Interferometry and metrology

Quantum states and dynamics of photons

Interferometry and metrology

Suppose you are given a phase shifter of unknown $\phi$:

\noindent How accurately can you determine $\phi$, given a certain time, and laser power? In this section, we consider this basic measurement problem, and show how the usual shot noise limit can be exceeded by using quantum states of light, reaching a quantum limit determined by Heisenberg's uncertainty principle. This limit is achieved using entanglement, which can be realized using entangled multi-mode photons, or by a variety of squeezed states. The physics behind such quantum measurement techniques generalizes to a wide range of metrology problems, but a common challenge the need to reduce loss.

Shot noise limit

The Poisson distribution of photon number in coherent (laser) light contributes an uncertainty of $\Delta n\sim {\sqrt {n}}$ to optical measurements. It is therefore reasonable to anticipate that with $n$ photons, the uncertainty $\Delta \phi$ with which an unknown phase $\phi$ can be determined might be bounded below by $\Delta \phi \geq 1/{\sqrt {n}}$, based on the heuristic that $\Delta n\Delta \phi \geq 1$. Such a limit is known as being due to shot noise, arising from the particle nature of photons, as we shall now see rigorously. Consider a Mach-Zehnder interferometer constructed from two 50/50 beamsplitters, used to measure $\phi$:

\noindent Let us analyze this interferometer, first by using a traditional quantum optics approach in the Heisenberg picture, and second by using single photons in the Schrodinger picture. Previously, we've defined the unitary transform for a quantum beamsplitter as being a rotation about the ${\hat {y}}$ axis, so as to avoid having to keep track of factors of $i$. For variety, let's now use a different definition; nothing essential will change. Let the 50/50 beamsplitter transformation be

$B={\frac {1}{\sqrt {2}}}\left[{\begin{array}{cc}{1}&{-i}\\{-i}&{1}\end{array}}\right]\,.$

This acts on $[a,b]^{T}$ to produce operators describing the output of the beamsplitter; in particular,

$\left[{\begin{array}{c}{a-ib}\\{b-ia}\end{array}}\right]=B\left[{\begin{array}{c}{a}\\{b}\end{array}}\right]\,.$

Similarly, the phase shifter acting on the mode operators performs

$P=\left[{\begin{array}{cc}{1}&{0}\\{0}&{e^{i\phi }}\end{array}}\right]\,.$

The Mach-Zehnder transform is thus

${\begin{array}{rcl}U&=&BPB\\&=&{\frac {1}{2}}\left[{\begin{array}{cc}{1}&{-i}\\{-i}&{1}\end{array}}\right]\left[{\begin{array}{cc}{1}&{0}\\{0}&{e^{i\phi }}\end{array}}\right]\left[{\begin{array}{cc}{1}&{-i}\\{-i}&{1}\end{array}}\right]\\&=&-ie^{i\phi /2}\left[{\begin{array}{cc}{\sin(\phi /2)}&{\cos(\phi /2)}\\{\cos(\phi /2)}&{-\sin(\phi /2)}\end{array}}\right]\,.\end{array}}$

The way we have defined these transformations here, the output modes of the interferometer, $c$ and $d$, are

$\left[{\begin{array}{c}{c}\\{d}\end{array}}\right]=U\left[{\begin{array}{c}{a}\\{b}\end{array}}\right]\,.$

We are interested in the difference between the photon numbers measured at the two outputs, $M=(d^{\dagger }d-c^{\dagger }c)/2$, where the extra factor of two is introduced for convenience. We find

${\begin{array}{rcl}c^{\dagger }c&=&\sin ^{2}{\frac {\phi }{2}}a^{\dagger }a+\cos ^{2}{\frac {\phi }{2}}b^{\dagger }b+\sin {\frac {\phi }{2}}(a^{\dagger }b+b^{\dagger }a)\\d^{\dagger }d&=&\cos ^{2}{\frac {\phi }{2}}a^{\dagger }a+\sin ^{2}{\frac {\phi }{2}}b^{\dagger }b-\sin {\frac {\phi }{2}}(a^{\dagger }b+b^{\dagger }a)\end{array}}$

The measurement result $M$ is thus

$M=(a^{\dagger }a-b^{\dagger }b)\cos \phi -(a^{\dagger }b+b^{\dagger }a)\sin \phi \,.$

Define $X=a^{\dagger }a-b^{\dagger }b$, and $Y=a^{\dagger }b+b^{\dagger }a$. Recognizing that $X$ is the difference in photon number between the two output arms, and recalling that this is the main observable result from changing $\phi$, we identify the signal we wish to see as being $X$. Ideally, the output signal should go as $\cos \phi$. The signal due to $Y$ goes as $\sin \phi$, and we shall see that this is the noise on the signal. The average output signal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle X{\rangle}} , as a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , looks like this:

\noindent Note that if our goal is to maximize measurement sensitivity to changes in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , then the best point to operate the interferometer at is around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=\pi/2} , since the slope Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \langle x{\rangle}/d\phi} is largest there. At this operating point, if the interferometer's inputs have laser light coming into only one port, then the outputs have equal intensity; thus, the interferometer is sometimes said to be "balanced" when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=\pi/2} . What is the uncertainty in our measurement of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , derived from the observable $M$? By propagating uncertainties, this is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\langle}\Delta\phi^2 \rangle = \frac{{\langle}\Delta M^2{\rangle}} {\left|\frac{\partial \langle M{\rangle}}{\partial\phi}\right|^2} \,, }

where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \langle M{\rangle}}{\partial\phi} = -X \sin\phi - Y\cos\phi \,. }

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=\pi/2} , such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=Y} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \partial M/\partial\phi = -X} . For a coherent state input, $|\psi _{in}\rangle =|\alpha {\rangle }|0{\rangle }$, we find

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \langle X \rangle &=& \langle 0|{\langle}\alpha|( a^\dagger a - b^\dagger b )|\alpha{\rangle}|0{\rangle} \\ &=& |\alpha|^2 \\ &=& n \,, \end{array}}

if we define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = |\alpha|^2} as the input state mean photon number. Also,

${\begin{array}{rcl}\langle Y\rangle =\langle 0|{\langle }\alpha |(a^{\dagger }b+b^{\dagger }a)|\alpha {\rangle }|0\rangle =0\,.\end{array}}$

This is consitent with our intuition: the signal should go as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim n} , and the undesired term goes as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , so it is good that is small on average. However, there are nontrivial fluctuations in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} , because

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle Y^2 \rangle = \langle a^\dagger b a^\dagger b + a^\dagger b b^\dagger a + b^\dagger a b^\dagger a + b^\dagger a a^\dagger b{\rangle} \,, }

and $\langle a^{\dagger }bb^{\dagger }a\rangle =\langle a^{\dagger }a(1+b^{\dagger }b){\rangle }$ is nonzero for the coherent state! Specifically, the noise in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} is

$\langle Y^{2}\rangle =|\alpha |^{2}=n\,,$

and thus the variance in the measurement result is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\langle}\Delta M^2 \rangle = \langle Y^2 \rangle - \langle Y{\rangle}^2 = |\alpha|^2 = n \,. }

From Eq.(\ref{eq:l7-dphi}), it follows that the uncertainty in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is therefore

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\langle}\Delta\phi \rangle = \frac{\sqrt{{\langle}\Delta M^2{\rangle}}} {\left|\frac{\partial \langle M{\rangle}}{\partial\phi}\right|} = \frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}} \,. }

This is a very reasonable result; as the number of photons used increases, the accuracy with which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} can be determined increases with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{n}} . The improvement arises because greater laser power allows better distinction between the signals in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and $d$. Another way to arrive at the same result, using single photons, gives an alternate interpretation and different insight into the physics. As we have seen previously, acting on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |01{\rangle}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |10{\rangle}} "dual-rail" photon state, a 50/50 beamsplitter performs a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_y(\pi/2)} rotation, and a phase shifter performs a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_z(\phi)} rotation. The Mach-Zehnder interferometer we're using can thus be expressed as this transform on a single qubit:

\noindent where the probability of measuring a single photon at the output is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . Walking through this optical circuit, the states are found to be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} |\psi_1 \rangle &=& \frac{|0{\rangle}+|1{\rangle}}{\sqrt{2}} \\ |\psi_2 \rangle &=& \frac{|0{\rangle}+e^{i\phi} |1{\rangle}}{\sqrt{2}} \\ |\psi_3 \rangle &=& \frac{1-e^{i\phi}}{2} |0 \rangle + \frac{1+e^{i\phi}}{2} |1{\rangle} \,, \end{array}}

such that

$P={\frac {1+\cos \phi }{2}}\,.$

Repeating this Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} times (so that we use the same average number of photons as in the coherent state case), we find that the standard deviation in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is

$\Delta P={\sqrt {\frac {p(1-p)}{n}}}={\frac {\sin \phi }{2{\sqrt {n}}}}\,.$

Given this, the uncertainty in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\phi = \frac{\Delta\phi} {\left| \frac{dP}{d\phi} \right|} = \frac{1}{\sqrt{n}} \,. }

This is the same uncertainty as we obtained for the coherent state input, but the physical origin is different. Now, we see the noise as being due to statistical fluctuations of a Bernoulli point process, one event at a time. The ${\sqrt {n}}$ noise thus comes from the amount of time the signal is integrated over (assuming a constant rate of photons). The noise is simply shot noise.

Heisenberg limit: entanglement

The shot noise limit we have just seen, however, is not fundamental. Here is a simple argument that something better should be possible. Recall that the desired signal at the output of our Mach-Zehnder interferometer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X= a^\dagger a - b^\dagger b } , and the noise is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y= a^\dagger b+ b^\dagger a} . If the inputs have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^\dagger a =n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^\dagger b =0} , and if $Y$ were zero, then the measured signal would be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\cos\phi} . And at the balanced operating point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi=\pi/2} ,

${\frac {|\Delta m|}{\Delta \phi }}=n\sin \phi \leq n\,.$

Thus, if the smallest photon number change resolvable is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta m=1} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\Delta\phi \geq 1} , from which it follows that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \phi \geq \frac{1}{n} \,. }

This is known as the "Heisenberg limit" on interferometry. There are some general proofs in the literature that such a limit is the best possible on interferometry. It governs more than just measurements of phase shifters; gyroscopes, mass measurements, and displacement measurements all use interferometers, and obey a Heisenberg limit. The argument above only outlines a sketch for why Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/n} might be an achievable limt, versus $1/{\sqrt {n}}$; it assumes that the noise Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} can be made zero, however, and does not provide a means for accomplishing this in practice. Many ways to reach the Heisenberg limit in interferometry are now known. Given the basic structure of a Mach-Zehnder interferometer,

\noindent one can consider changing the input state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi_{in}{\rangle}} , changing the beamsplitters, or changing the measurement. Common to all of these approaches is the use of entangled states. How entanglement makes Heisenber-limited interferometry possible can be demonstrated by the following setup. Let us replace the beamsplitters in the Mach-Zehnder interferometer with entangling and dis-entangling devices:

\noindent Conceptually, the unusual beamsplitters may be the nonlinear Mach-Zehnder interferometers we discussed in Section~2.3. They may also be described by simple quantum circuits, using the Hadamard and controlled-{\sc not} gate; for two qubits, the circuit is

\noindent Note how the output is one of the Bell states. For three qubits, the circuit is

\noindent This output state, $|000{\rangle }+|111{\rangle }$ (suppressing normalization) is known as a GHZ (Greenberger-Horne-Zeilinger) state. Straightforward generalization leads to larger "Schrodinger cat" states Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |00\cdots0{\rangle}+|11\cdots 1{\rangle}} , using one Hadamard gate and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} controlled-{\sc not} gates. Note that the reversed circuit unentangles the cat states to produce computational basis states. The important feature of such $n$-qubit cat states, for our purpose, is how they are transformed by phase shifters. A single qubit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}+|1{\rangle}} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}+e^{i\phi}|1{\rangle}} . Similarly, two entangled qubits in the state $|00{\rangle }+|11{\rangle }$, when sent through two phase shifters, becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |00{\rangle}+e^{2i\phi}|11{\rangle}} , since the phases add. And Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} qubits in the state $|00\cdots 0{\rangle }+|11\cdots 1{\rangle }$ sent through $n$ phase shifters becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |00\cdots 0{\rangle}+e^{ni\phi}|11\cdots 1{\rangle}} . When such a phase shifted state is un-entangled, using the reverse of the entangling circuit, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} controlled-{\sc not} gates leave the state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |00\cdots 0{\rangle}[ |0 \rangle + e^{ni\phi}|1 \rangle ]} , where the last Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-1} qubits are left in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0{\rangle}} , and the first qubit (the qubit used as the control for the {\sc cnot} gates) is

${\frac {|0\rangle +e^{ni\phi }|1{\rangle }}{\sqrt {2}}}\,.$

Compare this state with that obtained from the single qubit interferometer, Eq.(\ref{eq:l7-1qubitphase}); instead of a phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} , the qubit now carries the phase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\phi} . This means that the probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} of measuring a single photon at the output becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{1+\cos(n\phi)}{2} \,. }

The standard deviation, from repeating this experiment, on average, would be

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta P = \sqrt{P(1-P)} = \frac{\sin(n\phi)}{2} \,. }

Using $dP/d\phi =-n\sin(n\phi )/2$, we obtain for the uncertainty in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta\phi = \frac{\Delta\phi} {\left| \frac{dP}{d\phi} \right|} = \frac{1}{{n}} \,, }

which meets the Heisenberg limit.

Squeezed light interferometry

{{#lst:Squeezed light interferometry|content}}

Sensitivity to loss

Entangled states, while very useful for a wide variety of tasks, including interferometry and metrology, are unfortunately generally very fragile. In particular, entangled photon states degrade quickly with due to loss. Consider, for example, the two-qubit state $|00{\rangle }+|11{\rangle }$ (suppressing normalization). If one of theses photons goes through a mostly-transmitting beamsplitter, then the photon may be lost; let us say this happens with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} . If a photon is lost, the state collapses into one with one remaining photon, say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |10{\rangle}} . This is a product state -- no longer entangled. It is not even a superposition. No photon is lost with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\epsilon)} . Even worse, if both modes suffer potential loss of a photon, then no matter whcih mode looses a photon, the entangled state collapses; this happens even if just one photon is lost. Thus, the state retains some entanglement only with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\epsilon)^2} . And worst of all, if we have an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -photon cat state $|00\cdots 0{\rangle }+|11\cdots 1{\rangle }$, and all modes are subject to loss Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} , then useful entanglement is retained only with probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\epsilon)^n} . Due to such loss, the phase measurement uncertainty of an entangled state interferometer will go as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\langle}\Delta\phi \rangle \sim \frac{1}{n(1-\epsilon)^n} \,, }

which is clearly undesirable. Some physical systems, however, naturally suffer very little loss, and can keep entangled states intact for long times. Photons unfortunately do not have that feature, but certain atomic states, such as hyperfine transitions, can be very long lived. Thus, many of the concepts derived in the context of quantum states of light, actually turn out to be more useful when applied to quantum states of matter.