Generation and Applications of Squeezed Light

Generation and Applications of Squeezed Light



Abstract In this chapter we shall describe how the squeezing spectrum may be cal- culated for intracavity nonlinear optical processes. We shall confine the examples to processes described by an effective Hamiltonian where the medium is treated clas- sically. We are able to extend the treatment o squeezing in the parametric oscillator to the above threshold regime. In addition, we calculate the squeezing spectrum for second harmonic generation and dispersive optical bistability. We also consider the non degenerate parametric oscillator where it is possible to achieve intensity fluctu- ations below the shot-noise level for the difference in the signal and idle intensities. Two applications of squeezed light will be discussed: interferometric detection of gravitational radiation and sub-shot-noise phase measurements.



Parametric Oscillation and Second Harmonic Generation


We consider the interaction of a light mode at frequency ?1 with its second har- monic at frequency 2?1. The nonlinear medium is placed within a Fabry–Perot cav- ity driven coherently either at frequency 2?1 (parametric oscillation or frequency ?1 (second harmonic generation)). We shall begin by including driving fields both at frequency ?1 and 2?1 so that both situations may be described within the one formalism. We write the Hamiltonian as [1]


H = H1 + H2 ,

H1 = k?1a†a1 + 2h¯ ?1a†a2 + i



h¯ ?



(a†2a2 ? a2a†)+ ih¯ (E1a† e?i?1t )

1 2 2 1 1 2 1
? E?a1ei?1t)+ ih¯ (E2a†e?2i?1t ? E?a2e2i?1t ) ,
1 2 2
H2 = a1?† + a†?1 + a2?† + a†?2 ,
1 1 2 2

where a1 and a2 are the Boson operators for modes of frequency ?1 and 2?1, re- spectively, ? is the coupling constant for the interaction between the two modes and



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the spatial mode functions are chosen so that ? is real, ?1, ?2 are heat bath operators which represent cavity losses for the two modes and E1 and E2 are proportional to the coherent driving field amplitudes.
The master equation for the density operator of the two cavity modes after tracing out over the reservoirs is


?? = 1 [H , ? ]+ (L


+ L )? , (8.2)



where

? t ih¯ 1 1 2

Li? = ?i(2ai? a† ? a†ai? ? ? a†ai) ,
i i i
and ?i are the cavity damping rates of the modes.
This master equation may be converted to a c-number Fokker–Planck equation
in the generalised P representation. The generalised P representation must be used since the c-number equation would have a non-positive definite diffusion matrix if the Glauber–Sudarshan P representation were used. The result is

? . ?
P(a)=



(? ?


E ??†? )+ ?



(? ?†


E? ?? ?†)

? t ? ?1 1 1 ?
+ ? .

1 ? 1 2
? .

??† 1 1 ?
? .

1 ? 1 2
? .

?2?2 ? E2 +

?2 +

?2?† ? E? +

?†2

? ?2
1 . ? 2

2 1 ??†
? 2 ..

2 2 2 1

+ 2 ??2

(??2)+


??†2

(??†)

P(a) , (8.3)


where a = [?1, ?†, ?2 , ?†], and we have made the following transformation to the
1 2
rotating frames of the driving fields
?1 ? ?1 exp(?i?1t), ?2 ? ?2 exp(?2i?1t) .

In the generalized P representation ? and ?† are independent complex variables and the Fokker–Planck equation has a positive semi-definite diffusion matrix in an eight-dimensional space. This allows us to define equivalent stochastic differential equations using the Ito rules


? .?1 . .E1 + ??†?2 ? ?1?1 . .??2 0

.1/2

.?1(t).
, (8.4)

? t ?†

E? + ??1?† ? ?1?†

0 ??†

?†(t)

1 1 2 1 2 1

? 2
? .?2 . = . E2 ? 2 ?1 ? ?2?2 . , (8.5)

? t ?†

E? ? ? ?†2 ? ?2?†

2 2 2 1 2

where ?1(t), ?†(t) are delta correlated stochastic forces with zero mean, namely


(?1(t)) = 0
(?1(t)?1(t?)) = ?(t ? t?) (8.6)
(?1(t)?†(t?)) = 0 .