Optics in Our Time pp 435454  Cite as
Quantum Mechanical Properties of Light Fields Carrying Orbital Angular Momentum
Abstract
This Chapter presents a review of the quantum mechanical properties of spatially structured light fields, specifically those fields carrying orbital angular momentum (OAM). This review is concerned both with the conceptual understanding of the quantum features of these light fields and with the use of these features for applications. We describe how to produce spatially entangled light fields by means of the nonlinear optical process of parametric downconversion. We ask and provide a tentative answer to the question of how much information can be encoded into a single photon. As an example, we review a recent experiment that demonstrated the ability to discriminate among four target objects using only one photon for illumination. We also present a description of the concept of the OAM of light, and we describe means to generate and detect OAM. We then present a brief survey of some recent studies of the fundamental quantum properties of structured light beams. Much of this work is aimed at studying the nature of entanglement for the complementary variables of angular position and OAM. Finally, as a realworld application, we describe a secure communication system based on quantum key distribution (QKD). This key distribution system makes use of encoding information in the OAM modes of light and hence is able to transmit more than one bit of information per photon.
Keywords
Orbital Angular Momentum Light Field Spatial Light Modulator Entangle Photon Digital Micromirror Device17.1 Introduction
There is a growing appreciation of the importance of quantum mechanics and in particular of quantum information science both for understanding the nature of the world in which we live and in the development of new technologies for communication and imaging. It is in this spirit that this chapter is written. The chapter deals with structured light fields, especially fields that carry orbital angular momentum (OAM), and their application to communication systems.
The topic of structured light fields in the quantum domain is intimately related to the topic of quantum imaging [1]. Quantum imaging is a discipline that studies quantum aspects of image formation and that uses quantum properties of light to produce images. Imaging techniques inherently require the ability to encode massive amounts of information in a light field. The quantum aspects of optical images can therefore be a key resource for quantum information and communication systems.
In this chapter we will review several examples of the quantum properties of structured light field. In broad concept, there are two sorts of quantum states that appear in our examples. One sort is a “singlephoton” state, a state in which it is known that there is one and only one photon present in the field of interest. Such a state shows strong quantum properties because, for example, if you send such a beam onto a beamsplitter, the photon will emerge in one of the two output ports but not half and half in both. The other sort of quantum state of interest in this chapter is an entangled state of two photons. In fact, the concept of entanglement is one of the great mysteries of quantum mechanics. The term entanglement and the first explicit description of this phenomenon were introduced by Schrödinger in 1935 [2]. Here is quote from his paper:
“When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought.”
This concept of entanglement leads to what today is often called “quantum weirdness,” a term that arises from the seemingly paradoxical effects that can occur in an entangled system. Entanglement has, for instance, played a key role in compelling laboratory demonstrations [3, 4] of the nonlocality of quantum phenomena. However, this weirdness has in fact turned into an asset. This entanglement weirdness leads to effects such as quantum teleportation and certain forms of quantum communication with guaranteed security. One of the easiest ways to produce quantum entanglement uses nonlinear optical methods, and it is these methods that therefore give the field of photonics a special and elevated position in the arena of quantum technologies.
The concept of entanglement generation as applied to light fields can be visualized using the drawing shown in Fig. 17.1. Here a laser beam at frequency ω_{ p } excites a secondorder nonlinear optical crystal, whose nonlinear response can be characterized in terms of its secondorder susceptibility χ^{(2)}. Occasionally, a pump photon can be absorbed and generates two lowerfrequency photons of frequencies ω_{ s } and ω_{ i }, a process known as spontaneous parametric downconversion (SPDC) [5, 6, 7]. It can be shown that the rate at which photon pairs are created is proportional to the product [χ^{(2)}]^{2}L^{2}I_{ p }, where L is the length of the nonlinear crystal and I_{ p } is the intensity of the pump laser.
By conservation of energy, the condition \( {\omega}_p={\omega}_s+{\omega}_i \) must be satisfied, as illustrated in part (b) of the figure, furthermore these two new photons are generated at the same position (i.e., position correlated). However, photon momentum must also be conserved in this generation process, as illustrated in part (c) of the figure, and hence the two new photons are generated with opposite transverse momentum components (i.e., momentum anti correlated). It is these simultaneous conditions on position and momentum that lead to the paradox of Einstein et al. [8] and the concept of quantum entanglement [2].

position and transverse momentum

angular position and orbital angular momentum

time and energy

polarization in different measurement bases.
Examples of the first two types of entanglement will be presented later in this chapter. Here we present a brief discussion of the other two types of entanglement.
By timeenergy entanglement, one means that if one measures, for example, the energy of the signal photon, one is able to predict with certainty that the energy of the idler photon will be given by \( \hslash {\omega}_i=\hslash {\omega}_p\hslash {\omega}_s. \) However, if one instead chooses to measure the moment of time at which the signal photon is emitted, one will always find that the idler photon is emitted at exactly the same moment. It seems that the product of uncertainty in tightness in the correlation of energies multiplied the uncertainty in the correlation of times can be arbitrarily small and certainly smaller than the value \( \frac{1}{2}\hslash \) that one might have envisaged from the naive application of uncertainty relations [9, 10]. The situation is the essence of entanglement: the resolution of this seeming paradox is that a measurement that one performs on the signal photon results in a restriction of our ability to predict the properties of the idler photon, even if that idler photon is arbitrarily distant from the signal photon measurement location. This nonlocality which applies to entangled systems leads to the phrase “spooky action at a distance.”
Polarization entanglement can be similarly described. Under certain circumstances [6], each of the photons emitted by SPDC will be unpolarized, that is a complete statistical mixture of two orthogonal polarization states. However, for any one particular measurement the polarization of the signal photon will be found to have a defined value; one says that the measurement process projects the polarization state unto one of the polarization eigenstates. Furthermore, one finds that the idler photon will always be projected onto a polarization state that is orthogonal to that of the first photon (due to the conservation of angular momentum in the down conversion process). This type of entanglement is conveniently described in terms of the Bell states, which have played a key role in the understanding of many of the conceptual foundations of quantum mechanics. Details can be found in the excellent textbook of Gerry and Knight [11].
17.2 How Much Information Can One Photon Carry?
In classical optical telecommunication systems, many photons are required to transmit one bit of information. But it can be interesting to turn this question around and ask it differently: How much information can be carried by a single photon? Perhaps surprising to some is that research conducted over the last decade shows that there is no fundamental limit to the amount of information that can be carried by a single photon.
We start this section by giving a specific example of the ability to transfer many bits of information for each photon; this example will be developed in greater depth in the following sections. Laboratory procedures now exist for switching between singlephoton states in any one of the Laguerre–Gaussian modes (Eq. (17.3)) of light using, for example, liquid crystalbased spatial light modulators [12] or a digital micromirror devices (DMD) [13]. It is crucial to recall that the Laguerre–Gaussian modes constitute an infinite set of basis functions. Thus, to the extent that one can perform OAM encoding and decoding with high efficiency, there is no limit to the amount of information that can be carried by a single photon.
The ability to encode more than one bit per photon is, of course, not restricted to the Laguerre–Gaussian light beams. More generally, the transverse degree of freedom of the light field offers a means to carry and manipulate quantum information. An example of multibit information transfer relating to imaging is provided in an experiment performed by Broadbent et al. [14]. A schematic of this experiment is shown in Fig. 17.2. Part (a) of the figure shows a multiplexed hologram of objects A and B. By saying that the hologram is multiplexed, we mean that different writebeam directions are used to form the interference fringes for each object. Part (b) of the figure shows the readout stage. It makes use of entangled photons created by parametric downconversion in a BiBO crystal. One of these photons falls onto the trigger detector, which heralds the presence of the photon in the other arm. This photon falls onto an object in its arm, which could be either object A or B. This photon is diffracted from the hologram into the path of either detector A or B, depending on which object is placed in this arm. In this manner, one can determine with high reliability which object is located in this arm, even though only one photon is used to make this determination. Quantitative results are presented in the paper. It is shown that the likelihood of a misidentification (that is, for example, that the photon is detected by detector B when in fact object A is present) is less than 1 %.
The experiment just described shows that one can discriminate between two objects using singlephoton illumination. A subsequent experiment [15] demonstrated the ability to discriminate among four objects, again using only singlephoton illumination. For this experiment a “quantum ghostimaging” protocol [16] is used. The setup is shown schematically in Fig. 17.3. Spatially entangled photons are again created by the process of parametric downconversion. One of these photons illuminates one of the four test objects (only two are displayed in the diagram to avoid clutter) and the other falls onto a multiplexed hologram, where it is diffracted into one of four output ports. Coincidence events between the reference detector R and one of the detectors A, B, etc. are recorded. In this figure, DM denotes a dichroic mirror for blocking the pump laser and IF is an interference filter with a 10 nm bandwidth, centered at 727.6 nm.
The results of this experiment are shown in Fig. 17.4. Note that input object a produces counts predominantly in detector A, and similarly for the other three objectdetector combinations. The data are displayed using two different normalization conventions. In part (a), data for each objectdetector combination are normalized by the maximum coincidence count for the corresponding object. In part (b), the T/A ratio is calculated by dividing the total coincidences by the accidental coincidences for each objectdetector combination. Part (c) of the figure shows the four test objects. These results show that one can reliably discriminate among four objects even when they are illuminated with weak light at a single photon level. However, in this experiment the detection efficiency was low, and thus more than one photon needed to illuminate the object in order to make an unambiguous determination. In fact, for the sort of simple, multiplexed hologram used in this experiment the maximum detection efficiency (that is, assuming lossless optical elements and unit quantum efficiency detectors) is equal to 1∕N (where N is the number of objects). However, there seems to be no reason in principle [17] why a hologram could not be designed to give a maximum detection efficiency of unity.
We note that this sort of ghost imaging relies upon correlations between photon pairs. If detector R registers a photon, we know with certainly that this photon possessed the transverse mode structure given by the transmission function of the object in its path. Since this detected photon is entangled with the photon in the other arm, this detected photon must therefore acquire the same conjugate mode structure, and thus be diffracted by the hologram into a specific output port.
17.3 Light Beams that Carry Orbital Angular Momentum
We can understand why the field given by Eq. (17.1) carries angular momentum with the help of the sketch in part (a) of Fig. 17.5. We see that such a field possesses a wavefront structure in the form of a helix, and that the phase at each point advances in the azimuthal direction at a rate proportional to the value of l. One might well imagine that a small particle placed in such a beam would experience a radiation pressure in the direction of phasefront normal and hence a force with an azimuthal component that induces the object to begin to rotate around the beam axis, and in fact this is just what has been observed experimentally [19, 20].
Some methods for the production of beams that carry OAM are sketched in Fig. 17.5. The use of a spiral phase plate is shown in part (a) [19]. Another important method for generating OAM light beams (part b) is to impress a specially designed computergenerated hologram (CGH) taking the form of a forked diffraction grating [12, 20, 21, 22, 23] or a digital micromirror device (DMD) [13]. If a beam with nearly plane wave fronts, such as a Gaussian laser beam, is made to fall onto such a CGH, the diffracted light will acquire the desired form of a beam carrying OAM. Another means to form beams carrying OAM is through the use of a device known as a qplate [24, 25]. This device is a birefringent phase plate in which the orientation of the birefringent axes varies uniformly as a function of azimuthal position around the axis of the plate. Such a device acts as a spin angular momentum to OAM converter, that is, the OAM carried by the output beam depends on the polarization state of the input beam. A qplate can thus serve as a quantum interface between polarizationencoded quantum light states and OAMencoded quantum light states.
17.4 Fundamental Quantum Studies of Structured Light Beams
In Sect. 17.1 of this chapter we noted that the process of spontaneous parametric downconversion can lead to entanglement in several different degrees of freedom, including positionmomentum [26], timeenergy [9, 10], polarization [3, 4], and superpositions of OAM modes [27] or explicitly angleOAM [28]. In this section we provide a brief account of work aimed at studying these various types of entanglement.
In Einstein, Podolski, and Rosen’s (EPR’s) classic paper [8], they argued against the completeness of quantum mechanics. Their argument was based on the situation of two particles that were strongly correlated both in position and momentum. Later, David Bohm [29] restated this argument in terms of two particles entangled in their spin (or polarization), and it was this spinversion of the EPR paradox that was treated by John Bell in devising his celebrated Bell inequalities. In the ensuing decades, most subsequent work [3, 4] has concentrated on the polarization of the EPR paradox. In 1990 Rarity and Tapster [30] extended the Bell violation to one based on measurement of phase and momentum. Howell et al. [26] later performed an experimental investigation in which they studied the original (i.e., Einstein et al. [8]) positionmomentum version of the EPR paradox. Some of their results are summarized in Fig. 17.6. Their experimental procedure is as follows. Photons entangled in position and momentum were created by typeII parametric downconversion in a BBO nonlinear crystal, and the two photons were separated by a polarizing beamsplitter (PBS) and traveled over separate paths. Either the birthplace (i.e., position) of each photon or its transverse momentum could be determined by placing a slit followed by an area detector either in a focal plane of the crystal or in its far field, respectively. Coincidence counts between the two detectors were measured, and the conditional count rates associated with position and momentum are shown in the graphs on the righthand side of the figure. The measured conditional uncertainty product is found to be \( {\left(\varDelta {x}_2\right)}_{x_1}{\left(\varDelta {p}_2\right)}_{p_1}=0.1\hslash \), which is ≈ five times smaller than which might be expected for the uncertainty principle as applied to independent particles.
There has also been considerable interest in studies of timeenergy entanglement. For example, AliKhan et al. [10] have developed a protocol for quantum key distribution (QKD) that can encode as much as ten bits of information onto a single photon. In a separate study, Jha et al. [10] have studied timeenergy entanglement controlled by a geometrical (Berry) phase on the Poincare sphere instead of by using a dynamical phase. The ability to manipulate entanglement by means of a geometrical phase could have important consequences for quantum information technology, because polarization controllers can be much more stable than translation stages needed to actively control optical path lengths.
In a related experiment, Jha et al. [33] studied the correlations between the OAM values of two entangled photons after each had passed through separated doubleslit apertures. Strong, nonclassical, correlations were observed in the resulting interference pattern. These authors also showed that under their experimental conditions the visibility of this interference pattern was numerically equal to the concurrence of the twophoton state, a measure of the degree of entanglement between the two photons. A measured visibility of between 85 and 92 % quantifies the nonclassical entanglement of the photons produced by their twophoton source.
In still another related experiment, Leach et al. [28] performed an experiment that is the direct analog of the Einstein–Podolsky–Rosen (EPR) gedankenexperiment, but in the angleOAM degrees of freedom. Part of the motivation for this study is that, unlike the continuous and unbounded variables in positionmomentum entanglement considered by EPR, angular position is a periodic variable leading to a discrete OAM spectrum. As such, the analysis of EPR for angles involves subtleties similar to the issue of the existence of photonnumber photonphase uncertainty relation [34]. The details of this experiment are presented in Fig. 17.8.
There has also been great recent interest in harnessing the radial modes of Laguerre–Gaussian (LG) beams in addition to the azimuthal modes that we have primarily discussed up to now. One reason for this interest is to increase the information capacity of a light beam of a given restricted diameter. We note that the LG modes of Eq. (17.3) depend on two indices, the azimuthal index ℓ and the radial index p. But there are also further subtleties involved in exploiting the radial distribution, related to the fact that the radial coordinate ρ ranges from 0 to \( \infty \), unlike the azimuthal coordinate ϕ, which ranges from 0 to 2π. Recently, Karimi et al. [35] presented a theoretical analysis of the operator nature of the radial degree of freedom. Moreover, Karimi et al. [36] have studied the dependence of Hong–Ou–Mandel interference on the transverse structure of the interfering photons.
17.5 Secure Quantum Communication with More than One Bit Per Photon
We now turn to an application for the OAM of light in the field of cryptography and secure communication. To put this application topic in context, we first review the use of a onetime pad in cryptography (Shannon [37]). We consider the situation in which one party, A (or Alice), wants to communicate securely to another party, B (or Bob). We assume that by prearrangement the two parties share the same string of random binary digits known as the key, that no one else has access to this key, and that this string of digits is at least as long as the message that Alice wants to send. Alice encodes her message by first placing it into a binary format and then performing a binary add of the ith digit in her message with the ith digit in the onetime pad. This encoded message is then sent over a public channel. It is provably true that this message contains no useful information except to someone who has possession of the secret key. Bob then decodes the message by performing a binary add with his copy of the key, thus obtaining the original message that Alice sent.
A difficulty with implementing the onetime pad method is the distribution of the secret key between Alice and Bob. This is especially difficult if the two parties are not and cannot be in the same place, where the key can simply be handed from one to the other. When not in the same place, a procedure proposed by Bennet and Brassard in 1984 (know as the BB84 protocol) can be used to distribute the key in an entirely secure manner. In brief (some of the details are provided below), Alice sends the key one element at a time, and each digit is encoded in the quantum state of a single photon. If an eavesdropper (Eve) intercepts and measures this photon and then tried to send an exact replacement photon of it, she will certainly fail, because the laws of quantum mechanics prohibit her from determining full knowledge of the quantum state of a photon in a single measurement. The impossibility of doing so results from the celebrated “nocloning” theorem of quantum mechanics [38]. Secure communication through use of the BB84 protocol of quantum key distribution (QKD) is now a commercial reality (see, for example the website http://www.idquantique.com).
Work in which the present authors have participated involves extending the BB84 protocol so that more than one bit of information can be carried by each photon. Such a procedure may be used to increase the secure bit rate of a quantum communication system. To achieve the goal of transmitting more than one bit per photon, we encode information in the transverse degree of freedom of the light field. For the transverse degree of freedom one can choose any complete set of orthonormal modes. In keeping with the context of this chapter we consider encoding in OAM modes such as Laguerre–Gaussian (LG) modes. In the original QKD proposal of Bennett and Brassard, information is encoded in the polarization degree of freedom of an individual photon. As a result, only one bit of information could be impressed onto each photon. In contrast, when using OAM, there is no limit to how many bits of information can be impressed onto a single photon, as the LG modes span an infinitedimensional state space. As mentioned above, one motivation for doing this is that rate of data transmission is thereby increased. Another more subtle motivation is that the security of the protocol can be increased by encoding information within a higherdimensional state space.
The system that we envisage is illustrated in broad scope in Fig. 17.9. It consists of a sender, Alice, and a receiver, Bob. Alice impresses information onto the transverse degree of freedom of individual photons through the use of a spatial light modulator (SLM). Bob then randomly guesses which basis (OAM or angle) Alice might be using and makes a measurement of the quantum state of the received photon in this basis. The procedure for ensuring the security of the transmission is a generalization of that of the BB84 protocol and is described in the review of Gisin and Thew [39]. In the remainder of the present section we describe in more detail our laboratory procedure and present some laboratory data.
An example of the implementation of this protocol is shown in Fig. 17.11. In this example, Alice is attempting to send the string of numbers 13, 3, 2, 3, 15, 14, 16, 8, 24, 26 to Bob. For each transmitted photon, Alice chooses randomly between the LG basis and the AB basis. Also, for each transmitted photon Bob chooses randomly between the OAM and AB bases. After the transmission of the entire data train is complete, Alice and Bob publicly disclose the basis they used for each measurement. If they used different bases (which occurs on average half of the time), they discard the results of that measurement. The remaining data string is known as the sifted data, and this data should contain no errors. Any error in this data string could be the result of measurement errors or to the presence of an eavesdropper. For reasons of extreme caution, one must ascribe all errors to the presence of an eavesdropper. To test for errors, Alice and Bob sacrifice some fraction of their data for public comparison. If errors are detected, they conclude that an eavesdropper is present and take appropriate corrective measures.
We have implemented this BB84type protocol in our laboratory. Figure 17.11 shows how Alice forms each of the basis states. Basically, she programs a spatial light modulator (SLM) to convert an individual photon in a planewave state into one of the desired LG or AB modes [41]. The upper row shows the LG basis and the lower row shows the angular AB basis. The panel on the left shows representative examples of the pattern displayed on the SLM. The panels on the right show examples of the field distribution written onto the light field. These frames show actual laboratory results, although read out with intense classical light, not with single photons (Fig. 17.12).
Special considerations apply to the configuration of the receiver, or Bob. He is presented with a single photon and needs to determine its quantum state. Thus, he is allowed to perform only one measurement to determine in which of a large number of quantum states the received photon resides. This sorting task has eluded the scientific community until very recently, when Berkhout et al. [40] demonstrated a means for performing this task. Their approach is illustrated in Fig. 17.13. The key element of this approach is the ability to map the azimuthal phase distribution of an incident mode onto a linear phase distribution at the output of the device. Of course, a linear phase ramp in one cartesian dimension is simply a wavefront tilt, and leads to a shift in the position of the beam in the far field. It turns out that one can determine analytically the form of the phase function that needs to be applied to a light field to perform this mapping. In their original implementation of this sorting procedure, Berkhout et al. [42] applied this phase mapping through the use of an SLM. In a more recent work they have fabricated refractive elements that perform this same function but with much higher conversion efficiency than those based on diffraction from an SLM.
Some laboratory results validating the performance of this sorter are shown in Fig. 17.14. These results demonstrate our ability to discriminate among various quantum states in either the LG or angular basis. In each basis we include only four basis states. This limitation is due to the number of photodetectors (APDs) available to us. We see no fundamental limit to our ability to distinguish among all of the states in our protocol, 27 in this particular situation. We see that there is a small amount of crosstalk among the various channels.
Using the procedures described here, we recently performed a realistic demonstration of quantum key distribution based on OAM encoding [43]. Our experimental setup is shown in Fig. 17.15 and is composed of the various components described above. Alice prepares state to be sent by first carving out pulses from a highly attenuated HeNe laser through the use of an AOM. Then spatial mode information is impressed on these pulses with a digital micromirror device (DMD). Bob’s mode sorter and fanout elements map the OAM modes and the ANG modes onto separated spots that are collected by an array of fibers and sent to individual APD detectors.
Some of the results of this demonstration are shown in Fig. 17.16. The top row (left) shows the string of numbers sent by Alice and the top row right shows the string of numbers received by Bob. Note that the strings are not identical; due to various transmission and detection errors, some of the symbols are not detected as transmitted. In the figure errors are marked in red and are underlined. As a test of their system, Alice and Bob could publicly disclose these results to determine the fractional error rate. However, in an operating system, Alice and Bob would want to sacrifice only a small fraction of their data to test system security. Alice and Bob therefore employ an error correction algorithm and a method known as privacy amplification (which shortens the length of their shared string) to decrease the number of errors in the shared strings. They end up with a shared key that contains essentially no errors. (For this reason we show the string constituting the shared key only once.) As a graphic demonstration of the use of this procedure, we also show how it could be used for the secure transmission of an image. The image is separated into pixels which are then digitized and transmitted using the secret key shared by Alice and Bob. An eavesdropper who intercepted the signal would see only the noisy pattern that is also displayed.
17.6 Summary and Conclusions
In this chapter we have presented a review of the quantum mechanical properties of spatially structured light fields, paying special attention to light fields that carry orbital angular momentum (OAM). We have considered both the conceptual understanding of the quantum features of these light fields and the use of these quantum features for applications. We describe how to produce spatially entangled light fields by means of the nonlinear optical process of spontaneous parametric downconversion. We address the question of how much information can be encoded onto a single photon. As an example, we review a recent experiment that demonstrated the ability to discriminate among four target objects using only one photon for illumination. We also present a description of the concept of the OAM of light, and we describe means to generate and detect OAM. We then present a brief survey of some recent studies of the fundamental quantum properties of structured light beams. Much of this work is aimed at studying the nature of entanglement for the complementary variables of angular position and OAM. Finally, as a realworld application, we describe a secure communication system based on quantum key distribution (QKD). This key distribution system makes use of encoding information in the OAM modes of light and hence is able to transmit more than one bit of information per photon.
Notes
Acknowledgements
One of us (RWB) acknowledges support from the Canada Excellence Research Chair program, from the US Office of Naval Research, DARPA, and the Air Force Office of Scientific Research. Moreover, MJP acknowledges support from the Engineering and Physical Sciences Research Council of the UK and from the European Research Council.
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