Impossibility of steady squeezing for two-mode linear system without self-action

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Impossibility of steady squeezing for two-mode linear system without self-action

A. Bakasov, N. Bakasova, E. Bashkirov, V. Chmielowski

To cite this version:

A. Bakasov, N. Bakasova, E. Bashkirov, V. Chmielowski. Impossibility of steady squeezing for two-

mode linear system without self-action. Journal de Physique I, EDP Sciences, 1991, 1 (9), pp.1217-

1227. �10.1051/jp1:1991105�. �jpa-00246407�

Classification Physics Abstracts

03.658 12.90

Impossibility of steady squeezing for two-mode linear system

without self-action

A. A. Bakasov (I.

*),

N. V. Bakasova

(I),

E. K.

Bashkirovf)

and V.

Chnficlowskif)

(I) Intemational Centre for Theoretical Physics, P-O- Box 586, 34100 Trieste, Italy

f)

Joint Institute for Nuclear Research, Head Post Office, P-O- Box 79, 101000 Moscow, U-S-S-R-

(Received I June 1990, revised12 March 1991, accepted 30 May 1991)

Abs&act. A solution for the problem of the time evolution of quadrature component variances has been obtained for the two-mode system with the two-boson interaction. _{As a} result, the impossibility of steady squeezing is strictly established. Comparison with the single-mode model shows that the possibility of steady squeezing is due solely to self-action within the modes which is absent for the model considered. Thus, there cannot be cross-over between the model considered and the single-mode model.

1. Goals of the work.

In the

previous

_paper

ill

_we considered the

simplest microscopical

model of _a

single-mode

system ^with the two-boson interaction. The initial conditions for

steady squeezing

were

found,

and the

explicit

time

dependence

of the second-order moments was determined. In this article _we

designate

the

squeezing

which is constant in the _course of time _as

steady squeezing.

Naturally,

the next step ^{is to}

study

the two-mode model with the two-boson interaction between the modes. Besides

establishing

the

explicit dependence

^of the second-order

moments, and

including

the operators ^{of the}

quadrature

component ^variances upon time and initial

conditions,

it is also of interest to clear _up the

possibilities

of

steady squeezing

in this model and to construct a cross-over between this model and the model considered in

[I].

It should be

immediately

mentioned that the

investigations

described in this _paper have shown that neither

steady squeezing

in the two-mode model _nor the _cross-over between the

single-

mode

[I]

and the two-mode models _are

possible (see

discussions in Sects.

4, 5).

The

problem

of

explicit

time evolution of

quadrature

component ^variances was earlier considered for several systems

(e.g.

_see

[2-4]),

but the

problem

of

finding steady squeezing

was not

posed.

It should also be mentioned that, besides

solving

the operator

problem

in the

Heisenberg representation,

_{one can use} the

Schr6dinger representation

_on the

basis,

for

example,

of the results obtained in the

theory

of correlated states [5].

(*) The author all correspondence should be _sent to.

2. Hamiltouian and _some indhpemable formal relations.

Let _us first of all consider _a

problem

of time evolution of

quadrature

_component variances with in the framework of the two-mode model with the Hamiltonian

H=

w~a+ a+w~b+

b+

f*ab+ fa+

b+

(I)

Here a+

(a)

and b+

(b)

_are the creation

(annihilation)

operators ^of two different Bose-field modes with the

frequencies

_w~ and _w~

respectively, f

is the

time-independent coupling

constant.

Let _us define _the

quadrature

components ^{for the initial fields} :

xi=

(a+a+),

_x~=

(a-a+),

2 2i

x~=~(b+b+),

x~=

(b-b+), (2)

2 2i

It is also assumed that all commutators between the operators of the different modes _are

equal

to zero for the

coinciding

times.

The

non-degenerated

transfornlation into _new operators ^{has the form}

A=pa+vb+, B=va++pb, (p(~-(v(~#0, (3)

where _p and _v

generally

_are

complex

^numbers.

The

quadrature

components ^for new fields _are introduced in the _same way :

yi=((A+A+), y~=)(A-A+),

Y3"((B+B+), y4"~(B-B~), ₍₄₎

Let _us also define the

following

operator ^columns

X =

(Xi,

X2, X3,X4)

, ^~

(Yl,

Y2, Y3, Y4)

'

related _as

y "

ul,~

,

(5)

where the direct transformation is determined

by

matrix

Uj

:

Re ^(a

^Im

^{(a )}

^Re

^(v)

^Im

^(v)

u~

=

Im (11

)

Re

(11)

Im

(V)

Re

(V)

Re

(v)

Im

(v)

Re

(p )

Im

(p )

~6~

Im

(

_v Re

(

_v

)

Im

(a )

Re _p

)

The inverse transformation has the form _:

x ₌

Up ~y, ₍₇₎

where

Re(p) Im(p) -Re(v) Im(v)I

up1

~

l im

(p

Re

(p )

im

(v)

Re

( v>

~~~~

~g~

~~~~-b()~

©(~~ -~($~ t($~

The fluctuation operators ^{of the initial and}

auxiliary

fields _are

Ax;

= x~

(x~)

,

by = y~

(y;)

,

(9)

where

(. )

is the

averaging

_{over a} state _{or a}

density

matrix. To describe the time evolution of the variances of the

quadrature

_components of the initial fields it is convenient to introduce the

following

operator columns _:

Ax

=

(Ax/, Axj, Axj, ^Ax(, Axj

Ax~ + Ax~

Axj, Axj

Ax~ + Ax~

Axj,

Axj

Ax~ + Ax~

Axj,

_{Ax~ Ax~} + Ax~ Ax~, Ax~ Ax~ + Ax4 Ax2, Ax~ Ax4 + Ax4

Ax~)

,

(lo)

by

=

(Ay),

by

I,

by

], Ayl,

_{by, Ay2 + AY2 Ay,, by1 AY3 + AY3 AY}

i,

Ayj

Ay~ + Ay~

by

j, Ay~ Ay~ + by Ay~, Ay~ Ay~ + Ay~ Ay~,

by

~Ay~ + Ay~ Ay~

,

(ii)

which form _a

complete

set of operator

quantities

for _our

problem. Using (5)-(9)

_we obtain the relations between the operators of the second-order moments

(10)-(11)

_:

Ay~ ₌ U~

Ax~, (12)

Ax~

=

Uj

by^~

(l 3)

The

explicit expression

for the cumbersome 10 _x 10 matrix U~ ^is

given

in

Appendix

I.

Now,

following [I],

we can

formally

write down

Ax~(t)

_{as a} function of time and its initial conditions

Ax~(to)

^{in the}

following

_way. Let _{us assume} that the time evolution of

Ay~(t)

_{as a} function of the initial conditions

Ay~(to)

is determined

by

a matrix

U~(t, to)

_:

Ay2(t)

=

u~(t, to) Ay2(to) (14)

Since relations

(12), (13)

_are valid for all moments of time, one can

apply

them to both sides of

equality (14)

and obtain _a formal

expression

which determines the time evolution of

Ax~(t) through

^the matrices of direct and inverse transformation into the

auxiliary

operators :

Ax~(t)

=

Uj U~(t, to)

U~

Ax~(to). (15)

3.

Explicit

exprewions

determwng

the rime evolution of the variances of quadrature components.

In the

previous

section _we considered _a

general non-degenerated

transformation into

auxiliary

operators. Now _we should fix the Bose commutation relations for _new operators.

Let _us take the parameters ^of

Bogolubov

transformation

(3)

in the

following

form [6]

p ₌ cosh

(w ),

_v

=

sinh

(w exp(I#) (16)

where

tanh

(~>

= D

=

] _jw~

+ w~

~/(w~

₊

_w~)~ 4j /j~j

_#

= arg

~/). (17)

Since _{now p} ^~ _v ^~

=

l,

transformation

(3)

becomes canonical _:

IA,

A +

=

[B, B+]

= I,

[A, B]

=

[A,

^B+

= 0, and the transformation matrices

Uj

and U~ become

unitary.

The Hamiltonian

(I)

in the

representation (16), (17)

is _a

diagonal

bilinear bosonic form _:

H=fl~A+ A+flaB+ B+flo, (18)

where

fl I

j~

_~

~/(~

~ ~ )2

~jfj2j

A-j

a~ b~ _a b _,

Ha

=

(w~

w~

~/(w~

₊

w~)~

4

f)~)

,

(19)

2

flo= fl~ ~/(w~+w~)~-4(f)~.

D I

In this

parameterization

the matrix elements

Uj

^and U~ are

expressed through

the constants of the Hamiltonian

(I) by

the

following

relations _:

Re

(

_p

)

^' Im

(

_p 0 Re

(

_v

)

^{~ ~°~}

⁽

^~ Im _v

)

^{~ ~'~}

(

~

~fi~~

^~

~fi~ ~fi' (20)

For the

diagonal

Hamiltonian

(18)

the fluctuation operators evolve _as coordinates and momenta of two

non-interacting

harnlonic oscillators of the

auxiliary

fields _:

Ay,(t)

=

Ayi (to)

cos

(n A(t to))

₊

Ay~(to)

sin

(nA(t to)),

AY2(t> ₌

AYI(to>

Sin

(llA(t

to» +

AY2(to)

_C°s

(HA(t to>), Ay~(t)

=

Ay~(to)

_cos

(HB(t to))

₊

Ay4(to>

sin

(HB(t to))

,

~~~~

Ay~(t)

=

Ay~(to)

sin

(nB(t to))

+

Ay~(to)

cos

(nB(t to))

,

Using

formulae

(21),

one can

readily

find the evolution matrix

U~(t, to).

^The non-zero

elements of this matrix have the fornl

iU~(t, to)ii,

_{i "}

iU~(t,

to)12,_{2 =}

(1

+ _cos

(2 HA (t to)>)

,

iU~(t,

_{to>11,2 "}

iU~(t,

to)12,1 "

(1

_C°S

(2 ilA(t to)))

,

jU~(t, to)ij,

_{5 =}

jU~(t, to)i~,

_{5 =} ^sin

(2 £lA(t to>),

[U~(t,

to)]~,_j =

[U~(t, to)]~,~

=

sin

(2 nA(t to)),

~~~~'

_~0)15,5 CDS

(~ "A(t t0))

' ~~~)

[lfy(t,

_t0)13,3

"

[lfy(t, t0)14,4

"

(1

_{+ CDS}

(~ "B(t t0))),

iUy(t,

_to>13,4

=

iUy(t,

_to)14,3

"

(1-

_C°S

(2 llB(t to))),

[lfy(t> t0))3,10

_"

[l~y(t, t0))4,lO

" Sl~

(~ ilB(t t0))

>

[lfy(t>

t0))10,₃ "

[lfy(t, _t0))10,4

" Sl~

(~ ilB(t t0))

,

lUy(t, to)lie,

io ⁼ CDs

(2 na(t to))

Now,

using

formulae

(13), (16)

and

(22),

_{one can} find the final

expressions

for the time evolution of the operator

quantities Ax~(t).

After _a cumbersome calculation _we obtain the

following

relations for the variances of the

quadrature

components ^of the initial fields

Ax~(t)

_:

Ax)(t)

= Ko +

Kj

_cos

(2 HA (t to) )

+ K~ cos

(2 nB(t to) )

+

K~

sin

(2

n

~

(t to))

₊

K~

sin

(2 nB(t to))

+

K~

sin

((n~

₊

nB)(t to))

₊

K~

sin

((n~ nB)(t to))

^~~~~

+

K~

_cos

((n~

+

nB)(t to))

_{+ K~ cos}

((n~ nB)(t to)), A4(t)

=

Po

₊

Pi

_cos

(2 fl~(t to) )

+ P2 _cos

(2 ilB(t to) )

+

P~

sin (2

fl~(t to))

₊

P~

sin

(2 flB(t to))

+

P~

sin

((fl

~ +

flB)(t to))

₊

P~

sin

((fl

~

flB)(t to))

^~~~~

+

P~

_cos

((n~

₊

128)(t to))

₊

P~

cos

((n~ nB)(t to))

The

expressions

for the operators

Axj(t)

_and

Ax((t)

could be obtained

through

the simultaneous

replacements Ax/(to)

++

Axj(to), A4(to)

++

Ax((to)

and

n~

₊₊

nB

in the

expressions

for the operator coefficients

Ki

and P~ ^of

Appendix

2

respectively.

It is also _seen from the contents of

Appendix

2 that there exist _very

simple

relations between operators K~ and P~ ^due to the symmetry ^{of the} modes in the Hamiltonian

(I). Namely,

harnlonics with

double

frequencies

and with difference of the

frequencies

contribute in

expressions (23)-(24)

with

opposite signs

while the harmonics with _zero

frequency

and with _sum of

frequencies

contribute

similarly.

We have confined ourselves to the

explicit

form of the time

dependence

of

Ax/(t)

_{and omitted the}

expressions

for

Ax~(t)

_by

x(t)

₊

Axy(t)Ax;(t),

I # j. ^{We have} calculated them _as well, but the fornlulae _are too cumbersome. The interested reader _can do it himself

using

relations

(12), (15)

and

(22).

4. On the

impossibility

of the

Steady squeezing

in the quadratures of the iuidal fields.

Explicit expressions (23)-(24)

obtained for the time evolution of the variances of the

quadrature

_components

Ax/(t)

_can be used for the

rigorous study

of the

problem

of the

possibility

of the

stationary squeezing

in the initial fields vithin the model considered.

Steady squeezing

is

possible

if the _mean values of the operator coefficients K~ ^and

Pt,

I

= 1,

...,

8 _; I # 0

(see Appendix 2)

in the

expressions (23)-(24)

will be

equal

to zero. One

can

easily

_see that the _proper

degenerated algebraic

_system of the linear

equations

Mith respect to the _mean values

(Ax)(to))

^and

(J~(to),(to)+ ~(to) x;(to)),

I #j, _can be obtained

by equating

the averages

(K~)

and

(P~),

_I, j # 0, to _zero. Thus, the order of this system will be

equal

to 10. Yet, the

analytical

solution of such _a system ^is a difficult task On the other

hand,

there is _a somewhat

simpler

way of

finding

the conditions for the

stationary

behaviour of the variances of the

quadrature

components. ^In

fact,

we can find

by

_means of

calculations of _a

quite

reasonable

length

the column of the initial conditions

(y~(to))

which

win determine the

stationary

behaviour of the variances

(y~(t))

_{of the}

_quadrature

components ^{of the}

auxiliary

fields and then _use the

time-independent

transformation

(13).

As

a result _we obtain the initial conditions for the

quadrature

component variances which _ensure the

stationary

behavitour of the initial field variances. The next step vill be to

analyze

the

conditions in order to select the _ones which will

correspond

to a

steady-squeezing

evolution of the system.

The

equations defining

the

stationary

^{initial conditions} for the

auxiliary

field variances (~) have been found

by

_means of formulae

(21)

which have

yield

(AYi(to))

₌

(AYI(to)), _(AYI(to))

"

(AY](to)), (AYi(to)Ayj(to))

₊

+

(Ay~(t~) Ay~(i~)j

= o, I,j ; I,j

= 1, 2, 3, 4.

(25)

Now,

using

the transformation defined

by

the inverse of the matrix U~ ^{of the}

Appendix

I,

we obtain the initial conditions

corresponding

to the

stationary

evolution of the second-order moments for the initial fields :

(I~X~(t0))

"

(AX~(t0)), (hX~(t0))

_"

(I~X((t0)), (AXl(t0)hX2(t0))

+

(h~K2(t0)AXl(t0))

m

(AX3(to) AX4(to))

+

(AX4(to) AX3(to))

= 0

,

(l~Xl(t0)6X3(t0))

+

(1iX3(t0)1iXl(t0))

"

((hX2(t0)6X4(t0))

+

(6X4(t0) 6X2(t0))

=

~ ~ ~°~

~~ ⁽ (Ax/(

to) +

(Axj(to) ⁾

,

(26)

1+ D

(hXl (t0) hX4(t0))

+

(hX4(t0)

hXl

(t0))

~

(hX2(t0) 1~X3(t0))

+

(hX3(t0) hX2(t0))

~

~~ _~~~~~~

~

~~~~~~~~

~

In formulae

(26)

two initial values _are taken

arbitrarily,

and other

stationary

second-order moments _are

expressed through

them, _e.g.

through (Ax/(to))

and

(Axj(to))

5. Physical discussions.

The

analysis

of the above relations allows the

following

conclusions _:

I)

Variances of the two

quadrature

components

belonging

to the _same initial field _are equal in the

stationary

evolution. This _means that

steady squeezing

is

impossible

within the considered model because of the absence of _an asymmetry ^between the variance values _;

2)

In the

stationary

evolution

only

the fluctuation correlation functions between the different fields have _non-zero values. This _means that there is _a correlation

only

^between the

steady

fluctuations of the different fields but _not between the

quadrature

variances of the

same field. If the relative

phase

# is

equal

to zero then, _as seen from

(26),

the correlation becomes pure, I.e. between the

steady

fluctuations of the

generalized

coordinate of the first mode and

steady

fluctuations of the

generalized

coordinate of the second mode, and the _same would be for

generalized

_momenta of the modes. This is due to the structure of Hamiltonian

(I)

in which there _{are an} interaction between the modes and _no self-action inside the modes.

(I) It may be useful to note that the stationary initial conditions for the variances do not mean the

stationary initial conditions for other quantities, for example, for the field amplitudes [10]. Thus, the steady-squeezing wave functions [10], if they exist, do not generally coincide with the stationary _wave

functions which _are, by the known definition, the eigenstates of the Hamiltonian of the system.

It _seems worthwhile to make _a second remark.

Saying

_« self-action inside the modes _{» we}

mean here the terms such _as

(a+

_{)~ and} a~ _or

(b+

)~ ^{and b~ which}

keep

_a

linearity

of the system

f) by creating simultaneously

the interaction between the quanta ^{of the} same mode.

For the

single-mode

linear system,

obviously, only

self-action _can be _an interaction, and the

problem

_was solved

explicitly

ⁱⁿ reference

Ill.

On the other hand, _{we are}

considering

here the two-mode linear system ^vithout self-action _terns in the Hamiltonian

trying

to clear _up the

qualitative

difference between the self-action inside the modes and the interaction between the different modes. Of _course,

higher-order

terns, such _as

(a+

)~^{b and} so on are omitted

they destroy

the

linearity

and

simplicity

of the system. ^{We realize that such} a linear model is not very rich

physically

but everyone is _aware that

generally

it is very difficult _to obtain the

rigorous

and

explicit

results for realistic models. We have

finally

chosen the

rigour

of the results

bearing

in mind that it is

always

convenient _to have

rigorous

and

explidt

asymptotes for

sophisticated problems

^{for which this}

simpler

_{one may} be used _{as an}

exactly

soluble

ground.

3)

Reference

Ill

showed that

steady squeezing

is

possible

for the

single-mode

model with two-boson self-action. The

rigorous

consideration of the two-mode model with two-boson interaction in this _paper allows _one to see that in the limit _w~

- w~ the

regime

of

stationary

evolution of the variances of the

quadrature

_components in the model under consideration here does not coincide with the

corresponding regime

^for the model considered in reference

Ill,

since the

steady squeezing

is

impossible

^{in the former} case and

possible

in the latter _one.

From the formal mathematical

point

of view this is _an obvious fact since the limit w~ - w~ does _{not mean a}

change

of the commutation relations

ja, bi

=

ja,

^b+

= 0

,

(27)

I-e- in the

degeneracy

limit the operators

belonging

to the separate ^modes are still

commutating.

The true

degeneracy

limit should _mean the simultaneous

change

of the second

commutator in

(27)

_as follows

ja,

b+

-

ja,

a+

i

= ,

(28)

but this

procedure

is

singular

and cannot be done without the

singular changes

of the solution obtained in this article.

Summarizing 1)-3),

_{one can see} that _we have

rigorously

studied the

stationary regime

of the second-order _moments for the

simplest microscopic

model of two-boson interaction of _two field modes and obtained the

explicit

time evolution of these _{moments as a} function of the initial conditions.

Comparison

^{with the results known earlier}

[I]

revealed the role of self- action inside the field modes _{as a} factor that determines the

possibility

of the

steady squeezing.

It is evident that introduction of self-action in _at least _one of the field modes leads to

steady squeezing

in the

quadrature

components ^{of the} system.

Realization of the

steady-squeezing

states

II,

10] which

obviously

differ from the familiar

dynamic squeezed

states

[7-9] having

the

non-stationary

evolution of the

quadrature

variances in the _course of time is _more

probable

for the systems with ndxed

photon-phonon

interaction

[10-13]

than for the quantum

optical

systems. ^The main _reason is that for quantum

optical

models the

coupling

in the Hamiltonian Eke

(I)

is

time-dependent [2-4]

while for

photon- phonon

interaction this

coupling

_can be chosen _as

time-independent

[I1-13].

Photon-phonon

(2) ^{I-e- the} linearity of the Heisenberg equations of motion with respect ^to the field operators.

interacfiion in the

crystals,

for instance with rocksalt space-group symmetry, can be described at the lowest orders _over the field

strength by

the Hamiltonian like

(I) Ill,

13, 14].

Finally,

it should be

emphasized

that the

topic

of this paper can also be considered _as not

belonging

to quantum

optics

at all. The motivation of

writing

this article _was that

squeezed

states are of

general

interest, since the bosonic systems ^{for which the}

(dynamic) squeezed

states have been

already

observed 18, 9] are not

only

present ⁱⁿ quantum

optics

but also, for instance, in the

theory

of solids

Ill

_], in the

theory

of

superfluidity II

5, 16], ^{in nuclear}

physics II?, 18],

in

high-energy-physics [19, 20],

etc. The distinctive feature of the enumerated

systems ^{should be stressed that}

they

_possess

interesting physical properties

also in

equilibrium

or in

quasi-equilibrium

states with

steady

behaviour of fluctuations. Such _a

steady

behaviour of fluctuations is, of _course, reflected

by steady

evolution in time of the _mean values of the

variance operators

Ax~(t) confirming

the

importance

of the

topic

^of our work.

6. Briefly about the steady-squeezing wave fuucfiom.

An

independent interesting question

arises about the

explicit parameterization

of the _wave functions

realizing steady squeezing by

the _constants of the Hamiltonian. There is _no

steady squeezing

for the model considered above. _At the _same

time,

the

single-mode

model

ill

possesses the

steady-squeezing

behavitour. For this model the

explicit

forn1of

possible steady- squeezing

_wave functions has been obtained in

[10].

These _wave functions tumed _out to be

non-gaussian

and

phase-unsensitive hinting

at the

possible

elimination from the

experimental

set-up ^{of the}

bulky phase-sensitive

methods. In addition _we note that the

explicit

form of _a

steady-squeezing

wave function

essentially depends

on the form of _a Hamiltonian because

just

the Hamiltonian determines

properties

of _a system ⁱⁿ the _course of time.

Acknowledglnents.

We _are

grateful

to the referees of _{our paper,}

especially thanking

that referee who

proposed

in detail how _we could

improve

the

presentation

of the results.

Appendix

I.

The

explicit

fornl of the matrix L§.

We note once more that

only

condition

imposed

_on the transformations

(5)-(8)

and

(12)-(13), connecting

the

original

Bose field modes with the

subsidiary

_operators, is its

non-degeneracy.

In vitnue of

that,

all relations obtained in section 2 also remain valid in the _case when the transformations _are

non-unitary

because the

dissipation

or

stochasticity

is

effectively

taken into account in such transformations. In reference [2

Ii

the _so called _«

absorption

matrix _» has been introduced which in _{our case can} be written down _as H§ ₌ ^I L~

UT

^{' This matrix}

measures the

degree

of the

dissipation

or

gain

of the field

characteristics,

I-e- the

degree

of the

non-unitarity

of the evolution of the system. In fact, in real devitces there is

always

_some

dissipation. Examples

in quantum

optics

are dielectric

mirror,

Nicol

prism, space-frequency

filter and _{so on.} Another kind of

non-unitarity

system is quantum

amplifier

which has

negative dissipation,

etc.

[21].

Denoting

_a

= Re

(p ), p

= Im

(p ),

_y

=

Re

(v),

8

= Im

(v),

_we can write down the

general

matrix U~ for the transformation of the second-order _moments of the field in the form of the

following

table _:

~~

P~

_Y~ ~~ ~~fl _"Y

a3 -fly -fl3 y3

fl~ a~ 3~ Y~ all fl3 -~y a3 -ay -Y3

Y~ 3~

a ~ P~ _{Y3 ay fly a3 pa ap}

3~ y~ fl~ ^a~ -y3 pa ^a3 -fly _-ay ap

2ap-2ap 2y3-2y3 a~-fl~ a3+PY -aY+P3 ay-p3 a3+PY -Y~+3~

2ay -2 pa 2ay -2py ap y3

a~+y~

_-all

+ y3 -ap ₊ y3 p~+ 3~ a3 fly

2a3 2py 2 fly 2a3 _-ay- pa all + y3 a~+3~ -fl~-yl -all -y3 _ay + pa

2fly 2a3 2a3 2fly ay+fl3

all+Y(

-p2-y2 a2+a2 _{-a~-ya -ay-~a}

2P3-2aY 2pa -2ay

aa-fly~ ^fl~+3

a~-ya ap-ya a2+y2 aa-py

2Y3-2Y3 2aP-2aP -Y~+3 a3+PY ay-fla -ay+fla aa+fly a~-fl~

It may be

interesting

to _see what is the

explicit

form of the matrix _U~ for _some

simply calculating

_case, This matrix _can be

essentially simplified

for the

special

_case of the canonical

transformation with _zero relative

phase. Recalling

formulae

(20)

and

putting

#

= 0, we

obtain

0

D~

_{0 0} D 0 0 0 0

0 0

D~

0 0 0 0 -D 0

D~

_{0 0 0 D 0 0 0 0}

o

D2

_{o i o} 0 0 0 -D 0

0 0 0 0 0 -D D 0 -D~

~2~g

_{2D 0 2D 0 0}

_(1+D~)

_{0 0}

0 0

o o 0 0 -D 0

-D~

_{0 D}

0 0 0 0 D o

-D~

_{0 -D}

0 -2D 0 -2D 0 o o o

(1+D2~

_~

° ° ° 0

-D~

_o

D -D 0

where N

=

I

D~.

Thus,

the matrix obtained _can be

applied

for _any

problem

where _a need arises to calculate the second-order _moments of the field in the _course of the

unitary (I.e. non-dissipative)

evolution while the

general

forn1of the matrix U~ is

applicable

for _{any open} system.

It is

worthy

to _wam

against putting

#

=

0

directly

in the Hamiltonian

(I)

because the presence of _non-zero relative

phase

_can tum out to be crucial for the _appearance of

steady squeezing

_as it _was the _case for the

single-mode

model

ill. Only

after

convincing

oneself that the

steady squeezing

is

impossible

for _an

arbitrary

value of # one can put ^{the relative}

phase equal

to zero to

simplify

the calculation.

The alert reader, when

checking

the

unitarity

of the matrix U~,

shquld

also bear in mind that U~ ^is not

formally unitary

_as 10 _x 10 matrix in Euclidean _space

R~°

_because it acts not _on c-number columns but _on the operator colunms Ax~ _and

by

~. As _a matter of

fact,

this matrix is

to

defined _on the direct _{sum 3C}

= q~ _3C; often identical Hilbert spaces _3C~ wherein the operators

I i

(a+ )

_a and

(b+ )

b _are determined.

Appendix

2.

The

explicit

^{term of the} operator coefficients in fiwmulae (23) and (2/l) ^for the time evolution of

A@(t)

and

&4(t).

In this

Appendix

_we introduce the

designations

_:

p~

~2(1-D~)~'

Ax;~ m

Ax; (to)

_Axy

(to)

+ Ax~

to)

Ax,

(to),

I ~ j Ax

)

m

Ax/(

_to)

Then the coefficients in

(23)

_can be written down _as follows _:

Ko/N

=

Po/N

₌

(I

₊

D~)(Ax/

₊

Axj)

₊ 2 D~(Ax~ +

Ax()

₊

+ D

(I

₊

D~) (cos (ji)(fixj

₊ Ax~~) + sin

( pi (Axj4

_{+ Ax~~}

Kj IN

=

Pi IN

=

hi Axj

_{+ D} ^~

cos

(2

ji

)(Axj Ax()

₊

+ D

(cos (ji)(Axj~

₊ Ax~~) + sin #i

)(Axj~

Ax~~) + D sin

(2 ji)

_Ax~4;

K~/N

=

P~/N

=

D~(Ax/ Axj)

₊

^D~

_cos

(2 ji)(Axj _Ax()

+

+ D

~(cos ^(# )(Axj~

₊ Ax~~) + sin

(ji )(Axj~

Ax~~) +

D~

_sin

(2

ji Ax~~; K~,/_fir =

P~/N

=

D~

_sin

(2

#

) (Ax) Ax()

_{+ Axj~ +}

+ D

(sin (ji) (Axj~

₊ Ax24) + cos

(# )(Ax~~ Axj~))

D

~cos (2

dr Ax~4I

K~/N

=

P~/N

=

D~

_sin

(2 #)(Axj _Ax()

_D ^~_Axi~

D ^~

(sin (# (Axj~

₊ Ax~~) + cos

(ji)(A~~ Axj~))

₊ D

~cos (2

_{#i Ax~~;}

K~/N

=

P~/N

= D(I D~)

(sin (ji)(Axjj-

Ax~~) cos

(ji )(Ax~~

+

Axj~)) K~/N

₌

P~/N

=

2

D~

_sin

(2

_#i

)(Axj _Ax()

_2d~

_~Axj~

_D

_(I

+

D~)

_x

x

(sin

tk

) (Axj~

+ Ax~~) + cos #i

)(Ax~~ Axj~))

+ 2 D^~ _cos

(2

_{#i Ax~~}

K~/~Af =

P~/N

= 2

D~(Ax/

₊

Axj

₊

Axj

₊

Ax()

D

(I

+ D~) x

X

(Sin (#f)(fiX14

_{+ 1~X23) + CDS}

(#i)(hX13

_1~X24)

,

K~/N

=

P~/N

= 2 D

~(Ax) Axj)

_{2 D} ^~ _{cos (2 #i}

Axj Ax()

D

(I

+ D ~) x

x

(sin (#i (Axj~

Ax~~) + cos

(#i (Axj~

+ Ax~~) ²

D~

_sin

(2

_#i

) Ax~~.

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