Let us consider the problem of two back-to-back photons in a total spin zero state. Measurements of this system are sensitive to the basic rules of QM. See, for example, the work of J.G. Rarity for a more detailed discussion of actual experiments. Other topics that are often connected in these discussions include, Bells Theorem and EPR.

In any review of Quantum Mechanics the questions of interpretation arises. There are two types of questions:

 

There are many articles written on the subject of QM. This short review will list a few points that help address the questions.

 

Some important points:

  1. Complete orthonormal basis.

 

 

States will be written as sums over basis vectors.

 

 

  1. Only amplitudes of the same basis vector interfere.

Consider a two state system with basis |1>, |2>

 

 

These states will interfere as follows:

 

 

The probability of finding the final state in the condition |1> is

 

 

Two quantum states will not interfere unless they have common amplitudes for particular basis vectors. For the case that a photon is emitted in the two slit descriptions of Feynman, the basis might be:

 

 

The states cannot interfere with the state because they are orthonormal. The basis states above need to be developed as a two state basis (composite system) as treated below.

 

  1. Two systems that interact or that make up a composite quantum state are treated as the product of the separate states as follows:

 

System 1 basis |a>

System 2 basis |ά>

 

Composite System |a>|ά>

 

As you build states from the above basis:

 

 

One finds that in general the final composite systems cannot be decomposed into a product of the form:

 

 

The two systems can become entangled.

 

The often treated two particle system is the two photons emitted back to back in a total spin zero state.

 

 

When the photons are separated by a significant distance the entanglement means that an interaction of photon 1 with another system may be dependent on photon 1 polarization.

 

 

 

 

 

Remember that a photon can be represented just like the Electric field vector for a traveling E&M wave. One can break the field into two components perpendicular to the motion then specify the x and y components. One can also use a complex coefficient, which rotates the field around the direction of motion. The two components can then be resolved as a right handed or left handed spinning field. Similarly the QM photon has either linear polarization or circular polarization. For a photon represented as a spin 1 helicity state |1,1>, |1,-1> the field rotates occupying all orientation in the two dimensional plane. Notice that having a component in the x direction merely specifies that the amplitude of a sine wave at its maximum value and that the field oscillates through + to orientation. Two photons with 180o phase difference but the same polarization can cancel so that the overall Electric field (polarization) is zero. Two x component photons therefore can combine to provide 0 spin and two y component photons can combine to provide 0 spin. [See http://beige.ovpit.indiana.edu/B679/node70.html for a more extended derivation of the relationship between linear and circular.]

 

Authors describing the two-photon systems often choose linear polarization states. An apparatus similar to the Stern-Gerlach type filter that Feynman describes can be made for photons. Although magnetic fields cannot be used to split the photon into its two basis states, there are mirrors that transmit one polarization but reflect the opposite polarization (beam splitter cubes often reflect and transmit orthogonal polarization). One could separate and then recombine using optical elements. Whatever the mechanism we imagine that photon 1 enters the apparatus, splits based on polarization, travels both paths (A and B) and reassembles to the identical QM state at the output. If a path is blocked (e.g. the y component path) then the emerging particle is polarized 100% in the x direction. Not all photons will emerge from a blocked filter but some will.

 

For the two photon state, we can represent the the |0,0> overall spin 0 state as:

 

 

Spin 0 is always a symmetric combination of the two polarization states no matter what coordinate axes are chosen (x-y or xy).

 

Now if the |Ξ> state finds a filter (system 3 with path A and B) it will add an additional component to the wf |3,0>. The system for transporting the photon along two paths is labeled as 3 and its state is a neutral state labeled 0. Assume that the original system |X1> is written with respect to a filter oriented at an angle θ.

 

 

Since the photon travels through system 3 without changing the state it contributes just a common factor and does not become entangled in the quantum state. Of the four terms (1,3) travel along path A and (2,4) along B. If path A is blocked.

 

 

The photon2 state must be in the same polarization at photon 1. The probability for observing this is of course . An arbitrary polarization state for photon 1 will be measured to be |1Y> half of the time. When you observe the photon leaving you can say with 100% confidence that photon 2 is also polarized in this direction |2Y>. Let us look at this problem a different way. Imagine that there is a small amount of gas along path A and that the photon with an amplitude α to Compton or Raleigh scatter so that system 3 changes. The state now becomes |3,1> rather than |3,0>. This is an interesting case because it is similar to slit electron diffraction with a light source to observe the slit. It also perhaps is easier to see that the photon that travels down A is still part of the system but that it is now in an orthogonal state due to the presence of a scattered part of system 3. The above example illustrated the effect of the orientation of the filter with respect to the initial choice of basis. It would simplify the problem if we simply chose to express the state with respect to that orientation.

 

Note these final states are entangled and one cant factor out system 3. This forces the quantum states to be labeled by the presence of a changed system 3. One can consider the case α=1, β=0. This is the same as blocking the (a) path since the final state always contains a new state |3,1> and is therefore orthogonal.

 

 

In this situation if you measure the result of scattering |3,1> you get one polarization state for photon 2 and if there is no change in the measurement apparatus (system 3) you get another.

 

When considering the implications of quantum mechanics it is sometime confusing to sort and arrange the various ideas. Critical to the understanding is the assertion that composite systems behave by multiplying the states of each system as shown above. When the new composite states are formed they cannot always be written as a separated product of two isolated systems. The entangled states are those that cant be written in the form |1,A> |2,α> where all the pieces of the 1 state can be collected together and multiplied by all the pieces of the 2 state. The two-photon state shown above is, of course, an example of entanglement. The QM predictions for these states are somewhat counterintuitive. A measurement of photon 1 somehow determines the state of photon 2. Without the measurement tests on photon 2 would conclude that the photon was in an arbitrary state of polarization. Any Polaroid filter would see a 50-50 split at all orientations. The QM wave function used to describe photon 2 would be 1/√2 (|2X> + |2Y>) as far as observers could tell. Somehow however should observers of photon 1 ever compare notes with observers of photon 2 they would find 100% correlation between the photons.