Popper ’ s Experiment Two

16.4 Popper ’ s Experiment

16.4.2 Popper ’ s Experiment Two

Once again, the demonstration of Popper’s experiment calls our attention to the important message: the physics of the entangled two-particle system must inherently be very different from that of individual particles.

correlation measurement. Again, the question of Popper is: Do we expect to observe a diffraction pattern that satisfies Δp_yΔy>h? To answer this question, we again make two measurements following Popper’s suggestion. Measurement-I is illustrated in the upper part of.Fig.16.25. In this measurement, we place slit-B, which has the same width as that of slit-A, coincident with the 1:1 ghost image of slit-A and measure the diffraction pattern by scanning D_B along the y-axis in far-field. In this measurement, we learn the value ofΔp_ydue to the diffraction of a real slit ofΔy. Measurement-II is illustrated in the lower part of.Fig.16.25. Here, we open slit-B completely, scanningD_Bagain along the same y-axis to measure the

“diffraction”pattern of the 1:1 ghost image with the same width as slit-A. By comparing the observed pattern width in measurement-II with that of measurement-I, we can examine Popper’s prediction.

The experimental details are shown in . Fig.16.26. The light source is a standard pseudo-thermal source, consisting of a He-Ne laser beam and a rotating ground glass (GG). A 50/50 beamsplitter (BS) is used to split the pseudo-thermal light into two beams. One of the beams illuminates a single slit, slit-A, of width D ¼0. 15 mm locateddA400 mm from the source. A“bucket”photodetector D_A is placed right behind slit-A. An equal-sized ghost image of slit-A is then observable from the positive-negative photon-number fluctuation correlation measurement between the “bucket” detector D_A and the transversely scanning point-like photodetectorD_B, ifD_Bis scanned on the ghost image plane located at d_B¼d_A ¼400 mm. In this experiment, however,D_Bis scanned on a plane that is locateddB⁰ 900 mm behind the ghost image plane, to measure the“diffraction”

DB

nA

tA

tB

Synchronized

0

NN PN 0 NP PP

Neg Pos

Pos-Neg Identifier

Event Timer A Slit B

' d A

Y

He-Ne Laser

Slit A BS

GG d B

d B

Event Timer B

DA

Pos-Neg Identifier

Neg Pos nB

. ^{Fig. 16.26} Schematic of the experimental setup. A rotating ground glass (GG) is employed to produce pseudo-thermal light. BS is a 50/50 non-polarizing beam splitter. After BS, the transmitted beam passes through slit-A (0.15 mm) and collected by a“bucket”detectorDAwhich is put right after the slit. The reflected beam passes slit-B, which can be adjusted to be the same width as that of slit-A or wide open, and then reaches the scanning detectorDB. The distances from slit-A and slit-B to the source are the same (dA¼dB¼400 mm). The distance from the scanning fiber tip ofDBto the plane of slit-B isdB0¼900 mm. A PNFC protocol is followed to evaluate the photon-number fluctuation correlations from the coincidences betweenDAandDB

428 Y.H. Shih

16

pattern of the ghost image. The output pulses from the two single-photon counting detectors are then sent to a PNFC circuit, which starts from two Pos-Neg identifiers follow two event-timers distinguish the “positive-fluctuation” Δn^þ, from the“negative-fluctuation”Δn, measured byD_AandD_B, respectively, within each coincidence time window. The photon-number fluctuation-correlations of D_A-D_B: ΔRAB¼ hΔnAΔnBi is calculated, accordingly and respectively, based on their measured positive-negative fluctuations. The detailed description of the PNFC circuit can be found in [30].

The experiment was performed in two steps after confirming the 1:1 ghost image of slit-A. In measurement-I, we place slit-B (D¼0. 15 mm) coincident with the ghost image and moveD_Bto a plane at d⁰_B900 mm to measure the diffraction pattern of slit-B. In measurement-II, we keep the same experimental condition as that of measurement-I, except slit-B is set wide open.

. Figure 16.27 reports the experimental results. The circles show the normalized photon-number fluctuation correlation from the PNFC protocol against the position of D_B along the y-axis for Popper’s measurement-I. As expected, we observed a typical single-slit diffraction pattern giving us the uncer- tainty in momentum,Δp^real_y . The squares show the experimental observation from the PNFC for Popper’s measurement-II, when slit-B is wide open. The measured curves agree well with our theoreticalfittings. We found the width of the curve representing no physical slit is much narrower than that of the real diffraction pattern, which agrees with Popper’s prediction.

Similar to our early analysis in Bell state and in quantum eraser, we chose the coherent state representation for the calculation of the joint photodetection counting rate ofD_AandD_Bwhich is proportional to the second-order coherence functionG_AB⁽²⁾:

G^ð2Þ_AB ¼ hDE^^ðÞð~ρ_A,zA,tAÞE^^ðÞð~ρ_B,zB,tBÞ E^^ðþÞð~ρ_A,zB,tBÞE^^ðþÞð~ρ_A,zA,tAÞi_QME

Es, ^(16.84)

where E^ðþÞð~ρ_j,zj,tjÞ (E^ðÞð~ρ_j,zj,tjÞ) is the positive (negative) field operator at space-time coordinate ð~ρ_j,zj,tjÞ, j¼A, B, with ð~ρ_j,zj,tjÞ the transverse, . ^{Fig. 16.27} The observed diffraction patterns.Circles: slit-A and slit-B are both adjusted for 0. 15 mm.Squares: slit-A is 0.15 mm, slit-B is wide open. The width of the curve without the slit is almost three times narrower than that of the curve with slit, agreeing well with the theoretical predictions from Eqs. (16.89) and (16.91)

longitudinal, and time coordinates of the photodetection event ofD_AorD_B. Note, in the Glauber-Scully theory [25, 36], the quantum expectation and classical ensemble average are evaluated separately, which allows us to examining the two-photon interference picture before ensemble averaging.

Thefield at each space-time point is the result of a superposition among a large number of subfields propagated from a large number of independent, randomly distributed and randomly radiating sub-sources of the entire chaotic-thermal source,

E^^ðÞð~ρ_j,zj,tjÞ ¼ X

m

E^^ðÞð~ρ_0m,z_0m,t_0mÞg_mð~ρ_j,zj,tjÞ

X

m

E^^ðÞ_m ð~ρ_j,zj,tjÞ, ^(16.85)

where E^^ðÞð~ρ_0m,z0m,t0mÞ is the mth subfield at the source coordinate ð~ρ_0m,z_0m,t_0mÞ, and g_mð~ρ_j,z_j,t_jÞ is the optical transfer function that propagates themth subfield from coordinate ð~ρ_0m,z_0m,t_0mÞto ð~ρ_j,z_j,t_jÞ. We can write the field operators in terms of the annihilation and creation operators:

E^^ðþÞ_m ð~ρ_j,zj,tjÞ ¼C ð

dk^amðkÞg_mðk;~ρ_j,zj,tjÞ, ^(16.86) C is a normalization constant, g_mðk;~ρ_j,zj,tjÞ, j¼A, B, is the optical transfer function for modekof themth subfield propagated from themth sub-source to the jth detector, and ^a_mðkÞ is the annihilation operator for the modek of themth subfield.

Substituting the field operators and the state, in the multi-mode coherent representation, into Eq. (16.84), we then writeG_AB⁽²⁾in terms of the superposition of a large number of effective wavefunctions, or wavepackets:

G^ð2Þð~ρ_B,z_B,t_B;~ρ_B,z_B,t_BÞ

¼ X

m,ⁿ,^p,^qψ^∗_mð~ρ_A,z_A,t_AÞψ^∗_nð~ρ_B,z_B,t_BÞψpð~ρ_B,z_B,t_BÞψqð~ρ_A,z_A,t_AÞ

* +

Es

¼ X

m,ⁿ

j

ψmð~ρ_A,z_A,t_AÞψnð~ρ_B,z_B,t_BÞ þψnð~ρ_A,z_A,t_AÞψmð~ρ_B,z_B,t_BÞ

j

²

* +

Es

¼ X

m

j

ψmð~ρ_A,zA,tAÞ

j

^2X

n

j

ψnð~ρ_B,zB,tBÞ

j

²

*

þX

m6¼n

½ψ^∗_mð~ρ_A,z_A,t_AÞψmð~ρ_B,z_B,t_BÞψnð~ρ_A,z_A,t_AÞψ^∗_nð~ρ_B,z_B,t_BÞ +

Es

hnAihnBi þ hΔnAΔnBi:

(16.87)

with

ψsð~ρ_j,zj,tjÞ ¼ ð

dkαsðkÞe^iφ0sg_sðk;~ρ_j,zj,tjÞ,

wheres¼m, n, p,q,j¼A, B, and the phase factore^iφ0s represents the random initial phase of themth subfield. In Eq. (16.87), we have completed the ensemble average in terms of the random phases of the subfields, i.e. φ0s, and kept the nonzero terms only. Equation (16.87) indicates the second-order coherence func- tion is the result of a sum of a large number of subinterference patterns, each

430 Y.H. Shih

16

subpattern indicates an interference in which a random pair of wavepackets interfering with the pair itself. For example, themth and thenth wave packets have two different yet indistinguishable alternative ways to produce a joint photodetection event, or a coincidence count, at different space-time coordinates:

(1) themth wavepacket is annihilated atD_Aand thenth wavepacket is annihilated at D_B; (2) the mth wavepacket is annihilated atD_B and the nth wavepacket is annihilated atD_A. In quantum mechanics, the joint detection probability ofD_A andD_Bis proportional to the normal square of the superposition of the above two probability amplitudes. We name this kind of superposition“nonlocal interfer- ence.”The superposition of the two amplitudes for each random pair results in an interference pattern, and the addition of these large number of interference patterns yields the nontrivial correlation of the chaotic-thermal light.

The cross interference term in Eq. (16.87) indicates the photon-numberfluc- tuation correlationhΔnAΔnBi:

ΔnAð~ρ_A,zA,tAÞΔnBð~ρ_B,zB,tBÞ

Es

¼ X

m6¼n

ψ^∗_mð~ρ_A,zA,tAÞψnð~ρ_A,zA,tAÞ

ψmð~ρ_B,zB,tBÞψ^∗_nð~ρ_B,zB,tBÞ

* +

Es

’ X

m

ψ^∗_mð~ρ_A,zA,tAÞψmð~ρ_B,zB,tBÞX

n

ψnð~ρ_A,zA,tAÞψ^∗_nð~ρ_B,zB,tBÞ

* +

Es

: (16.88)

In measurement-I, the optical transfer functions that propagate thefields from the source toD_AandD_Bare

g_mð~κ,ω;~ρ_A,zA¼dAÞ ¼iωe^iðω=cÞzA 2πcd_A

ð

d~ρ_sfð~ρ_sÞe^i~κ~ρsGð j~ρ_s~ρ_ojÞ_½ω=ðcdAÞ, and

g_nð~κ,ω;~ρ_B,z_B¼d_Bþd⁰_BÞ

¼ω²e^iðω=cÞzB ð2πcÞ²d_Bd⁰_B

ð d~ρ_s

ð

d~ρ_ifð~ρ_sÞe^i~κ~ρsGðj~ρ_s~ρ_ijÞ_½ω=ðcdBÞ tð~ρ_iÞGðj~ρ_i~ρ_BjÞ_½ω=ðcd⁰

BÞ,

where~ρ_sis defined on the output plane of the source andfð~ρ_sÞdenotes the aperture function of the source. We also assumed a perfect“bucket”detectorD_A, which is placed at the object plane of slit-A (~ρ_A¼~ρ_o), in the following calculation. ~ρ_i is defined on the ghost image plane, which coincides with the plane of slit-B, and~ρ_B is defined on the detection plane ofD_B,tð~ρiÞis the aperture function of slit-B. The functionG(jαj)[β]is the Gaussian functionGðjαjÞ_½β¼e^iβ2jαj2. The measuredfluctu- ation correlation can be calculated from Eq. (16.88)

ΔRAB¼ ð

d~ρ_ojtð~ρ_oÞj²sinc²½ω0D ~ρ_B

2cd⁰_B C⁰sinc²½ω0D ~ρ_B

2cd⁰_B, (16.89) where tð~ρ_oÞ is the aperture function of slit-A. The above calculation indicates a product between a constantC⁰, which is from the integral on the“bucket”detector D_A, and afirst order diffraction pattern of slit-B. With our experimental setup, the width of the diffraction pattern is estimated to be4 mm, which agrees well with the experimental observation, as shown in.Fig.16.27.

In measurement-II, with slit-B wide open, thefield atD_Bbecomes g_nð~κ,ω;~ρ_B,z_BÞ ¼iωe^iðω=cÞzB

2πczB

ð

d~ρ_sfð~ρ_sÞe^i~κ~ρsGðj~ρ_s~ρ_BjÞ_½ω=ðczBÞ: Wefirst check if a ghost image of slit-A is present when scanningD_Bin the ghost image plane ofd_B¼d_A. The photon-numberfluctuation correlation is calculated to be

ΔRAB¼ ð

d~ρ_ojtð~ρ_oÞj²sinc² ω0a

cdAj~ρ_o~ρ_Bj

¼ jtð~ρ_oÞj²sinc² ω₀a cdA

j~ρ_o~ρ_Bj

jtð~ρ_BÞj²:

(16.90)

Note, we have placedD_Aright behind slit-A and thus~ρ_A¼~ρ_o. This suggests an equal-sized 100%visibility ghost image on the plane ofd_B ¼d_A.

When we moveD_Baway from the ghost image plane to the far-field plane ofd_B +d_B⁰, the photon-numberfluctuation correlation becomes:

ΔRAB¼ ð

d~ρ_ojtð~ρ_oÞj²F~s2

ð

m~ρ_o~ρ_B:¼ jtð~ρ_oÞj²F~s2ðm~ρ_o~ρ_BÞ,(16.91) where F~s is the Fourier transform of the defocused pupil function Fs¼fð~ρ_sÞe^{iðω0=2cμÞ~ρs2} and μ, m are defined as 1/μ¼1/d_A 1/(d_B + d_B⁰), m¼(d_B +d_B⁰)/d_A, respectively. The measured result of measurement-II is thus a convolution between the aperture function of slit-A, tð~ρ_oÞ, and the correlation function F~s

ð

m~ρ_o~ρ_B:, resulting in a “blurred” image of slit-A. With our experimental setup, the width of the “diffraction” pattern is estimated to be 1:4 mm, which is almost three times narrower than the diffraction pattern of measurement-I and agrees well with the experimental observation, as shown in .Fig.16.27. Compared with the Kim-Shih experimental result [13], we can see that although the number varies due to different experimental parameters, we have obtained a very similar result: the measured width of the“diffraction pattern”in measurement-II is much narrower than that of the diffraction pattern in measure- ment-I.

The above analysis indicates that the experimental observations are reasonable from the viewpoint of the coherence theory of light. The important physics we need to understand is to distinguish thefirst-order coherent effect and the second- order coherent effect, even if the measurement is for thermal light. In Popper’s measurement-I, the fluctuation correlation is the result offirst-order coherence.

The joint measurement can be “factorized” into a product of two first-order diffraction patterns. After the integral of the “bucket”detector, which turns the diffraction pattern of slit-A into a constant, the joint measurement between D_A and D_B is a product between a constant and the standardfirst-order diffraction pattern of slit-B. There is no question the measured width of the diffraction pattern satisfies Δp_yΔyh. In Popper’s measurement-II when slit-B is wide open or removed, the measurement can no longer be written as a product of single-photon detections but as a non-separable function, i.e., a convolution between the object aperture function and the photon-number fluctuation correlation function of randomly paired photons, or the second-order coherence function of the thermal field. We thus consider the observation of Δp_yΔy<h the result of the second- order coherence of thermal field which is caused from nonlocal interference: a randomly paired photon interferes with the pair itself at a distance by means of a joint photodetection event betweenD_AandD_B. The result of nonlocal two-photon interference does not contradict the uncertainty principle that governs the

432 Y.H. Shih

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behavior of single photons. Again, the observation of this experiment is not a violation of the uncertainty principle. The observation ofΔp_yΔy<hfrom thermal light, however, may reveal a concern about nonlocal interference.