Quantized electromagnetic eld and the

Jaynes-Cummings 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

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

rE =

@B

@t

rB =

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 full

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 eective 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] = 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 so-called 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

<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 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 time-dependence 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 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).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 o-diagonal 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

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 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 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 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 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

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) 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,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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