محاظرة 10

Chapter 10
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 201

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


If we further restrict the state of the field to include only a single mode, with frequency ?0; perhaps using a high Q optical resonator, we arrive at the Jaynes–
Cummings hamiltonian,


H = h¯ ?0b†b +

h¯ ?a
2


?z + h¯ (gb?+ + g?b†?



) (10.30)


coupling a single harmonic oscillator degree of freedom to a two-level system, which might well be called the standard model of quantum optics [4]. The coupling constant g can vary from a few kHz to many MHz. An example is provided by the experiment of Aoki et al. [5] in which a cesium atom interacts with the toroidal whis- pering gallery mode of a micro-resonator as it falls under the action of gravity from
a magneto-optical trap. The atomic resonance at is the 6S1/2; F = 4 ? 6P3/2; F ? = 5
transition in cesium. A coupling constant as large as g/2? = 50 MHz was achieved.
On resonance, ?a = ?c = ?, we see that the interaction Hamiltonian HI =

h¯ g(b?+ + b†?

) (with g chosen as real), commutes with the free Hamiltonian,

H0 = h¯ ?(b†b + 1 ?z), so that the eigenstates of the full Hamiltonian can be writ- ten as a linear combination of the degenerate eigenstates of H0. Defining |n, s) =
|n)b ? |s), where b†b|n) = n|n), the degenerate eigenstates of the free Hamiltonian are |n, 2), |n + 1, 1). Within this degenerate subspace, the state at time t may be writ- ten |?n(t)) = cn,2(t)|n, 2) + cn+1,2(t)|n + 1, 1), and the Schro¨ edinger equation in the
interaction picture is

. c?n,2 . =

. c
i? ?

.
(10.31)

c?n+1,1

? n x

cn+1,1

where ?n = g?n + 1. The eigenvalues of this system of linear equations are ±i?n,
corresponding to the eigenstates of HI

1
|n, ±) = ?2 (|n, 2)± |n + 1, 1)) (10.32)

which are often referred to as the dressed states. The splitting of the degeneracy is depicted in Fig. 10.1.
Thus the general solution is
cn,2(t)= cn,2(0) cos ?nt ? icn+1,1(0) sin ?nt (10.33)
cn+1,1(t)= cn+1,1(0) cos ?nt ? icn,2(0) sin ?nt (10.34)

If the atom is initially in the excited state and the cavity field has exactly n photons, the probability for finding the atom in the same state at time t > 0 is

1

pe(t)= |(n, 2|?n(t))|2 =

2 (1 + cos2?nt) (10.35)





(|n + 1>|g >,|n >|e >) .
.
.

|n, +>

2g n + 1
|n, –>




(|2 >|g >, |1>|e >)

.
2 2 g



|2, +>

|2, –>





(|1>|g >, |0>|e >)

|1,+>
2g
|1,–>


Fig. 10.1 The dressed states for the energy eigenstates of the Jaynes–Cummings interaction. On the left are show the degenerate states for zero interaction. When the interaction is turned on the degeneracies are lifted



The excitation oscillates backward and forth between the cavity and the electronic system with frequency ?n, the Rabi frequency. Note that for n = 0 the separation of
these eigenvalues is 2g, which is known as the vacuum Rabi splitting.
If the field is in an arbitrary pure state, |?) = ?n fn|n) and the atom is initially
excited, the probability to find the atom in the excited state at time t > 0 may be
written
1 . ? 2 ? .

pe(t)= 2

1 + ? | fn|
n=0

cos(2g

n + 1t)

(10.36)


This is a discrete superposition of harmonic oscillations with incommensurate fre- quencies. Thus it must exhibit quasiperiodic behaviour. If the initial photon number
distribution | fn |2 has narrow support on n, only a few frequencies are involved and
there is a beating between these different frequencies leading to what are known
as collapses and revivals. The collapse refers to the decay of oscillations at short times due to beating between the incommensurate frequencies. The revival refers to partial re-phasing of the oscillations at later times. In the case of the field initially
in a coherent state, |?), the initial number distribution is Poissonian with standard deviation in number given by the root mean, n¯1/2|?|. An approximate evaluation of
the sum valid for times such that gt < n¯1/2 gives [6]


1 .
pe(t)= 2


1 + e

g 2t 2n¯
? 2(n+1)


cos(2g

??n¯ + 1t .


(10.37)


There is an average Rabi oscillation frequency under a Gaussian envelope. The char- acteristic time for the collapse of the oscillation is thus

10.3 Spontaneous Emission from a Two-Level Atom 203

Fig. 10.2 (a) The experimen- tal observation of collapse and revival of the oscillations in the occupation of the excited state of a two-level atom inter- acting with a microwave field initially in a coherent state with mean photon number
n¯ = 0.85. In (b) is the Fourier
transform of the oscillations with the Rabi frequencies
?n , n = 0, 1, 2, 3 marked
(from [7])














1
tcol ? g (10.38)

A more accurate evaluation using the Laplace summation formulae shows that the oscillations first revive at a t time


trev ?

2? n¯1/2
g


(10.39)


Thus a quasi periodic burst of Rabi oscillations occurs every n¯ Rabi periods. The collapse and revival has been seen experimentally using an atom excited to a Ryberg ground state interacting with the microwave field in a superconducting cavity [7]. The results of the experiment are shown in Fig. 10.2.



Spontaneous Emission from a Two-Level Atom


Spontaneous emission can also be treated using a master equation. In this case the system is a two-level electronic system, with ground state |g) of energy h¯ ?1 and ex- cited state |e) with energy h¯ ?2, representing an electric dipole transition, coupled to
the many modes of the radiation field in the dipole and rotating wave approximation.
The master equation is

d? =
dt


i
? h¯ [H, ?]+ ?(n¯ + 1)D [??]? + ?n¯D [?+]? (10.40)


where n¯ is the thermal occupation of the radiation field mode at the atomic resonance frequency ?a = ?2 ? ?1. We have neglected a small term which gives rise to a
shift in the atomic transition frequency and which contributes to the Lamb shift. At optical frequencies, n¯ ? 0. In the case of a free two-level atom, H = h¯ ?a ?z the probability to find the atom in the excited state, pe(t)= (e|?|e) satisfies the equation


dpe =
dt


?? pe(t) (10.41)


with the solution pe(t) = pe(0)e??t , which describes spontaneous emission. The dipole polarisation is proportional to the atomic coherence, (e|?|g) = (??) which
obeys





with the solution

d(??) =
dt

? .i?a + ? .

(??) (10.42)

(??(t)) = (??(0))e?

(?/2+i?a )t

(10.43)

The dipole oscillates at the transition frequency and decays, as it radiates.
The radiated field is related to the input field and the local source through an in- put/output relation in analogy with the case of a cavity discussed above. The positive frequency components of the field operator takes the form


E(+


(+)

2 ?x . ?x

o (?x,t)= Ei (?x,t) ? 4?? c2r

?? × r

× r ??(t ? x/c) (10.44)


where r = |?x| is the distance from the source to the point?x and ?? is the atomic dipole
moment.



Phase Decay in a Two-Level System


Spontaneous emission is not the only irreversible process involved in the absorption and emission of light. In an atomic vapour, atomic collisions are also a source of de-
coherence and cause a decay of the atomic polarisation, ?x + i?y, without changing
the decay of the inversion, ?z. We can model this process by a coupling between the
inversion and a high temperature heat bath,

Hcol = ?z?c(t) (10.45)

where ?c(t) is a bath operator describing the collisions. This Hamiltonian com- mutes with ?z and thus does not contribute to the decay of the inversion. It appears
like a fluctuating detuning in the Bloch equations and thus will effect the atomic


polarisation. The corresponding master equation, in the interaction picture and in- cluding spontaneous emission, is

d? = ? (2? ??


o ? ?


?? ? )


? [? , [? [, ?]] (10.46)

dt 2

? + ?

+ + ?

+ + ? p z z


The Bloch equations now become
d(?z) =




?( ?




+ 1) (10.47)

dt ? ( z)
d(?x ) = . ? .


dt
d(?y ) =
dt

? + ?p
. ? + ? .
2

(?x) (10.48)

(?y) (10.49)


In the presence of collisions the decay time for the polarisation, T2, is no longer given by twice the decay time for the inversion, T1 = ??1, but rather T2 < 2T1.



Resonance Fluorescence


If the atom is driven by a classical radiation field, the Hamiltonian becomes (see (10.30) and replace b ?? ?)

h¯ ?a
H = ?z + ?(?+e?
2


i?Lt


+ ??e


i?Lt


) (10.50)


where ? = g? is the Rabi frequency and ?L is the carrier frequency of the driving field. The master equation in an interaction picture at the frequency ?L is

d? =
dt


i ?? [? , ?] i?[? + ? , ?]+ ?D [? ]? (10.51) 2

where the detuning is ?? = ?a ? ?L. The resulting Bloch equations for the atomic moments are linear
d(??) = . ? .

dt ?

+ i??
2

(??) + i?(?z) (10.52)

d(?z) =


?( ?


+ 1)


2i?( ?


? ) (10.53)

dt ?

( z)

? ( +)? ( ?)


These inhomogeneous equations can be written as homogeneous equations as

d
dt ((??(t))? (??)ss)= A((??(t))? (??)ss) (10.54)



where


?? . ? . ?

2 ? i??

0 ?i?

A = ?

0 ? . ? + i??. i? ? (10.55)
?2i? 2i? ??
T

with (??) = ((?+), (??), (?z))

and the steady state solutions are

1 + ?2

(?z)ss = ? 1 + ?2 + Z2 (10.56)
i Z(1 + i?)




with

(?+)ss = ?2 1 + ?2 + Z2 (10.57)

2?2?
Z =
?


, ? =

2??
?


(10.58)

The solutions for resonance (?? = 0), with the atom initially in the ground state, are

8?2 .

3?t/4 .

3? ..

(?z(t)) = ?2 + 8?2

1 ? e?

cosh ?t + sinh ?t
4?

? 1 (10.59)

? . 3?t/4 .

. ? 3? . ..

(?+(t)) = 2i? ?2 + 8?2

1 ? e?

cosh ?t +

+
? 16?

sinh ?t

(10.60)


where



. 2
2

? = 2

4 ? 16?

(10.61)

Clearly there is a threshold at ? = ?/8 below which the solutions monotonically
approach the steady state and above which they are oscillating. A similar threshold occurs in the solutions for the two-time correlation function (?+(t)??(t + ?))t??
which determines the spectrum of the scattered light.
The stationary spectrum, as measured by a monochromatic detector at the point
?x is defined by [9]


S(?x, ?)= lim

1 ¸ ?


(E(?)(?x,t)E(+)(?x,t + ?))d? (10.62)

t?? 2? ??

the Fourier transform of the stationary two-time correlation function
(E(?)(t)E(+)(t + ?)) which using (10.44) is given by


S(?x, ?)=

I0 (?x) ¸ ?
2? ??


d?e?


i??


G(?) (10.63)



where

. ?2 .

x . x .2

I0(?x)= . 0

? ?
?? × × .

(10.64)



and

. 4??0c2r

r r .



with

G(?)= lim(?+(t)??(t + ?))? (?+??(?))ss (10.65)
?

(?+(t)??(t + ?)) = tr

.
??e

?(t)?+.

(10.66)

The equation of motion for G(?) couples in many other moments. If we define the correlation matrix
(?+?+(?))ss (???+(?))ss (?z?+(?))ss?
G (?)= ?(?+??(?))ss (????(?))ss (?z??(?))ss? (10.67)
(?+?z(?))ss (???z(?))ss (?z?z(?))ss

The quantum regression theorem indicates that G (?) as a function of ? obeys the same equations of motion as (??(?))? (??)ss,
G (?) = AG (?) (10.68)
d?

The initial conditions are simplified due to the algebra of the Pauli matrices, for
2

example ?+?? = (?z + 1)/2 and ?

= 0, and may thus be written in terms of the

stationary solutions in (10.57). On resonance we find that in the Schro¨ dinger picture,



G(?)=

4?2

. ?2


e?i?at +


1 e?(?/2+i?a)?

?2 + 8?2

?2 + 8?2 2

1 . ?2

3?/4 + ?

?/2

?/4 + ? .

? 2 ?2 + 8?2

? ? ? ? 2?

exp{?(3?/4 ? ? + i?a)?}

1 . ?2
+

3?/4 ? ?
?

?/2
?

?/4 ? ? .

.
exp{?(3?/4 + ? + i?a)?}

2 ?2 + 8?2 ?

? 2?

(10.69)

with ? ? 0. The corresponding spectrum has a single Lorentzian peak for weak driving fields, 4? << ?2/16,

4?2
S(?x, ?)= I0(r) ?2 + 8?2 ?(? ? ?a) (10.70)

which corresponds to elastic scattering. For very strong driving fields, ? >> ? we find that the spectrum acquires three Lorentzian peaks at ? = ?a and ? = ?a ± 2?.
The spectrum, including the elastic term, is

I0(r) .


4?2


1 ?/2

S(?x, ?)= 2?

2? ?(? ? )+
?2 + 8?2 2 ?2/4 + (? ?

?a)2

+ 1 3?/4

+ 1 3?/4 .

4 (3?/4)2 + [? ? (? + 2?)]2

4 (3?/4)2 + [? ? (? ? 2?)]2
(10.71)


This is the Mollow spectrum [10].
The light scattered by a two-level atom also exhibits photon anti-bunching. Con- sider the conditional probability that given a photon is counted at time t another
photon will be counted a time ? later. This is proportional to the second order corre-
lation function
G(2)(t, ?)= (a†(t)a†(t + ?)a(t + ?)a(t)) (10.72)
Usually we are interested in a stationary source so we let t ? ? and we normalise this by the intensity squared to define



g(2)(?)= lim

G(2)

(t, ?)
2


(10.73)

t?? (a†(t)a(t))

Using the result in (10.44) we can express this directly in terms of correlation func- tions for the atomic polarisation. As the equations of motion for the atomic variables
are linear, the stationary correlation function (?+(t)?+(t + ?)??(t + ?)??(t))t?? is
given by the quantum regression theorem. We then find that


g(2)(?)= 1 ? e?3?t/4

.
cosh ?? +

3? .
sinh ??
4?



(10.74)


The result g(2)(? = 0)= 0 indicates photon anti-bunching, as the probability to count a second photon, immediately after a first one has been counted, vanishes. This is a direct result of the emission process of the source. Photons are emitted when an excited atom relaxes back to the ground state. If a photon is counted, the atom is likely to be in ground state and thus a finite time must elapse before it is re-excited and capable of emitting another one. The probability to find the atom in the excited
state at time ? given that it starts in the ground state at ? = 0 is

4?2 .

3??/4 .

3? ..

Pe(?)= ?2 + 8?2

1 ? e?

cosh ?? + sinh ??
4?

(10.75)


Comparison with (10.74) indicates this interpretation is correct. This prediction, first made by Carmichael and Walls [8], was one of the earliest examples of how quantum optics would differ from a semiclassical description of light. In Fig. 10.3
we plot g(2)(?) for two values of the Rabi frequency.
The first observation of photon antibunching was made by Kimble et al. in 1977 on atomic beams [11]. They saw a positive slope for g(2)(?) which is consistent
with the predictions of the theory, however fluctuations from atomic numbers in the beam made a detailed comparison with the single atom result impossible. Ion traps


Fig. 10.3 The second order correlation function of the fluorescent light, given by
(10.74) versus delay time ?.
The solid line corresponds to ? = 2.5, while the dashed line corresponds to ? = 0.25. In both cases ? = 1.0







g(2)(?)


2


1.5


1


0.5



2 4 6 8 10
?



(see Chap. 17) provided a means to observe photon antibunching from a single atom [12]. In Fig. 10.4 we show the results of a measurement of the second order corre- lation function performed on a single trapped mercury ion by Walther’s Garching group [13].






Fig. 10.4 The second order correlation function of the fluorescent light form a single mercury ion in a trap versus delay, ?. (a)? = ?2.3?, ? = 2.8?. (b)? = ?1.1?, ? = ?, (c)? = ?0.5?, ? = 0.6? (from [13])


Exercises


In the Jaynes–Cummings model, show that if the atom begins in the ground state and the field begins in the state |?) = ? fn |n), the probability to find the
atom in the excited state at time t > 0 is given by


?
pe(t)= ? | fn|2 sin2(?n
n=1



?1t) (10.76)


In the Jaynes–Cummings model, show that if the atom begins in the ground state and the field begins in the state |?) = ? fn|n), the state at time t > 0 is
the entangled state





where

|?(t)) = |?g(t))|g) + |?e(t))|e) (10.77)


|?g(t)) = ? fn cos(?n?1t) (10.78)
n
|?e(t)) = i ? fn sin(?n?1t) (10.79)
n


In the case of the field initially in a coherent state, plot the Q-functions for
|?e(t)) and |?e(t)) at times equal to half way to the first revival and at the first
revival.
Compute the dressed states, and the corresponding eigenvalues, for the case in which the field mode is detuned from the atomic resonance, ? = ?a ? ?c ?= 0.
Define the trace-preserving density operator map on the state of a single two- level system,
? ?? E (?)= ?z??z (10.80)
Sow that this leaves unchanged the diagonal elements of ? in the eigenstates of ?z, but changes the phase of the off-diagonal elements by ?.
Calculate the second order correlation function g(2)(?) for resonance fluores-
cence in the presence of atomic dephasing (see (10.46).