Classification of squeezed states Squeezed coherent state

1 classification of squeezed states

1.1 based on number of modes

1.1.1 single-mode squeezed states
1.1.2 two-mode squeezed states


1.2 based on presence of mean field

classification of squeezed states
based on number of modes

squeezed states of light broadly classified single-mode squeezed states , two-mode squeezed states, depending on number of modes of electromagnetic field involved in process. recent studies have looked multimode squeezed states showing quantum correlations among more 2 modes well.

single-mode squeezed states

single-mode squeezed states, name suggests, consists of single mode of electromagnetic field 1 quadrature has fluctuations below shot noise level , orthogonal quadrature has excess noise. specifically, single-mode squeezed vacuum (smsv) state can mathematically represented as,

|


smsv

⟩
=
s
(
ζ
)

|

0
⟩


{\displaystyle |{\text{smsv}}\rangle =s(\zeta )|0\rangle }

where squeezing operator s same introduced in section on operator representations above. in photon number basis, can expanded as,

|


smsv

⟩
=


1

cosh
⁡
r




∑

n
=
0


∞


(
−
tanh
⁡
r

)

n





(
2
n
)
!



2

n


n
!




|

2
n
⟩


{\displaystyle |{\text{smsv}}\rangle ={\frac {1}{\sqrt {\cosh r}}}\sum _{n=0}^{\infty }(-\tanh r)^{n}{\frac {\sqrt {(2n)!}}{2^{n}n!}}|2n\rangle }

which explicitly shows pure smsv consists entirely of even-photon fock state superpositions. single mode squeezed states typically generated degenerate parametric oscillation in optical parametric oscillator, or using four-wave mixing.

two-mode squeezed states

two-mode squeezing involves 2 modes of electromagnetic field exhibit quantum noise reduction below shot noise level in linear combination of quadratures of 2 fields. example, field produced nondegenerate parametric oscillator above threshold shows squeezing in amplitude difference quadrature. first experimental demonstration of two-mode squeezing in optics heidmann et al.. more recently, two-mode squeezing generated on-chip using four-wave mixing opo above threshold. two-mode squeezing seen precursor continuous-variable entanglement, , hence demonstration of einstein-podolsky-rosen paradox in original formulation in terms of continuous position , momentum observables. two-mode squeezed vacuum (tmsv) state can mathematically represented as,

|


tmsv

⟩
=

s

2


(
ζ
)

|

0
⟩
=
exp
⁡
(

ζ

∗





a
^






b
^



−
ζ




a
^




†






b
^




†


)

|

0
⟩


{\displaystyle |{\text{tmsv}}\rangle =s_{2}(\zeta )|0\rangle =\exp(\zeta ^{*}{\hat {a}}{\hat {b}}-\zeta {\hat {a}}^{\dagger }{\hat {b}}^{\dagger })|0\rangle }

,

and in photon number basis as,

|


tmsv

⟩
=


1

cosh
⁡
r




∑

n
=
0


∞


(
tanh
⁡
r

)

n



|

n
n
⟩


{\displaystyle |{\text{tmsv}}\rangle ={\frac {1}{\cosh r}}\sum _{n=0}^{\infty }(\tanh r)^{n}|nn\rangle }

if individual modes of tmsv considered separately (i.e.,




|

n
n
⟩
=

|

n
⟩

|

n
⟩


{\displaystyle |nn\rangle =|n\rangle |n\rangle }

, tracing on or absorbing other mode), remaining mode left in thermal state.

based on presence of mean field

squeezed states of light can divided squeezed vacuum , bright squeezed light, depending on absence or presence of non-zero mean field (also called carrier), respectively. interestingly, optical parametric oscillator operated below threshold produces squeezed vacuum, whereas same opo operated above threshold produces bright squeezed light. bright squeezed light can advantageous quantum information processing applications obviates need of sending local oscillator provide phase reference, whereas squeezed vacuum considered more suitable quantum enhanced sensing applications. adligo , geo600 gravitational wave detectors use squeezed vacuum achieve enhanced sensitivity beyond standard quantum limit.