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| Diagram 1: Mach-Zehnder interferometer |
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.
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| Diagram 2: Path amplitudes |
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]
[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:
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.
= 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) = [eiφ 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°) = [ei*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]
[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.
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