Output Formulation of Optical Cavities

Input–Output Formulation of Optical Cavities

Abstract In preceding chapters we have used a master-equation treatment to calcu- late the photon statistics inside an optical cavity when the internal field is damped. This approach is based on treating the field external to the cavity, to which the sys- tem is coupled, as a heat bath. The heat bath is simply a passive system with which the system gradually comes into equilibrium. In this chapter we will explicitly treat the heat bath as the external cavity field, our object being to determine the effect of the intracavity dynamics on the quantum statistics of the output field. Within this perspective we will also treat the field input to the cavity explicitly. This approach is necessary in the case of squeezed state generation due to interference effects at the interface between the intracavity field and the output field.
An input–output formulation is also required if the input field state is specified as other than simply a vacuum or thermal state. In particular, we will want to discuss the case of an input squeezed state.



7.1 Cavity Modes


We will consider a single cavity mode interacting with an external multi-mode field. To being with we will assume the cavity has only one partially transmitting mirror that couples the intracavity mode to the external field. The geometry of the cav- ity and the nature of the dielectric interface at the mirror determines which output modes couple to the intracavity mode. It is usually the case that the emission is strongly direction. We will assume that the only modes that are excited have the same plane polarisation and are all propagating in the same direction, which we take to be the positive x-direction. The positive frequency components of the quan- tum electric field for these modes are then


?
E(+)(x,t)= i ?
n=0

. h¯ ?n .1/2
2?0V


bne?i?n(t?x/c) (7.1)


In ignoring all the other modes, we are implicitly assuming that they remain in the vacuum state.

127


Let us further assume that all excited modes of this form have frequencies cen- tered on the cavity resonance frequency and we call this the carrier frequency of ? >> 1. Then we can approximate the positive frequency components by



E(+)(x,t)= i

. h¯ ?n

.1/2 . c


?
? bne?i?n(t?x/c) (7.2)

2?0Ac

L n=0


where A is a characteristic transverse area. This operator has dimensions of electric field. In order to simplify the dimensions we now define a field operator that has di-
mensions of s?1/2. Taking the continuum limit we thus define the positive frequency
operator for modes propagating in the positive x–direction,


?
b(x,t)= e?i?(t?x/c) 1 ¸
?2?
??


d?b(?)e?i?(t?x/c) (7.3)

where we have made a change of variable ? ?? ? + ?? and used the fact that ? >> 1 to set the lower limit of integration to minus infinity, and
[b(?1), b†(?2)] = ? (?1 ? ?2) (7.4)
In this form the moment n(x,t)= (b†(x,t)b(x,t)) has units of s?1. This moment determines the probability per unit time (the count rate) to count a photon at space- time point (x,t).
Consider now the single side cavity geometry depicted in Fig. 7.1. The field op- erators at some external position, b(t)= b(x > 0,t)ei?t and b†(t)= b†(x > 0,t)e?i?t
can be taken to describe the field, in the interaction picture with frequency ?. As the cavity is confined to some region of space, we need to determine how the field out- side the cavity responds to the presence of the cavity and any matter it may contain. The interaction Hamiltonian between the cavity field, represented by the harmonic oscillator annihilation and creation operators a, a†, and the external field in the ro- tating wave approximation is given by (6.14). Restricting the sum to only the modes of interest and taking the continuum limit, we can write this as


?
¸
V (t)= ih¯
??


d?g(?)[b(?)a† ? ab†(?)] (7.5)





Fig. 7.1 A schematic representation of the cavity field and the input and output fields for a single- sided cavity

Cavity Modes 129

with [a, a†]= 1 and g(?) is the coupling strength as a function of frequency which is typically peaked around ? = 0 (which corresponds to ? = ? in the original non- rotating frame). In fact g(?) is the Fourier transform of a spatially varying coupling constant that describes the local nature of the cavity/field interaction (see [1]). If the cavity contains matter, the field inside the cavity may acquire some non trivial dynamics which then forces the external fields to have a time dependence differ- ent from the free field dynamics. This leads to an explicit time dependence in the frequency space operators, b(t, ?), in the Heisenberg picture.
We now follow the approach of Collett and Gardiner [1]. The Heisenberg equa-
tion of motion for b(t, ?), in the interaction picture, is

b? (t, ?)= ?i?b(?)+ g(?)a (7.6) The solution to this equation can be written in two ways depending on weather we choose to solve in terms of the initial conditions at time t0 < t (the input) or
in terms of the final conditions at times t1 > t, (the output). The two solutions are
respectively


t
b(t, ?)= e?i?(t?t0 )b0(?)+ g(?)
0
where t0 < t and b0(?)= b(t = t0, ?), and

t1
b(t, ?)= e?i?(t?t1 )b1(?) ? g(?)
t


e?i?(t?t? )a(t?)dt? (7.7)




e?i?(t?t? )a(t?)dt? (7.8)



where t < t1 and b1(?)= b(t = t1, ?). In physical terms b0(?) and b1(?) are usually specified at ?? and +? respectively, that is, for times such that the field is simply
a free field, however here we only require t0 < t < t1.
The cavity field operator obeys the equation


i
a? = ? h [HS , a] ?


?
¸
d? g(?)b(t, ?) (7.9)

??
where HS is the Hamiltonian for the cavity field alone. In terms of the solution with initial conditions, (7.7), this equation becomes


i
a? = ? h¯ [HS , a] ?


?
¸
d? g(?)e?


i?(t?t0 )



b0(?)

??
? t
¸ d? g(?)2



e?i?(t?t? )a(t?) (7.10)

?? t0


We now assume that g(?) is independent of frequency over a wide range of fre- quencies around ? = 0 (that is around ? = ? in non rotating frame). This is the first approximation we need to get a Markov quantum stochastic process. Thus we set

g(?)2 = ?/2? (7.11)

We also define an input field operator by


?
1 ¸
aIN(t)= ? 2?
??


d?e?


i?(t?t0 )



b0(?) (7.12)


(the minus sign is a phase convention: left-going fields are negative, right-going fields are positive). Using the relation

?
d?e?i?(t?t? ) = 2?? (t ? t?) (7.13)
??

the input field may be shown to satisfy the commutation relations
[aIN(t), a† (t?)= ? (t ? t?) (7.14) When (7.13) is achieved as the limit of an integral of a function which goes smoothly
to zero at ±? (for example, a Gaussian), the following result also holds


t t1
¸ ¸
f (t?)?(t ? t?)dt? =
t0 t


1
f (t? )?(t ? t?)dt? = 2 f (t), (t0 < t < t1) (7.15)


Interchanging the order of time and frequency integration in the last term in (7.10) and using (7.15) gives
i ? ?

a?(t)= ? h¯ [a(t), HSYS] ? 2 a(t)+

?aIN(t) (7.16)


Equation (7.16) is a quantum stochastic differential equation (qsde) for the intra- cavity field, a(t). The quantum noise term appears explicitly as the input field to the cavity.
In a similar manner we may substitute the solution in terms of final conditions,
(7.8) into (7.10) to obtain the time-reversed qsde as
i ? ?

a?(t)= ? h¯ [a(t), HSYS]+ 2 a(t) ?

where we define the output field operator as

?aIN(t) (7.17)

Linear Systems 131
?
1 ¸

aOUT(t)= ?
2?
??

d?e

?i?(t?t1 )

b1(?) (7.18)


(Note that the phase convention between left going and right going external fields re- quired for the boundary condition has been explicitly incorporated in the definitions of aIN, aOUT). The input and output fields are then seen to be related by
aIN(t)+ aOUT(t)= ??a(t) (7.19)

This represents a boundary condition relating each of the far field amplitudes outside the cavity to the internal cavity field. Interference terms between the input and the cavity field may contribute to the observed moments when measurements are made on aOUT.