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.

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► Category:Quantum Light

► Category:Optical Bloch Equations

► Category:Bose-Einstein condensates

► Category:Ion traps and quantum information

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► Category:Photon-atom interactions

  Introduction to Atomic Physics

  Quantum light: states and dynamics

  Optical Bloch equations

  Photon-atom interactions

  Frontiers in Atomic Physics

  Ion traps and quantum information

  Quantum gases

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 ${\vec {B}}$ are the electric and magnetic fields, and $\epsilon _{0}$ and $\mu _{0}$ 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,

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

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

${\vec {E}}=i{\vec {e}}{\sqrt {\frac {\hbar \omega }{\epsilon _{0}V}}}\left[{ae^{i(kr-\omega t)}-a^{\dagger }e^{-i(kr-\omega t)}}\right]\,,$

where ${\vec {e}}$ is the polarization vector, and $V$ 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:

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

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 $|n{\rangle }$, where $H|n\rangle =n\hbar \omega |n{\rangle }$, such that the energy levels may be depicted as a ladder of equally spaced rungs starting at $|0{\rangle }$, as shown in this figure:

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

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

A good way to reconstruct these is to remember just that $a|0\rangle =0$, and $a^{\dagger }|0\rangle =|1{\rangle }$. 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,

$\rho _{\rm {th}}={\frac {e^{-\beta H}}{\mathcal {Z}}}={\frac {e^{-{\frac {\hbar \omega n}{k_{B}T}}}}{\mathcal {Z}}}|n\rangle \langle n|\,,$

where $\beta =1/k_{B}T$, $k_{B}$ is Boltzmann's constant, $T$ is temperature, and

${\mathcal {Z}}=\sum _{n}e^{-{\frac {n\hbar \omega }{k_{B}T}}}={\frac {1}{1-e^{-\hbar \omega /k_{B}T}}}$

is the usual partition function normalization. This gives us

$\rho _{\rm {th}}=e^{-n\hbar \omega /k_{B}T}(1-e^{-\hbar \omega /k_{B}T})|n\rangle \langle n|\,.$

The mean number of photons in a thermal state is

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

If we define the probability distribution $P_{n}^{th}$ by $\rho _{\rm {th}}=P_{n}^{th}|n\rangle \langle n|$, then

$P_{n}^{th}={\frac {{\bar {n}}^{n}}{(1+{\bar {n}})^{n+1}}}\,.$

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

$\sum _{n}n(n-1)P_{n}^{th}=2{\bar {n}}^{2}\,,$

and thus the variance is ${\langle }\Delta n^{2}\rangle =\langle n^{2}{\rangle }-\langle n{\rangle }^{2}={\bar {n}}^{2}+{\bar {n}}$. 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 $Q(\alpha )$ Representation

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

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

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

$|\alpha \rangle =\sum _{n}|n\rangle \langle n|\alpha {\rangle }\,.$

Let $c_{n}=\langle n|\alpha {\rangle }$. By definition, $|\alpha {\rangle }$ is an eigenstate of $a$, and thus

${\begin{array}{rcl}a|\alpha \rangle &=&\sum _{n}c_{n}a|n{\rangle }\\&=&\sum _{n}c_{n}{\sqrt {n}}|n-1{\rangle }\,.\end{array}}$

This gives a recursion relation for $c_{n}$; 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:

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

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

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

What physical state does the coherent state represent? The diagonal elements of $|\alpha {\rangle }{\langle }\alpha |=\sum _{n}p_{n}|n\rangle \langle n|$ have a Poisson distribution,

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

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 ${\bar {n}}$. 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 $\alpha =|\alpha |e^{i\phi }$, $|\alpha |$ is its amplitude, and $\phi$ 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

$Q_{\rho }(\alpha )={\langle }\alpha |\rho |\alpha {\rangle }\,,$

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

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

independent of $\rho$. In this sense, we may interpret $Q_{\rho }(\alpha )/\pi$ as a probability distribution.

Vacuum state

For example, consider the $Q_{\rho }(\alpha )$ plot for $\rho =|0\rangle \langle 0|$, the "vacuum" state. This is $Q_{|0{\rangle }}(\alpha )=|{\langle }\alpha |0{\rangle }|^{2}=e^{-|\alpha |^{2}}$, a gaussian centered at the origin in the ${{\mathcal {I}}m}(\alpha )$, ${{\mathcal {R}}e}(\alpha )$ plane:

Thermal state

Another example is provided by the thermal state $\rho _{th}=\sum _{n}P_{n}^{th}|n\rangle \langle n|$, 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 $P_{n}^{th}$, we find

$Q_{th}(\alpha )={\frac {1}{1+{\bar {n}}}}e^{-{\frac {|\alpha |^{2}}{1+{\bar {n}}}}}\,,$

This again is a gaussian, centered at the origin:

Coherent state

A third example is given by the coherent state itself; let $\rho =|\beta {\rangle }{\langle }\beta |$. The $Q(\alpha )$ 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 $\beta$:

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

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

In other words, a coherent state parameterized by $\alpha$ evolves with time to become a coherent state parameterized by $\alpha e^{i\omega t}$. In terms of the $Q(\alpha )$ representation, this means that the time evolution of a coherent state centered at some initial point $\beta$ 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 $Q(\alpha )$ 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,

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

and using the fact that for a coherent state $|\alpha {\rangle }$, ${\langle }\alpha |a|\alpha \rangle =\alpha$, and ${\langle }\alpha |a^{\dagger }|\alpha \rangle =\alpha ^{*}$, we find that

${\begin{array}{rcl}{\langle }\alpha |P|\alpha \rangle &=&i{\sqrt {\frac {\hbar \omega }{2}}}(\alpha ^{*}-\alpha )\\{\langle }\alpha |P^{2}|\alpha \rangle &=&{\frac {\hbar \omega }{2}}({\alpha ^{*}}^{2}-2|\alpha |^{2}+\alpha ^{2}-1)\\{\langle }\alpha |Q|\alpha \rangle &=&{\sqrt {\frac {\hbar }{2\omega }}}(\alpha ^{*}+\alpha )\\{\langle }\alpha |Q^{2}|\alpha \rangle &=&{\frac {\hbar }{2\omega }}({\alpha ^{*}}^{2}+2|\alpha |^{2}+\alpha ^{2}+1)\,.\end{array}}$

These give

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

so that $\Delta P\,\Delta Q=\hbar /2$. 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 second order temporal coherence function, commonly denoted $g^{(2)}(\tau )$. 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 $g^{(2)}(\tau )$, but instead of ${\bar {I}}(t)$, operators must be used, which act on $\rho$. Letting $\langle x{\rangle }$ denote ${\rm {Tr}}(x\rho )$ as usual, the quantum mechanical definition of the second order temporal coherence function is

$g^{(2)}(0)={\frac {\langle a^{\dagger }a^{\dagger }aa{\rangle }}{\langle a^{\dagger }a{\rangle }^{2}}}={\frac {\langle n^{2}{\rangle }-\langle n{\rangle }}{\langle n{\rangle }^{2}}}\,,$

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

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

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

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

The Single Photon

So far, we have considered a variety of important multi-photon states, in which typically ${\bar {n}}>1$. What are the properties of a state of a single photon? First, keep in mind that $|\alpha {\rangle }$ with $\alpha =1$ 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, $|\alpha =0.1{\rangle }$ is mostly the vacuum state $|0{\rangle }$. The true single photon state $|1{\rangle }$ is an eigenstate of $a^{\dagger }a$ with eigenvalue $n=1$. This state can be generated, for example, by a single atom emitting a photon into a cavity. The $Q(\alpha )$ plot of this state is a ring, since

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

Here is a plot for a 3 photon state:

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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, $\tau =0$, and when the incident light is coherent, the correlation is measured to be $1$; when it is chaotic, as from a thermal source, it is measured to be $2$, as given in the table above. Classically, however, $g^{(2)}(0)$ can never be less than $1$.

On the other hand, 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 $g^{(2)}(0)=0$, as predicted by the quantum formula, but not allowed by the classical expression.

Here are some examples of experimental $g^{(2)}(0)$ measurements Single photon generation - experimental results.