Photons and statistics

Quantum states and dynamics of photons

We now turn to a study of the quantum properties of light, specifically, single-mode monochromatic light. We begin by considering the states in which quanta of the electromatic field, photons, may exist. We describe how these states are mathematically represented, and how they transform under simple physical operations, such as propagation through free space, and through optical beamsplitters. We also consider how two modes of light may relate to each other, in particular through entanglement, a purely quantum-mechanical property which can be a useful resource. Throughout this study of the quantum nature light, we develop an intuition and a language for quantum states and behaviors which may be applied not just to light, but also, to analogous states of matter.

Photons and statistics

What is quantum about light -- is it the wave behavior of electromagnetic waves? Or is it the particle behavior of photons? The truth is much deeper than either, and much more interesting! We begin with a brief review of the formalism of the simple harmonic oscillator, used to model single modes of light. Light from incandescent and other thermal sources can be described using this model. But more important to us will be coherent light from lasers, which is described by coherent states of light. These, and other more quantum states of light can be usefully depicted using a phase-space representation, as we demonstrate with $Q(\alpha )$ plots. Plots of thermal, coherent, and photon number states illustrate unique quantum properties which can lead to non-classical field fluctuations and statistical properties, particularly at the level of a single photon.

Starting point: electromagnetism

Recall that the energy of an electromagnetic field may be expressed as the Hamiltonian

$H={\frac {1}{2}}\int \,d^{3}r\left[{\epsilon _{0}|{\vec {E}}|^{2}+{\frac {1}{\mu _{0}}}|{\vec {B}}|^{2}}\right]\,,$

where ${\vec {E}}$ 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 \vec{B}} are the electric and magnetic fields, 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 \epsilon_0} 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 \mu_0} are the permitivity and permeability of free space. Second quantization of this Hamiltonian led to a new expression for this energy, in terms of quantum operators,

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 H = \frac{1}{2} (P^2 + \omega^2 Q^2) \,, }

where

${\begin{array}{rcl}P&=&i{\sqrt {\frac {\hbar \omega }{2}}}(a^{\dagger }-a)\\Q&=&{\sqrt {\frac {\hbar }{2\omega }}}(a^{\dagger }+a)\,,\end{array}}$

and the quantized electric field 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 \vec{E} = i \vec{e}\sqrt{\frac{\hbar\omega}{\epsilon_0 V}} \left[ { a e^{i(kr- \omega t)}- a^\dagger e^{-i(kr- \omega t)} } \right] \,, }

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 \vec{e}} is the polarization vector, 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 V} is the quantization volume. The bottom line of the QED Hamiltonian is that we may model a single mode of the electromagnetic field using this Hamiltonian:

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 H = \hbar\omega \left( { a^\dagger a + \frac{1}{2}} \right) \approx \hbar\omega a^\dagger a \,. }

This is the Hamiltonian of a simple harmonic oscillator. Everything we are interested in about the quantum properties of light will come from this Hamiltonian, and small perturbations of it!

Number and Thermal States

The simple harmonic oscillator Hamiltonian has eigenstates labeled 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{\rangle}} , where $H|n\rangle =h\hbar \omega |n{\rangle }$, such that the energy levels may be depicted as a ladder of equally spaced rungs starting at 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}} , as shown in this figure:

In terms of these energy eigenstates, known as number states, the Hamiltonian can be written 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 H = \hbar\omega |n \rangle \langle n|} . Note that the eigenstates are orthogonal: 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 n|m \rangle = \delta_{nm}} . 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 |n{\rangle}} is said to be 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 state. The ladder operators acting on these energy eigenstates, 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} 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 a^\dagger } , are known as the anihilation and creation operators, respectively. They satisfy the following properties, which are well worth remembering:

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} a|n \rangle &=& \sqrt{n} |n-1{\rangle} \\ a^\dagger |n \rangle &=& \sqrt{n+1} |n+1{\rangle} \\\, [a, a^\dagger ] &=& 1 \end{array}}

A good way to reconstruct these is to remember just 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 a|0 \rangle = 0} , 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 a^\dagger |0 \rangle = |1{\rangle}} . Light produced by thermal emission sources, such as resistive filaments, gas discharge, or other radiating black bodies, is a statistical mixture of number states, known as a thermal state, and sometimes called "chaotic light." This mixture is described by a density matrix characterizing a Boltzman distribution of number 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 \rho_{\rm th} = \frac{e^{-\beta H}}{\mathcal{Z}} = \frac{e^{-\frac{\hbar \omega n}{k_BT}}}{\mathcal{Z}} |n \rangle \langle n| \,, }

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 \beta=1/k_B T} , 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 k_B} is Boltzmann's constant, 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 T} is temperature, 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 {\mathcal{Z}} = \sum_n e^{-\frac{n\hbar\omega}{k_BT}} = \frac{1}{1+e^{-\hbar \omega /k_BT}} }

is the usual partition function normalization. This gives us

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 \rho_{\rm th} = \frac{e^{-n\hbar \omega /k_BT}}{1-e^{-n\hbar \omega /k_BT}} |n \rangle \langle n| \,. }

The mean number of photons in a thermal state is

${\bar {n}}={\rm {Tr}}(a^{\dagger }a\rho _{\rm {th}})=\sum _{n}{\frac {ne^{-n\hbar \omega /k_{B}T}}{1-e^{-n\hbar \omega /k_{B}T}}}={\frac {1}{e^{\hbar \omega /k_{B}T}-1}}\,.$

If we define the probability distribution 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_n^{th}} 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 \rho_{\rm th} = P_n^{th} |n \rangle \langle n|} , 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 P_n^{th} = \frac{\bar{n}^n}{(1+\bar{n})^{n+1}} \,. }

This distribution, together with some mode volume considerations, gives Planck's law of blackbody radiation. Note that $\sum _{n}P_{n}^{th}=1$, as expected for a properly normalized probability distribution. A quick calculation gives

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 \sum_n n(n-1) P_n^{th} = 2\bar{n}^2 \,, }

and thus the variance 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 n^2 \rangle = \langle n^2{\rangle}- \langle n{\rangle}^2 = \bar{n}^2+\bar{n}} . Constrast this with the Poisson distribution, for which average and variance are the same. Laser light, as is well known, has such a Poisson distribution of photon number; it is thus quite distinct from thermal light, from the standpoint of number statistics.

Coherent States and 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 Q(\alpha)} Representation

Definition & properties. A coherent state $|\alpha {\rangle }$ is defined to be an eigenstate of the anihilation operator 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} :

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|\alpha \rangle = \alpha |\alpha{\rangle} \,, }

with normalization 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}\alpha|\alpha \rangle = 1} . It is important to keep in mind 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 \alpha} is a complex number. In the number basis 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{\rangle}} , a coherent state has the representation

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 |\alpha \rangle = \sum_n |n \rangle \langle n|\alpha{\rangle} \,. }

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 c_n = \langle n|\alpha{\rangle}} . By definition, $|\alpha {\rangle }$ is an eigenstate 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 a} , and 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} a|\alpha \rangle &=& \sum_n c_n a|n{\rangle} \\ &=& \sum_n c_n \sqrt{n} |n-1{\rangle} \,. \end{array}}

This gives a recursion relation for 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_n} ; from this, and the normalization of $|\alpha {\rangle }$, it follows that

$|\alpha \rangle =e^{-{\frac {|\alpha |^{2}}{2}}}\sum _{n}{\frac {\alpha ^{n}}{\sqrt {n!}}}|n{\rangle }\,.$

In contrast to basis states usually used for Hilbert spaces, coherent states are not orthogonal to each other:

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}\beta|\alpha \rangle = e^{-\frac{|\alpha|^2 + |\beta|^2}{2} + \beta^*\alpha } \,. }

However, coherent states are often still useful as an {\em overcomplete} basis, taking advantage of this resolution of identity:

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{1}{2\pi} \int |\alpha{\rangle}{\langle}\alpha| \, d^2\alpha = { \mathcal I} \,. }

What physical state does the coherent state represent? The diagonal elements 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 |\alpha{\rangle}{\langle}\alpha| = \sum_n p_n |n \rangle \langle n|} have a Poisson distribution,

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_n = e^{-|\alpha|^2} \frac{\alpha^{2n}}{n!} \,. }

It follows from this that the average photon number ${\bar {n}}=\langle n\rangle ={\langle }\alpha |n|\alpha \rangle =|\alpha |^{2}$. Similarly, the variance in the photon number 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 \bar{n}} . This is distinct from the thermal state we studied above, and agrees with the well known Poisson statistics of photon number for laser light. The coherent state represents a mode of the electromagnetic field. For 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 \alpha = |\alpha|e^{i\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 |\alpha|} is its amplitude, 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 \phi} its phase. In fact, we will see that a coherent state provides an excellent quantum-mechanical model of classical states of light. Visualization: $Q_{\rho }(\alpha )$. A good way to visualize this coherent state (and other quantum states of light, as we shall see) is in terms of something known as "quasi-probability" distributions. Let us 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 Q_\rho(\alpha) = {\langle}\alpha |\rho |\alpha{\rangle} \,, }

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 \rho} is the density matrix of the state which we wish to visualize, 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 \alpha} is a complex number, 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 Q_\rho(\alpha)} may be represented by non-negative, real-valued two-dimensional plot. Note that $Q_{\rho }(\alpha )$ is normalized:

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 \int Q_\rho(\alpha) \,d^2\alpha = \pi \,, }

independent 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 \rho} . In this sense, we may interpret 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 Q_\rho(\alpha)/\pi} as a probability distribution. For example, consider 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 Q_\rho(\alpha)} plot for $\rho =|0\rangle \langle 0|$, the "vacuum" state. 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 Q_{|0{\rangle}}(\alpha) = |{\langle}\alpha|0{\rangle}|^2 = e^{-|\alpha|^2}} , a gaussian centered at the origin in 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 {\mathcal Im}(\alpha)} , 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 {\mathcal Re}(\alpha)} plane:

Another example is provided by the thermal 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 \rho_{th} = \sum_n P_n^{th}|n \rangle \langle n|} , for which

${\begin{array}{rcl}Q_{th}(\alpha )&=&\sum _{n}P_{n}^{th}|{\langle }\alpha |n{\rangle }|^{2}\\&=&e^{-|\alpha |^{2}}\sum _{n}{\frac {|\alpha |^{2n}}{n!}}P_{n}^{th}\,.\end{array}}$

Inserting Eq.(\ref{eq:thermal_prob}) for 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_n^{th}} , 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 Q_{th}(\alpha) = \frac{1}{1+\bar{n}} e^{-\frac{|\alpha|^2}{1+\bar{n}}} \,, }

This again is a gaussian, centered at the origin:

A third example is given by the coherent state itself; 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 \rho = |\beta{\rangle}{\langle}\beta|} . 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 Q(\alpha)} representation of this is

${\begin{array}{rcl}Q_{\beta }(\alpha )=|{\langle }\alpha |\beta {\rangle }|^{2}=e^{-|\alpha -\beta |^{2}}\,.\end{array}}$

This is a gaussian centered at 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 \beta} :

Recall that the Hamiltonian for a mode of light as it propagates 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 H = \hbar\omega a^\dagger a} . A coherent state thus evolves to become

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 e^{iHt/\hbar} |\alpha \rangle = e^{- \frac{|\alpha|^2}{2}} \sum_n \frac{\alpha^n}{\sqrt{n!}} e^{i n\omega t} |n{\rangle} = |\alpha e^{i\omega t}{\rangle} \,. }

In other words, a coherent state parameterized 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 \alpha} evolves with time to become a coherent state parameterized by $\alpha e^{i\omega t}$. In terms of 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 Q(\alpha)} representation, this means that the time evolution of a coherent state centered at some initial 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 \beta} is depicted as rotation about the origin, as the phase of its electric field evolves:

Note that this understanding allows us to visualize the thermal state as being a mixture of coherent states with random phase.

Fluctuations and Noise

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 Q(\alpha)} depiction of a state of light is particularly useful because it allows direct visualization of the statistical properties of quantum states versus the classical coherent states. Recall that the Heisenberg uncertainty principle restricts the product of fluctuations in two conjugate observables, such as position and momentum:

$\Delta P\,\Delta Q\geq {\frac {\hbar }{2}}\,.$

For a simple harmonic oscillator,

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 = i \sqrt{\frac{\hbar\omega}{2}} ( a^\dagger - a) ~~~~~~~~~~~ Q = \sqrt{\frac{\hbar}{2\omega}} ( a^\dagger + a) \,. }

and using the fact that for a coherent 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 |\alpha{\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 {\langle}\alpha|a|\alpha \rangle = \alpha} , and ${\langle }\alpha |a^{\dagger }|\alpha \rangle =\alpha ^{*}$, we find 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 \begin{array}{rcl} {\langle}\alpha|P|\alpha \rangle &=& i \sqrt{\frac{\hbar\omega}{2}} (\alpha^* - \alpha) \\ {\langle}\alpha|P^2|\alpha \rangle &=& \frac{\hbar\omega}{2} ({\alpha^*}^2 -2|\alpha|^2 + \alpha^2 - 1) \\ {\langle}\alpha|Q|\alpha \rangle &=& \sqrt{\frac{\hbar}{2\omega}} (\alpha^* + \alpha) \\ {\langle}\alpha|Q^2|\alpha \rangle &=& \frac{\hbar}{2\omega} ({\alpha^*}^2 +2|\alpha|^2 + \alpha^2 + 1) \,. \end{array}}

These give

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} \Delta Q = \sqrt{{\langle}\Delta Q^2{\rangle}} &=& \sqrt{\frac{\hbar}{2\omega}} \\ \Delta P = \sqrt{{\langle}\Delta P^2{\rangle}} &=& \sqrt{\frac{\hbar\omega}{2}} \end{array}}

so 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 P \,\Delta Q = \hbar/2} . Thus, the coherent state is a minimum uncertainty state of light. This distinction is apparent from the width of the coherent state in the $Q(\alpha )$ plot:

Another useful measure of the statistical properties of a state of light is provided by a measure known as the {\em second order temporal coherence} function, commonly denoted 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 g^{(2)}(\tau)} . The definition of this function, for a classical state of light described by a stationary intensity versus time distribution ${\bar {I}}(t)$, is

$g_{cl}^{(2)}(\tau )={\frac {{\langle }{\bar {I}}(t){\bar {I}}(t+\tau ){\rangle }}{{\langle }{\bar {I}}{\rangle }^{2}}}\,.$

The numerator of this expression is a familiar autocorrelation function; moreover, since $\langle ({\bar {I}}(t)-{\langle }{\bar {I}}(t)\rangle )^{2}\rangle \geq 0$, it follows that $g_{cl}^{(2)}(0)\geq 1$. For a state of light described by some density matrix $\rho$, we may write a similar expression for 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 g^{(2)}(\tau)} , but instead 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 \bar{I}(t)} , operators must be used, which act 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 \rho} . Letting $\langle x{\rangle }$ denote 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 {\rm Tr}(x \rho)} as usual, the quantum mechanical definition of the second order temporal coherence function 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 g^{(2)}(\tau) = \frac{ \langle a^\dagger a^\dagger a a{\rangle}}{ \langle a^\dagger a{\rangle}^2} = \frac{ \langle n^2{\rangle}- \langle n{\rangle}}{ \langle n{\rangle}^2} \,, }

where we have used 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= a^\dagger a} and $[a,a^{\dagger }]=1$. Note that this expression, which is given for a single mode, has a value that is independent 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 \tau} . It is obtained by substituting quantum operators for the electric field into the classical definition 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 g^{(2)}(\tau)} . In contrast to the classical case, the quantum expression allows 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 g^{(2)}(0)<1} . In particular, the values of $g^{(2)}(0)$, and the Fano factor (a measure of photon number fluctuations relative to the Poisson distribution),

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 F = \frac{ \langle n^2{\rangle}- \langle n{\rangle}^2}{ \langle n{\rangle}}-1 \,, }

are given for several important states of light in this table:

%

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 \langle n^2{\rangle}}	Fano factor	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 g^{(2)}(0)}
Thermal	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 \bar{n}(1+2\bar{n})}	${\bar {n}}$     Super-poissonian	2
Coherent	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 \bar{n}(1+\bar{n})}	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}         Poissonian	1
Number 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{\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 n^2}	$-1$     Sub-poissonian	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})/{n}}

The Single Photon

So far, we have considered a variety of important multi-photon states, in which typically 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 \bar{n}>1} . What are the properties of a state of a single photon? First, keep in mind 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 |\alpha{\rangle}} 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 \alpha=1} is not a single photon state. That is a coherent state with an average of one photon. In fact, the probabilities of finding $n$ photons in this state are:

\noindent Thus, for example, 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 |\alpha=0.1{\rangle}} is mostly the vacuum 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 |0{\rangle}} . The true single photon 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 |1{\rangle}} is an eigenstate of $a^{\dagger }a$ with eigenvalue 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} . This state can be generated, for example, by a single atom emitting a photon into a cavity. 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 Q(\alpha)} plot of this state is a ring, since

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 Q_{|1{\rangle}}(\alpha) = |{\langle}\alpha|1{\rangle}|^2 = |\alpha|^2 e^{-|\alpha|^2} \,. }

The single photon state is one of the most non-classical photon states which can be created in the laboratory. Generating single photons at a specific desired time, and with a desired frequency, is a challenge at the forefront of much atomic physics today. One important experimental signature of a single photon is obtained by performing a Hanbury Brown-Twiss experiment, which measures the $g^{(2)}(\tau )$ intensity-intensity correlation function of photons from a single source, split by a 50/50 beamsplitter:

Classically, photons incident on the beamsplitter are split equally between the two paths. For equal distances between the beamsplitter and the two detectors, 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 \tau=0} , and when the incident light is coherent, the correlation is measured 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 1} ; when it is chaotic, as from a thermal source, it is measured 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 2} , as given in the table above. Classically, however, $g^{(2)}(0)$ can never be less than 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} . However, when a single photon sourced into the beamsplitter, it exits in a superposition of being in one or the other of the two paths, and only one of the two detectors will ever click. This gives 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 g^{(2)}(\tau) = 0} , as predicted by the quantum formula, but not allowed by the classical expression.