Quantum indistinguishability by path identity and with undetected photons

Two processes of photon-pair creation can be arranged such that the paths of the emitted photons are identical. The path information is thereby not erased but rather never born in the first place due to this path identity. In addition to its implications for fundamental physics, this concept has recently led to a series of impactful discoveries in the fields of imaging, spectroscopy, and quantum information science. Here the idea of path identity is presented and a comprehensive review of recent developments is provided. Specifically, the concept of path identity is introduced based on three defining experimental ideas from the early 1990s. The three experiments have in common that they contain two photon-pair sources. The paths of one or both photons from the different sources overlap such that no measurement can recognize from which source they originate. A wide range of noteworthy quantum interference effects (at the single- or two-photon level), such as induced coherence, destructive interference of photon pairs, and entanglement generation, are subsequently described. Progress in the exploration of these ideas has stagnated and has gained momentum again only in the last few years. The focus of the review is the new development in the last few years that modified and generalized the ideas from the early 1990s. These developments are overviewed and explained under the same conceptual umbrella, which will help the community develop new applications and realize the foundational implications of this sleeping beauty.

Figure 1

Quantum imaging by path identity, without detecting the photons that interact with the object ([139]). The yellow beam coherently pumps two nonlinear crystals such that one of them creates a photon pair. The photon pair is in a superposition of being created in the first or second crystal. When the green photon path is made identity, the red photon is in a superposition of being in the upper or lower path, which leads to interference at a beam splitter. By introducing an object (a picture of a cat) into the overlapped green path, the origins become partially distinguishable, which can be observed in the resulting interference pattern. The image of the cat is constructed without detecting any of the green photons. This concept is entirely different than conventional ghost imaging, where both photons need to be detected, and which can be performed classically ([17]).

Figure 2

Papers that explore quantum indistinguishability by path identity. After the Zou-Wang-Mandel experiment and its variations was studied for a decade, research nearly stopped. We can observe an awakening of the research interest in 2014, with exciting new applications including quantum imaging and microscopy, quantum spectroscopy, and quantum information. To create this list, we use all publications that cite the original Zou-Wang-Mandel experiment ([244]). From there, we select those papers that actually explore the underlying ideas and concepts rather than just mentioning them. A paper or concept with this behavior is called “sleeping beauty” ([104]).

Figure 3

Three historic experiments initiated the research in path identity. (a) The experiment by Zou, Wang, and Mandel first showed that identifying the path of photons can induce coherence in an unexpected way ([224]; [244]). (b) [82] identified that two-photon interference phenomena occur when it is fundamentally indistinguishable in which crystal the photon pair was created. (c) [79] proposed the mode shifting and identification of two paths as a method for the generation of quantum entanglement. The source has become a cornerstone of photonic quantum entanglement experiments ([127]; [172]; [239]). The review concerns these ideas and their generalizations and applications during the last quarter of a century.

Figure 4

Zou-Wang-Mandel experiment. $A$ and $B$ are two identical photon-pair sources. $A$ and $B$ emit a photon pair in paths $(S_a,I_a)$ and $(S_b,I_b)$, respectively. Paths $I_a$ and $I_b$ are made identical. An object with the amplitude transmission coefficient $T$ is placed on $I_a$ between $A$ and $B$. A single-photon interference pattern is observed at both outputs of the beam splitter (BS) if $S_a$ and $S_b$ are superposed. Visibility is proportional to the modulus of the amplitude transmission coefficient.

Figure 5

Nonclassicality of the Zou-Wang-Mandel experiment. (a) Experimental data show that the visibility of the single-photon interference pattern is linearly proportional to $|T|$. Adapted from [244]. (b) $g^{(1)}(1,2)$ refers to visibility and $t$ is the same as $|T|$ in the text and in (a). The theoretical curves show that when the effect of stimulated emission is prominent, the linear dependence is no longer observed. Adapted from [229].

Figure 6

Degree of polarization and path identity. (a) Schematics of the experiment. (b) Experimentally observed dependence of the degree of polarization on $|T|$ for various values of $\gamma$. The solid lines represent theoretical curves considering experimental imperfections. Adapted from [131].

Figure 7

An extension of the ZWM experiment to three crystals allows for more general tests of the complementarity principle. From [84].

Figure 8

Three crystal setups. (a) If the coherence lengths are long enough, the reflectivity of optical sample OS1 affects the interference pattern between NL2 and NL3. This is possible because the two samples OS1 and OS2 form an optical cavity. (b) Intuitively, if the reflectivity of OS1 is high, an idler photon from NL2 has many tries to pass through OS2 because idler photons that are not transmitted by OS2 remain inside the cavity and can still be transmitted to NL3 later. This increases the effective transmission and thus the visibility observed at D23. From [138].

Figure 9

(a) Imaging setup. NL1 and NL2 are two identical nonlinear crystals pumped coherently by 545 nm laser light. Each crystal can produce a signal (810 nm) and idler (1550 nm) photon by collinear spontaneous parametric down-conversion. Dichroic mirrors separate signal and idler light and are sent to different paths. The alignment of $I_1$ and $I_2$ gives the path identity. The object is placed on $I_1$ between NL1 and NL2. $4f$ lens systems are used in the signal and idler arms. (b) $4f$ lens system on an idler path. An idler wave vector is focussed at a point ${\mathit{\rho }}_o$ on the object plane. The emerging spherical wave from ${\mathit{\rho }}_o$ is converted back to a plane wave. (c) Detection system. The corresponding signal wave vector is focused at a point ${\mathit{\rho }}_c$ on the image plane (camera). Adapted from [135].

Figure 10

(a) The two outputs of the beam splitter. The image of the cat shows the area where interference occurs. (b) The absorptive object: the cat. (c) The sum of the outputs of the beam splitter contains no image. (d) Subtraction of the outputs leads to enhancement of the image contrast. Adapted from [139].

Figure 11

Midinfrarer microscopy with undetected photons. Two separate experiments demonstrated how a magnification system in the idler photon can lead to a microscope where the magnified image can be observed with standard infrared cameras. (a) Characterization of an imaging arrangement. The yellow curves indicate the unmagnified setup, while the green one shows the magnified one. The system shows a magnification of roughly 10 times. From [124]. (b) Resolution test measured with different focusing lenses. From [178].

Figure 12

Ghost imaging is fundamentally different from imaging by path identity. In ghost imaging, the photons interacting with the object must be detected, and coincidence or an equivalent measurement must be performed. Adapted from [184].

Figure 13

Principle scheme of spectroscopy with undetected photons. Two nonlinear crystals probabilistically emit identical and frequency correlated photon pairs. Identifying the paths of the respective photon pairs then allows one to observe interference in the signal photon. The acquired phase in the interference pattern depends on all three photons involved, the pump, signal, and idler, which are all at different frequencies. The interference visibility is governed by the transmission of the sample placed in the idler beam. Thus, at an absorption line of the sample under study the visibility in the spectrogram detected at the camera disappears. From Jean-Pierre Wolf.

Figure 14

Experimental details of spectroscopy with undetected photons. Two lithium niobate nonlinear crystals are employed to create the nondegenerate but perfectly frequency correlated photon pairs. A quasicollinear emission scheme is used to identify their respective paths, and the emission angle $\theta$ of the down-conversion is small relative to the pump beam waist $a$. A vacuum chamber is used to host the two nonlinear crystals and the sample of interest, which is ${\mathrm{CO}}_2$ here. A lens images the down-converted photons onto a slit to select a specific angular emission spectrum. The following spectrograph enables precise determination of the signal wavelength. Finally, a two-dimensional spectrogram in the angular-wavelength dimensions is recorded by a camera. Adapted from [103].

Figure 15

Experimental results. (a) Calibration measurement in a quasivacuum regime (inset). Selecting a specific wavelength cut and fitting Eq. (18) yield the absorption coefficient for the vacuum and a phase reference. (b) Spectrograph with a filled carbon dioxide chamber. The decrease of visibility and the phase shift is due to the absorption and refractive index change between the vacuum in (a) and the carbon dioxide in (b). The data displayed in (c) and (d) show the wavelength dependence of the absorption and refractive index coefficient in the vicinity of the resonance for carbon dioxide at a pressure of 10.5 torr. The blue squares represent experimental measurements, and the red curve is a theoretical calculation using HITRAN data for (c) and a Kramers-Kronig relation for (d). Data figures from [103].

Figure 16

Terahertz quantum sensing using frustrated photon-pair generation. (a) A conceptual depiction shows that the signal $s_i$ goes through the nonlinear crystal (NL) and is directly reflected back. The idler $i_i$ is reflected at an indium tin oxide (ITO) coated glass and interacts with the object before it is reflected back to the crystal. Two-photon quantum interference there allows one to measure the depth information of the object. (b) Concrete plot of the experimental setup, with pump laser, wave plates ($\lambda /2$), a nonlinear periodically poled lithium niobate crystal (commonly known as PPLN), volume Bragg grating (VBG), transmission grating (TG), and spatial filters (SFs). Adapted from [122].

Figure 17

Classical optical coherence tomography and optical coherence tomography with undetected photons. (a) Classical OCT is typically implemented with light of a short coherence length in a Michelson interferometer. The reflection from different layers within the object produces an interference pattern if recombined with light that has traveled the corresponding distance in the second path. The sample can be laterally resolved by tuning this distance using the mirror $M$. (b) The specified implementation of OCT with undetected photons uses nondegenerate photon pairs produced by a nonlinear crystal. The signal (idler) beam emitted toward the right is reflected back through the crystal via the mirror $M$ (via the sample). Consequently, interference in the rate of detected photon pairs can be observed. Only the signal beam is detected toward the left of the source. The observed interference is affected by the properties of the sample if the coherence length requirements are met. This is the case if the idler photon is reflected at a particular depth of the object that corresponds to the length of the signal path. It is thus possible to probe different depths of the sample. (c) Example showing the detected rate of signal photons (at 582 nm) as the signal path length is scanned by translating the mirror $M$ in the $z$ direction. The sample in the idler beam (at 3011 nm) in this case is a Si window. Reflections at both the front and back surfaces as well as multiple reflections result in interfering signal photon rates at the corresponding mirror positions. Adapted from [176].

Figure 18

Fourier-domain optical coherence tomography (FDOCT). (a)(i) Classical FDOCT. (a)(ii) The quantum version with undetected photons. In the experiment of [215], they produced a midinfrared photon that interacted with the sample. The information is obtained through a spectrometer that detects the 800 nm signal photon. FDOCT has no moving parts and thus promises significantly higher sampling rates and stability. Adapted from [215].

Figure 19

(a) Top panel: no image is obtained when the object is imaged by 810 nm light. Bottom panel: both outputs of the beam splitter contain the image when the object is placed in the idler beam (1550 nm). (b) The phase object that is invisible to 810 nm light. Adapted from [139].

Figure 20

Interference fringes produced with undetected photons. The spatial interference pattern observed in the superposed beam can be controlled by manipulating a third beam, which is not detected. The appearance of the pattern is analogous to that obtained if the same manipulations were performed on one of the interfering beams in a traditional two-path interferometer. From [88].

Figure 21

Wavelength dependence of circular interference fringes. The first two lines depict classical interferometers and circular fringes obtained by shifting the lens by a fixed distance. In (a) the light is at the signal wavelength (810 nm), whereas in (b), it is at the idler wavelength (1550 nm). (c) The interference fringes obtained in the ZWM setup, where the spacing of the fringes corresponds to the equivalent wavelength ${\lambda }_{\mathrm{eq}}={\lambda }_S^2/{\lambda }_I$. From [87].

Figure 22

Measuring the transverse momentum correlation of a photon pair with and without coincidence detection. (a),(b) In the traditional method, each of the two photons is registered by a detector in a way such that its individual momentum can be inferred. By comparing the coincidence detection events at different pairs of momenta, the correlation is quantified. (a) A sharp correlation (implemented by a large pump focus) leads to a narrow coincidence peak at a particular relative momentum, whereas (b) weakly correlated photons (small pump waist) lead to a similar amount of coincidence counts in a wider range of momenta ([161]). Using the concept of path identity, two copies of the photon-pair source are arranged in a ZWM configuration. By introducing a spatially dependent phase shift on the undetected idler beam between the two sources, a single-photon interference pattern is observed in the superposed signal beam. The visibility of the pattern depends on how strongly signal and idler photons are correlated. (c) In the case of a loosely focused pump beam, the photons are highly correlated in momentum, and spatial features of an interference pattern are visible. (d) The opposite is true of weakly correlated photons. The visibility is determined from measurements on only one of the photons and can be used to quantitatively reconstruct the correlation between the two photons. Adapted from [87].

Figure 23

Experimental results of the correlation measurement between two photons when only one of them is detected. The correlation between the signal and idler photons was tuned by changing the waist ($w_P$) of the pump beam in both crystals simultaneously. The fringe pattern observed in the interference of the two signal beams gradually exhibits a higher visibility as the correlation between the signal and idler photons is increased. This allows one to numerically reconstruct the variance of the conditional probability distribution of the momentum of an idler photon, given the momentum of a signal photon, without relying on coincidence detection. The results are shown compared to the theoretical prediction. Adapted from [89].

Figure 24

New scheme to create two-photon high-dimensional entangled photon pairs using path identity, which is denoted as entanglement by path identity ([116]). Consecutively stacked indistinguishable photon-pair sources [nonlinear crystals (NLs)] coherently emit photon pairs. Identifying their paths leads to a coherent superposition of possible origins of a photon pair. Placing spiral phase plates between two NLs that add one quantum of orbital angular momentum to the incoming photons results in a $d$-dimensional entangled quantum state $|\psi ⟩$.

Figure 25

Detailed experimental setup for creating three-dimensional entangled photon pairs using path identity. A continuous-wave laser centered at 405 nm is split with polarizing beam splitters (PBSs) and is guided to three NLs made from periodically poled potassium titanyl phosphate, which is better known as KTP. For experimental simplicity, the spiral phase plate (SPP) is placed within the pump beam instead of the down-converted photons (as depicted in Fig. 24). A lens (Lf) is used to focus the pump beam onto the NL. Using a $4f$ optical imaging system between two consecutive NLs ensures the spatial indistinguishability of the down-converted photon pairs. Utilization of an actively stabilized Mach-Zehnder interferometer using a proportional-integral-derivative (PID) controller ensures the interferometric stability between two NLs. The phase between two NLs can be set using the quarter-half-quarter (QHQ) wave plates of the stabilization laser depicted in green. The photon pairs are deterministically separated using a PBS in the detection part. Arbitrary superposition projections can be measured with a spatial light modulator (SLM) in combination with a single-mode fiber. From [128].

Figure 26

Coincidence interference fringes of two consecutive nonlinear crystals. The relative phase ${\phi }_1$ between NL1 and NL2 has been altered. The coincidences are measured without a SPP; thus, the interference is occurring between the fundamental Gaussian modes (denoted as 0) and the corresponding quantum state reads $|0,0{⟩}_{\mathrm{NL}1}+e^{i\phi 1}|0,0{⟩}_{\mathrm{NL}2}$. If the observed visibility reaches 1, then the two photon-pair sources are identical, meaning that $\mathrm{NL}1=\mathrm{NL}2$. In this experiment, a visibility of $0.971\pm 0.005$ is observed. From [128].

Figure 27

Multiphoton Greenberger-Horne-Zeilinger (GHZ) state creation. (a) The commonly used technique ([172]) to create GHZ states is to overlap two maximally entangled two photons in the state $|{\varphi }^{+}⟩=(|H,H⟩+|V,V⟩)/\sqrt{2}$, with $H$ and $V$ denoting the polarization state of the photon. Since the PBS transmits only horizontally ($H$) polarized photons and reflects only vertically ($V$) polarized photons, simultaneous $2n$-fold detection of photons at detectors $\{d1,\dots ,d2n\}$ results in a maximally and genuine multiphoton ($2n$) GHZ state. (b) Creation of multiphoton GHZ states using path identity. Displayed are two rows of $n$ crystals, where all $2n$ crystals can coherently emit $n$ photon pairs. The lower row creates vertically polarized photons, while the upper row solely creates horizontally polarized photon pairs. Identifying the paths of the photons in the lower row with the ones in the upper row (as previously indicated) and conditioning upon $2n$-fold photon detection result in a GHZ state. Because of the path identification, a $2n$-fold photon detection event can occur only if all crystals of the upper row emit photon pairs or if all crystals in the lower row emit photon pairs.

Figure 28

Genuine multiphoton and high-dimensional quantum entanglement using path identity. Six NLs labeled with roman numbers where each NL can emit photon pairs in the transverse spatial Gaussian mode (denoted by $|0,0⟩$) are utilized. Each photon pair emitted by one of the six crystals is connected to a pair of detectors $\{A,B,C,D\}$ as indicated on the right in such a way that overlapping paths of different crystals are identified. Simultaneous four-photon events where each detector registers a photon can occur in three possible ways: The lower, middle, or top row emits two pairs simultaneously. None of the other combinations can occur because of the specific path identification and routing of the photon paths. Inserting spiral phase plates between the different layers of crystals adds one quantum ($+1\hslash$) of orbital angular momentum (OAM) to the incoming photons. Since all photon-pair emissions occur coherently, the three possibilities can be written in a coherent superposition yielding a genuinely four-photon and three-dimensional entangled GHZ state $|\psi ⟩$.

Figure 29

Scheme of generating and controlling many-particle entangled states. Two identical $N$-particle sources emit particles $(1,2,\dots ,N)$ into paths $(b_1,b_2,\dots ,b_N)$ and $(b_1^{\prime },b_2^{\prime },\dots ,b_N^{\prime })$, respectively. Path identity is applied for $M$ particles $(N-M+1,\dots ,N)$. The rest of the particles $(1,2,\dots ,N-M)$ produce many-particle interference patterns and many-particle entangled states when their paths are superposed by beam splitters. Adapted from [130].

Figure 30

Three-dimensional GHZ state via entanglement by path identity. (a) A quantum optical experiment for the generation of a three-dimensional GHZ state. (b) Its correspondence to a graph. Every vertex is a photon path, and every edge corresponds to a nonlinear crystal. Colors represent the mode numbers. (c) A resulting state, which arises conditioned on fourfold coincidence clicks, corresponds to perfect matchings of the graph.

Figure 31

Design of quantum experiments using graph theory. The connection to graph theory is an ideal descriptive tool of quantum experiments and can be used to design experiments in order to design specific quantum states. In this example, a general recipe for the generation of $W$ states is given. From [74].

Figure 32

Quantum random networks ([179]) have interesting critical properties, such as the emergence of certain quantum states when the edge probability $p_c(N)>N^{-2}$, where $N$ are the vertices. (a) A specific random graph with 14 edges connecting eight vertices is a quantum random network. (b) The corresponding setup consists of 14 crystals that are coherently pumped and output into $N=8$ paths. The SPDC probability $p$ corresponds to the edge probability. From [115].

Figure 33

Graphical approach to linear-optical quantum experiments developed by [7], [8]. (a) Depiction of the ZWM experiment including the pump beam. (b) The translation to a graphical model. The idea is to represent photons as creation operators. The transformation of every linear-optical element as well as nonlinear crystals for the creation of photon pairs can be understood in photon transformations of the photon’s paths. From [8].

Figure 34

Interference of photons from different sources. (a) A beam of a local oscillator (a laser beam with a defined coherence relation to the pump beam of the SPDC) overlaps with the output of a SPDC crystal. There are two different possibilities how a pair of photons could be generated: either (b) via SPDC or (c) from the weak local oscillator. Resch, Lundeen, and Steinberg were able to make these two possibilities indistinguishable, and therefore observed interference between them. In (d), the solid circles represent coincidence counts, with a fringe visibility of more than 50%. The open squares stand for the single counts, in which one can also see statistically significant modulation, as the phase between the two possibilities varies. Adapted from [190].

Figure 35

Constructive and destructive quantum interference based on path identity ([75]). (a) Four crystals aligned such that the emission of fourfold coincidence clicks in all four detectors $a$, $b$, $c$, and $d$ can happen only when crystals I and II emit a pair of photons or crystals III and IV emit a pair of photons each. Here all photons are indistinguishable. These two possibilities lead to two terms in the quantum state, which are coherently superposed. A phase shifter in one of the arms changes the relative phase between these terms, thus leading to either an increased or a decreased rate of fourfold counts. (b) An interpretation of using graph theory where weighted edges lead to phases between the perfect matchings, which can cancel each other. This interpretation will help one to find and understand follow-up applications. (c) Depiction of the rate of twofold counts in detectors $a$ and $b$ and fourfold counts in all four detectors when the phase $\varphi$ is rotated. While the pair counts do not change, the fourfold counts can vanish.

Figure 36

Frustrated pair creation was observed on an integrated silicon photonic chip by [166] in the Rarity group. The source of photon pairs is a spontaneous four-wave mixing process of a ${\chi }^3$ nonlinearity. Two sources can each create a pair of photons. As the origin of the pair is undefined, it is in a coherent superposition of being created in either. Thus, Ono et al. observed constructive and destructive interference of the resulting photon pair.

Figure 37

Multiphotonic interference can be exploited for special-purpose quantum computation ([75]). (a) Path identified photons produced in a random network of crystals. The network has six output modes and is pumped in such a way that events of more than four photons can be ignored. (b) Four-photon count rates. Every experimental setup corresponds to a weighted undirected graph $G_H$. A fourfold coincidence count (for example, in outputs $a$, $b$, $c$, and $e$) corresponds to a subgraph $G_{H_S}$ of those vertices in $\mathbf{C}$. The count rate in these output modes can be calculated as the coherent sum of all weighted perfect matchings of $G_{H_S}$. This is equivalent to calculating the matrix function Hafnian applied on the adjacency matrix of the graph $G_{H_S}$ seen in $\mathbf{D}$ (a problem known to be extremely difficult to calculate).

Figure 38

(a) Quantum circuit describing a frustrated SPDC ([82]). (b) Each mode is interpreted as a three-level state. The first level is the state of a nonexistent photon, while states 2 and 3 encode the usual computational qubit. The pair creation of NL1 and NL2 (nonlinear crystals 1 and 2) are described by two CNOT gates and one controlled-unitary gate. If no qubit exists in the mode, it creates one in the computational state $|1⟩$. If a qubit with state $|1⟩$ exists, it will annihilate the qubit. Formally, the mode can be considered as a qutrit (three-level system). The path alignment between the crystals is governed by two SWAP gates. From [4].