Quantized electromagnetic eld and the Jaynes-Cummings Hamiltonian

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Nov 15, 2013 (3 years and 4 months ago)

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Quantized electromagnetic eld and the
Jaynes-Cummings Hamiltonian
a lecture in Quantum Informatics the 17th and 20th of September 2012
Goran 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
Jaynes-Cummings Hamiltonian is derived.
1 Quantizing a single electromagnetic mode
Consider a one dimensional cavity along the z-axis,with perfectly conducting
walls at z = 0 and z = L.Maxwell's source-free equations
rE =
@B
@t
rB = 
0
"
0
@E
@t
r B = 0
r E = 0
The electric eld is polarized along the x-axis,E(r;t) = e
x
E
x
(z;t) and to full
the boundary conditions of vanishing E-eld 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 eective volume of
the cavity,and q(t) is a time-dependent 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] = ih
^
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 so-called creation (^a
y
) and annihilation (^a) operators
^a =
!^q +i^p
p
2h!
and ^a
y
=
!^q i^p
p
2h!
;
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
<q|0>
<q|1>
<q|2>
<q|3>
<q|4>
<q|5>
<q|6>
<q|7>
Energy,￿<q|n>
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 zero-point 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"non-classical"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 454,310-314 (2008),"Resolving
photon number states in a superconducting circuit",D.I.Schuster et al.,Na-
ture 445,515-518 (2007),and"Quantumjumps of light recording the birth and
death of a photon in a cavity",S.Gleyzes,et al.,Nature 446,297-300 (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 so-called 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 denition of coherent states and refer the student being
more interested in coherent states to a quantum optics textbook.
The coherent states ji are eigenstates of the annihilation operator with the
complex eigenvalue ,
^aji = ji;hj^a
y
= 

hj:
4
The eigenvalue is not real since ^a is not Hermitian,i.e.not an observable.
Expanded in the number basis we nd
ji = e
jj
2
=2
1
X
n=0

n
p
n!
jni:
Consider the time-dependence of the coherent state
j(t)i = e
jj
2
=2
1
X
n=0

n
e
in!t
p
n!
jni = e
jj
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
= jhnjij
2
= je
jj
2
=2

n
p
n!
j
2
= e
n
n
n
n!
;
which is the Poisson distribution with average n = hj^nji = jj
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 jj 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
suciently 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 so-called Rydberg atoms are used,where highly excited electron states,
following almost classical orbits.The coupling strength is 51 kHz,leaving room
for 6000 atom-eld oscillations in the decay time of the cavity.
Another system where the interaction between a two-level 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 qubit-cavity 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 Jaynes-Cummings Hamiltonian
Starting from the Rabi hamiltonian
^
H =
^
H
0

^
d  E(t);
we simply replace E(t)!
^
E(t).Dening
^
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 dening d  hej
^
djgi and
we get hgj
^
djei = d

,we nd that
^
d has only o-diagonal elements.Without loss
o generality we assume a real d and get
^
d = d
x
.Dening 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 

de-excites 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 Jaynes-Cummings 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 Jaynes-Cummings 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 dierence h!
0
,i.e.E
gn
= nh!h!
0
=2 and E
en
=
nh!+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 o-diagonal elements are
hejhnjg(^a
+
+^a
y


)jgijn +1i = hgjhn +1jg(^a
+
+^a
y


)jeijni = g
p
n +1:
So in the two-dimensional subspace spanned by jeijni and jgijn + 1i,the
Jaynes-Cummings Hamiltonian looks like
^
H
n
= h!

n +
1
2

^
1 
h(!
0
!)
2

z
+g
p
n +1
x
:
Thus,in this two-level system the detuning maps onto a magnetic eld in the
z-direction and the atom-eld 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 dened 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 z-direction in the mapping
to the Bloch sphere.The atom-eld coupling will make the state rotate around
the x-axis 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
2g￿n+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) atom-photon 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,hjj 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
!;hjj  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 eective dispersive Hamiltonian
^
H
disp
= 
1
2

h!
0
+
g
2
h


z
+

h!
g
2
h

z

^a
y
^a:
This implies that the eective 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