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Coherence and mode decomposition of weak twin beams

Properties of weak spatio-spectral twin beams in paraxial approximation are analyzed using the decomposition into appropriate paired modes. Numbers of paired modes as well as numbers of modes in the signal (or idler) field in the transverse wave-vector and spectral domains are analyzed as functions of pump-beam parameters. Spatial and spectral coherence of weak twin beams is described by auto- and cross-correlation functions. Relation between the numbers of modes and coherence is discussed.


Introduction
Generation of photon pairs by spontaneous parametric down-conversion [1] belongs to the most frequently studied nonlinear processes in optics. The reason is entanglement of photons constituting a photon pair. It represents a purely quantum property. Entanglement leads to nonclassical cross-correlation functions between the signal and idler fields [2] that influence many physical effects. Nonclassical multiple coincidence-count rates used, e.g., for tests of the Bell inequalities [3,4] or quantum teleportation [5] represent the most important manifestation of these correlations. Unusual entangled two-photon absorption with its entanglement-induced transparency [6,7] and virtual state two-photon spectroscopy [8] is another example of the large impact of entanglement of photons. Also ghost imaging has to be mentioned here [9]. At present, there already exist several applications based upon entangled photon pairs including quantum key distribution [10], ultra-fast measurements [11] and absolute detector calibration [12,13].
This wide use of entangled photon pairs has naturally stimulated investigations of a detailed structure of photon pairs. It has been shown that their properties are fully described by two-photon (spectral, temporal, spatial) amplitudes at the level of individual photon pairs [14,15]. Moreover, the Schmidt decomposition of a two-photon amplitude [16,17] has been found crucial for the quantification of entanglement by the Schmidt number. We note that no physical quantity is directly related to entanglement but entanglement influences even qualitatively the physical behavior of photon pairs. As the experimental determination of the Schmidt number [18,19] and profiles of the modes [20,21,22] is difficult, an alternative approach based on the measurement of field width and width of the corresponding intensity cross-correlation function has been developed giving the Fedorov ratio [23,24,25] as a good quantifier of entanglement.
In the last years, a greater deal of attention has been devoted to fields composed of many photon pairs arising in parametric down-conversion [26,27,28]. Properties of these twin beams reflect those of the individual photon pairs [29], at least for not very intense twin beams [30]. For characterizing twin beams, auto-correlation functions are important as well as the cross-correlation functions [31]. Auto-correlation functions then give, according to statistical optics [32], the information about the number of modes constituting the signal (or idler) part of the twin beam. In analogy with the definition of Fedorov ratio, this number of modes can be determined by the ratio of field width and width of the corresponding auto-correlation function. Considering weak twin beams, the autocorrelation functions can even be derived from the twophoton amplitudes describing photon pairs.
In this contribution, we determine side by side the spatial and spectral auto-and cross-correlation functions as they depend on pump-field parameters. Using these functions, we obtain the number of modes in the signal (or idler) field and compare it with the Schmidt number giving the number of paired modes [24].
The paper is organized as follows. Theory suitable for describing weak twin beams is presented in Sec. 2. Sec. 3 is devoted to the properties of twin beams in the transverse wave-vector plane. Spectral properties of the twin beams are discussed in Sec. 4. Sec. 5 brings conclusions.

Theory of weak twin beams
Parametric down-conversion in a medium with tensor d of the second-order nonlinear coefficients is characterized by momentum operatorĜ int written as follows [33,34]: denotes the positivefrequency part of the classical pump electric-field amplitude andÊ ] means the negativefrequency part of the signal-[idler-] field operator amplitude. Permittivity of vacuum is denoted as ǫ 0 , symbol : is shorthand for tensor shortening with respect to its three indices and h.c. replaces the Hermitian conjugated term.
Amplitudes of all three interacting fields can be decomposed into harmonic plane waves with wave vectors k a and frequencies ω a : In the considered paraxial approximation, the plane wave with wave vector k a is conveniently parameterized by its frequency ω a and transverse wave vector k ⊥ a . The quantum signal and idler spectral electric-field am-plitudesÊ (−) a (k ⊥ a , ω a ) can then be expressed in terms of the appropriate creation operatorsâ † (k ⊥ a , ω a ): h is the reduced Planck constant and c is the speed of light in vacuum; k a = |k a |.
Correlations in the transverse wave-vector planes of the signal and idler fields are described by the following function T L depending on the pump-field transverse spatial spectrum E ⊥ p (k ⊥ p ): sinc(x) ≡ sin(x)/x and δ stands for the Dirac δfunction. In Eq. (4), |k ⊥ a | 2 = k 2 a,x + k 2 a,y and L denotes the crystal length.
We assume that the emitted signal and idler fields have the radial symmetry. This is a good approximation for sufficiently narrow spatial spectral profiles E ⊥ p of the pump field. The pump field in the considered type-I nonlinear interaction propagates as an extraordinary wave and so its index n p of refraction changes linearly with the radial emission angle ϑ p [ϑ p = arcsin(|k ⊥ p |/k p )] in the plane containing the propagation direction and the crystal optical axis. This dependence introduces anisotropy into the generated signal and idler fields [35,36,37]. Assuming the Gaussian transverse pulse profile with radius w p (E ⊥ p (|k ⊥ p |) = w p / √ 2π exp[−w 2 p |k ⊥ p | 2 )/4]) the anisotropy is well developed for narrow transverse pump profiles with wide spatial spectra. Anisotropy considerably modifies the emitted signal and idler fields for the pump-field radii w p comparable or smaller than w a p (for more details, see [35]): x e ≈ 2.2 [sin(x e )/x e = 1/e]. The assumed radial symmetry qualitatively simplifies the description as it allows to decompose the function T L into the dual Schmidt basis in both radial and azimuthal directions in the transverse planes: Whereas the functions u a,ml describe the radial parts of the fields, the harmonic functions exp(imϕ a )/ √ 2π are appropriate for the azimuthal parts of the fields, a = s, i. Symbols λ ⊥ ml denote the Schmidt numbers and t ⊥ is the normalization constant.
Unitary transformations of the field operators, then allow us to write the first-order perturbation solution of the Schrödinger equation as follows: |vac is the incident vacuum state. The twophoton spectral amplitude F L introduced in Eq. (8) is determined by the formula where E p stands for the pump-field spectrum and d eff denotes an effective nonlinear coupling constant. The Schmidt decomposition of two-photon amplitude F L , and introduction of new field operatorsâ a,mlq , allows to rearrange the output state |ψ out into the form: Formula (12) represents the output state |ψ out decomposed into independent paired spatial and spectral modes. We note that there occurs degeneracy in the signal and idler mode structure for m = 0 owing to the radial and spectral symmetry of the signal and idler fields. Due to this degeneracy, we cannot distinguish a signal and an idler photons in modes with m = 0 as both photons are created in one spatiospectral mode. This considerably modifies properties of the generated paired fields in the collinear geometry [38]. On the other hand, non-collinear geometries are practically unaffected by this degeneracy as the relative contribution of such photon pairs to the structure of a twin beam is negligible. The emitted signal (or idler) field is characterized by its intensity profiles along variables k ⊥ a , ϕ a and ω a , a = s, i.
Its internal correlations are described by amplitude correlation functions in these variables. The mutual signal and idler correlations are characterized by intensity cross-correlation functions in these variables. In detail, intensity profile n s,k (expressed in photon-number density) of the signal field in the radial wavevector direction is derived from the function T L given in Eq. (7): Using Eq. (9) the signal-field intensity spectrum n s, The amplitude signal-field correlations in their and spectral [A a s,ω ] variables are described by the corresponding auto-correlation functions given as: The intensity correlations between the signal and idler fields in their radial : , symbol N :: mean the normal ordering of field operators], azimuthal [C s,ϕ ] and spectral [C s,ω ] variables are characterized by the following cross-correlation functions: Similar quantities as written in Eqs. (13)(14)(15)(16) for the signal field can be defined also for the idler field. The number of effectively populated paired modes in a twin beam is determined by the Schmidt number K defined as [39,40] using eigenvalues λ q of the Schmidt decomposition of two-photon amplitude normalized such that q λ 2 q = 1.
On the other hand, the number K ∆ b of modes constituting field b, b = s, i, in a given variable is quantified by the ratio of appropriate intensity width ∆n b of field b and width ∆A a b of the amplitude autocorrelation function introduced in Eq. (15):

Spatial properties of weak twin beams
We consider a BBO crystal 8-mm long cut for non-collinear type-I process (eoo) for the spectrallydegenerate interaction among the wavelengths λ 0 p = 349 nm and λ 0 s = λ 0 i = 698 nm (ϑ BBO = 36.3 deg). The pump field is provided by the third harmonics of the Nd:YLF laser at the wavelength 1.047 µm. Both the Gaussian transverse profile with radius w p and the Gaussian spectrum of the pump pulse with duration are considered in the calculations. Assuming the pump field at normal incidence, the signal and idler fields at the central frequencies ω 0 s and ω 0 i propagate outside the crystal under the radial emission angles ϑ s = ϑ i = 8.45 deg. As this configuration is symmetric with respect to the exchange of the signal and idler fields, we restrict our discussion to only the signal field. We assume in the discussion that the conditions are such that the spectral and spatial properties of the twin beams factorize.
We first pay attention to the properties of twin beams in the wave-vector transverse plane. We assume in accord with the developed model that the twin beam has the rotational symmetry around the z axis and so it is stationary in the azimuthal angle ϕ. Formula (5) suggests that the rotational symmetry is roughly observed for the pump-field radius w p larger than w a p that equals 270 µm for the 8-mm long crystal and the considered geometry [n p = 1.658, |dn p /dϑ p | = 0.123]. The pump-field radius w p is the crucial parameter that determines the properties of twin beams in the transverse plane. It gives the number K kϕ of independent transverse modes comprising the twin beam. In our configuration, the greater the value of radius w p the greater the number K kϕ of independent modes, as documented in Fig. 1. The number K kϕ of overall modes in the transverse wave-vector plane can approximately be factorized into the number K k of modes in the radial direction and the number K ϕ of modes in the azimuthal direction. This factorization arises from the fact that the number K of radial modes determined for a fixed value of the azimuthal number m depends only weakly on the number m. The  number K ϕ of azimuthal modes is much larger than the number K k of radial modes as these modes cover the whole circle. There typically occur hundreds or thousands of independent azimuthal modes. On the other hand, from several up to several tens of radial modes are needed for the description of a typical twin beam. Whereas the azimuthal modes take the form of harmonic functions, damped exponential functions modified by oscillating polynomials [41] approximate the radial modes. The first couple of radial modes for m = 0 are shown in Fig. 2 together with the radial intensity profile n s,k for the pump field 1-mm wide. It holds that the intensity profile of an l-th radial mode has l zeroes and l + 1 local peaks. Also the larger the value of l, the wider the mode. However and importantly, all modes are distributed along the whole intensity profile n s,k (k s ). We note that, in Fig. 1, for the values of pump-field radius w p smaller than 0.270 µm the generated twin beams are influenced by the crystal anisotropy. Anisotropy leads in general to (a) (b) Figure 3. (a) Widths ∆A s,k (FWHM, full width at half maximum) of signal-field intensity auto-correlation function (plain curve) and ∆C s,k of intensity cross-correlation function (dashed curve) in radial direction as they depend on pump-field radius wp.
more strict phase-matching conditions which results in greater values of the Schmidt numbers K compared to those shown in Fig. 1. Anisotropy and its role in photon-pair generation has been discussed in detail in [35]. Numbers K ∆ s of modes in the signal field determined along Eq. (18) are compared with the numbers K of paired modes from the Schmidt decomposition in Fig. 1. This comparison reveals that the values of K ∆ are only approx. by 40% lower than the values of K. The numbers K ∆ of effectively populated signal-field modes are in general smaller than the corresponding Schmidt numbers K as the widths of auto-correlation functions occurring in definition (18) are not able to fully take into account the complex structure of entanglement present in a twin beam [18,19]. In other words, numbers K of paired modes and numbers K ∆ s of signal-field modes are defined differently. However, their comparison done in Fig. 1 clearly reveals that both of them are suitable for quantifying the dimensionality of a twin beam.
As the range of radial angles ϑ s belonging to the emitted photons practically does not change with the pump-field radius w p and the allowed azimuthal emission angles ϕ s lie in interval (0, 2π), an increase of the number K ∆ s,kϕ of signal-field modes with the radius w p is caused by the decrease of the radial and azimuthal widths of effective modes given by the extension of amplitude auto-correlation functions A a s,k and A a s,ϕ defined in Eqs. (15). The dependence of widths ∆A s,k and ∆A s,ϕ of their intensity counterparts on the radius w p is shown in Figs. 3(a) and 4(a), respectively, and confirms this behavior. Profiles A s,k and A s,ϕ giving the signal-field intensity correlations for the pump field 1-mm wide are drawn in Figs. 3(b) and 4(b). The generation of signal and idler fields by photon pairs strongly correlated in the transverse plane results (a) (b) Figure 4.
(a) Widths ∆As,ϕ (FWHM) of signal-field intensity auto-correlation function (plain curve) and ∆Cs,ϕ of intensity cross-correlation function (dashed curve) in azimuthal direction depending on pump-field radius wp.
in strong cross-correlations between the intensities of the signal and idler fields. Also these cross-correlations described by the cross-correlation functions C s,k and C s,ϕ given in Eqs. (16) depend strongly on the pump-field radius w p . The larger the radius w p the smaller the widths ∆C s,k and ∆C s,ϕ of intensity cross-correlation functions in the radial and azimuthal directions, respectively [see Figs. 3(a) and 4(a)] [42,31]. The comparison of widths ∆C s,k and ∆A s,k (or ∆C s,ϕ and ∆A s,ϕ ) plotted in Figs. 3(a) and 4(a) reveals that the intensity cross-correlation functions are narrower than the intensity auto-correlation functions. This behavior can qualitatively be explained as follows.
Cross-correlations are formed by individual photon pairs with certain correlations between the signal and idler photons. As these correlations reflect the geometry of the nonlinear process, blurring of the ideal point-like correlations (in the transverse wavevector space) occurs. On the other hand, intensity auto-correlations in the signal (or idler) field are created at the quantum level by at least two photon pairs. Roughly speaking, the first emitted photon pair provides a signal photon and its correlated idler twin. The annihilation of the second photon pair using the already present idler photon then creates photonphoton auto-correlations in the signal field. As we need two photon pairs in this reasoning, the overall blurring is larger compared to that of one photon pair. As a consequence, the intensity auto-correlation functions are wider than the intensity cross-correlation functions. Also similarity of the profiles of intensity auto-correlation functions A s,k and A s,ϕ and intensity cross-correlation functions C s,k and C s,ϕ , respectively, can be deduced from this reasoning. The curves plotted in Figs. 3(b) and 4(b) confirm this behavior.

Spectral properties of weak twin beams
The behavior of twin beam in the spectral domain and under the considered conditions is more complex compared to the spatial domain. The number K ω of paired spectral modes attains its minimum when considered as a function of the pump-field spectral width ∆λ p [24]. The number K ω of paired spectral modes is larger (K ω ≈ 70) even in this minimum reached for the pump pulse with the spectrum approx. 0.2 nm wide, as documented in Fig. 5. It holds also in the spectral domain that the number K ∆ ω of signal-field modes determined from the ratio defined in Eq. (18) is smaller than the number K ω of paired modes arising from the Schmidt decomposition (see Fig. 5). The comparison of curves in Fig. 5 confirms that both the number K ω of paired modes and the number K ∆ ω of signal-field modes are suitable for quantifying dimensionality of the twin beam in a broad range of pump-field spectral widths.
The dependence of the number K ∆ ω of signalfield modes on the pump-field spectral width ∆λ p as drawn in Fig. 5 can be explained by the behavior of spectral intensity width ∆n s,ω and width ∆A s,ω of intensity auto-correlation function. Whereas the spectral intensity width ∆n s,ω monotonically increases as a function of ∆λ p [see Fig. 6(a)], the width ∆A s,ω of intensity auto-correlation function increases for lower values of ∆λ p and then it saturates [see Fig. 6(b)]. This results in an increase of the values of K ∆ ω for larger values of ∆λ p observed in Fig. 5. Modes of the spectral decomposition behave in the same manner as the modes in the radial direction. Thus, the intensity profile of a q-th mode has q zeros and q + 1 peaks and extends over all frequencies found in the spectrum [see Fig. 7(a)]. As a consequence of the conservation law of energy, the frequencies of the emitted signal and idler photons are correlated within the spectral interval given by the pump-field spectral width. However, (a) (b) Figure 6. (a) Width ∆ns,ω (FWHM) of signal-field intensity spectrum and (b) widths ∆As,ω (FWHM) of signal-field intensity auto-correlation function (plain curve) and ∆Cs,ω of intensity cross-correlation function (dashed curve) as they depend on pump-field spectral width ∆λp.
the interval of correlations is also influenced by the phase-matching conditions that limit its extension. It follows that a certain width of this interval is reached with the increasing pump-field spectral width ∆λ p and no further increase is possible [see Fig. 6(b)]. This determines correlations between the intensities of the signal and idler fields as well as among the intensities inside the individual fields. The same argumentation as that used in the transverse wave-vector domain shows that the intensity auto-correlation function A s,ω has to be broader than the intensity cross-correlation function C s,ω [compare the curves in Fig. 6(b)] and also that profiles of these correlation functions plotted in Fig. 7(b) are similar. The picture that the overall spectrum is composed of adjacent independent 'local' modes is useful in predicting the behavior of numbers K ω of paired modes and K ∆ s,ω of signal-field modes with respect to spectral filtering. It suggests that the narrower the filter width is, the lower the numbers K ω and K ∆ s,ω of modes needed in the description of a twin beam. This behavior has been confirmed numerically. A sufficiently strong filtering then allows to reach a twin beam with the number of modes approaching one [43]. The same conclusions are valid for the geometric filtering that reduces the numbers K kϕ and K ∆ s,kϕ of modes in the transverse wave-vector plane.

Conclusions
Weak spatio-spectral twin beams have been analyzed in paraxial approximation using the perturbation solution of the Schrödinger equation and its decomposition into the spatial and spectral paired modes. Applying these modes, coherence properties of weak twin beams have been studied using both the auto-and cross-correlation functions. Numbers of paired modes revealed by the Schmidt decomposition and numbers of modes constituting the signal (or idler) field and given by the ratio of field width and width of the appropriate amplitude auto-correlation function have been found mutually proportional in the broad area of the analyzed pump-field parameters. This justifies both of them as appropriate quantifiers of dimensionality of weak twin beams.
Whereas the number of transverse modes increases with the increasing pump-field radius in our configuration, the number of spectral modes considered as a function of pump-field spectral width has a well-formed minimum.
This behavior has been explained analyzing the widths of appropriate correlation functions. It has been shown that the spectral and transverse wave-vector cross-correlation functions are broader than the corresponding autocorrelation functions.