PHYSICAL REVIEW A 73, 023814 共2006兲
Experimental demonstration of a method to realize weak measurement of the arrival time of a single photon Qin Wang,* Fang-Wen Sun, Yong-Sheng Zhang, Jian-Li, Yun-Feng Huang, and Guang-Can Guo† Key Laboratory of Quantum Information, Department of Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China 共Received 23 June 2005; published 17 February 2006兲 We present a realization of weak measurement for the arrival time of a single photon in virtue of the simultaneous characteristic of biphotons. Our experimental setup is based on a Hong-Ou-Mandel interferometer. A birefringent crystal is used to perform weak measurement between a preselected and a postselected measurement by two polarizers. The extreme weak values lie well outside of the eigenvalues of the observable. DOI: 10.1103/PhysRevA.73.023814
PACS number共s兲: 42.50.Xa, 03.65.Ta, 03.67.⫺a
The concept of weak measurement was first presented by Aharonov, Albert, and Vaindman 关1兴 共AAV兲, but AAV’s analysis led to the controversial result; such a measurement procedure may yield values for an observable lying well outside its associated eigenvalues. However, the controversial result was resolved in a theoretical sense by Duck et al. 关2兴 共DSS兲, who showed that such surprising outcomes of weak measurement involve nothing outside ordinary quantum mechanics, but the result of superposition of quantum wave functions. In addition, they gave the general criteria that must be satisfied to observe the AAV effect and also proposed an optical experiment setup to verify these effects. Thereafter, many other arguments and suggestions are given 关3–8兴. An optical experiment realization was performed by Ritchie et al. 关9兴, and then a similar one was finished by Parks et al. 关10兴. However, what they used is a classical intense laser beam and aspects of their experiments can be understood in terms of classical electromagnetic theory. Recently, there appeared another experiment performed with a quantum mechanical method 关11兴, but the setup is quite complicated and hard to control. Here, we present an optical experiment in a much easier way, which is similar to the proposal of Ahnert 关8兴. In this experiment, we use entangled photons to measure the weak values of the arrival time of single photons. And it is a completely quantum optical process and cannot be explained in any classical physics theory. Different to Ahnert’s proposal, we use a piece of birefringent crystal which can separate the two components of horizontal and vertical polarization by a small distance to perform weak measurement. Our experiment setup is based on a Hong-OuMandel 共HOM兲 interferometer 关12兴. We use one of the biphotons to pass through weak measurement and the other as a reference photon. The HOM interference will occur when the two photons arrive at the beam splitter 共BS兲 simultaneously, so the weak value can be marked by the shift of the HOM dip. In the following, we will introduce some basic theory of weak measurement 关1,2兴 and describe the main process of our experiment briefly.
*Electronic address:
[email protected] †
Electronic address:
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1050-2947/2006/73共2兲/023814共5兲/$23.00
In weak measurement, the coupling between the measuring device and the observable is so weak that the uncertainty in a single measurement is much larger compared with the separation between the eigenvalues of the observable. Therefore, the eigenvalues are not resolved by the measuring device, i.e., the system is not left in an eigenstate, but in a complex superposition of variable states. After carrying out a postselected procedure, it can yield surprising outcomes because of the interference between quantum wave functions. Consider a system with an observable Aˆ with corresponding eigenvalues an and eigenstates 兩A = an典. The initial state of the system can be in the term 兩⌿i典 = 兺nan兩A = an典. Immediately after the weak measurement of Aˆ, a 共strong兲 measurement of some other observable Bˆ is performed, thus the final state is an eigenstate of Bˆ, which can be expressed as some combination of the eigenstates of Aˆ : 兩⌿ f 典 = 兩B = b典 = 兺nan⬘兩A = an典. Then the weak values can be defined as Aw =
具⌿ f 兩Aˆ兩⌿in典 , 具⌿ f 兩⌿in典
共1兲
when the two selected states are nearly orthogonal, these values can be greater than the largest eigenvalue of Aˆ. Figure 1 is the schematic diagram of the weak measurement on single photons. Polarizer 1 and analyzer 2 are set at angle ␣ and  to the x axis 共horizontal兲, respectively. When a single photon passes through polarizer 1, it will be preselected in the state 兩⌿i典 = cos ␣兩H典 + sin ␣兩V典,
共2兲
after analyzer 2, it will be postselected in the state 兩⌿ f 典 = cos 兩H典 + sin 兩V典.
共3兲
A birefringent quartz plate with an optical axis aligned along the x axis is placed between polarizer 1 and analyzer 2. It can introduce a time delay between the two components corresponding to horizontal polarization and vertical polarization. We assume that the initial photon location has a Gaussian distribution due to energy-time uncertainty, and the phase shift caused by the quartz plate is a multiple of 2. The processes above can be expressed as follows 关8兴:
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©2006 The American Physical Society
PHYSICAL REVIEW A 73, 023814 共2006兲
WANG et al.
allow only the horizontal component to pass in order to obtain the HOM interference at the beam splitter兲 can be written as 兩⌿⬘f 典 =
冋
+ sin ␣ sin 
FIG. 1. A schematic diagram of the weak measurement of single photons. Single photons pass through the polarizer 1, quartz plate, and analyzer 2 sequentially. Polarizer 1 with rotation angle ␣ and analyzer 2 with rotation angle  are used to generate a preselected state 兩⌿i典 and postselected state 兩⌿ f 典, respectively. The quartz plate with the optic axis aligned along the x axis is used to perform weak measurement.
冉
共y 0 − ct兲 1 1/4 exp − 共 兲 22 2
冉
2
冊
+ sin ␣
冉
冊
Aˆ = 兩V典具V兩,
dyy
−⬁
=
冉
共6兲
The probability for a success measurement is 兩具⌿ f 兩 ⌿i典兩2 = cos2共␣ − 兲. If ␣ = 0 and  = 0, Aw = 0, for there is no V component and no delay; if ␣ = / 2 and  = / 2, Aw = , which is just the delay between the two polarizations; if ␣ = / 4 and  = / 4, Aw = / 2, which is half of the delay ; and when ␣ =  + / 2 + ␦, Aw may be much larger than if ␦ is very small. So that the delay can be “magnified” by choosing a suitable preselected state and postselected state at the cost of low probability sin2 ␦. The weak value deduced in the frame of weak measurement can also be derived with another method. The state of the photons after the polarization analyzer 共which is set to
兩H典, 共7兲
冋
冋
冉 冊 冊
冉
1 共y − 兲2 1/2 exp − 共 兲 2 2
y 2 + 共y − 兲2 22
冊册
1 共2兲1/2
sin2 ␣ sin2  + sin ␣ sin  cos ␣ cos  N2
冉 冊册
⫻exp −
2 42
.
共8兲
When the delay is much less than , 共which is the condition of weak measurement兲 exp共−共2 / 42兲兲 → 1, then N2 → cos2共␣ − 兲 and
共5兲
具⌿ f 兩Aˆ兩⌿i典 sin ␣ sin  = 具⌿ f 兩⌿i典 cos ␣ cos  + sin ␣ sin 
冊册
1 1 y2 2 2 cos ␣ cos  exp − N2 共2兲1/2 2
+ sin2 ␣ sin2 
共4兲
and the weak measurement of the delay time for the preselected state 兩⌿i典 and postselected state 兩⌿ f 典 is
. cot ␣ cot  + 1
+⬁
⫻exp −
1 共y 0 − ct − 兲2 exp − 兩V典, 共2兲1/4 22
=
冕
+ 2 sin ␣ sin  cos ␣ cos 
where = c and = 2 / 共2⌬冑2 ln 2兲, is the wavelength of the down conversion photons and ⌬ is the full width at half maximum 共FWHM兲 of the interference filters in front of the single photon detectors in our experiment, and the delay is much less than to satisfy the condition of weak measurement. The measurement operator can be chosen as
Aw =
具⌿⬘f 兩y兩⌿⬘f 典 =
→
冊
冉
1 共y − 兲2 exp − 共2兲1/4 22
where y = y 0 − ct, and N = 关cos2 ␣ cos2  + sin2 ␣ sin2  + 2 exp关−共2 / 42兲兴sin ␣ sin  cos ␣ cos 兴1/2. The expectation value of the photon’s arrival time is proportional to y, and the expectation value of y in the exact solution is
共cos ␣兩H典 + sin ␣兩V典兲Aˆ
1 共y 0 − ct兲2 cos ␣ 2 1/4 exp − 兩H典 共 兲 22
冉 冊
1 1 y2 cos ␣ cos  2 1/4 exp − 2 共 兲 N 2
具⌿⬘f 兩y兩⌿⬘f 典 →
sin ␣ sin  = Aw . cos ␣ cos  + sin ␣ sin 
共9兲
Weak values of the delay time can be measured using a HOM interferometer 关12兴. In a HOM interferometer, if biphotons with the same linear polarization arrive at a 50/ 50 nonpolarizing BS simultaneously, they will always emerge in the same output port. As a result, a dip will appear in the coincidence curve, therefore the weak value can be characterized by the shift of the dip in time 共or position兲 axis. The experiment setup is shown in Fig. 2. The pump source is a mode locked Ti:Sapphire laser, which has a pulse width of 200 fs and a central wavelength of 390 nm. The power of the pump is 450 mW and the repetition rate is 76 MHz. A piece of type I BBO 共-BaB2O4兲 crystal 共1 mm兲 is used to produce spontaneous parametric down-conversion 共SPDC兲 photon pairs in horizontal polarization 共represented by 兩HH典兲. M1 and M2 are right-angle prisms, and the position of M2 can be controlled by a stepmotor. Half wave plates 共HWP1 and HWP2兲 are placed before and after a quartz plate 共Q兲 to change the polarization of incoming photons, and the displacement between the horizontal and vertical polarization caused by the quartz plate is 19 times the
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EXPERIMENTAL DEMONSTRATION OF A METHOD TO¼
FIG. 2. The experimental setup. It is based on a Hong-OuMandel interferometer. The birefringent quartz plate 共Q兲 may produce a time delay 共 ⬇ 49.40 fs, i.e., ⬇ 14.82 m between the two components corresponding to horizontal and vertical polarization. The half wave plate 共HWP1兲 placed before the quartz plate can generate a suitable preselected state, the other half wave plate 共HWP2兲 and the polarization analyzer 共H兲 make up the postselection. After meeting at a nonpolarizing beam splitter 共BS兲, photon pairs pass through irises and interference filters 共IF兲, and at last they are registered by single photon detectors D1 and D2.
wavelength of down conversion photons, i.e., = 14.82 m. H is a Glan-Thompson polarizer which is set to transmit the horizontal polarized photons only. Then photon pairs meet at the beam splitter and at last are detected by the single photon detector D1 and D2, respectively. The diameter of our irises is 2 mm. It should be emphasized that the full width at half maximum of our interference filters 共IF兲 is 1 nm, with a coherence length of photon pairs ⬇ 258.36 m, which is much larger than , so that the condition of weak measurement is satisfied. The HOM interference curves of our experiment are shown in Fig. 3. From them we can find that, the centroids of curves d, e, and f which correspond to nearly orthogonal weak measurements are shifted by a much larger distance than curve b or c which correspond to the aligned polarizers 关13兴. Furthermore, it even appears as a much larger shift in curve f in the opposite direction to the delay by the quartz, which seems like a superluminal effect. It should be pointed out that, weak measurements that yield weak values are intrinsically consistent with relativistic causality because they obey two rules. On the one hand, they are weak, hence they hardly disturb the measured system, on the other hand, they depend on the uncertainty in the position of initial photons which is defined with the Gaussian-type wave function. It can be said that, if there is no initial uncertainty, there is no magnified value. In fact, all these amazing phenomena are only the results of constructive and destructive interference between quantum wave functions, and they are consistent with formula 共6兲. However, the absolute weak values of our experiment indicated in captions are still smaller than those indicated in the following brackets calculated with formula 共6兲. What caused it? We think the main reason may be that, the actual optical path difference between the H and V polarization caused by the quartz is not exactly a multiple of the wavelength as expected. So it can introduce an additive relative
FIG. 3. 共Color online兲 Hong-Ou-Mandel interference curves with coincidence counts vs the position of the stepmotor L. The points are experimental data, and the solid lines are their Gaussian type fits. 共a兲 ␣ = 0,  = 0, which corresponds to the delay ⌬ = 0. 共b兲 ␣ = 45°,  = 45°. 共c兲 ␣ = 90°,  = 90°. 共b兲 and 共c兲 correspond to the aligned polarizers, and their centroid of curves are shifted by a small distance / 2 and , respectively. 共d兲 ␣ = 45°,  = −共45° + 10° 兲. 共e兲 ␣ = 45°,  = −共45° + 6 ° 兲. 共f兲 ␣ = 45°,  = −共45° −6 ° 兲. 共d兲, 共e兲, and 共f兲 correspond to the measurement of weak values, and their centroid of curves are shifted by 41.88 共49.43兲 m, 54.00 共77.91兲 m, and −36.88 共−63.09兲 m, respectively. The values before the brackets are experimental results, and those inside the brackets are the ideal ones calculated by formula 共6兲.
phase ei between the H and V polarization, then the state after the quartz plate should be modified as
冋
cos ␣
冉 冊 冉
1 y2 兩H典 1/4 exp − 共 兲 22 2
+ ei sin ␣
After the postselection by analyzer 2,
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冊 册
1 共y − 兲2 兩V典 . 1/4 exp − 共 兲 22 2
共10兲
PHYSICAL REVIEW A 73, 023814 共2006兲
WANG et al.
兩⌿⬘f 典 =
冋
冉
冉 冊
1 1 共y − 兲2 1 y2 cos ␣ cos  2 1/4 exp − 2 + ei sin ␣ sin  2 1/4 exp − 共 兲 共 兲 N⬘ 2 22
冊册
兩H典,
共11兲
where N⬘ is the normalization constant. Furthermore, the weak value of Aw can be modified as Aw =
sin2 ␣ sin2  + sin ␣ sin  cos ␣ cos  cos . sin2 ␣ sin2  + cos2 ␣ cos2  + 2 sin ␣ sin  cos ␣ cos  cos
共12兲
Now, the shifting values of our experimental curves 共d ⬃ 41.88 m, e ⬃ 54.00 m, f ⬃ −36.88 m兲 are put back into Eq. 共12兲, and their corresponding values of 共d = ± 0.1679, e = ± 0.1516, f = ± 0.1628兲 are inversely deduced out. Obviously, they are in good agreement with each other within the error range. Therefore, our assumption above has been proven to be true 关14兴. Of course, there still exist other sources that caused errors. Since the delay has been magnified by choosing a nearly orthogonal preselected state and postselected state at a cost of low probability, and the coincidence count rate is so low, that we have to enlarge the detecting time. Then the dark counts and random coincidence will be increased inevitably. As a result, the visibility is decreased and the shape of the coherence curve d, e, and f are changed in some degree. Also the error may come from the inaccuracy of the angles changed by half and quarter wave plates. In addition, the accuracy of our stepmotor is 1.25 m per step, which may bring in an instrumental error. In conclusion, we have presented a quantum optical experiment of weak measurement on the arrival time of single
photons in virtue of the simultaneous characteristic of biphotons. In addition, we found that a quartz plate with a nonideal length may decrease the weak values in some degree. Anyway, the experiment results can still verify weak measurement theory in principle. Nowadays, weak measurement has attracted more and more attentions because of its wide applications in quantum mechanics fields such as photon polarization interference 关2,3,9,10,15兴, barrier tunneling times 关16–18兴, photon arrival times 关5,8兴, anomalous pulse propagation 关19–21兴, correlations in cavity QED experiments 关6兴, nonclassical aspects of light 关22,23兴, communication protocols 关24兴, and retrodiction “paradoxes” of quantum entanglement 关25–27兴. We hope our experiment may have more useful effects on its development.
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corresponding 共ideal兲 weak value lying far outside the range of spectrum of eigenvalues by tens and a hundred times. However, for the low coincidence counting rate in our experiment, we choose ␦ = + 10°, +6°, −6°, with corresponding ideal weak values; Aw ⬇ + 3.3, +5.3, and −4.3, respectively. In fact, it was verified later that the phase shift between the H and V polarization caused by the quartz is not a multiple of 2 in another experiment. N. Brunner, A. Acin, D. Collins, N. Gisin, and V. Scarani, Phys. Rev. Lett. 91, 180402 共2003兲. A. M. Steinberg, Phys. Rev. Lett. 74, 2405 共1995兲. A. M. Steinberg, Phys. Rev. A 52, 32 共1995兲. Y. Aharonov, N. Erez, and B. Reznik, J. Mod. Opt. 50, 1139 共2003兲. D. Rohrlich and Y. Aharonov, Phys. Rev. A 66, 042102 共2002兲. D. R. Solli, C. F. Mc Cormick, R. Y. Chiao, S. Popescu, and J. M. Hickmann, Phys. Rev. Lett. 92, 043601 共2004兲. N. Brunner, V. Scarani, M. Wegmuller, M. Legre, and N. Gisin, Phys. Rev. Lett. 93, 203902 共2004兲. L. M. Johansen, Phys. Lett. A 329, 184 共2004兲.
This work was funded by the National Fundamental Research Program 共Grant No. 2001CB309300兲, National Natural Science Foundation of China 共Grants No. 10304017 and No. 10404027兲, and the Innovation Funds from Chinese Academy of Sciences.
关14兴
关15兴 关16兴 关17兴 关18兴 关19兴 关20兴 关21兴 关22兴
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