12

Introduction
\Truth and clarity are complementary."
N. Bohr
In the _rst part of these lectures we will focus our attention on some aspects
of the notion of a photon in modern quantum optics and a relativistic description
of single, localized, photons. In the second part we will discuss in great detail the
\standard model" of quantum optics, i.e. the Jaynes-Cummings model describing
the interaction of a two-mode system with a single mode of the second-quantized
electro-magnetic _eld and its realization in resonant cavities in terms of in particular
the micro-maser system. Most of the material presented in these lectures has
appeared in one form or another elsewhere. Material for the _rst set of lectures can
be found in Refs.[1, 2] and for the second part of the lectures we refer to Refs.[3, 4].
The lectures are organized as follows. In Section 2 we discuss some basic quantum
mechanics and the notion of coherent and semi-coherent states. Elements form
the photon-detection theory of Glauber is discussed in Section 3 as well as the
experimental veri_cation of quantum-mechanical single-photon interference. Some
applications of the ideas of photon-detection theory in high-energy physics are also
briey mentioned. In Section 4 we outline a relativistic and quantum-mechanical
theory of single photons. The Berry phase for single photons is then derived within
such a quantum-mechanical scheme. We also discuss properties of single-photon
wave-packets which by construction have positive energy. In Section 6 we present
the standard theoretical framework for the micromaser and introduce the notion of a
correlation length in the outgoing atomic beam as was _rst introduced in Refs.[3, 4].
A general discussion of long-time correlations is given in Section 7, where we also
show how one can determine the correlation length numerically. Before entering the
analytic investigation of the phase structure we introduce some useful concepts in
Section 8 and discuss the eigenvalue problem for the correlation length. In Section 9
details of the di_erent phases are analyzed. In Section 10 we discuss e_ects related
to the _nite spread in atomic velocities. The phase boundaries are de_ned in the
limit of an in_nite ux of atoms, but there are several interesting e_ects related to
_nite uxes as well. We discuss these issues in Section 11. Final remarks and a
summary is given in Section 12.
2 Basic Quantum Mechanics
\Quantum mechanics, that mysterious, confusing
discipline, which none of us really understands,
but which we know how to use"
M. Gell-Mann
Quantum mechanics, we believe, is the fundamental framework for the description
of all known natural physical phenomena. Still we are, however, often very
1
often puzzled about the role of concepts from the domain of classical physics within
the quantum-mechanical language. The interpretation of the theoretical framework
of quantum mechanics is, of course, directly connected to the \classical picture"
of physical phenomena. We often talk about quantization of the classical observables
in particular so with regard to classical dynamical systems in the Hamiltonian
formulation as has so beautifully been discussed by Dirac [5] and others (see e.g.
Ref.[6]).
2.1 Coherent States
The concept of coherent states is very useful in trying to orient the inquiring mind in
this jungle of conceptually di_cult issues when connecting classical pictures of physical
phenomena with the fundamental notion of quantum-mechanical probabilityamplitudes
and probabilities. We will not try to make a general enough de_nition
of the concept of coherent states (for such an attempt see e.g. the introduction of
Ref.[7]). There are, however, many excellent text-books [8, 9, 10], recent reviews [11]
and other expositions of the subject [7] to which we will refer to for details and/or
other aspects of the subject under consideration. To our knowledge, the modern
notion of coherent states actually goes back to the pioneering work by Lee, Low and
Pines in 1953 [12] on a quantum-mechanical variational principle. These authors
studied electrons in low-lying conduction bands. This is a strong-coupling problem
due to interactions with the longitudinal optical modes of lattice vibrations and in
Ref.[12] a variational calculation was performed using coherent states. The concept
of coherent states as we use in the context of quantum optics goes back Klauder [13],
Glauber [14] and Sudarshan [15]. We will refer to these states as Glauber-Klauder
coherent states.
As is well-known, coherent states appear in a very natural way when considering
the classical limit or the infrared properties of quantum _eld theories like quantum
electrodynamics (QED)[16]-[21] or in analysis of the infrared properties of quantum
gravity [22, 23]. In the conventional and extremely successful application of perturbative
quantum _eld theory in the description of elementary processes in Nature
when gravitons are not taken into account, the number-operator Fock-space representation
is the natural Hilbert space. The realization of the canonical commutation
relations of the quantum _elds leads, of course, in general to mathematical di_culties
when interactions are taken into account. Over the years we have, however, in
practice learned how to deal with some of these mathematical di_culties.
In presenting the theory of the second-quantized electro-magnetic _eld on an
elementary level, it is tempting to exhibit an apparent \paradox" of Erhenfest theorem
in quantum mechanics and the existence of the classical Maxwell s equations:
any average of the electro-magnetic _eld-strengths in the physically natural numberoperator
basis is zero and hence these averages will not obey the classical equations
of motion. The solution of this apparent paradox is, as is by now well established:
the classical _elds in Maxwell s equations corresponds to quantum states with an
2
arbitrary number of photons. In classical physics, we may neglect the quantum
structure of the charged sources. Let j(x; t) be such a classical current, like the
classical current in a coil, and A(x; t) the second-quantized radiation _eld (in e.g.
the radiation gauge). In the long wave-length limit of the radiation _eld a classical
current should be an appropriate approximation at least for theories like quantum
electrodynamics. The interaction Hamiltonian HI (t) then takes the form
HI (t) = ??
Z
d3x j(x; t) _ A(x; t) ; (2.1)
and the quantum states in the interaction picture, jtiI , obey the time-dependent
Schrodinger equation, i.e. using natural units (_h = c = 1)
i
d
dtjtiI = HI (t)jtiI : (2.2)
For reasons of simplicity, we will consider only one speci_c mode of the electromagnetic
_eld described in terms of a canonical creation operator (a_) and an annihilation
operator (a). The general case then easily follows by considering a system
of such independent modes (see e.g. Ref.[24]). It is therefore su_cient to consider
the following single-mode interaction Hamiltonian:
HI (t) = ??f(t)
_
a exp[??i!t] + a_ exp[i!t]
_
; (2.3)
where the real-valued function f(t) describes the in general time-dependent classical
current. The \free" part H0 of the total Hamiltonian in natural units then is
H0 = !(a_a + 1=2) : (2.4)
In terms of canonical \momentum" (p) and \position" (x) _eld-quadrature degrees
of freedom de_ned by
a =
r
!
2
x + i
1
p2!
p ;
a_ =
r
!
2
x ?? i
1
p2!
p ; (2.5)
we therefore see that we are formally considering an harmonic oscillator in the
presence of a time-dependent external force. The explicit solution to Eq.(2.2) is
easily found. We can write
jtiI = T exp
_
??i
Z t
t0
HI (t0)dt0
_
jt0iI = exp[i_(t)] exp[iA(t)]jt0iI ; (2.6)
where the non-trivial time-ordering procedure is expressed in terms of
A(t) = ??
Z t
t0
dt0HI (t0) ; (2.7)
3
and the c-number phase _(t) as given by
_(t) =
i
2
Z t
t0
dt0[A(t0);HI (t0)] : (2.8)
The form of this solution is valid for any interaction Hamiltonian which is at most
linear in creation and annihilation operators (see e.g. Ref.[25]). We now de_ne the
unitary operator
U(z) = exp[za_ ?? z_a] : (2.9)
Canonical coherent states jz; _0i, depending on the (complex) parameter z and the
_ducial normalized state number-operator eigenstate j_0i, are de_ned by
jz; _0i = U(z)j_0i ; (2.10)
such that
1 =
Z
d2z
_ jzihzj =
Z
d2z
_ jz; _0ihz; _0j : (2.11)
Here the canonical coherent-state jzi corresponds to the choice jz; 0i, i.e. to an
initial Fock vacuum state. We then see that, up to a phase, the solution Eq.(2.6)
is a canonical coherent-state if the initial state is the vacuum state. It can be
veri_ed that the expectation value of the second-quantized electro-magnetic _eld
in the state jtiI obeys the classical Maxwell equations of motion for any _ducial
Fock-space state jt0iI = j_0i. Therefore the corresponding complex, and in general
time-dependent, parameters z constitute an explicit mapping between classical
phase-space dynamical variables and a pure quantum-mechanical state. In more general
terms, quantum-mechanical models can actually be constructed which demonstrates
that by the process of phase-decoherence one is naturally lead to such a
correspondence between points in classical phase-space and coherent states (see e.g.
Ref.[26]).
2.2 Semi-Coherent or Displaced Coherent States
If the _ducial state j_0i is a number operator eigenstate jmi, where m is an integer,
the corresponding coherent-state jz;mi have recently been discussed in detail in the
literature and is referred to as a semi-coherent state [27, 28] or a displaced numberoperator
state [29]. For some recent considerations see e.g. Refs.[30, 31] and in
the context of resonant micro-cavities see Refs.[32, 33]. We will now argue that
a classical current can be used to amplify the information contained in the pure
_ducial vector j_0i. In Section 6 we will give further discussions on this topic. For a
given initial _ducial Fock-state vector jmi, it is a rather trivial exercise to calculate
the probability P(n) to _nd n photons in the _nal state, i.e. (see e.g. Ref.[34])
P(n) = lim
t!1 jhnjtiI j2 ; (2.12)
4
0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
< n > = jzj2 = 49
P(n)
n
Figure 1: Photon number distribution of coherent (with an initial vacuum state jt = 0i =
j0i - solid curve) and semi-coherent states (with an initial one-photon state jt = 0i = j1i
- dashed curve).
which then depends on the Fourier transform z = f(!) =
R
1
??1 dtf(t) exp(??i!t).
In Figure 1, the solid curve gives P(n) for j_0i = j0i, where we, for the purpose
of illustration, have chosen the Fourier transform of f(t) such that the mean value
of the Poisson number-distribution of photons is jf(!)j2 = 49. The distribution
P(n) then characterize a classical state of the radiation _eld. The dashed curve in
Figure 1 corresponds to j_0i = j1i, and we observe the characteristic oscillations.
It may be a slight surprise that the minor change of the initial state by one photon
completely change the _nal distribution P(n) of photons, i.e. one photon among a
large number of photons (in the present case 49) makes a di_erence. If j_0i = jmi one _nds in the same way that the P(n)-distribution will have m zeros. If we sum
the distribution P(n) over the initial-state quantum number m we, of course, obtain
unity as a consequence of the unitarity of the time-evolution. Unitarity is actually
the simple quantum-mechanical reason why oscillations in P(n) must be present.
We also observe that two canonical coherent states jtiI are orthogonal if the initialstate
_ducial vectors are orthogonal. It is in the sense of oscillations in P(n), as
described above, that a classical current can amplify a quantum-mechanical pure
state j_0i to a coherent-state with a large number of coherent photons. This e_ect
is, of course, due to the boson character of photons.
It has, furthermore, been shown that one-photon states localized in space and
5
time can be generated in the laboratory (see e.g. [35]-[45]). It would be interesting
if such a state could be ampli_ed by means of a classical source in resonance with
the typical frequency of the photon. It has been argued by Knight et al. [29] that
an imperfect photon-detection by allowing for dissipation of _eld-energy does not
necessarily destroy the appearance of the oscillations in the probability distribution
P(n) of photons in the displaced number-operator eigenstates. It would, of course,
be an interesting and striking veri_cation of quantum coherence if the oscillations
in the P(n)-distribution could be observed experimentally.
3 Photon-Detection Theory
\If it was so, it might be; And if it were so,
it would be. But as it isn t, it ain t. "
Lewis Carrol
The quantum-mechanical description of optical coherence was developed in a
series of beautiful papers by Glauber [14]. Here we will only touch upon some
elementary considerations of photo-detection theory. Consider an experimental situation
where a beam of particles, in our case a beam of photons, hits an ideal
beam-splitter. Two photon-multipliers measures the corresponding intensities at
times t and t + _ of the two beams generated by the beam-splitter. The quantum
state describing the detection of one photon at time t and another one at time
t+ _ is then of the form E+(t+ _ )E+(t)jii, where jii describes the initial state and
where E+(t) denotes a positive-frequency component of the second-quantized electric
_eld. The quantum-mechanical amplitude for the detection of a _nal state jfi
then is hfjE+(t+_ )E+(t)jii. The total detection-probability, obtained by summing
over all _nal states, is then proportional to the second-order correlation function
g(2)(_ ) given by
g(2)(_ ) =
X
f
jhfjE+(t + _ )E+(t)jiij2
(hijE??(t)E+(t)jii)2)
= hijE??(t)E??(t + _ )E+(t + _ )E+(t)jii
(hijE??(t)E+(t)jii)2 :
(3.1)
PHere the normalization factor is just proportional to the intensity of the source, i.e.
f jhfjE+(t)jiij2 = (hijE??(t)E+(t)jii)2. A classical treatment of the radiation _eld
would then lead to
g(2)(0) = 1 +
1
hIi2
Z
dIP(I)(I ?? hIi)2 ; (3.2)
where I is the intensity of the radiation _eld and P(I) is a quasi-probability distribution
(i.e. not in general an apriori positive de_nite function). What we call classical
coherent light can then be described in terms of Glauber-Klauder coherent states.
These states leads to P(I) = _(I ?? hIi). As long as P(I) is a positive de_nite function,
there is a complete equivalence between the classical theory of optical coherence
6
and the quantum _eld-theoretical description [15]. Incoherent light, as thermal light,
leads to a second-order correlation function g(2)(_ ) which is larger than one. This
feature is referred to as photon bunching (see e.g. Ref.[46]). Quantum-mechanical
light is, however, described by a second-order correlation function which may be
smaller than one. If the beam consists of N photons, all with the same quantum
numbers, we easily _nd that
g(2)(0) = 1 ??
1
N
< 1 : (3.3)
Another way to express this form of photon anti-bunching is to say that in this
case the quasi-probability P(I) distribution cannot be positive, i.e. it cannot be
interpreted as a probability (for an account of the early history of anti-bunching see
e.g. Ref.[47, 48]).
3.1 Quantum Interference of Single Photons
A one-photon beam must, in particular, have the property that g(2)(0) = 0, which
simply corresponds to maximal photon anti-bunching. One would, perhaps, expect
that a su_ciently attenuated classical source of radiation, like the light from a pulsed
photo-diode or a laser, would exhibit photon maximal anti-bunching in a beam
splitter. This sort of reasoning is, in one way or another, explicitly assumed in many
of the beautiful tests of \single-photon" interference in quantum mechanics. It has,
however, been argued by Aspect and Grangier [49] that this reasoning is incorrect.
Aspect and Grangier actually measured the second-order correlation function g(2)(_ )
by making use of a beam-splitter and found this to be greater or equal to one even for
an attenuation of a classical light source below the one-photon level. The conclusion,
we guess, is that the radiation emitted from e.g. a monochromatic laser always
behaves in classical manner, i.e. even for such a strongly attenuated source below
the one-photon ux limit the corresponding radiation has no non-classical features
(under certain circumstances one can, of course, arrange for such an attenuated
light source with a very low probability for more than one-photon at a time (see
e.g. Refs.[50, 51]) but, nevertheless, the source can still be described in terms of
classical electro-magnetic _elds). As already mentioned in the introduction, it is,
however, possible to generate photon beams which exhibit complete photon antibunching.
This has _rst been shown in the beautiful experimental work by Aspect
and Grangier [49] and by Mandel and collaborators [35]. Roger, Grangier and Aspect
in their beautiful study also veri_ed that the one-photon states obtained exhibit
one-photon interference in accordance with the rules of quantum mechanics as we,
of course, expect. In the experiment by e.g. the Rochester group [35] beams of
one-photon states, localized in both space and time, were generated. A quantummechanical
description of such relativistic one-photon states will now be the subject
for Chapter 4.
7
3.2 Applications in High-Energy Physics
Many of the concepts from photon-detection theory has applications in the context
of high-energy physics. The use of photon-detection theory as mentioned in Section
3 goes historically back to Hanbury-Brown and Twiss [52] in which case the
second-order correlation function was used in order to extract information on the
size of distant stars. The same idea has been applied in high-energy physics. The
two-particle correlation function C2(p1; p2), where p1 and p2 are three-momenta of
the (boson) particles considered, is in this case given by the ratio of two-particle
probabilities P(p1; p2) and the product of the one-particle probabilities P(p1) and
P(p2), i.e. C2(p1; p2) = P(p1; p2)=P (p1)P(p2). For a source of pions where any
phase-coherence is averaged out, corresponding to what is called a chaotic source,
there is an enhanced emission probability as compared to a non-chaotic source over
a range of momenta such that Rjp1 ?? p2j 1, where R represents an average of
the size of the pion source. For pions formed in a coherent-state one _nds that
C2(p1; p2) = 1. The width of the experimentally determined correlation function of
pions with di_erent momenta, i.e. C2(p1; p2), can therefore give information about
the size of the pion-source. A lot of experimental data has been compiled over the
years and the subject has recently been discussed in detail by e.g. Boal et al. [53]. A
recent experimental analysis has been considered by the OPAL collaboration in the
case of like-sign charged track pairs at a center-o_-mass energy close to the Z0 peak.
146624 multi-hadronic Z0 candidates were used leading to an estimate of the radius
of the pion source to be close to one fermi [54]. Similarly the NA44 experiment at
CERN have studied _+_+-correlations from 227000 reconstructed pairs in S + Pb
collisions at 200 GeV=c per nucleon leading to a space-time averaged pion-source
radius of the order of a few fermi [55]. The impressive experimental data and its
interpretation has been confronted by simulations using relativistic molecular dynamics
[56]. In heavy-ion physics the measurement of the second-order correlation
function of pions is of special interest since it can give us information about the
spatial extent of the quark-gluon plasma phase, if it is formed. It has been suggested
that one may make use of photons instead of pions when studying possible
signals from the quark-gluon plasma. In particular, it has been suggested [57] that
the correlation of high transverse-momentum photons is sensitive to the details of
the space-time evolution of the high density quark-gluon plasma.