Interferometric control of the photon-number distribution
H. Esat Kondakci, Alexander Szameit, Ayman F. Abouraddy, Demetrios N. Christodoulides, Bahaa E. A. Saleh
IInterferometric control of the photon-number distribution
H. Esat Kondakci, ∗ Alexander Szameit, Ayman F. Abouraddy, Demetrios N. Christodoulides, and Bahaa E. A. Saleh CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, Florida 32816, USA Institute for Physics, University of Rostock, 18059 Rostock, Germany
We demonstrate deterministic control over the photon-number distribution by interfering twocoherent beams within a disordered photonic lattice. By sweeping a relative phase between two equal-amplitude coherent fields with Poissonian statistics that excite adjacent sites in a lattice endowed withdisorder-immune chiral symmetry, we measure an output photon-number distribution that changesperiodically between super-thermal and sub-thermal photon statistics upon ensemble averaging.Thus, the photon-bunching level is controlled interferometrically at a fixed mean photon-numberby gradually activating the excitation symmetry of the chiral-mode pairs with structured coherentillumination and without modifying the disorder level of the random system itself.Optical interferometers implement deterministic field trans-formations that trace an interferogram by sweeping a phase –which enables applications across all of optics and photonics .The detected intensity is modulated, but the underlying photonnumber distribution P n does not change: the Poisson statisticsassociated with coherent light remain Poissonian, and the Bose-Einstein statistics that are the hallmark of thermal light remainBose-Einstein. Introducing dynamical randomness in the inter-ferometer can change the photon statistics. An early example inneutron interferometry implemented a chopper in one arm of atwo-path interferometer , while an optical ‘stochastic interfer-ometer’ devised by De Martini et al. changes the photon statisticsby inserting a randomly varying optical element . Along adifferent vein, coherent light traversing a random medium maygradually acquire chaotic behavior and under certain conditionsmay even attain Bose-Einstein statistics. This phenomenon hasbeen observed in several systems, including weakly transmittingbarriers embedded in a waveguide , quasi one-dimensional dis-ordered samples , and optically dense slabs containing multiplescatterers . In these approaches, changing the photon statis-tics is predicated on varying the system disorder level – with theamount of fluctuations (photon bunching) typically proportionalto the disorder level.We present here a different strategy for tuning photon statis-tics in a random structure that does not require modifying itsdisorder level. Instead, the ensemble-averaged P n is varied deter-minstically by an external element in the form of a relative phasebetween two input channels of a random network – analogouslyto a traditional two-path interferometer. Reconstructing P n re-veals deterministic interferometric tuning of photon-bunchingacross the super-thermal and sub-thermal regimes with fixed meanphoton-number ¯ n . This effect has its origin in the smooth and de-terministic breaking of the excitation symmetry in certain randomlattices, which can be achieved by sculpting the lattice excitation.Our experiment makes use of a one-dimensional (1D) disorderedphotonic lattice of evanescently coupled waveguides with off-diagonal disorder . Additionally, we measure two statisticalquantities: the normalized second-order photon correlation g ( ) ,which does not depend on ¯ n , and Mandel’s Q -parameter, whichdoes . In previous work , we measured the statistical parameter g ( ) . Although g ( ) can be computed from the photon-numberdistribution, the converse is not true. The counting distributionhas features that are not captured by a single parameter, its nor-malized second-order moment g ( ) . The novelty in our work liesin revealing the tunability of the shape of the distribution itself,by demonstrating the transition between a Poisson distribution,to a Bose-Einstein-like distribution, and the emergence of otherintermediate photon counting distributions. The concept of interferometric control over P n is depicted inFig. 1. We start by considering coherent light with a Poissonian dis-tribution P n = ¯ n n e − ¯ n / n !, n = ··· , that is fed into a single chan-nel of a disordered lattice consisting of an array of coupled opticalelements (waveguides , resonators , or fiber loops ).The output P n can be changed by varying the lattice disorder level ∆ (to be defined below). Surprisingly, high-order coherences do not decline while increasing ∆ [Ref. ]. If the lattice is endowedwith disorder-immune ‘chiral symmetry’ , a photonic thermal-ization gap emerges upon ensemble averaging: the regime ofsub-thermal photon statistics is forbidden at any disorder level,while super-thermal statistics are inaccessible to lattices lackingchiral symmetry . We make use of the normalized second-orderphoton correlation g ( ) = (cid:104) n ( n − ) (cid:105) / (cid:104) n (cid:105) ( (cid:104) . (cid:105) denotes ensemble av-eraging over both the quantum state and the lattice disorder)as a scalar measure of randomness to delineate the sub-thermal1 < g ( ) < g ( ) > . Instead of a mono-tonic trend towards ‘thermalization’ with increasing ∆ in latticescharacterized by chiral symmetry (such as those with off-diagonal disorder ), P n exhibits super -thermal statistics with a gradualdecline towards thermal statistics upon increasing ∆ [Fig. 1(a)].Such a lattice can thus tune the photon statistics in the super -thermal regime. Alternatively, in lattices lacking chiral symmetry(such as those with diagonal disorder ), P n exhibits sub-thermalstatistics with a gradual decline towards Poisson statistics uponincreasing ∆ [Fig. 1(b)]. This lattice can thus tune the photonstatistics in the sub -thermal regime.To tune the photon statistics without altering ∆ , we sculpt theexcitation over multiple lattice sites; e.g., by varying the relativephase between two sites. In traditional interferometry, a relativephase θ is introduced between two fields before they are super-posed to trace an intensity interferogram . In the interferometricscheme introduced here, a relative phase θ is introduced betweenadjacent channels of a random network with chiral symmetry fedwith coherent light. The two fields superpose within the networkand P n ( θ ) measured in a single output channel reveals a deter-ministic tuning of P n ( θ ) between super-thermal and sub-thermalregimes while sweeping θ [Fig. 1(c)].The above-described effect stems from the properties of theeigenmodes and eigenvalues of a lattice endowed with chiral sym-metry. To appreciate the underlying physics, consider a generictight-binding lattice model with nearest-neighbor-only coupling.The complex field amplitude E x ( z ) at site x after traveling along z is described by a set of coupled differential equations, − i d E x d z = β x E x + C x , x + E x + + C x , x − E x − , (1)where β x is the wave number for site x and C x , x ± the couplingcoefficient between site x and site x ± a r X i v : . [ phy s i c s . op ti c s ] J un Figure 1 | The concept of interferometric control over P n . (a,b) Coherent light with Poissonian statistics is fed into a channel of a random network while varying the disorder level ∆ . The emerging light (a) spansthe regime of super-thermal statistics when the lattice is endowed with chiral symmetry, or (b) spans the sub-thermal regime when the network lackschiral symmetry. (c) Coherent light is split into two paths and a relative phase θ is introduced before being launched into a random network with chiralsymmetry at fixed ∆ . Modulating θ breaks the mode-excitation symmetry and enables spanning the sub- and super-thermal regimes. (d) When the chiralsymmetric eigenmode pairs in a lattice with of off-diagonal disorder (corresponding phasors in a single lattice site depicted in blue and red) are activatedwith equal weights (symmetric excitation), the phasor sum (depicted in gray) takes on either real or imaginary values depending on the lattice site. Thissymmetry is absent when the chiral symmetry is not activated (asymmetric excitation) or absent (diagonal disorder). Thus, the phasor sum is alwayscomplex (not only real or imaginary). random variables when the lattice is disordered. We define aHermitian coupling matrix or Hamiltonian H for the lattice where { β x } correspond to the diagonal elements and { C x , x ± } occupythe two next diagonals. The eigenvalues { b j } and eigenfunctions { ψ ( j ) x } of H are defined through H ψ ( j ) x =( ¯ β + b j ) ψ ( j ) x , where ¯ β is theaverage wave number. Since H is real and symmetric, { ψ ( j ) x } areall real. If H can be recast into a block-diagonal form after setting¯ β = off-diagonal disorder( β x = ¯ β and C x , x ± are random) satisfy chiral symmetry, whereaslattices featuring diagonal disorder ( C x , x ± = ¯ C and β x are random)do not. Henceforth, we focus our attention on lattices with off-diagonal disorder. A consequence of chiral symmetry is that b − j = − b j and ψ ( − j ) x =( − ) x ψ ( j ) x . However, for the impact of this skew-symmetry to be manifested, the members of the skew-symmetricpaired modes with indices ± j must be excited with equal weights,which we refer to as ‘activating’ the chiral symmetry . Tovisualize the activation of chiral symmetry, we associate a phasorwith each mode at each lattice site. Then, in the rotating framethat is common to all modes, the eigenmodes appear in pairswhose members are equal magnitude counter-rotating phasors.This does not occur when the chiral symmetry is not activated orabsent [Fig. 1(d)].This can be understood as follows. High-order coherences de-pend on the expected values of cross-correlations of the excitedmodal coefficients. In a disordered lattice, these typically all van-ish except for the auto-correlation terms. However, in the presenceof skew-symmetric modes in each realization of the random ensem-ble, the cross-correlation of their excitation coefficients survivestatistical averaging, thereby adding a contribution to photonbunching and resulting in super-thermal statistics. This can beseen at both low ∆ via a perturbation analysis around the peri-odic lattice solutions, and high ∆ where localization reduces thenumber of lattice modes coupled to the excitation and thus alsoreduces the photon bunching . One can also understand thisbehavior by examining the statistics of the field quadratures .Under symmetric excitation configurations, the average field isconstrained to a single quadrature in the complex plane. As aresult, the intensity distribution approaches to chi-squared distri- bution with one degree of freedom corresponding to modifiedBose-Einstein photon number distribution .An input optical field E x ( z = ) can be analyzed in a basisof lattice eigenmodes, E x ( )= ∑ j c j ψ ( j ) x , where c j = ∑ x ψ ( j ) x E x ( ) is the amplitude of the j th mode ψ ( j ) x . The field subsequentlyevolves after a distance z into E x ( z )= ∑ j c j ψ ( j ) x e ib j z . In the spe-cial case of a single-site excitation at the input E x ( )= δ x ,0 , then | c j | = | c − j | = | ψ ( ± j ) | for all j , so that chiral symmetry is activated.This is not necessarily the case for more general field excita-tions. For example, when two adjacent sites are excited equally E x ( )= δ x ,0 + δ x ,1 , the modal coefficients are c ± j = ψ ( j ) ± ψ ( j ) , andchiral symmetry is activated | c j | = | c − j | only when the relativephase is θ = ± π /2. Gradually varying the phase θ for fixed rela-tive amplitudes ( | A | = | A | in our experiment) tunes the chiral-symmetry breaking: maximal symmetry breaking at θ = π ,and symmetry activation at θ = ± π /2.The photonic lattice we utilized consists of an array of 101 iden-tical 35-mm-long waveguides with nearest-neighbor-only evanes-cent coupling . The average separation between the waveguidesis 17 ¯m, resulting in an average coupling coefficient ¯ C ≈ − at a wavelength of 633 nm. The coupling coefficients are selectedindependently from a uniform probability distribution functionwith mean ¯ C and half-width ∆ in units of ¯ C ; our sample has ∆ ≈ P n is illustrated in Fig. 2. A single-mode coherent beam from a He-Ne laser is attenuated by a neutraldensity filter and split into two paths via a beam splitter. A relativephase shift θ is introduced via a delay in one path varied in 20-nmsteps. The two beams are then brought together by a second beamsplitter in parallel but closely spaced paths, which are imagedto two neighboring waveguides. The output facet of the arrayis imaged with a magnification of 8 × via a lens (focal length f = P n is reconstructed using three photon-detection time windows: 20,40, and 60 µ s. The input intensity level is reduced to low levels Figure 2 | Experimental setup for interferometric control of the photon-number distribution.
The input laser is attenuated, split into two paths (beam splitter 1), a relative phase θ is introduced, the two beams are brought together in parallel paths(beam splitter 2), and then imaged into two adjacent waveguides in the array sample. The intensity distribution along the photonic lattice is plotted (inlogarithmic scale) for a single realization of lattice disorder. The inset shows the coupling scheme, where E and E = E e i θ are the coherent fieldamplitudes coupled to waveguides and , respectively. Furthermore, a beam splitter at the output directs light that is imaged to a CCD camera, whichhelps ensure coupling into two neighboring waveguides for each disorder realizations (the camera and beam splitter are omitted for clarity). so that only a few photons are detected within these windowswhile minimizing the accidental arrivals of multiple photons. Wegenerate different disorder realizations by moving the array alongthe x -direction and coupling into a new pair of waveguides . Wemeasure single realizations of P n ( θ ) (averaged over 10 shots ofthe detection window) at the central lattice site x =
0, and thenaverage P n ( θ ) over an ensemble of 15 disorder realization for eachvalue of θ by shifting the input excitation site.We present in Fig. 3 our measurements confirming the determin-istic interferometric tuning of P n ( θ ) in the excitation waveguide( x =
0) while varying θ . As θ is swept, P n ( θ ) varies periodically (pe-riod π ) between sub-thermal to super-thermal statistics [Fig. 3(a)].The measured mean photon-number ¯ n ( θ ) does not vary with θ ( ¯ n ≈ ≈ n remains con-stant, the photon-number distribution itself varies with θ , achiev-ing maximal bunching when the chiral symmetry is fully activated θ = π /2,3 π /2, and minimal bunching at θ = π when chiral sym-metry is dormant. To better examine the salient changes in thephoton statistics with θ , we plot in Fig. 3(b,c) P n ( θ ) for the extremaat θ = π /2. The increase in the probability of higher pho-ton numbers at the distribution tail when θ = π /2 as compared tothat when θ = n ≈ n , thesymmetric-excitation statistics exhibit higher probabilities thanthe Bose-Einstein distribution, whereas the broken-symmetric ex-citation has lower. Our experimental data [Fig. 3(c)] is in excellentagreement with the simulations [Fig. 3(b)].Further analysis of the measured distributions P n ( θ ) helps bringabout the changes that take place in the photon statistics. First,we examine a quantity extracted from P n ( θ ) that does not dependon ¯ n for the field considered here: the normalized second-orderphoton correlation function g ( ) ( θ ) [Ref. ]. We plot in Fig. 4(a) g ( ) ( θ ) and note clearly that it varies sinusoidally with θ , betweenthe sub-thermal and super-thermal regimes. The statistics aretuned from super-thermal ( g ( ) >
2) to sub-thermal ( g ( ) < n fixed with θ [Fig. 4(a), inset]. Although ¯ n changes according to the photon-counting window ( ¯ n ≈ µ s, respectively), the three inter-ferometric traces of g ( ) ( θ ) are indistinguishable.Next we consider a quantity that does indeed depend on ¯ n :Mandel’s Q -parameter, Q = Var ( n ) / (cid:104) n (cid:105)− = ¯ n ( g ( ) − ) , which isthus linear in ¯ n for the field considered here . Varying θ at a fixed¯ n modulates Q ( ¯ n ; θ ) between two limits identified by the dashedinclined lines in Fig. 4(b); we identify in Fig. 4(b) only the values Figure 3 | Deterministic interferometric tailoring of P n . (a) The measured P n ( θ ) while varying the phase θ between two coherentbeams fed into adjacent sites of a disordered photonic lattice. The meanphoton-number ¯ n ( θ ) ≈ is independent of θ (with a standard deviation ≈ ). (b) Simulated P n corresponding to θ = and π /2 obtained for ¯ n = and utilizing the physical parameters of the lattice. (c) Measured P n ,extracted from (a) and corresponding to the simulations in (b). (a,b)Black-dotted line is Bose-Einstein distribution for ¯ n = corresponding to θ = π /2, which are the extrema of thisoscillation between the super-thermal and sub-thermal regimes.Increasing ¯ n (longer counting windows) leads to a linear growthin the two limits of Q modulation.In conclusion, we have demonstrated that the photon-numberdistribution P n – and hence any photon statistic such as g ( ) or Mandel’s Q -parameter – can be tuned deterministically byvarying a relative phase between two equal-amplitude beamslaunched into adjacent sites of a disordered photonic lattice. Suchinterferometric control over P n is possible by judicious excitation- Figure 4 | Deterministic interferometric control over g ( ) ( θ ) andMandel’s Q -parameter. (a) g ( ) as a function of θ obtained from the measured P n ( θ ) forphoton-counting windows of 20, 40, and 60 µ s. The g ( ) -interferogramsshow no dependence on ¯ n . The inset shows ¯ n ( θ ) for the differentphoton-counting windows: ¯ n increases with the detection window but isindependent of θ . (b) Mandel’s Q -parameter as a function of ¯ n at θ = (minimally activated chiral mode pairs) and θ = π /2 (maximally activatedchiral mode pairs) obtained from the measured P n . The data pointscorrespond to photon-counting windows in (a). The blue dashed line andthe red dot-dashed lines are linear fits that fall in the super-thermal(blue-black color scheme) and sub-thermal (white-red color scheme)regimes, respectively. The border between the two color schemesdemarcated by a solid gray line corresponds to Bose-Einstein statistics(thermal light). The dashed inclined lines correspond to horizontal dashedlines in (a). symmetry breaking of the skew-symmetric chiral mode pairs indisordered lattices that exhibit such symmetries.Coupling light into a single site is a highly symmetric configura-tion. By sculpting the complex spatial distribution of the excitationover multiple sites, one may gradually break the excitation sym-metry, which facilitates a smooth transition across the edge of thethermalization gap. This interferometric control strategy can begeneralized in multiple ways. One may vary the relative ampli-tude of the fields exciting the two lattice sites while maintaininga fixed relative phase, or one may excite an extended section ofthe lattice while alternating the relative phases between adjacentsites. The latter approach has the advantage of producing thesame phase-tunable P n with fixed ¯ n across the lattice.Finally, we note that multiple experiments have been reportedin which non-classical light is coupled into ordered or disorderedphotonic lattices . 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