Quantum optics

Multimode radiation field states: localized single photon state

M. Brune, A. Aspect

Homework of Lesson 4

Introduction

In the lesson you learned how to quantize the multimode radiation field. The method is based
on the decomposition of an arbitrary classical electromagnetic field into plane waves modes by
using Fourier transform. Each of these modes is then quantized independently of all the others as
a single harmonic oscillator. In this homework, we will use this multimode field description for-
malism in order to explicitly describe the state of a single photon wavepacket and its propagation.
We will introduce the concept of a localized mode of the radiation field.
For the sake of simplicity, we will consider here a one-dimensional wavepacket with a well
defined linear polarization ε~. Generalizing this approach to 3D would provide a full description
of the propagation of any quantum state of the radiation field. In particular it would include
diffraction effects that are obviously not describes by our simplified 1D approach.

1 Single photon wavepacket

P
One considers the most general, normalized one-photon state : |1iwp = ` c` |1` i. The index `
represents a plane wave propagating in the Ox direction with the wave vector k` = k` ex , where
ex is a unit vector defining the propagation direction, and a fixed polarization ε~. The state |1` i
represents a one-photon state in mode ` and 0 photons in any other mode.
We recall the expression of the electric field operator for a 1D radiation field :
r
E` = 2εh̄ωLS
 
(1) † −ik` ·x (1) `
X
ik` ·x
Ê(x) = i E` ε~ â` e − â` e with ω` = ck` (1)
0
`

where S is the transverse cross section of the beam and L the length of the 1D quantization
volume. In our simplified 1D model, the field amplitude is assumed to be constant over the surface
(1)
S and to vanish outside S, as seen in video 2.2. Introducing Ê (+) (x) = i ` E` ε~ â` eik` ·x and
P
 †
Ê (−) (x) = Ê (+) (x) one has

Ê(x) = Ê (+) (x) + Ê (−) (x). (2)

1. Express the field state |ψ1 (t)iwp at time t for the initial state |ψ1 (0)iwp = |1iwpP
. Be careful
while taking into account the contribution of vacuum whose energy is EV = ` 1/2 h̄ω` .

In all the following parts of the problem, we will use the Hamiltonian

h̄ω` â†` â`

X
0
ĤR = ĤR − EV = (3)
`

1
 instead of ĤR . These two Hamiltonian only differ by the constant energy EV . You can
check that the amplitude of electric field vacuum fluctuations are not affected by this
redefinition of the vacuum energy. Most physical effects related to vacuum fluctuations
0 .
are well described by ĤR

0 instead of Ĥ .
2. Rewrite |ψ1 (t)iwp by using the Hamiltonian ĤR R
3. Give the explicit expression of the vectors â` eik` ·x |ψ1 (t)iwp .
(+) (+)
4. Show that Ê (+) (x)|ψ1 (t)iwp = Ewp (x, t)|ψR i. Give the expression of the function Ewp (x, t)
and of the state |ψR i.
5. Give the average value of the electric field operator hE(x, t)i = wp hψ1 (t)|Ê(x)|ψ1 (t)iwp .
6. We introduce the operator Âwp (t) = ` c∗` (t) â` . Calculate the commutator [Âwp (t), †wp (t)].
P

Express |1iwp as a function of the operator †wp (t) and of the vacuum state |0i. Give the
physical interpretation of the operator Âwp (t).

2 Gaussian single photon wavepacket

For all the following parts of the problem, we now define

1 (k` −k0 )2
c` = N √ e− 4∆k2 , (4)
(2π)1/4 ∆k
corresponding to a gaussian wavepacket centered around k0 . The variance of the probability
distribution of the wavevector k` is ∆k 2 .
q q
(1) (1)
We assume that ∆k  k0 so that one will consider that E` = 2εh̄ω `
0 LS
' h̄ω0
2ε0 LS = E ω0
with ω0 = ck0 .
1. Express the norm of |1iwp as a discrete sum over the modes. By transforming the discrete
sum into an integral (see note below), give the explicit expression of N as a function of
the size of the quantization volume L.

Note :
- The rule for transforming a discrete sum over k` in one dimension into an integral over
the continuous variable k is
L +∞
X Z
[f (k` )] → dk [f (k)] , (5)
2π −∞
k`

where f (k) is a function of k.

- We remind the expression of a normalized Gaussian probability distribution of a conti-

nuous variable y centered at y0
1 (y−y0 )2
PG (y) = √ e− 2σ2 (6)
2πσ
where σ is the standard deviation of the variable y.
(+)
2. Calculate the explicit expression of Ewp (x, t) for the considered gaussian wave packet.
R +∞
Note : Using the definition g(X) = √12π −∞ dk f (K) eiKX , the Fourier transform of the
K2 X2
gaussian f (K) = e− 2∆K 2 is the function g(X) = 1
∆X e− 2∆X 2 with ∆X = 1/∆K.

3. Give the explicit expression of the one photon detection rate w(1) (x, t) = s k Ê (+) (r)|ψ1 (t)iwp k2
(1)
as a function of X = x − ct, Eω0 , L and ∆k. The parameter s is a coefficient characte-
rizing the detector efficiency. Does this expression depend on L ?

2
 4. What is the position of the wavepacket at time t ? What is the spatial width ∆x of the
wavepacket. Does it depends on time ?
5. Using creation and annihilation operators defined in question (1.6), we define Âwp =
Âwp (0) and = †wp = †wp (0). We also define a dimensionless function F (x) describing
(+) (1)
the spatial mode structure of the wavepacket by Ewp (x, 0) = i ε~ Eω0 F (x). We introduce
the electric field operator
 
† ∗
Êwp (x) = iEω(1)
0
~
ε Âwp F (x) − Â wp F (x) . (7)

We now use this single mode expression for describing the electric field corresponding to
the gaussian mode wavepacket considered above. Give the expression of F (x).
Using directly the properties of the operators Âwp and †wp , calculate
hÊwp (x)ivac = vac (x)2 in the vacuum state |0i
wp h0|Êwp (x)|0iwp and ∆Ewp wp as a function
of Eω(1) and F (x). Do these result depends on L ?
0

3 Advanced complementary question

The first order spatial correlation function of the radiation field in the state |ψi is defined by
g (1) (x, x0 ) = hψ|Ê (−) (x) Ê (+) (x0 )|ψi.
This correlation function can be measured with a Michelson interferometer. The single photon
detection rate at the output of the interferometer has an interference term which is proportional
to the real part of hψ|Ê (−) (x, t)Ê (+) (x, t−∆L/c)|ψi ∝ g (1) (x, x−∆L/c), where ∆L is the optical
path difference between the two arms of the interferometer.
For the one-photon radiation field considered above, we define
(1)
gwp (x, x0 ) = wp h1|Ê
(−)
(x) Ê (+) (x0 )|1iwp (8)

(1) (1) 2
Show that gwp (x, x0 ) = C Eω0 f (x, x0 ), where C is a normalization constant that one will not
calculate. Give the explicit expression of f (x, x0 ). What is its value for x = x0 = 0.

Concluding remarks

We have seen here how to use the quantum formalism describing a multimode radiation
field in order to explicitly describe the propagation of a single photon wavepacket. Indeed, the
final expression of |ψ(t)iwp generalizes to the description of an arbitrary n photon state in the
considered gaussian mode, which can be defined as
1
n
|niwp = √ †wp |0i. (9)
n!
Note that this formalism automatically describes the spatial shape of the wavepacket as well as
its propagation with time. To get a real feeling of the power of this formalism, you should try to
express the same state in the number state basis of plane wave modes ! You find it too difficult,
it is normal !