Interferometer 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 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 1/\sqrt{n}} , based on the heuristic inequality 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 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 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} :

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.

Sensitivity limit for Mach-Zehnder interferometer

Previously, we've defined the unitary transform for a quantum beamsplitter as being a rotation about 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 \hat{y}} axis, so as to avoid having to keep track of factors 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 i} . For variety, let's now use a different definition; nothing essential will change. Let the 50/50 beamsplitter transformation 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 B = \frac{1}{\sqrt{2}} \left[ \begin{array}{cc}{1}&{-i}\\{-i}&{1}\end{array}\right] \,. }

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

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 \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

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

The Mach-Zehnder transform is thus

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} U &=& B P B \\ &=& \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] \\ &=& -i e^{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 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} , 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, 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 = ( 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 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} is thus

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

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 X = a^\dagger a - b^\dagger b } , and $Y=a^{\dagger }b+b^{\dagger }a$. Recognizing 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 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 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} .

Ideally, the output signal should go as $\cos \phi$. The signal due to 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} 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:

Note that if our goal is to maximize measurement sensitivity to changes in $\phi$, 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 $d\langle x{\rangle }/d\phi$ 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 $\phi$, derived from the observable 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} ? By propagating uncertainties, this is

${\langle }\Delta \phi ^{2}\rangle ={\frac {{\langle }\Delta M^{2}{\rangle }}{\left|{\frac {\partial \langle M{\rangle }}{\partial \phi }}\right|^{2}}}\,,$

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 $M=Y$, 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} .

Limit for coherent state input

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 $n=|\alpha |^{2}$ as the input state mean photon number. Also,

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 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 $\sim n$, 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 $Y$, 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 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 a^\dagger b b^\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

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 = |\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 $c$ 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 d} .

Limit for single photons

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:

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

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\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 $P$ 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 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 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}} 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.