Interaction of Radiation with Atoms

Interaction of Radiation with Atoms


Abstract The preceding chapters have been concerned with the properties of the radiation field alone. In this chapter we turn to the interaction between radiation and matter. This is of course the domain of quantum electrodynamics, however in quantum optics we are usually only concerned with low energy systems of bound electrons which simplifies matters considerably. We will use the occupation num- ber representation for bound many-electron systems to quantize the electronic de- grees of freedom, following the approach of Haken [1] and also Cohen-Tannoudji et al. [2].



10.1 Quantization of the Many-Electron System


In the full theory of QED, the interaction between the electromagnetic field and charged matter is described by coupling between the vector potential and the Dirac spinor field. In quantum optics we only need the low energy (non relativistic) limit of this interaction. This is given by the minimal coupling Hamiltonian [3]

1
H = (?p ? e?A)2 + eV (?x)+ Hrad (10.1)

where ?p is the momentum operator for a particle of charge e moving in a Coulomb potential V (?x). The vector potential is quantised in a box of volume V as

.
?A(?x,t)= ?


h¯
2? ? V ?en,?

. i(?kn.?x??nt)



an,? + e


?i(?kn.?x??nt)

† .
n,?



(10.2)

n,? o n

where ?en,? are two orthonormal polarisation vectors (? = 1, 2) which satisfy ?kn ·
?en,? = 0, as required for a transverse field, and the frequency is given by the dis- persion relation ?n = c?kn|. The positive and negative frequency Fourier operators,

respectively an,? and a†

, satisfy



197


[an,? , an?,?? ]= ???? ?nn? (10.3)

The last term, Hrad is the Hamiltonian of the free radiation field given by
Hrad = ?h¯ ?k a†ak (10.4)
k

where we have subsumed polarisation and wave vectors labels into the single sub- script k.
We now use an occupation number representation in the antisymmetric sector of the many body Hilbert space for the electronic system based on a set of single
particle states |? j ), with position probability amplitudes, ? j (?x), which we take as the
bound energy eigenstates of the electronic system without radiation. They could for
example be the stationary states of a atom, the quasi bound states of a single Cooper pair on a mesoscopic super-conducting metal island, or the bound exciton states of semiconductor quantum dot. We then define the electronic field operators
?ˆ (?x)= ?c j? j (?x) (10.5)
j

where the appropriate commutations relations for the antisymmetric sector are the fermionic forms

ckc† + cl c† = ?kl (10.6)
l k
ckcl + cl ck = c†c† + c†c† = 0 (10.7)
k l l k

In the occupation number representation the Hamiltonian may be written as the sum of three terms, H = Hel + HI + Hrad where the electronic part is given by

¸
Hel =

.
d3?x?ˆ †(?x) ?


h¯ 2

.
?2 + eV (?x)


?ˆ (?x)= ?E j c†c j (10.8)

2m j j

The interaction part may be written as the sum of two terms HI = HI, 1 + HI, 2 where

¸
Hl, 1 =

d3?x?ˆ †(?x) .?

e (?A(?x).?p + ?p.?A(?x)). ?ˆ (?x) (10.9) 2m

¸
Hl, 2 =

d3?x?ˆ †(?x)

. e2
2m

.
(?A(?x)2

?ˆ (?x) (10.10)


Unless we are dealing with very intense fields for which multi-photon processes are important, the second term HI, 2 may be neglected.
The dominant interaction energy may then be written as

† †

HI = h¯

? g?k,n,m(b?k + b?k )cncm (10.11)
?k,n,m


where the interaction coupling constant is

Quantization of the Many-Electron System 199

e . 1

.1/2 ¸
3 ?

. i?k.?x .

g?k,n,m = ? m

2? h¯ ? V

d ?x?n(?x) e

?p ?m(?x) (10.12)

0 k

We now proceed by making the dipole approximation. The factor ei?k?x varies on a spatial scale determined by the dominant wavelength scale, ?0, of the field state. At optical frequencies, ?0 ? 10?6 m. However the atomic wave functions, ?n(?x) vary on a scale determined by the Bohr radius, a0 ? 10?11 m. Thus we may remove the
oscillatory exponential from the integral and evaluate it at the position of the atom
?x = ?x0. Using the result
[?p2, ?x]= ?i2h¯?p (10.13)
we can write the interaction in terms of the atomic dipole moments

¸ d3?x??(?x) .ei?k.?x?p. ?


m
(?x)= i ?

ei?k.?x0 ¸


d3?x??(?x)(e?k)?


(?x) (10.14)

n m

where ?nm = (En ? Em)/h¯ .

e nm n m

In the interaction picture the interaction Hamiltonian becomes explicitly time
dependent,


H?I(t)= h¯

? g?k,n,m(b?ke?
?k,n,m

i?(?k)t

+ b†ei?(?k)t
?k


)c†cme

i?nmt


(10.15)



where the tilde indicates that we are in the interaction picture. If the field is in state for which the dominant frequency is such that ?(?k0) ? ?nm, the field is resonant
with a particular atomic transition and we may neglect terms rotating at the very high frequency ?(?k)+ ?nm. This is known as the rotating wave approximation. This
assumes that the field strength is not too large and further that the state of the field does not vary rapidly on a time scale of ??1 i.e. we ignore fields of very fast strong
pulses. As a special case we assume the field is resonant (or near-resonant) with a single pair of levels with E2 > E1. The interaction picture Hamiltonian in the dipole and rotating wave approximation is then given by

H?I = h¯ ?c†c2b†g e?i(?(?k)??21)t + h.c (10.16)



where

1 ?k k
?k

.



.?1/2





i?k.?x0



and

g?k = ?i

2h¯ ?0 ?(?k)V

?a ?21e

(10.17)


with ?a = ?2 ? ?1.

?21 = (?n|e?x|?m) (10.18)

It is conventional to describe the operator algebra of a two level system in terms
of pseudo-spin representation by noting that the Pauli operators may be defined as


?z = c†c2 ? c†c1 (10.19)
2 1

?x = c†c1 + c†c2 (10.20)
2 1
?y = ?i(c†c1 ? c†c2) (10.21)
2 1
?+ = ?† = c†c1 (10.22)
? 2
The operators s? = ??/2 (with ? = x, y, z) then obey the su(2) algebra for a spin half system. In terms of these operators we may write the total Hamiltonian for the system of field plus atom in the dipole and rotating wave approximation as
h¯ ?a

H = ?h¯ ?(?k)b†b? +

?z + h¯ ?g? b? ?+ + h.c . (10.23)

?k k 2 k k
?k ?k

The free Hamiltonian for the two-level electronic system is
h¯ ?a

Hel =

2 ?z (10.24)

Denoting the ground and excited states as |1) and |2) respectively, we see that
h¯ ?a
H s = ( 1) s s = 1, 2 (10.25)
2

The action of the raising and lowering operators on the energy eigenstates is:
2

?+|1) = |2) and ??|2) = |1), while ?

= 0. We now relabel the ground state and

excited state respectively as |1)? |g), |2)? |e). If the state of the system at time t is
? , the probability to find the electronic system in the excited state and ground state
are, respectively,

pe(t)= (2|?|2) = (?+??) (10.26)
pg(t)= (1|?|1) = (???+) (10.27)

The atomic inversion is defined as the difference between these two probabilities and is given by
pe(t) ? pg(t)= (?z) (10.28)
While the atomic coherences are defined by
?12 ? (1|?|2) = (?+) (10.29)
with ?21 = ?? .

Interaction of a Single Two-Level Atom with a Single Mode Field