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VOLUME 84, NUMBER 4

P HYS ICAL REVIEW LETTERS

24 J

ANUARY

2000

Dispersion-Independent High-Visibility Quantum Interference in Ultrafast Parametric Down-Conversion
Mete AtatÝre,1 Alexander V. Sergienko,2 Bahaa E. A. Saleh,2 and Malvin C. Teich
1
2

1,2

Quantum Imaging Laboratory, Department of Physics, Boston University, 8 Saint Mary's Street, Boston, Massachusetts 02215 Quantum Imaging Laboratory, Department of Electrical and Computer Engineering, Boston University, 8 Saint Mary's Street, Boston, Massachusetts 02215 (Received 8 April 1999) We demonstrate the effective removal of intrinsic distinguishability between entangled-photon pairs in femtosecond spontaneous parametric down-conversion. High-visibility quantum interference is recovered (an increase to 96% from 17%) while preserving the high photon-flux density associated with the use of long nonlinear crystals. This new technique is expected to serve as a basic component in the preparation of multiphoton entangled states.
PACS numbers: 42.50.Dv, 03.67. ­ a, 42.65.Ky, 42.65.Re

The nonlinear-optical process of continuous-wave (cw) spontaneous parametric down-conversion [1] has long provided a useful source of nonclassical light for carrying out practical tests of the foundations of quantum mechanics. The nonlinear process takes place in a second-order nonlinear crystal pumped by a laser, and generates photon pairs in an entangled quantum state, as originally defined by SchrÆdinger [2]. Recent experiments with multiphoton entangled states in the areas of quantum teleportation [3,4], entanglement swapping [5], and tests of local realism (GHZ states) [6,7] have imposed an additional requirement -- that independent photon pairs must be generated synchronously in different nonlinear crystals. To satisfy this requirement, ultrafast pump lasers (sub-psec regime) with pulse widths much shorter than the time of coherent nonlinear interaction in the crystal (which increases proportionally with crystal thickness) must be employed for the generation of multiphoton entangled states through parametric down-conversion. Unfortunately, however, quantum-interference experiments display strong decoherence effects [8,9] under these conditions. None of the ultrafast type-II experiments reported thus far have exhibited visibility that is independent of crystal length, as a result of significant linear dispersion in the crystal arising from the different group velocities of the down-converted photons and the pump pulse. In spite of this, researchers have managed to obtain high-visibility fringes by using narrow-band interference filters placed before the detectors. This spectral postselection technique artificially increases the limited coherence time of the pairs produced by the brief pump pulses, and therefore restores the quantum interference to a degree. However, this benefit is accompanied by a drastic reduction of the photon-flux density reaching the detectors. The observation of multiphoton entangled states is greatly facilitated by a large photon-flux density. In this Letter we demonstrate a new method which effectively removes the intrinsic distinguishability present 618 0031 - 9007 00 84(4) 618(4) $15.00

in photon pairs generated via femtosecond downconversion in long nonlinear crystals. This is achieved by drawing on the dual polarization and energy-time entanglement available in type-II optical parametric downconversion [10]. While the rate of the photon-pair generation rate scales linearly with nonlinear crystal length, we demonstrate that the pattern of the measured coincidence rate is completely defined by the spectrum of the pump pulse and does not depend on the dispersion parameters of the crystal, nor on its length. The validity of our technique is experimentally demonstrated by the revival of highvisibility quantum interference (from 17% to 96%) in a photon-coincidence experiment without the use of spectral filtering, thereby preserving high photon-flux density at the detectors. Cascaded transformations of the two-photon state. -- Consider the quantum-interference experiment illustrated in Fig. 1. The two-photon state at the output of the b -barium borate (BBO) nonlinear down-conversion crystal is developed on the basis of one-dimensional (along the z axis) collinear type-II parametric down-conversion. The perturbed state (after evolution from initial time t0 to time t ) at the output of the nonlinear crystal of thickness L assumes a form that strongly depends on the region of coherent nonlinear interaction deff via the partial decoherence function W z 2 j , which is

FIG. 1. Schematic of the polarization quantum-interference experimental setup.

© 2000 The American Physical Society


VOLUME 84, NUMBER 4

P HYSICAL REVIEW LETTERS

24 JANUARY 2000

localized about an arbitrary position j in the nonlinear crystal [9]: i Zt 0 2Z0 2 jCj j0 2 dt x dz W z 2 j h t0 ¯ 2L ^2 ^ 3 Ep1 t 0 , z es Es2 t 0 , z ei Ei t 0 , z 1 H.c. j0 . (1) Here x 2 is the second-order susceptibility of the nonlinear medium; ej is the unit vector denoting the polarization ^2 state of the j th field Ej t 0 , z ; the subscripts p , s, and i represent the pump, signal, and idler, respectively; the pump field is assumed to be a propagating classical transform-limited pulse whose amplitude is slowly varying: sµ z 10 0 0 Ep t , z exp i kp z 2 vp t 0 . (2) Ip t 0 2 yp The quantity Ip t 0 , z repeats the intensity profile of the pump at time t 0 and position z along the propagation axis with a full width at half maximum (FWHM) denoted tp . Because of the characteristic of our laser beam, we specifically chose it to have a Gaussian q p z form, Ip t 0 , z I0 exp 2 t 0 2 yp 2 2s 2 with p s tp 2 ln2 , though the theory is more general and can accommodate any pulse shape. The central wave number, carrier frequency, and group velocity of the pump (at the carrier frequency inside the nonlinear crystal) are 0 0 denoted in Eq. (2) by kp , vp , and yp , respectively. The explicit form of W z 2 j is taken to be a half-Gaussian with argument z 2 j and width deff ; its magnitude decreases with increasing argument as a consequence of the effective gradual separation of the pump pulse from the down-converted fields in the nonlinear crystal arising from linear dispersion. This function reflects the distinguishability not only between the down-converted photons of each pair but also between pairs born at different locations as the femtosecond pump pulse traverses the nonlinear crystal. We now consider the timing diagram shown in Fig. 2 in conjunction with the schematic of our experimental apparatus presented in Fig. 1. Once an orthogonally polarized photon pair in the laboratory polarization basis [H (horizontal), V (vertical)] is generated at time t0 around position j , it propagates through the nonlinear downconversion crystal, acquiring a relative time delay fj imparted by the birefringence of the nonlinear crystal. The two-photon state in the H , V basis, jC
2 j

FIG. 2. A cross-polarized photon pair generated in the vicinity of position j within the nonlinear down-conversion crystal acquires a relative time delay fj . These photons are projected to a new basis X , Y that is tilted by 45± with respect to the original basis H , V , and the Y -projected photon experiences a relative optical-path delay t with respect to the X -projected photon as a result of the birefringence of the delay element. (Projection and optical-path delay operations are displayed separately for clarity.) Although there appear to be four possible configurations that the photon pair can assume before arriving at the beam splitter (far right), the bottom two (dotted) vanish because of their coefficients. The remaining two contributions (solid) preserve their relative delay fj , so that the interfering terms are the XX and YY terms with a delay t , whatever the value of fj .

Fig. 2 for clarity). As a result of the projection, the two-photon state in the rectilinear X , Y basis becomes jC
2 j

jX

t

0

1 jY

t

0

jX

t0 1f

j

2 jY

t0 1f

j

.

(4)

Finally, the longitudinal optical-path delay t along the Y axis with respect to the X axis gives rise to jC
2 j

jX t0 jX 1 jY

t0 1f

j

2 jY
t0 1f
j

t0 1t

jY

t0 1fj 1t t0 1fj 1t

t0 1t

jX

2 jX t0 jY

,

(5)

which is illustrated at the far right of Fig. 2. This then arrives at a nonpolarizing beam splitter, as shown in Fig. 1, which injects photons into the two arms of the polarization interferometer (denoted in the following equations as subscripts 1 and 2). The polarization analyzers at the ends of the arms, being the final elements in the polarization interferometer, transform the two-photon state once more, ^ via the operator P jQ 1 jQ 2 2 Q j1 Q j, which is constructed in the polarization analyzer states Q1 and Q2 . Taking account of the transformations described above, one can calculate the two-photon detection probability amplitude 2 ^1^1 Aj t1 , t2 0jE1 E2 jCj , (6) where t1 t2 denotes the event registration time at the detector located at the first (second) arm of the interferometer. For an arbitrary region about the position j in the crystal, Eq. (6) takes the explicit form [9] sµ j L Aj T , t t2 Ip T 2 Aj t1 , t2 D yp µ t 3 W t rect 2L,0 1j D 0 0 3 exp i kp j 2 vp T , (7) 619

jV

t

0

jH

t0 1fj

,

(3)

then propagates through a polarization delay line made of birefringent material (such as crystalline quartz) oriented so that its fast X (slow Y ) axis is rotated by 45± with respect to the H V polarization axis (projection and optical-path-delay operations are displayed separately in


VOLUME 84, NUMBER 4

P HYSICAL REVIEW LETTERS Bj ,t T , t

24 JANUARY 2000

where T t1 1 t2 2, t t1 2 t2 , L 1 yp 2 1 2 1 ys 1 1 yi , and D 1 yi 2 1 ys . The function rect 2L,0 z has a unity value when its argument is in the range 2L,0 and vanishes otherwise, and W t is the partial decoherence function discussed above. When we carry out the above-described set of transformations, the rotated optical delay line causes multiple possibilities of coincidence detection, thereby yielding an overall two-photon detection probability amplitude that is the superposition of these possibilities. However, as a consequence of the cascaded transformations, the possibilities originating from the two right-most terms in Eq. (5) (shown as dotted in Fig. 2) vanish, leaving only two surviving terms that contribute to the overall two-photon amplitude: Z
0

µ µ p p 2 u1 cos 2 u 2 Aj T , t cos 4 4 µ µ p p 2 sin 2 u1 sin 2 u2 4 4 3 Aj T 1 t , t . (8)

Finally, using Eq. (8) in the coincidence-count rate equation [1] provides Rt Z
0

dj
2L1deff

Z

` 2`

Z

`

dT dt jB

2`

j ,t

T , t j2 ,

(9)

from which we determine the explicit form of the coincidence-count rate at the detectors, as a function of polarization-analyzer angles u1 , u2 and optical-path relative delay time t , to be

d jF j
eff

R u1 , u2 , t

Gp 0

2L1d

cos2 q1 cos2 q2 1 sin2 q1 sin2 q2 2 2 sinq1 sinq2 cosq1 cosq
2

where Gp t and qi 2` dt Ip t Ip t 1 t 1, 2. All of the j -dependent terms in p 4 2 ui , i Eq. (10) are collected in the function F j so that the interference term (in square brackets) is generation-region independent. This reflects the successful liberation of entangled photon pairs from any influence of dispersion associated with the nonlinear crystal which results in the width of the quantum-interference pattern being solely determined by the pump-pulse duration tp . Quantum-interference experiments with the new cascaded-transformation state. -- Experiments were carried out using the experimental arrangement shown in Fig. 1. An actively mode-locked Ti:sapphire laser (pumped by a cw Ar-ion laser) emitted pulses of light at 830 nm. This radiation was frequency doubled to provide 80 fsec pulses (FWHM) at lp 415 nm, with a repetition rate of 82 MHz and an average power of 15 mW. This sequence of femtosecond pulses was delivered to a BBO crystal, where it underwent type-II spontaneous parametric down-conversion in a collinear degenerate 0 0 0 v1 v2 vp 2 configuration. The collinear beam of down-converted photons was selected by a 2.5 mm circular aperture located about 7 cm beyond the crystal. After the residual pump pulses were separated from the signal and idler beams using a fused-silica dispersion prism, the down-converted photons were sent through the first stage of the cascaded-transformation apparatus which was composed of a birefringent material (crystalline quartz) whose fast axis was oriented at 45± with respect to the H , V laboratory coordinate system. Relative optical delay was introduced by changing the thickness of the birefringent material. The down-converted photons continued to a nonpolarizing beam splitter. The light at each output port of the beam splitter was then directed toward 620

R`

q

Gp t 0 cos vp t Gp 0

, (10)

a Glan-Thompson polarization analyzer, and thence to a convex lens for focusing onto an actively quenched Peltier-cooled avalanche photodiode photon-counting detector. The detector event registrations were conveyed to a coincidence circuit with a 3 ns coincidence-time window. Corrections for accidental coincidences were not necessary. Discussion. -- The experimental data (solid squares) illustrated in the insets of Fig. 3 display the coincidence rates (CR) as a function of polarization-analyzer angle u2 when u1 is fixed at 0±. Coincidence patterns are shown for two representative relative opticalpath delays: 296 fsec (left inset) and 0 fsec (right inset). The visibility of the coincidence pattern, V CRmax 2 CRmin CRmax 1 CRmin , at 0 fsec delay turns out to be 96%, which is just a hair short of the theoretical maximum of 100%. This is to be compared with the rather anemic visibility of 17% that was observed in a similar configuration using a nonrotated polarization delay line [9] (time-delay fringes and polarization-rotation fringes are interchangeable, as has recently been established for femtosecond downconversion [11]). Coincidence patterns such as the representative ones shown in the insets were collected for various values of the relative optical-path delay t . The open squares in the main plot of Fig. 3 represent the maximum and minimum values of the coincidence rate (connected by a vertical dashed line) for various values of t . The coincidence-rate visibility is seen to decrease as t deviates from 0 fsec. The width of the coincidence-rate pattern, which is about 80 fsec, is determined solely by the pump-pulse duration and demonstrates a crystal-dispersion-independent profile


VOLUME 84, NUMBER 4

P HYSICAL REVIEW LETTERS

24 JANUARY 2000

Note added in proof. -- Since completing this work, a recently published paper [12] describing an experiment that achieves a visibility of 64% using a different technique has come to our attention.

FIG. 3. Maxima and minima (open squares) of coincidencerate quantum-interference patterns plotted as a function of relative optical-path delay time t . Each pair of data points (connected by a dashed vertical line) corresponds to the maximum and minimum coincidence rates CR, sec21 obtained from a polarization experiment at a particular value of t (examples are shown in the insets) in which the angle u2 of the second polarization analyzer is varied while u1 is fixed at 0±. The visibility of the interference pattern is measured to be 96% at 0 fsec relative path delay and the width of the path-delay interference pattern nicely reveals the 80 fsec pump-pulse duration used in these experiments. This experiment was conducted using a 3 mm thick BBO crystal.

that remains identical for different thicknesses of the nonlinear down-conversion crystal. To conclude, we have successfully co-opted the dual energy time and polarization entanglement inherent in type-II parametric down-conversion to implement a new scheme for eliminating the effects of decoherence in femtosecond parametric down-conversion. Despite our use of long nonlinear crystals, which permits high photon-flux densities to be obtained, we have obtained a high-visibility quantuminterference pattern without the use of spectral filtering. We expect that this new technique will find use in a range of quantum-optics applications. We are grateful to Bradley M. Jost and Jan Perina, Jr. for useful discussions. This work was supported by the National Science Foundation and the Boston University Photonics Center.

[1] D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970); J. Pe rina, Z. Hradil, and B. Jurco, Quantum Optics and Fundamentals of Physics (Kluwer, Boston, 1994); L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995), Chap. 22. [2] E. SchrÆdinger, Naturwissenschaften 23, 807 (1935); 23, 823 (1935); 23, 844 (1935) [translation in Quantum Theory and Measurement, edited by J. A. Wheeler and W. H. Zurek (Princeton University Press, Princeton, NJ, 1983)]. [3] C. H. Bennett, G. Brassard, C. CrÈpeau, R. Jozsa, A. Peres, and W. Wootters, Phys. Rev. Lett. 70, 1895 (1993); D. Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Phys. Rev. Lett. 80, 1121 (1998). [4] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature (London) 390, 575 (1997). [5] J.-W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998). [6] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell's Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer, Dordrecht, The Netherlands, 1989); D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990). [7] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999). [8] G. Di Giuseppe, L. Haiberger, F. De Martini, and A. V. Sergienko, Phys. Rev. A 56, R21 (1997); W. P. Grice, R. Erdmann, I. A. Walmsley, and D. Branning, Phys. Rev. A 57, R2289 (1998); W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997); T. E. Keller and M. H. Ru bin, Phys. Rev. A 56, 1534 (1997); J. Perina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 59, 2359 (1999). [9] M. AtatÝre, A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 83, 1323 (1999). [10] T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, Phys. Rev. A 51, 3495 (1995); Y. H. Shih and A. V. Sergienko, Phys. Lett. A 191, 201 (1994). [11] A. V. Sergienko, M. AtatÝre, Z. Walton, G. Jaeger, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 60, R2622 (1999). [12] D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, Phys. Rev. Lett. 83, 955 (1999).

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