Single Photon Quantum Interference

Joel Peter Wagner and Isaiah Gray

University of Minnesota

Methods of Experimental Physics Spring 2013

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

Abstract

We demonstrate single photons and their self-interference using a source of correlated photon pairs. We first establish the existence of individual photons, showing that they violate the classical prediction for wavelike behavior by 54 standard deviations. However, when these photons are directed into an interferometer, they exhibit wavelike properties and form a sinusoidal interference pattern. Furthermore, we demonstrate the phenomenon of quantum erasure, in which interference can be destroyed and recovered depending on the photon path information.

Introduction

In this experiment we investigate the quantum mechanical behavior of light, beginning with an examination of the evidence for the existence of photons. Although the photoelectric effect is commonly cited as proof of the quantization of light, it can also be explained semiclassically by assuming that light is a classical wave impinging on atoms with quantized energy levels [1]. The incident light can be treated as a sinusoidal perturbation to the ground state. Using Fermi’s golden rule, a transition probability can be obtained from the ground state to the ionized free state that grows linearly with time. This transition rate is proportional to the intensity of the incident wave, and reproduces all the characteristics of the photoelectric effect [2].

To more explicitly show that light is quantized, we perform a test based on the 1986 experiment of Grangier, Roger, and Aspect, in which they examined correlations between photodetections at the transmission and reflection outputs of a beamsplitter. Because a single photon can only be detected once, a photon passing through a beamsplitter should be transmitted or reflected, but not both. In contrast, classical electromagnetism predicts that some portion of the incoming beam will always be reflected and some transmitted [3]. However, sending low intensity laser light through a beamsplitter and measuring the detection rates is insufficient to show that light is quantized. Such an experiment cannot distinguish between the classical and quantum mechanical predictions because lasers output light in bunches which look very similar to classical waves. Therefore, instead of a laser we use as our source a birefringent β-Barium Borate (BBO) crystal, which converts input photons to pairs of correlated photons. In this arrangement, observations at either side of the beamsplitter are conditional upon an observation of a correlated photon, forcing the photons into individual states rather than a superposition of states [4].

The photon detection experiment can be modified to observe single photon interference by directing one of the photons in each downconverted pair through an interferometer. By placing a detector outside the interferometer and measuring the coincidence rates as a function of the difference in path length between the arms of the interferometer, we can observe an interference pattern for single photons. This pattern can be destroyed by tagging the photon polarizations perpendicular to each other depending on which arm of the interferometer they pass through, thereby adding information which could be used to reconstruct the photon path. Finally, the interference can be regained by inserting a linear polarizer between the detector and the interferometer, effectively erasing the tagged polarization information [4, 5].

I. Single Photon Detection

Theory and Apparatus

Shining low intensity laser light on a beamsplitter and measuring uncorrelated detections between the transmitted and reflected beams may seem sufficient to show that light is quantized. However, this experiment would actually observe a correlation between the outputs of the beamsplitter [6]. Such a measurement would be consistent with a semiclassical theory in which the laser emits waves that divide at the beamsplitter, but the light falls on photodetector atoms with quantized energy levels, causing them to trigger randomly and independently [1, 4]. It would also be consistent with a quantum mechanical model in which light is composed of single-photon states, because laser light is a coherent superposition of photon number states with a Poissonian distribution [5]. The two theories give exactly the same predictions, so the experiment would not be able to determine if the laser beam consisted of photons or classical electromagnetic waves [6].

Instead of using laser light shined directly on a beamsplitter, a light source is needed that consistently produces light with a single photon number state. The source used in our experiment is a BBO crystal, which uses a process known as spontaneous parametric downconversion to convert high energy pump photons into a pairs of correlated low energy photons. The pairs exit the crystal at the same time at a conic angle of approximately 3° from the incident beam [4]. Theories describing downconversion claim that electrons in an asymmetric BBO crystal are driven into nonlinear oscillations when illuminated by a pump photon, resulting in the annihilation of the pump photon and the creation of two downconverted photons [7]. Energy conservation requires that a pump photon’s angular frequency is equal to the sum of the frequencies of the output photons, and momentum conservation requires that the wave vectors of the emitted photons sum to the wave vector of the pump photon [4].

Our final photon detection configuration is shown in Figure 1. In this setup, detectors B and B' are only measured conditional upon measurements at detector A. If light is quantized, we would therefore expect to observe correlations between A and B and between A and B', but not between B and B'.

Figure 1: Experimental schematic for photon detection. An 85 mW pump laser outputs light at a peak wavelength of 405 nm to the BBO crystal, which converts incident photons to correlated pairs of lower energy photons. Measurements of these downconverted photons are recorded at detectors B and B', conditional upon measurements at detector A. [Image: Pearson and Jackson, Figure 3].

In any coincidence counting experiment consisting of two detectors with the same pulse width τ, the total coincidence window is τ(c) = 2τ. If the detectors measure signals at rates R1 and R2, the accidental coincidence rate from two random and independent sources is given by [4]:

In keeping with the notation introduced by Grangier et al. in their 1986 paper on photon detection, we define an “anticorrelation parameter” α, which is the ratio of the probability of measuring a coincidence count to the product of the probabilities of measuring a count in each detector [2]. The anticorrelation parameter for two detectors can be expressed in terms of measurable count rates as the ratio of the measured coincidence rate to the accidental coincidence rate, where N is the number of counts at a particular detector. According to this definition, will be greater than one for correlated sources, one for uncorrelated sources, and less than one for anticorrelated sources [4]:

For our measurement of single photons, we perform a correlation test between three detectors. If NAB, NAB’, and NABB’ are the numbers of counts measured at AB, AB', and ABB', and RAB, RAB’, and RABB’ are the corresponding rates, then the anticorrelation parameter for three detectors is [4]:

A measurement of α(3d) ≥ 1 indicates that photons simultaneously reflect and transmit through the beamsplitter in accordance with classical electrodynamics, but a value close to zero indicates that photons are either reflected or transmitted but not both. However, the measured value cannot be exactly zero, because there will always be some rate of accidental coincidences for the Poissonian-distributed photon observations within a given coincidence window [6].

Procedure and Results

Our first step before confirming the existence of photons was to measure counts from direct downconversion without the beamsplitter was to verify that the beams were correlated. We began by aligning the beam from an ultraviolet 405 nm pump laser and inserting the BBO crystal. The downconverted light, of wavelength 810 nm, had an unfiltered intensity of approximately one million counts per second. Measuring a 3° angle on the optical bench and mounting the A and B fiber-coupled detectors at the appropriate locations along a rail, we roughly placed the detectors in the paths of the invisible downconverted light beams. At this point, photon counts could be observed with a single photon counting module (SPCM). These counts were then maximized by adjusting the horizontal and vertical detector angles, the height and position of the detectors, and the angle of the BBO crystal. Repeating the placement procedure for detector B, we completed the alignment by maximizing the coincidence counts between it and detector A. Theoretically, the two detector anticorrelation parameter for this configuration would have been infinite because downconverted photons are perfectly correlated. However, due to a nonzero coincidence window, accidental coincidences at a rate proportional to will always be measured, resulting in lower values of for longer coincidence windows. In our setup, we observed values for as high as 24.4 ± 0.2. This value is significantly greater than one, indicating a strong correlation between the downconverted photons.

After completing the initial test for photon correlation, we performed the three-detector experiment, adding a polarizing beamsplitter so that detector B received transmitted light while detector B' received reflected light. A half-wave plate was also used to control the intensities of the two components. Repeating a similar placement and adjustment procedure as before, we observed counts at all three detectors and measured for downconverted light.

Our data for the single photon experiment, shown in Table 1, consisted of count rates handled by the SPCM. This module contained four silicon avalanche photodiodes that converted the weak current signal from the detectors to measurable voltage pulses. The pulses and coincidences from the SPCM were counted by a NEXYS 3 FPGA board, using adjustable coincidence windows. Data from this board was then sent via a serial adapter to a computer where the information was processed and displayed in LabVIEW.

Table 1: Two sets of three detector data. The first set was taken with a low attenuation neutral density filter in front of the laser and a bandpass filter in front of detector A, while the second set was taken with high attenuation neutral density filter in front of the laser and no filter in front of detector A.

We recorded two sets of data – one with a low attenuation neutral density filter in front of the laser and a bandpass filter in front of detector A, and one with a high attenuation filter in front of the laser and no filter in front of detector A. Our best result, α(3d) = 3.3 ± 1.8, is approximately 54 standard deviations smaller than the classical prediction of α(3d) ≥ 1. As in the initial test for correlation, lengthening the coincidence window increased the rate of accidental coincidences, but this time the effect was to increase the value of α(3d), since the rate of accidental coincidences is in the numerator for α(3d) and the denominator for α(2d). Our measurement of α(3d) was reduced by a factor of two when we reduced the number of counts into the detectors, consistent with the expectation that the number of accidental coincidences is proportional to the square of the detector count rates and linearly proportional to the coincidence rates.

II. Single Photon Interference

Theory and Apparatus

To observe interference patterns produced by single photons, we used a setup similar to the one for photon detection, except that we directed the single photons into a Mach-Zehnder interferometer, consisting of two half-silvered beamsplitters as shown in Figure 2. By adjusting mirror M2, a pattern of interference fringes can be observed at detector B, indicating that the photons somehow behave as waves, as if they are sampling both paths. A subtle difference between this setup and the previous experiment is that in this case the two beams are recombined before observation, so that we cannot know along which of the two possible paths the photon has travelled.

Figure 2: The addition of an interferometer to the setup shown in Figure 1 allows observation of interference pattern, showing that the quantized photons also have wave-like properties. [Image: Adapted from Pearson and Jackson, Figure 4].

In this experiment, the probability of a detection at detector B for the open cavity configuration follows a sinusoidal form, corresponding to an interference pattern [4]:

This prediction seems to imply that light in one path of the interferometer interferes with light from the other path. However, since the path length of the interferometer was approximately 30 cm, each photon took about one nanosecond to pass through. We recorded data with a coincidence window of 10 ns, so every time a coincidence was registered between A and B there was a dead time of 10 ns, guaranteeing that two photons entering the interferometer at the same time would not be registered. Any measured interference would therefore be self-interference from single photons, even though the previous experiment showed that photons cannot simultaneously transmit and reflect through a beamsplitter.

In our setup, all photons are initially polarized perpendicular to the plane of the BBO crystal, but by inserting a half-wave plate in each arm of the interferometer, we can tag the photons so that they have different polarizations depending on their path. Since the beamsplitters are non-polarizing, the path of the photons could then be reconstructed by examining their polarizations. This extra information forces the photons into a particular polarization state and destroys the interference pattern, with a detection probability for detector B that can be shown to be one half [4].

By inserting an external linear polarizer between the photodetector and beamsplitter in an arrangement called a quantum eraser, all photons leaving the interferometer are given an identical polarization, removing the polarization information added by the half-wave plates [5]. This polarizer acts as a filter that passes photons of a particular polarization without rotating the polarization. It introduces a superposition of horizontal and vertical states, and the interference pattern returns, even though the orientation of the half-wave plates has not been altered. For optimal interference, we align the polarizer at 45Ëš between both half-wave plates. Assuming a perfectly efficient polarizer, the detection probability at detector B will therefore be [4]:

As shown in this equation, the amplitude of the interference will be reduced by at least a factor of two. In reality, the polarizer is not perfectly efficient and is not aligned exactly with the half-wave plates, so the amplitude will be further reduced.

Procedure and Results

To observe the interference of individual photons, we used an interferometer consisting of a non-polarizing beamsplitter aimed at two adjustable mirrors, with a second beamsplitter at the intersection point of the two paths. Because downconverted light is less coherent than laser light, the path length difference between the two interferometer arms had to be shorter than the coherence length of the light, approximated as a function of the bandwidth Δf of the filters used in the setup [8]:

For our experiment, the coherence length was approximately 65μm, so we had to align the interferometer to obtain a significantly smaller path length difference. We accomplished this by first adjusting the mirrors until we could see constructive and destructive fringes from a He-Ne alignment laser. Using a spectrometer, we were then able to resolve interference fringes in the blackbody spectrum of white light, as shown in Figure 3. At this point the ultraviolet pump laser was turned on and detector B was placed at one of the interferometer outputs. After optimizing the coincidences between detectors A and B, we connected mirror M2 to a piezoelectric controller, which adjusted the mirror’s position so that the interference pattern could be plotted as a function of path length difference.

Figure 3: White light interference, showing fringes superimposed on the blackbody spectrum of an incandescent light bulb. Further adjustment resulted in fringes large enough to be seen with the naked eye.

We automated our interference data acquisition with LabVIEW, adjusting the voltages of the piezoelectric controller at 0.2 V intervals between 0 and 7 V. At each voltage level, 10 data sets were recorded, each of which used 10 ns coincidence windows averaged over 5 seconds. Because the coincidence rates tended to fluctuate for several seconds after adjusting the path length, we excluded from our analysis the first 5 data sets at each voltage. The final measurement of the coincidence rate at a particular voltage was thus taken as the average of the remaining five data points, with an uncertainty approximated as the standard deviation of the measurements.

Ideally, we would have recorded our data as a function of path length differences measured with a strain gauge. However, software for communication with the strain gauge was unavailable, so we were forced to use manual strain gauge readings to calibrate the path length differences. Although the plot of the calibration shown in Figure 4 appears to show a directly proportional relationship, a quadratic fit was needed to obtain an acceptable reduced χ2 value. In fact, even the quadratic approximation was not sufficient because higher order systematics still showed up in the residual plots, resulting in poor fits for the interference data. It was necessary to take the calibration to third order to sufficiently absorb the systematics and give a significantly improved sinusoidal interference fit.

Figure 4: Plot of the calibration of path length difference as a function of piezoelectric controller voltage. The fitted polynomial was: Δx = -0.0039(10) + 0.2035(18) V + 0.01214(83) V^2 - 0.00097(11) V^3. Also shown are the residuals from this fit, indicating that there were still some systematics at higher voltages.

Once the path length difference had been calibrated as a function of voltage, interference data was recorded for the open cavity, tagged, and erased configurations, and each data set was fitted to a sinusoid. However, as shown in Figure 5, the geometry of a Mach-Zehnder interferometer requires a correction of

in the measured displacements to account for the angle θ between the incident light and the translational axis of the mirror. Our translational axis was oriented at an angle of 45° to the beam, so in this case the path length correction was √(2)Δx. Any propagated uncertainty in Δl was small and could be safely ignored.

Figure 5: Close-up of the translational mount for mirror M2 of the Mach-Zehnder interferometer, illustrating how the path length difference is a function of Δx and θ.

Final results for our open cavity, tagged, and erased data are shown along with their sinusoidal fits in Figure 6. Because the ratio of the period of an interference pattern to the wavelength of the light is expected to be one, these results are plotted as a function of Δl/λ [4].

Figure 6: The rate of AB coincidence counts as a function of the path length difference between the interferometer arms normalized to the wavelength of light.

Figure 7: Residual plots from the sinusoidal fits of open cavity and erased interference data. The residual plot for the tagged data was evenly distributed and is not shown. The open cavity residuals follow a fairly even distribution, but the erased residuals reveal obvious systematics in the fit.

The equations for the sinusoidal fits of the open cavity and erased data were found to be:

As shown by the reduced χ2 values and residual errors for the open cavity and erased fits, plotted in Figure 7, the fits were not statistically ideal. Although no signal was expected for the tagged data, a low amplitude interference pattern was still observed because the two half-wave plates were not exactly perpendicular to each other. As for the large phase shift between the open cavity and erased data, it was probably caused by the linear polarizer as the light crossed the boundary between media of different indices of refraction.

The most perplexing phenomena seen in our data were the variations in period, since the normalized fringe width was expected to be one. Although the period of the open cavity fit remained constant to within 1.8σ of the expectation, the erased period was much longer and increased proportionally to Δl, as seen in the plot of the residuals in Figure 7. One explanation for these fluctuations was that the path length difference would have been better calibrated as a function of the sum of a sinusoid and a polynomial, because the position of mirror M2’s translational stage was adjusted by the rotation of a screw that may have been imperfectly aligned. However, this was not the cause, because although the same calibration was used for all three interference configurations, only the erased data showed significant variations in period. Another idea was that the erased data had some component of the low amplitude tagged interference superimposed onto it. However, a fit to the sum of two sinusoids was not better than the fit to a single sinusoid, and the Fourier transform of the erased data showed a peak at only one frequency. Therefore, the most reasonable explanation was that the variations in period came from a combination of several mechanical systematics, including a calibration of path length difference that did not remain constant over time and interference fringes which drifted in location over the course of the data runs. Indeed, although the interference measurements were repeated several times, the period variations did not appear to be consistent. We found that coincidence rates recorded at a particular voltage level drifted gradually with time, and that the position measured by the strain gauge sometimes tended to increase by a few nanometers per minute for a given piezoelectric controller voltage, an effect which may have been caused by electrostatic relaxation in the piezoelectric mount.

#SecConclusions

Conclusions

Our results for single photon detection and interference were qualitatively acceptable. However, several improvements to our procedure could be considered for future experiments. For the single photon detection procedure, an improvement would be to mount the detectors on three-axis stages to allow for more methodical adjustments of the detector positions. For the interference experiment, an improvement would be to record the strain gauge data directly, instead of calibrating the path length differences as a function of voltage. Future experiments might also consider offsets in the coincidence data caused by accidental coincidences and dark currents in the SPCM.

A future experiment based upon our work would be to perform a test of local realism, showing that even though the photon pairs emitted by a BBO are spatially separated, they are entangled in a single coherent state. For downconverted light, local realism would assume that the photon pairs have a distinct location and well-defined polarization. However, a modification of the single photon detection setup could be used to demonstrate a quantum mechanical contradiction of local realism [9]. By changing the polarization angle and phase of two orthogonal BBO crystals, measurements of pairs of coincidences lead to an inequality which contradicts the predictions of local realism. Such a violation implies that neither photon has a well-defined polarization before measurement, and that an observation of the polarization of one photon instantaneously determines the polarization of the other [10, 11].

Other extensions to our work would include experimenting with the coherence length and time correlation of downconverted photons. To investigate coherence length, the interferometer path lengths could be distorted so that the interference disappears. Because downconverted photons are entangled, it would be interesting to test whether filtering the photons at detector A would affect the coherence of the twin photons and restore the interference. To investigate the effects of offsetting the photon correlations, a long segment of fiber optic cable could be added to one of the detectors, effectively extending the time for light to reach the photodetector. In this case, we would expect to no longer observe photon coincidences, because the two photons would not arrive at the photodetectors within a single coincidence window.

Although there is room for improvement in our measurements, in this experiment we have demonstrated the wave-particle duality of light, showing that photons are quantized by measuring coincidence counts between photodetectors at the ends of a beamsplitter, and showing that light behaves as a wave by sending the same photons through an interferometer and observing an interference pattern. We have also further demonstrated the quantum mechanical behavior of light by tagging the photon polarizations, adding path information that destroyed the interference pattern. Inserting a linear polarizer erased the tagged polarizations and removed the path information, allowing interference to reappear.

#SecReferences

References

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#SecAcknowledgments

Acknowledgments

We would like to thank Kurt Wick for his advice and Alex Card for his technical assistance.