Using a basic Mach-Zehnder interferometer, we demonstrate experimentally the measurement of the photonic de Broglie wavelength of an entangled photon pair (a biphoton) generated by spontaneous parametric down-conversion. The observed interference manifests the concept of the photonic de Broglie wavelength. The result also provides a proof-of-principle of the quantum lithography that utilizes the reduced interferometric wavelength.
Correlations between photons are interesting for a number of applications and concepts in metrology in particular for resolution improvements in different methods of quantum imaging. Since Fock-states of N-photons of wavelength lambda in interference schemes acquire N-times the phase shift of single photons these states can appear as if they had a de Broglie wavelength of lambda/N. A biphoton beam diffracted at a blazed grating shows this reduced de Broglie wavelength. This experiment can be seen as a purification of biphotons of a certain correlation strength on the one hand. On the other han…
We verified operational approach based on direct measurement of entanglement degree for bipartite systems. In particular spectral distributions of single counts and coincidence for pure biphoton states generated by train of short pump pulses have been measured and entanglement quantifier calculated. The approach gives upper bound of entanglement stored in total biphoton states, which can reach extremely high value up to $10^{4}-10^{5}$.
We show that the wave packet of a biphoton generated via spontaneous parametric down conversion is strongly anisotropic. Its anisotropic features manifest themselves very clearly in comparison of measurements performed in two different schemes: when the detector scanning plane is perpendicular or parallel to the plane containing the crystal optical axis and the laser axis. The first of these two schemes is traditional whereas the second one gives rise to such unexpected new results as anomalously strong narrowing of the biphoton wave packet measured in the coincidence scheme and very high degr…
Spontaneous parametric down-conversion (SPDC) is a widely used source for photonic entanglement. Years of focused research have led to a solid understanding of the process, but a cohesive analytical description of the paraxial biphoton state has yet to be achieved. We derive a general expression for the spatio-temporal biphoton state that applies universally across common experimental settings and correctly describes the non-separability of spatial and spectral modes. We formulate a criterion on how to decrease the coupling of the spatial from the spectral degree of freedom by taking into acco…
As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we …
We experimentally study two-photon coherence in plasmon-assisted transmission with a two-photon Mach-Zehnder (MZ) interferometer. Two collinear photons of identical or orthogonal polarization are simultaneously incident on one optically thick metal film, perforated with a periodic array of subwavelength holes. The de Broglie wavelength of plasmon-assisted transmitted photons is measured, which shows that two photons are re-eradiated by the plasmons and the quantum coherence of biphoton is preserved in the conversion process of transforming biphoton to plasmons and then back to biphoton.
Analytical expressions are derived for the distribution rates of spatial coincidences in the counting of photons produced by spontaneous parametric down conversion (SPDC). Gaussian profiles are assumed for the wave function of the idler and signal light created in type-I SPDC. The distribution rates describe ellipses on the detection planes that are oriented at different angles according to the photon coincidences in either horizontal-horizontal, vertical-vertical, horizontal-vertical or vertical-horizontal position variables. The predictions are in agreement with the experimental data obtaine…
We calculate the properties of a biphoton source based on resonant backward-wave spontaneous parametric down-conversion. We show that the biphotons are generated in a single longitudinal mode having a subnatural linewidth and a Glauber correlation time exceeding 65 ns.
We show that broadband biphoton wavepackets produced via Spontaneous Parametric Down-Conversion (SPDC) in crystals with linearly aperiodic poling can be easily compressed in time using the effect of group-velocity dispersion in optical fibres. This result could foster important developments in quantum metrology and lithography.
Spectral and temporal coincidence and single-particle photon wave packets are described and their widths and durations are found. The degree of entanglement is characterized by the experimentally measurable parameter R defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration. This function is shown to have a minimum and even in the minimum, at rather natural conditions, the parameter R is found to be very high (R=73>>1). The Schmidt number K is found for both short and long pum…
We utilized the all-copropagating scheme, which maintains the phase-match condition, in the spontaneous four-wave mixing (SFWM) process to generate biphotons from a hot atomic vapor. The scheme enables our biphotons not only to surpass those in the previous works of hot-atom SFWM, but also to compete with the biphotons that are generated by either the cold-atom SFWM or the cavity-assisted spontaneous parametric down conversion. The biphoton linewidth in this work is tunable for an order of magnitude. As we tuned the linewidth to 610 kHz, the maximum two-photon correlation function, $g_{s,as}^{…
We demonstrate one- and two-photon diffraction and interference experiments utilizing parametric down-converted photon pairs (biphotons) and a transmission grating. With two-photon detection, the biphoton exhibits a diffraction-interference pattern equivalent to that of an effective single particle that is associated with half the wavelength of the constituent photons. With one-photon detection, however no diffraction-interference pattern is observed. We show that these phenomena originate from the spatial quantum correlation between the down-converted photons.
We analyze spontaneous parametric down-conversion in various experimentally feasible 1D quadratic nonlinear waveguide arrays, with emphasis on the relationship between the lattice's topological invariants and the biphoton correlations. Nontrivial topology results in a nontrivial "winding" of the array's Bloch waves, which introduces additional selection rules for the generation of biphotons. These selection rules are in addition to, and independent of existing control using the pump beam's spatial profile and phase matching conditions. In finite lattices, nontrivial topology produces single ph…
We describe spontaneous parametric down-conversion experiments in which induced coherence across two coupled interferometers results in high-visibility single-photon interference. Opening additional photon channels allows "which-path" information and reduces the visibility of the singlephoton interference, but results in nearly perfect visibility when photons are counted in coincidence. A simplified theoretical model accounts for these complementary observations and attributes them directly to the relations among the vacuum fields at the different crystals.
We report a scalable linear optical scheme for generating entangled states of multiple ququarts in which the individual single-ququart state is prepared with the biphoton polarization state of frequency-nondegenerate spontaneous parametric down-conversion. The output state is calculated with the full consideration of the higher order effect (double-pair events) of spontaneous parametric down-conversion. Scalability to multiple-ququart entanglement is demonstrated with examples: linear optical entanglement of three and four individual biphoton ququarts.
We propose experimental measurements of the logarithmic negativity, which quantifies quantum correlations using Gouy phase measurements in an asymmetric double-slit interference experiment for twin photons. This is possible because both quantities have analogous dependence with the spatial confinement by the slits and enables one to manipulate the portion of entanglement by the Gouy phase. In order to obtain those measurements, we need to work in a regime where the position correlations between particles are strong, therefore we investigate such correlations for biphotons. Since we would like …
The nonlinear optical process of spontaneous parametric down-conversion (SPDC) is widely studied for applications in quantum information science due to its ability to produce two photons that can be entangled in many degrees of freedom. For applications in quantum communication, two metrics of this process are particularly important: heralding efficiency and total joint rate. Here, we derive expressions for both quantities for a variety of different beam geometries and frequencies. We pay specific attention to the spectrum of both biphotons and individual photons. We reveal the underlying mech…
We systematically studied the temporal profile of biphotons, i.e., pairs of time-correlated single photons, generated from a hot atomic vapor via the spontaneous four-wave mixing process. The measured temporal width of biphoton wave packet or two-photon correlation function against the coupling power was varied from about 70 to 580 ns. We derived an analytical expression of the biphoton's spectral profile in the Doppler-broadened medium. The analytical expression reveals that the spectral profile is mainly determined by the effect of electromagnetically induced transparency (EIT), and behaves …
In spontaneous parametric down-conversion photons are known to be created coherently and with equal probability over the entire length of the crystal. Then, there is no particular position in the crystal where a photon pair is created. We make the seemingly contradictory observation that we can control the time delay with the crystal position along the propagation direction. We resolve this contradiction by showing that the spatio-temporal correlations critically affect the temporal properties of the pair of photons, when using a finite detector size. We expect this to have important implicati…
We measure the transverse entanglement of photon pairs on their propagation from the near to the far field of spontaneous parametric downconversion (SPDC). The Fedorov ratio, depending on the widths of conditional and unconditional intensity measurements, is shown to be only able to characterize entanglement in the near and far field zones of the source. Therefore we also follow a different approach. By evaluating the first-order coherence of a subsystem of the state we can quantify its entanglement. Unlike previous measurements, which determine the Fedorov ratio via intensity correlations, ou…
Encoding quantum information in continuous variables is intrinsically faulty. Nevertheless, redundant qubits can be used for error correction, as proposed by Gottesman, Kitaev and Preskill in Phys. Rev. A \textbf{64} 012310, (2001). We show how to experimentally implement this encoding using time-frequency continuous degrees of freedom of photon pairs produced by spontaneous parametric down conversion. We experimentally illustrate our results using an integrated AlGaAs photon pairs source. We show how single qubit gates can be implemented and finally propose a theoretical scheme for correcting…
Spontaneous parametric down conversion has been shown to be a reliable source of entangled photons. Amongst the wide range of properties that have been shown to be entangled, it is the orbital angular momentum that is the focus of our study. We investigate, in particular, the bi-photon state generated using a Gaussian pump beam. We derive an expression for the simultaneous correlations in the orbital angular momentum and radial momentum of the down-converted Laguerre-Gaussian beams. Our result allows us, for example, to calculate the spiral bandwidth with no restriction on the geometry of the …
The idea about a quantum nature of Planck's blackbody radiation law is deeply rooted in minds of most physicists. Einstein's work, in which the coefficients of spontaneous and induced emission were introduced, has always been regarded as a proof that quantum energy discreteness of an atom plays a crucial role in the derivation of this law. In our paper we avoid this standpoint. It may be shown that the de Broglie wave assigned to every material particle is a result of interaction of the particle with zero-point vibrations of electromagnetic ground field. The energetic spectrum of a harmonic os…
We provide a toolbox for continuous variables quantum state engineering and characterization of biphoton states produced by spontaneous parametric down conversion in a transverse pump configuration. We show that the control of the pump beam's incidence spot and angle corresponds to phase space displacements of conjugate collective continuous variables of the biphoton. In particular, we illustrate with numerical simulations on a semiconductor device how this technique can be used to engineer and characterize arbitrary states of the frequency and time degrees of freedom.
The photonic de Broglie wave, in which an ensemble of $N$ identical photons with wavelength $λ$ reveals $λ/N$ interference fringes, has been known to be a unique feature exhibited by the photon number-path entangled state or the NOON state. Here, we report the observation of the photonic de Broglie wave for a pair of photons, generated by spontaneous parametric down-conversion, that are not photon number-path entangled. We also show that the photonic de Broglie wave can even be observed for a pair of photons that are completely separable (i.e., no entanglement in all degrees of freedom) and di…
We study the theory of linearly chirped biphoton wave-packets produced in two basic quasi-phase-matching configurations: chirped photonic-like crystals and aperiodically poled crystals. The novelty is that these structures are considered as definite assembles of nonlinear layers that leads to detailed description of spontaneous parametric down-conversion (SPDC) processes through the discrete Gauss sums. We demonstrate that biphoton spectra for chirped photonic crystals involving a small number of layers consist from definite well-resolved spectral lines. We also discuss the forming of broadban…
In this paper, we investigate the influence of the symmetry of the biphoton wavefunction on the coincidence measurement of the generalized Mach-Zehnder (MZ) interferometer, where there are a temporal and frequency shift operations between the two beam-splitters. We show that the generalized MZ interferometer is the measurement of the short-time Fourier transform of the function modeling the energy conservation of a spontaneous parametric down-conversion process if the full biphoton state is symmetric, and of the symmetric characteristic distribution of the phase-matching function if the state …
We report a proof-of-principle experiment demonstrating that appropriately chosen set of Hermite-Gaussian modes constitutes a Schmidt decomposition for transverse momentum states of biphotons generated in the process of spontaneous parametric down conversion. We experimentally realize projective measurements in Schmidt basis and observe correlations between appropriate pairs of modes. We perform tomographical state reconstruction in the Schmidt basis, by direct measurement of single-photon density matrix eigenvalues.
Spectrally uncorrelated biphoton state generated from the spontaneous nonlinear optical process is an important resource for quantum information. Currently such spectrally uncorrelated biphoton state can only be prepared from limited kinds of nonlinear media, thus limiting their wavelengths. In order to explore wider wavelength range, here we theoretically study the generation of spectrally uncorrelated biphoton state from 14 isomorphs of potassium dihydrogen phosphate (KDP) crystal. We find that 11 crystals from the `KDP family' still maintain similar nonlinear optical properties of KDP, such…
We study first-order interference in spontaneous parametric down-conversion generated by two pump pulses that do not overlap in time. The observed modulation in the angular distribution of the signal detector counting rate can only be explained in terms of a quantum mechanical description based on biphoton states. The condition for observing interference in the signal channel is shown to depend on the parameters of the idler radiation.
We consider a double Gaussian approximation to describe the wavefunction of twin photons (also called a biphoton) created in a nonlinear crystal via a type-I spontaneous parametric downconversion (SPDC) process. We find that the wavefunction develops a Gouy phase while it propagates, being dependent of the two-photon correlation through the Rayleigh length. We evaluate the covariance matrix and show that the logarithmic negativity, useful in quantifying entanglement in Gaussian states, although Rayleigh-dependent, does not depend on the propagation distance. In addition, we show that the two-p…
We investigate the atom-optical analog of degenerate four-wave mixing of photons by colliding two Bose-Einstein condensates (BECs) of metastable helium and measuring the resulting momentum distribution of the scattered atoms with a time and space resolved detector. For the case of photons, phase matching conditions completely define the final state of the system, and in the case of two colliding BECs, simple analogy implies a spherical momentum distribution of scattered atoms. We find, however, that the final momenta of the scattered atoms instead lie on an ellipsoid whose radii are smaller th…
We theoretically investigated spectrally uncorrelated biphotons generated in a counter-propagating spontaneous parametric downconversion (CP-SPDC) from periodically poled MTiOXO4 (M = K, Rb, Cs; X = P, As) crystals. By numerical calculation, it was found that the five crystals from the KTP family can be used to generate heralded single photons with high spectral purity and wide tunability. Under the type-0 phase-matching condition, the purity at 1550 nm was between 0.91 and 0.92, and the purity can be maintained over 0.90 from 1500 nm to 2000 nm wavelength. Under the type-II phase-matching con…
Quantum correlations encoded in photonic Laguerre-Gaussian modes were shown to be related to the Gouy phase shifts (D. Kawase et al., Phys. Rev. Lett. 101, 050501 (2008)) allowing for a non-destructive manipulation of photonic quantum states. In this work we exploit the relation between phase space correlations of biphotons produced by spontaneously parametric down conversion (SPDC) as encoded in the logarithmic negativity (LN) and the Gouy phase as they are diffracted through an asymmetrical double slit setup. Using an analytical approach based on a double-gaussian approximation for type-I SP…
Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $sinc(LΔk/2)$-function, where $L$ is the length of the nonlinear medium and $Δk$ the phase mismatch between the pump and down-converted fields. In theoretical investigations, the \textit{sinc} behavior of the phase mismatch has often been approximated by a Gaussian function $\exp{(-αx^2)}$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literatur…
We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation of the degree of entanglement and polarization is shown to occur and to be characterized b…
We investigate spontaneous four wave mixing (SFWM) in a single-channel side-coupled integrated spaced sequence of resonators (SCISSOR). Analytic expressions for the number of photon pairs generated, as well as the biphoton wave function (joint spectral amplitude) describing the pairs, are derived and numerically computed for different pump pulse durations and numbers of ring resonators. In the limit of a long input pump pulse, we show a strong analogy between super-linear scaling of generation efficiency with respect to the number of rings in the structure and Dicke superradiance. More general…
In this work we study the transverse spatial correlation of the pair of photons generated via the process of spontaneous parametric frequency down-conversion, in periodically poled non-linear crystals illuminated by a pulsed laser beam. It is well known that the two-photon state generated in quasi-phase matching (QPM) configurations depends explicitly on the characteristics of the pump beam, on the crystal modulated non-linearity, and on the detection geometry. This has allowed the development of several techniques for controlling the biphoton spectral and spatial properties. Here we discuss a…
We investigate the transverse spatial profile of down-converted light produced by noncollinear, degenerate, Type-I spontaneous parametric down-conversion in two types of nonlinear crystals. We find that the pattern produced by one crystal, beta barium borate (BBO), produces a circular down-conversion pattern while the other crystal, bismuth triborate (BiBO) produces an elliptical pattern. We show this difference is due to the angle-independent refractive index experienced by the daughter photons in BBO, while they experience an angle-dependent refractive index in BiBO. We image the transverse …
Extended phase-matching conditions for spontaneous parametric down-conversion are examined. By augmenting the conventional phase-matching conditions, they permit the creation of a class of frequency-entangled states that generalizes the usual twin-beam biphoton state. An experimental characterization of these states is possible through interferometric coincidence counting.
As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in Spontaneous Parametric Down-Conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro…
The requirement on the symmetry of the biphoton wave function results in a double peak structure of the coincidence spectra of Spontaneous Parametric Down Conversion (SPDC) [1]. In this work we report the experimental observation of this effect by careful tailoring parameters of the SPDC crystal, the ultra-fast pump laser, and the measurement setup. The results are shown to be relevant to quantum state engineering of ultra-fast polarization-, or frequency-entangled mixed states.
A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate co-linear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic 'curves'in Hilbert space. Explicit expressions for Pancharatnam, dynamical and geometrical phases are derived for the transformations produced by linear phase-converters. The problem of gauge invariance is treated along all the article.
We demonstrate Einstein-Podolsky-Rosen (EPR) entanglement by detecting purely spatial quantum correlations in the near and far fields of spontaneous parametric down-conversion generated in a type-2 beta barium borate crystal. Full-field imaging is performed in the photon-counting regime with an electron-multiplying CCD camera. The data are used without any postselection, and we obtain a violation of Heisenberg inequalities with inferred quantities taking into account all the biphoton pairs in both the near and far fields by integration on the entire two-dimensional transverse planes. This ensu…
We demonstrate a new class of frequency-entangled states generated via spontaneous parametric down-conversion under extended phase matching conditions. Biphoton entanglement with coincident signal and idler frequencies is observed over a broad bandwidth in periodically poled KTiOPO$_4$. We demonstrate high visibility in Hong-Ou-Mandel interferometric measurements under pulsed pumping without spectral filtering, which indicates excellent frequency indistinguishability between the down-converted photons. The coincident-frequency entanglement source is useful for quantum information processing an…
We derive frequency correlation and exit probability expressions for photons generated via spontaneous parametric downconversion (SPDC) in nonlinear waveguides that exhibit linear scattering loss. Such loss is included within a general Hamiltonian formalism by connecting waveguide modes to reservoir modes with a phenomenological coupling Hamiltonian, the parameters of which are later related to the usual loss coefficients. In the limit of a low probability of SPDC pair production, the presence of loss requires that we write the usual lossless generated pair state as a reduced density operator,…
We propose an experiment that permits observation of the de Broglie two-photon wave packet behavior for a pair of photons, using a Mach-Zehnder interferometer. It is based on the use of pulsed lasers to generate pairs of photons via spontaneous parametric down-conversion and the post-selection of events. It differs from previous realizations by the use of a third time-correlated photon to engineer the state of the photons. The same technique can give us which-path information via an ``interaction-free'' experiment and can be used in other experiments on the foundations of quantum mechanics rel…
We derive an analytical form of the Schmidt modes of spontaneous parametric down-conversion (SPDC) biphotons in both Cartesian and polar coordinates. We show that these correspond to Hermite-Gauss (HG) or Laguerre-Gauss (LG) modes only for a specific value of their width, and we show how such value depends on the experimental parameters. The Schmidt modes that we explicitly derive allow one to set up an optimised projection basis that maximises the mutual information gained from a joint measurement. The possibility of doing so with LG modes makes it possible to take advantage of the properties…
