Photons and statistics

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 ${\vec {B}}$ 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 $H|n\rangle =h\hbar \omega |n{\rangle }$, 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 $H=\hbar \omega |n\rangle \langle n|$. 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

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

The mean number of photons in a thermal state is

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

If we define the probability distribution $P_{n}^{th}$ 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:

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

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,

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

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. 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:

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:

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 ph