Order-Invariant Two-Photon Quantum Correlations in PT-Symmetric Interferometers

Multiphoton correlations in linear photonic quantum networks are governed by matrix permanents. Yet, surprisingly few systematic properties of these crucial algebraic objects are known. As such, predicting the overall multiphoton behavior of a network from its individual building blocks typically defies intuition. In this work, we identify sequences of concatenated two-mode linear optical transformations whose two-photon behavior is invariant under reversal of the order. We experimentally verify this systematic behavior in parity-time-symmetric complex interferometer arrangements of varying compositions. Our results underline new ways in which quantum correlations may be preserved in counterintuitive ways, even in small-scale non-Hermitian networks.


■ INTRODUCTION
Quantum correlations in linear optical networks are a crucial resource for quantum information processing.−9 Furthermore, the lack of tangible permanent properties also has effects when considering small-scale networks.There, it hinders intuitive prediction of the behavior of composite systems from their building blocks.This especially holds true for non-Hermitian systems that incorporate losses.−15 Among the wide variety of non-Hermitian settings, systems obeying parity-time (PT) symmetry 16 are of particular interest since they can still possess entirely real-valued spectra, despite violations of energy conservation.PT-symmetric systems are described by Hamiltonians that are invariant under the combined operation of parity-inversion and time-reversal. 17hrough tuning of a single physical parameter, they can undergo a symmetry-breaking phase transition at an exceptional point 18,19 where the eigenvalue spectrum becomes complex.−27 Along these lines, light-based realizations have been instrumental for the observation of many features of PT symmetry in the classical domain, 28,29 with potential applications ranging from sensors with enhanced sensitivity 30,31 to efficient lasers with robust single-mode characteristics. 32,33−36 Straightforwardly extending classical PT-symmetric systems is not possible, as gain-induced quantum noise would inevitably break PT symmetry. 37Thus, realizing PTsymmetric photonic systems in the quantum domain necessitates a shift to passive systems. 38Such passive PT systems recently enabled the very first observations of PTsymmetric quantum interference 39 and the quantum simulation of coupled PT-symmetric Hamiltonians 40 on a photonic platform, which opened up the exploration of quantum correlations in larger non-Hermitian networks.
Here, we systematically identify types of sequences of general two-mode systems that perform distinct linear optical transformations, whereas their permanents remain invariant under the reversal of the entire sequence's order.In other words, we present composite systems of different arrange-ments, which may be based on non-Hermitian building blocks, that exhibit identical two-photon correlations.We illustrate this behavior on systems obeying PT symmetry and experimentally verify the discovery by probing and comparing the two-photon correlations in PT-symmetric interferometers of wildly different compositions.Our results demonstrate that quantum correlations in non-Hermitian photonic networks may be preserved in a counterintuitive manner.

=
. Here, the elements m ij represent the transmission and reflection coefficients between the two channels of the system.In the quantum domain, we now excite this system with two single photons in a 11 Fock state, having a single photon in each input.The probability that the two photons also emerge from separate output modes in a 11 state, which is signaled by observing a coincidence between the outputs, is given by P 11 = |perm M| 2 , 1 where perm M denotes the permanent of the matrix M. Notably, the same holds true for the row-and-column-reversed transpose of the transformation M obtained by

=
, where ( ) is the exchange matrix and T denotes transposition.
As the transmission matrices of these two systems do not commute, concatenating them can result in two different transformations, depending on the specific order, as illustrated in Figure 1a: In other words: while these two systems behave differently when probed with classical light or single photons, their twophoton behavior is strictly identical.This fact is independent of the (in)distinguishability of the impinging photons, as also the probabilities for distinguishable photons are equal according to , where the absolute square represents the Hadamard product of a matrix and its complex conjugate |A| 2 = A•A*.
Notably, this peculiar effect is not rooted in trivial properties of permanents such as their invariance to transposition or the permutation of rows or columns, 2 but instead arises from the equal antidiagonal elements of M and XM T X.From a physical point of view, it can be understood by tracing all possible exchange paths between input and output states in the two modes (Figure 1a).To observe the state 11 at the output when 11 is injected at the input, both photons either need to remain in their modes, or whenever a photon would exchange modes, one needs to return.For systems with identical antidiagonals, this exchange always results in a probability of m 12 m 21 , regardless of where the flip occurs.As a result, this property straightforwardly extends to longer sequences, as for any sequence of complex-valued 2 × 2 matrices with equal antidiagonals, multiplication to either the left or the right results in matrices with identical diagonals and permanents.Thus, the permanent of any arbitrary sequence of transformations M and XM T X, and therefore its two-photon behavior, is strictly invariant under reversal of the entire sequence (Figure 1b).Note that this invariance does not directly translate to systems with a larger number of modes or photons.Instead, e.g., in a three-mode system of concatenated transformations M and N, where N has the same entries as M, the only matrices whose permanents or subpermanents do not depend on the order are related through N = PMP, with a permutation matrix P. In those cases, the (sub)permanents of NM and MN, and thus their multiphoton behavior, are trivially equal.Whether order-invariant photon correlations systemati- cally occur in larger systems or may be enforced by certain symmetries remains an open question.
When we interpret the transformation M as the result of the evolution of a time-dependent Hamiltonian H(z), where represents the time-ordering operator, its counterpart XM T X corresponds to evolving H(z) in reverse order while also exchanging modes.Note that this is distinct from a conventional time reversal as it involves no complex conjugation.The concatenations (XM T X) M and M (XM T X) then correspond to the evolution of a Hamiltonian which reverses its order and exchanges modes midway, having point symmetry in the x−z plane, and its permanent is invariant to reversing the order of the first and second halves.The invariance of the permanent holds for any complex-valued 2 × 2 transmission matrix M and thus naturally includes non-Hermitian systems.For the subset of unitary systems, the transformations (XM T X) M and M (XM T X) only differ in external phases that photon-counting statistics are fundamentally agnostic toward, so that losses are in fact essential to distinguish nontrivially invariant two-photon correlations.

■ PT-SYMMETRIC INTERFEROMETERS
In a photonic context, such Hamiltonians can be readily mapped onto a system of two interacting waveguides with complex on-site potentials, all of which may be varying along the propagation coordinate z in ways that obey the symmetry condition, as sketched in Figure 1c.In the following, we will therefore turn to customized waveguide circuits to probe this invariance in PT-symmetric interferometer arrangements.PT-symmetric systems are non-Hermitian systems, whose Hamiltonians are invariant under the combined operation of parity-inversion and time-reversal.For PT-symmetric systems, the Hamiltonian H commutes with the PT operator: [H,PT] = 0. Here, P denotes the parity operator, ( ) for our two-mode system, and T is the timereversal operator, represented by complex conjugation.A twom o d e P T -s y m m e t r i c H a m i l t o n i a n i s g i v e n b y z z z It describes a system of two modes that are coupled with a coupling constant κ, of which the first/ second mode experiences loss/gain at a rate γ/2.For γ/κ < 2, the eigenvalue spectrum of the system is entirely real (unbroken PT phase), whereas for γ/κ > 2 the eigenvalues turn complex (broken PT phase).While the presence of gain is incompatible with quantum photonics as it introduces thermal noise, PT symmetry can be implemented in a fully passive system by introducing an additional common loss γ/2 to both modes. 38he fundamental building block of our interferometers is a passive PT coupler that consists of two waveguides interacting with a coupling constant κ over a certain length l, while one of the guides is subject to losses at a rate of γ.Evolution in the system is governed by the effective Hamiltonian: For γ/κ < 2, PT symmetry is unbroken and the Hamiltonian has real eigenvalues when viewed in the codamped reference frame. 38The full quantum-mechanical evolution of the non-Hermitian system may be described using a noise-operator approach, 41 using a quantum master equation in Lindblad form, 36,42,43 or by unitary dilation. 44When one probes the two-photon behavior by postselecting the cases where neither of the photons is lost, the probability amplitudes directly follow from the classical propagator U iH l exp( ) eff = , which now describes a lossy (i.e., nonunitary) transformation.
We now construct interferometers of two concatenated PT couplers that are either aligned or inverted, with respect to their loss distribution (Figure 2a).The aligned sequence yields a single PT coupler of twice the length, i.e., length 2l.The inverted arrangement with the opposite loss profile (green) nevertheless remains PT symmetric at each point along z.Additionally, we consider both of these systems in a rotated basis 45 realized by placing them between a pair of Hermitian directional 50:50 couplers (yellow and magenta, respectively).As it turns out, each of these four systems can be equivalently described in terms of a transformation M = U ̃R t h a t c o n s i s t s o f a 5 0 : 5 0 d i r e c t i o n a l c o u p l e r ( ) = −iX, two successive 50:50 couplers bring about a full population transfer, effectively swapping modes while adding an otherwise inconsequential global phase.Figure 2b displays the four configurations of the systems thus expanded to the same overall interaction length: M (XM T X), M M T , M T M, and (XM T X) M, respectively.These representations reveal that sandwiching the aligned or inverted PT coupler sequences in between two directional couplers indeed establishes the aforementioned permanent-preserving symmetry condition between those otherwise very different arrangements and, by extension, imbues them with identical two-photon behavior.This can readily be verified by calculating the visibility of twophoton quantum interference, defined as V = P indist /P dist − 1, with P indist and P dist being the coincidence probability for indistinguishable and distinguishable photons, respectively, as a function of the normalized loss γ/κ and interaction length κl, as shown in Figure 3. Figure 3a,d shows the two-photon visibility for the first configuration of aligned PT couplers, M (XM T X), that is equivalent to a single PT coupler of length 2l.In the lossless case (γ/κ = 0), the visibility periodically oscillates between −1 and 0 as a function of length as the effective splitting ratio of the coupler varies, with the first Hong-Ou-Mandel (HOM) dip 46 occurring at κl = π/8.Introducing loss into the system (γ/κ < 2) systematically alters these dynamics by simultaneously slowing down the overall oscillation period and shifting the first minimum toward shorter interaction lengths. 39urthermore, while the minima remain at −1, the visibility now oscillates between negative (bunching) and positive (antibunching) as a function of length: Around the odd-numbered maxima of V, the visibility now turns positive and thus crosses zero twice, indicating that two-photon interference actually increases the survival probability of the photon pair.Whenever one of the transmission/reflection coefficients of the system crosses zero and changes sign, the visibility reaches zero as well, because the quantum interference vanishes, and likewise swaps sign afterward.At even maxima of V, two coefficients simultaneously cross zero such that the sign of the visibility remains unaffected.In contrast, above the PT-symmetry threshold (γ/κ > 2), the system's oscillation period turns imaginary, erasing all but the first HOM dip as V monotonically approaches +1 for all larger interaction lengths κl.Likewise, the (XM T X) M configuration of inverted PT couplers between two directional couplers obeys the permanentpreserving symmetry and therefore displays identical two-photon behavior for any choice of effective gain or interaction distance.
Note that antibunching behavior (V > 0) is precluded in the lossless system and can be achieved only in the non-Hermitian context.Even though PT-symmetric directional couplers and, more generally, lossy directional couplers, are both non-Hermitian systems that can exhibit superficially similar oscillations in the two-photon visibility, two distinct underlying mechanisms are at work.In lossy directional couplers, the sign and magnitude of V depend on both the amplitudes of the transmission/reflection coefficients u ij | | as well as their phase relation, 10−12  .Zero visibility, for example, can therefore arise from the absence of interference, when one of the coefficients reaches zero, or dynamically occur through interference for specific values of the internal phase.In contrast, PT symmetry restricts the value of the internal phase such that the magnitude of V solely depends on the amplitudes of the coefficients: zero visibility thus always corresponds to the complete suppression of interference.
Figure 3b,e illustrates the two-photon visibility in inverted PT coupler sequences (M M T ).Qualitatively, the influence of loss on the system is similar to that of the aligned case.However, the dependence on the section length is different, as the visibility now turns positive every four maxima, starting from the second one.Upon closer inspection, the locations of the even maxima and the sign of the visibility correspond to those of a single PT coupler (cf. Figure 3a, but scaled by a factor of 2 in horizontal direction for a single coupler of length l).This can be understood by considering that zeros in the transmission/reflection coefficients of a single coupler directly map into zero coefficients and thus vanishing V in the inverted sequence.Moreover, additional (odd) maxima of zero visibility appear at exactly those lengths where full HOM bunching (minimum visibility) occurs in a single coupler, as the second coupler reverses the transformation of the first, leading to pairs of zero coefficients and vanishing interference.
Finally, two aligned PT couplers, or, equivalently, a single one of twice the length, sandwiched between two 50:50 directional couplers, M T M, exhibits entirely different behavior (Figure 3(c,f)).In the unbroken phase (γ/κ < 2), the visibility stays strictly negative despite oscillating at similarly increasing periods, remains identically zero regardless of the interaction length at the PT-breaking threshold (γ/κ = 0), and turns globally positive in the broken phase (γ/κ > 2).As such, this type of arrangement allows for the PT-broken phase to be unambiguously identified directly from its quantum correlations. 47

■ EXPERIMENTAL OBSERVATIONS
To experimentally test the predictions of our model, we fabricate non-Hermitian waveguide circuits via the femtosecond-laser direct writing technique. 48The desired losses are introduced into the waveguides by rapidly undulating their out-of-plane positions. 49Figure 4a schematically illustrates the waveguide geometry used to implement the (XM T X) M configuration.We then probe the two-photon dynamics in these systems by injecting photon pairs generated by spontaneous parametric down-conversion and registering coincidence counts between the channels.
As reference for the non-Hermitian arrangements, Figure 4b shows the experimentally observed visibility of two-photon quantum interference as a function of the length in a conventional Hermitian coupler (γ/κ = 0): a clear sequence of identical HOM dips unfolds as the effective splitting ratio of the coupler varies with its length.Note that this conventional coupler is the limiting case of all four complex configurations for vanishing losses.As soon as losses are introduced, differences in the visibility dynamics become apparent.Figure 4c shows the observed behavior for a loss coefficient of γ/κ = 0.38 + 0.19i.While all four PT-symmetric configurations retain certain similarities at such low loss levels and V stays negative for M T M (yellow), M (XM T X) (cyan) and (XM T X) M (magenta) both begin to display positive visibility around the first maximum, whereas M M T (green) turns positive at the second one.
As shown in Figure 4d, the qualitative differences in the behavior of the four configurations become more prominent for increased losses (γ/κ = 0.83 + 0.41i), as M T M (yellow) and M M T (green) systematically diverge from M (XM T X) (cyan) and (XM T X) M (magenta), whose two-photon visibility dynamics are both governed by the same matrix permanent.The measurements (data points) are in good agreement with the predicted behavior as well as calculations based on system parameters retrieved in a classical calibration (curves), with minor deviations attributable to additional coupling in the fanning sections (shading) and inadvertent detunings of the propagation constant incurred by the undulating loss regions.
The latter results in complex-valued loss coefficients γ/κ which gradually reduce the interference contrast, such that V tends to zero at longer propagation distances.Notably, even under these imperfect conditions, the two-photon visibilities of (XM T X) M and M (XM T X) are kept in lockstep by the permanent-preserving symmetry existing between them.

■ CONCLUSION
In summary, we have identified a new type of symmetry transformation that preserves the two-photon interference properties in sequences of non-Hermitian two-mode systems.From an algebraic point of view, this is a consequence of a property of matrix permanents, which remain invariant when transforming complex sequences of 2 × 2 matrices in line with this type of symmetry transformation.We have experimentally verified these findings in PT-symmetric interferometers of varying composition by demonstrating that two-photon correlations are indeed preserved by this symmetry.Their non-Hermitian nature is in fact essential to distinguish the nontrivially invariant two-photon correlations.Whether networks with a larger number of modes may support similar order-invariant correlations remains an open question.Nevertheless, our results emphasize that even in deceptively simple two-mode systems, non-Hermitian quantum correlations may be governed by highly nonintuitive mechanisms.
The PT-symmetric interferometers investigated here pave the way for the incorporation of non-Hermitian elements into larger quantum photonic networks.While losses may seem detrimental at first glance, they in fact introduce new freedom that enables new functionality.Along these lines, we hope that our work will inspire a new approach to the design of linear optical networks that harness non-Hermiticity for advanced quantum information processing and sensing applications.
■ METHODS Waveguide Fabrication.Waveguide circuits are fabricated via the femtosecond-laser direct writing technique. 48The individual channels are inscribed by focusing 270 fs laser pulses from an ultrafast fiber laser amplifier (Coherent Monaco, wavelength 512 nm) at a repetition rate of 333 kHz and an average power of 70 mW through a 50× microscope objective (NA = 0.6) into fused silica (Corning 7980).The waveguide trajectories are defined by the motion of a high-precision translation stage (Aerotech ALS180) at a speed of 100 mm/min.At the design wavelength of 814 nm, the propagation losses of these waveguides are below 0.12 dB cm −1 , which is negligible compared to the desired loss in the modulated sections.In the interaction regions of our couplers, the waveguides are separated by 20 μm, corresponding to a coupling coefficient of κ = 0.85 cm −1 .The desired losses are introduced into the waveguides by rapidly undulating their out-of-plane positions following a cosine trajectory 49 with a 0.15 cm period and amplitudes of 1.0 and 1.5 μm, resulting in excess loss coefficients of γ = 0.32 and 0.70 cm −1 , respectively.
Photon-Pair Generation.Horizontally polarized wavelength-degenerate photon pairs at 814 nm are generated by type-I spontaneous parametric down-conversion from a continuous-wave pump at 407 nm in a bismuth borate crystal.The degree of indistinguishability of the photons was characterized by observing HOM interference, resulting in a visibility of 96%.

■ AUTHOR INFORMATION
Corresponding Authors

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CONCATENATED TWO-MODETRANSFORMATIONSLet us consider an arbitrary two-mode linear optical system.The way the two input and output modes of this linear optical system are related is described by its 2 × 2 transmission matrix

Figure 1 .
Figure 1.Schematic representation of sequences of two-mode transformation M and its row-and-column-reversed transpose XM T X with their transmission and reflection coefficients m ij .Exciting sequences of (a) two or (b) more copies of M and XM T X, concatenated in either forward or backward order, with a 11 state gives rise to identical two-photon coincidences.(c) Photonic implementation of the transformations (XM T X) M and M (XM T X) as two coupled waveguides with z-dependent propagation constants β 1 , β 2 (shading), and coupling constant κ (distance).

Figure 2 .
Figure 2. Schematic drawing of (a) concatenated PT-symmetric directional couplers (length l) and conventional 50:50 directional couplers (γ = 0, length π/4κ) and (b) their equivalent descriptions in terms of the transformation M (right), consisting of a 50:50 directional coupler R, followed by a PT coupler U ̃. Shading indicates lossy waveguide sections.The color of the lossy section is used to distinguish the four system geometries: M (XM T X) (cyan), M M T (green), M T M (yellow), (XM T X) M (magenta).

Figure 3 .
Figure 3. Calculated two-photon visibility as a function of normalized length κl and loss coefficients γ/κ in the non-Hermitian systems with geometries of (a) M (XM T X) and (XM T X) M, (b) M M T , and (c) M T M. Spontaneous PT-symmetry breaking occurs at γ/κ = 2. Negative visibility corresponds to relative bunching of photons, with complete HOM interference at V = −1, and positive visibility corresponds to antibunching behavior.(d−f) Cross sections of the two-photon visibility in (a)−(c) as a function of normalized length for constant loss values γ/κ = 0, 1, 2, 2.5.

Figure 4 .
Figure 4. (a) Experimental implementation of (XM T X) M, implementing the desired loss through modulation of the waveguide trajectory.(b−d) Measured (points) and calculated (curves) twophoton visibility as a function of length l in the non-Hermitian systems, for increasing loss coefficients of (b) γ/κ = 0 (lossless), (c) γ/κ = 0.38 + 0.19i, and (d) γ/κ = 0.83 + 0.41i.Color indicates the system geometry: M (XM T X) (cyan), M M T (green), M T M (yellow), and (XM T X) M (magenta).Error bars are based on the square root of the number of observed coincidences.The shaded areas indicate the effect of additional coupling in the fanning sections.
which can be captured in an internal phase