Quantum theory of observation/Examples of measurements

Observation of quantum superpositions with the Mach-Zehnder interferometer


This classical experiment is one of the simplest to actually observe quantum superpositions.

A beam of light encounters a first balanced beam splitter which splits it into a transmitted beam and a reflected beam. On the trajectories of these two beams, mirrors are placed so as to make them meet on a second beam splitter. On the trajectories of the beams transmitted and reflected by this second plate, photon detectors are placed.

A very simplified model of a beam splitter consists in assigning to the incident photon two quantum states $$ | 0 \rangle $$ and $$ | 1 \rangle $$ which correspond to the two perpendicular directions according to which it may encounter the beam splitter. If transmitted, it retains its direction and remains in the same state. If it is reflected it switches to the other state. It is very simplified, of course, but it is enough to account for the experiment.

With this simplification, a balanced beam splitter can be described by a Hadamard gate. It is a one-qubit gate defined by:

$$ H |0 \rangle = \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle) $$

$$ H | 1 \rangle = \frac {1} {\sqrt 2} (| 0 \rangle - | 1 \rangle) $$

Between the two beam splitters the evolution of the photon is described by a simple phase shift, which depends on the length of the followed path:

$$ U_ {int} | 0 \rangle = e ^ {i \delta_0} | 0 \rangle $$

$$ U_ {int} | 1 \rangle = e ^ {i \delta_1} | 1 \rangle $$

The propagation of the photon in the interferometer is thus described by the operator:

$$ U_ {MZ} = H U_ {int} H $$

If $$ | 0 \rangle $$ is the initial state of the photon its state when leaving the second beam splitter is:

$$U_{MZ}|0\rangle = \frac{1}{2}[(e^{i\delta_0}+e^{i\delta_1})|0\rangle + (e^{i\delta_0}-e^{i\delta_1})|1\rangle ]$$

If the difference between $$ \delta_0 $$ and $$ \delta_1 $$ is a multiple of $$ \pi $$ the output path of a photon is always the same. One can therefore predict with certainty which detector it will encounter. This conclusion is surprising because it obliges us to suppose that a photon follows at the same time the two intermediate paths between the two plates. If it took only one of these paths one could not predict its way out.

A detector preceded by a beam splitter is therefore capable of detecting a quantum superposition. In the Mach-Zehnder experiment, one of the detectors detects the photons whose intermediate state is $$ \frac {1} {\sqrt 2} (| 0 \rangle + | 1 \rangle)$$, the other detects the photons whose intermediate state is $$ \frac {1} {\sqrt 2} (| 0 \rangle - | 1 \rangle) $$. Such detection is not an ideal measurement because the photon is destroyed when detected.

In the Mach-Zehnder interferometer, the first beam splitter prepares a photon in a non-localized quantum superposition, the second beam splitter followed by a detector enables us to observe this superposition.

When the optical paths are equal, we can ignore the effect of $$ U_ {int} $$:

$$ U_ {MZ} = H ^ 2 = Id $$

That the Hadamard gate is its own inverse summarizes the principle of Mach-Zehnder interferometry.

An ideal measurement: the CNOT gate
The CNOT gate is a quantum gate (Nielsen & Chuang 2010) with two qubits determined by the following evolution operator:

$$ CNOT | 00 \rangle = | 00 \rangle $$

$$ CNOT | 01 \rangle = | 01 \rangle $$

$$ CNOT | 10 \rangle = | 11 \rangle $$

$$ CNOT | 11 \rangle = | 10 \rangle $$

The state change of the second qubit is controlled by the state of the first qubit. That is why they are called the target qubit and the control qubit.

$$ | 00 \rangle $$ is an abbreviated form of $$ | 0 \rangle \otimes | 0 \rangle $$.

If the initial state of the target qubit is $$ | 0 \rangle $$ or $$ | 1 \rangle $$ (but not a superposition of the two) then the CNOT gate makes an ideal measurement of the control qubit by the target qubit. The target qubit of a CNOT gate is the most simple example of a quantum detector. The pointer states are the states $$ | 0 \rangle $$ and $$ | 1 \rangle $$ of the target qubit. The measurement results are $$ 0 $$ and $$ 1 $$. If the detector's initial state is $$ | ready \rangle =| 0 \rangle $$ then the eigenstate of the detected qubit associated with the result $$ i $$ is $$ | i \rangle $$.

A non-ideal measurement: the SWAP gate
The SWAP gate is a quantum gate with two qubits determined by the following evolution operator:

$$SWAP|00\rangle = |00\rangle$$

$$SWAP|01\rangle = |10\rangle$$

$$SWAP|10\rangle = |01\rangle$$

$$SWAP|11\rangle = |11\rangle$$

Each qubit passes in the state of the other. Each one is therefore sensitive to the other. This is why the SWAP gate can be interpreted as a quantum measurement of one qubit by the other. For example, the second qubit can be interpreted as a detector for which we have chosen the initial state $$ | ready \rangle = | 0 \rangle $$ and the pointer states $$ | 0 \rangle $$ and $$ | 1 \rangle $$. The measurement results are $$ 0 $$ and $$ 1 $$ which respectively have as eigenstates the states $$ | 0 \rangle $$ and $$ | 1 \rangle $$ of the first qubit. If the initial state of this qubit is $$ \alpha | 0 \rangle + \beta | 1 \rangle $$, the final state after the measurement is:

$$SWAP(\alpha|0\rangle + \beta|1\rangle)|ready\rangle = SWAP(\alpha|00\rangle + \beta|10\rangle) = \alpha|00\rangle + \beta|01\rangle)$$

The SWAP gate is not an ideal measurement, because one of the eigenstates of the observed system is disturbed by the measurement.

Experimental realization of quantum gates
The interaction of an atom with a cavity enables us to construct quantum gates.

One can make a qubit with an atom. For example, the state $$ | 0 \rangle_A $$ is defined as a state $$ | g \rangle_A $$ of the atom, the ground state or an excited one, and $$ | 1 \rangle_A $$ is another excited state $$ | e \rangle_A $$. Similarly, a cavity can be used as a qubit. The state $$ | 0 \rangle_C $$ is its state empty of photons, the state $$ | 1 \rangle_C $$ is its state with a single photon. If the atom passes through the cavity in such a way that they exchange their energy, the following interaction can be obtained (Haroche & Raimond, 2006, p.282):

$$ U | 0 \rangle_C | 0 \rangle_A = | 0 \rangle_C | 0 \rangle_A $$

$$ U | 0 \rangle_C | 1 \rangle_A = | 1 \rangle_C | 0 \rangle_A $$

$$ U | 1 \rangle_C | 0 \rangle_A = - | 0 \rangle_C | 1 \rangle_A $$

$$ U | 1 \rangle_C | 1 \rangle_A = | 1 \rangle_C | 1 \rangle_A $$

It is not the SWAP gate but it looks like it. For example,

$$ U | 0 \rangle_C (\alpha | 0 \rangle_A + \beta | 1 \rangle_A) = (\alpha | 0 \rangle_C + \beta | 1 \rangle_C) | 0 \rangle_A $$

$$ U (\alpha | 0 \rangle_C + \beta | 1 \rangle_C) | 1 \rangle_A = | 1 \rangle_C (\alpha | 0 \rangle_A + \beta | 1 \rangle_A) $$

The exchange of energy can also lead to a change of state:

$$ U | 1 \rangle_C (\alpha | 0 \rangle_A + \beta | 1 \rangle_A) = (- \alpha | 0 \rangle_C + \beta | 1 \rangle_C) | 1 \rangle_A $$

$$ U (\alpha | 0 \rangle_C + \beta | 1 \rangle_C) | 0 \rangle_A = | 0 \rangle_C (\alpha | 0 \rangle_A - \beta | 1 \rangle_A) $$

By modifying the parameters of the experiment, and with another excited state $$|i\rangle_A$$ of the atom, it is also possible to arrange for the passage of the atom through the cavity to be described by (pp.320-322):

$$ U '| 0 \rangle_C | g \rangle_A = | 0 \rangle_C | g \rangle_A $$

$$ U '| 0 \rangle_C | i \rangle_A = | 0 \rangle_C | i \rangle_A $$

$$ U '| 1 \rangle_C | g \rangle_A = - | 1 \rangle_C | g \rangle_A $$

$$ U '| 1 \rangle_C | i \rangle_A = | 1 \rangle_C | i \rangle_A $$

If we redefine the states $$ | 0 \rangle_A $$ and $$ | 1 \rangle_A $$ by:

$$ | 0 \rangle_A = \frac {1} {\sqrt 2} (| g \rangle_A + | i \rangle_A) $$

$$ | 1 \rangle_A = \frac {1} {\sqrt 2} (- | g \rangle_A + | i \rangle_A) $$

then the above-mentioned $$ U '$$ operator determines a CNOT gate, where the cavity is the control qubit and the atom the target qubit. In this way one is able to detect a photon without destroying it.

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