Quantum-state engineering using nonlinear optical Sagnac loops

The Kerr nonlinearity of an optical-fibre Sagnac loop can be utilized to engineer a variety of two-photon quantum states. These include correlated, identical photon pairs as well as degenerate, maximally entangled states—both of which are used in quantum information processing. In fact, their underlying principle—the reverse Hong–Ou–Mandel effect—can also be applied to free-space, down-conversion-based analogues of either identical or entangled photon-pair sources. Due to their simple structure, such versatile devices are expected to find widespread applications in quantum-state engineering.


Introduction
The optical-fibre Sagnac loop (OFSL) has enjoyed numerous applications since its introduction in 1976 [1], which have expanded from a rotation measuring instrument to a very versatile sensing tool. Such applications range from the fibre-loop mirror [2] to Sagnacbased intruder alarms, hydrophones, geophones and current measuring systems [3]. In classical nonlinear optics, the OFSL is a powerful tool in various all-optical operations. For example, an asymmetric (i.e. non-50 : 50 splitting ratio) OFSL [4]- [6] or an OFSL embedded with an amplifying medium [7]- [9] has demonstrated usefulness in optical switching and optical storage [10,11], and a symmetric (i.e. 50 : 50 splitting ratio) OFSL has been shown to be useful for optical frequency conversion and phase conjugation with high pump suppression [12,13]. In the continuous-variable quantum domain, soliton squeezing has been demonstrated using an asymmetric OFSL [14]- [16]. In this paper, we discuss the use of a symmetric OFSL as a discrete-variable quantum-stateengineering device. We specifically consider strategies for engineering sources of both identical photon pairs and polarization-entangled photon pairs using the OFSL. Because the fundamental principle underlying OFSL's many quantum-state-engineering functionalities is the reverse Hong-Ou-Mandel (HOM) effect-a phenomenon which exists in both fibre optics and free-space optics-it is possible to extend these fibre-optic techniques to free-space down-conversion-based sources of identical and entangled photon pairs. This paper is organized as follows. After briefly reviewing four-wave mixing (FWM), down-conversion and the concepts of identical and entangled photons, we introduce the deterministic quantum splitter (QS), a device based on a 50 : 50 OFSL. Its basic operating principle is explained first using an intuitive Feynman-path approach and then compared with a standard quantum-mechanical calculation. This analysis is then extended to a more general investigation of the forward and reverse HOM effect, particularly as it relates to polarizationentangled input states. Finally, we envision the application of the reverse HOM effect to engineer 3 both identical and polarization-entangled sources of photon pairs (along with a few other exotic states, such as the generation of the 'noon' state [17]).

FWM
The nonlinear process employed in an OFSL for quantum-state-engineering applications is called FWM, a third-order process mediated by the Kerr (χ (3) ) nonlinearity of an optical fibre to produce correlated photon pairs. Lacking a χ (2) nonlinearity due to its centrosymmetry, an optical fibre's first appreciable nonlinearity is χ (3) , which, although weak in itself, can lead to greatly enhanced nonlinear interactions due to the fibre's single spatial mode and long interaction length. Spontaneous Raman scattering, although widely used as a stimulated process in fibre-optic communications to make Raman fibre amplifiers, is the predominant source of noise in χ (3) processes, producing uncorrelated photons which degrade the purity of any fibrebased correlated-photon source. Fortunately, this noise process can be suppressed by either cooling the fibre to a low temperature (e.g. 77 K in [18,19]) to deplete the phonon bath, or pumping the fibre in its normal dispersion regime [20,21] so that the phase-matched photon pairs reside outside the primary Raman band (peaked at 13 THz detuning from the pump). In FWM, a pair of correlated photons, commonly denoted as signal (s) and idler (i), are produced at the expense of two pump photons ( p 1 and p 2 ) 2 . Energy and momentum conservations are obeyed during the FWM process: ω p 1 + ω p 2 = ω s + ω i and k p 1 + k p 2 = k s + k i , where ω j and k j represent the frequency and wave-vector of the jth photon ( j = p 1 , p 2 , s, i). The FWM process can occur for either co-polarized or cross-polarized pump photons [22]: x yx y −→ x s y i . (cross-polarized). (1) Here, x and y denote orthogonal polarization states. Because co-polarized FWM gain is proportional to (χ (3) x x x x ) 2 and cross-polarized FWM gain is proportional to (χ (3) x yx y ) 2 ≡ 1 9 (χ (3) x x x x ) 2 , the co-polarized process is much stronger than the cross-polarized process.

Spontaneous parametric down-conversion (SPDC)
SPDC is an analogous nonlinear process employed in some nonlinear bulk crystals (or waveguides) for the production of photon pairs, in this case mediated by the χ (2) nonlinearity. Unlike FWM, the SPDC process involves the transformation of a single pump photon into a signal and idler photon pair. As above, this process conserves both energy and momentum: ω p = ω s + ω i and k p = k s + k i . Like FWM, SPDC can create both co-polarized (type-0, type-I) and cross-polarized (type-II) daughter photons: x p χ (2) x x y −→ x s y i . (type-II). 2 The terms signal (idler) photon and Stokes (anti-Stokes) photon are sometimes interchangeable. For the remainder of this paper, we restrict our discussion to type-I and type-II sources, though in principle all type-I designs could also be adapted to type-0 waveguides without significant changes.

Identical and entangled photon pairs
This manuscript will focus on two types of photon-pair sources: identical and entangled photon pairs. In linear optical quantum computing and many other quantum information protocols, daughter photons from photon-pair sources are required to be indistinguishable. In other words, the photon states must be identical in all degrees of freedom (spatial, temporal/frequency and polarization). The type of entangled photons that we shall consider in this manuscript are closely related to identical photon pairs, but they are entangled in polarization (such as the Bell state 1 √ 2 (|αβ + |α ⊥ β ⊥ )) while being indistinguishable in all other degrees of freedom. We note that polarization-entangled but frequency-nondegenerate photons have also been extensively studied in the context of fibre-based photon sources [23]- [26], but they are outside the scope of this paper.
Traditionally, indistinguishable photons are generated from a χ (2) correlated-photon source that relies on SPDC [27] with degenerate phase matching. For most applications it is preferred to have the indistinguishable photons deterministically separated from each other. In SPDC, this can be achieved using collinear type-II phase matching, in which case degenerate photons can be deterministically separated by means of their orthogonal polarization and afterwards made indistinguishable by rotating one photon's polarization by 90 • [28]. Alternatively, the degenerate photons can also be deterministically separated using non-collinear type-I phase matching, wherein the photon pairs are emitted into different spatial modes and are therefore easily separable [29]. However, the coupling of SPDC photons into single-mode optical fibres remains technically challenging, since these photons are naturally emitted into a large number of correlated spatial and spectral modes.
Fibre-based photon sources that rely on FWM provide an elegant solution to the aforementioned coupling problem, because all the correlated photons are born with the same single spatial mode supported by the standard single-mode optical fibre. The challenge is how to deterministically separate these identical photons, which are enforced by the fibre geometry to be necessarily collinear. One way is to utilize a cross-polarized dual-frequency pump to excite the χ x yx y component of the nonlinear susceptibility tensor [30,31], which produces cross-polarized degenerate photons and can thus be deterministically separated in a similar way to type-II collinear SPDC photons. As mentioned above, this process is generally much less efficient than the co-polarized FWM process because of the intrinsically weaker nature of χ x yx y [22]. If, however, one decides to use co-polarized FWM to generate indistinguishable photons, one immediately realizes that the output photons are not readily separable in a deterministic way. The situation is analogous to a collinear type-I SPDC configuration [32]. In the past, researchers have simply used a regular 50 : 50 non-polarizing beam splitter (BS) to probabilistically separate the identical photons [32]- [34]. The input and output states are shown below (refer to figure 1): . For instance, as shown in [32], the HOM-dip [35] visibility attainable with such a source would be limited to only 50%, making it unattractive for practical quantum information applications.
On the other hand, if the output wavefunction | out only consists of the 11 component (i.e. deterministically separated indistinguishable photons), the HOM-dip visibility can in principle reach 100%. We have described in our previous publications [36,37], mainly from an experimental viewpoint, how this obstacle can be circumvented by using a new type of co-polarized identicalphoton source, namely the QS. Demonstrations of a near-unity HOM dip visibility [36] and a telecom-band quantum controlled-not gate [37] have shown the practical usability of the QS. In this paper, we focus on quantum-state engineering with the QS. But first we show how to account for the operating principle of the QS theoretically, using an intuitive Feynman-path approach followed by a standard quantum mechanical calculation.

The nonlinear OFSL
The OFSL is well-known for its role as a total reflector (TR) [2] or a total transmitter (TT).
Here we explore its use as a QS, essentially an OFSL set to its previously unexplored 50 : 50 state in which the OFSL equally transmits and reflects light. The difference between the TT, TR and QS configurations of an OFSL lies with different settings of the intraloop fibre polarization controller (FPC), which effectively results in different relative phase shifts between the clockwise (CW) and the counter-clockwise (CCW) paths. The situation is illustrated in figure 2, which explains the classical interference between two equally-split pump pulses, as well as the quantum interference between the FWM photon-pair amplitudes copropagating with each pump pulse. As shown in figure 2, the pump is injected from port a into the Sagnac loop, which is composed of a 50 : 50 non-polarizing fibre BS, a piece of dispersion-shifted fibre (DSF) and an FPC. The peak pump power is denoted by P, which is then equally split into two equally-powered pulses (P/2) by the BS. The two pump pulses traverse the DSF in a counterpropagating manner, each of which probabilistically scatters co-polarized FWM photon pairs, denoted pictorially by signal and idler in figure 2. The probability of generating FWM photon pairs that are orthogonally polarized to the pump (i.e. excitation of the nonlinear susceptibility tensor component χ yx x y ) is neglected due to its smallness. We also neglect the probability that both pump pulses undergo FWM scattering, as this corresponds to a higher-order process of multi-photon generation, whose probability is vanishingly small when the pump power is low.
We start off by showing that the classical counter-propagating fields in the Sagnac loop can achieve a directional phase difference 3 while remaining co-polarized after each has traversed the entire loop (before being recombined at the BS). Interested readers should refer to [2] for a formal mathematical treatment of the Sagnac loop. Here, we follow a simplified schematic shown in figure 3. The Sagnac loop (denoted as 'Fibre' in figure 3) is pictorially straightened out, and the birefringence effects of the fibre and the FPC are modelled by a half-wave plate (λ/2) and a quarter-wave plate (λ/4). For simplicity, we further assume that the fibre performs no polarization rotation; λ/2 is set to 45 • , rotating H (horizontal polarization) to V (vertical polarization) and V to H; λ/4 is set to 0 • , leaving H and V unchanged except for a relative phase. For horizontally polarized CW and CCW light, the following evolution takes place: where only relative phase shifts are recorded above. Note there is a directional phase difference between the CW and CCW fields (which in this case is π/2), while their final polarization states are still the same. In a more general scenario, one can substitute any fractional waveplate for the λ/4-plate in figure 3, which results in a continuously tunable directional phase difference. We note that in practice careful alignment techniques have to be applied in order to ensure copolarization of the counter-propagating light fields before their recombination at the BS.
We are now ready to come back to figure 2 and explain its working principle. Note that φ CW and φ CCW are symbols that we use to track the classical phases of the CW and CCW pump pulses, respectively. To be consistent, the π/2 phase shift gained upon reflection from the BS has already been included in φ CW . We first consider the conventional FWM case, namely, degenerate pumps ( p 1 = p 2 ) and non-degenerate signal/idler (s = i). A similar analysis can be applied to the reverse degenerate FWM case, i.e. non-degenerate pumps ( p 1 = p 2 ) and degenerate signal/idler (s = i). We have already shown that, the counter-propagating pumps arriving at the BS before being recombined can be made co-polarized with a variable phase difference φ ≡ φ CCW − φ CW . The corresponding photon-pair states (|s |i ) are co-polarized with each pump, and have a relative phase difference of δ = 2 φ. 4 Using the terminology shown in figure 2, we arrive at the following Feynman-path summations of indistinguishable signal/idler photon scattering outcomes for three distinct operational modes of the OFSL (TR, TT, 50 : 50/QS): (i) Totally reflective (TR) • Phase tracking: φ = π 2 ; δ = π . • Quantum amplitudes: |CW • Normalized output state: (ii) Totally transmissive (TT) • Phase tracking: φ = − π 2 ; δ = −π .
• Quantum amplitudes: |CW • Normalized output state: The subscripts a and b label the spatial modes of the output photon, and all the output wavefunctions are normalized with their global phases ignored. From equations (4) and (5), we can see that the photon pairs bunch together when the Sagnac loop is operated as a TR/TT for the pump. And from equation (6), the photon pairs split up when the Sagnac loop functions as a 50 : 50 mirror for the pump. The above analysis, when applied in a similar fashion to the reverse degenerate FWM case, leads to the discovery of the QS source. The results are exactly the same as before, with the substitution of s = i and the reinterpretation of φ as the counter-propagating phase difference for a classical optical field at the degenerate signal/idler frequency 5 .

The reverse HOM effect
A convenient way to summarize the above results is to use the quantum mechanical description of a 50 : 50 BS (two input modes and two output modes) with an arbitrary relative phase between the two inputs. This derivation, detailed below, reproduces the same phenomena as in a recent time-reversed HOM experiment [36], an analogy which will be particularly useful when this derivation is extended to Bell-state outputs. In addition, the derivations below apply equally well to FWM or SPDC photon pairs, and will therefore be applicable to both fibre-based and free-space photon-pair sources. We remark that the reverse HOM effect is also relevant in some other contexts, for instance in number-path entanglement [38] and for creating an anti-bunched light beam [39].

Producing separable photon pairs through the reverse HOM effect
As shown in figure 4, the input modes are labelled a and b, while the output modes are labelled c and d in consistency with figure 1 (note that the output modes are labelled differently in figure 2). As before, we start with the case of a non-degenerate signal/idler photon pair ( figure 4(a)), the end result of which can be easily adapted to the degenerate case ( figure 4(b)). The input state in figure 4(a) is written as where a † s (a † i ) is the creation operator for the signal (idler) photons in the a-mode, and likewise b † s (b † i ) is the creation operator for the signal (idler) photons in the b-mode. The 50 : 50 BS's action on these operators are where the π/2 phase shift upon reflection from a BS is represented by the additional i in front of the reflected operators. Plugging equation (8) into equation (7), we immediately obtain For the degenerate signal/idler case shown in figure 4(b), equations (7) and (9) are rewritten as: One can verify the agreement between the above two approaches by setting δ to π , −π, and 0 in equations (9) and (11), which correspond to the three operational modes of the OFSL-TR, TT and 50 : 50, respectively. Therefore, as emphasized in [36,37], the 50 : 50 OFSL, or QS, can indeed be interpreted as a manifestation of time-reversed HOM interference.

Non-ideal BSs and the reverse HOM effect
An experimentally relevant question is: how would a non-ideal 50 : 50 BS (i.e. R + T = 1, R = T , where R (T ) stands for intensity reflectivity (transmissivity)), or a non-optimal alignment of the 50 : 50 OFSL (i.e. δ = 0), affect the output wavefunction. Detailed calculations can be done by writing, in analogy to equation (7), the input state in the non-ideal case as In contrast to the case shown in figure 2, we note that here the input pump power is split into two unequal parts, with P CW = T P and P CCW = R P (neglecting loss of the BS). The two-photon state generated by each pump has a coefficient proportional to the pump power [40], hence the different coefficients in front of the a-mode (CW) and b-mode (CCW) creation operators. Here we further assume that where α represents a small deviation from an ideal 50 : 50 BS with |α| 1.
The BS transformation on the input operators can be written in analogy to equation (8) as Plugging equation (14) into equation (12), we obtain The fidelity F, defined as the inner product between the output state given by equation (15) and the one from the ideal 50 : 50-BS case (i.e. , which is the same as equation (6) with appropriate changes in the mode symbols), is given by where in the last step we have substituted in equation (13) for the T and R. The first-order approximation of equation (16) can be easily derived. Namely, when α 2 1 and δ 2 1, we have F ≈ 1 − 8α 2 − δ 2 /4, which shows the robustness of the output wavefunction against small perturbations in α and δ. Note that F = 1 when α = δ = 0, as expected. the mid-frequency wavelength of the two pump wavelengths (λ p1 and λ p2 ). Pure 11 state is expected to emerge from the output ports of this device at the degenerate wavelength λ m .
Abundant quantum-state-engineering applications exist for the OFSL source, when we consider various pump inputs in conjunction with various operational modes of the Sagnac loop. We summarize all envisioned applications of the OFSL in a compact matrix shown in figure 6, which also includes the quantum functionalities of a straight piece of fibre for comparison. The rows of the matrix correspond to 'straight fibre', 'Sagnac loop in TT/TR mode', and 'Sagnac loop in 50 : 50 mode', respectively, whereas the columns of the matrix correspond to 'single-frequency pump', 'dual-frequency co-polarized pump', and 'dual-frequency crosspolarized pump', respectively. The working principles for each matrix element can be worked out in a straightforward way following the QS example. It is worth noting that in addition to the other deterministically split states, a certain kind of noon state [17] (here N = 2) can also be generated (matrix element {2, 2}). Furthermore, the versatility of the OFSL source allows it to be easily configured to perform experiments done in [42] with its matrix element {2, 1}, and experiments in [43] with its matrix element {3, 1}. The matrix elements {2, 3} and {3, 3} turn out to be functionally equivalent and are therefore absorbed into one entry, dubbed the 'quantum buncher'. This device outputs pairs of orthogonally-polarized signal and idler photons into a superposition of both propagating in mode a or both propagating in mode b.
By extending the same principles to SPDC, it is also possible to create sources of identical photons in free space. Although the pump accumulates phase at only half of the rate of a FWM pump (because only one photon at a time down-converts, but two pump photons are required for FWM), it is also approximately half of the wavelength of the daughter photons. These two factors balance, and although individual dispersive effects must be taken into account for specific implementations, in general the previous analysis can be directly extended to the χ (2) case. Figure 7 is a diagram of the OFSL analogue for SPDC. Waveplates replace the FPC from the OFSL, allowing easy tuning of the phase difference δ.

Sources of entangled photon pairs
A spatially-separated polarization-entangled identical photon-pair source can also be realized by utilizing the configuration shown in figure 5, but with a twist: polarization multiplexes two such processes and makes them indistinguishable from each other. Three operationally equivalent designs of such a source are shown in figure 8. All designs utilize dual-frequency orthogonallypolarized input pumps (H-pump and V-pump). In the first case, figure 8(a), these two pumps are temporally separated before the Sagnac loop, and are launched into the 50 : 50 OFSL from the same input port. In the second case, figure 8(b), a single diagonally-polarized dual-frequency input pump is temporally separated into two orthogonally-polarized components by polarization maintaining (PM) fibres (or some other polarization dependent time-delay device) inside the OFSL. In the third case, figure 8(c), a single diagonally-polarized dual-frequency pump is split by a polarizing BS (PBS) into two orthogonally-polarized, time-delayed pumps, which are launched into opposite ends of the 50 : 50 OFSL. All cases utilize a fixed amount of time The OFSL analogue for the down-conversion sources, using a type-I down-conversion process. Waveplates provide the phase difference δ, allowing one to tune the source to a QS. delay τ between the two pump pulses. Each pump independently and probabilistically scatters spatially-separated identical photons in its time epoch that are co-polarized with it. Polarization entanglement of the form |H s H i + |V s V i is generated for signal/idler photons at the degenerate middle frequency (wavelength λ m ) when all the distinguishing timing information between the two processes is erased. In figure 8(a), the reflected photon is picked up by a circulator, and two pieces of PM fibre with suitable lengths are introduced, one in each photon's path, to precisely remove the delay τ between the FWM processes driven by the H-pump and the V-pump. In figure 8(b), the distinguishing temporal information is erased by the exact same PM fibres which separated the counter-propagating pumps into two components each. In order to remove this time delay, rather than exacerbate it, a 90 • rotation is necessary between the fast and slow axes of the two lengths of PM fibre. This second model utilizes the reverse HOM effect for entangled states (cf equation (19)), allowing us to generate a deterministically-separated polarization-entangled state using the OFSL. Figure 8(c) bears some resemblance to the doubleloop scheme introduced in [44]. The apparent difference between the two is the operational mode of the OFSL-50 : 50 in the former and totally reflective in the latter. As in the doubleloop scheme, FPC1 and FPC2 are configured to restore the polarization states of both OFSLtransmitted and reflected photons to their original values [44]. The identical-photon amplitudes following the common path (i.e. both transmitted by the OFSL) come out of the PBS's O-port with no relative time delay. Their twins (i.e. both reflected by the OFSL) out of the PBS's I-port are displaced relatively in time and are picked up by a circulator. A piece of PM fibre with a suitable length is inserted to remove the relative time delay between them, which is 2τ in this case. Finally, all three schemes in figure 8 should be contrasted with that from [33], in which such a spatially-separated polarization-entangled state is generated 50% of the time.
The reverse HOM principle, which allows the second fibre-based entangled photon source to operate, can be extended to both type-I and type-II SPDC sources of entangled photons, as  [44] is also capable of generating deterministically-separated entangled photons.
shown in figure 9. Both of these sources rely on the entangled reverse HOM effect to transform a superposition of CW-and CCW-propagating probability amplitudes into a deterministicallysplit pair of entangled photons. When constructing these sources, care must be taken to control In both cases, an additional phase (e.g. from a dispersive optic (not shown)) introduced between the CW and CCW SPDC amplitudes will change these devices from a QS into a quantum buncher, and vice versa.
the relative phase between the CW and CCW amplitudes. Dispersive or birefringent elements may cause the pump to experience a different optical phase than the SPDC pairs, and so in practice a variable-thickness dispersive element (such as a tilted glass slide (not shown in figure 9)) should be inserted inside the Sagnac loop to tune between the QS and the quantum buncher configurations.
We note that other free-space Sagnac-loop designs have been used to generate entangled photon pairs [45,46]. In [45], a type-I down-converter embedded in a free-space Sagnac loop is used; however, the Sagnac loop is not operated in the QS mode and the setup only post-selects polarization entanglement. In comparison, in figure 9(a) we use two type-I down-converters within a QS, and thus create genuine polarization entanglement (i.e. without any post-selection). In [46], a polarization Sagnac loop (rather than a regular Sagnac loop) is used together with a type-II down-converter. We believe the scheme depicted in figure 9(b) offers a practical alternative with comparable simplicity and robustness to the scheme demonstrated in [46].

Conclusion
The nonlinear optical Sagnac loop is a versatile device, usable for a number of quantum information applications. Here, we have presented designs for both fibre-based and free-space sources of identical and entangled photons. These new sources, in addition to being immediately useful for quantum-state-engineering applications, show the power and versatility of a new tool for experimental quantum information.