Modeling quantum interference (Part 2)

Diagram 1: Mach-Zehnder interferometer

In my previous post I described how to model a Mach-Zehnder interferometer by taking into account all paths that a photon can take and calculating the amplitudes. In this post, I will elaborate on the mathematical model further to create a visual sense of what is going on.

The interferometer and the photon travelling within it can be considered together as a quantum system. At the beginning of the experiment the system has a single quantum state that contains all the information about the system. This quantum state has a complex value associated with it called an amplitude which can be visualized as an arrow that can rotate around a center point (like a clock hand or compass arrow).^[1]

In our model, the amplitude for the initial quantum state is 1. As the photon travels toward the first beam splitter, the quantum state continually changes and this change is reflected in the amplitude. Using the clock analogy, the clock hand is continually rotating as the photon travels. When the photon arrives at the beam splitter, the quantum state is transformed into two distinct quantum states which are in superposition - one representing a reflected photon and one representing a transmitted photon. Each quantum state has its own amplitude as determined by the photon interaction with the beam splitter. There are now three distinct quantum states. One state represents a photon travelling on the upper (green) path, one state represents a photon travelling on the lower (red) path and the global state that now represents those two component states in superposition.

To help conceptualize this, we can think of each quantum state as representing a real system. So we can refer to the photon on the green path, the photon on the red path, and the global photon that is in superposition, with each referent indexed to a distinct quantum system.^[2]

Mathematically, the complex amplitude that represents a quantum state is a vector and it allows us to model the superposition of the two quantum states. Consider an arrow pointing north-east. This can be thought of as a vector that is the combination of two basis vectors - one pointing north and one pointing east. Suppose the arrow is 1 unit long. The arrow therefore has a length of 1/√2 in the north direction and a length of 1/√2 in the east direction (per the Pythagorean theorem). So we can represent this as a linear equation using bra-ket notation:

  |north-east> = 1/√2|north> + 1/√2|east>

In the same notation, when the photon passes through the first beam splitter the new quantum state is:

  |blue_aftersplitter> = -1/√2|green> + 1/√2|red>

This captures all the information about the system at this point. |blue_aftersplitter> is the main quantum state that is in superposition with an amplitude of 1 (implied). The state for the green (reflected) path has an amplitude of -1/√2 and the state for the red (transmission) path has an amplitude of 1/√2. See diagram 2 below for the path amplitudes.

Diagram 2: Path amplitudes

Note that the squares of the two component amplitudes sum to 1. If a measurement were performed at this point (i.e., the detectors were placed on those two path segments), there would be an equal probability of finding the photon in either state.^[3]







Evolving from the initial (blue) quantum state to the superposition state after the beam splitter requires a transformation operation which is represented by a matrix. The matrix required for the beam splitter is:^[4]

  bs = [1/√2  1/√2] = 1/√2[1  1]
       [1/√2 -1/√2]       [1 -1]

The beam splitter has two input ports - one at the front (indicated by the dot) and one at the rear. The matrix columns describes the rear port and front port behavior respectively. The matrix rows describe the transformations on the transmitted beam and reflected beam respectively. (For example, the bottom-right cell describes the transformation on a photon that enters the beam splitter through the front port and is reflected.)

Since the photon initially passes through the front port, the initial (blue) state is described by the following vector which specifies an amplitude of 0 for the photon passing through the rear port and an amplitude of 1 for the photon passing through the front port:^[5]

  |blue> = [0]

           [1]

We can now see what happens when the initial (blue) state is transformed by the beam splitter matrix (i.e., when the vector is multiplied by the matrix).

  bs|blue> = 1/√2[1  1][0] = 1/√2[0*1 +  1*1] = 1/√2[ 1] = [ 1/√2]

                 [1 -1][1]       [0*1 + 1*-1]       [-1]   [-1/√2]

The top cell of the resulting vector describes the transmitted (red) state amplitude and the bottom cell describes the reflected (green) state amplitude. That is, the amplitude for the global (blue) state is now distributed between two component states (red and green). This can be equivalently expressed as:

  bs|blue> = 1/√2(-|green> + |red>)

           = -1/√2|green> + 1/√2|red>

To explain the above equation, we know the front port behavior is described by the second column of the bs matrix. Then the bottom cell applies to the reflected beam state (which, in this case, is green) so it is multiplied by -1. The top cell applies to the transmitted beam state (which, in this case, is red) so it remains unchanged. Finally, the 1/√2 coefficient applies to both states so they are both multiplied by it.

The transformation matrix required for the two mirrors is:

  mi = [-1  0]

       [0  -1]

which inverts the phase of both states (i.e., multiplies each state by -1). The second beam splitter has the same matrix as the first. Note that, in this case, a beam will pass through the rear of the beam splitter as well as a beam through the front.

The entire evolution from the initial quantum state is:

  bs mi bs|blue>
    = bs mi (-|green> + |red>)/√2
    = bs (-(-|green) + -(|red>))/√2
    = bs (|green - |red>)/√2
    = ((|detector1> + |detector2>) - (-|detector1> + |detector2>))/2
    = (|detector1> + |detector2> + |detector1> - |detector2>)/2
    = (2|detector1> + 0|detector2>)/2
    = |detector1>

  P(|detector1>) = |1|² = 1 = 100%

Thus the photon always ends up at detector 1. To explain the mathematics, the photon in the initial (blue) state passes through the front of the beam splitter so its reflected beam (green) is inverted. Both beams are then inverted by the mirrors. Finally, the green state itself becomes a superposition of two states representing the photon heading towards each detector. Similarly for the red state. But note that the photon in the red path passes through the front of the beam splitter, so its reflected beam to detector 1 is inverted. The detector 2 states destructively interfere (since they are indistinguishable) while the detector 1 states constructively interfere (since they are also indistinguishable) which finally results in a single quantum state at detector 1 with an amplitude of 1.

Note that it is also possible to insert a phase shifter into one of the paths (emulating a sample or change in path length) which will change the probability of the photon arriving at each detector. The transformation matrix for a phase shifter in the lower (red) path is:

  ph(φ) = [e^iφ 0]

          [0   1]

where φ is the phase angle. A phase shift of 180° means the photon will always arrive at detector 2 while a phase shift of 90° means the photon will be found at either detector with equal probability. To illustrate this, adding a phase shift of 90° gives:

  ph(90°) = [e^i*pi/2 0] = [i 0]

            [0      1]   [0 1]

The state evolution is now:

  bs mi ph(90°) bs|blue>
    = bs mi ph(90°) (-|green> + |red>)/√2
    = bs mi (-|green> + i|red>)/√2
    = bs (-(-|green) + -(i|red>))/√2
    = bs (|green - i|red>)/√2
    = ((|detector1> + |detector2>) - i(-|detector1> + |detector2>))/2
    = (|detector1> + |detector2> + i|detector1> - i|detector2>)/2
    = (1+i|detector1> + 1-i|detector2>)/2
    = 0.5+0.5i|detector1> + 0.5-0.5i|detector2>

  P(|detector1>) = |0.5+0.5i|² = 0.5 = 50%

  P(|detector2>) = |0.5-0.5i|² = 0.5 = 50%

One final interesting effect occurs when two separate (but otherwise indistinguishable) photons are directed into each port of the first beam splitter simultaneously. There are four possible combinations: (1) the front port photon is reflected and the rear port photon is transmitted, (2) both photons are reflected, (3) both photons are transmitted and (4) the front port photon is transmitted and the rear port photon is reflected. This results in a superposition of four states as represented below:

  1/√2(-|greengreen> - |greenred> + |redgreen> + |redred>)

However the two states with a photon on each path are actually the same state since the photons are indistinguishable. And, since their amplitudes sum to 0, they destructively interfere. Thus, if a measurement were performed at this point by adding detectors, there would be an equal probability of finding either two photons on the green path or two photons on the red path but never one photon on each path.^[6]

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[1] The complex plane is analogous to a clock face or compass face. The real number line is horizontal, with 3pm or East representing the number 1 and 9pm or West representing the number -1. The imaginary number line is vertical, with 12am or North representing the imaginary number i and 6am or South representing the imaginary number -i. The origin is the center point which is 0.

[2] The global photon in superposition could be considered an abstraction similar to a university that is an abstraction over its distinct buildings or campuses.

[3] This is in accordance with the Born rule. The probability that the photon will be observed on a particular path is given by the square of the amplitude.

[4] This is known as the Hadamard matrix. It also represents the Hadamard gate in quantum computing which can be used to transform a qubit into a superposition state.

[5] The vector can be regarded as a qubit that has been prepared in state |1> (i.e., by directing the photon towards the front port of the beam splitter). The beam splitter transforms the qubit in state |1> into the superposition 1/√2(|0> - |1>) which later results in the selection of both ports of the second beam splitter.

[6] This is the Hong-Ou-Mandel effect.

Modeling quantum interference

Diagram 1: Mach-Zehnder interferometer

In this post, I'm going to model a device that exhibits quantum behavior in a simple but striking way.

The device pictured at the left is called a Mach-Zehnder interferometer. The beam splitter splits a beam of light into two paths. 50% of the beam is reflected towards mirror 1 and 50% of the beam is transmitted towards mirror 2. When the beams reach the second beam splitter, each beam is split again and is reflected or transmitted towards the detectors.

Intuitively, it would seem that half the light should end up at detector 1 and half at detector 2.^[1] However, assuming the two paths are the same length, all the light actually ends up at detector 1 at the right and none at the top detector.

This result is due to quantum interference at the second beam splitter where light heading towards the top detector destructively interferes and light heading towards the right detector constructively interferes. In true quantum style, this result always occurs even if only a single photon of light is emitted towards the first beam splitter.

In quantum mechanical terms, the photon is in a superposition of travelling along both paths simultaneously. At the second beam splitter, each path forms a further superposition (again with one path reflecting and one transmitting - see the four arrows heading towards the detectors in the diagram). The two paths heading towards detector 2 destructively interfere (i.e., they are 180° out of phase) and thus cancel each other out. Whereas the two paths heading toward detector 1 constructively interfere and so the photon is always detected there.

So how does the device actually work? The mathematics is actually fairly straightforward. The basic strategy is to model each path that the photon can take and combine identical paths at the end. Each path segment has a complex value associated with it called an amplitude which can be visualized as an arrow that can rotate around a center point (like a clock hand).^[2]

The initial (blue) path amplitude is 1 (see Diagram 2 below which specifies the calculated amplitudes for each path segment). The basic rule at the beam splitter is that the path splits into two paths and each path takes the amplitude of the source path value and multiplies it by 1/√2 (this is the normalization condition - the squares of the amplitudes in a superposition of paths must always sum to 1, i.e., 1/2 + 1/2 = 1).^[3] Also, the path of the reflected beam additionally multiplies the amplitude by -1 which represents a phase change of 180°. So the upper (green) path has an amplitude of -1/√2 (-0.707) and the lower (red) path has an amplitude of 1/√2 (0.707).

(Note: If the photon passes through the rear of the beam splitter, the result is the same except that the phase change rule does not apply.^[4] This is the case for the upper beam path when it reaches the second beam splitter. The front of each beam splitter is indicated by the dot.)

At each mirror, the amplitude for each path is multiplied by -1 (i.e., a phase change of 180°). So the upper path now has a value of 1/√2. The lower path now has a value of -1/√2. At the second beam splitter, the upper (reflection) path itself splits into reflection and transmission paths toward the two detectors. The upper beam reflection path value is 0.5 (1/√2 * 1/√2) and the upper beam transmission path value is also 0.5 (1/√2 * 1/√2). The lower beam reflection path value is 0.5 (-1/√2 * 1/√2 * -1) and the lower beam transmission path value is -0.5 (-1/√2 * 1/√2).

Diagram 2: Path amplitudes

This is where the quantum magic happens. The upper beam reflection path and the lower beam transmission path coincide. They are both directed towards detector 2. So the paths merge and the amplitudes are added to give a value of 0 (0.5 + -0.5) which is destructive interference. Similarly, the upper beam transmission path and the lower beam reflection path also coincide. They are both directed towards detector 1. So the paths merge and the amplitudes are added to give a value of 1 (0.5 + 0.5) which is constructive interference.

The probability of finding the photon at a particular detector is given by the amplitude squared, which is 100% for detector 1. Thus the photon always ends up at detector 1.

Note that this result depends on the physical configuration of the interferometer. In this case, the two paths between the beam splitters are the same length. However if the length of one of the paths is changed, the results also can change such that the photon is instead always found at detector 2 (i.e., change a path phase by 180° by multiplying by -1 and recalculate the subsequent path values), or found at either detector with equal probability (i.e., change a path phase by 90° by multiplying by the imaginary number i and recalculate), or any other probabilistic combination.

For further interferometer fun, see Part 2.

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[1] If the second beam splitter were removed, the light would be distributed between both detectors. In the case of one emitted photon, the photon would be observed at one detector or the other with 50% probability. No interference between the photon paths would occur since the paths are different (one is directed towards detector 1 and one is directed towards detector 2 when they cross).

[2] The amplitude actually continually changes as the photon travels (i.e., the arrow rotates). To simplify the example, the path segments are of lengths that are multiples of the wavelength. So a photon that leaves the beam splitter with a particular phase angle will have the same phase angle when it arrives at the mirror. Also, the top path and lower path are the same length.

[3] This is in accordance with the Born rule. The probability that the photon will be observed on a particular path is given by the square of the amplitude.

[4] There is a phase change for a reflection at a surface with a higher refractive index which is true at the front of the beam splitter (the glass refracts more than the air the photon is travelling in) but not at the rear (where the photon is already travelling in the glass before it reflects).