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Characterization of sp ectral entanglement of sp ontaneous parametric-down conversion biphotons in femtosecond pulsed regime
G. Brida, V. Caricato, M. V. Fedorov, M. Genovese, M. Gramegna and S. P. Kulik EPL, 87 (2009) 64003

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September 2009
EPL, 87 (2009) 64003 doi: 10.1209/0295-5075/87/64003 www.epljournal.org

Characterization of spectral entanglement of spontaneous parametric-down conversion biphotons in femtosecond pulsed regime
G. Brida1 , V. Caricato
1 2 3 4 1, 2

, M. V. Fedorov3 , M. Genovese

1 (a)

, M. Gramegna1 and S. P. Kulik

4

I.N.RI.M. - Istituto Nazionale di Ricerca Metrologica - Turin, Italy, EU Dipartimento di Fisica, Politecnico di Torino - Turin, Italy, EU A.M. Prokhorov General Physics Institute, Russian Academy of Science - Moscow, Russia Faculty of Physics, M.V. Lomonosov Moscow State University - Moscow, Russia received 4 June 2009; accepted in final form 12 September 2009 published online 9 October 2009
PACS PACS PACS

42.65.Lm ­ Parametric down conversion and production of entangled photons 03.65.Ud ­ Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.) 03.67.Mn ­ Entanglement measures, witnesses, and other characterizations

Abstract ­ We verified the operational approach based on the direct measurement of the entanglement degree for bipartite systems. In particular spectral distributions of single counts and coincidence for pure biphoton states generated by a train of short pump pulses have been measured and the entanglement quantifier has been calculated. The approach gives an upper bound of entanglement stored in total biphoton states, which can reach extremely high values up to 104 ­105 .
Copyright c EPLA, 2009

Entanglement, once an argument of debate on foundations of quantum mechanics, is now a resource for developing quantum technologies [1], such as quantum communication, q -calculus, q -imaging, q -metrology, etc. In this sense a precise and easy-implementable method for characterizing entanglement properties is a fundamental tool. There are several measures that quantify entanglement of quantum bipartite states both in discrete and continuous variables [2,3] as Schmidt rank, entropy, concurrence, etc. From a theoretical point of view all these quantifiers can be easily evaluated from the density matrix describing a given state of a bipartite system, thus there is no problem to find the meaning of a chosen entanglement measure for a certain density matrix. From an experimental side and following the same paradigm it would be necessary to perform a quantum tomography procedure (either complete or reduced) over the state [4] under study and then to extract the entanglement measure from the reconstructed density matrix. However, this scheme is not optimal in the case of high-dimensional systems since it requires a number of measurements growing quadratically
( a)

E-mail: m.genovese@inrim.it

with the dimension of Hilbert space. An alternative is exploiting particular links between entanglement and measurable characteristics of the state, just measuring some auxiliary parameters of the state. In particular Fedorov's parameter [5] Rq (where q denotes an arbitrary space, e.g. frequency or spatial one), defined for pure biphoton states as the ratio of the single-particle and coincidence distributions widths in q -space, can be rather easily measured at variance with all other entanglement quantifiers. R can be understood qualitatively from the entropy approach to entanglement: high entanglement leads to a better knowledge about composite bipartite systems (narrower coincidence distribution) and a worse knowledge about individual subsystem(s) (wider singleparticle distribution). Both clear physical meaning and operationability return R to an extremely useful tool for entanglement control. It has been proved [5] that for double-Gaussian bipartite states the parameter R coincides exactly with the Schmidt number K , which is the most fundamental entanglement measure [6]. Moreover their values remain quite close even for special classes of non­double-Gaussian wave functions like those describing Spontaneous Parametric Down-Conversion (SPDC). This

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G. Brida et al. approach can be used both for continues and discrete variables, so the parameter R has rather a universal sense: earlier it has been successfully applied for demonstrating strong entanglement anisotropy in spatial distributions of biphotons [7]. The present paper is devoted to the verification of the operational entanglement quantifier R for two-photon states entangled in a frequency domain as an efficient alternative to other quantifiers (for example in specific experiments the visibility of the interference pattern can provide some knowledge about entanglement [8]). Indeed such states belong to the multi-dimensional Hilbert space and can posses an extremely high entanglement degree (up to several hundreds) that makes them very perspective ob jects for quantum information and quantum communication. In this work, both in theory and experiment, we restrict our consideration to purely spectral entanglement. This means that only photons propagating along the pump axis are important and taken into account. In experiment this condition is realized with the help of very thin slits in front of detectors in the focal plane of photons emitted from a crystal (see further details below). An opposite case, when the photon frequencies are given but directions of their propagation can vary, has been investigated earlier [7]. The pump is assumed to have the form of a sequence of relatively short Fourier-limited pulses, which provides conditions for generating pure biphoton states characterized by a wave function. For the type-I degenerate collinear phase matching the wave function depending only on frequencies of the emitted signal and idler photons 1 and 2 is given by [9,10] (1 + 2 )2 2 (1 , 2 ) exp - 8ln 2 L (1 - 2 )2 âsinc A(1 + 2 ) - B 2c p argument cannot be substituted by any Gaussian function and, for this reason, the wave function (1) is considered as a non­double-Gaussian one. The wave function (1) can serve for determining coincidence and single-particle biphoton spectra. These spectra are significantly different in the cases of short an long pump pulses. The control parameter separating the regions of short and long pulses is given by [9] = 2 c 1 sinc , = 2 ( ( 1 pump AL L/vgp) - L/vgo) (3)

i.e., it is equal to the ratio between the double pumppulse duration and the difference of time required for the the pump and idler/signal photons for traversing all the crystal. Pump pulses are short if 1 and long if 1 and, typically, 1 at 1 ps. In the two asymptotic cases 1 and 1 the pump spectral amplitude as a function of 1,2 is, correspondingly, much narrower or much wider than the sinc-function in (1). Evidently, the parameter (3) accumulates factors to be rather easily controlled in experiment, namely length of the crystal and pump-pulse duration. Also the longitudinal walk-off effect can be evolved via A by choosing a crystal with appropriate dispersion properties as a source of biphotons. In the case of short pump pulses ( 1) the FWHM of the coincidence and single-particle spectra were found analytically to be given by [9] c = 5.56 c , AL s = 2A ln(2) p . B (4)

The first of these two formulae follows directly from (1), whereas the second one requires for its derivation a preliminary integration of |(1 , 2 )|2 , e.g., over 2 , and the integration is greatly simplified owing to significantly , (1) different widths of the pump spectral amplitude and the sinc-function [9]. where the Gaussian exponent characterizes the pump In the case of long pump pulses ( 1), the coincidence spectral amplitude, is the pump-pulse duration, L is and single-particle spectral widths can be found in a the length of the crystal, 1 and 2 are deviations of similar way and have the form [9] frequencies of the signal and idler photons 1, 2 from the (0) (0) 4ln 2 2.78 c 0 central frequencies 1 = 2 = p /2, |1, 2 | p , p is , s = . (5) c = the central frequency of the pump spectrum, A and B are LB the temporal walk-off and dispersion constants Equations (4) and (5) can be used for finding the parameter R( ) in regions of short and long pulses. Its expres1 1 - (o) , sion in the intermediated region ( 1) can be determined A = c kp ( ) = - k1 ( )|=p /2 = c (p) p vg vg approximately with the help of a simple quadratic interc B = p k1 ( )|=0 /2 , (2) polation [9] 4
2 2 (p) (o) R( ) Rshort + Rlong = vg and vg are the group velocities of the pump and ordinary waves and k1 and kp are the wave vectors of signal A L 1 1 (6) 0.75 2 + = 55 2 + and pump photons. B 0 In a general case, both linear and quadratic terms in the argument of the sinc-function in (1) are important and together with the Schmidt number K ( ) [K ( )]. The none of them can be dropped. A sinc-function with such an latter was calculated numerically by Silberhorn and

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Characterization of spectral entanglement of SPDC biphotons etc.

Fig. 1: (Colour on-line) Sketch of the experimental set-up: a duplicated titanium sapphire laser beam pumps a LiIO3 crystal producing collinear SPDC. After a Half-Wave Plate (HWP) the biphotons are split on a beam-splitter (BS) and fed to SPAD detectors, whose output feeds a Time-to-Amplitude Converter (TAC), counters and a Multi-Channel Analyzer (MCA).

Mauerer [11], who also showed that, under the conditions we consider, the phase of the wave function does not affect the Schmidt number calculated in the regions of short and long pulses [12]. The functions R( ) and K ( ) are seen to be rather close to each other, which confirms the applicability of the experimentally measurable parameter R for quantifying the degree of entanglement of biphoton states characterized by a non­double-Gaussian wave function of the form (1). The degree of spectral entanglement reaches rather high values in the whole range of pump-pulse durations and is especially high in the case of sufficiently short and long pulses (far from the minimum that localizes at 1 or 1 ps). The experiment (fig. 1) was performed in fs pulsed regime using a Mode-Locked Titanium-Sapphire laser at a working wavelength of I R = (795.0 ± 0.1) nm with a I R = (5.9 ± 0.1) nm, ( 2 mm waist). After doubling in frequency to p = (397.5 ± 0.2) nm and p = (1.8 ± 0.1) nm, corresponding to a pulse duration of = (186 ± 30) fs, the pump beam is addressed to a LiIO3 crystal where type-I collinear SPDC is produced. After eliminating the UV pump, biphotons are split on a beam-splitter and fed to two photodetection apparatuses (consisting of red glass filters and SPAD detectors). In front of each detector a monochromator with a variable spectral resolution is placed. Light is focused by a lens (f = 20 cm, 135 cm far from the crystal) inside each monochromator (50 µm slits width) and a couple of lenses with the same focal distance are placed on the output of the monochromators for focalizing the output packet spread by all the optics of the apparatus and by the slits. It is worth mentioning that in such a configuration a single spatial mode was collected. Following the general idea of the R -quantifier method all experiments were performed as a consecution of corresponding measurements of spectral distributions both in single counts and coincidences. To measure the coincidence distribution one of the two monochromators (by

Fig. 2: (Colour on-line) Normalized coincidence distribution for the 10 mm length LiIO3 sample. Points with error bars correspond to the measurements while the solid line shows the fitting by a Gaussian function with FWHM = (0.29 ± 0.03) nm.

convention selecting idler wavelength) is fixed at the central wavelength of SPDC (795 nm) whilst the other (signal) scans in a range around this value. In order to check the R -dependence on the control parameter the measurements have been repeated for crystals with different length (but with the same orientation), namely L = 10 mm and L = 5 mm. First of all we have performed measurements with a 10 mm LiIO3 crystal. To measure the coincidences distribution we have used monochromators with 0.2 nm resolution. We scanned the signal wavelength with steps of 0.2 nm (the resolution of the monochromator) and for each wavelength we got three measurements of real and accidental coincidence distributions. Each measurements was performed in an acquisition time of 100 s and we selected on the multi-channel analyzer a window of 670 channels for both distributions. After this we substracted the accidental counts and we evaluated the average and the standard deviation. Both average and standard deviation were corrected taking into account the efficiency of the detectors and the plot was finally normalized. Subsequently, we fitted the experimental points with a Gaussian distribution in order to have for the distribution an expected value for the FWHM with its uncertainty. A narrow peak with c = (0.29 ± 0.03) nm has been obtained (fig. 2). It is clearly seen that the pump width is 6.2 times larger than the coincidence spectrum. The spectral distribution of single counts was performed using a monochromator in the transmission arm with a spectral resolution of 1 nm. In this case for each scanned wavelength we got five measurements of the counts registered on one detector for a temporal window of 5 s and similarly for the background. Then we subtracted the background from single counts

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G. Brida et al.

Fig. 3: Normalized single counts distribution for the 10 mm length LiIO3 sample. Table 1: Results for the 10 mm LiIO3 sample. The pump parameters and p are measured independently by using a cross-correlator (with first harmonic) and a spectrometer.

Fig. 4: Quantifiers calculated analytically (R [9]) and numerically (K [11]) for a LiIO3 crystal of a length L = 0.5 cm and a pump wavelength p = 400 nm. Table 2: Results for the 5 mm LiIO3 sample.

Experimental results Pump p = 397.5nm p = 397.5 ± 0.2nm p = 1.25 nm p = (1.8 ± 0.1)nm = 186 fs = 186 ± 30 fs Coincidence distribution c = 795 nm c = 795.0 ± 0.2nm c = 0.32 nm c = (0.29 ± 0.03) Single counts distribution s = 100 nm s = (101 ± 1) nm R-quantifier R R 316 R R (349 ± 43)

Theory model

Theory model Experimental results Coincidence distribution c = 0.63 nm c = (0.64 ± 0.06) Single counts distribution s = 100 nm s = (115 ± 1) nm R-quantifier R R 158 R R (179 ± 18) Then the same measurement has been performed with a crystal of 5 mm length obtaining s = (115 ± 1) nm and c = (0.64 ± 0.06) nm, corresponding to R = (179 ± 18). Considering that the pulse duration of the pump did not change we can observe that in this case the pump is 2.8 times greater than the coincidence spectrum whereas the single counts distribution is 64 times larger than the width of the pump. The corresponding results are listed in table 2. Certainly, the most comprehensive contribution to the experimental verification of the R-quantifier validity in frequency domain would be testing the dependence R on the control parameter through all available parameters. Agreement between the measured and predicted shapes of the curve shown in fig. 4 would confirm completely the adequacy of the R-quantifier approach. Looking at (3), (4) it is seen that the "working" parameters might be and L. However, it is difficult to change the pump-pulse duration in a wide range with the same laser (from hundreds of fs to dozens of ps) and pass through the minimum of the curve R( ) varying this parameter only. That is why we took into account the fact of the linear dependence of R on the sample length L through the coincidence distribution width (4). So doubling the sample length leads to doubling the entanglement degree for the short pumppulse regime of SPDC. This fact is clearly illustrated by the obtained results, demonstrating the validity of the

for each wavelength and evaluated the uncertainty with usual uncertainty propagation formula. Also in this case each value was corrected taking into account the efficiency of the detector for different wavelengths and the plot was normalized. After that we estimated the FWHM of the distribution from the experimental points and we associated the resolution of the CVI monochromator as the uncertainty of this measure. In this way we find a single counts spectral width s = (101 ± 1) nm (fig. 3). The asymmetry of the right wing in the measured spectrum might be caused by falling spectral sensitivity of monochromator in the longwavelength range. Thus the distribution of single counts is 56 times larger than that of the pump, corresponding to an experimental ratio between the widths of the two distributions R = (349 ± 43). The complete results are collected in the table 1.

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Characterization of spectral entanglement of SPDC biphotons etc. conceptual scheme even if, unfortunately, the subsequent decrease of the sample length reduces the biphoton flux too much for measuring widths s and reaching the minimum of R( ). It is fair to mention that in some works (see, e.g., [13]) the 2D-photon distribution was measured and plotted in the (1 , 2 )-plane. Such 2D-distributions are very useful for making the entanglement analysis of biphoton states in the context of conditional pure state preparation, dispersion broadening [14,15], etc. This would eventually allow evaluating the R-quantifier directly. We would like to mention also that a measurement of spectral widths can be done by transforming the spectrum in time difference through a fiber [16]. In conclusion we have verified the approach of ref. [5] addressed to estimate the entanglement degree in the frequency domain for pure biphoton states generated by short pump pulses. An approach that can find widespread application in quantum technologies exploiting the entanglement of biphotons. The generalization to the mixed two-photon states will be discussed elsewhere. This work has been supported in part by MIUR (PRIN 2007FYETBY), Regione Piemonte (E14), "San Paolo foundation", NATO (CBP.NR.NRCL 983251) and RFBR (08-02-12091). Additional remark : We would like also to acknowledge the paper [17] where a measure of Fedorov's ratio in different conditions (type-II PDC, more critical estimation of overlap function by HOM interference) is described and that appeared after this article was completed.
REFERENCES [1] Genovese M., Phys. Rep., 413 (2005) 3197. [2] Bengtsson I. and Zyczkowski K., Geometry of Quantum States (Cambridge University Press) 2006. [ [3] Brida G. et al., arXiv:0907.4117 and references therein. [4] Lvovsky A. I. et al., Phys. Rev. Lett., 87 (2001) 050402; Zavatta A. et al., Phys. Rev. A, 70 (2004) 053821; Allevi A. et al., Phys. Rev. A, 80 (2009) 022114. [5] Fedorov M. V., Efremov M. A., Volkov P. A. and Eberly J. H., J. Phys. B: At. Mol. Opt. Phys., 9 (2006) S467. [6] Sperling J. and Vogel W., arXiv:0908.3974. [7] Fedorov M. V., Efremov M. A., Volkov P. A., Moreva E. V., Straupe S. S. and Kulik S. P., Phys. Rev. Lett., 99 (2007) 063901; Phys. Rev. A, 77 (2008) 032336. [8] Kim Y. et al., Phys. Rev. A, 62 (2000) 043820; JeronimoMoreno Y. and U'Ren A. B., Phys. Rev. A, 79 (2009) 033839. [9] Mikhailova Yu. M., Volkov P. A. and Fedorov M. V., Phys. Rev. A, 78 (2008) 062327. 10] Keller T. E. and Rubin M. H., Phys. Rev. A, 56 (1997) 1534. 11] Mauerer W. and Silberhorn C., Quantum Communication Measurement and Computing (QCMC'08) AIP Conf. Proc., 1110 (2009) 220. 12] Mauerer W. et al., private communication. 13] Mosley P. J., Lundeen J. S., Smith B. J., Wasylczyk P., U'Ren A. B., Silberhorn Ch. and Walmsley I. A., Phys. Rev. Lett., 100 (2008) 133601; Valencia Alejandra, Cer Alessandro, Shi Xiaojuan, Molina-Terriza Gabriel and Torres Juan P., Phys. Rev. Lett., 99 (2007) 243601; Avenhaus M. et al., arXiv:0810.0998. 14] Baek S. Y., Kwon O. and Kim Y. H., Phys. Rev. A, 77 ( 2 0 0 8 ) 0 1 3 8 2 9 ; 78 ( 2 0 0 8 ) 0 1 3 8 1 6 . 15] Erdmann R., Branning D., Grice W. and Walmsley I. A., Phys. Rev. A, 62 (2000) 053810. 16] Brida G., Chekhova M., Genovese M., Gramegna M. and Krivitsky L., Phys. Rev. Lett., 96 (2006) 143601; Brida G., Genovese M., Krivitsky L. A. and Chekhova M. V., Phys. Rev. A, 75 (2007) 015801; Avenhaus M., Eckstein A., Mosley P. J. and Silberhorn C., arXiv:0902.3364 (2008). 17] Avenhaus M., Chekhova M. V., Krivitsky L. A., Leuchs G. and Silberhorn C., Phys. Rev. A, 79 (2009) 043836.

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