CQED

CQED

Abstract In this chapter we will discuss two different physical systems in which a single mode of the electromagnetic field in a cavity interacts with a two-level dipole emitter. In the first example, the system is comprised of a single two-level atom inside an optical cavity. The study of this system is often known as cavity quantum electrodynamics (cavity QED) as it may be described using the techniques of the previous chapter.
The second example is comprised of a superconducting Cooper pair box inside a co-planar microwave resonator. The description of this system is given in terms of the quantisation of an equivalent electronic circuit and thus goes by the name of circuit quantum electrodynamics (circuit QED).
In both cases we are typically interested in the strong coupling regime in which the single photon Rabi frequency g (the coupling constant in the Jaynes–Cummings
model) is lager than both the spontaneous decay rate, ?, of the two-level emitter and
the rate, ? , at which photons are lost from the cavity.


11.1 Cavity QED


The primary difficulty we face in cavity QED is finding a way to localise a single two-level atom in the cavity mode for long time intervals. One approach, pioneered by the Caltech group of Kimble [1], is to first trap and cool two-level atoms in a magneto-optical trap (MOT) (see Chap. 18) and then let them fall into a high finesse cavity placed directly below the MOT. If the geometry is correctly arranged then at most one atom will slowly fall through the cavity at a time. Another approach is to use constraining forces to trap a single atom in the optical cavity. This can be done using the light shift forces of a far off resonant laser field on a two-level atom[2, 3] (see Chap. 18), or it can be done using an ion trap scheme [4] (see Chap. 17). A very novel way to get atoms from the MOT into the cavity deterministically has been pioneered by the Chapman group in Georgia [5]. They use an optical dipole standing wave trap as a kind of atomic conveyor belt to move atoms from the MOT into the cavity.



213


?




ao(t )
e

g

a ai (t )

?

Fig. 11.1 A cavity QED scheme: a single two-level dipole emitter is fixed at a particular location inside a Fabry-Perot cavity. The dipole is strongly coupled to a single cavity mode, a, but can emit
photons at rate ? into external modes. Photons are emitted from the end mirror of the cavity at
rate ?


Consider the scheme in Fig. 11.1. The interaction Hamiltonian between a single two-level atom at the point?x in a Fabry–Perot cavity is given by




where

HI = g(?x)a†?+ + g?(?x)a?


(11.1)

. ?2?c .1/2

g(?x)=

2h¯ ?0V

U (?x) ? g0U (?x) (11.2)


This is obtained from (10.17) with the traveling wave mode function replaced by a cavity standing wave mode function, U (?x). Here ? is the dipole moment for the two-level system and V is the cavity mode volume defined by V = ¸ sin |U (?x)|2d3x.
Let us consider the interaction between a single cavity mode and a two-level sys- tem. For the present we neglect the spatial dependance of g(?x). The master equation, in the interaction picture, for a single two-level atom interacting with a single cavity mode, at optical frequencies, is

d? =


i? [a†a, ? ]

i ? [? , ? ]


i[??a + ?a†, ? ]


ig[a?


+ a†?


, ? ]

dt ?
?

? 2 z ?
?

? + ?

+ (2a? a† ?a†a? ?? a†a)+

2 (2??

??+

? ?+??

? ? ??+??

) (11.3)


where ? represents a classical coherent laser field driving the cavity mode at fre- quency ?L, the detuning between the cavity field and the driving field is ? = ?c ? ?L and ? = ?a ? ?L is the detuning between the two-level system and the driving field.
From this equation we can derive equations for first order field/atom moments;


d(a) =
dt d(??) =
dt

. ? + i? 2
. ? + i? 2

.
(a)? i? ? ig(??) (11.4)
.
(?? ) + ig(a?z) (11.5)

d(?z) =


?( ?



+ 1)


2ig( a?


a†?



) (11.6)

dt ?

( z)

? ( +)? ( ?)


Looking at these equations we see that we do not get a closed set of equations for the first order moments, for example the equation for (?? ) is coupled to (a?z). A
number of procedures have been developed to deal with this. If there are many atoms interacting with a single mode field, an expansion in the inverse atomic number can be undertaken and we will describe this approach in Sect. 11.1.3. However a good idea of the behaviour expected can be obtained simply by factorising all higher order moments. This of course neglects quantum correlations and is thus not expected to be able to give correct expressions for, say, the noise power spectrum of light emitted from the cavity. Nonetheless it is often a good pace to start as it captures the underlying dynamical structure of the problem. We thus define the semiclassical equations as
??
?? = ? 2 ? ? i? ? igv (11.7)
??
v? = ? 2 v + ig?z (11.8)
z? = ?2ig(?v? ? ??v) ? ?(z + 1) (11.9)

where the dot signifies differentiation with respect to time and

?? = ? + 2i? (11.10)
?? = ? + 2i? (11.11)

The first thing to consider is the steady state solutions, ?s , zs, vs which are given implicitly by
. n .?1

zs = ?

1 +
n (1 + ? 2)

(11.12)

0
?
2i?

1
2C(1 + i? )?1(1 + i?1)?1

??1

?s = ? ??

2ig

?1 +

1 + n ?
n0(1+? 2)

(11.13)




where

vs =

?? ?szs (11.14)

n = |?s |2 (11.15)

is the steady state intracavity intensity and

? = 2? /? (11.16)
?1 = 2? /? (11.17)
?2
n0 = 8g2 (11.18)


2g2
C =
??



(11.19)


The parameter n0 sets the scale for the intracavity intensity to saturate the atomic inversion and is know as the critical photon number. The parameter C is sometimes
defined in terms of the critical atomic number, N0, as C = N?1. Why this name is
appropriate is explained in Sect. 11.1.3 where we consider cavity QED with many
atoms.
We can now determine how the steady state intracavity intensity depends on the driving intensity. We first define the scaled driving intensity and intracavity intensity by

4?2
Id = ? 2n


(11.20)

n Ic =
0

The driving intensity and the intracavity intensity are then related by


(11.21)

.. 2C .2 .

2C?1 .2.

Id = Ic

1 +
1 + ? 2 + Ic

+ ? ?

1 + ? 2 + Ic

(11.22)


The phase ?s of the steady state cavity field is shifted from the phase of the driving
field (here taken as real) where


tan ?s = ?

? ? 2?1C/(1 + ? 2 + Ic)
1 + 2C/(1 + ? 2 + Ic)



(11.23)


Equation (11.22) is known as the bistability state equation, a name that makes sense when we plot the intracavity intensity versus the driving intensity, see Fig. 11.2. It can be shown that the steady state corresponding to those parts of the curve with



100

(a)

100 (b)


100

(c)


80 80 80

60 60 60
Ic
40 40 40

20 20 20


20 40 60 80 100 120 140
Id

20 40 60 80 100 120 140
Id

20 40 60 80 100 120 140
Id


Fig. 11.2 The intracavity intensity versus the driving intensity, as given implicitly by (11.22), for various values of the detuning between the atom and the driving field. In all cases we assume the
driving is on resonance with the cavity so that ? = 0 and C = 9. (a) ?1 = 0, (b) ?1 = 2, (c) ?1 = 3


negative slope are unstable. Clearly there are regions for which two stable steady states coexist for a given driving intensity.
Cavity QED requires that we are in the strong coupling limit in which g0 > ?, ? .
Furthermore a necessary condition for strong coupling is that (n0, N0) << 1. In this limit a single photon in the photon can lead to significant dynamics. One way to
make g0 large is to use a very small mode volume V and a large dipole moment. In recent years, with optical Fabry-Perot cavities, it has been possible to achieve n0 ? 10?3–10?4 and N0 ? 10?2–10?3. The mirrors of these cavities are highly reflective,
with reflectivity coefficients greater than 0.999998. This means that the inter mirror spacing can be made very small giving a small mode volume. Typical parameters for the Caltech group of Kimble, using atomic cesium, are [1]
(g0, ? , ?)= (34, 2, 1.25)MHz (11.24)

which give critical parameters n0 = 0.0029 and N0 = 0.018. Even better perfor- mance is possible using microtoroidal resonators, again implemented by the Caltech group [6], or excitonic dipoles in quantum dots integrated into photonic band gap materials, implemented by the Imamoglu group in Zurich [7]. A very different ap- proach is to use Rydberg atoms, which have very large dipole moments, in super- conducting microwave cavities. This approach has been pioneered by the group of Haroche in Paris [8].