7 November 1994 PIIYSlCS LETTERS A
ELSEVIER
PhysicsLetters A 194 (1994) 272278
Parasupersymmetry in quantum optics V.A. A n d r e e v 1 P.N. Lebedev Physical Institute, Academy of Sciences of Russia, Leninsky Prospect 553, 117924 Moscow B333, Russian Federation
Received 17 June 1994;acceptedfor publication 1 September 1994 Communicatedby A.R. Bishop
Abstract
A quantum optical realization of the RubakovSpiridonov parasupersymmetrical model is found. The twolevel systems interacting with one or two boson modes are considered as examples. It is shown that three such related systems, described by the JaynesCummings Hamiltonians, can form a parasupersymmetric system. The simplest way to construct parasupersymmetric systems is to use the chains of supersymmetric systems.
1. Introduction
Recently there has been a lot of interest in the transformations that mix bosonic and parafermionic degrees of freedom. Such transformations have been introduced by Rubakov and Spiridonov and were called parasupersymmetry (PSS) [ 1 ]. They constructed the PSS quantum mechanics model which is the direct generalization of Witten's supersymmetric (SS) quantum mechanics [ 2 ]. In Ref. [ 1 ] the onedimensional quantum mechanical realization of the second order PSS was constructed. The higher order PSS quantum mechanics was investigated in Ref. [ 3 ], its connection with SS quantum mechanics was found in Ref. [4]. A lot of works are devoted to the investigation of concrete quantum mechanical systems with PSS and their applications [ 512 ]. In a series of works we have constructed the realization of Witten's SS quantum mechanics scheme in quantum optics [ 1318 ]. We considered the twolevel systems interacting with one or two bosonic modes. They are described by the JaynesCummings HamEmail address:
[email protected].
iltonian and its generalizations. It was shown that some of such Hamiltonians form SS pairs and can be considered components of one SS Hamiltonian. In this work a similar quantum optical realization of the RubakowSpiridonov PSS model is found. For simplicity we consider mostly, like in Ref. [ 1318 ], the twolevel systems interacting with bosonic modes. But perhaps the manylevel systems are a more suitable object for PSS models. We hope to investigate this problem in the future.
2. RubakovSpiridonov model In this section we recall the basic features of onedimensional second order PSS quantum mechanics [1]. Here, three onedimensional models are considered, //j=½P2+~(x),
j=1,2,3. P =   I .~d.
(1)
Their potentials ~ ( x ) can be formed with the help of two functions W ~ ( x ) , W z ( x ) in the following manner,
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3. Quantum optical realization of the RubakovSpiridonov model
v, =1(w~, + w~ + w 2 x  w , x ) + w , x , v2=¼(w~, + w~ + W2xW,~,) ,
v~ = I ( w,~ + w~ + W2x w , ~ )  W2x.
(2)
It is also assumed that the following conditions,
w, + w2 ~o, ( w ~  w ~ ) x + (w2 + W~)x~=O ,
(3)
are satisfied. Then one can construct the PSS Hamiltonian
We will consider now the situation when Hamiltonians H~, the components of Hpss, describe some quantum optical systems. As usual, they will be generalized JaynesCummings Hamiltonians of twolevel systems interacting with one or two bosonic modes. Let us consider charge operators in the form Q1 =
0
0+ ,
B (4) 0 iB
Q2 =
Two charges are connected with the Hamiltonian,
Q= p+'Wl
Q+=
0 p+iWz
i piW1 0 0
,
0 ) piWz .
(5)
(6)
0
They satisfy the following algebra
Q2Q+ +QQ+ Q+Q+ QZ=4QHess , Q+ZQ+Q+QQ+ +QQ+Z=4Q+Hpss ' [Hpss, Q] = [Hpss, Q+ ] = 0 .
(7)
In terms of the Hermitian charges
QI=½(Q++Q), Qz=½i(QQ+),
(8)
(9)
This constriction is connected with the second order parafermionic algebra. It is characterized by the conditions
aa+a=2a, a3=0,
a2a++a+aZ=2a.
.
(11)
Here A, B, A +, B + are some operators. Sometimes operators in pairs (A, A + ) and (B, B + ) are conjugated to each other, but we do not demand this for all models considered. The only condition we require is that the compositions of these operators AA +, A +A, BB +, B +B be the Hermitian operators. But the charge operators Q1Q2 ( 11 ) can be nonHermitian. This fact depends on the explicit form of the operators A, A +, B, B ÷, which must be specified for concrete models. Let us now find the analog of Eqs. (3). They can be derived from the condition that the offdiagonal part of the operator Q2 must be annulated by the operator QI. We also require that along with the conditions (9)
Q3=a, Hess, Q3z=Q2Hess ,
(12)
the additional conditions
the relations (7) take a simpler form,
Q3=QIHpss, Qa2=QzHpss.
iB + 0
(10)
In the PSS model of order n the PSS Hamiltonian consists of n + 1 onedimensional Hamiltonians Hj [ 3 ]. Their potentials V: can be constructed with the help ofn auxiliary functions Wj, satisfying the system of equations, which is the generalization of system (3).
Q31=HpssQl, Q3 =HpssQ2 ,
(13)
must also be satisfied. Condition ( 12 ) for QI can be written in the form Q3=
o
B
0
0 / \ BA
.
0
BB + ,I (14)
Since the PSS Hamiltonian Hess is diagonal one can represent the matrix Q2 in the form
Q2=npss +C,
(15)
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274
where QIC=O. We are looking for the operator C in the form
and the Hamiltonian
2A +A AA ++ B +B
Hpss =  C=
0
0
.
Condition ( 15 ) gives the equations for X and Y
AX+B+BA=O, AA+B+ +B+Y=O.
(17)
Now consider condition ( 13 ) for Q~,
o
\ BA
0
BB +
0
Eqs. (17), ( 18 ) are the analogs of Eqs. (3). They can be found also from the condition [Hess, Ql] = 0 .
(19)
y)
Now the PSS Hamiltonian takes the form
.
BB +
(20)

Choosing the specific X, Y one gets the concrete PSS Hamiltonian and the concrete form of Eqs. (17), (18). There are two cases to consider in this work:
(i)X=B+B,
(ii) X=A+A,
Y =  A A +, Y =  B B
+
[A+,B+B]=O,
(22)
.
[B, AA+] = 0 .
(23)
AA + + B +B
.
AA+ +BB + (24) In case (ii) we have the equations
(AA+B+B)A=O, A+(AA+B+B)=O,
(AA
+
1 i)
f+=
0 0
?+=
o 0
° i)
, f=
0 1
o)
1,7=
,
o
0
1
i)
(27)
The operators (27) satisfy the algebra
f + f f + = 2 f +, f + f f + = 2 f
+, f f + f = 2 f ,
ff+~2~ f + f f + = 2 f + ,
[B+,AA+I=O,
Hess =
( 1 ) PSS twomode oscillator. The system consists of two bosonic modes and one parafermionic mode. The bosonic modes are described by the conventional creationannihilation operators a +, a2, a~, a2 with the frequencies wl and to2. The parafermionic mode is described by the ordertwo parafermionic operators.
(21)
In case (i) Eqs. (17), (18) and Hamiltonian (20) take the form
[A,B+BI=O,
In the case of onedimensional quantum mechanics conditions (3) impose some constraints on the form of the potentials of the Hamiltonians H,, H2, //3, forming the PSS Hamiltonian (4). In our quantum field model conditions (17), (18), or more specific (23), (25), define some connection between the operators A, A +, B, B +.
4. Examples
(18)
AA++B+B
(26)
.
XA++A+B+B=O, BAA++YB=O.
\/'A+Ax
. 2BB +
B
The condition CQ1 = 0 gives the equations
Hess=[
\
(16)
BAO
)
f+2y+~f+2=2f+,
]'+af+ff+2=2f+, f2f++f+f2=2f+,
f +2f+ff +/=2f + ,
fzf++f+fz=2f+,
?2f++f+i'2=_2f, ? g + + y + y 2 = _ 2 y + ,
f3 ~.~73= f
+3 = f
+3
=fff=f~f=fff=fff
=fj"f=fff =fj7]'=o ,
B+B)B+ =O,
B(AA+B+B)=O,
f+zf+ff+z=2f+,
f +f +f +=f +f +f +=~ +f +f + (25)
=f +y +y +=y +f +s,+=? 7 +f +=o .
(28)
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KA. Andreev / PhysicsLettersA 194 (1994) 272278
These operators describe the transitions in the threelevel system. We will take operators A, A +, B, B + in the form A=x/r~ al, a2,
1
i
[ v / ~ OC+ff)al + ~
[~
OCY)ax
(f+Y)al + ~
(ff)a2
The PSS Hamiltonian (24)
Hess
=(to,a~a,+to2a~a2og,a,a~+to2a+a2 co,a,a~+cota2a~) (29) can be represented in the form Hess= (tol a ~al + toEa + a2)I+ ½09,( f f + + f f + ) + j r f + ) .
(30)
This system is the direct generalizations of the SS oscillator [ 19 ]. (2) The other examples can be constructed with the help of conditions (25). The simplest way to satisfy them is to assume that (31)
AA+=_B+B.
\[2A÷A )
Now the PSS Hamiltonian (26) is Hess=
2AA +
•
(32)
(33)
H=PIP2 =ql q2 •
Then one can associate B+=ql,
every three successive members of it form PSS Hamiltonian. Therefore each SS chain system from Ref. [ 18 ] can be regarded as an example of a PSS system as well. Of course the infinite chains of SS Hamiltonians [2022 ] can be used for constructing the PSS systems too. But, unfortunately, I do not know any quantum optical infinite SS chain. One of these systems is a twolevel system that interacts with an electromagnetic field with two modes tol and 092. Four different processes can exist, depending on the relation between transition frequency and field frequency. The processes are described by the Hamiltonians HI =tola~+ al + t o a f a2 + (to1092)63 + g ( a l a f a+ + a ~ a E a _ ) , n a = t o l a + a l +toaa+a2 + (COl+092)63 +g(ala20+ +a+ a f 0_) , H3 =tola~al + t o a a f a2 + ( 091 + to2)03 +g(a ~ a20+ + a t a + 0_ ) ,
+g(a 1+ a f 0+ + al aEa_ ) .
Such a model can be realized when some scalar Hamiltonian H can be factorized in different ways,
A+=P2,
(36)
114 =tola+al + toza~ a2 + (  t o t  to2)a3
2BB +
A=P1,
(35)
•..~Hi_l .~Hi*ni+l ~Hi+ 2 ~...,
_ x / r ~ ( f + +jr + ) a t  x / ~ 2 ( f + ]" + ) a f ].
+ ¼toE(ff + + f f +  f f
H2=2PIP2=2AA + ,
The scalar Hamiltonians (35) form the PSS Hamiltonian (4). The Hamiltonians with different factorizations have been discussed in Ref. [ 18 ] in connection with the problem of closed chains of SS Hamiltonians. It is obvious that if one has such a chain,
8+
+ x / ~ l ( f + +jr +)a~ +x/c~2 ( f +  j r + ) a + ] , Q2 = ~
HI=2P2PI=2A+A,
//3 =2q2ql =2BB +
A +=v/~ a + ,
The charge operators ( 11 ) are Q, = ~
and construct three related Hamiltonians
(37)
Here a +, al, a f are boson creationannihilation operators of the tol and 092 modes: [al,a~]=[aE, af]=l,
0+=(00
10), 0=(01
[ai~, a 2 ] = 0 , 00), 03=1(10
O1).
Hamiltonian H 1 describes the process of induced B=q2
(34)
combinational scattering (Raman scattering), i.e. the
V.A.Andreev/ PhysicsLettersA 194 (1994)272278
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process of system transition from a low to a high energy state in which an energy quant with frequency 091 is adsorbed and an energy quant with frequency 092 is emitted. Hamiltonian HE describes the twophoton adsorption process: two photons with frequencies (D1 and 092 are being adsorbed during the transition of the system from a low energy level to a higher energy level. Hamiltonians//3 and H4 describe the interaction of the modes al, a2 with holes in a state space of a twolevel system. The matrixes a+ and a_ are the creation and annihilation hole operators, the energy of the twolevel system increases if a hole is created and the energy of the field is decreased. The pairs of Hamiltonians (37) form the SS Hamiltonians
H12 
o)
o)
H2 ,
H3
,
H34=(H3 04), H4I=(H4 O1). Each of the factorizations
Hamiltonians
(37)
(38) allows
two
s2kux/~2a2
_( ~22a~ (v/~ /u)al~ ql\(x/(D1/u) a+ x~2a2 }' q _[ V/~:a2 u~r~l a,'~ :kUx~l a + ~ a~ } '
a,
(
a~
]'
)' I.)a:
fl=\(d(Dl///)al
~2aJ
(xf(D2 a~ uv/~ a?
al
)'
)'
al
]
x / ~ a : }.
(40)
In all equations the parameter u is a solution of the equation u+ 1 U 
g d(D1O)
1 "
The Hamiltonians (37) form a closed chain (36), ~H1 ,HE ~ H 3 ~ H 4 ~ H l + . Any two neighboring elements in this chain create an SS pair, and each three of them can be considered as a component of the PSS Hamiltonian (4). (3) The next example is more specific for the parasupersymmetry. Like in the previous case the twolevel system and two bosonic modes will be studied. We will consider three systems, described by the Hamiltonians
((Dlala ++(D2a2a+ ~ a l a 2 ' ~ H,=\ ~ a + a ~ (D2a~a2 ]'
(41)
,[(Dla+al +(Dza2a~ ~ a:a2 + (D2a~a2 iF, (42)
.
where
~l
~
(39)
n4=f2fl=sis2,
191  \ ( d ( D 2 /tl)a~
(~/~21u)a~'~,
( ~ 2 /u)a2
H2 =F
H1 =qlq2=sis2, HE=q2ql =PIP2 , H3=p2pl=flf2,
x/~ a+
sl=(
3=
F'{k ~ l
a,+)F '
a3F lOJlala+ +(D2a2a2 ]
(43)
4F2=(Dlala~ +4(D2a2a~
.
(44)
The Hamiltonians (41), (42) coincide with the Hamiltonians (37), (38) of stimulated Raman scattering and two photon absorption. The only difference is that in the Hamiltonians (41), (42) the coupling constant g takes a concrete form: g=x/~l(D2. The Hamiltonian (43) describes the onemode JaynesCummings model coupled with another boson mode (D2 by Eq. (44). The Hamiltonians ( 4 1 )  ( 4 3 ) can be factorized
HI=hlhzh3=A+A, hl=A +, hEh3=A, H2=hzh3hI=AA+~B+B, h z = B +, h3hl=B, H3=h3hlhE=BB+ . (45)
V.A.Andreev/ PhysicsLettersA 194 (1994)272278 Here
have zero energy
{ ~ 1alE ~20a2F ) h' =~,v/~2 a~ F
E~
h{ ½Fl~la+ 21~ e'x/~2 af
3+ ( (wlm+o~2n)l/2lml,nl) ) ~ S n= +(OOl(m+l)+ogzn)l/2[m,n_l)
10)
277
=0.
The Hamiltonian (43) has the eigenfunctions
'
(53)
0
(46)
I
with the eigenvalues One can see that in this model the operators in pairs (A, A + ) , (B, B ÷) are not Hermitian conjugate to each other, but the Hamiltonians constructed by the formulas (45) are Hermitian. The Hamiltonians H, are defined at the Fock space with vectors
3 + = lWlm+w2n+ ½x/~m2+4o91o92mn E~,n
The ground state with zero energy is absent. The lower eigenstate 3 0 ~vd,° = ( i 0, 0 ) ) has the energy Eo3,o= 092. This level is nondegenerate.
The eigenfunctions of the Hamiltonian (41) are 5. Conclusion 1,+
~"n=
(Inl,
nl))
/3+_Im,n)
'
/3+_=  X/091m / 4o92n + ~/1 +o91m / 4o92n , m, n = 1, 2, ...,
(47)
with the eigenvalues E , 1,+_ .,.=t
1 2 2 + 4o91 co2mn. ~oJlm+r.Ozn+_iX/~lm
We have constructed some examples of PSS quant u m optical systems. Their properties will be investigated in detail in another work. These examples show that the simplest way to construct the PSS q u a n t u m optical system is to use the chain of SS q u a n t u m optical systems.
(48)
The zero eigenvalue is degenerate. All eigenstates References
7'~,~ = ( 1 m 0 0 ) )
(49)
have the energy Em,o l, = 0 . Hamiltonian//2 (42) has the eigenfunctions
~~+ ( (Ogl(m+l)+4o92n)1/Zlm, n1 7aFn=~,/3+(oglm+4ogz(n+ l ) )1/Zlm1, )n) ) ' /3+_=x/cnlm/4oJ2n+_x/l+og~m/4ogzn
(50)
and eigenvalues
2 + = lo91m +o9~ n+_x/oo~mZ+4o91ogzmn ET~,~
(51)
As in the previous case the ground state is degenerate. All eigenfunctions ~ r ~ = (1 m01, 0 ) )
(52)
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