology (one of the most common applications of interferometers), and in Section 4, we will show a generalization of the single-mode description to a multi-mode one, allowing the description of imaging with undetected photons. Those familiar with the basics of quantum optics and the main statistical states of light (coherent, number, etc.) can proceed to those sections. Those wishing an introduction or refresher can see Appendix A at the end of this paper. Experimentalists who do not plan on performing detailed calculations can skip ahead to Section 5.

We point out that in our theoretical expositions in this tutorial, we use the language of squeezed correlated fields, such as states produced in parametric downconversion. However, the interferometers we analyze and the mathematical formalism we introduce can also be applied to a variety of other systems, such as atomic spin waves [67], superconducting microwave cavities [68,69], and four wave mixing [70].

3. THEORY OF PHASE METROLOGY WITH UNDETECTED PHOTONS

A typical task in interferometry is to determine the relative phase delay between two (or more) modes of the device. For example, in a MZI (Fig. 1), we can measure the delay between the two modes that yields information about the difference of path lengths between modes A and B (or, perhaps, the index of refraction of some intervening transparent material, or some other similar thing). In the simplest case, this information can be abstracted out as a phase shift, and in fundamental studies of interferometers, the actual mechanism of the shift is typically ignored. Devices are then characterized by the minimum phase shift that can be observed—corresponding to the most sensitive configuration.

The fundamental limit for classical interferometers (typically classified as those that use coherent light only) is called the “standard quantum limit” (SQL), which itself is the combination of two effects.

The first is radiation-pressure noise, which is a result of the light beam imparting a fluctuating amount of momentum into the mirror. Obviously, the imparted momentum causes “jitter,” fowling up the very sensitive phase measurements an interferometer might otherwise perform. This noise increases as the power of the light increases. Conversely, it decreases as the mass of the mirror increases. Much of the research into reducing radiation-pressure noise is concerned with mirror stabilization.

The second is shot noise, which is a result of the photon number fluctuations from “shot to shot.” The relative shot noise decreases as the intensity increases, in contrast to radiation-pressure noise. This is the noise source that is typically the limiting factor. In principle, the mass of the mirrors in an interferometer may be made very large, such that radiation pressure becomes small when compared to shot noise. Though in practice this may be very difficult, there are a number of systems where the dominant source of noise is indeed shot noise, and it is often the case that the terms SQL and shot noise limit (SNL) are used interchangeably.

By using simple arguments about the statistics of coherent states, the limiting case for classical interferometry may be found to be

(14)$$\Delta {\phi _{{\min}}} = \frac{1}{{\sqrt {\bar n}}},$$

which is the SNL on the minimum detectable phase shift. This is the best that can be done classically. Factors $\Delta {\phi _{{\min}}}$ and $\bar n$ are the minimum detectable phase shift and average photon number, respectively.

Now the obvious questions are, “Can this be improved upon using quantum resources? And, if so, what new fundamental limit constrains quantum devices?” The answers to these questions are commonly agreed to be “yes” and “the Heisenberg limit,” respectively.

The Heisenberg limit presumes to be the absolute limit on the phase sensitivity of an interferometer. Unlike the SNL in Eq. (14), it makes no assumptions about the specific kind of light being used. Instead, the Heisenberg limit draws upon the fundamental laws of quantum mechanics to place a bound on how accurately we may measure a light field’s (or matter wave’s) phase. However, it must be noted that it is not straightforward to provide a rigorous derivation of the Heisenberg limit for phase measurement, which is accepted by all. The root of the problem lies in the definition of the phase operator. Below we provide a brief and lucid description of the issue.

A phase operator was originally introduced by Dirac in his celebrated paper on quantum theory of radiation [71]. To understand Dirac’s approach, let us take the annihilation operator acting on a coherent state:

(15)$$\hat a|\alpha \rangle = \alpha |\alpha \rangle = {e^{{i\phi}}}|\alpha ||\alpha \rangle = {e^{{i\phi}}}\sqrt {\bar n} |\alpha \rangle .$$

The annihilation operator is a purely quantum mechanical object with no classical analog. However, it can be decomposed into quantities that are familiar in classical optics: the average intensity, $\bar n$, and phase, $\phi$, of an optical field. Dirac took this to mean that the creation and annihilation operators could be factored into Hermitian observables as

(16a)$$\hat a = {e^{i\hat \phi}}\sqrt {\hat n} ,$$(16b)$${\hat a^\dagger} = \sqrt {\hat n} {e^{- i\hat \phi}},$$

where the second equation is obtained by simply taking the conjugate transpose of the first. He thus defined the phase operator $\hat \phi$. This combined with the commutation relation $[\hat a,{\hat a^\dagger}] = 1$ yields $[\hat n,\hat \phi] = i$.

Dirac thus concluded that the photon number and phase are conjugate (canonical) observables. Therefore, to become more certain about one, we must become less certain about the other. This relationship can be made quantitative by employing the generalized Heisenberg uncertainty principle for non-commuting operators: $\Delta A\Delta B \ge \frac{1}{2}|\langle [\hat A,\hat B]\rangle |$. Using this, we have $\Delta n\Delta \phi \ge 1/2$. So to know as much as we can about the phase, we must reduce by as much as possible our knowledge of the number of photons in the field. Since we are concerned with fundamental limits, we will take the case where the photon number is as uncertain as possible: when $\Delta n = \bar n/2$. The uncertainty cannot be made any larger than this because then there would be a non-zero probability of detecting a negative number of photons in the field (which is physically meaningless). Therefore, we get the following expression for the Heisenberg limit:

(17)$$\Delta {\phi _{{\min}}} = \frac{1}{{\bar n}}.$$

However, there is a flaw in this argument. The problem exists in the definition of the phase operator that was later shown to be non-Hermitian [72,73]. In fact, if we attempt to write down an eigenstate for this phase operator, we can visualize the issue.

Eigenstates of the number operator (representing a light field with an exactly known number of photons but a completely undefined phase) are well defined as a ring in quadrature space with a diameter equal to the intensity of the field and a finite area (and thus energy); see Fig. 14. However, in quadrature space, a phase eigenstate is a wedge radiating out from the origin to infinity (all of the space that exists at a particular angle). Such a state has infinite area and thus infinite energy, and thus is not normalizable (see Fig. 14 again).

Since the problem with Dirac’s phase operator was pointed out, there have been numerous discussions and proposals on this issue (see, for example, [73–76]). Furthermore, there exists another approach to understand the achievable precision of phase measurement from the perspective of quantum estimation theory [77,78]. Nevertheless, the Heisenberg limit ($\Delta {\phi _{{\min}}} = 1/\bar n$) is widely used and commonly regarded as an approximate bound, in the limit of high photon numbers [72,76]. The Heisenberg limit remains a useful and common goalpost for studies in interferometry.

Now suppose we wish to consider a specific device that probes the abstract phase shift by imprinting it on some measurable quantity. Mathematically, this means we have some quantity that is a function of the phase $M(\phi)$. So we ask the question, “Given that we are measuring $M$, what is the smallest change we can detect in $\phi$?” To answer this question, start with the Taylor series expansion of a function $M$ with the variable as $\phi$ about a point ${\phi _o}$:

(18)$$M(\phi) = M({\phi _0}) + (\phi - {\phi _0}){\left. {\frac{{\partial M}}{{\partial \phi}}} \right|_{\phi \to {\phi _0}}} + \quad ...,$$

where $\partial (\phi - {\phi _0}) = \partial \phi$ since ${\phi _0}$ is a constant. The smallest detectable phase shift would be equal to the smallest we could make $\phi - {\phi _0}$. Since we are considering this quantity to be very small, we can truncate the series after the second term and recast the above as

(19)$$\frac{{M(\phi) - M({\phi _0})}}{{{{\left. {\frac{{\partial M}}{{\partial \phi}}} \right|}_{\phi \to {\phi _0}}}}} = \phi - {\phi _0} = \Delta {\phi _{{\min}}}.$$

If we take ${\phi _0}$ to be the average value of the phase, then $M(\phi) - M({\phi _0})$ gains the interpretation of being the variance of $M$ for a single measurement, as $M(\langle \phi \rangle) = \langle M(\phi)\rangle$—and the derivative becomes the derivative with respect to the phase of $\langle M\rangle$. However, we want the statistically averaged variance for a series of measurements (standard deviation) of $M$, so we take ${(M - \langle M\rangle)^2}$ and average it (which is the standard deviation squared), which yields $\langle {M^2}\rangle - {\langle M\rangle ^2}$.

Then, squaring both sides of Eq. (19), making the substitutions above, and identifying $\langle {M^2}\rangle - {\langle M\rangle ^2}$ as the square of the variance, we arrive at

(20)$$\Delta {\phi _{{\min}}} = \frac{{\Delta \hat M}}{{\left| {\frac{{\partial \langle \hat M\rangle}}{{\partial \phi}}} \right|}},$$

where we have promoted $M$ to a quantum mechanical observable and taken the root of both sides. This is the minimum detectable phase shift. To calculate this, we need both a choice of measurement operator, and the quantum-mechanical state that operator works on (to take the expectation values).

Now, recall the limit of a standard MZI with coherent light input—Eq. (14). We wish to use this new formula to calculate the sensitivity of this device to changes in the abstract phase $\phi$. We need the relationship between the input modes and output modes given by Eq. (A21). These will allow us to write the operators at the detection end of the MZI in terms of the operators at the input end. We then take the expectation values of these operators at the input end.

The most important question, with regard to our sensitivity formula and the MZI, is the choice of the detection scheme $\hat M$, which for a MZI is the difference of the intensities at the bright port and dark port:

(21)$$\hat M = \hat b_f^\dagger {\hat b_f} - \hat a_f^\dagger {\hat a_f},$$

where the subscript indicates that these operators act on the final state. This corresponds to an intensity difference measurement between modes A and B in Fig. 1. Using this information and Eq. (20), we find for this setup $\Delta {\phi _{{\min}}} = 1/|\alpha | = 1/\sqrt {\bar n}$, which is, unsurprisingly, the SNL in Eq. (14).

An analysis of the phase sensitivity has also been recently performed for the ZWMI by some of us and others [79]. Several interesting effects are found. First, as might be expected for a “highly quantum” device, the minimum detectable phase shift reaches below the classical bound of the SNL, meaning that the device is “super-sensitive.” This effect is maintained regardless of gain regime (at least in principle). Furthermore, when the initial crystal ${Q_1}$ is seeded with a strong coherent (laser) beam, the sensitivity is further increased (“boosted”) into the bright-light regime while still maintaining some aspects of super-sensitive scaling. Though the general equations produced by this calculation are very large, a simple case of the minimum detectable phase shift (squared) can be presented for coherent light injection into one of the modes (in this case, the one that does not pass through the sample, corresponding to mode ${S_1}$ in Fig. 2) and intensity difference subtraction measurement between the two detected modes (modes ${S_1}$ and ${S_2}$ in the same), when the gains are very large (and equal to each other), and with the probe phase (and all other phases) set to zero:

(22)$$\Delta \phi _{{\min}}^2 = \frac{{{e^{- 2r}}}}{{4(1 + {\beta ^2})}}.$$

Note, as a reminder, that $r$ represents the modulus of the squeezing parameter $\xi$ ($\xi = r{e^{{i\theta}}}$) and $\beta$ the coherent pump field. Also, this equation is not optimal—rather, it is presented because of its tractability (for detailed discussion, see [79]). From this equation, it is clear that both the squeezing due to nonlinearities and the coherent light injection improve the sensitivity. The improvement is exponential for the gain and inverse-squared for the coherent light injection.

The SU(1,1) interferometer, Fig. 3, has become a commonly studied device [50,51,54,61,80] due to the fact that it allows super-sensitive detection with bright light. Recently, these interferometers have also been modified so that a MZI is nested inside [81], which has some conceptual similarity to the ZWMI configuration in the sense that both BS and squeezing operations are performed.

An argument against the use of all nonlinear interferometers could be paraphrased as, “If we need a bright laser to pump the nonlinearity, would it not just be better to use that bright light in a MZI?” To confront this criticism, any light that is used to pump a nonlinear source is added to the MZI “light budget,” making the comparison “fair.” In Fig. 5, the phase sensitivity of the ZWM, SU(1,1), and MZIs are compared. This graph uses the concept of a “fair comparison state.” We see that, though the SU(1,1) configuration performs best, the ZWMI type also displays super-sensitivity, beating the MZI after even a modest nonlinear gain.

Fig. 5. Metrology with undetected photons. Figure taken from [79]. The minimum detectable phase-shift squared of several fair comparison interferometric setups and detection schemes as a function of gain [of the first crystal for ZWMI and of both crystals for SU(1,1)]. Here we display the boosted ZWMI setup with intensity detection at mode $B$ (green), intensity difference detection between modes $B$ and $C$ (brown), the boosted SU(1,1) setup (black), and a standard coherent-light-seeded MZI with the extra light needed to create the aforementioned squeezings added to the initial input (red). The latter is equivalent to the shot-noise limit. All other parameters are numerically optimized at each point. The circular points (upper set) represent injected coherent light of about the same intensity as would be needed for a high-gain nonlinearity, and the square points (lower set) represent a much brighter coherent input.

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Surprisingly, this super-sensitive phase detection is available regardless of which of the two initial input modes is “boosted” into the bright-light regime (corresponding to injecting coherent light either into mode ${S_1}$ or into mode $I$ in Fig. 2). Therefore, one can choose not to shine the extra laser through the sample and achieve the same sensitivity increase as if it had been. Likewise, one can shine the light through the sample into the mode that is discarded and thus avoid hitting the detectors. This technique could prove very useful in cases where either the detectors or sample is sensitive to bright coherent light. Furthermore, when contrasted with the SU(1,1) configuration, which requires adaptive intensity measurements (where the signal is produced by summing the intensities of the two output modes) or homodyning for detection, the ZWMI geometry uses intensity subtraction, making it a more stable and thus at least in some cases more experimentally desirable.

The ZWMI has also been studied from the perspective of the signal-to-noise ratio (SNR). In [82], the authors theoretically study a ZWMI with a variable-field transmittance BS inserted in the idler path between two nonlinear sources, with different pump intensities for the nonlinear sources as well. They look at the SNR and visibility of the output as a function of the gain in the nonlinearities and the field transmittance of the aforementioned BS, paying special attention to the qualitative and quantitative difference between various regimes of gain. They also find that the visibility of the system may be optimized by proper choice of the field transmittance of the BS.

So far we have examined only “single-mode” descriptions of the ZWMI and the SU(1,1). That is, we do not take into account the real momentum and frequency distributions. We have assumed that fields can differ only in path. In the following section, we build on the previous to create a more complete picture, especially as it relates to imaging.

4. THEORY OF IMAGING WITH UNDETECTED PHOTONS

Amplitude and phase imaging using a ZWMI was introduced in [22] using collinear SPDC, as shown in Fig. 2(b). Two absorptive objects (a cardboard cut-out and an etched silicon plate) and a phase object (an etched fused silica plate) were placed in the undetected idler arm with wavelength 1550 nm. The images of these objects were retrieved in the interference pattern of the combined 810 nm signal field using a low-light sensitive camera, although there is no requirement of single-photon detection.

In this section, we present a rigorous theoretical description of the image formation in the ZWMI. To have a thorough idea of the imaging, we must consider the multi-mode structure of optical fields.

QIUP relies on transverse spatial correlations between signal and idler photons. Currently, there exist two complementary setups. In one of them, lenses in the idler path between the crystals are used such that the object is in the far field relative to ${Q_1}$ and ${Q_2}$ [22,23,64,65,83]. The object is imaged onto the camera also using lenses in the signal fields. In that case, the imaging is enabled by momentum correlation between the twin photons. In the other case, the object and camera can be placed in the near field (source plane) relative to the twin photon sources and, in this case, the imaging is enabled by the position correlation between twin photons [24,84,85]. (Evanescent fields play no role in this configuration and, therefore, must not be confused with conventional near-field imaging.) Images generated in these two cases have distinct features. We discuss the two configurations separately in Sections 4.B and 4.C. We stress here that the theory is not restricted to twin photons generated by SPDC; it applies to spatially correlated twin photons generated by any source, provided one considers only pure states.

A. Multi-mode Twin-Photon States

Throughout the analysis, we assume that photons propagate as paraxial beams and are always incident normally on both the object and the detector. Under these assumptions, the two-photon quantum state can be written as (see, for example, [30])

(23)$$|\tilde \psi \rangle = \int {\rm d}{{\textbf{q}}_s}\;d{{\textbf{q}}_I}\;C({{\textbf{q}}_s},{{\textbf{q}}_I})|{{\textbf{q}}_s}{\rangle _s}|{{\textbf{q}}_I}{\rangle _I},$$

where $|{{\textbf{q}}_s}{\rangle _s} \equiv \hat a_s^\dagger ({{\textbf{q}}_s})|vac\rangle$ denotes a signal photon Fock state labeled by the transverse component ${{\textbf{q}}_s}$ of the wave vector ${{\textbf{k}}_s}$. Similarly, $|{{\textbf{q}}_I}{\rangle _I} \equiv \hat a_I^\dagger ({{\textbf{q}}_I})|vac\rangle$ denotes an idler photon Fock state labeled by the transverse component ${{\textbf{q}}_I}$ of the wave vector ${{\textbf{k}}_I}$. The complex quantity $C({{\textbf{q}}_S},{{\textbf{q}}_I})$ ensures that $|\psi \rangle$ is normalized, i.e.,

(24)$$\int {\rm d}{{\textbf{q}}_s}d{{\textbf{q}}_I}|C({{\textbf{q}}_s},{{\textbf{q}}_I}{)|^2} = 1.$$

Imaging enabled by both momentum correlation (Section 4.B) and position correlation (Section 4.C) can be described by the quantum state given by Eq. (23). Such a quantum state is usually generated by SPDC at a nonlinear crystal. However, the theoretical analysis applies to any source that can generate such a state.

B. Imaging Enabled by Momentum Correlation

1. General Theory

It is evident from Eq. (23) that the joint probability density of detecting a signal photon with transverse momentum $\hbar {{\textbf{q}}_s}$ and idler photon with transverse momentum $\hbar {{\textbf{q}}_I}$ is given by

(25)$$P({{\textbf{q}}_s},{{\textbf{q}}_I}) \propto |C({{\textbf{q}}_s},{{\textbf{q}}_I}{)|^2}.$$

This probability density characterizes the momentum correlation between the twin photons. We now show that in the far-field configuration, this momentum correlation enables image formation.

The experimental setup is illustrated in Fig. 6. There are two sources, ${Q_1}$ and ${Q_2}$, each of which can emit a photon pair. ${Q_1}$ emits the signal and idler photons into beams ${S_1}$ and ${I_1}$, respectively. Likewise, ${S_2}$ and ${I_2}$ represent the beams into which the signal and idler photons are emitted by ${Q_2}$. The two sources (${Q_1}$ and ${Q_2}$) almost never emit simultaneously and almost never produce more than two photons individually. Furthermore, the two sources emit coherently. Under these circumstances, the quantum state of light generated by the two sources is given by the superposition of the states generated by them individually, i.e., by

(26)$$\begin{split}|\psi \rangle &= \int {\rm d}{{\textbf{q}}_s}d{{\textbf{q}}_I} C({{\textbf{q}}_s},{{\textbf{q}}_I}) \\&\quad\times \left[{{\alpha _1}\hat a_{{s_1}}^\dagger ({{\textbf{q}}_s})\hat a_{{I_1}}^\dagger ({{\textbf{q}}_I}) + {\alpha _2}\hat a_{{s_2}}^\dagger ({{\textbf{q}}_s})\hat a_{{I_2}}^\dagger ({{\textbf{q}}_I})} \right]|vac\rangle,\end{split}$$

where ${\alpha _1}$ and ${\alpha _2}$ are complex numbers satisfying the condition $|{\alpha _1}{|^2} + |{\alpha _2}{|^2} = 1$, and $|vac\rangle$ represents the vacuum state.

Fig. 6. Quantum imaging with undetected photons. (a) In a collinear non-degenerate ZWMI [86], non-degenerate photon pairs are emitted along the propagation axis of a laser at each source. Wave plates and a polarizing beam splitter (PBS) in the pump are used to control the relative phases and amplitudes of the two-photon states generated in sources ${Q_1}$ and ${Q_2}$. Dichroic mirrors or long pass filters can be used to separate the pump from the daughter fields after each crystal. A lens ${L_0}$ is used to control the pump waist at the crystals, which affects the twin photon transverse momentum correlations, which in turn affects the image resolution, as shown in Section 4.B.3. Imaging system $A$ (e.g., a lens or lens system) ensures a good overlap of the combined signal fields, and optical system $C$ (again a lens or lens system) is used to image the plane of the object with spatial features $T(\boldsymbol\rho)$ onto the plane of the camera or scanning detector. (b) In the case of imaging enabled by momentum correlation, optical systems $B$ and $B^\prime $ (e.g., also a lens or lens system) guarantee that the object $T(\boldsymbol\rho)$ is at the Fourier plane of sources ${Q_1}$ and ${Q_2}$. An effective positive lens with focal length ${f_c}$ associates the plane on the camera with the Fourier plane of sources ${Q_1}$ and ${Q_2}$. A plane wave vector ${q_s}$ makes an angle $\theta$ with the optical axis and is focused along a circle of radius $|{\boldsymbol\rho _c}|$. (c) In the case of imaging enabled by position correlation, optical systems $B$ and $B^\prime $ are imaging systems; a point ${\boldsymbol\rho _s}$ on ${Q_j}(j = 1,2)$ is imaged onto ${\boldsymbol\rho _c} = {M_s}{\boldsymbol\rho _s}$ on the camera by $A$.

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As shown in Fig. 6, lens systems are used to place both the object and the camera in the far field (Fourier plane) of the sources. These lens systems must also ensure that source ${Q_1}$ is imaged onto ${Q_2}$ and the camera is on the image plane of the object. For the best possible alignment of idler beams, it is important that ${Q_1}$ is imaged onto ${Q_2}$ with unit magnification. Here we present a generic treatment that does not consider any specific lens systems and provides a full understanding of the imaging mechanism. The theory can certainly be tailored to specific systems, and this may result in only incremental differences such as a different sign of image magnification.

A lens system $B$ that places the object in the Fourier plane of ${Q_1}$ can be effectively modeled by a single positive lens, where ${Q_1}$ and the object are located on the back and front focal planes of this positive lens, respectively. Similarly, the lens system $B^\prime $ that places ${Q_2}$ on the Fourier plane of the object can be modeled by another positive lens. For ${Q_1}$ to be imaged onto ${Q_2}$ with unit magnification, the focal length these two lenses must be equal, and shall here be denoted effective focal length, ${f_I}$. Since a transverse wave vector ${{\textbf{q}}_I}$ is focused on a point ${\boldsymbol\rho _o}$ on the object, within the paraxial approximation, we obtain from lens rules

(27)$${\boldsymbol\rho _o} = \frac{{{\lambda _I}{f_I}}}{{2\pi}}{{\textbf{q}}_I},$$

where ${\lambda _I}$ is the mean wavelength of the idler photon.

The interaction of the idler field with the object can be represented by the transformation made by a BS, and we, therefore, have the following expression [23]:

(28)$${\hat a_{{I_2}}}({{\textbf{q}}_I}) = {e^{i{\phi _I}}}\left[{T({\boldsymbol\rho _o}){{\hat a}_{{I_1}}}({{\textbf{q}}_I}) + R({\boldsymbol\rho _o}){{\hat a}_0}({{\textbf{q}}_I})} \right],$$

where ${\phi _I}$ is the phase due to propagation of the idler beam from ${Q_1}$ to ${Q_2}$, operator ${\hat a_0}$ represents the vacuum field at the unused port of the BS (object), $T({\boldsymbol\rho _o})$ is the amplitude transmission coefficient of the object at a point ${\boldsymbol\rho _o}$ that is related to ${{\textbf{q}}_I}$ by Eq. (27), and $|T({\boldsymbol\rho _o}{)|^2} + |R({\boldsymbol\rho _o}{)|^2} = 1$. The quantity, $R({\boldsymbol\rho _o})$, can be interpreted as the amplitude reflection coefficient at the same point while illuminated from the other side.

The quantum state of light generated by the system is obtained by combining Eqs. (26) and (28). We first determine $\hat a_{{I_2}}^\dagger ({{\textbf{q}}_I})$ (Hermitian conjugate of ${\hat a_{{I_2}}}({{\textbf{q}}_I})$) from Eq. (28). We then substitute for $\hat a_{{I_2}}^\dagger ({{\textbf{q}}_I})$ into Eq. (26). Finally, we use the relations

(29a)$$\hat a_{{s_{\!j}}}^\dagger ({{\textbf{q}}_s})\hat a_{{I_1}}^\dagger ({{\textbf{q}}_I})|vac\rangle = |{{\textbf{q}}_s}{\rangle _{{s_j}}}|{{\textbf{q}}_I}{\rangle _{{I_1}}},\quad j = 1,2,$$(29b)$$\hat a_0^\dagger ({{\textbf{q}}_I})|vac\rangle = |{{\textbf{q}}_I}{\rangle _0},$$

and find that the quantum state of light generated by the system is given by

(30)$$\begin{split} |\Psi \rangle & = \int {\rm d}{{\textbf{q}}_s}d{{\textbf{q}}_I} C({{\textbf{q}}_s},{{\textbf{q}}_I})[{\alpha _1}|{{\textbf{q}}_s}{\rangle _{{s_1}}} + {e^{- i{\phi _I}}}{\alpha _2}{T^ *}({\boldsymbol\rho _o})|{{\textbf{q}}_s}{\rangle _{{s_2}}}]|{{\textbf{q}}_I}{\rangle _{{I_1}}}\\& \quad+ \int {\rm d}{{\textbf{q}}_s}d{{\textbf{q}}_I} C({{\textbf{q}}_s},{{\textbf{q}}_I}){e^{- i{\phi _I}}}{\alpha _2}{R^ *}({\boldsymbol\rho _o})|{{\textbf{q}}_s}{\rangle _{{s_2}}}|{{\textbf{q}}_I}{\rangle _0}.\end{split}$$

Since the camera is placed in the combined signal field, at the far field relative to the sources, we can once again use the concept of an effective positive lens, as shown in Fig. 6(b). Suppose that the focal length of this lens is denoted by ${f_c}$. Following an argument similar to the one used to obtain Eq. (27), we find that point ${\boldsymbol\rho _c}$ on the camera is related to the transverse signal wave vector ${{\textbf{q}}_s}$ by the following formula:

(31)$${\boldsymbol\rho _c} = \frac{{{\lambda _s}{f_c}}}{{2\pi}}{{\textbf{q}}_s},$$

where ${\lambda _s}$ is the mean wavelength of the signal photon. The quantized field at point ${\boldsymbol\rho _c}$ on the camera plane can now be represented by

(32)$$\hat E_s^{(+)}({\boldsymbol\rho _c}) \propto {e^{i{\phi _{{s_1}}}}}{\hat a_{{s_1}}}({{\textbf{q}}_s}) + i{e^{i{\phi _{{s_2}}}}}{\hat a_{{s_2}}}({{\textbf{q}}_s}),$$

where ${\phi _{{s_1}}}$ and ${\phi _{{s_2}}}$ are phases due to propagation of the signal beams from ${Q_1}$ and ${Q_2}$, respectively, to the camera. The single-photon counting rate (intensity) at point ${\boldsymbol\rho _c}$ on the camera can be determined by the standard formula ${\cal R}({\boldsymbol\rho _c}) \propto \langle \Psi |\hat E_s^{(-)}({\boldsymbol\rho _c})\hat E_s^{(+)}({\boldsymbol\rho _c})|\psi \rangle$. It now follows from Eqs. (30) and (32) that

(33)$${\cal R}({\boldsymbol\rho _c}) \propto \int {\rm d}{{\textbf{q}}_I} P({{\textbf{q}}_s},{{\textbf{q}}_I})\big[{1 + | {T({\boldsymbol\rho _o})} |\cos ({\phi _{{in}}} - {\arg}\{T({\boldsymbol\rho _o})\})} \big],$$

where ${\phi _{{in}}} = {\phi _{{s_2}}} - {\phi _{{s_1}}} - {\phi _I} + {\arg}\{{\alpha _2}\} - {\arg}\{{\alpha _1}\} + \pi /2$, arg represents the argument of a complex number, and ${\boldsymbol\rho _o}$, ${{\textbf{q}}_I}$ and ${\boldsymbol\rho _c}$, ${{\textbf{q}}_s}$ are related by Eqs. (27) and (31), respectively, and we have assumed $|{\alpha _1}| = |{\alpha _2}|$ for simplicity.

It is evident from Eq. (33) that the information about the object (both magnitude and phase of the amplitude transmission coefficient) appears in the interference pattern observed on the camera, even though the photons probing with the object are not detected by the camera. The presence of $P({{\textbf{q}}_s},{{\textbf{q}}_I})$ in Eq. (33) shows that the momentum correlation between the twin photons enables image acquisition. For example, when there is no correlation between the momenta, $P({{\textbf{q}}_s},{{\textbf{q}}_I})$ can be expressed in the product form $P({{\textbf{q}}_s},{{\textbf{q}}_I}) = {P_s}({{\textbf{q}}_s}){P_I}({{\textbf{q}}_I})$. It can be checked from Eq. (33) that in this case, no interference pattern will be observed and the information of the object will be absent in the photon counting rate measured by the camera. Therefore, for the imaging scheme to work, there must be some correlation between the momenta of the twin photons. Furthermore, the momentum correlation also determines image quality. In fact, we will see in Section 4.B.3 that this momentum correlation limits the image resolution.

In the ideal scenario, when the momenta of the twin photons are perfectly correlated, the probability density $P({{\textbf{q}}_s},{{\textbf{q}}_I})$ can be effectively replaced by a Dirac delta function. Consequently, it follows from Eq. (33) that

(34)$${\cal R}({\boldsymbol\rho _c}) \propto 1 + \left| {T\left({{\boldsymbol\rho _o}} \right)} \right|\cos [{\phi _{{in}}} - {\arg}\{T({\boldsymbol\rho _o})\}].$$

The phase ${\phi _{{in}}}$ is varied experimentally, and consequently, an interference pattern is observed at each point ${\boldsymbol\rho _c}$ on the camera. It is evident from Eq. (34) that the information of a point (${\boldsymbol\rho _o}$) on the object appears in the interference pattern observed at a point (${\boldsymbol\rho _c}$) on the camera. Extraction of this information results in imaging.

We illustrate the imaging by first considering an absorptive object for which we can set ${\arg}\{T({\boldsymbol\rho _o})\} = 0$. It is seen from Eqs. (2) and (34) that the visibility of the single-photon interference pattern at a point $({\boldsymbol\rho _c})$ on the camera is given by [24]

(35)$${\cal V}({\boldsymbol\rho _c}) = \left| {T({\boldsymbol\rho _o})} \right|.$$

Clearly, the spatially dependent visibility provides an image of the object.

Alternatively, one can acquire the image of an absorptive object by subtracting the minimum intensity from the maximum intensity, i.e., by determining the quantity

(36)$$G({\boldsymbol\rho _c}) = {{\cal R}_{{\max}}}({\boldsymbol\rho _c}) - {{\cal R}_{{\min}}}({\boldsymbol\rho _c}).$$

In [22], images were acquired using this method. In Figs. 7-IA and 7-IB, interference is seen in the body of the cat, corresponding to regions of the idler field that are transmitted through a cardboard cutout. No interference is seen outside the cat, because the corresponding idler modes are blocked by the cardboard. If one sums the two outputs, the cat disappears (Fig. 7-ID), and the Gaussian profile of the signal field is seen. This shows that the total signal field intensity is not affected by the absorptive object. Subtracting the two outputs results in a high-contrast absorption image of the sample (Figs. 7-IC and 7-IIC).

Fig. 7. Absorption and phase imaging enabled by momentum correlations. I(A–D) and II(A-B) have been adapted from [86], which used the setup in Fig. 6(a) and realized imaging enabled by momentum correlations. In IA and IB are shown two intensity signal outputs of a collinear non-degenerate ZWMI. The detection wavelength is $810 \pm 1.5\;{\rm{nm}}$; the sample is a cardboard cutout placed at the Fourier plane of the sources and illuminated by an idler beam with wavelength centered at 1550 nm. The difference (sum) of those two outputs is shown in IC (ID). Phase imaging of an etched silica plate using the same setup is shown in IIA and IIB. Momentum correlation enabled absorption (IIC) and phase (IID) images (adapted from [64]) from a sample of a mouse heart. The setup is shown in Fig. 3(c) with the addition of lenses and an off-axis parabolic mirror. The detection and illumination central wavelengths are 0.8 µm and 3.8 µm, respectively.

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We call $G({\boldsymbol\rho _c})$ the image function. It can be readily checked from Eq. (34) that when the momenta of the twin photons are maximally correlated, $G({\boldsymbol\rho _c}) \propto | {T({\boldsymbol\rho _o})} |$.

Since ${\arg}\{T({\boldsymbol\rho _o})\}$ appears in Eq. (34), phase imaging is also possible using this scheme (Figs. 7-IIA, 7-IIB, and 7-IID). For objects with relatively simple phase distributions as the ones considered in [22], the image can be obtained by intensity subtraction.

2. Image Magnification

It follows from Eqs. (34) and (35) that point ${\boldsymbol\rho _o}$ on the object is imaged at point ${\boldsymbol\rho _c}$ on the camera. Therefore, the image magnification ($M$) is equal to the ratio $|{\boldsymbol\rho _c}|/|{\boldsymbol\rho _o}|$. It now follows from Eqs. (27) and (31) that

(37)$$M \equiv \frac{{|{\boldsymbol\rho _c}|}}{{|{\boldsymbol\rho _o}|}} = \frac{{{f_c}{\lambda _s}|{{\textbf{q}}_s}|}}{{{f_I}{\lambda _I}|{{\textbf{q}}_I}|}}.$$

We now note that image blurring must be neglected to define the magnification. Therefore, we must consider only the ideal case in which the momenta of the twin photons are perfectly correlated. As mentioned above, in this case, the probability density governing the momentum correlation can be effectively replaced by a Dirac delta function. In fact, Eqs. (34) and (35) are obtained with this condition. In particular, we consider twin photons generated by SPDC, for which $P({{\textbf{q}}_s},{{\textbf{q}}_I}) \propto \delta ({{\textbf{q}}_s} + {{\textbf{q}}_I})$. Consequently, to determine image magnification, we need to use the condition ${{\textbf{q}}_s} = {{\textbf{q}}_I}$. Applying this condition to Eq. (37), we find that image magnification is given by

(38)$$M = \frac{{{f_c}{\lambda _s}}}{{{f_I}{\lambda _I}}}.$$

An interesting feature of QIUP in the far-field configuration (i.e., enabled by momentum correlation) is that image magnification depends on the wavelengths of twin photons. This fact is illustrated by Fig. 8, which shows experimental observations presented in [83]. The wavelength dependence of magnification is also observed in the experiments reported in [22,64,65,83].

Fig. 8. Image magnification in momentum correlation enabled QIUP. The same object is imaged for two sets of values of ${\lambda _s}$ and ${\lambda _I}$, while other parameters such as focal lengths and distances are unchanged. Higher value of the ratio ${\lambda _s}/{\lambda _I}$ resulted in larger image magnification (right). (Adapted from Fig. 2 of [83].)

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Thus far, we have not considered the sign of the magnification. This is because the sign will depend on the details of the lens systems used in the setup. For example, if the lens system is exactly as chosen in [22], the image magnification will have a positive sign, i.e., the image will be erect. A detailed analysis of the image magnification for this case is presented in [23].

3. Spatial Resolution

The resolution limit of momentum correlation enabled QIUP can be studied by applying the theory discussed in Section 4.B.1. A detailed description of this topic can be found in [83]. Here we discuss one resolution measure, namely, the edge-spread function (ESF). We will consider another resolution measure in Section 4.C.3 when we will discuss the resolution limit of position correlation enabled QIUP.

We pointed out in Section 4.B.1 that the image of an absorptive object can be obtained by determining the image function, $G({\boldsymbol\rho _c})$. It follows from Eqs. (33) and (36) that

(39)$$G({\boldsymbol\rho _c}) \propto \int {\rm d}{{\textbf{q}}_I} P({{\textbf{q}}_s},{{\textbf{q}}_I})\left| {T({\boldsymbol\rho _o})} \right|,$$

where ${\boldsymbol\rho _o}$, ${{\textbf{q}}_I}$ and ${\boldsymbol\rho _c}$, ${{\textbf{q}}_s}$ are related by Eqs. (27) and (31), respectively. We will use the image function to determine the resolution because the mathematical analysis becomes simpler. We stress that the results remain the same if one obtains the image from visibility.

It is evident from Eq. (39) that when the momenta of the twin photons are not perfectly correlated, information about a range of points on the object plane appears at a single point on the camera. The broader the probability distribution $P({{\textbf{q}}_s},{{\textbf{q}}_I})$, the larger the range of the points on the object plane. Therefore, it can be readily guessed that a weaker momentum correlation results in reduced resolution.

To study the resolution quantitatively, we need to know the form of the probability density function, $P({{\textbf{q}}_s},{{\textbf{q}}_I})$. To this end, we consider twin photons generated through SPDC and assume that the pump beam has a Gaussian profile. In this case, the probability density function can be approximated in the following form (see, for example, [30,87]):

(40)$$P({{\textbf{q}}_s},{{\textbf{q}}_I}) \propto {\exp}\! \left({- \frac{1}{2}|{{\textbf{q}}_s} + {{\textbf{q}}_I}{|^2}w_p^2} \right),$$

where ${w_p}$ represents the waist of the Gaussian pump beam. Clearly, the standard deviation of the probability distribution is inversely proportional to the pump waist. Consequently, a larger pump waist (${w_p}$) results in a narrower probability distribution, i.e., enhanced momentum correlation.

To determine the ESF, a knife edge can be used as an object. The image, which turns out to be a blurred edge, effectively represents the ESF. Without any loss of generality, we assume that the knife edge is placed parallel to the ${y_o}$ axis and along the line ${x_o} = x_0^\prime $, such that the idler field is blocked form ${x_o} \le {x_0^{\prime}}$. Therefore, we can write

(41)$$T({\boldsymbol\rho _o}) \equiv T({x_o},{y_o}) = \left\{{\begin{array}{*{20}{l}}0&\,\,\,{{x_o} \le {{x^\prime_0}},}\\1&\,\,\,{{x_o} \gt {{x^\prime_0}},}\end{array}} \right.\quad \forall \;{y_o}.$$

It now follows from Eqs. (39)–(41) that the ESF is given by

(42)$${\rm{ESF}}({x_c}) \propto G({\boldsymbol\rho _c}) \propto {\rm{Erfc}}\left({\frac{{\sqrt 2 \pi {w_p}}}{{{f_c}{\lambda _s}}}\left({{x_c} - Mx_0^\prime} \right)} \right),$$

where Erfc is the complementary error function, and $M$ is the wavelength-dependent image magnification given by Eq. (38). We stress that to determine the ESF, one can also measure the position-dependent visibility instead of the image function.

Figure 9(a) shows an experimental observer image of a knife edge [83]. The experimental results are in full accordance with the theoretical predictions made by Eq. (42). A measure of image blurring is how steeply the complementary error function representing the ESF rises. A sharper rise means less blurring. Mathematically, the blurring can be quantified by the inverse of the coefficient of ${x_c}$ inside the Erfc in Eq. (42), i.e., by

(43)$$\sigma = \frac{{{f_c}{\lambda _s}}}{{\sqrt 2 \pi {w_p}}}.$$

Fig. 9. Edge-spread function (ESF) and resolution. (a) The image of a knife edge is obtained by measuring the position-dependent visibility on the camera (left). The visibility measured along an axis (${x_c}$) is fitted with error function to experimentally determine the edge-spread function (right). The blurring ($\sigma$) is determined from the ESF. (b) Experimentally measured values (data points) of $\sigma$ are compared with theoretical prediction (solid lines) for two sets of wavelengths, ${\lambda _I} = 1550 \;{\rm{nm}}$, ${\lambda _s} = 810 \;{\rm{nm}}$ (red) and ${\lambda _I} = 780 \;{\rm{nm}}$, ${\lambda _s} = 842\; {\rm{nm}}$ (blue). Since the detected wavelengths are close to each other, the blurring appears to be almost equal despite a wide difference between the illuminating (undetected) wavelengths. (c) The resolution ($\sigma /M$) is measured experimentally (data points) and compared with theoretical results (solid curves) for the same sets of wavelengths. Shorter illumination wavelength results in higher resolution. The resolution enhances with increasing pump waist (${w_p}$), i.e., with stronger momentum correlation between twin photons. (Adapted from Fig. 4 of [83].)

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This quantity can be determined from the experimentally obtained ESF [Fig. 9(a)]: one can check from the properties of the complementary error function that $\sigma$ is the distance for which the value of ESF rises from 24% to 76% of the maximum attainable value.

Equation (43) shows how the resolution depends on the momentum correlation between the twin photons. As mentioned below Eq. (40), a larger value of the pump waist (${w_p}$) implies a stronger momentum correlation between the twin photons. It follows from Eq. (43) that a larger value of ${w_p}$ results in a smaller value of $\sigma$, i.e., less blurring implying higher resolution. Figure 9(b) shows the experimentally measured values of $\sigma$ for two experimental setups [83]. The solid lines represent theoretical predictions made from Eq. (43). Figure 10(a) demonstrates how the image of a collection of three slits gets blurred when the momentum correlation between twin photons is reduced.

Fig. 10. Resolution of momentum correlation enabled QIUP. (a) Resolution enhances as momentum correlation becomes higher. A set of slits is imaged for five values of pump waist (${w_p}$) in decreasing order (left to right). A bigger value of ${w_p}$ means a stronger momentum correlation between the twin photons, which results in higher resolution. (Wavelengths are kept the same for each measurement.) (b) A smaller value of undetected wavelength (${\lambda _I}$) results in better resolution. The same set of slits is imaged for ${\lambda _I} = 1550 \;{\rm{nm}}$ (left) and ${\lambda _I} = 780 \;{\rm{nm}}$ (right), while the pump waist is kept the same. (Adapted from Figs. 3b and 5b of [83].)

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We now discuss the wavelength dependence of the resolution. Equation (43) shows that $\sigma$ does not depend on the wavelength (${\lambda _I}$) of the undetected photon that interacts with the object; it instead depends on the wavelength (${\lambda _s}$) of the detected photon that never interacts with the object. However, it must not be concluded from this observation that the resolution depends on the detected wavelength. This is because $\sigma$ is the blurring measured in the camera coordinates, and the resolution is basically the minimum resolvable distance on the object place. Therefore, a more accurate measure of resolution is obtained if one divides $\sigma$ by the magnification, $M$, which essentially results in expressing $\sigma$ in object coordinates. The division by magnification is essential for understanding the wavelength dependence of the resolution because in this imaging configuration, magnification depends on wavelength. It now follows from Eqs. (38) and (43) that

(44)$${\rm{res}} = \frac{\sigma}{M} = \frac{{{f_I}{\lambda _I}}}{{\sqrt 2 \pi {w_p}}},$$

which is a measure of resolution of this imaging scheme. It follows from Eq. (44) that the resolution depends only on the undetected wavelength, i.e., the wavelength that probes the object. Here we note that the resolution depends on the momentum correlation in the same way $\sigma$ does. Therefore, our conclusions regarding the dependence of resolution on the momentum correlation remains unchanged.

The wavelength dependence of resolution has been verified experimentally by building two experimental setups for which the values of detected wavelength are very close (810 nm and 842 nm), whereas the undetected wavelengths are widely separated (1550 nm and 780 nm) [83]. The experimental results are displayed in Fig. 9(b), which confirms the prediction made by Eq. (43): the values of $\sigma$ for the two setups are very close to each other because the values of detected wavelength are also very close. However, the experimentally obtained value of $\sigma /M$ shows that the resolution of the two setups are significantly different [Fig. 9(c)]. This is because the undetected wavelengths for the two setups are widely separated. The wavelength dependence of the resolution has also been experimentally tested by using a 1951 USAF resolution test chart and it has been found that the results match accurately with theoretical predictions [83]. In Fig. 10(b), we show images of the same collection of slits for two values of undetected wavelength while the pump waist (i.e., momentum correlation) is kept fixed. It is evident from this figure that the resolution is higher for a shorter undetected wavelength.

In [64,65], the same imaging resolution Eq. (44) was verified for the $SU(1,1)$ interferometer with illumination photons at the mid-infrared ($3\;{{\unicode{x00B5}{\rm m}}}$) and detection wavelength suitable for silicon-based cameras.

4. Resolution and Field of View

A useful parameter to calculate is the number of spatial modes per direction, also sometimes referred to as “number of spatial modes,” which is typically estimated by the field of view (FoV) divided by the spatial resolution. The FoV is straightforwardly given by the emission angle of the downconverted idler light that defines the size of the illuminating area:

(45)$${{\rm{FOV}}_{{\rm{MC}}}} = 2{f_I}\tan ({\theta _I}) \approx 2{f_I}{\theta _I},$$

where ${f_I}$ denotes the focal length of the collimating optical element adjacent to the crystal, and ${\theta _I}$ is the idler divergence angle. Defined as half-width at half-maximum (HWHM), it is given by

(46)$${\theta _I} = {\lambda _I}\sqrt {\frac{{2.78{n_s}{n_I}}}{{\pi L({n_s}{\lambda _I} + {n_I}{\lambda _s})}}} ,$$

where ${n_s}\!({n_I})$ is the index of refraction of the signal (idler) field in the crystal (see supplementary material in [64]).

The number of spatial modes per direction can therefore be estimated as

(47)$${m_{{MC}}} = \frac{{{{\rm{FOV}}_{{\rm{MC}}}}}}{{{\rm{re}}{{\rm{s}}^{{\rm{FWHM}}}}}} \propto {w_p}\sqrt {\frac{n}{{L({\lambda _I} + {\lambda _s})}}} ,$$

where ${\rm{re}}{{\rm{s}}^{{\rm{FWHM}}}} = 2\sqrt {ln2} \;{\rm{res}}$ and $n = {n_s} \approx {n_I}$. Unsurprisingly, the number of spatial modes per direction does not depend on ${f_I}$ or the magnification ($M$)—provided that the intermediate optics features a sufficiently large numerical aperture.

We conclude this section by summarizing the key features of the resolution of momentum correlation enabled QIUP. The resolution enhances if the momentum correlation between twin photons becomes stronger. For photons generated by SPDC, the resolution is linearly proportional to the standard deviation of the probability distribution that governs the momentum correlation. The resolution is linearly proportional to the wavelength of the undetected photon (i.e., the photon that probes the object). Therefore, a shorted undetected wavelength results in a higher correlation.

Finally, the method described in this section applies to any twin-photon state. Here we considered a Gaussian probability distribution. For other forms of probability distribution, an exact mathematical expression may not be obtained, and resorting to numerical simulation may be necessary.

C. Imaging Enabled by Position Correlation