week ending 9 MAY 2014
PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014)
Delayed-Choice Quantum Eraser with Thermal Light Tao Peng,* Hui Chen, and Yanhua Shih Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA
Marlan O. Scully Princeton University, Princeton, New Jersey 08544, USA and Texas A & M University, College Station, Texas 77843, USA Baylor University, Waco, Texas 76706, USA (Received 24 February 2014; published 5 May 2014) We report a random delayed-choice quantum eraser experiment. In a Young’s double-slit interferometer, the which-slit information is learned from the photon-number fluctuation correlation of thermal light. The reappeared interference indicates that the which-slit information of a photon, or wave packet, can be “erased” by a second photon or wave packet, even after the annihilation of the first. Different from an entangled photon pair, the jointly measured two photons, or wave packets, are just two randomly distributed and randomly created photons of a thermal source that fall into the coincidence time window. The experimental observation can be explained as a nonlocal interference phenomenon in which a random photon or wave packet pair, interferes with the pair itself at distance. DOI: 10.1103/PhysRevLett.112.180401
PACS numbers: 03.65.Ta, 42.50.Dv
Wave-particle duality, which Feynman called the basic mystery of quantum mechanics [1], says that there is always a trade-off between the knowledge of the particlelike and wavelike behavior of a quantum system. In slightly different words, Bohr suggested a complementarity principle in 1927: one can never measure the precise position and momentum of a quantum simultaneously [2]. Since then, complementarity has often been superficially identified with the “waveparticle duality of matter.” How quantum mechanics enforce complementarity may vary from one experimental situation to another. In a Young’s double-slit interferometer, the common “understanding” is that the position-momentum uncertainty relation makes it impossible to determine which slit a photon or wave packet passes through without at the same time disturbing the photon or wave packet enough to destroy the interference pattern. However, it has been shown that under certain circumstances this common understanding may not be true. In 1982, Scully and Drühl showed that a “quantum eraser” may erase the which-path information even after the annihilation of the quantum itself and determine its early behavior of wavelike or particlelike [3,4]. Since then, several quantum eraser experiments have been demonstrated [5–7]. One particular quantum eraser demonstrated in 2000 by Yoon-Ho Kim et al. [7] combined with “delayed choice” [8–13] has been considered a full demonstration of the original scheme of Scully and Drühl. In that experiment, an entangled signal-idler photon pair is generated simultaneously in a spontaneous parametric down-conversion process (SPDC). The nonlinear interaction mechanism of SPDC guarantees that the signal-idler pair is created together within only one slit of a Young’s interferometer. This peculiar internal property of the entangled photon pair was used to 0031-9007=14=112(18)=180401(5)
mark the “which-path” of the interferometer and guaranteed the disappearance of the interference pattern on the signal side. However, the interference pattern reappeared when the which-slit information was erased on the idler side. In this Letter we wish to report a quantum eraser using two random photons, or wave packets, of thermal light. The schematic of the experiment is illustrated in Fig. 1. In an experimental setup similar to Kim’s 2000 experiment, this quantum eraser measures the photon-number fluctuation correlation of thermal light [14], instead of coincidence counts of entangled photon pairs. Thermal light has a peculiar “spatial coherence” property: the fluctuation of the measured photon numbers, or intensities, correlated within its spatial coherence area only. When the measurements are beyond its coherence area, the photon-numbers fluctuation correlation vanish: hΔnA ΔnB i ∝ jGð1Þ ð~ρA ; ρ~B Þj2 ;
(1)
where Δnj , j ¼ A, B, is the photon-number fluctuation at (~ρj , tj ) of the double-slit plane, Gð1Þ ð~ρA ; ρ~B Þ is the first-order spatial coherence function of the thermal field. The spatial coherence of thermal light guarantees the photon numbers fluctuate correlatively only when j~ρA − ρ~B j < lc , where lc is the spatial coherence length. In this experiment, we choose j~ρA − ρ~B j ≫ lc . Under this condition, we have achieved hΔnA ΔnA0 i ≠ 0, hΔnB ΔnB0 i ≠ 0 but hΔnA ΔnB i ¼ 0. Note, again, here (~ρj ) is on the double-slit plane; see Fig. 1. This peculiar property of thermal light together with the photonnumber fluctuation-correlation measurement between D0 and D3 (or D4 ) provides the which-path information. It is interesting, the which-slit information is erasable in the fluctuationcorrelation measurement between D0 and D1 (or D2 ).
180401-1
© 2014 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014) d0
D0
x
GG A
f MB
B
He-Ne Laser
dBS
D4
BSB to D2
BSA
BS to D3
MA
Pos-Neg Identifier Neg
t4
0
Pos
R04
Event Timer 4
n0
Event Timer 0
n1
Event Timer 1
PP NN PN 0 Pos-Neg Identifier
NP
Neg
t0
0
Pos
R01
n4
D1
NN
PN
0
NP
PP
t1
Neg Pos
Pos-Neg Identifier
FIG. 1 (color online). Schematic of delayed choice quantum eraser. The He-Ne laser beam spot on the rotating ground glass has a diameter of ∼2 mm. A double-slit, with slit-width 150 μm and slit-separation 0.7 mm, is placed ∼25 cm away from the GG. The spatial coherence length of the pseudothermal field on the double-slit plane is calculated from Eq. (1), lc ¼ λ= Δθ ∼ 160 μm, which guarantees the two fields EA and EB are spatially incoherent. Under this experimental condition, the photon number fluctuates correlatively only within slit A or slit B. All beam splitters are nonpolarizing and 50=50. The two fields from the two slits may propagate to detector D0 which is transversely scanned on the focal plan of lens f for observing the interference patten of the double-slit interferometer; and may also pass along a Mach-Zehnder-like interferometer and finally reach D1 or D4 (D2 or D3 ). The rise time of detector D0 is less than 1 ns, ðdBS − d0 Þ= c ≈ 5 ns ensures that a “delayed choice” is made at BSB . A positivenegative fluctuation-correlation protocol is followed to evaluate the photon-number fluctuation correlations from the coincidences between D0 -D1 and D0 -D4 (or D0 -D2 and D0 -D3 ).
The experimental setup in Fig. 1 can be divided into four parts: a thermal light source, a Young’s double-slit interferometer, a Mach-Zehnder-like interferometer, and a photon-number fluctuation-correlation measurement circuit. (i) The light source is a standard pseudothermal source [15] which consists of a He-Ne laser beam (∼2 mm diameter) at wavelength λ ¼ 633 nm and a rotating ground glass (GG). Within the ∼2 mm diameter spot, the ground glass contains millions of tiny diffusers. A large number of randomly distributed subfields, or wave packets, are scattered from millions of randomly distributed tiny diffusers with random phases. The pseudothermal field then passes a double-slit which is about 25 cm away from the GG. The spatial
week ending 9 MAY 2014
coherence length of the pseudothermal field on the double slit plane is calculated from Eq. (1), lc ¼ λ=Δθ ∼ 160 μm, which guarantees the spatial incoherence of the two fields EA and EB that pass through slit A and slit B, respectively. Under this experimental condition, the photon numbers fluctuate correlatively only within slit A or slit B. We therefore learn the which-slit information in a photon number-fluctuation correlation measurement. (ii) The double slit has a slit width a ¼ 150 μm, and a slit separation d ¼ 0.7 mm (distance between the center of two slits). A lens f is placed following the double slit. On the focal plane of the lens a scannable pointlike photodetector D0 is used to learn the which-slit information or to observe the Young’s double-slit interference pattern. (iii) The Mach-Zehnder-like interferometer and the photodetectors D1 , D2 , are used to “erase” the which-slit information. Simultaneously, the joint detection between D0 and D3 or D4 is used to “learn” the which-slit information. All five photodetectors are photon-counting detectors working at the single-photon level. The Mach-Zehnder-like interferometer has three beam splitters, BS, BSA , and BSB , all of them are 50=50 nonpolarizing beam splitters. Moreover, the detectors are fast avalanche photodiodes with rise time less than 1 ns, and the optical path difference between BSA or BSB to the slits(dBS ), and D0 to the slits(d0 ) is dBS − d0 ≈ 1.5 m which ensure that, at each joint-detection measurement, when a photon is reflected (read which-way) or transmitted (erase which-way) at BSA or BSB, it is already 5 ns later than the annihilation of its partner at D0 . Comparing the 1 ns rise time, we are sure this is a delayed choice made by that photon. (iv) The photon-number fluctuation-correlation circuit consists of five synchronized “event timers” that record the registration times of D0 , D1 , D2 , D3 , and D4 . A positive-negative fluctuation identifier follows each event timer to distinguish “positive-fluctuation” Δnþ , from “negative-fluctuation” Δn− , for each photodetector within each coincidence time window. The photon-number fluctuation correlations of D0 -D1 : ΔR01 ¼hΔn0 Δn1 i and D0 -D4 : ΔR04 ¼ hΔn0 Δn4 i are calculated, accordingly, and, respectively, based on their measured positive-negative fluctuations. The detailed description of the photon-number fluctuation-correlation circuit can be found in Ref. [14]. Assuming a random pair of subfields, such as the mth and nth wave packets scatter from the mth and the nth subsources located at transverse coordinates ρ~0m and ρ~0n of the ground glass and fall into the coincidence timewindows of D0 -D1 and D0 -D4 , the mth wave packet may propagate to the double-slit interferometer and the nth wave packet may pass through the Mach-Zehnder-like interferometer, or vice versa. Under the experimental condition of spatial incoherence between EA and EB , thewhich-slit information is learned from the photonnumber fluctuation-correlation measurements ΔR04 ¼ hΔn0 Δn4 i ¼ hΔnB0 ΔnB4 i of D0 -D4 , and no interference is observable by scanning D0 . It is interesting that the which-slit information is erasable in the photon-number fluctuationcorrelation measurements of ΔR01 ¼ hΔn0 Δn1 i of D0 -D1 ,
180401-2
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PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014) 0.10
wave vector. jαm ðkÞi is an eigenstate of the annihilation operator with an eigenvalue αm ðkÞ,
Read
0.08
aˆ m ðkÞjαm ðkÞi ¼ αm ðkÞjαm ðkÞi: 0.06 04
Thus, we have aˆ m ðkÞjΨi ¼ αm ðkÞjΨi:
0.04
0.02
0.00
1.0
0.5
0.0
0.5
1.0
D0 Position mm
FIG. 2 (color online). The measured ΔR04 by scanning D0 on the observation plane of the Young’s double-slit interferometer. The black dots are experimental data, the red line is the theoretical fitting with Eq. (12).
resulting in a reappeared interference pattern as a function of the scanning coordinate of D0 . The experimental observation of ΔR04 is reported in Fig. 2. The data exclude any possible existing interferences. This measurement means the coincidences that contributed to ΔR04 must have passed through slit B. Figure 3 reports a typical experimental result of ΔR01 : a typical double-slit interference-diffraction pattern. The 100% visibility of the sinusoidal modulation indicates complete erasure of the which-slit information. Here we give a simple two-photon interference model to explain the observation. In this model, we assume the thermal source contains a large number of independent and randomly radiated point subsources. We model the chaoticthermal field in coherent state representation [16,17]: Y Y jΨi ¼ jfαm gi ¼ jαm ðkÞi; (2) m
m;k
where m labels the mth subfield that is scattered from the mth subsource of the pseudothermal source, and k is a
Eˆ ðþÞ ðr0 ; t0 Þ ≡
X ðþÞ Eˆ m ðr0 ; t0 Þ m
ðþÞ ðþÞ ¼ Eˆ A ðr0 ; t0 Þ þ Eˆ B ðr0 ; t0 Þ X ðþÞ ðþÞ ¼ ½Eˆ mA ðr0 ; t0 Þ þ Eˆ mB ðr0 ; t0 Þ m
¼
XZ
dkaˆ m ðkÞ½gm ðk; rA ; tA ÞgA ðk; r0 ; t0 Þ
m
þ gm ðk; rB ; tB ÞgB ðk; r0 ; t0 Þ;
(5)
where gm ðk; rs ; ts Þ is a Green’s function which propagates the mth subfield from the mth subsource to the sth slit (s ¼ A, B). gs ðk; r0 ; t0 Þ is another Green’s function that propagates the field from the sth slit to detector D0. It is easy to notice that, although there are two ways a photon can be detected at D0 , due to the first order incoherence of EA and EB , there should be no interference at the detection plane. D4 (D3 ) in the experiment can only receive photons from slit B (slit A), so the field operator is then Eˆ ðþÞ ðr4 ;t4 Þ ¼
Erase
0.15
(4)
The above model is reasonable for the experiment: the laser beam is scattered by millions of tiny diffusers which are randomly distributed in the ground glass. The coherent beam is scattered to all directions denoted by k. The scattered fields also acquire random phases. jαm ðkÞi describes the state of a scattered field that is scattered by the mth diffuser with a vector k. The field operator at detector D0 can be written in the following form in terms of the subfields:
0.20
X ðþÞ Eˆ mB ðr4 ;t4 Þ m
¼ 01
(3)
XZ
dkaˆ m ðkÞgm ðk;rB ;tB ÞgB ðk;r4 ;t4 Þ: (6)
m 0.10
The detector D1 (D3 ), however, can receive photons from both slit A and slit B through the Mach-Zehnder-like interferometer, so the field operator has two terms:
0.05
0.00
1.0
0.5
0.0
0.5
Eˆ ðþÞ ðr1 ; t1 Þ ¼
1.0
X ðþÞ ðþÞ ½Eˆ mA ðr1 ; t1 Þ þ Eˆ mB ðr1 ; t1 Þ m
D0 Position mm
FIG. 3 (color online). The measured ΔR01 as a function of the transverse coordinate of D0 . The black dots are experimental data, the red line is the theoretical fitting with Eq. (13).
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¼
XZ
dkaˆ m ðkÞ½gm ðk; rA ; tA ÞgA ðk; r1 ; t1 Þ
m
þ gm ðk; rB ; tB ÞgB ðk; r1 ; t1 Þ:
(7)
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PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014)
Based on the state of Eq. (2) and the field operators of Eqs. (5)–(7), we apply the Glauber-Scully theory [16,18] to calculate the photon-number fluctuation correlation or
the second-order coherence function Gð2Þ ðr0 ; t0 ; rα ; tα Þ from the coincidence measurement of D0 and Dα , (α ¼ 1, 2, 3, 4):
Gð2Þ ðr0 ; t0 ; rα ; tα Þ ¼ hhΨjEð−Þ ðr0 ; t0 ÞEð−Þ ðrα ; tα ÞEðþÞ ðrα ; tα ÞEðþÞ ðr0 ; t0 ÞjΨiiEs X ð−Þ X ð−Þ X ðþÞ X ðþÞ ¼ hhΨj Em ðr0 ; t0 Þ En ðrα ; tα Þ Eq ðrα ; tα Þ Ep ðr0 ; t0 ÞjΨiiEs m
n
q
p
X X X ¼ ψ m ðr0 ; t0 Þψ m ðr0 ; t0 Þ ψ n ðrα ; tα Þψ n ðrα ; tα Þ þ ψ m ðr0 ; t0 Þψ n ðr0 ; t0 Þψ n ðrα ; tα Þψ m ðrα ; tα Þ m
n
n≠m
¼ hn0 ihnα i þ hΔn0 Δnα i;
(8)
where the subscript Es denotes the ensemble average which is done after the quantum average. ψ m ðrα ; tα Þ is the effective wave function of the mth subfield at (rα , tα ). In the case of α ¼ 1, 2
hΔn0 Δnα i ¼
ψ m ðrα ; tα Þ ¼ ψ mAα þ ψ mBα Z ¼ dkαm ðkÞ½gm ðk; rA ; tA ÞgA ðk; rα ; tα Þ þ gm ðk; rB ; tB ÞgB ðk; rα ; tα Þ:
(11) (9)
(10)
since ψ mA4 ¼ 0. ΔR01 ∝ hΔn0 Δn1 i ∝
X ψ m ðr0 ; t0 Þψ n ðr0 ; t0 Þψ n ðrα ; tα Þψ m ðrα ; tα Þ: n≠m
This shows that the measured effective wave function ψ m ðrα ; tα Þ is the result of a superposition between two alternative amplitudes in terms of path A and path B, ψ mα ¼ ψ mAα þ ψ mBα . When α ¼ 4 (or α ¼ 3), the effective wave function has only one amplitude: ψ m ðr4 ; t4 Þ ¼ ψ mB4 Z ¼ dkαm ðkÞgm ðk; rB ; tB ÞgB ðk; r4 ; t4 Þ;
From Eq. (8) and the measurement circuit in Fig. 1, it is easy to find that what we measure in this experiment is the photon-number fluctuation correlation:
We thus obtain ΔR04 ∝ hΔn0 Δn4 i ¼
X ψ mB0 ψ nB0 ψ nB4 ψ mB4 n≠m
¼ hΔnB0 ΔnB4 i ∝ sinc2 ðxπa=λfÞ; (12) where x is the transverse coordinate of the scanning detector D0. This equation indicates a diffraction pattern which agrees with the experimental observation of Fig. 2. In the case of α ¼ 1, 2, we obtain
X ½ψ mA0 ψ nA0 ψ nA1 ψ mA1 þ ψ mB0 ψ nB0 ψ nB1 ψ mB1 þ ψ mA0 ψ nB0 ψ nB1 ψ mA1 þ ψ mB0 ψ nA0 ψ nA1 ψ mB1 n≠m
≡ hΔnAA0 ΔnAA1 i þ hΔnBB0 ΔnBB1 i þ hΔnAB0 ΔnBA1 i þ hΔnBA0 ΔnAB1 i ∝ sinc2 ðxπa=λfÞcos2 ðxπd=λfÞ;
(13)
which agrees with the experimental observation in Fig. 3. In conclusion, we have realized a delayed-choice quantum eraser by using thermal light. The experimental observation can be explained as a nonlocal interference phenomenon in which a random photon or wave packet pair, of thermal light interferes with the pair itself at distance [19]. The authors wish to thank R. French, J. P. Simon, and J. Sprigg for helpful discussions. This work was supported, in part, by AFOSR, ARL, the National Science Foundation
Grant No. PHY-1241032 (INSPIRE CREATIV), and the Robert A. Welch Foundation (A-1261).
*
[email protected] [1] R. P. Feynman, R. B. Leighton, and M. L. Sands, Lectures on Physics (Addison-Wesley, Reading, MA, 1965). [2] N. Bohr, Naturwissenschaften 16, 245 (1928). [3] M. O. Scully and K. Druhl, Phys. Rev. A 25, 2208 (1982). [4] M. O. Scully, B. Englert, and H. Walther, Nature (London) 351, 111 (1991).
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PHYSICAL REVIEW LETTERS
[5] T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, 3034 (1995). [6] S. P. Walborn, M. O. Terra Cunha, S. Padua, and C. H. Monken, Phys. Rev. A 65, 033818 (2002). [7] Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000). [8] J. A. Wheeler, in Mathemaltical Foundations of Quantum Mechanics, edited by J. A. Wheeler and W. H. Zurek (Princeton Univ. Press, Princeton, NJ, 1984). [9] A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, Nature (London) 433, 230 (2005). [10] V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, Science 315, 966 (2007). [11] J. Tang, Y. Li, X. Xu, G. Xiang, C. Li, and G. Guo, Nat. Photonics 6, 602 (2012). [12] A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’ Brien, Science 338, 634 (2012).
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[13] F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012). [14] H. Chen, T. Peng, and Y. Shih, Phys. Rev. A 88, 023808 (2013). [15] W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964). [16] R. J. Glauber, Phys. Rev. Lett. 10 84 (1963); Phys. Rev. 130, 2529 (1963). [17] Y. H. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC Press, Taylor & Francis, 2011), 1st ed. [18] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997). [19] This interference happens at different space-time points, (r0 , t0 ) and (r1 , t1 ) [or (r2 , t2 )]. Imagine the two detectors are separated in light years, we may ask a similar question as Einstein used to ask: how long does it take for this superposition to complete? Following Einstein, we name it nonlocal interference.
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PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014)
Delayed-Choice Quantum Eraser with Thermal Light Tao Peng,* Hui Chen, and Yanhua Shih Department of Physics, University of Maryland, Baltimore County, Baltimore, Maryland 21250, USA
Marlan O. Scully Princeton University, Princeton, New Jersey 08544, USA and Texas A & M University, College Station, Texas 77843, USA Baylor University, Waco, Texas 76706, USA (Received 24 February 2014; published 5 May 2014) We report a random delayed-choice quantum eraser experiment. In a Young’s double-slit interferometer, the which-slit information is learned from the photon-number fluctuation correlation of thermal light. The reappeared interference indicates that the which-slit information of a photon, or wave packet, can be “erased” by a second photon or wave packet, even after the annihilation of the first. Different from an entangled photon pair, the jointly measured two photons, or wave packets, are just two randomly distributed and randomly created photons of a thermal source that fall into the coincidence time window. The experimental observation can be explained as a nonlocal interference phenomenon in which a random photon or wave packet pair, interferes with the pair itself at distance. DOI: 10.1103/PhysRevLett.112.180401
PACS numbers: 03.65.Ta, 42.50.Dv
Wave-particle duality, which Feynman called the basic mystery of quantum mechanics [1], says that there is always a trade-off between the knowledge of the particlelike and wavelike behavior of a quantum system. In slightly different words, Bohr suggested a complementarity principle in 1927: one can never measure the precise position and momentum of a quantum simultaneously [2]. Since then, complementarity has often been superficially identified with the “waveparticle duality of matter.” How quantum mechanics enforce complementarity may vary from one experimental situation to another. In a Young’s double-slit interferometer, the common “understanding” is that the position-momentum uncertainty relation makes it impossible to determine which slit a photon or wave packet passes through without at the same time disturbing the photon or wave packet enough to destroy the interference pattern. However, it has been shown that under certain circumstances this common understanding may not be true. In 1982, Scully and Drühl showed that a “quantum eraser” may erase the which-path information even after the annihilation of the quantum itself and determine its early behavior of wavelike or particlelike [3,4]. Since then, several quantum eraser experiments have been demonstrated [5–7]. One particular quantum eraser demonstrated in 2000 by Yoon-Ho Kim et al. [7] combined with “delayed choice” [8–13] has been considered a full demonstration of the original scheme of Scully and Drühl. In that experiment, an entangled signal-idler photon pair is generated simultaneously in a spontaneous parametric down-conversion process (SPDC). The nonlinear interaction mechanism of SPDC guarantees that the signal-idler pair is created together within only one slit of a Young’s interferometer. This peculiar internal property of the entangled photon pair was used to 0031-9007=14=112(18)=180401(5)
mark the “which-path” of the interferometer and guaranteed the disappearance of the interference pattern on the signal side. However, the interference pattern reappeared when the which-slit information was erased on the idler side. In this Letter we wish to report a quantum eraser using two random photons, or wave packets, of thermal light. The schematic of the experiment is illustrated in Fig. 1. In an experimental setup similar to Kim’s 2000 experiment, this quantum eraser measures the photon-number fluctuation correlation of thermal light [14], instead of coincidence counts of entangled photon pairs. Thermal light has a peculiar “spatial coherence” property: the fluctuation of the measured photon numbers, or intensities, correlated within its spatial coherence area only. When the measurements are beyond its coherence area, the photon-numbers fluctuation correlation vanish: hΔnA ΔnB i ∝ jGð1Þ ð~ρA ; ρ~B Þj2 ;
(1)
where Δnj , j ¼ A, B, is the photon-number fluctuation at (~ρj , tj ) of the double-slit plane, Gð1Þ ð~ρA ; ρ~B Þ is the first-order spatial coherence function of the thermal field. The spatial coherence of thermal light guarantees the photon numbers fluctuate correlatively only when j~ρA − ρ~B j < lc , where lc is the spatial coherence length. In this experiment, we choose j~ρA − ρ~B j ≫ lc . Under this condition, we have achieved hΔnA ΔnA0 i ≠ 0, hΔnB ΔnB0 i ≠ 0 but hΔnA ΔnB i ¼ 0. Note, again, here (~ρj ) is on the double-slit plane; see Fig. 1. This peculiar property of thermal light together with the photonnumber fluctuation-correlation measurement between D0 and D3 (or D4 ) provides the which-path information. It is interesting, the which-slit information is erasable in the fluctuationcorrelation measurement between D0 and D1 (or D2 ).
180401-1
© 2014 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014) d0
D0
x
GG A
f MB
B
He-Ne Laser
dBS
D4
BSB to D2
BSA
BS to D3
MA
Pos-Neg Identifier Neg
t4
0
Pos
R04
Event Timer 4
n0
Event Timer 0
n1
Event Timer 1
PP NN PN 0 Pos-Neg Identifier
NP
Neg
t0
0
Pos
R01
n4
D1
NN
PN
0
NP
PP
t1
Neg Pos
Pos-Neg Identifier
FIG. 1 (color online). Schematic of delayed choice quantum eraser. The He-Ne laser beam spot on the rotating ground glass has a diameter of ∼2 mm. A double-slit, with slit-width 150 μm and slit-separation 0.7 mm, is placed ∼25 cm away from the GG. The spatial coherence length of the pseudothermal field on the double-slit plane is calculated from Eq. (1), lc ¼ λ= Δθ ∼ 160 μm, which guarantees the two fields EA and EB are spatially incoherent. Under this experimental condition, the photon number fluctuates correlatively only within slit A or slit B. All beam splitters are nonpolarizing and 50=50. The two fields from the two slits may propagate to detector D0 which is transversely scanned on the focal plan of lens f for observing the interference patten of the double-slit interferometer; and may also pass along a Mach-Zehnder-like interferometer and finally reach D1 or D4 (D2 or D3 ). The rise time of detector D0 is less than 1 ns, ðdBS − d0 Þ= c ≈ 5 ns ensures that a “delayed choice” is made at BSB . A positivenegative fluctuation-correlation protocol is followed to evaluate the photon-number fluctuation correlations from the coincidences between D0 -D1 and D0 -D4 (or D0 -D2 and D0 -D3 ).
The experimental setup in Fig. 1 can be divided into four parts: a thermal light source, a Young’s double-slit interferometer, a Mach-Zehnder-like interferometer, and a photon-number fluctuation-correlation measurement circuit. (i) The light source is a standard pseudothermal source [15] which consists of a He-Ne laser beam (∼2 mm diameter) at wavelength λ ¼ 633 nm and a rotating ground glass (GG). Within the ∼2 mm diameter spot, the ground glass contains millions of tiny diffusers. A large number of randomly distributed subfields, or wave packets, are scattered from millions of randomly distributed tiny diffusers with random phases. The pseudothermal field then passes a double-slit which is about 25 cm away from the GG. The spatial
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coherence length of the pseudothermal field on the double slit plane is calculated from Eq. (1), lc ¼ λ=Δθ ∼ 160 μm, which guarantees the spatial incoherence of the two fields EA and EB that pass through slit A and slit B, respectively. Under this experimental condition, the photon numbers fluctuate correlatively only within slit A or slit B. We therefore learn the which-slit information in a photon number-fluctuation correlation measurement. (ii) The double slit has a slit width a ¼ 150 μm, and a slit separation d ¼ 0.7 mm (distance between the center of two slits). A lens f is placed following the double slit. On the focal plane of the lens a scannable pointlike photodetector D0 is used to learn the which-slit information or to observe the Young’s double-slit interference pattern. (iii) The Mach-Zehnder-like interferometer and the photodetectors D1 , D2 , are used to “erase” the which-slit information. Simultaneously, the joint detection between D0 and D3 or D4 is used to “learn” the which-slit information. All five photodetectors are photon-counting detectors working at the single-photon level. The Mach-Zehnder-like interferometer has three beam splitters, BS, BSA , and BSB , all of them are 50=50 nonpolarizing beam splitters. Moreover, the detectors are fast avalanche photodiodes with rise time less than 1 ns, and the optical path difference between BSA or BSB to the slits(dBS ), and D0 to the slits(d0 ) is dBS − d0 ≈ 1.5 m which ensure that, at each joint-detection measurement, when a photon is reflected (read which-way) or transmitted (erase which-way) at BSA or BSB, it is already 5 ns later than the annihilation of its partner at D0 . Comparing the 1 ns rise time, we are sure this is a delayed choice made by that photon. (iv) The photon-number fluctuation-correlation circuit consists of five synchronized “event timers” that record the registration times of D0 , D1 , D2 , D3 , and D4 . A positive-negative fluctuation identifier follows each event timer to distinguish “positive-fluctuation” Δnþ , from “negative-fluctuation” Δn− , for each photodetector within each coincidence time window. The photon-number fluctuation correlations of D0 -D1 : ΔR01 ¼hΔn0 Δn1 i and D0 -D4 : ΔR04 ¼ hΔn0 Δn4 i are calculated, accordingly, and, respectively, based on their measured positive-negative fluctuations. The detailed description of the photon-number fluctuation-correlation circuit can be found in Ref. [14]. Assuming a random pair of subfields, such as the mth and nth wave packets scatter from the mth and the nth subsources located at transverse coordinates ρ~0m and ρ~0n of the ground glass and fall into the coincidence timewindows of D0 -D1 and D0 -D4 , the mth wave packet may propagate to the double-slit interferometer and the nth wave packet may pass through the Mach-Zehnder-like interferometer, or vice versa. Under the experimental condition of spatial incoherence between EA and EB , thewhich-slit information is learned from the photonnumber fluctuation-correlation measurements ΔR04 ¼ hΔn0 Δn4 i ¼ hΔnB0 ΔnB4 i of D0 -D4 , and no interference is observable by scanning D0 . It is interesting that the which-slit information is erasable in the photon-number fluctuationcorrelation measurements of ΔR01 ¼ hΔn0 Δn1 i of D0 -D1 ,
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PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014) 0.10
wave vector. jαm ðkÞi is an eigenstate of the annihilation operator with an eigenvalue αm ðkÞ,
Read
0.08
aˆ m ðkÞjαm ðkÞi ¼ αm ðkÞjαm ðkÞi: 0.06 04
Thus, we have aˆ m ðkÞjΨi ¼ αm ðkÞjΨi:
0.04
0.02
0.00
1.0
0.5
0.0
0.5
1.0
D0 Position mm
FIG. 2 (color online). The measured ΔR04 by scanning D0 on the observation plane of the Young’s double-slit interferometer. The black dots are experimental data, the red line is the theoretical fitting with Eq. (12).
resulting in a reappeared interference pattern as a function of the scanning coordinate of D0 . The experimental observation of ΔR04 is reported in Fig. 2. The data exclude any possible existing interferences. This measurement means the coincidences that contributed to ΔR04 must have passed through slit B. Figure 3 reports a typical experimental result of ΔR01 : a typical double-slit interference-diffraction pattern. The 100% visibility of the sinusoidal modulation indicates complete erasure of the which-slit information. Here we give a simple two-photon interference model to explain the observation. In this model, we assume the thermal source contains a large number of independent and randomly radiated point subsources. We model the chaoticthermal field in coherent state representation [16,17]: Y Y jΨi ¼ jfαm gi ¼ jαm ðkÞi; (2) m
m;k
where m labels the mth subfield that is scattered from the mth subsource of the pseudothermal source, and k is a
Eˆ ðþÞ ðr0 ; t0 Þ ≡
X ðþÞ Eˆ m ðr0 ; t0 Þ m
ðþÞ ðþÞ ¼ Eˆ A ðr0 ; t0 Þ þ Eˆ B ðr0 ; t0 Þ X ðþÞ ðþÞ ¼ ½Eˆ mA ðr0 ; t0 Þ þ Eˆ mB ðr0 ; t0 Þ m
¼
XZ
dkaˆ m ðkÞ½gm ðk; rA ; tA ÞgA ðk; r0 ; t0 Þ
m
þ gm ðk; rB ; tB ÞgB ðk; r0 ; t0 Þ;
(5)
where gm ðk; rs ; ts Þ is a Green’s function which propagates the mth subfield from the mth subsource to the sth slit (s ¼ A, B). gs ðk; r0 ; t0 Þ is another Green’s function that propagates the field from the sth slit to detector D0. It is easy to notice that, although there are two ways a photon can be detected at D0 , due to the first order incoherence of EA and EB , there should be no interference at the detection plane. D4 (D3 ) in the experiment can only receive photons from slit B (slit A), so the field operator is then Eˆ ðþÞ ðr4 ;t4 Þ ¼
Erase
0.15
(4)
The above model is reasonable for the experiment: the laser beam is scattered by millions of tiny diffusers which are randomly distributed in the ground glass. The coherent beam is scattered to all directions denoted by k. The scattered fields also acquire random phases. jαm ðkÞi describes the state of a scattered field that is scattered by the mth diffuser with a vector k. The field operator at detector D0 can be written in the following form in terms of the subfields:
0.20
X ðþÞ Eˆ mB ðr4 ;t4 Þ m
¼ 01
(3)
XZ
dkaˆ m ðkÞgm ðk;rB ;tB ÞgB ðk;r4 ;t4 Þ: (6)
m 0.10
The detector D1 (D3 ), however, can receive photons from both slit A and slit B through the Mach-Zehnder-like interferometer, so the field operator has two terms:
0.05
0.00
1.0
0.5
0.0
0.5
Eˆ ðþÞ ðr1 ; t1 Þ ¼
1.0
X ðþÞ ðþÞ ½Eˆ mA ðr1 ; t1 Þ þ Eˆ mB ðr1 ; t1 Þ m
D0 Position mm
FIG. 3 (color online). The measured ΔR01 as a function of the transverse coordinate of D0 . The black dots are experimental data, the red line is the theoretical fitting with Eq. (13).
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¼
XZ
dkaˆ m ðkÞ½gm ðk; rA ; tA ÞgA ðk; r1 ; t1 Þ
m
þ gm ðk; rB ; tB ÞgB ðk; r1 ; t1 Þ:
(7)
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PHYSICAL REVIEW LETTERS
PRL 112, 180401 (2014)
Based on the state of Eq. (2) and the field operators of Eqs. (5)–(7), we apply the Glauber-Scully theory [16,18] to calculate the photon-number fluctuation correlation or
the second-order coherence function Gð2Þ ðr0 ; t0 ; rα ; tα Þ from the coincidence measurement of D0 and Dα , (α ¼ 1, 2, 3, 4):
Gð2Þ ðr0 ; t0 ; rα ; tα Þ ¼ hhΨjEð−Þ ðr0 ; t0 ÞEð−Þ ðrα ; tα ÞEðþÞ ðrα ; tα ÞEðþÞ ðr0 ; t0 ÞjΨiiEs X ð−Þ X ð−Þ X ðþÞ X ðþÞ ¼ hhΨj Em ðr0 ; t0 Þ En ðrα ; tα Þ Eq ðrα ; tα Þ Ep ðr0 ; t0 ÞjΨiiEs m
n
q
p
X X X ¼ ψ m ðr0 ; t0 Þψ m ðr0 ; t0 Þ ψ n ðrα ; tα Þψ n ðrα ; tα Þ þ ψ m ðr0 ; t0 Þψ n ðr0 ; t0 Þψ n ðrα ; tα Þψ m ðrα ; tα Þ m
n
n≠m
¼ hn0 ihnα i þ hΔn0 Δnα i;
(8)
where the subscript Es denotes the ensemble average which is done after the quantum average. ψ m ðrα ; tα Þ is the effective wave function of the mth subfield at (rα , tα ). In the case of α ¼ 1, 2
hΔn0 Δnα i ¼
ψ m ðrα ; tα Þ ¼ ψ mAα þ ψ mBα Z ¼ dkαm ðkÞ½gm ðk; rA ; tA ÞgA ðk; rα ; tα Þ þ gm ðk; rB ; tB ÞgB ðk; rα ; tα Þ:
(11) (9)
(10)
since ψ mA4 ¼ 0. ΔR01 ∝ hΔn0 Δn1 i ∝
X ψ m ðr0 ; t0 Þψ n ðr0 ; t0 Þψ n ðrα ; tα Þψ m ðrα ; tα Þ: n≠m
This shows that the measured effective wave function ψ m ðrα ; tα Þ is the result of a superposition between two alternative amplitudes in terms of path A and path B, ψ mα ¼ ψ mAα þ ψ mBα . When α ¼ 4 (or α ¼ 3), the effective wave function has only one amplitude: ψ m ðr4 ; t4 Þ ¼ ψ mB4 Z ¼ dkαm ðkÞgm ðk; rB ; tB ÞgB ðk; r4 ; t4 Þ;
From Eq. (8) and the measurement circuit in Fig. 1, it is easy to find that what we measure in this experiment is the photon-number fluctuation correlation:
We thus obtain ΔR04 ∝ hΔn0 Δn4 i ¼
X ψ mB0 ψ nB0 ψ nB4 ψ mB4 n≠m
¼ hΔnB0 ΔnB4 i ∝ sinc2 ðxπa=λfÞ; (12) where x is the transverse coordinate of the scanning detector D0. This equation indicates a diffraction pattern which agrees with the experimental observation of Fig. 2. In the case of α ¼ 1, 2, we obtain
X ½ψ mA0 ψ nA0 ψ nA1 ψ mA1 þ ψ mB0 ψ nB0 ψ nB1 ψ mB1 þ ψ mA0 ψ nB0 ψ nB1 ψ mA1 þ ψ mB0 ψ nA0 ψ nA1 ψ mB1 n≠m
≡ hΔnAA0 ΔnAA1 i þ hΔnBB0 ΔnBB1 i þ hΔnAB0 ΔnBA1 i þ hΔnBA0 ΔnAB1 i ∝ sinc2 ðxπa=λfÞcos2 ðxπd=λfÞ;
(13)
which agrees with the experimental observation in Fig. 3. In conclusion, we have realized a delayed-choice quantum eraser by using thermal light. The experimental observation can be explained as a nonlocal interference phenomenon in which a random photon or wave packet pair, of thermal light interferes with the pair itself at distance [19]. The authors wish to thank R. French, J. P. Simon, and J. Sprigg for helpful discussions. This work was supported, in part, by AFOSR, ARL, the National Science Foundation
Grant No. PHY-1241032 (INSPIRE CREATIV), and the Robert A. Welch Foundation (A-1261).
*
[email protected] [1] R. P. Feynman, R. B. Leighton, and M. L. Sands, Lectures on Physics (Addison-Wesley, Reading, MA, 1965). [2] N. Bohr, Naturwissenschaften 16, 245 (1928). [3] M. O. Scully and K. Druhl, Phys. Rev. A 25, 2208 (1982). [4] M. O. Scully, B. Englert, and H. Walther, Nature (London) 351, 111 (1991).
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[5] T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, 3034 (1995). [6] S. P. Walborn, M. O. Terra Cunha, S. Padua, and C. H. Monken, Phys. Rev. A 65, 033818 (2002). [7] Y.-H. Kim, R. Yu, S. P. Kulik, Y. Shih, and M. O. Scully, Phys. Rev. Lett. 84, 1 (2000). [8] J. A. Wheeler, in Mathemaltical Foundations of Quantum Mechanics, edited by J. A. Wheeler and W. H. Zurek (Princeton Univ. Press, Princeton, NJ, 1984). [9] A. Zeilinger, G. Weihs, T. Jennewein, and M. Aspelmeyer, Nature (London) 433, 230 (2005). [10] V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier, A. Aspect, and J. F. Roch, Science 315, 966 (2007). [11] J. Tang, Y. Li, X. Xu, G. Xiang, C. Li, and G. Guo, Nat. Photonics 6, 602 (2012). [12] A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’ Brien, Science 338, 634 (2012).
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[13] F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and S. Tanzilli, Science 338, 637 (2012). [14] H. Chen, T. Peng, and Y. Shih, Phys. Rev. A 88, 023808 (2013). [15] W. Martienssen and E. Spiller, Am. J. Phys. 32, 919 (1964). [16] R. J. Glauber, Phys. Rev. Lett. 10 84 (1963); Phys. Rev. 130, 2529 (1963). [17] Y. H. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC Press, Taylor & Francis, 2011), 1st ed. [18] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, England, 1997). [19] This interference happens at different space-time points, (r0 , t0 ) and (r1 , t1 ) [or (r2 , t2 )]. Imagine the two detectors are separated in light years, we may ask a similar question as Einstein used to ask: how long does it take for this superposition to complete? Following Einstein, we name it nonlocal interference.
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