22 May 2000
Physics Letters A 270 Ž2000. 27–40
www.elsevier.nlrlocaterpla
General multimode difference-squeezing
Nguyen Ba An
a
a,)
, Vo Tinh
b
Asia Pacific Center for Theoretical Physics, 207-43 Cheongryangry-dong, Dongdaemun-gu, Seoul 130-012, South Korea
b
Physics Department, Hue UniÕersity, 32 Le Loi, Hue, Viet Nam
Received 5 January 2000; received in revised form 8 February 2000; accepted 17 April 2000
Communicated by P.R. Holland
Abstract
We consider a quantum process in a nonlinear medium in which a number of input modes interact to generate an output
mode with a general difference-frequency. We introduce for the input modes a concept of general multimode differencesqueezing whose relationship with normal squeezing of the output mode is established. We then analyze the conditions for
the input to be multimode difference-squeezed in dependence on the individual modal states. Finally, we study possible
connection of the general multimode difference-squeezing with a symmetry group. q 2000 Elsevier Science B.V. All rights
reserved.
PACS: 42.50.D
1. Introduction
Squeezed light Žseec, e.g., Refs. w1x. with a noise level below the shot noise limit promises intriguing
applications in future low-noise optical communication networks w2;3x, high-precision measurements w4x, optical
waveguide tapping w5x, etc. A single-mode field E with frequency v is formulated in quantum language through
the annihilation and creation bosonic operators a, aq as
E Ž t . A a eyi Ž v tq w . q aq e iŽ v tq w .
Ž 1.
where t is the time and w the phase. Alternatively, we can rewrite Ž1. in the form
E Ž t . A Qw cos Ž v t . q Qwq p r2 sin Ž v t .
Ž 2.
where the operators
Qw s 12 Ž a eyi w q aq e i w .
)
Corresponding author. Permanent address: Institute of Physics, P.O. Box 429 Bo Ho, Hanoi 10000, Vietnam.
E-mail address: nbaan@netnam.org.vn ŽN.B. An..
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 2 9 2 - 9
Ž 3.
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
28
satisfy the commutation relation
Qw ,Qwq p r2 s
i
2
.
Ž 4.
From Ž2. it follows that a field may always be decomposed into two pr2-dephased components. The Qw are
therefore called quadrature operators of the field. Because of Ž4., the uncertainty relation for the two
pr2-dephased quadratures reads
VQw VQwq p r2 G 161
Ž 5.
where V means the variance: V . . . ' ² Ž . . . .
squeezed along a direction w if
2
:y ² . . . :2
VQw y 14 - 0.
with ² . . . : the quantum average. A state is said to be
Ž 6.
Of course, when VQw is squeezed, VQwq p r2 should accordingly be stretched to not violate the relation Ž5.
which was solidly set by quantum mechanics. The squeezed state associated with Ž6. is referred to as normal
squeezed state which is obviously single-mode and first-order.
Multimode first-order squeezing Žsee, e.g., w6;7x. was also considered for which the ‘‘quadrature’’ operator is
defined as a linear superposition Ži.e., a sum. of the modes’ operators.
Single-mode higher-order squeezed states are meaningful as well. Their first version was suggested in w8x
where higher-powered variances are dealt with instead of the conventional variance. Another version of
single-mode higher-order squeezing was defined in w9;10x where amplitude-squared squeezing was introduced.
This was afterward extended to amplitude-cubed w11x and amplitude-K-powered Žsee, e.g. w12;13x. squeezing. A
nice nontrivial unified single-mode higher-order squeezing operator w14x based on non-Hermitian realizations of
the relevant algebras’ operators by means of generalized multiboson operators w15x was also constructed.
The so-called sum- and difference-squeezing were proposed for the first time in w16x for two modes which are
in fact the simplest versions of multimode higher-order squeezing. These concepts have recently been
generalized to include three modes for sum- w17x and difference-squeezing w18x as well as an arbitrary number of
modes for sum-squeezing w19x. Sum- as well as difference-squeezing are multimode and, at the same time,
higher-order since their underlying ‘‘quadrature’’ operator is defined in terms of a product Žnot a sum. of the
modes’ operators.
Like sum-squeezing, it is natural to generalize the two-mode w16x and three-mode w18x difference-squeezing
to the most general multimode case. That is the purpose of this paper. In Section 2 we define the most general
multimode difference-squeezing and study its nonclassical property. In Section 3 we establish the relation
converting the input general multimode difference-squeezing to the output normal squeezing. Section 4
elucidates the condition for the input system to be multimode difference-squeezed in dependence on the modal
squeezings. Possible formal identification between the general difference-squeezing operators and the generators
of a symmetry group is made in Section 5. In the final section we contrast the general multimode sum-squeezing
w19x with difference-squeezing and conclude.
2. General multimode difference-squeezed state
Let us consider a multiwave frequency-conversion process occurring via a nonlinear medium through which
N input modes with frequencies v 1 , v 2 , . . . , v N interact to create an output mode with frequency V given by
P
Vs
N
Ý vk y Ý
ks1
jsPq1
vj ) 0
Ž 7.
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
29
where 1 F P - N Ž N G 2.. This physical process is described by the following Hamiltonian Ž " s 1.
N
Hs
Ý v q aqq a q q V bq b q F Ž bq aqN aqNy1 . . . aqPq1 a P a Py1 . . . a2 a1 q h.c. . .
Ž 8.
qs1
Ž q .
Ž
.
In Ž8. h.c. means hermitic conjugate, aq
q ,a q b ,b are bosonic operators for the input modes output mode and
F is the real effective interaction constant.
Since in practice the interaction constant is much smaller than the frequencies involved, i.e. F < V , v q , we
may decompose the operators as
a q Ž t . s A q Ž t . exp Ž yi v q t . ,
b Ž t . s B Ž t . exp Ž yi V t .
Ž 9.
where A q Ž t . and B Ž t . vary in time much slower than expŽyi v q t . and expŽyi V t .. By the motivation which
will be seen later, we now define a collective operator Gw Ž t . in form of product of the input modal operators as
P
Gw Ž t . s 12 exp Ž yi w .
N
Ł Ak Ž t . Ł
ks1
Aq
j Ž t . q h.c.
Ž 10 .
jsPq1
with w being some angle made by that operator with the real axis in the complex plane. It is easy to derive from
Ž10. the following commutation relation
Gw Ž t . ,Gwq p r2 Ž t . s
i
2
MŽ t.
Ž 11 .
where M Ž t . is a Hermitian operator given by
P
M Ž t . s Mq Ž t . s
N
Ł Ž1 q n k Ž t . . Ł
ks1
jsPq1
P
nj Ž t . y
N
Ł n k Ž t . Ł Ž1 q n j Ž t . .
ks1
Ž 12 .
jsPq1
Ž . Ž .
Ž .
with n q Ž t . s Aq
q t A q t being the number operator of mode v q . As a consequence of 11 , the uncertainty
relation for two ‘‘orthogonal’’ variances reads
VGw Ž t . VGwq p r2 Ž t . G
²MŽ t. : 2
16
.
Ž 13 .
All the relation Ž13. requires is that the product VGw Ž t .VGwq p r2 Ž t . and not its factors be bounded from below.
Nothing in principle prevents squeezing one of the factors at the price of stretching the other factor accordingly,
or vice versa. When
VGw Ž t . y
²MŽ t. :
4
- 0,
Ž 14 .
the state of the input modes v 1 , v 2 , . . . , v N is said being general multimode difference-squeezed in the
direction w .
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
30
In Glauber–Sudarshan P-representation Žsee, e.g., w21x. of the density matrix the variance VGw has the form
VGw Ž t . s
²MŽ t. :
P
q
4
HPŽ a , . . . ,a
1
N
ž
i Ž V ty w .
. Re e
2
N
Ł ak Ł
ks1
jsPq1
a j)
/
y Re² Gw Ž t . :
N
Ł d 2a q Ž 15.
qs1
where P Ž a 1 , . . . , a N . with complex a q stands for the so-called multimode quasi-probability distribution
function and the operator M Ž t . is given by
P
MŽ t. s
P
N
P
Ł Ž1 q n k Ž t . . y Ł n k Ž t . Ł
ks1
ks1
nj Ž t . q
jsPq1
N
N
Ł n k Ž t . Ł Ž1 q n j Ž t . . y Ł
ks1
jsPq1
nj Ž t .
jsPq1
Ž 16 .
whose averaged value is evidently always positive
is equal to
² M Ž t . :) 0.
It is also followed from Ž12. and Ž16. that
² M Ž t . :y ² M Ž t . :
¦
2
P
N
Ł nk Ž t .
ks1
ž
N
Ł Ž1 q n j Ž t . . y Ł
jsPq1
nj Ž t .
jsPq1
;/
)0
Ž 17 .
if ² M Ž t . :) 0 and to
¦ž
2
P
P
Ł Ž1 q n k Ž t . . y Ł n k Ž t .
ks1
ks1
N
/
Ł
jsPq1
;
nj Ž t . )0
if ² M Ž t . :- 0. This indicates that ² M Ž t . : is always greater than
when
VGw Ž t . -
²MŽ t. :
4
.
Ž 18 .
² M Ž t . : while Eq. Ž15. implies that P - 0
Ž 19 .
Classically, a probability distribution function must be definitely non-negative. Thus, states for which the
inequality Ž19. holds have no classical counterparts and belong to the family of so-called nonclassical states.
From Ž17., Ž18. and Ž14. one sees that the above-defined general multimode difference-squeezed state is clearly
nonclassical. However, because of ² M Ž t . :) ² M Ž t . : , the domain of difference-squeezed states lies entirely
inside the domain of nonclassical states. In other words, the general multimode difference-squeezed state is a
special kind of squeezing having no classical states nearby Žcompare with other kinds of squeezing, e.g. normal
squeezing or multimode sum-squeezing w19x which have a common boundary with classical states.. This
peculiar property was found out in w16x for N s 2 but was skipped for N s 3 w18x. Here we have confirmed it for
any N G 2.
3. Relation with normal squeezing of the output mode
In this section we shall examine normal squeezing of the output mode V . Since squeezing as a statistical
property can be transferred between modes when these are coupled to each other w20x, one could expect from the
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
31
Hamiltonian Ž8. that general multimode difference-squeezing of the input modes be related to normal squeezing
of the output mode. To justify this let us use Ž8. to set up the equations of motion for the operators of interest.
These are of the form
P
E Al Ž t .
A˙l Ž t . '
s yiFB Ž t .
Et
N
Aq
k Ž t.
Ł
ks1, k/l
Aj Ž t .
Ł
Ž 20 .
jsPq1
for l s 1 % P, while
P
E Al Ž t .
A˙l Ž t . '
s iFB Ž t .
Et
N
Ł Aqk Ž t .
ks1
Aj Ž t .
Ł
Ž 21 .
jsPq1, j/l
for l s P q 1 % N, and
Ḃ Ž t . '
E BŽ t.
Et
P
N
s yiF Ł A k Ž t .
ks1
Ł
Aq
j Ž t. .
Ž 22 .
jsPq1
The second time-derivative of B Ž t . is obtained from Ž22. as
B̈ Ž t . '
E 2BŽ t .
Et
2
E
s yiF
Et
P
ž
N
Ł Ak Ž t .
ks1
/
Ł
P
Aq
j Ž t. q
jsPq1
E
Ł Ak Ž t . E t
ks1
N
ž
Aq
j Ž t.
Ł
jsPq1
/
.
Ž 23 .
Similarly to the little-tricky derivation given in w19x, we get
E
Et
P
P
žŁ /
A k Ž t . s yiFB Ž t .
ks1
P
žŁ
Ž1 q n k Ž t . . y
ks1
Ł nk Ž t .
ks1
N
/
Ł
Aj Ž t .
Ž 24 .
jsPq1
and
E
Et
N
ž
Ł
jsPq1
P
/
Aq
j Ž t . s yiFB Ž t .
Ł Aqk Ž t .
ks1
N
ž
Ł
N
nj Ž t . y
jsPq1
Ł Ž1 q n j Ž t . .
jsPq1
/
.
Ž 25 .
Substituting Ž24. and Ž25. into Ž23. leads to the simple equation
B¨ Ž t . s yF 2 B Ž t . M Ž t .
with M Ž t . nothing else but that appeared in Ž11..
Ž 26 .
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
32
Since the interaction time is practically very short, we can apply Taylor’s expansion and confine ourselves to
terms up to second order in t. The time dependence of the operator B Ž t . is then
t2
B Ž t . s B q tB˙ q
2
¨
B.
Ž 27 .
In Ž27. and from now on operators evaluated at t s 0 are written without arguments. That is, B ' B Ž0., B˙
' E B Ž t . rE t ts0 , . . . Use of Ž22. and Ž26. in Ž27. gives
P
N
B Ž t . s B y iFt Ł A k
ks1
Ł
Aq
j y
F2t2
2
jsPq1
BM.
Ž 28 .
Eq. Ž28. indicates that B Ž t . depends on Ft rather than on t revealing the much slower variation of B Ž t . as
compared to the fast variation of bŽ t . A expŽyi V t ., a fact already used in the decompositions Ž9..
Let
Qw Ž t . s 12 B Ž t . exp Ž yi w . q Bq Ž t . exp Ž i w .
Ž 29 .
be a quadrature operator of the output mode V . Introducing Ž28. into Ž29. and assuming that the input and
output modes are not correlated at t s 0 we obtain the variance of the output quadrature in the form
Qw Ž t . s VQ
Qw Ž 1 y F 2 t 2² M :. q F 2 t 2 VGwq p r2 .
VQ
Ž 30 .
Assuming further that the output mode is initially in the vacuum or in a coherent state we can simplify Ž30. to
Qwy p r2 Ž t . y 14 s F 2 t 2 VGw y
VQ
² M:
4
.
Ž 31 .
The ² M : will be positive, or the same, ² M :s ² M : , if the averaged numbers of the input modes are such that
N
P
Ł ²1 q n j :
Ł ²1 q n k :
ks1
P
Ł ² nk :
ks1
)
jsPq1
N
.
Ž 32 .
Ł ² nj:
jsPq1
In this case, i.e. when Ž32. is satisfied, Eq. Ž31. is equivalent to
Qwy p r2 Ž t . y 14 s F 2 t 2 VGw y
VQ
² M:
4
.
Ž 33 .
Comparing Ž33. with Ž6. and Ž14. leads to the following conclusion: the Hamiltonian Ž8. converts general
multimode difference-squeezing of the input modes to normal squeezing of the output mode. More precisely, the
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
33
output mode cannot evolve into a squeezed state if the input modes are not multimode difference-squeezed. On
the other hand, if the input modes are multimode difference-squeezed along some direction w at t s 0, then the
output mode will be normal squeezed along the perpendicular direction w y pr2 at an immediate later time
t ) 0. This is the important relation between the input multimode difference-squeezing and the output normal
squeezing we wish to establish.
In passing we note that if the inequality Ž32. fails then ² M :F 0 and, consequently, the left-hand-side of Ž33.
cannot be negative. This implies that the output mode cannot evolve into a squeezed state for whatever states of
the input modes. The estabilshed relation Ž33. between the input and output squeezing is therefore bound to
fulfillment of the condition Ž32. which is imposed on populations of the input modes. In particular, when N s 2
and v 1 ) v 2 it requires that ² n 2 :) ² n1 :, a result coincident with that of w16x. The population-dependent
conditions were not derived in w18x for N s 3. According to Ž32., when N s 3 we get
² n1 :)
² n 2 :² n 3 :
1 q ² n 2 :q ² n 3 :
Ž 34 .
for P s 1 and,
² n 3 :)
² n1 :² n 2 :
1 q ² n1 :q ² n 2 :
Ž 35 .
for P s 2. Note that in Ž34. v 1 ) v 2 q v 3 whereas in Ž35. v 3 - v 1 q v 2 .
It is the relation Ž33. that motivated the introduction of the collective operator Gw , Eq. Ž10., and the
definition of the multimode difference-squeezing in the previous section. The relation Ž33. says that in order to
generate the output mode in a normal squeezed state one should prepare the input modes so that they are
multimode difference-squeezed. The multimode difference-squeezed state is therefore an intermediate useful
state through which it is easy to analyze how the input modes should be prepared to produce a squeezed output
mode. Such an analysis will be done in the next section.
4. Dependence on states of the individual input modes
From Ž33. we have learnt that the output normal squeezing is originated from the input multimode
difference-squeezing. Since there are numerous modes in the input a natural question arises: for which modal
states the input would be multimode difference-squeezed? This question will be answered in this section.
Under the fulfillment of Ž32. we use Ž10. and Ž12. to reformulate the squeezing condition Ž14. for
uncorrelated input modes in the more explicit form as
P
½
Re e 2 i w
ks1
P
yŁ
ks1
N
P
N
2
2
Ł ² A2k : Ł ² Aq2
Ł ² Aqj :
j :y Ł ² A k :
jsPq1
N
² A k : 2 Ł ² Aqj :
ks1
2
- 0.
jsPq1
5
P
q
N
Ł ² nk : Ł
ks1
jsPq1
ž 1 q² n :/
j
Ž 36 .
jsPq1
It is a simple matter to check that the condition Ž36. is broken when either there is a mode being in a Fock
state or all the modes are coherent. It is therefore necessary to have one or more squeezed modes among the
remaining coherent modes.
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
34
We first examine the condition Ž36. in the case when there are modes v Pq 1 , v Pq2 , . . . , v PqK with
1 F K F N y P being normal squeezed and all the remaining modes are coherent. Making use of the properties
of coherent states we are able to reduce the condition Ž36. to
PqK
PqK
2
Ł ² Aq2
Ł ² Aqp :
p :y
cos Ž 2 w q F q Q .
psPq1
PqK
qU
psPq1
PqK
ž
psPq1
Ł
1 q² n p : y
/
Ł ² Aqp :
2
-0
Ž 37 .
psPq1
where
P
F s arg
Q s arg
ž
ž
N
Ł a k2
ks1
jsPqKq1
PqK
Ž 38 .
PqK
2
Ł ² Aq2
Ł ² Aqp :
p :y
psPq1
psPq1
N
Ł
Us
/
a j) 2 ,
Ł
jsPqKq1
N
ž1 q
aj
2
/
,
Ž 39 .
/
)1
aj
Ł
Ž 40 .
2
jsPqKq1
and a q s r q expŽ i uq . is a complex number characterizing the coherent state whose state-vector is a q :s
)
.
expŽ a q Aq
a p , z p :s
q y a q A q 0 :. Using the explicit expression for the state-vector of a squeezed mode
q
.
x
Ž
.
Ž
.
:
expŽ a p A p y a p)A p .expwŽ z p) A2p y z p Aq2
r2
0
with
z
s
s
exp
i
x
in
37
casts
it
into
p
p
p
p
PqK
cos Ž 2 w q F q Q .
Ł
psPq1
ž
r p2 ey2 i u p y
PqK
qU
1
2
PqK
eyi x p sinh Ž 2 s p . y
/
Ł
r p2 ey2 i u p
psPq1
PqK
Ł Ž 1 q r p2 q sinh 2 Ž s p . . y Ł
psPq1
r p2 - 0.
Ž 41 .
psPq1
Formally, the inequality Ž41. is satisfied if and only if
PqK
Ł
U - Uc s
psPq1
ž
r p2 ey2 i u p y
1
2
PqK
eyi x p sinh Ž 2 s p . y
/
Ł
PqK
r p2 ey2 i u p q
psPq1
PqK
Ł
psPq1
The function Uc in the limit r p
Ł
psPq1
r p2
.
Ž 42 .
Ž 1 q r p2 q sinh 2 Ž s p . .
™ 0 is
PqK
™0
lim Uc s
rp
Ł
psPq1
tanh Ž s p . - 1,
Ž 43 .
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
35
Fig. 1. Uc versus r s r p for up s x p s 0, s p s 0.5 and a. K s1; b. K s 3; c. K s 5.
whereas in the limit r p
™ ` it is
™`
lim Uc s 1.
Ž 44 .
rp
In particular, when K s 1 we have for any a and s
Uc s 1 y
1 q ey2 s
- 1.
2
2 Ž 1 q a q sinh 2 Ž s . .
Ž 45 .
The above limiting cases signal that Uc is always less than unity and this is confirmed in Fig. 1. The constraint
Ž42., U - Uc , is therefore incompatible with the fact that U ) 1 by the definition Ž40. and, hence, multimode
difference-squeezing cannot appear, despite the presence of squeezed modes in the input.
We next examine the condition Ž36. in the case when there are modes v 1 , v 2 , . . . , v K with 1 F K F P
being normal squeezed and all the remaining modes are coherent. In this case the condition Ž36. simplifies to
cos Ž 2 w q F X q Q X .
K
K
K
K
ps1
ps1
ps1
ps1
2
Ł ² A2p :y Ł ² A p : q V Ł ² n p :y Ł ² A p :
2
-0
Ž 46 .
where
P
F X s arg
ž
ž Ł²
N
a k2
Ł
ksKq1
Ł
K
Q X s arg
K
A2p :y
ps1
/
a j) 2 ,
jsPq1
2
Ł ² Ap:
ps1
/
Ž 47 .
Ž 48 .
and
N
Ł
Vs
jsPq1
N
ž1 q
Ł
jsPq1
aj
2
/
) 1.
aj
2
Ž 49 .
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
36
Using the analytic expression for the state-vector of squeezed modes we cast Ž46. into the more explicit form as
K
cos Ž 2 w q F X q Q X .
Ł
ps1
K
qV
ž
a p2 y
1
2
K
e i x p sinh Ž 2 s p . y
K
2
a p q sinh 2 Ž s p . y
Łž
/
/
ps1
Ł a p2
ps1
2
a p - 0.
Ł
Ž 50 .
ps1
Formally, the inequality Ž50. is satisfied if and only if
K
Ł
V - Vc s
ps1
ž
a p2 y
1
2
K
e i x p sinh Ž 2 s p . y
/
K
Łž
2
ps1
a p q sinh 2 Ž s p .
ps1
K
Ł a p2
q
Ł
ps1
ap
2
.
Ž 51 .
/
When K s 1 we have for any a and s
Vc s 1 q
eys sinh Ž s .
2
a q sinh 2 Ž s .
™
which is clearly greater than unity. However, for K ) 1 it is not so straightforward. The Vc in the limit r p
and r p ` are
Ž 52 .
™0
K
™0
lim Vc s
rp
Ł coth Ž s p . ) 1
Ž 53 .
ps1
and
™`
lim Vc s 1.
Ž 54 .
rp
1 ixp
K
2
K
Yet, for Ł ps
sinh Ž 2 s p . . s Ł ps1
a p2 the function Vc undergoes a minimum which is less than
1Ž ap y 2 e
unity Žsee Fig. 2.. In view of the above analysis we arrive at the conclusion that the input modes may or may
Fig. 2. Vc versus r s r p for up s x p s 0, s p s 0.5 and a. K s1; b. K s 3; c. K s 5.
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
37
not be multimode difference-squeezed when modes v 1 , v 2 , . . . , v K with 1 F K F P are normal squeezed and
all the remaining modes are coherent. For the input modes to be multimode difference-squeezed the additional
constraint Ž51., V - Vc , must be obeyed. This constraint is compatible with V ) 1 when K s 1 but may be
incompatible for intermediate values of r p around the minimum when K ) 1. Interestingly to notice that V
depends only on the coherent modes v j with j g w P q 1, N x while Vc depends only on the squeezed modes v p
with p g w1, K x. The constraint V - Vc thus expresses the role played by coherent modes in generating
multimode difference-squeezing. Likewise, not only squeezed modes but also coherent modes are important.
When multimode difference-squeezing exists the maximal squeezing degree is found to be along the direction
determined by
wmax s 12 Ž p y F X y Q X . .
Ž 55 .
Since F X and Q X are given by Ž47. and Ž48. the angle wmax depends in general on both the phases and the
amplitudes of the individual modes.
We finally examine the condition Ž36. in the case when modes v 1 , v 2 , . . . , v K with 1 F K F P and modes
v Pq 1 , v Pq2 , . . . , v PqK X with 1 F K X F N y P are normal squeezed while all the remaining modes are coherent
Žfor K s P and K X s N y P all the input modes are squeezed.. The condition Ž36. in this case becomes
K
XX
PqK
A2p
XX
cos Ž 2 w q F q Q .
qW
K
Ł² : Ł ²
X
ps1
K
X
p sPq1
X
PqK
X
Aq2
p
X
:y Ł ² A p : Ł ² Aqp :2
ps1
p sPq1
X
X
K
Ł ²np: Ł
PqK
2
2
PqK
X
qX
p
Ł ²A : Ł ²A :
ž 1 q² n :/ y ps1
p sPq1
p sPq1
p
X
ps1
X
p
X
2
-0
Ž 56 .
where
P
F XX s arg
N
žŁ
ž Ł²
a k2
ksKq1
Ł
K
XX
Q s arg
/
a j) 2 ,
X
jsPqK q1
PqK
A2p
ps1
Ž 57 .
X
K
X
Aq2
p
: Ł ²
p sPq1
X
2
PqK
X
:y Ł ² A p : Ł ² Aqp :2
ps1
p sPq1
X
X
/
Ž 58 .
and
N
Ł
Ws
ž1 q
X
jsPqK q1
N
Ł
aj
2
/
) 1.
aj
X
Ž 59 .
2
jsPqK q1
Formally, the inequality Ž56. is satisfied if and only if
K
Xq
W - Wc s
Ł ² Ap:
ps1
K
PqK
Ł ²np: Ł
ps1
X
PqK
2
X
Ł ² Aqp :
2
X
X
p sPq1
Ž 60 .
X
p sPq1
ž 1 q² n :/
p
X
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
38
X
X
X
X
X
X
X
Fig. 3. Wc versus r s < a < s < a < for K s K s1, u s u s x s x s 0 and a. ss 0.5, s s1; b. ss s s 0.5; c. ss1, s s 0.5.
with
K
Xs
PqK
A2p
X
K
Ł² : Ł ²
ps1
X
X
Aq2
p
p sPq1
PqK
2
X
:y Ł ² A p : Ł ² Aqp :2 .
ps1
p sPq1
Ž 61 .
X
X
Similarly to the preceding case, here we also come to the fact that the input modes may or may not be
multimode difference-squeezed. For simplicity, it suffices to show this by an explicit example of K s K X s 1 in
which case Wc reduces to
X
Wc s
X
Ž a 2 y 12 e i x sinh Ž 2 s . . Ž a ) 2 y 12 eyi x sinh Ž 2 sX . y a 2a ) 2
Ž
2
X
q aa X
2
.
2
a q sinh2 Ž s . .Ž 1 q a X q sinh2 Ž sX . .
Ž 62 .
In Fig. 3 we plot Wc after Ž62. as a function of r s < a < s < a X < for u s u X s x s x X s 0 and different s and sX . As
seen from the figure, for s G sX the Wc is less than or equal to unity in the whole range of r leading to the
absence of multimode difference-squeezing. However, the multimode squeezing may appear for small r if
s - sX because for such r the function Wc becomes greater than unity and the additional constraint Ž60.,
W - Wc , for multimode difference-squeezing turns out to be possible. The difference-squeezing, if exists, occurs
maximally along w s wmax s Žp y F XX y Q XX .r2.
A counterintuitive result of this last case is that the input modes may not be difference-squeezed Žhence, the
output mode is unsqueezed. even when all modes of the input are individually squeezed.
5. Connection to a symmetry group
As learnt from above, the general multimode difference-squeezing is characterized by three operators Gw ,
Gwq p r2 and M. In this section we shall examine the question whether or not these characteristic operators form
a group. The commutator between Gw and Gwq p r2 was given by Ž11. which is proportional to M. We now
derive the commutator between Gw and M. As a result, we obtain
P
ž
Gw , M s 12 W1 Ł A k
ks1
N
Ł
jsPq1
P
yi w
Aq
q W2 Ł Aq
j e
k
ks1
N
Ł
jsPq1
A j eiw
/
Ž 63 .
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
39
where the operators W1,2 were found to be of the form
P
W1 s
N
P
N
Ł Ž 2 q n k . Ł Ž n j y 1. y 2 Ł Ž 1 q n k . Ł
ks1
jsPq1
P
ks1
N
W2 s 2 Ł n k
ks1
P
P
nj q
jsPq1
N
Ł n k Ł Ž1 q n j . ,
ks1
N
P
N
Ł Ž1 q n j . y Ł Ž n k y 1. Ł Ž2 q n j . y Ł Ž 1 q n k . Ł
jsPq1
ks1
jsPq1
Ž 64 .
jsPq1
ks1
nj.
Ž 65 .
jsPq1
If the right-hand-side of Ž63. were proportional to Gwq p r2 then the three operators Gw , Gwq p r2 and M form a
group. This, on account of Ž10. and Ž63., will be the case if W1 q W2 s 0. From Ž64. and Ž65. we get
P
W1 q W2 s 3
ž
N
P
N
Ł n k Ł Ž1 q n j . y Ł Ž 1 q n k . Ł
ks1
jsPq1
P
y Ł Ž n k y 1.
ks1
ks1
jsPq1
P
/
nj q
N
Ł Ž 2 q n k . Ł Ž n j y 1.
ks1
jsPq1
N
Ł Ž2 q n j . .
Ž 66 .
jsPq1
A direct inspection of Ž66. shows that there are only two circumstances under which W1 q W2 s 0. The first one
corresponds to the case when all the modes are identical and P s Nr2 with N an arbitrary even number. This
circumstance is however not physically relevant because it leads to V s 0 which is incompatible with Ž7.. The
remaining circumstance is realized for two nonidentical modes Ž P s 1 and N s 2. in which W1 s y2 and
W2 s 2 for any v 1 ) v 2 providing V ) 0. In this case it is easy to verify that the three operators Gw , Gwq p r2
and M are proportional to the three generators of the suŽ2. Lie algebra, a fact already discovered in w16x. The
merit of our effort in this section is that we show explicitly that the general multimode difference-squeezing for
N G 3 cannot be connected to any symmetry groups. This property qualitatively distinguishes the general
multimode difference-squeezing from the general multimode sum-squeezing whose characteristic operators
always form a representation of a symmetry group, the suŽ1,1. Lie algebra, for any number N G 2 of modes
involved w19x.
6. Discussion and conclusion
We now compare the results obtained here for the general multimode difference-squeezing with those
reported in w19x for the general multimode sum-squeezing. On one side, these are two different kinds of the
same class of squeezing, the multimode higher-order squeezing. They are thus similar in the sense that both of
them are a necessary input phase to produce a single-mode squeezed output. As a consequence, they can be
indirectly detected experimentally as follows w16x. One sends light beams through a medium with the proper
nonlinearity responsible for generation of sum- or difference-frequency and homodyne-detects the output. If the
output is detected in a normal squeezed state, then the original input was sum- or difference-squeezed. On the
other side, being governed by different specific Hamiltonians, they exhibit qualitatively different properties.
First, the condition for multimode sum-squeezing coincides with that for a state to be nonclassical, while the
condition for multimode difference-squeezing is more stringent than that for a state to be nonclassical. In other
words, sum-squeezed states have a common boundary with classical states but difference-squeezed states are
well separated from any classical state. Second, a combination of one squeezed mode with otherwise coherent
40
N.B. An, V. Tinh r Physics Letters A 270 (2000) 27–40
modes is sufficient Žnot sufficient. for a multimode state to be sum-squeezed Ždifference-squeezed.. Third, the
operators characterizing sum-squeezing for any number of modes form a representation of a symmetry group
Žnamely, the suŽ1,1. Lie algebra., but those characterizing N-mode difference-squeezing are connected to the
generators of a group representation Žnamely, the suŽ2. Lie algebra. only for N s 2.
In conclusion, together with the investigations reported in w17–19x this work has completed the problem of
multimode higher-order squeezing originally introduced for the first time w16x in terms of two-mode sum- and
difference-squeezing. It is emphasized that both the sum- and difference-squeezed states are not necessarily be
observed directly. They serve as an inside-medium-bridge connecting the input to the output and their
introduction is helpful in preparating the input modes for a desired output mode. The idea of multimode
higher-order squeezing can be used for low-density elementary excitations in solid state physics and atomic
physics as well since the word ‘‘modes’’ here is quite generic implying bosons in general, not necessarily
photons.
Acknowledgements
One of us ŽN.B.A.. is grateful to the APCTP for a financial support and hospitality during his stay in Seoul.
This work was also partially supported by a national basic research program KT-04.
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