-- Main.khanx169 - 16 May 2014

Single Photon Quantum Interference

Asad Khan and Quynh Nguyen

University of Minnesota

Methods of Experimental Physics Spring 2014

Note: This paper focuses on the theory and results. For a more detailed description of the equipment and alignment procedures, please visit the procedure documentation page.

Abstract

We show that light is composed of single photons by carrying out a correlation experiment exploiting Spontaneous Parametric Down-Conversion using Type-I BBO crystal. Our best result violates classical prediction of wave-like behavior for light by 74 standard deviations

Introduction

The first arguments in favor of light quantization were given by Einstein to explain the photoelectric effect, but semi-classical theories which treat light as an electromagnetic wave and only view the detector atoms as composed of quantized energy levels are equally successful at explaining the photoelectric effect [1][2]. A more rigorous demonstration of the existence of single photons can be given by passing a dim beam of laser light through a beam splitter and examining the intensities of the reflected and transmitted beams with sensitive detectors. If light behaves like a wave, then classical physics predicts that some portion of the incident beam will be reflected and some transmitted simultaneously. However, if light is composed of single photons, and since a photon under observation cannot go through two different paths at the same time [3], we should never detect light at the two detectors simultaneously (see Fig. 1).

Theory

Although in principle shining a dim laser beam through a beam splitter and measuring the reflected and transmitted intensities should show the existence of single photons, in practice this methodology fails. This is because the laser light is a Poissonian distribution of photon number states, i.e; photons are emitted very close together in time, resulting in bunches, which behave very similar to a classical wave [4]. Hence to definitively demonstrate the quantization of light, we shine our laser onto a β-barium borate (BBO) crystal and use the type-I spontaneous parametric down-converted output beam as our source of light. Type-I spontaneous parametric down-conversion is a non-linear optical effect, which converts a single photon at frequency into two lower energy photons at frequencies and , such that [4]. Down-conversion occurs in a non-linear crystal (the BBO crystal in our case), and the two downconverted photons simultaneously exit the crystal in a cone about the input axis of the incident laser beam (henceforth also referred to as the ‘pump beam’) as shown in Figure. 2.

The essential feature of down-conversion that makes it possible to show the quantization of light is that the two down-converted photons are in an entangled state, i.e, the two photons are correlated, so that Page | 3 if we place two 100% efficient detectors intercepting the down-converted cone in the same plane and at the same distance from the BBO, then a measurement of a photon in one of the detectors provides absolute certainty that a corresponding photon will be simultaneously measured at the other detector [8] (see Figure. 3).

If we use the down-converted arm B as our light source in the beam-splitter correlation experiment and condition our observations at either side of the beam splitter upon the measurement of the corresponding correlated photon in detector A (see figure. 4), we effectively force the photons into a “heralded” single-photon state [4] .

Now, if light is composed of single photons, then whenever light is measured at A, there’d also be light measured at either B or B’, but not both. However, if light behaves like a classical wave, then light may be measured at both B and B` simultaneously with A.

Experimental Setup

Our final experimental configuration is essentially similar to the one depicted in Figure 4. We use a 405 nm ultraviolet diode laser as our pump beam, a polarizing beam splitter (PBS), and the β-barium borate (BBO) crystal cut at 29o so that the 810 nm down-converted photons (henceforth also referred to as ‘twin photons’) are emitted along a conic angle of 3o with respect to the initial pump direction [4]. Our single photon detectors convert an incoming photon to an electronic pulse. The pulse width depends on the type and quality of the detector, but typical pulse widths range from tens of nanoseconds to milliseconds [4]. The detector output signals are sent through an AND gate. Hence for the counting circuit to register a coincidence, some portion of the pulse from the first detector must overlap with some portion of the pulse from the second detector. If the two pulse detectors have the same pulse width , then the total coincidence window is defined as (1) In our set-up, we had four choices for ranging from 10ns to 60ns. The data for the count rates was handled by a Single Photon Counting Module (SPCM), which consisted of four silicon avalanche photodiodes. The signals from the SPCM were counted by an FPGA board, and the data from the FPGA board was sent to a computer through a serial port. Finally, a LABVIEW program processed the data and displayed it.

Experimental Technique

Anti-correlation Parameter for Three Detector Set-up In order to quantify the correlation between measurements made at any two detectors, it is convenient to define an anti-correlation parameter α as (2) where PC is the probability of measuring a coincidence count between the two detectors, and Pi is the probability of measuring an event at the ith detector [4]. Since we perform a correlation experiment between detectors B and B’, conditioned on measurements made at detector A (see figure 4.), the probability of measuring an event at detector B is given by (3) where, is the number of measured coincidence counts between detectors A and B, and is the number of measured counts at detector A. The probability of measuring an event at detector B’ is given similarly, with B’ replacing B in (3). Finally, the probability of measuring a coincidence count between detectors B and B’ is given as (4) where is the number of triple coincidence counts between detectors A, B and B’. Substituting these probabilities into equation (2), we see that the anti-correlation parameter for B and B’ in the three detector setup (fig. 4) is where the subscript 3d signifies that the measurement is for three detector set-up. Dividing each term in (5) by the time T for which the observations were made, we see (6) where R represents the corresponding count rates. Anti-correlation Parameter for Two Detector Set-up For the two detector set-up (fig. 3), the anti-correlation parameter changes into (7) where is the rate of coincidence between detectors A and B, is the rate of counts at detector A, is the rate of counts at detector B, and is the coincidence window as defined in (1). [4]. If instead of two correlated beams, the detectors A and B were to measure two independent and random events, then any coincidences between the two detectors would be purely accidental. Hence, if the average count rates at detectors A and B are respectively, then the rate of accidental coincidences is given by (8) Hence, (7) changes into (9) From (9), it is clear that for correlated sources, which produce more coincidences between the two detectors than accidental, the anti-correlation parameter is greater than one, while for anti-correlated sources, the anti-correlation parameter is less than one.

Experimental Method Before confirming the existence of single photons, we verify that the two down converted beams are in fact correlated by first performing the experiment without adding a beam splitter, and merely measuring the count rates at detector A and detector B (see fig. 3). The BBO crystal is added perpendicularly to pump beam’s path, and since the down converted beams are emitted at a conic angle of 3o relative to the incident beam, the detectors A and B are placed in a horizontal plane at the rough appropriate angles. The position of the crystal and of the detectors is then adjusted until is maximized. The maximum values for various coincidence windows, after the proper alignment was achieved, are summarized in table 1. (ns) HIGH ATTENUATION 10 14 800 16 700 224 2.47 20 14 600 16 300 231 4.77 40 14 300 16 000 239 9.15 60 14 200 15 800 244 13.5 LOW ATTENUATION 10 46 100 51 400 718 23.7 20 45 600 50 700 758 46.3 40 44 300 49 200 803 87.2 60 43 800 48 600 844 128 Table 1. Correlation results for two detector set-up using type I spontaneous parametric down-converted light source. The results quoted are average of twenty 30s runs.

It can be clearly seen from table 1 that is greater than one in all cases, showing that the two down-converted beams are correlated. Although theoretically the anti-correlation parameter should be infinite for the two detector set-up since the two down converted beams are perfectly correlated, in reality, the non-zero coincidence window , and low efficiency of optics and alignment result in finite values comparable to those in table 1. [4] After confirming the correlation between the down-converted light beams, we move on test the existence of single photons. A polarizing beam splitter (PBS) is added into the B arm (see fig. 4), so that detector B measures the transmitted beam, and a new detector B’ is introduced to measure the reflected beam. A half wave plate (HWP) is then placed before the PBS, and tuned so that the transmitted and reflected beam intensities at B and B’ are nearly equal. After achieving the desired alignment, we run the experiment for the three detector set-up and record the count rates for all three detectors twice; once with a low attenuation neutral density filter in front of the laser, and once with a high attenuation neutral density filter.

Results

After getting the three detector set-up aligned, we run the experiment and record count rates at all three detectors together with the relevant coincidence rates. is then calculated using equation (6). The experiment was performed for two different attenuations, and four different timing windows . For each combination of attenuation and timing window , the experiment was run for about 20 times, with each run lasting 30s. The 20 readings were then averaged, and are displayed in table 2.

Since a wave can simultaneously reflect and transmit at the beam splitter, a classical theory of light predicts . Hence, measuring demonstrates that light is composed of single photons [4]. As can be seen from table 1, we found in all of our experimental runs. Our best value, found for high attenuation and , violates classical prediction of by 74 standard deviations.

Analysis

Each value quoted in Table 2. is an average of twenty readings, with the uncertainty given by the standard deviation of those twenty readings. A number of interesting patterns emerge in Table 1 and Table 2. These are discussed in some detail below.

1) We see from table 1 that decreases as the coincidence window increases, even though the down-converted photons are perfectly correlated and hence the coincidence rates should have been independent of . This shows that a portion of our measured coincidences are in-fact accidental. From equation (8) it is clear that the accidental coincidence counts can be made negligibly small by reducing the overall count rates, as well as by reducing the coincidence window . This is consistent with what we observe in Table 1. 2) If light is composed of single photons, then ideally we should expect = 0 . Since we see that in table 2, it is clear from equation (6) that the rate of threefold coincidences, (i.e. there are some coincidences between B and B’, which must be purely accidental, if light is quantized). There are two sources for these three fold coincidences; one possibility is that three uncorrelated photons get registered at detectors A, B and B’ within , and hence get measured as a coincidence. These threefold random accidentals are given by (10) Since these accidentals depend on the square of , their rate is negligibly small [4]. A second possibility is that of a two-fold random accidental between a random B’ count (or a random B count) and a real AB coincidence (or a real AB’ coincidence). These accidentals dominate the threefold coincidences , and are given by (11) The presence of explains why . Hence reducing improves the results in two ways. Firstly, it reduces (see eq. 8), which in turn increases (see eq. 9)). Hence the correlation between arm A and arm B improves (see fig. 3). Secondly, reducing reduces (see eq. 11), and hence . This results in a lower value, and hence classical prediction for light is violated more strongly. Another feature that stands out in table 1 and table 2 is that increases and decreases as we reduce intensity of pump beam (i.e. use higher attenuation and hence decrease the twin production rate ). This is because in the two detector set-up, the accidental coincidence rate depends quadratically on , while the measured coincidence rate depends linearly on . [4]

Discussion and Conclusions

Our results show that , which amounts to an experimental evidence in favor of the quantized nature of light, which can either reflect or transmit at a beam splitter, but not both simultaneously. Our best result of was found for , with high attenuation neutral density filter in front of the pump beam. This result violates classical prediction of wave-like behavior for light by 74 standard deviations. Some pertinent sources of systematic error include alignment and detector inefficiencies. Since the single photons counting module (SPCM) is not 100% efficient, whenever there is a photon detected at detector A, there’s a significant probability that the corresponding correlated photon will go undetected in detector B (see fig. 3). In addition, the detectors are not perfectly aligned; hence the two detectors A and B don’t intercept exactly similar bandwidths from the BBO crystal. So in addition to the two down converted twin pairs, each detector also registers some non-twin photons coming from the crystal. The effect of both of these inefficiencies is to increase the accidental count rates [4]. Hence, the results of the experiment can be improved in the future by first trying to get a numerical estimate of the efficiency of the detectors and the SPCM, and by trying to achieve higher precision alignment.

A natural future extension of this experiment is to perform a test of local realism based on Hardy’s version of Bell’s inequality known as Bell-Clauser-Horne Inequality [5]. Locality is the assumption that the results of measurements in one location are not affected by the results of measurements in another location, while reality means that measurable quantities have definite values prior to their measurement. Violating bell’s inequality using polarization entangled photons produced in spontaneous parametric down conversion with two BBO crystals stacked back to back (with their axes perpendicular to each other) would mean that the down-converted photons do not have definite polarization angles prior to their measurement, and that the measurement of one photon’s polarization instantaneously determines the polarization of the other [6, 7].

Acknowledgements

We would like to extend my gratitude to the following people: Kurt Wick for his critical insight and feedback, and indispensable help at every stage of the experiment, and Professor Gregory Pawlosky and Professor Clement Pryke for reviewing and criticizing certain sections of early drafts of this paper.

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