Quantized electromagnetic eld and the
JaynesCummings Hamiltonian
a lecture in Quantum Informatics the 17th and 20th of September 2012
Goran Johansson and Thilo Bauch
This lecture deals with the quantization of a single mode electromagnetic
eld.The interaction between this mode and an atom is discussed,and the
JaynesCummings Hamiltonian is derived.
1 Quantizing a single electromagnetic mode
Consider a one dimensional cavity along the zaxis,with perfectly conducting
walls at z = 0 and z = L.Maxwell's sourcefree equations
rE =
@B
@t
rB =
0
"
0
@E
@t
r B = 0
r E = 0
The electric eld is polarized along the xaxis,E(r;t) = e
x
E
x
(z;t) and to full
the boundary conditions of vanishing Eeld at z = 0 and z = L,
E
x
(z;t) =
2!
2
V"
0
1=2
q(t) sin(kz);
where k
m
= m=L for an integer m> 0,!
m
= ck
m
.V is the eective volume of
the cavity,and q(t) is a timedependent amplitude with dimension length.From
the second of Maxwell's equations we nd the magnetic eld B(r;t) = e
y
B
y
(z;t)
B
y
(z;t) =
0
"
0
k
2!
2
V"
0
1=2
_q(t) cos (kz);
and the classical eld energy,i.e.the Hamiltonian is
H =
1
2
Z
dV
"
0
E
2
(r;t) +
1
0
B
2
(r;t)
=
1
=
1
2
Z
dV
"
0
E
2
x
(z;t) +
1
0
B
2
y
(z;t)
=
=
1
2
p
2
+!
2
q
2
;
which is the Hamiltonian of a harmonic oscillator of frequency!,with coordinate
q and momentum p = _q.
2 Quantized harmonic oscillator  photons
It is now well established that the electromagnetic eld in general,and a single
mode eld in particular,is quantized.Each mode is a quantum mechanical
harmonic oscillator,and we promote the coordinate q(t) and momentum p(t) to
operators
^
H =
1
2
^p
2
+!
2
^q
2
;
where ^q and ^p obey the canonical commutation relation
[^q;^p] = ih
^
1:
The electric eld of the mode is now an operator
^
E
x
(z;t) =
2!
2
V"
0
1=2
^q(t) sin(kz);
and of course also the magnetic eld
^
B
y
(z;t) =
0
"
0
k
2!
2
V"
0
1=2
^p(t) cos (kz) =
r
2
0
V
^p(t) cos (kz):
2.1 Creation and annihilation operators
It is useful to introduce the socalled creation (^a
y
) and annihilation (^a) operators
^a =
!^q +i^p
p
2h!
and ^a
y
=
!^q i^p
p
2h!
;
obeying the commutation relation
[^a;^a
y
] =
^
1:
The Hamiltonian now has the familiar form
^
H = h!
^a
y
^a +
1
2
= h!
^n +
1
2
;
where ^n = ^a
y
^a is the number operator,counting the number of excitations,i.e.
photons,in the mode.The electric eld operator is
^
E
x
(z;t) = E
0
(^a +^a
y
) sin(kz);
2
<q0>
<q1>
<q2>
<q3>
<q4>
<q5>
<q6>
<q7>
Energy,<qn>
q
h
Figure 1:Quantum harmonic oscillator.The Eigenenergies of the harmonic os
cillator are equidistantly spaced resulting in the spectrumE
n
= h!(n+1=2).The
wavefunctions hqjni for the number states jni for n = 0 to n = 7 are visualized
on top of the corresponding energy levels.Picture partly from Wikipedia.
and similarly for the magnetic eld operator
^
B
y
(z;t) = iB
0
(^a ^a
y
) cos (kz);
where E
0
=
p
h!="
0
V and B
0
=
p
0
h!=V represents the electric and magnetic
elds"per photon".The quotation mark indicate that this somewhat inexact,
as the average elds of photon number states are exactly zero.But it is a useful
measure of e.g.the uctuations of the eld.
2.2 Photon number states
So each mode of the electromagnetic eld is a quantum mechanical harmonic
oscillator,and both the electric and magnetic eld are operators.The photon
number state jni is an eigenstate to the Hamiltonian,
^
Hjni = h!
n +
1
2
jni;
with energy (n+1=2)h!.The ground state j0i is also called the vacuumand has
the energy h!=2.The vacuumenergy is also called the zeropoint energy,and the
uctuations of the electromagnetic eld in its vacuum state are called vacuum
uctuations.Fig.1 displays the eigenenergies and corresponding eigenstates jni
3
(represented as wave functions hqjni) of the harmonic oscillator Hamiltonian.
The creation operator adds a photon to the eld
^a
y
jni =
p
n +1jn +1i;
while the annihilation operator destroys one photon
^ajni =
p
njn 1i;
and gives zero acting on the vacuum state
^aj0i = 0:
The number states are normed and orthogonal hnjmi =
nm
,and form a com
plete basis for the eld
1
X
n=0
jnihnj =
^
1:
In nature,the only pure number state commonly observed is the vacuum.The
other number states are"nonclassical"and the creation and detection of num
ber states is a research topic ("Generation of Fock states in a superconducting
quantum circuit",M.Hofheinz et al.,Nature 454,310314 (2008),"Resolving
photon number states in a superconducting circuit",D.I.Schuster et al.,Na
ture 445,515518 (2007),and"Quantumjumps of light recording the birth and
death of a photon in a cavity",S.Gleyzes,et al.,Nature 446,297300 (2007)).
E.g.the average electric eld of a number state is zero
hnj
^
E
x
(z;t)jni = E
0
sin(kz)hnj(^a +^a
y
)jni = 0;
and similarly for the magnetic eld.For a highly excited oscillator this is a very
"unusual"state,as we know classically the eld performs harmonic oscillations
in time.
3 Coherent States
The eld states that most resembles classical states are the socalled coherent
states (R.J.Glauber,Phys.Rev.131,2766 (1963)),for which Roy Glauber
got the Nobel Prize in 2005.If the mode contain a large number of photons
N,the uctuations around the average eld values are small 1=
p
N,so the
classical description is a very good approximation.However,in this quantum
informatics course we will mostly work with elds on the single photon level,so
we will only give a brief denition of coherent states and refer the student being
more interested in coherent states to a quantum optics textbook.
The coherent states ji are eigenstates of the annihilation operator with the
complex eigenvalue ,
^aji = ji;hj^a
y
=
hj:
4
The eigenvalue is not real since ^a is not Hermitian,i.e.not an observable.
Expanded in the number basis we nd
ji = e
jj
2
=2
1
X
n=0
n
p
n!
jni:
Consider the timedependence of the coherent state
j(t)i = e
jj
2
=2
1
X
n=0
n
e
in!t
p
n!
jni = e
jj
2
=2
1
X
n=0
e
i!t
n
p
n!
jni = j(0)e
i!t
i;
and now evaluate the average electric eld
h(t)j
^
E
x
(z)j(t)i = E
0
sin(kz)h(t)j(^a +^a
y
)j(t)i =
= E
0
sin(kz)((0)e
i!t
+(0)
e
i!t
) = E
0
sin(kz)j(0)j cos (
0
!t);
where (0) = j(0)je
i
0
,i.e.a classical eld with amplitude j(0)j and phase
0
.The probability to nd the eld in a state with n photons is
P
n
= jhnjij
2
= je
jj
2
=2
n
p
n!
j
2
= e
n
n
n
n!
;
which is the Poisson distribution with average n = hj^nji = jj
2
.The standard
deviation of this distribution is n =
p
h^n
2
i h^ni
2
=
p
n giving that the
fractional width of the photon distribution n=n goes towards zero as 1=
p
n for
large amplitude coherent states jj 1,i.e.the classical limit.
4 Interaction between a single mode eld and
an atom
Recently it has become possible to create cavities supporting a single mode of
suciently high quality factor Q to investigate the dynamics of a single atom
interacting with a single photon in a single mode.The quality factor indicates
the lifetime of a single photon in the cavity Q=!,where!is the photon fre
quency.In the group of Serge Haroche in Paris (see Gleyzes et al.),a high Q
microwave cavity with resonance frequency 51.1 GHz was fabricated.The decay
time of a eld in this cavity is 0.13 s,giving a quality factor of Q 6 10
9
.The
light travels 39.000 km before leaving the cavity.This is necessary since the
coupling of the single photon eld to an atom is indeed weak,mainly due to the
smallness of the atom,compared to the wavelength of the light.To increase the
coupling socalled Rydberg atoms are used,where highly excited electron states,
following almost classical orbits.The coupling strength is 51 kHz,leaving room
for 6000 atomeld oscillations in the decay time of the cavity.
Another system where the interaction between a twolevel system and a sin
gle photon can be studied is a superconducting qubit placed in a superconduct
ing stripline cavity.The coplanar stripline cavity can be viewed as a squashed
5
coaxial cable.The length of the cavity corresponds to the microwave wave
length,i.e.millimeters.On the contrary,the width of the cavity is only a few
micrometers,making even the single photon electric eld comparably strong.
The qubit can be designed to have a rather large"dipole moment".In the rst
realization of this system,by the group of Rob Schoelkopf at Yale university (A.
Wallra et al.,Nature (London) 431 162 (2004)),the resonance frequency was 6
GHz,the qubitcavity coupling strength was 11.6 MHz,and the cavity damping
rate was 0.8 MHz (Q 7500).Thus a moderate quality factor is compensated
by the cavity geometry increasing the electric eld strength.
5 The JaynesCummings Hamiltonian
Starting from the Rabi hamiltonian
^
H =
^
H
0
^
d E(t);
we simply replace E(t)!
^
E(t).Dening
^
d =
^
d e
x
;
we get
^
H =
^
H
0
E
0
(^a +^a
y
) sin(kz)
^
d:
Noting that hej
^
djei = hgj
^
djgi = 0 because of parity,and dening d hej
^
djgi and
we get hgj
^
djei = d
,we nd that
^
d has only odiagonal elements.Without loss
o generality we assume a real d and get
^
d = d
x
.Dening g = E
0
sin(kz)d
we arrive at
^
H =
^
H
0
+g
x
(^a +^a
y
):
In the spirit of the RWA performed in the analysis of the Rabi oscillations,
we rst write
x
=
0 1
0 0
+
0 0
1 0
+
+
;
where
+
excites and
deexcites the atom.The coupling has four terms
^a
+
;^a
y
;^a
;and ^a
y
+
:
In the near resonance case (j!!
0
j !
0
),the two rst terms are nearly energy
conserving,exciting the atomwhile annihilating a photon or deexciting the atom
while creating a photon.These terms loosely correspond to the slow terms in
the Rabi analysis.Neglecting the two terms ^a
and ^a
y
+
is also known as the
RWA,and we arrive at the JaynesCummings hamiltonian
^
H =
h!
0
2
z
+h!^a
y
^a +g(^a
+
+^a
y
):
Note that we removed the term
h!
2
^
1,which implies a new common reference of
energy.The same procedure was done when deriving the Hamiltonian of the
charge qubit (see Eq.15 and subsequent paragraph on page 4 of the lecture
notes"A quantum bit { in theory and one realization")
6
6 Discussion of the JaynesCummings dynamics
The Hamiltonian acts in the Hilbert space spanned by the vectors jgijni and
jeijni.The uncoupled spectrum (g = 0) is given by two equidistant ladders
shifted by the atomic energy dierence h!
0
,i.e.E
gn
= nh!h!
0
=2 and E
en
=
nh!+h!
0
=2 (see Fig.2).These are the diagonal elements of the Hamiltonian
in this product basis.
Due to the RWA the remaining interaction term conserves the excitation
number,i.e.
^
N
e
= ^n + jeihej = ^n + (
^
1
z
)=2,which commutes with the
Hamiltonian [
^
H;
^
N
e
] = 0.This implies that the interaction can only couple the
two states jeijni and jgijn +1i.Thus,the Hamiltonian is block diagonal with
two by two blocks.The odiagonal elements are
hejhnjg(^a
+
+^a
y
)jgijn +1i = hgjhn +1jg(^a
+
+^a
y
)jeijni = g
p
n +1:
So in the twodimensional subspace spanned by jeijni and jgijn + 1i,the
JaynesCummings Hamiltonian looks like
^
H
n
= h!
n +
1
2
^
1
h(!
0
!)
2
z
+g
p
n +1
x
:
Thus,in this twolevel system the detuning maps onto a magnetic eld in the
zdirection and the atomeld coupling maps onto a magnetic eld in the x
direction.From our experience with the charge qubit we can immediately write
down the eigenenergies
E
n
= h!
n +
1
2
q
h
2
2
+4g
2
(n +1)
2
;
where =!
0
!is the detuning,and the eigenstates are
jn;+i = cos
n
2
jeijni +sin
n
2
jgijn +1i;
jn;i = sin
n
2
jeijni +cos
n
2
jgijn +1i;
where the mixing angle is dened through
n
= arctan
2g
p
n +1
h
:
We see that at resonance!=!
0
,the eigenstates are equal superpositions of
jeijni and jgijn +1i,which is similar to the charge qubit at degeneracy.
To compare with the Rabi dynamics we can start with the systemin the state
jgijn+1i.This corresponds to a state pointing in the zdirection in the mapping
to the Bloch sphere.The atomeld coupling will make the state rotate around
the xaxis with angular velocity
(n) = 2g
p
n +1=h,which is sometimes called
a quantum electrodynamic Rabi frequency,or a single photon Rabi nutation
frequency.
The states jeijni and jgijn + 1i are often called the bare states of the JC
Hamiltonian,while the states jn;i are called the dressed states.
7
Energy
0>
1>
2>
3>
n+1>
0>
0>
1>
1>
2>
2>
n>
0>
1>
2>
n>
2g
2gn+1
g>
e>
g>
e>
h g/h
2
h g/h+
2
=
0
=
0
(a) (b)
0
n>
Energy
0
h+(2n+1)g/h
0
2
Figure 2:(a) Energy spectrum of the uncoupled (left and right) and coupled
(center) atomphoton states for zero detuning (!=!
0
).The degeneracy of the
two states jeijni and jgijn +1i,with total excitation number n +1,is lifted by
2g
p
n +1 (b) Energy spectrum in the dispersive regime,hjj g,(long dashed
lines).To second order in g=h,the level separation is independent of n,but
depends on the state of the atom.From A.Blais et al.,Phys.Rev.A 69 062320
(2004)
7 The dispersive regime
In the regime of large detuning ( =!
0
!;hjj g) the coupling between
the eld and the atom cannot induce any real transitions,but still the coupling
will renormalize the energies of the system.Doing perturbation theory in the
parameter g= one nd the eective dispersive Hamiltonian
^
H
disp
=
1
2
h!
0
+
g
2
h
z
+
h!
g
2
h
z
^a
y
^a:
This implies that the eective cavity frequency now depends on the atom state,
and equivalently the atomic energy splitting depends on the number of photons
in the cavity (see Fig.2).The term proportional to
z
^a
y
^a is called the ac
Stark shift,while the"vacuum"shift of the atomic energy splitting (
z
g
2
=h)
is called the Lamb shift.From the above Hamiltonian,which is diagonal in the
base representation jgijni and jeijni,we can directly determine the eigenenergies
of the system:
E
gn
=
1
2
h!
0
+
g
2
h
+n
h!
g
2
h
;
E
en
=
1
2
h!
0
+
g
2
h
+n
h!+
g
2
h
:
8
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