Lab Report: Entangled photons and the Bell inequality ...joveriab.weebly.com/uploads/9/9/5/1/9951095/report_bell_inequailty...Lab Report: Entangled photons and the Bell inequality

Lab Report: Entangled photons and the Bell inequality Submitted by: Pei-Ying Lin, Joveria Baig

Introduction: Despite of Albert Einstein’s earlier contribution to quantum mechanics with his interpretation of the photon electric effect, he stayed a big critic of quantum mechanics until his death. Einstein was convinced that “god does not throw dice”. When discussing with Bohr, he designed a series of experiments to disprove the formulation of quantum mechanics. Whereas some of them, for example, the photon scale could be shown by Bohr to be correct with the framework of general relativity and quantum mechanics, a seemingly paradox formulated by Einstein, Podolsky and Rosen (ERP) in 1935 remained unanswered. Quantum mechanics allowed formulating states where the measurement of a property of the state at one point in the universe instantaneously decides the property of the state at another point in the universe. Einstein saw his views about causality violated and concluded that quantum mechanics must be incomplete. However, Bohr disagreed, he regarded quantum mechanics as complete, but neither Bohr nor Einstein could prove their hypothesis. The problem remained unsolved until both they died. When the physicist John Bell studied the ERP paradox, he realized there is an inequality in certain situations. In this genial step, John Bell had found a weapon to find an answer of the dispute of Bohr and Einstein. Finally, in 1982, a group directed by Alain Aspect demonstrated the first convincing violation of Bell’s inequality. Quantum mechanics had succeeded and hidden variable theories were strong hit. In this report, an experimental resembling the famous Bell’s inequality is performed. For this purpose, a detailed discussion of the theoretical basics will be given. After this, the used apparatus and the adjustment procedure were described and eventually the Bell’s inequality is demonstrated. Theoretical Background:

1. Polarization states As a basis for this report, the quantum description of polarization states of photons has to be known. For example, the state can be described in the form of

|ψ>=𝑐!|𝑉 > +𝑐!|𝐻 > The photon is in a superposition state of the two states vertical polarization and horizontal polarization with the complex coefficients 𝑐! and 𝑐! . The probabilities 𝑃!,  𝑃! of measuring the photon in the vertical or horizontal direction are given by |𝑐!|2 and |𝑐!|2 respectively. Therefore, |𝑐!|! + |𝑐!|! = 1 has to be fulfilled. As an example of such a superposition state a photon rotated in the clockwise direction with an angle 𝛼 with respect to the vertical axis can be considered. A photon can be described by

|𝜓 >!"#= 𝑐𝑜𝑠𝛼|𝑉 > +𝑠𝑖𝑛𝛼|𝐻 > and a comparison with equation above leads to probability of 𝑐𝑜𝑠𝛼!for a detection along the vertical direction. As an example with complex coefficients, a left circular polarized photon can be taken

|𝜓 >!"#!=12|𝑉 > −𝑖|𝐻 >

giving a probability of !

! for detection in the vertical direction.

Although the considerations made before it has to be pointed out that the state in first equation can be written in any possible orthogonal basis |𝑉! >, |𝐻! > constructed by an rotation of an angle 𝛼 in the anticlockwise direction of the original basis |𝑉 >, |𝐻 >. This is a special feather of polarization state, that is, it cannot be found in superposition states of atomic levels. In the new basis, the corresponding probability amplitudes will change. The new vector of basis can describe the rotation of the basis.

|𝑉! >= 𝑐𝑜𝑠𝛼|𝑉 > −𝑠𝑖𝑛𝛼|𝐻 |𝐻! >= 𝑠𝑖𝑛𝛼|𝑉 > +𝑐𝑜𝑠𝛼|𝐻

and so the horizontal and vertical states of the non-rotated system are written in the new basis by

|𝑉 >= 𝑐𝑜𝑠𝛼|𝑉! > +𝑠𝑖𝑛𝛼|𝐻! > |𝐻 >= −𝑠𝑖𝑛𝛼|𝑉! > +𝑐𝑜𝑠𝛼|𝐻! >

so the probability of detecting a former vertical state |𝑉 > along the |𝑉! > direction in the new basis is 𝑐𝑜𝑠𝛼!. In case of |𝐻 > a probability of 𝑠𝑖𝑛𝛼! is found.

2. Polarization-entangled photon pairs In this report pairs of polarization-entangled photons will be studied. These states, also called ERP states, have the following form

|𝜓 >!"#=12(|𝑉 >! |𝑉 >!+ 𝐻 >! |𝐻 >!>

The state is non-factorable, for example it is not possible to give the photons a polarization the other photon has to have vertical polarization as well. Since this state does not include any parameter of time or space, the photons can be infinitely separated. Nevertheless, this does not allow any transmission of information. This is sue to the face that the probability of measuring the vertical state is ½ formula – so it is completely random if a vertical or a horizontal state is measured and only random results for the other photon appear, no information can be transmitted. The same is true if the basic of the polarization states is changed. Now the more general case of the joint probabilities is considered, i.e. the probability of both detecting the first photon with the rotated polarization |𝑉! >!and the second photon with |𝑉! >!. The probability is written as

𝑃  (𝑉! ,𝑉!) = | < 𝑉!|! < 𝑉!|!|𝜓 >!"# |!

Hence, by mapping the polarization states from above on the EPR-state and applying the trigonometric relation (𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽)! = 𝑐𝑜𝑠!(𝛼 − 𝛽) , one obtains 𝑃  (𝑉! ,𝑉!) =

!!𝐶𝑂𝑆!(𝛼 − 𝛽). For 𝛼 = 𝛽, the probability is ½. Thus, the probability of

finding the first photon in the state |𝑉! >! is 1/2 and one can conclude that when one measures one of the photons in a certain polarization, it is sure that the other photon has the same polarization. To understand this property better, one can rewrite the state |𝜓 >!"# in any other basis |𝑉! >, |𝐻! >. By applying the transformations, one sees that

|𝜓 >!"#=12( 𝑉 >! 𝑉 >!+ 𝐻 >! 𝐻 >!)

In this sense, if a Bell state was prepared in one coordinate system it stays a Bell state in any other coordinate system. It is directly obvious that measuring polarization of one photon in one direction gives the knowledge of the polarization of the other photon. Einstein regarded such a form of direct correlation as impossible. He assumed that quantum mechanics was incomplete and proposed the existence of supplementary parameters in addition to quantum theory. In such a way correlations similar to classical situation could be established, where the physical property of both systems is known before the measurement, thus being both real and local. Nevertheless, due to randomness of the result of each measurement, it seemed to be impossible to prove that such “hidden variables” exist - both theories, quantum theory with and without supplementary parameter could account for any found correlations and the discussions between Einstein and Bohr were considered as mere philosophical questions. This opinion changed dramatically when John Bell showed in 1964 that any correlations with supplementary parameter theories have to obey an inequality, it was now possible to find an experimental answer to dispute of Einstein and Bohr. In the following, the Bell inequality will be written in the formulation give by C.H.S.H., thus allowing an easy access to the experimental application. For this purpose, a quantity called degree of correlation is defined by:

𝐸(𝑉! ,𝑉!) = 𝑃(𝑉! ,𝑉!) + 𝑃(𝐻! ,𝐻!) − 𝑃(𝐻! ,𝑉!) − 𝑃(𝑉! ,𝐻!) By calculating each single coefficient and finally using the trigonometric expression 𝑐𝑜𝑠!(𝛼 − 𝛽) − 𝑠𝑖𝑛!(𝛼 − 𝛽) = cos  [2(𝛼 − 𝛽)], one obtains

𝐸(𝑉! ,𝑉!) = cos  [2(𝛼 − 𝛽)] this expression takes values form 𝐸(𝑉! ,𝑉!) = 1 to 𝐸(𝑉! ,𝑉!!!") and has a zero at 𝐸 𝑉! ,𝑉!!!" . With the defined degree of correlation, the following parameter can be defined:

𝑆!"##(𝑎, 𝑎′, 𝑏, 𝑏′) = 𝐸(𝑉! ,𝑉!) + 𝐸(𝑉! ,𝑉! ′) + 𝐸(𝑉! ′,𝑉!) + 𝐸(𝑉! ′,𝑉! ′)

This number, which called Bell’s parameter, was shown to have necessarily a value of less than two if the correlations could be explained by a supplementary parameter theory. In contrast, by choosing values of polarizations shown in following Figure

Figure1. Angles of analyzer of the Bell inequality

Quantum mechanics can predict values of the Bell parameter of 2 2. Thus by performing an experiment with the chosen angles of the polarization entangled photons; Bell’s inequality can be tested. If Bell’s inequality is correct, supplementary parameters are missing in quantum mechanics thus shaking the fundaments of theory, if the inequality is violated quantum mechanics seems to be complete but

either the concept of locality or reality has to be abandoned. Clauser and Freedman gave the first test of Bell’s inequality, in 1972. Nevertheless, due to experimental imperfections and opposite results of a different group with a similar setup, and strong doubts remained. In the beginning of the 80’s, a group of Alain Aspect performed a series of experiments giving the first convincing proof of the violation of the bell inequalities, in the following, an experiment will be performed similar to one of the experiments by Aspect and his colleagues. Experimental Part:

The setup shows like the following figure2. Vertical polarized laser light at 405nm is directed onto a nonlinear BBO crystal crating photons at a wavelength of 810nm by spontaneous down conversion. The generated light is directed onto avalanche photodiodes able to count single photons. ½ plate at 405nm is placed in the violet pump light in order to rotate its polarization. Polarization beam splitters combined with ½ plates at 810nm are used as analyzers for the generated photons. Later on, a Babinet compensator is placed in the pump beam. For the first adjustment of the system, the ½ plate and beam splitters are taken out of the system.

Figure2. Experimental setup

The other part is for adjustment of the coincidence counters. As a prerequisite for the experiment the coincidence counters had to be calibrated in order to make sure that the two independent photons accidentally in coincidence (𝑛!, 𝑛!) were detected in the gate window 𝜏!. Knowing that the number of number coincidence per second 𝑛! is given 𝑛! = 𝜏!𝑛!𝑛! , the counts 𝑛! , 𝑛! and 𝑛! were measured and recorded in LabVIEW, and the experimental gate window was calculated 𝜏!. The coincidence gate depend on the switches SW16 and SW17, thus the theoretical and calculated results are shown in Table1. Once the coincidence circuit was checked, the gate window was set in the LabVIEW program to the calculated window size, in order to achieve more accurate results. For our purposes, the window size chosen for all the measurements performed in the lab from this point on was the smallest window that is roughly 9.3ns.

Table1. Calculated window time of 𝜏! for each switch pair SW16 SW17 𝝉𝒇 Calculated ON ON 9.299 ns ON OFF 15.63ns OFF ON 19.28ns OFF OFF 77.49ns

BBO  Crystal PBS

Mirror

Laser  diode  at  405nm

APD  in  single   Photon  Counting  mode ½  plates  at  405nm

For parametric down conversion, BBO is a negative uniaxial nonlinear crystal, which produces photon pairs by spontaneous parametric down-conversion. The BBO crystal used for this experiment was cut to fulfill a type I matching phase condition, that means, when a fully horizontal or vertical polarized photon of wavelength is incident on the extraordinary crystal axis, two photons of wavelength emerge from the crystals were placed side by side, and rotated 90 degrees with respect to each other around the axis of the pump. Therefore, for vertical polarized photons, while the photon in the horizontal polarization, pass straight through. Phase correction: The system comprising of half wave plate and a polarization beam splitter is placed in each arm to act as an analyzer, as shown in figure 1. When the axis of the half wave plate is vertical, it leads to the detection of photons in the horizontal state |H> state. When the half plate is kept at an angle of 45 with respect to its axis, it leads to the detection of photons in the vertical state |V>.

Figure 3. Analyzer setup

In order to obtain a good source of photon pairs in the state |H1>|H2>, the wave plate at 405 nm is kept vertical. However, when this wave plate is rotated by an angle of 45°, it detects photons in the state |V1>|V2>. When the wave plate at 405nm is kept at an angle of 22.5° precisely, it leads to a diagonal polarization and hence same number of coincidences are detected when both analyzers are horizontal or vertical as was verified by the experimental measurements. However, in order to obtain the EPR state, it is important to adjust the polarization state of the pump beam. The reason for implementing this stems from the dephasing that occurs between the processes of dispersion and birefringence. This dephasing results in an expression for the joint probability of finding photon in state |V45> and photon 2 in state|V45> given by:

P(V!",V!") =14(1 + cos ∅ )

For an EPR state, we expect this probability to be equal to half since we expect equal contribution from the horizontal and vertical states. Hence, it can be seen that in order to obtain this value of half, it is necessary to have Ø=0 which corresponds to the case where we introduce a dephasing in our pump beam in such a way that it compensates the dephasing introduced by the difference between the dispersion and birefringence of the crystal.

|𝛹 !"#$ = |𝑉 + exp 𝑖∅ |𝐻                                                                                      𝑓𝑜𝑟  ∅!"#! = −∅

In order to implement this, a babinet compensator was used. After aligning the babinet with the beam by ensuring that the rate of coincidences is always equal when the analyzers are kept vertical and horizontal in the two paths, the babinet can introduce a dephasing proportional to the displacement of the babinet. With the analyzer kept at 45° in both arms, the rate of coincidence is recorded, the results are presented in the next section.

Results: The displacement of the babinet and the rate of coincidence were recorded. The rate of coincidence was then plotted as a function of the babinet displacement. The results are as shown in figure 1, for the case when the babinet axis was not aligned and when the babinet axis was aligned with the neutral axis. The results show a cosine behavior between the dephasing and the joint probability given by:

P V!",V!" =141 +m  cos ∅ − ∅!"#$ =

141 +m  cos ∅ − 2π

zz!

From this graph, it can be seen that at the maximum value of the cosine function where net dephasing is equal to zero, one observes a probability given by !"#

!"##= 0.41

which is close to the desired value of 0.5 considering the losses introduced by the crystal and the babinet. In the case where ∅ − ∅!"#$ =

!!, using the equation, we

would expect to have a probability of !!. In this case, the modulating term containing

the cosine part becomes zero and hence has no effect on the probability. This results in obtaining the anti EPR state.

Displacement of Babinet

Without the babinet axis correction

With babinet axis correction

10 520 460 20 480 630 30 190 220 40 100 98 50 350 300 60 540 630 70 400 430 80 150 160 90 145 130

100 419 380 110 550 650 120 330 350 130 115 110 140 180 160 150 500 470 Table2. Parameters of babinet’s displacement

Figure 4. Relationship of the coincidence rate with the babinet displacement. (Red): With the babinet axis aligned. (Blue): without the axis aligned

Bell's Inequality and Measurement of Bell's parameter: The next step of the experiment was to verify the bell's inequality and to measure the bell parameter. In order to maximize the violation of the bell parameter, the babinet was placed at the displacement which gives the purest EPR state, i.e. at the maximum of the cosine curve described above. The bell parameter is given by:

𝑆!"##(𝑎, 𝑎′, 𝑏, 𝑏′) = 𝐸(𝑉! ,𝑉!) + 𝐸(𝑉! ,𝑉! ′) + 𝐸(𝑉! ′,𝑉!) + 𝐸(𝑉! ′,𝑉! ′)

where,

𝐸 𝑉! ,𝑉! =𝑛 𝛼,𝛽 + 𝑛 𝛼 + 90,𝛽 + 90 − 𝑛 𝛼,𝛽 + 90 − 𝑛(𝛼 + 90,𝛽)𝑛 𝛼,𝛽 + 𝑛 𝛼 + 90,𝛽 + 90 + 𝑛 𝛼,𝛽 + 90 + 𝑛(𝛼 + 90,𝛽)

Results: Using the above formulation, the measurements made are summarized in figure 2. Using the calculations shown, the bell parameter was calculated to be equal to 𝑆!"## = 1.53191071 and the standard deviation was calculated to be equal to 0.01616283. A graph of the coincidence rate in different configuration of analyzer is presented in Appendix 2.

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0   20   40   60   80   100   120   140   160  

rate  of  coincidence  

Babinet  displacement(mm)  

Discussion: In the EPR argumentation, the bell parameter S is predicted to be less than 2 by the bell inequality. However, in this experiment, the aim was to objectively prove that S is greater than 2 which conforms to the quantum mechanics prediction of a value of 2 2. Due to experimental errors such as inaccuracy in measurements of coincidence, we could not verify this value and prove the violation of bell's inequality. However, this experimental setup can objectively prove that Bell's inequality is violated and should give a bell parameter of greater than 2, which is predicted by quantum mechanics. Although this value should be greater than 2 and contradict Bell's inequality, it will deviate from the quantum mechanical prediction of 2 2 due to the presence of noise within the experiment and inability to isolate the purest EPR state possible.

NA NB NC Nc corr 𝝈Ni*dS/dNi

aV,bV 74000 86000 0 5437 0 aV,bH 74000 86000 510 1119 0.001658847

aH,bV 74000 86000 810 2754 0.002090565

aH,bH 74000 86000 6150 959 0.005760481

E1= 0.245690914 aV,b'V 74000 86000 600 950 0.003156493 aV,b'H 74000 86000 3250 5367 0.007346338

aH,b'V 74000 86000 5550 5400 0.009600099

aH,b'H 74000 86000 1350 1210 0.00473474

E2= -0.66581573 a'V,bV 74000 86000 3050 2600 0.003578045 a'V,bH 74000 86000 2350 3724 0.003140725

a'H,bV 74000 86000 1010 1650 0.002059001

a'H,bH 74000 86000 4610 4906 0.004398923

E3= 0.16552795 a'V,b'V 74000 86000 4110 4800 0.00264814

a'V,b'H 74000 86000 1160 1854 0.001406854

a'H,b'V 74000 86000 2210 1743 0.001941852

a'H,b'H 74000 86000 3410 4800 0.00241211

E4= 0.454876108 Table3. Measurements for calculation of Bell Parameter

Appendix 1 Graph illustrating the coincidence rate for different configurations of the analyzer

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6000  

aV,bV  

aV,bH  

aH,bV  

aH,bH  

aV,b'V  

aV,b'H  

aH,b'V  

aH,b'H  

a'V,bV  

a'V,bH  

a'H,bV  

a'H,bH  

a'V,b'V  

a'V,b'H  

a'H,b'V  

a'H,b'H  

Coincidence  Rate  for  different  conOigurations  

Nc  corr