Bose Einstein Statistics Centenary Special: Photon Bunching and Bose Einstein Statistics
Have you ever wondered, how do we differentiate between smallest of
quanta from each other? Like aren't they all the same? Well, kinda
yes, but when you talk about them in terms of their degrees of freedom
like their energy, momentum, angular momentum etc., they show very
peculiar property. How come nature come to excite us in adventure
of finding secrets about our "favourite'' boson aka photons?
Bunching of paths imposed by "friends"
Has it ever happened to you that you're returning back from classes
to have lunch and in the meanwhile you met with a friend who is also
as ambiguous as you to which mess they should eat today, so you guys
happen to choose your paths "collectively'' this time, however
on regular lonely day, you'd have chosen a totally random mess to eat?
 |
| En route to mess at H5 or H7 @IISERM |
Well, this same analogy can capture, when 2 photons of matching degrees
of freedom happen to "take'' decision "collectively'' in which
path to choose. When they were "strangers" to each other, they were
altogether going random pathways but when their degrees of freedom
match, they happen to "bunch'' around to any random path. This
phenomenon is called Bunching!
With the centenary of Bose Einstein Statistics, we would
embark on journey of photon statistics of how they come to show Bose
Einstein Statistic when they are "bunched'' and discrete Poissonian
statistics when they are completely distinguishable!
Generalised Particle Statistics
It has got its applications in many fields of current research like
quantum metrology, quantum computing etc. Generally, statistical distribution
of particles can be described as
$P_{\epsilon}=\frac{1}{e^{\frac{\epsilon}{k_{B}T}}-S}$
where $\epsilon$ represents energy, $k_{B}$ represents Boltzmann's
Constant and $T$ is absolute temperature.\cite{Sun2017} For indistinguishable
fermions (particle with half integer spins) $P_{\epsilon}$ represents
Fermi Dirac Statistics with $S=-1$ while for boson (particle with
integer spins), it shows Bose Einstein Statistics with S=1. You can
wonder, what is S then, how does S vary?
Hong Ou Mandel Interferometer
Remember the analogy of 2 friends going to mess to eat collectively
together? Well, replace them with photons now, voila! Here is your
Hong Ou Mandel Interferometer!! Ta-da!
Excited? No, because I have really skipped a lot of nuances to fill
in, right? So, starting with definition: Hong Ou Mandel (HOM) Interferometery
is a two photon interference in which we test distinguishability of
two photons of several degrees of freedom (e.g. spatial time delay
$\tau$, frequency $\nu$ etc.,), by injecting them into two input
ports (paths 1 and 2) of 50/50 beamsplitter with exactly same DOFs
such that the two photon pairs (under correct resolution) follow only
one path at a time i.e. either path 3 or 4 together.
 |
(a) A balanced beam splitter with input ports designated a and b, and output ports designated c and d. (b) Four possibilities of interaction of two photons at the input ports of BS |
This non classical bunching phenomenon can be explained by Symmetrization
postulate and Spin statistics theorem.\cite{LoFranco2021}
The symmetrization postulate states that the wavefunction of a system
of identical particles must be either symmetric or antisymmetric under
the exchange of any two particles. For two particle system,
$\ket{\chi_{1},\chi_{2}}=e^{i\phi_{ex}}\ket{\chi_{2},\chi_{1}}$
where particle exchange phase $\phi_{ex}$ is zero for bosons and
$\pi$ for fermions.
Distinguishability and Indistinguishability (by transform pulses)
Generally, indistinguishability of photons comes from entanglement
with outer system. However, in some cases like in SPDC process, property
of entanglement can be achieved back through phase matching condition!
In frequency degree of freedom (i.e, photon can have different energies)
photon state is a mixed state of transform-limited{*} pulses with
different center frequencies.
For a pulse to be transform limited, relation between duration of
pulse and range of frequencies a single photon contains, is at its
theoretical minimum. So, broader the range of frequencies (also energies),
the shorter the pulse duration.
A transform limited pulse ensures the photon temporal profile is "tightest''
minimizing any uncertainty in arrival time!
\ket{\omega}=\intop_{-\infty}^{+\infty}dvg_{\omega}(v)a^{\dagger}(v)\ket{vac}
where $a^{\dagger}(v)$ is the single photon creation operator, $\left|g(v)\right|^{2}$
is the spectrum of transform limited pulse with center frequency $\omega$
and width $\Delta_{g}$ (intrinsic width). Considering interaction of external environment
is similar during photon generation so width of other independent
transform limited pulses, say $\ket{\omega_{j}}$, same.
Therefore, indistinguishability as we know of it is related to identicality
of these separate transform limited pulses. Indistinguishability of
2 independent transform-limited pulse is $K_{ij}^{TL}=\left|\braket{\omega_{i}\left|\omega_{j}\right.}\right|^{2}$ i.e,
when those photons are totally distinguishable when $\left|\omega_{i}-\omega_{j}\right|\gg\Delta_{g}$
and indistinguishable for $\omega_{i}=\omega_{j}$.\cite{Sun2009}
Also single photon may get entangled with outer environment thus center
frequencies have the distribution $f(\omega)$ such that $\int_{-\infty}^{+\infty}d\omega f(\omega)=1$
with extrinsic width $\Delta_{f}$. State of photon could be written
as:
\rho=\int_{-\infty}^{+\infty}d\omega f(\omega)\ket{\omega}\bra{\omega}
.png) |
Illustration of total single photon pulse (red dashed curve, width $\Delta_{s}$) composed of transform-limited pulses (grey bold curves, width $\Delta_{g}$)\cite{Sun2009}} |
The total spectrum $S(v)=\int_{-\infty}^{+\infty}d\omega f(\omega)\ket{\omega}\bra{\omega}$
is broadened to $\Delta_{s}\geq\Delta_{g}$ because of distribution
$f(\omega)$. From figure 3, when $\Delta_{s}=\Delta_{g}$ is satisfied
single photon pulse become transform limited and "indistinguishable''.
For two photons, the indistinguishability of two independent single
photons can be described as
K=tr(\rho\otimes\rho)=\int\int_{-\infty}^{+\infty}d\omega_{i}d\omega_{j}f(\omega_{i})f(\omega_{j})\left|\braket{\omega_{i}|\omega_{j}}\right|^{2}
When state $\rho$ is a pure state i..e, it does not entangle with
outer environment i.e $\Delta_{f}=0$ and $\Delta_{s}=\Delta_{g}$,
K becomes 1 and single photon states are indistinguishable!\cite{Sun2009}
Extending it further....
Consider a multi photon state from N separated emitters can be described
as
\begin{equation}
\rho_{N\thinspace photon}=C_{0}\otimes_{k=1}^{N}(\ket{vac}\bra{vac}+c_{k}\rho_{k})\label{4}
\end{equation}
where each state $\rho_{k}$ describes a quantum state of a single
photon so $tr(\rho_{k})=1$, $C_{0}$ is a normalisation constant.
We take same assumption as before that all emitters are under same
environment during photon generation process i.e, $c_{k}=c$ and $\rho_{k}=\rho$.
We define indistinguishability of n photons as $K_{n}=tr\rho^{n}$where
in our general analogy or normal HOM Interferometer, we would get
concerned about two photon interference. When both $g_{\omega}(\nu)=\frac{e^{-\frac{(\nu-\omega)^{2}}{(2\pi\sigma_{g}^{2})^{4}}}}{2\pi\sigma_{f}^{2}}$and
$f(\omega)=\frac{e^{-\frac{(\omega-\omega_{c})^{2}}{2\sigma_{f}^{2}}}}{\sqrt{2\pi\sigma_{f}^{2}}}$
are Gaussian function with widths $\sigma_{g}$and $\sigma_{f}$,
respectively we can obtain $K=\frac{\sigma_{g}}{(\sqrt{\sigma_{g}+\sigma_{f}})}$.
As before above equation to find $K_{n}$ we would have to calculate
K_{ij}=\left|\braket{\omega_{i}\left|\omega_{j}\right.}\right|^{2}=\int_{-\infty}^{+\infty}(g_{\omega_{i}}^{*}(v)g_{\omega_{j}}(v))^{2}dv=e^{-\frac{(\omega_{i}-\omega_{j})^{2}}{4\sigma_{g}^{2}}}\label{5}
Integrate it over tensor product of n photon state of which elaborate
calculation is discussed in \cite{Sun2017}'s paper of which value
of $K_{n}$ can be tabulated as:
 |
| indistinguishability of 2 photons |
It was shown that value of $K_{n}$ decays with increase in photon
numbers. Also, it is well fitted by exponential decay rate of $\alpha(K)$
i.e, \[K_{n}(K)=e^{-\alpha(K)n}\]
Because the nonzero K will induce photon bunching, the photon-number
distribution of $\rho_{Nphoton}$ strongly depends on value of $K_{n(n>1)}.$
Formally, the photon state can be written as:
\[ \rho_{Nphoton}=C\sum_{n=0}^{N}\left(\begin{array}{c}N\\n\end{array}\right)B_{n}c^{n}\{n\}\]
where C is a new normalization constant, \{n\} describes the state
with the photon number of n, and $B_{n}$ is an indistinguishability-
$(K_{n(n>1)}>0)$ induced photon-bunching coefficient.
.png) |
The way photon can permute among themselves induces bunching effect, thus a significant property of boson! |
Boson Permutation Symmetry induces photon bunching effect. As per
permutation of n photons to obtain $B_{n}$ of n photon state viz,
\[B_{n}=\sum_{k=2}^{n}D_{n,n-k}K_{k}+1\]
where $D_{n,n-k}=\frac{n!}{(n-k)!}$$\sum_{i=2}^{k}\frac{(-1)^{i}}{i!}$
are recontres numbers, which show number of permutations of n photons
with (n-k) photons without permutations.\ref{fig:4}
In this way, Recontres Number are defined as number of derangements
with k fixed points when $0\leq k\leq n$. Derange all those point
which are unfixed!
- For totally indistinguishable states, $K_{n}=1$, $B_{n}=\sum_{k=2}^{n}D_{n,n-k}+1=n!$
shows n photon-bunching result, and \{n\}=$\ket{n}\bra{n}$ is a n-photon
Fock (Number) state.
- For totally distinguishable state with K=0, $K_{n}=0$ and $B_{n}=1$.
- For partially indistingusihable photons, $1<$$B_{n}<n!$. When n$\gg1,$
\[\frac{B_{n+1}(K)/(n+1)!}{B_{n}(K)/n!}\rightarrow\frac{K_{n+1}}{K_{n}}=e^{-\alpha(K)}\]
It shows that there is exponential decay rate of $\frac{B_{n}(K)}{n!}$
with decay rate of $\alpha(K)$. \cite{Sun2017}
Signature of Bose Einstein Statistics while Bunching
- For distinguishable states, $B_{n}=1$ photon bunching doesn't occur
so they follow random path across different ports (as in case of beam
splitter) i.e, $\rho_{Nphoton}$ show a classical state with a binomial
distribution which converts to Poissonian distribution when $N$$\gg1$.
- For all indistinguishable states with $K_{n}=1$ and $B_{n}=n!$,
photon number distribution is
\[\rho_{Nphoton}\simeq(1-Nc)\sum_{n=0}^{N}(Nc)^{n}\ket{n}\bra{n}=\sum_{n=0}^{N}P_{n}\ket{n}\bra{n}\]
when $Nc<1$ and $N\gg1$. It can be described by Bose Einstein Statistics
with
\[P_{n}=\frac{\bar{n}^{n}}{(1+\bar{n})^{n+1}}=P\frac{e^{-n\epsilon/k_{B}T}}{e^{\epsilon/k_{B}T}-1}\]
where $Nc=e^{-\epsilon/k_{B}T},P=e^{\epsilon/k_{B}T}+e^{-\epsilon/k_{B}T}-2$
and $\bar{n}=Nc/(1-Nc)=1/(e^{\epsilon/k_{B}T}-1)$ is the mean photon
number.
- However, for photons with partial indistinguishability ($0<K_{n}<1$),
the photon state should be
\[\rho_{Nphoton}\simeq(1-Nce^{-\alpha(K)})\sum_{n=0}^{N}(Nce^{-\alpha(K)})^{n}\ket{n}\bra{n}=\sum_{n=0}^{N}P_{n}(K)\ket{n}\bra{n}\]
When $Nc<1$ and $N\gg1$, a modified Bose Einstein statistics can
be represented as:
\[P_{n}(K)=P(K)\frac{e^{-n[\epsilon/k_{B}T+\alpha(K)]}}{e^{\epsilon/k_{B}T}-S}\]
where $P(K)=e^{\epsilon/k_{B}T}+e^{-\epsilon/k_{B}T-2\alpha(K)}-2e^{-\alpha(K)}$and
mean photon number is $\bar{n}=\frac{Nce^{-\alpha(K)}}{1-Nce^{-\alpha(K)}}=\frac{1}{e^{\epsilon/k_{B}T+\alpha(K)}-1}$
and S is indistinguishability induced bunching factor.
Statistical Transition during Bunching
We apply second order correlation function $g^{(2)}(\tau)$ to probe
photon statistical transition from Poissonian Statistics to Bose Einstein
Statistics. What is exactly $g^{(2)}(\tau)$?
Suppose a Gaussian wavepacket of light with longitudinal spatial width
$\sigma$ is propagating in a given direction and another
gaussian wavepacket of light with longitudinal spatial width $\sigma$
having the same spectrum of frequencies is propagating in the same
direction. The peak of the two gaussians is separated by a distance
$\tau$ . As long as $\tau$>>$\sigma$ a high degree of interference will occur. If
$\sigma$>>$\tau$ a low degree of interference will occur. Maximum interference obviously
occurs when $tau$ = 0, when the wavepackets have maximum
overlap.
From single photon state, c is photon emission probability from an
emitter and Nc is number of photons from N emitters without photon
bunching.
- When $Nc\ll1$, $g^{(2)}(0)=1+K$
- When $Nc\gg1$ and $K>0$, bunching effect dominates the quantum statistics!
When n$\gg1$, photons condense into n photon Fock (Number) state
with $g^{(2)}(0)\rightarrow1$. We can infer behaviour of photon statistical
transitions from $g^{(2)}(0)=1+K$ to $g^{(2)}(0)\rightarrow1$ with
an increase in photon number Nc. \cite{Sun2017}
.png) |
You can see transition from Bose Einstein Statistics to (usual) Poissonian Statistics of Laser light \cite{Sun2017} |
For an indistinguishable photon state with Bose-Einstein statistics,
the transition occurs at Nc = 1. The transition is largely contributed
indistinguishability induced bunching effect.\cite{Sun2017}
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