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A linear polarizer is an anisotropic flat plate with a transmission axis in its plane. In the classical domain, in which light behaves as a transverse electromagnetic wave, linearly polarized light passes undiminished through an ideal polarizer if its axis is parallel to the electric field in the wave. However, no light is transmitted if the electric field and polarizer directions are perpendicular. The meaning of “no light” can be extended to the quantum realm, where it means “no photons” are transmitted. Thus it is possible to regard the polarizer as a filter for quantum states.

In Figure 3, light travels along the experiment’s axis of symmetry, shown here as the z-axis, with a linear polarizer in the xy plane. Capital letter X will denote the photon state transmitted by a polarizer aligned along the x-axis, and similarly Y for the y-axis. The general polarization state Ψ for a single photon is a linear combination, or coherent mixture, of these states: Ψ = a x X + a y Y , (1) where a_x and a_y are amplitudes (in general, complex) that tell how much of each state is present in the admixture.

Quantum theory provides two ways for the state of a system to change in time:

(1) The Time-dependent Schrödinger Equation is central to quantum mechanics. Solving it yields a wave function or state vector Ψ(t) that describes the system and its continuous evolution between measurements.

(2) The Reduction Postulate determines how Ψ changes when a measurement is made. A measurement leads to a sudden collapse, or projection, of Ψ onto the observed state.

The state Ψ for a photon therefore changes discontinuously as a result of a polarization measurement. If the photon described by Eq. 1 is detected after passing through a polarizer with its transmission axis along x, that constitutes a measurement. In this case the reduction postulate requires that the Y term must drop out. Only the observed-state X term remains in Ψ after the measurement.

The experiment in Figure 2 explores the reduction postulate in a two-photon system.

Photon spin states for light traveling in the z-direction can be expressed in terms of linear polarization states X and Y (as above), or in terms of helicity. The two sets of basis states are related as follows (with normalization factors not shown):

Spin parallel to photon momentum: Positive helicity   Ψ + = X + i   Y

Spin antiparallel to photon momentum: Negative helicity   Ψ − = X   −   i   Y

Conservation laws for angular momentum and parity play a central role in quantum correlation phenomena. For the three-level radiative cascade in Figure 1, the initial atomic state A and final state C both have zero total angular momentum and even parity. Therefore, the two-photon final state Ψ must also satisfy the conditions of zero angular momentum and even parity: Ψ = Ψ 1 + · Ψ 2 + + Ψ 1 ‐ · Ψ 2 ‐ . (2)

If the same z-axis is used for both photons and is directed to the right in Figure 2, the states for Photon 1 require i → - i. Then Ψ = X 1   − i   Y 1 ·   X 2 + i   Y 2 + X 1 + i   Y 1 ·   X 2   −   i   Y 2 , (3) in which the X₁·Y₂ and Y₁·X₂ terms drop out, yielding (without normalization factor) a simple and elegant result Ψ = X 1 · X 2 + Y 1 · Y 2 (4) for the two-photon system, before either photon is detected. The reasoning in Eqs 2, 3, leading to Eq. 4, gives it a sense of universality, as no consideration is given to the internal structure of the source atom or its interaction with a quantized radiation field. Before any measurements have been made, each photon has a potential to pass through a linear polarizer with any orientation.

The two-particle quantum state Ψ is not a simple product, as it would be for photons having no common history. Instead it is a sum of products representing an entangled state. Entanglement of the photons is evident in Eq. 4, where neither photon has an independent identity. In each of the two terms, the amplitude for one of the photons is a wave function for the other.

Since the orientation for the x- and y-axes around z is arbitrary, the form of Eq. 4 will remain unchanged if the xy coordinate system is rotated through any angle about the z-axis.

If the first photon (green) passes through a linear polarizer transmitting the state X₁, the reduction postulate removes the second term, containing Y_1, from Ψ in Eq. 4. Only the first term remains, leaving the second photon (violet) unambiguously in polarization state X₂. More generally, if one of the photons passes through a linear polarizer at any orientation, the remaining photon will then be in the same polarization state, pending future measurements.

Quantum theory makes specific predictions for the experiment shown in Figure 2.

(1) If both polarizers are aligned with their axes parallel, coincidence counts will be observed.

These predictions may seem counterintuitive, bizarre, or weird, especially because there is no known evidence for physical transmission of information from one detector to the other. This question is addressed further in Section 6.

Additional remarks:

(a) Taken separately, the green and violet beams are unpolarized.

A photomultiplier detector is an evacuated and sealed glass tube with a light-sensitive cathode on a window at one end. As Einstein first realized, the energy of a detected photon is conveyed to a single photoelectron from the cathode. This electron is accelerated toward a positively charged metal dynode, where additional electrons are knocked loose. This process is repeated at additional dynodes, producing a negative pulse that can be counted with standard electronics. Quantum efficiencies (output pulse probability per photon) are of order 10% (green) to 20% (violet). Photoelectrons released from different locations on the cathode travel a range of distances in reaching the first dynode, introducing some loss of time resolution, typically several nanoseconds. In addition, thermal processes can release electrons randomly from the cathode, resulting in spurious output pulses, or “dark noise.”

The oven, shown in Figure 4, is 6.5 cm in length and machined from tantalum, a nonreactive refractory metal. It is heated by an electric current through internal resistive coils. A cylinder of calcium metal is loaded into the well. When the oven is installed in a vacuum chamber and heated, monatomic calcium vapor evaporates from the solid and comes out through an opening in the front, forming a beam.

A thermocouple junction, set into a small hole in the oven, reads out the temperature. At 1000 K, an oven load of Ca (12 g) will empty in about 25 h, and a typical beam velocity for a Ca atom is about 10⁵ cm/s. Since the radiative cascade requires about 10^–8 s, the atom moves only about 10^–3 cm—a negligible distance—while the photons are being emitted.

The oval at the center of Figure 2, labeled “Ensemble of Atoms,” represents a cross section of the atomic beam from this oven.

The two-stage cascade, as shown in Figure 1, requires excitation of calcium atoms from the 4S ground state to the 6S excited state. This is a challenging problem, since the direct 4S → 6S transition is “forbidden” for single-photon absorption. (Dipole matrix elements are zero.) An acceptable alternative would be to optically excite the 6P state (also shown in Figure 1) by an allowed transition, 4S → 6P. The 6P can decay to 6S by emitting an infrared photon that is not observed, and then the desired cascade can take place. The 4S → 6P excitation requires a 228 nm ultraviolet light source.

A minor complication is that while the 6P state can decay to 6S, it can also decay to 5S and several D-states (in total about 8 times as likely as 6S). All of these return to the ground state via the 4P state, producing a Photon 2 not time-correlated with a Photon 1. Detector pulses from unpaired violet photons can trigger false coincidences and are a source of noise.

In 1964 it was a major challenge to find an acceptable 228 nm UV excitation source for the 4S → 6P transition. No tunable lasers, UV lasers, or UV LEDs were available. (There were also no pocket calculators and no lab computers. It was still the slide rule era.)

A calcium discharge lamp could not be considered as a 228 nm source, as it would produce intense 423 nm (violet) radiation that could not be effectively blocked from reaching the Photon 2 detector. Electron impact excitation would pose a similar problem. A third possibility was a continuum source of ultraviolet light, in conjunction with a 228 nm bandpass interference filter. A high-pressure mercury lamp was considered, but even this produced far too much visible light.

Then I read about the UV continuum emitted by molecular hydrogen, with wavelengths spanning the range from about 180 nm to 450 nm. No suitable lamps were commercially available, so I designed and built a cylindrical low-voltage high-current H₂ arc lamp in a brass chamber, using a porous tungsten dispenser cathode and a continuous flow of H₂ gas. (Kocher, 1967b). This turned out to be essential to the eventual success of the experiment. The lamp operated at 17 V, 30 amps, with the discharge produced between the cathode and anode in the cross-sectional view of Figure 5. Fused quartz transmits 228 nm radiation, so a quartz focusing lens is mounted between the lamp and the excitation region.

If the broadband UV light were applied perpendicular to the Ca beam, only atoms within the natural linewidth (about 30 MHz) for the 4S → 6P transition could be excited, and all the useful radiation would be absorbed near the edge of the atomic beam. Atoms beyond this edge would not be excited. However, there is a spread in atomic velocities from a thermal oven, and the Doppler-broadened linewidth (1,000 MHz) exceeds the natural linewidth by a factor of about 30. In the experimental plan I therefore introduced the calcium beam at 45° relative to the observation z-axis, from lower left to upper right in Figures 2 and 6. With this configuration the much larger number of atoms in the Doppler-broadened absorption line can potentially be excited to the 6P state. The oblique angle between the atomic beam and the detector axis also effectively eliminates trapping and multiple scattering of the emitted 423 nm violet photons.

It would not be wise to proceed with a complex experiment unless the signal and noise levels can be estimated. Therefore, before the major construction of a vacuum chamber and dealing with pumps, hoses, ion gauges, etc., an effort was made to calculate the coincidence counting rate under reasonable experimental conditions.

For each detector I considered the fractional solid angle of intercept, together with the quantum efficiency and the filter transmission, and found that about 10⁶ cascade-emitting atoms are needed for each observable coincidence count—without the polarizers.

The atomic beam oven holds about 10²³ atoms of Ca, but only 1 atom in 10³ would pass through the excitation region.

I used a radiation thermopile to measure the intensity of the H₂ arc lamp in conjunction with a 228 nm bandpass filter. I also searched the literature for transition rates in calcium and determined the branching ratios for the transitions.

It was a complex process putting these pieces together, attempting to identify every limitation and concern. In the end I estimated 1 coincidence per second (with the polarizers removed), with an uncertainty factor of about 5.

Under these conditions, reasonable statistics—and a clear experimental result—might be obtained with a multi-hour observation. It would be a difficult undertaking, requiring considerable care, patience, a long observation time, and some courage.

The experimental plan employs a vacuum system with two diffusion-pumped brass chambers and removable flanges for access. A water-cooled source chamber holds the calcium beam oven. The excitation chamber, pumped to a lower pressure (10^–6 Torr), contains the interaction region, where UV light from the H₂ lamp would excite Ca atoms in the atomic beam, and from which the green and violet photons would be detected by photomultiplier assemblies outside the chamber.

Details of the experiment are shown in Figure 6 and in a photograph, Figure 7.

It took more than a year to reach the point where all the parts of the experiment could be assembled. The components were tested separately, to the extent possible, and then in tandem.

Two flags, shown in Figure 6, can be controlled from outside the chamber. One can block the calcium beam, and the other can block the UV radiation from the lamp so it cannot reach the beam. This flexibility made it possible to monitor and optimize the counting rate for each detector separately. It was then possible to determine the sources of extraneous coincidence counts, of which there were many, including stray light from the oven heating coils and visible-light fluorescence due to the UV from the lamp. I installed light-blocking baffles and applied lampblack to the chamber walls. The improvements were slow and incremental.

After this was done, photomultiplier “dark noise,” which had always been present, became noticeable at room temperature. To address this problem, I cooled the photomultipliers by soldering a helix of copper tubing around each brass photomultiplier enclosure and installing a refrigeration compressor that could circulate refrigerant through the tubing. The photocathode temperatures were cooled to −15°C, reducing the dark noise significantly.

Instead of using a simple coincidence circuit, I recorded coincidence counts versus the time delay between the pulses from the two detectors, using a time-to-pulse-height converter and a multichannel pulse-height analyzer, as in Figure 8.

The time-to-height converter produces an output pulse with an amplitude proportional to the time delay between the “start” pulse (from the Photon 1 detector} and the “stop” pulse (from the Photon 2 detector). A pulse-height analyzer stores these counts in an array of magnetic-core memory channels corresponding to a span of delay times. Each memory channel effectively represents a separate coincidence circuit, for which the time spread can be varied by changing the ramp rate. The time offset, corresponding to sliding the distribution to the right or left, can be adjusted by varying a delay line (a length of coaxial cable) on the Photon 2 side.

If the pulse pairs are from the same atom, they contribute to a central peak in the distribution, as viewed on an oscilloscope. Pulse pairs may also be due to photons from different atoms, or to stray light. In these cases the time intervals are random, contributing to a background signal, with fluctuating statistics, along the entire horizontal time scale.