New Differential Formulae Related to Hermite Polynomials and their Applications in Quantum Optics

0.33 MB·English

by  Sun Yun

New Differential Formulae Related to Hermite Polynomials and
their Applications in Quantum Optics
Sun Yun1, Wang Dong1, Wu Jian-guang1, and Tang Xu-bing1,2†
1
School of Mathematics & Physics Science and Engineering,
Anhui University of Technology, Ma’anshan 243032, China
5 2Institute of Intelligent Machines, Chinese Academy of Sciences,
1
Hefei 230031, China
0
2 †[email protected]

n
a January 27, 2015
J

6
2 Abstract

] In this work, based on quantum operator Hermite polynomials and Weyl’s mapping rule, we find a
h generationfunctionofthetwo-variableHermitepolynomials. Andthen,notingthattheWeylorderingis
p invariantunderthesimilartransformations, weobtainanothergeneralizeddifferentialexpressionrelated
-
to theHermite polynomials. Those identites can be applied to investigate the nonclasscial properties of
t
n quantumoptical fields.
a
u Keywords: Weyl mapping rule; Hermite polynomials; similar transformation
q PACS numbers: 42.50.Dv, 03.67. a, 42.50.Ex
[

1
1 Introduction
v
6
3 As the ”language” of quantum mechanics, Dirac’s bra-kets have come to represent a quantum world of
2 abstractideasanduniversalconceptsandcangetafarbetterunderstandingofquantummechanics. Because
6
conceptionsinquantummechanicsquitedifferfromthoseinclassicalmechanics,itisinevitablethatquantum
0
mechanics must have its own mathematical symbols which are endowed with special physical meaning. For
.
1 instance, in math [1], an inhomogeneous Fredholm equation (FE) of the first kind is written as g(t) =
0
b
k(t,s)f(s)ds, K(t,s) is the continuous kernel function. In quantum mechanics [2, 3], we introduced
5 a
1 a−n operator Fredholm equation defined as G a,a = b K a,a ,q F (q)dq, in which the kernel function
v: RK a,a ,q isaquantumoperator,qisarealvaria†ble,aa−nada den†otetheannihilationandcreationoperator
† (cid:0) (cid:1) R †(cid:0) (cid:1)
i ofaquantizedradiationfield. Asiswellknown,integrationsovertheoperatorsoftype cannotbedirectly
X
(cid:0) (cid:1) |ih|
performed by Newton–Leibniz integration rule.
r
a In Ref. [4], Fan proposed the technique of integration within an ordered product (IWOP) of operators
which enables Newton–Leibniz integration rules directly working for Dirac’s ket–bra operators with contin-
uumvariables. The technique ofIWOPshowsthatthe operatorFredholmequation(OFE)candirectlyper-
formintegrationifK a,a ,q isanorderedproductoperator. AnexampleoftakingK a,a ,q =: exp(q Q): ,
† †
−
we have
(cid:0) ∞ d(cid:1)q : exp (q Q)2 : f(q)=: G(Q): , Q= a+a†,(cid:0) (cid:1) (1)
√π − − √2
Z−∞ h i
where : exp (q Q)2 : is the integral kernel, a commutes with a within the normally ordered symbol
†
− −
”: : ”. Notinhg that 1 :iexp (q Q)2 : = q q and the completeness relation ∞ dq q q = 1, we
√π − − | ih | | ih |
can see −∞
h i R
∞ dq : exp (q Q)2 : f(q)= ∞ dq q q f(q)= ∞ dq q q f(Q)=f(Q), (2)
√π − − | ih | | ih |
Z−∞ h i Z−∞ Z−∞
where q = π 1/4exp q2/2+√2qa a 2/2 0 denotes the coordinate representation with its eigen-
− † †
| i − − | i
function Q q =q q , and : G(Q): is the normally ordered expansion of f(Q). When f(q)=H (q), we
n
| i | i (cid:0) (cid:1)
1
have
∞ dq : exp (q Q)2 : H (q)=2n: Qn: , (3)
n
√π − −
Z−∞ h i
where 2n: Qn: is the normally ordered form of the operator Hermite polynomials H (Q). H (q) is the
n n
single-variable Hermite polynomials with its generation function
∞ qn
H (t)=exp 2tq q2 (4)
n
n! −
n=0
X (cid:0) (cid:1)
andHn(q)spansanorthonormalandcompletefunctionspace,namely ∞ √dqπe−q2Hn(q)Hm(q)=2nn!δnm.
−∞
Notingtheannihilationoperatora=(Q+iP)/√2,andhq|a† =2−1/2Rq− ddq hq|,wecanderivethematrix
element of Hn(Q) in the coordinate representation and the vacuum st(cid:16)ate (cid:17)
q H (Q) 0 =2n q : Qn: 0 ,
n
h | | i h | | i
and then it follows
Hn(q) = 2n/2eq2/2 q : a†+a n : 0 =2n/2eq2/2 q a†n 0 =eq2/2 q d ne−q2/2
h | | i h | | i − dq
(cid:18) (cid:19)
= ( 1)neq2 dn e(cid:0)q2, (cid:1) (5)
−
− dqn
which is just the differential expression of single-variable Hermite polynomials. The above derivation shows
that Dirac’s symbol and its own arithmetic rule can promote the development of the basic quantum theory.
In Ref.[5], author introduced a two-variable Hermite polynomials in complex space
min(m,n)
m!n!
Hm,n(α,α∗)= ( 1)lαm−lα∗n−l, α=q+ip, (6)
l!(m l)!(n l)! −
l=0 − −
X
whose generating function is
∞ tmτn
H (α,α )=exp( tτ +tα+τα ). (7)
m,n ∗ ∗
m!n! −
m,n=0
X
Two-variable Hermite polynomials can be applied in many fields of physics. For instance, H (α,α ) is
m,n ∗
proved to be the eigenmode of the complex fractional Fourier transform [6, 7, 8], so it may be observed
in the light propagation in graded index (GRIN) medium, and this eigenmode is also the mechanism for
two-dimensional Talbot effect demonstrated in GRIN medium [9].
Due to there exists some similarities between the generation function of single-variable Hermite polyno-
mialsH (x)andthatoftwo-variablecomplexHermitepolynomialsH (α,α ),itisinterestingtoseethat
n m,n ∗
thedifferentialexpressionofH (α,α )issimilartothatofH (x). Inthiswork,byvirtueofWeyl’s map-
m,n ∗ n
ping rule and quantum operator Hermite polynomials, we obtain the differential expression of H (α,α )
m,n ∗
and another generalized forms. So our work is arranged as follows. In Sec. 2, we briefly introduce Weyl
correspondence rule, which is related to the Weyl ordering and the technique of integrationwithin Weyl or-
dered product(IWWP) of operators. We revealthat quantum correspondenceoperatorof classicalfunction
f(p,q)candirectlybeobtianedbyreplacingq andpinf(p,q)byQandP withthefunctionforminvariant.
In Sec. 3 we introduce the two-variable Hermite function. By formulizing the Weyl correspondence, we
derive a differentialexpressionofH (α,α ). Noting that the Weylorderingis invariantunder the similar
m,n ∗
transformations, the generalized differential expressions are obtained in Sec. 4. Enlightened by those new
identities, in Sec. V we investigate the nonclassical properties of an excited squeezed vacuum state, which
can be generated in quantum systems with restricted dimensions.
2 Weyl Ordering and Weyl Correspondence Rule
As is well known, the Weyl correspondence rule [10, 11], i.e.
F (P,Q)= dqdpf(p,q)∆(q,p), (8)
Z Z
2
isarecipeforquantizingaclassicalfunctionf(p,q)definedinclassicalphasespaceasaquantumcorrespond
operator F (P,Q). ∆(q,p) is the Wigner operator difined as
∆(p,q)= ∞ dueipu q+ u q u . (9)
2π 2 − 2
Z−∞ (cid:12) ED (cid:12)
(cid:12) (cid:12)
When f(p,q)=qmpr, from Eq. (8) its quantum corresp(cid:12)onding operator(cid:12)s is
qmpr → 21 m m l!(mm! l)!Qm−lPrQl = 21 m m l!(mm! l)! ˙... Qm−lPrQl ˙... = ˙... QmPr ˙... , (10)
(cid:18) (cid:19) l=0 − (cid:18) (cid:19) l=0 −
X X
˙. ˙.
in which . . denotes the Weyl ordering symbol, and Eq. (10) tell us that the Weyl ordered correspond
. .
operator of classcial function f(p,q) is obtained by just replacing p,q in f(p,q) by P,Q with the function
forminvariant. Asoneofthe definite operatororderings(suchasnormalordering,anti-normalorderingand
˙. ˙.
Weyl ordering), Weyle ordering is a useful one and within Weyl ordering symbol . . the Bose operators
. .
arepermutable. EnlightedbythetechniqueofIWOP,[13]proposedthetechniqueofintegrationwithinWeyl
ordered product (IWWP) of operators. From (9) we obtain the Weyl ordering form of the Wigner opertor
˙. ˙.
∆(p,q)= . δ(p P)δ(q Q) . . (11)
. − − .
Thereforeonecaneasilyobtainquantumcorrespondoperatorofclassicalfunctionh(p,q)byreplacingq Q,
→
p P, i.e.
→
˙. ˙. ∞
F(P,Q)= . h(P,Q) . = dpdqh(p,q)∆(p,q). (12)
. .
Z Z−∞
Eq. (1) can tell us that the Weyl correspondence rule is also an OFE. Noting that
d2z
Tr[∆(p ,q )∆(p ,q )] = z [∆(p ,q )∆(p ,q )] z
1 1 2 2 1 1 2 2
π h | | i
Z
1
= δ(q q )δ(p p ), (13)
1 2 1 2
2π − −
it then follows that the reciprocal relation of the Weyl correspondence rule is
2πTr[F (P,Q)∆(q,p)]=2πTr dq dp h(p ,q )∆(p ,q )∆(p,q) =h(p,q). (14)
1 1 1 1 1 1
(cid:20)Z Z (cid:21)
In many cases, taking α=(q+ip)/√2, the Wigner operator in (9) can be rewritten as
d2z
∆(p,q) ∆(α,α∗)= α+z α z eαz∗−α∗z
→ π2 | ih − |
Z
1 1 ˙. ˙.
= π: exp −2 a†−α∗ (a−α) : = 2 .. δ a†−α∗ δ(a−α) .. . (15)
(cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)
It then follows that the Weyl correspondence formula in Eq. (12) can be recast to
˙. ˙.
G a,a† = .. f a,a† .. =2 dqdpf(α,α∗)∆(α,α∗), (16)
Z Z
(cid:0) (cid:1) (cid:0) (cid:1)
with its reciprocal relation
˙. ˙.
2πTr G a,a† ∆(α,α∗) =2πTr .. f a,a† .. ∆(α,α∗) =f(α,α∗). (17)
(cid:20) (cid:21)
(cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)
Especially, when G a,a =ρ a,a describes a density of states for an interesting quantum system, from
† †
the above reciprocal relation we can see
(cid:0) (cid:1) (cid:0) (cid:1)
2πTr ρ a,a† ∆(α,α∗) =W (α,α∗), (18)
W (α,α ) denotes a quasi-probability d(cid:2)ist(cid:0)ributi(cid:1)on Wigner(cid:3)function. Thus Eq. (18) can also be called the
∗
Wigner-Weyl correspondence rule.
3
3 New Differential Expression of Two-variable Hermite Polyno-
mials H (α,α )
m,n ∗
In quantum theory, H (α,α ) is the generalized Bargmann representationof the two-mode Fock state in
m,n ∗
the bipartite entangled state representation [14], i.e.
a†mb†m 1 |ξ|2
m,n = 0,0 Hm,n(ξ,ξ∗)e− 2 ,
| i √m!n! | i→ √m!n!
and it spans an orthonormaland complete function space,
2 dπ2ξe−2|ξ|2Hm,n √2ξ,√2ξ∗ Hm∗′,n′ √2ξ,√2ξ∗ =√m!n!m′!n′!δm,m′δn,n′. (19)
Z Z (cid:16) (cid:17) (cid:16) (cid:17)
Eq.(19) indicates any one function f(α,α ) can be expanded by those orthogonalbasis,
∗
∞
f(α,α∗)= Cm,nHm∗,n √2α,√2α∗ . (20)
mX,n=0 (cid:16) (cid:17)
where C is a constant to be determined by follow derivation. From its generationfunction shown in (7),
m,n
we can also expand the normally ordered form of Wigner operator in Eq. (15) as
∆(α,α )= 1: exp 2 a α (a α) : = 1e 2α2: ∞ √2m+na†manH √2α,√2α : . (21)
∗ † ∗ − | | m,n ∗
π − − − π m!n!
(cid:2) (cid:0) (cid:1) (cid:3) mX,n=0 (cid:16) (cid:17)
Substituting (20) and (21) into (16), we have
G a†,a = 2 dπ2αe−2|α|2: ∞ √2mm+n!na!†manHm,n √2α,√2α∗ : ∞ Cm′,n′Hm∗′,n′ √2α,√2α∗
(cid:0) (cid:1) Z mX,n=0 (cid:16) (cid:17) m′X,n′=0 (cid:16) (cid:17)
= 2: ∞ ∞ C √2m+na†man: d2αe 2α2H √2α,√2α H √2α,√2α(22)
m′,n′ m!n! π − | | m,n ∗ m∗′,n′ ∗
mX,n=0m′X,n′=0 Z (cid:16) (cid:17) (cid:16) (cid:17)
∞
= : C √2m+na man: : F a ,a : .
m,n † †
≡
m,n=0
X (cid:0) (cid:1)
Taking the coherent state expectation values of (22), we have
α : F a ,a : α =F (α ,α)= α : ∞ C √2m+na man: α = ∞ C √2m+n(α )mαn,
† ∗ m,n † m,n ∗
h | | i h | | i
m,n=0 m,n=0
(cid:0) (cid:1) X X
and then
1 ∂m ∂n
C = F (α ,α) .
m,n m!n!√2m+n ∂α∗m∂αn ∗ (cid:12)α=0
(cid:12)
Therefore, from (20) we obtain (cid:12)
(cid:12)
∞ 1 ∂m ∂n
F (α,α∗)=mX,n=0Hm,n(cid:16)√2α,√2α∗(cid:17)m!n!√2m+n ∂α∗m∂αnF (α∗,α)(cid:12)(cid:12)α=0. (23)
(cid:12)
This is a new formula for deriving Weyl’s classical correspondence of normally ordered(cid:12)quantum operators.
For example, when G a ,a =: F a ,a : =: a man: , from Eq. (23) we have
1 † 1 † †
(cid:0) (cid:1) (cid:0) (cid:1) 1
g (α,α )= H √2α,√2α . (24)
1 ∗ m,n ∗
√2m+n
(cid:16) (cid:17)
And then considering the reciprocal relation in (17), we can see
1
g (α,α )=2πTr G a ,a ∆(α,α ) =2πTr a man∆(α,α ) = H √2α,√2α
1 ∗ 1 † ∗ † ∗ m,n ∗
√2m+n
(cid:2) (cid:0) (cid:1) (cid:3) (cid:2) (cid:3) (cid:16) (cid:17)
4
On the other hand, given any operator F a,a , its classical correspondence can also be obtained by
†
utilizing the following formular (see Appendix for details)
(cid:0) (cid:1)
f(α ,α)=e2α2 d2β β F a ,a β e2(αβ∗ α∗β). (25)
∗ | | † −
π h− | | i
Z
(cid:0) (cid:1)
Substituting G a ,a =: a man: into (25), we have
1 † †
g (α,α ) = (cid:0)e2α2(cid:1) d2β β G a ,a β e2(αβ∗ α∗β) =e2α2 d2β β : a man: β e2(αβ∗ α∗β)
1 ∗ | | 1 † − | | † −
π h− | | i π h− | | i
Z Z
= e2|α|2 d2β ( β∗)mβ(cid:0)ne−2(cid:1)|β|2+2(αβ∗−α∗β).
π −
Z
Making the substitution √2β β,√2β∗ β∗, the above equation can be derived as
→ →
( 1)me2α2 d2β
g1(α,α∗) = −√2 m+|n| π β∗mβnexp −|β|2+√2(αβ∗−α∗β)
Z h i
= (−(cid:0)1)m(cid:1)−ne2|α|2 ∂m ∂n d2β exp β 2+√2(αβ∗ α∗β) .
2m+n ∂αm∂α n π −| | −
∗ Z h i
Utilizing the following integral formula
d2α 1 sη
exp h α2+sα+ηα = exp , Re[h]<0,
∗
π | | h − h
Z h i h i
we can obtain
g1(α,α∗)= (−1)2mm−+nne2|α|2 ∂∂αmm∂∂αnn exp −2|α|2 . (26)
∗
(cid:16) (cid:17)
Comparing Eq. (26) and (24), and setting √2α α,√2α α we can derive
∗ ∗
→ →
Hm,n(α,α∗)=(−1)m−ne|α|2∂∂αmm∂∂αnn exp −|α|2 , (27)
∗
(cid:16) (cid:17)
which is just the differential form for the generation function of two-variable Hermite polynomials, and for
the especial case of m=n
∂2m
H (α,α )=exp α2 exp α2 .
m,m ∗ | | ∂α mαm −| |
∗
(cid:16) (cid:17) (cid:16) (cid:17)
4 Generalized Differential Expressions related to Hermite Poly-
nomials
In order to generate non-symmetric quantum mechanical representation, in Ref.[18] authors proposed a
non-unitary operator Uˆ, defined as
1 d2z µ ν z z
Uˆ =
√µ π σ τ z∗ z∗
Z (cid:12)(cid:18) (cid:19)(cid:18) (cid:19)(cid:29)(cid:28)(cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
= 1 : exp (cid:12)(cid:12) ν a†2+ 1 1 a†a+ σ a2(cid:12)(cid:12) : , (28)
√µ −2µ µ − 2µ
(cid:20) (cid:18) (cid:19) (cid:21)
whichisaquantumoperatorimageoftheclassicalsymplectictransformation(z,z ) (µz+νz ,σz+τz )
∗ ∗ ∗
→
z µ ν z
inphasespace,andwhere =exp za z a 0 denotesthecoherentstate[19]and =
z∗ †− ∗ | i σ τ z∗
(cid:12)(cid:18) (cid:19)(cid:29) (cid:12)(cid:18) (cid:19)(cid:18) (cid:19)(cid:29)
µz+νz (cid:12) (cid:0) (cid:1) (cid:12)
∗ =exp ((cid:12)µz+νz )a (σz+τz )a 0 . The invers of Uˆ reads as (cid:12)
σz+τz∗ (cid:12) ∗ †− ∗ | i (cid:12)
(cid:12)(cid:18) (cid:19)(cid:29)
(cid:12) (cid:2) (cid:3)
(cid:12) d2z z τ ν z
(cid:12) Uˆ−1 = √µ π z∗ σ −µ z∗
Z (cid:12)(cid:18) (cid:19)(cid:29)(cid:28)(cid:18) − (cid:19)(cid:18) (cid:19)(cid:12)
1 (cid:12)ν 1 σ (cid:12)
= : exp (cid:12)(cid:12) a†2+ 1 a†a a2 : . (cid:12)(cid:12) (29)
√τ 2τ τ − − 2τ
(cid:20) (cid:18) (cid:19) (cid:21)
5
From (28) and (29), we can find Uˆ =Uˆ 1 and Uˆ engenders a similar transformation
† −
6
UˆaUˆ−1 =µa+νa†, Uˆa†Uˆ−1 =σa+τa† (30)
and its invers transformation
Uˆ 1aUˆ =τa νa , Uˆ 1a Uˆ =µa σa, (31)
− † − † †
− −
where four complex parameters satisfies µτ νσ =1 for keeping µa+νa , σa+τa =1.Also it is impor-
† †
−
tant to note that the Weyl ordering is invariant under the similar transformations [4, 15, 16], i.e.
(cid:2) (cid:3)
UˆG a†,a Uˆ−1 =Uˆ ˙... g a†,a ˙... Uˆ−1 = ˙... g σa+τa†,µa+νa† ˙... . (32)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
4.1 Generalized Differential Expression Related to the Product of Two Single-
variable Hermite Polynomials
Supposing an operator G a ,a = a mUˆ 0 0 Uˆ 1an, and from Eqs. (17) and (31), we can express the
2 † † −
| ih |
classical Weyl correspondence of G a ,a as
(cid:0) (cid:1) 2 †
(cid:0) (cid:1)
g (α ,α) = 2πTr G a ,a ∆(α,α ) =2πTr a mUˆ 0 0 Uˆ 1an∆(α,α )
2 ∗ 2 † ∗ † − ∗
| ih |
= 2πTr(cid:2)Uˆ (cid:0)µa† (cid:1)σa m 0 (cid:3)0 τa νha† nUˆ−1∆(α,α∗) . i (33)
− | ih | −
h (cid:0) (cid:1) (cid:0) (cid:1) i
By virtue of the following formular
n
n fg f g
fa+ga = i : H i a+i a : ,
† n †
− r 2 ! s2g r2f !
(cid:0) (cid:1)
we can see
m
m σµ σ µ
µa σa = : H a a : (34)
† m †
− 2 − 2µ − 2σ
(cid:18)r (cid:19) (cid:18) r r (cid:19)
(cid:0) (cid:1) n
n ντ τ ν
τa νa† = : Hn a a† : . (35)
− 2 − 2ν − 2τ
(cid:18)r (cid:19) (cid:18) r r (cid:19)
(cid:0) (cid:1)
Therefore we have
m n
σa+µa 0 0 τa νa
† †
− | ih | −
m n
(cid:0) σµ (cid:1) ντ (cid:0) (cid:1) µ τ
= : H a exp a a H a : , (36)
m † † n
2 2 − 2σ − − 2ν
(cid:18)r (cid:19) (cid:18)r (cid:19) (cid:18) r (cid:19) (cid:18) r (cid:19)
(cid:2) (cid:3)
where 0 0 =: exp a a : . Utilizing the following formular (for details in Appendix)
†
| ih | −
(cid:2) ˙. (cid:3) ˙. d2z ˙. ˙.
F a†,a = .. f a†,a .. =2 π .. h−z|F a†,a |ziexp 2 az∗−a†z+a†a .. , (37)
Z
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
m n
we can obtain the Weyl ordering form of σa+µa 0 0 τa νa , namely
† †
− | ih | −
(cid:0) (cid:1) (cid:0)m (cid:1) n
m n σµ ντ
σa+µa† 0 0 τa νa† =2
− | ih | − 2 − 2
(cid:18)r (cid:19) (cid:18) r (cid:19)
(cid:0)× ˙... dπ2zH(cid:1) m 2µ(cid:0)σz∗ Hn (cid:1) 2τνz exp −|z|2−2a†z+2az∗+2a†a ˙... . (38)
Z (cid:18)r (cid:19) (cid:18)r (cid:19) h i
An explicit integral formula proved in Ref. [17]
d2z min[m,n] 22lm!n!
Hm(z∗)Hn(z)exp[ (z λ)(z∗ λ∗)]= Hm l(λ∗)Hn l(λ) (39)
π − − − l!(m l)!(n l)! − −
Z l=0 − −
X
6
can be utilized for the derivation of Eq.(38). Furtherly, we have
d2z min[m,n] (4αβ)lm!n!
Hm(αz∗)Hn(βz)exp[ (z λ)(z∗ λ∗)]= Hm l(αλ∗)Hn l(βλ), (40)
π − − − l!(m l)!(n l)! − −
Z l=0 − −
X
then it follows
m n
m n σµ ντ
σa+µa 0 0 τa νa =2m!n!
† †
− | ih | − − 2 − 2
(cid:18) r (cid:19) (cid:18) r (cid:19)
×(cid:0) ˙... minl=[m0,n(cid:1)]l!(m(cid:0)−−2pl(cid:0))!µσ(nτν(cid:1)−l l)!H(cid:1) m−l r2σµa†!Hn−l r2ντa!exp −2a†a ˙... . (41)
X (cid:2) (cid:3)
Noting that Weyl ordering is invariant under the similar transformations shown in Eq. (32), and using the
Weyl correspondence formula in Eq. (17), we can obtain the classical correspondence of G a ,a ,
2 †
g (α ,α) = 2πTr G a ,a ∆(α,α ) =2πTr a mUˆ 0 0 Uˆ 1an∆(α,α ) (cid:0) (cid:1)
2 ∗ 2 † ∗ † − ∗
| ih |
(cid:2) (cid:0) σµ(cid:1) m ν(cid:3)τ nmin[hm,n] 2 µτ l i 2µ
= 2m!n! − σν H (σα+τα ) (42)
m l ∗
(cid:18)−r 2 (cid:19) (cid:18)−r 2 (cid:19) l=0 l!(m(cid:0) −pl)!(n(cid:1)−l)! − "r σ #
X
2τ
Hn l (µα+να∗) exp[ 2(σα+τα∗)(µα+να∗)].
× − "r ν # −
From Eq. (25), the classical correspondence of G a ,a can also be expressed as
2 †
g (α ,α) = e2α2 d2β β a mUˆ 0 0 Uˆ (cid:0)1an (cid:1)β e2(αβ∗ α∗β)
2 ∗ | | † − −
π h− | | ih | | i
Z
= 1 e2|α|2 d2β ( β∗)mβnexp β 2 2α∗β+2αβ∗ σ β2 ν β∗2 (43)
√µτ π − −| | − − 2τ − 2µ
Z (cid:20) (cid:21)
e2α2 ( 1)m+n ∂m ∂n d2β σ ν
= | | − exp β 2 2α∗β+2αβ∗ β2 β∗2 .
√µτ 2m+n ∂αm∂α n π −| | − − 2τ − 2µ
∗ Z (cid:20) (cid:21)
And then using the following integral formula
d2α 1 hsη+s2g+η2f
exp h α2+sα+ηα∗+fα2+gα∗2 = exp − ,
π | | h2 4fg h2 4fg
Z h i − (cid:20) − (cid:21)
whose convergence conditions are Re(h f g)<0 and Rpe h2−4fg <0, we have
± ± h f g
± ±
(cid:16) (cid:17)
g2(α∗,α)=e2|α|2(−21m)+mn+n∂∂αmm∂∂αnn exp −4µτ|α|2−2ντα∗2−2σµα2 . (44)
∗
h i
Comparing Eqs. (42) and (44), we can see
∂m ∂n
exp 4µτ α2+2ντα 2+2σµα2 exp 4µτ α2 2ντα 2 2σµα2
∗ ∗
| | ∂αm∂α n − | | − −
∗
h i h i
min[m,n] l
= 2(2µσ)m2 (2τν)n2 m n l! 4µτ (45)
l l − σν
Xl=0 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) q (cid:19)
H 2µ(σα+τα ) H 2τ (µα+να ) .
× m−l σ ∗ n−l ν ∗
(cid:20)q (cid:21) (cid:20)q (cid:21)
whichis ageneralizeddifferentialformularelatedto theproductoftwosingle-variableHermite polynomials.
If taking τ =µ , σ =ν to meet µ2 ν 2 =1, Uˆ is unitary. Thus Eq. (45) reduces to
∗ ∗
| | −| |
∂m ∂n
exp 4 µ2 α2+2µ να 2+2µν α2 exp 4 µ2 α2 2µ να 2 2µν α2
∗ ∗ ∗ ∗ ∗ ∗
| | | | ∂αm∂α n − | | | | − −
∗
h i h i
min[m,n] l
m n m n 2 µ
= 2(2µν∗)2 (2µ∗ν)2 l! | | (46)
l l − ν
l=0 (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) | | (cid:19)
X
×Hm−l 2νµ∗ (ν∗α+µ∗α∗) Hn−l 2µν∗ (µα+να∗) .
(cid:20)q (cid:21) (cid:20)q (cid:21)
7
Especially, for m=n, Eq. (46) is
∂m ∂m
exp 4 µ2 α2+2µ να 2+2µν α2 exp 4 µ2 α2 2µ να 2 2µν α2
∗ ∗ ∗ ∗ ∗ ∗
| | | | ∂αm∂α m − | | | | − −
∗
h m i l h 2 i
m m 2 µ 2µ
= 2m+1 µm ν m l! | | Hm l (ν∗α+µ∗α∗) . (47)
| | | | Xl=0(cid:18)l (cid:19)(cid:18)l(cid:19) (cid:18)− |ν| (cid:19) (cid:12)(cid:12) − "rν∗ #(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4.2 Generalized Differential Expression R(cid:12) elated to Two-variab(cid:12)le Hermite Poly-
nomials
Followingweshallderiveanothergereralizeddifferentialexpressionrelatedtotwo-variableHermitePolynomi-
als. FromEqs. (34)and(35)wecanexpresstheclassicalcorrespondenceofG a ,a =a mUˆ 0 0 Uˆ 1anas
2 † † −
| ih |
g2(α∗,α) = 2πTr Uˆ µa† σa m 0 0 τa νa† nUˆ−1∆(α,α∗) (cid:0) (cid:1)
− | ih | −
h (cid:0)σµ m (cid:1) ντ n (cid:0) (cid:1) µ i τ
= 2π Tr UˆHm a† 0 0 Hn a Uˆ−1∆(α,α∗) .
− 2 − 2 2σ | ih | 2ν
(cid:18) r (cid:19) (cid:18) r (cid:19) (cid:20) (cid:18)r (cid:19) (cid:18)r (cid:19) (cid:21)
Due to
[m/2] ( 1)km!
Hm(x)= − (2x)m−2k,
k!(m 2k)!
k=0 −
X
we can obtain
k
g (α ,α) = 2π µ m τ n[m/2][n/2] −2σµ m! −2ντ ln!
2 ∗ −2 −2 k!(cid:16) (m(cid:17) 2k)!l! (n 2l)!
(cid:16) (cid:17) (cid:16) (cid:17) Xk=0 Xl=0 − (cid:0) −(cid:1)
Tr Uˆ m 2k n 2l Uˆ 1∆p(α,α ) . p (48)
− ∗
× | − ih − |
h i
Supposed a projection operator of the number state m n and from Eq. (37), its Weyl ordering reads as
| ih |
d2z ˙. ˙.
|mihn| = 2 π .. h−z|mihn|ziexp 2 az∗−a†z+a†a ..
Z
= 2 dπ2z ˙... (−√zn∗)!mm!zn exp −2(cid:2)|z(cid:0)|2+2 az∗−a†z(cid:1)+(cid:3) a†a ˙... . (49)
Z h (cid:0) (cid:1)i
By virtue of the following integral formula
d2ββ∗kβlexp h β 2+sβ+fβ∗ =( i)k+lh−k+2l−1eshfHk,l is , if
π − | | − √h √h
Z h i (cid:18) (cid:19)
we have
2 ˙. ˙.
|mihn|= √n!m! .. Hm,n 2a†,2a exp −2a†a .. . (50)
(cid:0) (cid:1) (cid:0) (cid:1)
Therefore, considering Eqs. (32) and (50), we can see
k
g2(α∗,α) = 4π −µ2 m −τ2 n[m/2][n/2]k(cid:16)!−(m2σµ(cid:17)2mk)!!l!−(n2ντ l2nl)!!
(cid:16) (cid:17) (cid:16) (cid:17) Xk=0 Xl=0 − (cid:0) −(cid:1)
˙. ˙.
×Tr .. Hm−2k,n−2l 2 σa+τa† ,2 µa+νa† exp −2 σa+τa† µa+νa† .. ∆(α,α∗) .
(cid:20) (cid:21)
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:0) (cid:1)(cid:3)
Weyl correspondence rule in (17) tells us that the classical correspondence of G a ,a is
2 †
µ m τ n (cid:0) (cid:1)
g2(α∗,α) = 2 exp[ 2(σα+τα∗)(µα+να∗)]
−2 −2 −
(cid:16) (cid:17) (cid:16) (cid:17) k
[m/2][n/2] −2σµ m! −2ντ ln!H [2(σα+τα ),2(µα+να )]. (51)
× k(cid:16)!(m (cid:17)2k)!l!(n 2l)! m−2k,n−2l ∗ ∗
k=0 l=0 − (cid:0) −(cid:1)
X X
8
Comparing Eqs. (51) and (44), we also derive a simplified equation
∂m ∂n
exp 4µτ α2+2ντα 2+2σµα2 exp 4µτ α2 2ντα 2 2σµα2
∗ ∗
| | ∂αm∂α n − | | − −
∗
h k i h i
= 2µmτn[mk=/02][nl=/02]k(cid:16)!−(m2σµ−(cid:17)2mk)!!l(cid:0)!−(n2ντ−(cid:1)l2nl)!!Hm−2k,n−2l[2(σα+τα∗),2(µα+να∗)]. (52)
X X
The right of (52) is a summation of two-variable Hermite Polynomials, while Eq. (45) is related to the
product of two single-variable Hermite Polynomials. Eqs. (45) and (52) are new generalized differential
expressions related to the Hermite Polynomials. For a special case of m=n, Eq. (52) reduces to
∂m ∂m
exp 4µτ α2+2ντα∗2+2σµα2 exp 4µτ α2 2ντα∗2 2σµα2
| | ∂αm∂α m − | | − −
∗
h k i h i
= 2(µτ)m[m/2] −2σµ m! −2ντ lm!H [2(σα+τα ),2(µα+να )].
k(cid:16)!(m (cid:17)2k)!l!(m 2l)! m−2k,m−2l ∗ ∗
k,l=0 − (cid:0) −(cid:1)
X
Foraspecialcaseofunitaryoperator,e.g. Uˆ =exp r a 2 a2 ,wehaveτ =µ =coshr,σ =ν =sinhr,
2 † − ∗ ∗
Eq. (52) can be simplified to be
(cid:2) (cid:0) (cid:1)(cid:3)
∂m ∂n
exp 4cosh2r α2+2sinhrcoshr α2+α 2 exp 4cosh2r α2 2sinhrcoshr α2+α 2
∗ ∗
| | ∂αm∂α n − | | −
∗
h [m/2][n/2] tanhr k+l(cid:0) m!n! (cid:1)i h (cid:0) (cid:1)i
= 2coshm+nr − 2 H [2(α coshr+αsinhr),2(αcoshr+α sinhr)].
m 2k,n 2l ∗ ∗
k!(m 2k)! l!(n 2l)! − −
k=0 l=0 (cid:0) −(cid:1) −
X X
5 Applications of Eq.(45) in the Study of Nonclassical Features of
Quantum Light Field
For a composite systemconsisting ofa multi-levelatom(or quantum dot)coupled to a cavityand drivenby
a weak coherent field, quantum-optical effects can be demonstrated in the interaction processes of photon
emissionandabsorbtionwithatombetweengroundandexcitedstates[20,21,22,23]. InRef. [24],authorhas
consideredaweakcoherentincidentfield,andthequantumstateofemittedlightcanbeexpressedasaseries
of excited coherent states C a m α or a superposition of the different Fock states ψ = C n
(due to α = αn n ). Hmeremw†e c|onisider a strong coupling case, and the initial stat|e oif the inncinde|nti
| i n √n!| i P P
source is a single-mode squeezed vacuum field. Only considering an effective measurement to optical field
P
(e.g. photon-statistics methods), this multi-photon processes originated from quantum nonlinearity can be
monitored by
ρ(r,n)=C 1a nS(r) 0 0 S 1(r)an. (53)
n− † | ih | −
which can exhibit similar behavior to that of an excited quantum state (e.g. a m ϕ ). C = Tr[ρ(r,n)]=
† n
n!coshnrP (coshr) is a normalized constant. P (coshr) is the expression of th|eiLegendre polynomials.
n n
S(r)= exp r a 2 a2 denotes a unitary operator with the following transformation identities
2 † −
(cid:2) (cid:0) (cid:1)(cid:3) S 1(r)aS(r) = acoshr+a sinhr,
− †
S 1(r)a S(r) = a coshr+asinhr. (54)
− † †
In order to investigate the non-classicalfeatures of ρ(r,n), Eq. (18) shows that its quasi-probability distri-
bution Wigner function can be written as
W (α,α ) = 2πTr[ρ(r,n)∆(α,α )]
∗ ∗
= 2πC 1Tr S(r) a coshr+asinhr n 0 0 acoshr+a sinhr nS 1(r)∆(α,α ) (.55)
n− † | ih | † − ∗
h (cid:0) (cid:1) (cid:0) (cid:1) i
9
Comparing (54) with (31), we can see µ =τ coshr, σ =ν sinhr. From Eqs. (55) and (33), thus we
→ →−
can obtain the Wigner function of ρ(r,n),
n
1 sinhr
W (α,α∗) = − exp 2 αsinhr α∗coshr 2
P (coshr) 2 − | − |
n (cid:18) (cid:19) h i
n n 2l( cothr)l 2 2
×Xl=0(cid:18) l (cid:19) (−n−l)! (cid:12)(cid:12)Hn−l"irtanhr (α∗coshr−αsinhr)#(cid:12)(cid:12) .
(cid:12) (cid:12)
(cid:12) (cid:12)
In Fig. 1, negative part in certain region of phase(cid:12)-space indicates an evidence of nonclas(cid:12)sicality of the
state generated by adding photon to a weak squeezed radiation field. The variance of photon-addition
can exhibit different nonclassical features with the fixed squeezing value of r = 0.2, which dominates the
quadrature distribution in the directions of X and Y. The top row with three pictures shows that even
− −
photon have been added to the weak squeezed radiation field, odd photon at bottom row.
6 Acknowledgements
This work has been supported in part by the Natural Science Foundation of China (NSFC, No.61205115
and No.11204004), Outstanding Young Talent Foundation of Anhui Province Colleges and Universities
No.2012SQRL040, and also Natural Science Foundation of Anhui Province Colleges & Universities under
GrantKJ2012Z035. TheauthorsacknowledgefruitfuldiscussionsaboutphysicsmeaningwithProfessorFan
Hong-yi.
7 Appendix: Derivation of Eq. (37)
The Glauber-Sudarshan P representation, as one of most important quantum phase space theory, is the
quasiprobability distribution in which observables are expressed in normal order. By virtue of the coherent
staterepresentation z =exp za z a 0 ,theP-representationofaquantumdensitymatrixρisdefined
† ∗
| i − | i
as
(cid:0) (cid:1) d2z
ρ= P (z,z ) z z . (56)
∗
π | ih |
Z
The inverse relation of Eq. (56) is
P(z,z∗)=e|z|2 d2β β ρ β exp β 2+zβ∗ z∗β , (57)
π h− | | i | | −
Z (cid:16) (cid:17)
which was first obtained by Mehta [25]. Substituting (57) into (56) and Utilizing Weyl ordering of the
coherent state projector, namely
˙. ˙.
|zihz|=2 .. exp −2 z∗−a† (z−a) .. ,
(cid:2) (cid:0) (cid:1) (cid:3)
we have
ρ = 2 dπ2ze|z|2 dπ2β h−β|ρ|βiexp |β|2+zβ∗−z∗β ˙... exp −2 z∗−a† (z−a) ˙...
Z d2β ˙. Z (cid:16) ˙. (cid:17) (cid:2) (cid:0) (cid:1) (cid:3)
= 2 π .. h−β|ρ|βiexp 2 aβ∗−a†β+a†a .. . (58)
Z
(cid:2) (cid:0) (cid:1)(cid:3)
which can conveniently recast operators into their Weyl ordering.
References
[1] See e.g. H. Hochstadt, Integral Equations, Wiley (1973) New York.
[2] H. Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R1–R17.
[3] H. Y. Fan and X. B. Tang, Commun. Theor. Phys. 49 (2008) 1169–1172.
10