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ISSN 0021 3640, JETP Letters, 2011, Vol. 94, No. 4, pp. 262­265. © Pleiades Publishing, Inc., 2011. Original Russian Text © K.G. Katamadze, A.V. Paterova, E.G. Yakimova, K.A. Balygin, S.P. Kulik, 2011, published in Pis'ma v Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 2011, Vol. 94, No. 4, pp. 284­288.

Control of the Frequency Spectrum of a Biphoton Field Due to the Electro Optical Effect
K. G. Katamadzea, A. V. Paterovaa, E. G. Yakimovab, K. A. Balyginc, and S. P. Kulik
a b a

Faculty of Physics, Moscow State University, Moscow, 119992 Russia Institute of Physics and Technology, Russian Academy of Sciences, Nakhimovskii pr. 34, Moscow, 117218 Russia c National Research Centre Kurchatov Institute, pl. Akademika Kurchatova 1, Moscow, 123182 Russia
Received June 29, 2011

A method for controlling the spectrum of spontaneous parametric down conversion has been implemented. This method is based on the application of the gradient of an electrostatic field to a nonlinear crystal where down conversion occurs. Due to the electro optical effect, i.e., the dependence of the refractive indices on the applied field, the phase matching conditions are different along the length of the crystal and, as a result, the spectrum of the emitted biphoton field is modified. DOI: 10.1134/S0021364011160089

1. INTRODUCTION Control of the parameters of nonclassical fields is one of the main problems of quantum optics. Polariza tion, angular and spectral distributions, and correla tion and time properties are among these parameters. Recently, a great deal of attention has been focused on the control of the spectrum of a biphoton field as one of the most popular representatives of the family of nonclassical field. For some applications such as quantum memory [1], transmission of quantum information through optical fibers [2], and the measurement of the time characteristics of single photon detectors [3], the biphoton with a narrow spectrum is required. Other applications, such as quantum optical coherent tomography [4], nonlinear microscopy [5], quantum interferometric optical lithography [6], entangled photon virtual state spectroscopy [7], and distant clock synchronization [8] require a broad spectrum. In addition, it is worth noting that an increase in the width of the spectrum of the biphoton field is accom panied by an increase in the degree of its entanglement [9]. 2. CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD The biphoton field is usually obtained in the spon taneous parametric down conversion of light [10], when a pump photon decays in a nonlinear medium into a pair of photons, which are traditionally called signal and idle. For spontaneous parametric down conversion, the following frequency and phase match ing conditions should be satisfied: p ­ s ­ i = 0 , (1)

k = kp ­ ks ­ ki 0 .

(2)

Here, subscripts p, s, and i refer to pump, signal, and idle photons, respectively. In the case of monochromatic pumping, the fre quencies of the signal and idle photons anticorrelate and it is convenient to perform the change s = s 0 + , i = i 0 ­ , (3) where s0 and i0 are the central frequencies in the spectra. In this case, the spectral state of the biphoton field has the form [11] | = |vac + d F ( ) a ( s 0 + ) a i ( i 0 ­ ) |vac ,




s



(4)

where a s and a i are the photon creation operators in the fixed signal and idle space polarized modes, respectively, and the generally complex function F() is the spectral amplitude of a biphoton, which describes the frequency spectrum of the biphoton field. In the case of a homogeneous crystal, the spectral amplitude is given by the expression F ( ) L exp [ ­ i k ( ) L / 2 ] sinc [ k ( ) L / 2 ] , (5) where L is the length of the crystal and k() is the frequency dependence of the phase detuning. Accord ing to this expression, the frequency distribution F() is narrower for a longer crystal. To reduce the spectral width, the crystal is placed in a cavity [12], a Bragg superlattice is imposed [13], or backward wave phase matching is used [14], when one of the photons of the pair propagates in the direction opposite to the pump direction. A less trivial problem consists of obtaining a

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Fig. 2. Layout of the experimental setup.

Fig. 1. Creation of an inhomogeneous field in the KDP crystal. (a) Configuration of electrodes. (b) Approximate distribution of the field along the crystal.

quecy of the grating [22]. Another method is based on the change in the refractive indices of a nonlinear crys tal subjected to the temperature gradient [23]. In this work, we analyze a similar method where the modula tion of the refractive indices is due to the electro opti cal effect. 3. IDEA OF THE METHOD We consider a nonlinear crystal where collinear spontaneous parametric down conversion occurs and refractive indices in the modes p, s, and i depend on the applied electrostatic field E, which is in turn a function of the longitudinal coordinate z. In this case, taking into account change (3), the phase detuning given by Eq. (2) has the form k ( , z ) = n p [ p, E ( z ) ] ­ n s [ s 0 + , E ( z ) ] c
p

broad spectrum. The simplest method involves a decrease in the length of the crystal [15] is accompa nied by a decrease in the intensity. Another method consists of creating conditions under which the first derivatives of the function k() are zero. This can be ensured by choosing an appropriate nonlinear medium and operating wavelengths [16] by using quasi phase matching in periodical structures [17], increasing the angular [18] or frequency [19] spectra of the pump, and using elements with angular dispersion [20]. However, in all cases, this can be made for no more than the first three derivatives of the function k(). The spectrum can be broadened using an inhomoge neous nonlinear medium. In this case, the mismatch k depends not only on the frequency , but also on the longitudinal coordinate z, which coincides with the direction of pump propagation. The spectral amplitude of the biphoton field has the form
L

s 0 + c ­ ­ n i [ i 0 ­ , E ( z ) ] i 0 . c In different regions of the crystal, phase achieved for different pairs of frequencies ~ ~ ~ ~ (z) and = ­ (z), where (z) is
i

(7)

matching is ~ s = s0 + the solution

i0

F ( ) dz exp [ i k ( , z ) z ] .
0



(6)

Thus, phase matching conditions (1) and (2) are sat isfied for different pairs of the frequencies of the signal and idle photons in different sections of the nonlinear medium. As a result, the superposition of the contri butions from all sections has a broad spectrum. Unfor tunately, in this case, the phase of F() depends strongly on the frequency and the spectrum is not Fourier limited. For this reason, to obtain a short cor relation time, it is necessary to apply special compres sion methods [21]. There are various inhomogeneous nonlinear media where the k(z) dependence is observed. Periodically polarized crystals can be used with the polarization period that increases so as to ensure the linear increase (chirp) in the spatial fre
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~ of the equation k( , z) = 0. If the field increases or decreases monotonically with an increase in z, the spectral width can be estimated as ~ ~ (8) = ( 0 ) ­ ( L ) , where L is the length of the crystal. 4. EXPERIMENT The experiment was performed with potassium dihydrogen phosphate 30 mm in length that is cut for collinear degenerate phase matching for a pump wavelength of 351 nm (Fig. 1). The upper and lower surfaces of the crystal were equipped with pairs of elec trodes to which a static voltage (up to 15 kV) can be applied. As a result, the electrostatic field along the crystal can be varied in the range from ­30 to


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Fig. 4. Width of the spectrum of spontaneous parametric down conversion in the nondegenerate regime versus the applied field. The points are experimental data and the line is the theoretical estimate by Eq. (8).

formed by means of the automated rotation of the prism of the spectrograph. The spectra thus detected were normalized to the pump power using a built in analog to digital converter to the input of which a sig nal from detector D1 was fed. We performed several experiments. First, the crys tal was oriented so that biphotons were generated in it in the nondegenerate regime. Thus, two peaks were observed in the spectrum of spontaneous parametric down conversion without field (see Fig. 3a). When the field was applied, each section of the crystal with elec trodes generates two additional peaks. As a result, the total spectrum is inhomogeneously broadened. Figure 4 shows the field dependence of the spectral width of each peak. The solid line shows the theoretical esti mate by Eq. (8). A broad spectrum was obtained in the degenerate regime in the next experiment, where the crystal was oriented so that the spectrum without field was weakly nondegenerate and that the inner peaks overlap after the application of the field (Fig. 3b). As a result, the total width of the spectrum is 168 nm (102 THz), whereas the width of the degenerate spec trum that can be obtained from the same crystal with out field was as small as 60 nm (37 THz). 5. NUMERICAL SIMULATION Analyzing the control of the spectrum by means of the electro optical effect, we simulated various situa tions for KDP and LiNbO3. The calculation was per formed using Eqs. (6) and (7). The direction of the field was perpendicular to the direction of radiation propagation and its magnitude was varied linearly from ­30 to 30 kV. The crystal was oriented so as to ensure the maximum spectral width. As a result, the maximum spectral width for the 3 cm long KDP crystal (Fig. 5) was 113 nm (69 THz). For comparison, we note that the maximum spectral width calculated for the 10 mm long lithium niobate crystal is 200 nm
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Fig. 3. Broadening of the spectrum of spontaneous para metric down conversion in the (a) nondegenerate and (b) degenerate regimes from (squares) the homogeneous crystal, (circles) the crystal with the same orientation where the field was varied from ­30 to 30 kV, and (trian gles) the homogeneous crystal in the degenerate regime. The intensity is normalized to the maximum value.

30 kV/cm. Figure 2 shows the layout of the experi mental setup. Pumping was ensured by an argon laser operating in a cw mode at a wavelength of 351.1 nm with an angular divergence of about 0.2 mrad. Using prism P, which separates a necessary spectral mode, and mirror M, the beam was guided to the optical sys tem. A fraction of the pump beam that was transmitted through the mirror was detected by photodiode D1 to control the power. After the transmission through the vertically oriented polarization prism V, laser radia tion was incident on the crystal where biphotons were generated. Filter F and horizontally oriented polariza tion prism H screen parasitic pump radiation and luminescence and transmit spontaneous parametric down conversion radiation. Then, objective O focused radiation on the input slit of an ISP 51 spec trograph. Detector D2, which is a Perkin Elmer silicon avalanche photodiode operating in the photon count mode, was placed in the focal plane of the chamber of the spectrograph. Scanning in the wavelength was per


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length of the crystal. As a result, the spectrum of the emitted biphoton field changes (is broadened). The method is convenient for crystals with large electro optical coefficients. This work was supported by the Ministry of Educa tion and Science of the Russian Federation (state con tract no. 02.740.11.0223), the Russian Foundation for Basic Research (project nos. 10 02 00204 and 10 02 90036), and the Dynasty Foundation (program for support of post graduate students and young scientists without scientific degrees). REFERENCES
1. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, Nature Lett. 452, 67 (2008). 2. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002). 3. Z. Y. Ou and Y. J. Lu, Phys. Rev. Lett. 83, 2556 (1999). 4. M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, et al., Phys. Rev. Lett. 91, 083601 (2003). 5. J. Squier and M. Muller, Rev. Sc. Instrum. 72, 2855 (2001). 6. A. N. Boto, P. Kok, D. S. Abrams, et al., Phys. Rev. Lett. 85, 2733 (2000). 7. B. E. A. Saleh, B. M. Jost, H. Fei, et al., Phys. Rev. Lett. 80, 3483 (1998). 8. A. Valencia, G. Scarcelli, and Y. Shih, Appl. Phys. Lett. 85, 2655 (2004). 9. G. Brida, V. Caricato, M. V. Fedorov, et al., Eur. Phys. Lett. 87, 64 003 (2009). 10. D. N. Klyshko, Physical Principles of Quantum Elec tronics (Nauka, Moscow, 1986) [in Russian]. 11. A. V. Belinsky and D. N. Klyshko, Laser Phys. 4, 663 (1994). 12. M. Scholz, L. Koch, and O. Benson, Phys. Rev. Lett. 102, 063603 (2009). 13. L. Yan, L. J. Ma, and X. Tang, Opt. Express 18, 5957 (2010). 14. Ch. S. Chuu and S. E. Harris, Phys. Rev. A 83, 061803 (2011). 15. E. Dauler, G. Jaeger, A. Muller, et al., J. Res. Nat. Inst. Stand. Tech. 104, 1 (1999). 16. A. Pe'er, Y. Silberberg, B. Dayan, et al., Phys. Rev. A 74, 053805 (2006). 17. K. A. O'Donnell and A. B. U'ren, Opt Lett. 32, 817 (2007). 18. S. Carrasco, M. Nasr, A. Sergienko, et al., Opt. Lett. 31, 253 (2006). 19. M. B. Nasr, G. D. Giuseppe, B. E. A. Saleh, et al., Opt. Commun. 246, 521 (2005). 20. M. Hendrych, A. V. X. Shi, and J. P. Torres, Phys. Rev. A 79, 023817 (2009). 21. G. Brida, M. V. Chekhova, I. P. Degiovanni, et al., Phys. Rev. Lett. 103, 193602 (2009). 22. M. B. Nasr, S. Carrasco, B. E. A. Saleh, et al., Phys. Rev. Lett. 100, 183601 (2008). 23. D. A. Kalashnikov, K. G. Katamadze, and S. P. Kulik, JETP Lett. 89, 224 (2009).

Fig. 5. Simulation of the broadening of the spectrum of spontaneous parametric down conversion in the KDP crystal. The dashed line is the spectrum of the homoge neous crystal, the dash­dotted line is the spectrum of the crystal in the weakly inhomogeneous regime, and the solid line is the spectrum of the crystal, where the field is linearly varied from ­30 to 30 kV. The intensity is normalized to the maximum value.

(53 THz). The difference of the numerical simulation results from the experimental data (cf. Figs. 3 and 5) can be caused by possible differences of the tabulated electro optical coefficients from real values. However, there is obvious qualitative agreement in the width of the spectrum and in the details of its structure. 6. DISCUSSION AND CONCLUSIONS The considered method of broadening the spec trum of biphoton radiation does not significantly differ from methods based on the use of chirped structures or temperature gradient; at the same time, it has a num ber of advantages and disadvantages. First, it is con structively simpler to create the necessary gradient of the electric field. In addition, the problem of the cal culation of the field inside the crystal is almost removed (unlike the temperature gradient method [23]). Second, similar to the thermo optical effect, the field distribution can be varied for various experimen tal tasks. A disadvantage of this method, as well as all broadening methods based on spatially inhomoge neous structures, is that the biphoton field at the out put of the crystal is significantly non Fourier limited. This complicates the creation of biphoton packets with a narrow second order function. In this work, a method for controlling the spectrum of spontaneous parametric down conversion has been proposed and implemented. This method is based on the application of the gradient of an electrostatic field to a nonlinear crystal where biphotons are generated. Due to the electro optical effect, the refractive indices depend on the magnitude of the applied field. The phase matching conditions are modified along the
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Translated by R. Tyapaev