Superposed Coherent and Squeezed Light

0.1 MB·English

by  Fesseha Kassahun

Superposed Coherent and Squeezed Light
Fesseha Kassahun∗
Department of Physics, Addis AbabaUniversity,P. O. Box 33761, AddisAbaba,Ethiopia
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(Submitted on 14 January 2012)
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n
a
J Abstract

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1 We first calculate the mean photon number and quadrature variance of superposed coherent

and squeezed light, following a procedure of analysis based on combining the Hamiltonians
]
h and using the usual definition for the quadrature variance of superposed light beams. This
p
procedureofanalysisleadstophysicallyunjustifiable meanphotonnumber ofthe coherentlight
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t and quadrature variance of the superposed light. We then determine both of these properties
n
employing a procedure of analysisbased onsuperposing the Q functions and applying a slightly
a
u modified definition for the quadrature variance of a pair of superposed light beams. We find
q the expected mean photon number of the coherent light and the quadrature variance of the
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superposed light. Moreover, the quadrature squeezing of the superposed output light turns out

1 to be equal to that of the superposed cavity light.
v
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1 Keywords: Q function, Mean photon number, Quadrature Squeezing, Superposition
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3
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1 1. Introduction
with the mean photon number of the coherent
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2 light andquadraturevarianceof thesuperposed
1
coherent and squeezed light, we undertake a
: It has been established that a one-mode sub-
v procedure of analysis based on combining the
i harmonic generator produces light with a max-
X Hamiltonians and using the usual definition for
imum quadrature squeezing of 50% below the
r thequadraturevarianceofthesuperposedlight.
a coherent-state level [1-4]. In addition, some
From the results of our analysis, we discover
authors [5-7] have studied the statistical and
thatthepresenceof thesqueezed light hassome
squeezingpropertiesofsuperposedcoherentand
effect on the mean photon number of the coher-
squeezed light, produced by harmonic and sub-
ent light. This must certainly be due to the
harmonic generations in the same cavity. These
procedure of analysis we have employed. More-
authors have carried out their analysis by com-
over, we find that the quadrature variance of
bining the pertinent Hamiltonians and apply-
the superposed coherent and squeezed light is
ingtheusualdefinition for quadraturevariance.
equaltothatofthesqueezedlightonly. Inother
But such procedure leads to physically unjusti-
words, the coherent light has no contribution to
fiablemeanphotonnumberofthecoherentlight
thequadraturevarianceofthesuperposedlight.
and quadrature variance of the superposed co-
The origin of this problem must be the use of
herent and squeezed light.
the usual definition for the quadrature variance
In order to see clearly the problems connected of the superposed light.
∗Email address: [email protected]
1
In order to avoid the aforementioned prob- then note that the noise operator associated
lems,wecalculatethemeanphotonnumberand with the vacuum reservoir has no effect on the
quadrature variance of the superposed coher- dynamics of the cavity mode operators. Hence
ent and squeezed light, applying a procedure of upon dropping this noise operator, the quan-
analysis based on superposing the Q functions tum Langevin equation for the operator aˆ can
and slightly modifying the usual definition for be written as
thequadraturevarianceofthesuperposedlight.
daˆ(t) 1
= κaˆ(t) i[aˆ(t),Hˆ], (4)
We also determine the quadrature squeezing of
dt −2 −
the superposed cavity and output light.
where κ is the cavity damping constant. There-
fore, on account of (4) and (3), we have
2. The Hamiltonian combining daˆ(t) 1
= κaˆ(t) ε aˆ†(t)+ε . (5)
2 1
procedure dt −2 −
Now applying the relation
2.1 The mean photon number d daˆ(t) daˆ(t)
aˆ(t)aˆ(t) = aˆ(t) + aˆ(t)
dth i (cid:28) dt (cid:29) (cid:28) dt (cid:29)
We seek to obtain the mean photon number (6)
and quadrature variance of superposed coher- along with Eq. (5), we get
ent and squeezed light, produced by harmonic d
aˆ(t)aˆ(t) = κ aˆ2(t) 2ε aˆ†(t)aˆ(t)
and subharmonic generations, in a cavity cou- 2
dth i − h i− h i
pledtoavacuumreservoirviaasingle-portmir- +2ε aˆ(t) ε . (7)
1 2
h i−
ror. We consider the case in which the coherent
Moreover, using the relation
and squeezed light beams have the same fre-
quency. The process of harmonic generation is d daˆ†(t) daˆ(t)
aˆ†(t)aˆ(t) = aˆ(t) + aˆ†(t)
described by the Hamiltonian dth i (cid:28) dt (cid:29) (cid:28) dt (cid:29)
(8)
Hˆ = iε (aˆ† aˆ), (1)
c 1 − together with Eq. (5), we find
whereε is proportionalto theamplitudeof the d
1 aˆ†(t)aˆ(t) = κ aˆ†(t)aˆ(t) ε aˆ2(t)
coherentlightdrivingthecavity mode. Andthe dth i − h i− 2h i
process of subharmonic generation is described ε aˆ†2(t) +ε aˆ(t)
2 1
− h i h i
by the Hamiltonian +ε aˆ†(t) . (9)
1
h i
iε
Hˆ = 2(aˆ2 aˆ†2), (2)
s 2 − The steady-state solutions of Eqs. (7) and (9)
have the form
with ε being proportional to the amplitude of
2
b
the coherent light pumping the nonlinear crys- aˆ2 = b aˆ†aˆ +a aˆ , (10)
tal. The analysis of the superposed coherent h i − h i h i− 2
and squeezed light is usually carried out em- b b a a
aˆ†aˆ = aˆ2 aˆ†2 + aˆ + aˆ† , (11)
ploying the Hamiltonian [5-7] h i −2h i− 2h i 2h i 2h i
where
iε
Hˆ = iε (aˆ† aˆ)+ 2(aˆ2 aˆ†2), (3) a = 2ε /κ (12)
1 1
− 2 −
and
which is the sum of the Hamiltonians given by
b = 2ε /κ. (13)
Eqs. (1) and (2). 2
In addition, the steady-state solution of the ex-
We will calculate the expectation values of only
pectation value of Eq. (5) turns out to be
normally-ordered operators pertaining to a cav-
ity mode coupled to a vacuum reservoir. We aˆ = b aˆ† +a. (14)
h i − h i
2
We also note that 2.2 Quadrature variance
aˆ† = b aˆ +a. (15)
We next wish to calculate the quadrature vari-
h i − h i
ance for the superposed coherent and squeezed
Upon adding and subtracting Eq. (15) to and
light. The variance of the quadrature operators
from Eq. (14), we arrive at
aˆ = aˆ†+aˆ (23)
a a +
aˆ aˆ† = ± . (16)
h i±h i 1+b and
aˆ = i(aˆ† aˆ), (24)
It then follows that − −
for the superposed light, is usually defined by
a
aˆ = aˆ† = . (17)
h i h i b+1 (∆a )2 = 1+ : aˆ ,aˆ : . (25)
± ± ±
h i
Furthermore, with the aid of (17), Eqs. (10) This can be put in the form
and (11) can be rewritten as
(∆a )2=1+2 aˆ†aˆ aˆ2 aˆ†2
±
h i±h i±h i
2a2 b(1+b) aˆ 2 aˆ† 2 2 aˆ† aˆ . (26)
aˆ2 = b aˆ†aˆ + − , (18) ∓h i ∓h i − h ih i
h i − h i 2(1+b)
Now employing Eqs. (17), (20), and (21), one
a2 can readily verify that
aˆ†aˆ = b aˆ2 + . (19)
h i − h i 1+b b
(∆a )2 = 1 (27)
Following the same procedure as the one lead- ± ∓1 b
±
ing to the result given by (17), we obtain from
and in view of (13), we have
Eqs. (18) and (19) that
2ε
(∆a )2 = 1 2 . (28)
aˆ2 = a2 + b (20) ± ∓κ±2ε2
h i (1+b)2 2(b2 1)
− This is just the quadrature variance of the
and squeezed light alone [4]. Contrary to our ex-
aˆ†aˆ = a2 + b2 . (21) pectation, the coherent light has no contribu-
h i (1+b)2 2(1 b2) tion to the quadrature variance of the super-
−
Now on account of (12) and (13), we see that posed light. We maintain standpoint that the
problems connected with the first term in Eq.
4ε2 2ε2 (22) and the quadrature variance described by
aˆ†aˆ = 1 + 2 . (22)
h i (κ+2ε2)2 κ2−4ε22 Eq. (28) could be resolved by carrying out the
pertinent analysis based on superposing the Q
Evidently the second term in Eq. (22) is the
functions, instead of combining the Hamiltoni-
mean photon number of the squeezed light.
ans, and by slightly modifying the usual defini-
However, the first term does not represent the
tion for the quadrature variance of superposed
mean photon numberof thecoherent light. The
light beams.
mean photon number of the coherent light is
just the first term with ε = 0. We clearly see
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thatthepresenceofthesqueezed light hassome 3. The Q function superposing
effect on the mean photon number of the coher-
procedure
ent light. There cannot be any physically valid
justification for this effect. The origin of the
problem connected with the first term in Eq. 3.1 The Q function
(22) must certainly be the procedure of analy-
sis we used, which is based on combining the In this section we first calculate the Q func-
Hamiltonians. tionsforthecoherentandsqueezedlightbeams.
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Then using these results, we determine the Furthermore, applyingEq, (31)andconsidering
Q function for the superposed coherent and the coherent light to be initially in a vacuum
squeezed light. state, one easily gets
aˆ′2(t) = aˆ′†(t)aˆ′(t) = 0. (39)
3.1.1 The coherent light Q function h i h i
Therefore, in view of this result, Eq. (38) takes
WenowseektoobtaintheQfunctionfortheco-
at steady state the form
herent light produced by harmonic generation.
To this end, upon setting ε2=0 in Eq. (5), we φa(z) = exp[ z∗z+a(z z∗)]. (40)
− −
have
daˆ(t) 1
= κaˆ(t)+ε . (29) Finally,uponintroducing(40)intoEq. (34)and
1
dt −2
carrying out the integration, the coherent light
The solution of this equation can be written as
Q function is found to be
aˆ(t)= a(1 e−κt/2)+aˆ′(t), (30)
− 1
Q(α∗,α) = exp[ α∗α+a(α+α∗) a2]. (41)
where π − −
aˆ′(t) = aˆ(0)e−κt/2 (31)
and a is given by (12). Then we easily get 3.1.2 The squeezed light Q function
daˆ′(t) 1
= κaˆ′(t) (32) We next proceed to determine the Q function
dt −2
forthesqueezed light producedby subharmonic
andwiththecoherentlightconsideredtobeini-
generation. Thus upon setting ε = 0 in Eqs.
1
tially in a vacuum state, we see that
(5), (20), and (21), we have
aˆ′(t) = 0. (33)
h i daˆ(t) 1
= κaˆ(t) ε aˆ†(t), (42)
The Q function for a single-mode light can be dt −2 − 2
expressed as
b
1 aˆ2 = , (43)
Q(α∗,α,t) = d2zφ (z,t)ez∗α−zα∗, (34) h i 2(b2 1)
π2 Z a −
and
in which
b2
aˆ†aˆ = . (44)
φ (z,t) = e−z∗aˆ(t)ezaˆ†(t) (35) h i −2(1 b2)
a h i −
UsingEq. (42)and assumingthe squeezed light
is the antinormally-ordered characteristic func-
to be initially in a vacuum state, one can read-
tion. With the aid of the identity
ily verify that aˆ(t) = 0. Now on realizing that
eAˆeBˆ = eAˆ+Bˆ+[Aˆ,Bˆ], (36) the operator aˆh(t) isia Gaussian variable with a
vanishingmeanandtakingintoaccount(43)to-
Eq. (35) can be put in the form
gether with (44), Eq. (37) can be put at steady
φ (z,t) = e−z∗z/2 ezaˆ†(t)−z∗aˆ(t) . (37) state in the form
a
h i
Now on substituting (30) into Eq. (37) and φ (z) = exp[ a z∗z+a (z2+z∗2)/2], (45)
observing that the operator aˆ′(t) is a Gaussian a − 1 2
variable with a vanishing mean, we have
in which
b2
φa(z,t)=exp[−z∗z+a(z−z∗)(1−e−κt/2) a1 = 1+ 2(1 b2), (46)
+1z2 aˆ′2(t) + 1z∗2 aˆ′†2(t) −
2 h i 2 h i b
z∗z aˆ′†(t)aˆ′(t) ]. (38) a2 = 2(b2 1). (47)
− h i −
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3.2 The mean photon number
Hence introducing (45) into Eq. (34) and car-
rying out the integration, the squeezed light Q
function is found to be
We now proceed to calculate the mean pho-
1 ton number of the superposed coherent and
[u2 v2]2
Q(α∗,α)= − exp[ uα∗α squeezed light. We recall that the expectation
π −
value of an operator function A(aˆ†,aˆ) can be
+v(α2+α∗2)/2], (48)
written as
where
1 b2/2 Aˆ = d2αQ(α∗,α)A (α∗,α), (55)
a
u = − (49) h i Z
1 b2/4
−
where A (α∗,α) is the c-number function cor-
and a
b/2 responding to Aˆ(aˆ†,aˆ) in the antinormal order.
v = . (50)
−1 b2/4 Now applying (53), one can put Eq. (55) in the
−
form
3.1.3 The superposed light Q function A
Aˆ = αd2exp[ uα∗α+v(α2+α∗2)/2
h i π Z −
We finally seek to calculate the Q function for +a(u v)(α+α∗)]Aa(α∗,α). (56)
−
the superposed coherent and squeezed light.
NowusingthefactthatA = α∗α 1, themean
This Q function is expressible as [8] a −
photon number can be written as
1
Q(α∗,α)= d2βd2γQ (β∗,α γ) A
π Z c − aˆ†aˆ = d2αexp[ uα∗α+v(α2 +α∗2)/2
Q (γ∗,α β)exp( α∗α β∗β h i π Z −
× s − − − +a(u v)(α+α∗)]α∗α 1, (57)
γ∗γ+α∗β+αβ∗ +α∗γ +αγ∗ − −
−
β∗γ βγ∗), (51) so that on carrying out the integration, there
− −
follows
whereQ (β∗,α γ) and Q (γ∗,α β) represent
c − s − b2
the Q functions for the coherent and squeezed aˆ†aˆ = a2+ . (58)
h i 2(1 b2)
light beams. With the aid of (41) and (48), one −
can put Eq. (51) in the form
Finally, in view of (12) and (13), we see that
1
[u2 v2]2 4ε2 2ε2
Q(α∗,α)= − d2βd2γexp[ α∗α aˆ†aˆ = 1 + 2 . (59)
π3 Z − h i κ2 κ2 4ε2
− 2
β∗β γ∗γ+[α∗+(u 1)γ∗
− − − As expected this is the sum of the mean pho-
vα]β +aβ∗+(α∗ a)γ
− − tonnumberofthecoherentlightandthatofthe
+(1 u)αγ∗ +aα a2 squeezed light.
− −
+v(α2 +β2+γ∗2)/2], (52)
We next wish to calculate the mean photon
number of the output light. The output mode
sothatoncarryingouttheintegration, weread-
operatoraˆ canbewrittenintermsofthecav-
ily arrive at out
ity mode operator aˆ and the input mode oper-
Q(α∗,α)=Aexp[ uα∗α+v(α2 +α∗2)/2 ator aˆin as
π −
+a(u v)(α+α∗)], (53) aˆ = √κaˆ aˆ , (60)
out in
− −
in which where κ is the cavity damping constant. When
calculating the expectation values of only
1
A= [u2 v2]2exp[a2(v u)]. (54) normally-ordered output operators, with the
− −
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cavity mode coupled to a vacuum reservoir, one Moreover, following the same procedure, one
can use the relation can easily verify that
aˆ = a. (70)
aˆout = √κaˆ. (61) h i
Now on account of Eq. (67) along with (58),
Then the mean photon number of the output
(69), and (70), the quadrature variance of the
light, defined by
superposed coherent and squeezed light turns
out to be
†
nout = haˆoutaˆouti, (62) (∆a )2 = 2 b . (71)
±
∓1 b
is expressible as ±
We observe that the quadrature variance given
by (71) is the sum of the quadrature variance
n = κ aˆ†aˆ . (63)
out
h i of the coherent light and that of the squeezed
light.
We then see that the mean photon number of
theoutputlightisjustκtimesthatofthecavity
One usually calculates the quadrature squeez-
light.
ing of a pair of superposed light beams relative
to the quadrature variance of a single coherent
light beam. But now this does not appear to be
3.3 Quadrature squeezing
a correct procedure. We then assert that the
quadrature squeezing of a pair of superposed
We finally discus the quadrature squeezing of light beams must be calculated relative to the
thesuperposedcoherentandsqueezedlight. We quadrature variance of a pair of superposed co-
define the variance of the quadrature operators herent light beams. Evidently, employing (67)
one easily finds the quadrature variance of a
aˆ+ = aˆ†+aˆ (64) pair of superposed coherent light beams to be
two. WeseefromEq. (71)thatthesqueezingof
and
the superposed coherent and squeezed light oc-
aˆ = i(aˆ† aˆ), (65)
− − curs in the plus quadrature. We then define the
for a pair of superposed light beams, by quadraturesqueezingof thesuperposedlightby
2 (∆a )2
(∆a )2 = 2+ :aˆ ,aˆ : . (66) S = − + (72)
± ± ±
h i 2
and in view of (71), we see that
This can be rewritten as
b
(∆a )2=2+2 aˆ†aˆ aˆ2 aˆ†2 S = . (73)
± 2(1+b)
h i±h i±h i
aˆ 2 aˆ† 2 2 aˆ† aˆ . (67)
This is just half of the quadrature squeezing of
∓h i ∓h i − h ih i
the squeezed light.
We next proceed to determine the expectation
Finally, we seek to determine the quadrature
valueoftheoperatoraˆ2. Thisexpectation value
squeezing of the output light. We define the
can be written employing (53) as
variance of the quadrature operators
aˆ2 =A d2αexp[ uα∗α+v(α2 +α∗2)/2 aˆo+ut = aˆ†out+aˆout (74)
h i π Z −
+a(u v)(α+α∗)α2. (68) and
− aˆout = i(aˆ† aˆ ), (75)
− out− out
Hence on carrying out the integration, we read- for the superposed coherent and squeezed out-
ily get put light, by
b
haˆ2i= a2+ 2(b2 1). (69) (∆ao±ut)2 = 2κ+ :aˆo±ut,aˆo±ut : , (76)
− h i
6
with 2κ being the quadrature variance of a pair both properties employing a procedure of anal-
of superposed coherent output light. On ac- ysis based on superposing the Q functions and
count of (61) and (66), one can put Eq. (76) applying a slightly modified definition for the
in the form quadraturevarianceofapairofsuperposedlight
beams. We have found the usual mean photon
(∆ao±ut)2 = κ(∆a±)2. (77) number of the coherent light and the quadra-
ture variance turned out be, as expected, the
We now realize that the quadrature variance
sum of the quadrature variance of the coherent
of the output light is just κ times that of the
lightandthatofthesqueezedlight. Inaddition,
cavity light. Moreover, we define the quadra-
our analysis shows that the quadrature squeez-
ture squeezing of the superposed coherent and
ingoftheoutputlight isexactly sameas thatof
squeezed output light by
the cavity light. It is also perhaps worth men-
tioning that the presence of the coherent light
2κ (∆aout)2
Sout = − + . (78) leads toanincrease inthemeanphoton number
2κ
and to a decrease in the quadrature squeezing.
Hence on account of Eq. (77), we see that
2 (∆a )2 References
Sout = − + , (79)
2
[1] G.J. Milburn and D.F. Walls, Phys. Rev.
whichisexactlythesameastheresultdescribed
A 27, 392 (1983).
by Eq. (72). We then see that the quadrature
[2] M.J. Collet and C.W. Gardiner, Phys. Rev.
squeezing of the output light is equal to that of
A 30, 1386 (1984).
the cavity light.
[3] J. Anwar and M.S. Zubairy, Phys. Rev.
A 45, 1804 (1992).
[4] B. Daniel and K. Fesseha, Opt. Commun.
4. Conclusion
151, 384 (1998).
[5] Berihu Teklu, Opt. Commun. 261, 310
Ourcalculation ofthemeanphotonnumberand (2006).
quadrature variance of the superposed coher- [6]Tewodros Y. Darge and Fesseha Kassahun,
ent and squeezed light, following a procedure of PMC Physics B, 1 (2010).
analysis based on combining the Hamiltonians [7] Misrak Getahun, PhD Dissertation (Addis
and using the usual definition for the quadra- Ababa University, 2009).
ture variance of superposed light beams, leads [8] Fesseha Kassahun, Fundamentals of Quan-
tophysically unjustifiablemean photon number tum Optics (Lulu Press Inc., North Car-
of the coherent light and quadraturevariance of olina, 2008).
the superposed light. We have then determined
7