The recently introduced notion of a quantum group is discussed conceptually and then related to deformed harmonic oscillators ('q-harmonic oscillators'). Two developments in applying q-harmonic oscillators are reviewed: q-coherent states and the q-symplecton

Biedenharn, L. C.

Lohe, M. A.

Nomura, Masao

English

NASA Technical Reports Server

N93-27320
WHICH Q-ANALOGUE OF THE SQUEEZED OSCILLATOR?
Allan I. Solomon
Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, United Kingdom
1 Introduction and Content
The noise (variance squared) of a component of the electromagnetic field - considered as a quan-
tum oscillator -in the vacuum is equal to one half, in appropriate units (taking Planck's constant
and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed
state is one for which the noise is less than the vacuum value - and the amount of squeezing is
determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed,
as they produce the same variance as the vacuum. However, it is not difficult to define states
analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent
states in the more general group sense but with respect to groups other than the Heisenberg-Weyl
Group which defines the Glauber states. The original, conventional squeezed state in quantum
optics is that associated with the group SU(1, 1). Just as the annihilation operator a of a single
photon mode (and its hermitian conjugate a t, the creation operator) generates the Heisenberg-
Weyl algebra, so the pair-photon operator a 2 and its conjugate generates the algebra of the group
SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the
automorphism group of the Heisenberg-Weyl algebra is SU(1, 1). Needless to say, each of these
viewpoints generalizes differently to the quantum group context. In this talk we shall discuss
both. The structure of the talk is as follows:
• Conventional Coherent and Squeezed States
• Eigenstate Definitions
• Exponential Definitions
• Algebra (Group) Definitions
• Automorphism Group Definition
• Example: Signal-to-Noise Ratio
• q-Coherent and q-Squeezed States
• M and P q-bosons
• Eigenstate Definitions
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https://ntrs.nasa.gov/search.jsp?R=19930018130 2020-03-17T06:07:20+00:00Z
• Exponential Definitions
• Algebra (q-Group) Definitions
• Example: Signal-to-Noise Ratio
• Automorphism q-Group Definition
2 Conventional Coherent and Squeezed States
The elementary treatment of (a single frequency) of the quantized electromagnetic field leads to
the identification of its components
E,-_x, B-,_p, [x,p]=i
in suitably chosen units. We may introduce boson operators b, bt by
x = (b + bt)/ vf2 p = (b - bt)/v_i
which then satisfy the Heisenberg-Weyl Algebra
[b,b t] = 1
[N,b ?] = b_ (1)
where N =btb. The interpretation of these operators is that they annihilate (resp. create)
photons; the vacuum state IO > satisfies
b]O>= O.
The quantum noise of the x-component (E-field) in the vacuum state is given by
(Ax) 2_<x 2>- <x>2 =1/2
with a similar result for the B-component. The vacuum signal ( < x >:) vanishes.
The conventional coherent states (Glauber [1] states) are defined as eigenstates of the operator
b,
blA >= AI_ >. (2)
For these states one readily evaluates
(Ax) 2 = 1/2 < x >2= (A + X)2/2.
An alternative, suggestive definition of the coherent states which readily lends itself to generaliza-
tion, is that they are obtained by the action of the realizations of the group corresponding to the
Heisenberg-Weyl Algebra generated by {b, bt, 1} on the vacuum, thus;
IA >= exp(Abt)10 >. (3)
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It is an important practicalproblemto maximizethe signal-to-noiseratio p for radiation states;
here we are of course only considering the quantum noise. What we see from the preceding is that
p vanishes for the vacuum; it attains the value 4Ns for a coherent state, taking (real) _2 = Ns
where N_ is the number of photons in the signal.
In a classic paper, Yuen [2] showed that for any radiation field the maximum signal-to-quantum
noise ratio p for fixed energy has the value P,n_z = 4N,(N, + 1), where N_ gives the upper
limit on the number of photons in the signal ( effectively a maximum power per unit frequency
constraint). He further showed that this value is attained by the squeezed states [3], two-photon
coherent light states generated as eigenstates of the operator #b + vb t where 1#12 - Iv] 2 = 1. The
only mathematical input to this result consists of the canonical commutation relations Equation
(1). The term "squeezed" derives from the fact that in these states the quantum dispersion may
attain values below the vacuum (or coherent) state value of 1/2. Such states have been produced
experimentally. These squeezed states may also be defined by the action on the vacuum (more
generally, on Glauber coherent states) of the group corresponding to the algebra generated by
{b2, (bt)_, (bb' +btb)}. (4)
Thus a typical squeezed state (up to normalization ) may be written
t_,z >- exp(_(b*)2)exp(zbt)lO > • (5)
The state [_, z > is an eigenstate of (b- _b t) with eigenvalue z, in agreement with the definition of
squeezed state above (it = 1, u = -_ and for convergence we require that ___1.) The operators
in Equation (4) satisfy the commutation relations of SU(1, 1)
[K+,K_] = -2Ko
[Ko, K±] = -I-K+. (6)
An alternative definition which results in states exhibiting squeezing is to define them as (normal-
ized) eigenstates of the of the lowering operator K_ = b2. These states have the form
i
_o _1 .
i=0
O_
[_2> = E _' i),12i+I >. (7)
=0{(2i+
An appropriate sum of these squeezed statesrecoversa Glauber coherent state.
A more basic definitionof squeezed statesarisesfrom the observationthat the automorphism
group of the H-W' algebra isSU(I, I);thus a unitarytransformationU on b gives
bH u b = + (s)
where Ipl _ -lul: - 1. The conventional squeezed state is then defined, exactly as above, as an
eigenstate of the transformed bose destruction operator
(#b + ubI)14 >= 414 > • (9)
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More generally,a conventionalsqueezedstate is definedas the action of the unitary operator
U(#,v) on a coherent state ]z >= D(z)lO >, thus:
Ig >= U(_,v)D(z)10 >= U(l,, v)lz > (10)
This definition is not only elegant but, by applying the inverse transformation, enables calculations
in squeezed states to be made as readily as in the coherent states.
For example; using
V(#,v) -1 b V(#,v) = _b- vb t
one may readily evaluate the dispersion of x in the squeezed state ]_ > to be
and the signal to be
(_x)2= ½1,- _1_
< x >2= {(, _ _)_ + (_ _ _)z)}2.
For real values of the parameters, the maximum of the signal-to-quantum noise ratio
p _=< z >2/(Az)2 = 42
may readily be seen to be attained at p,,,a, = 4N,(N8 + 1) as cited above [2].
3 q-Coherent and q-Squeezed States
A deformation aM of the standard boson operator b was introduced some years ago by Arik and
Coon [4]. Their deformed bosons satisfy
aMaM ? --qaM?aM = 1. (11)
More recently, the deformed q-boson operator ap satisfying the Quantum Heisenberg-Weyl Algebra
(H- Wq Algebra)
apapt _ qaptap = q-N
[N, ap t] = ap t (12)
has been introduced [5, 6]. ( I have used the subscript M to denote the relation to the mathe-
matician's classical q-analysis, a study which goes back at least as far as Gauss, in contrast to the
more recently introduced physicist's form, subscript P. The second equation of (12) is satisfied by
both forms. There is no need to subscript the operator N for the reason given below.)
In principle, either Equation (2) or Equation (3) can be used as a starting point for an eigenstate
definition of q-coherent states for both types of deformed bosons. It is easily shown that an attempt
to use Equation (3) does not lead to a normalizable state (for q _ 1) in either case. Starting from
Equation (2), q-coherent states for the deformed boson operator of Arik and Coon were constructed
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by these authors [4]; the same equation was used [6] for the q-bosons defined in Equation (12).
Both forms of q-boson lead to the q-coherent state
18 >q = N-lexpq(_at)10 > (13)
where a = aM or ap and
X 2= expq(l_12). (14)
The q-exponential is defined in both cases by
(15)
The symbol [r]q! is defined by [r]q! = [r]q[r- 1]q[r- 2]q..-[llq where, in the case of Equation (11),
we define
[x]q = (qX _ 1)/(q- 1) (16)
and in the case of Equation (12), we define
[xlq=(q_ q-X)/(q _ q-1). (17)
Equation (11) gives rise to the classical form of the q-exponential usually written as Eq(x),
which converges for Iql > 1, or for Ixl < I__-_lwhen Iql < 1. The form of q-exponential corre-
sponding to Equation (12) is convergent for all z and q. In both cases, limq._.l expq(x) = exp(z),
and when q = 1 the q-boson operators reduce to standard boson operators.
The q-bosons are related to the conventional bosons b as follows:
a=b [_ l_ (18)
where N = btb, using the appropriate definition of [N]q for "mathematical" bosons Equation (16)
[7] or "physical" bosons Equation (17) [8].
The q-coherent states defined above do not give rise to (time-independent) squeezing, just as
in the case of the conventional coherent states. In fact, it may be shown [7] that the term which
gives rise to squeezing is, in general,
< a 2 > -- < a >2
which is zero for eigenstates of a. However, Buzek [9] has shown that there is time-dependent
squeezing, by choice of a suitable analogue of the usual Hamiltonian; and this has also been found
by Celeghini, Rasetti and Vitiello [10].
It is not immediately clear how conventional squeezed states can be generalized to the quantum
group context. The most direct approach is to use a q-boson realization of the analogous suq(1, 1)
algebra; one may then attempt to define the analogous q-squeezed states by the exponential action
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of the resulting operators on the vacuum (or on the q-coherent states). A realization of suq(1,1)
given by Kulish and Damaskinsky [8] is
1 1
I(+=p(at) _ K_ =p(a) 2 Ko=_(n+-_) (19)
with p = (q + q-1)-1 and [K+, K_] = -[2K0]q2. However, the exponential action of the operators
of this algebra fail to give a normalizable state not only for the conventional exponential (which
was to be expected) but also for the q-exponential expq(x) defined above (and also for expq2(x)
which one would have thought to be the appropriate function here).
The eigenstates of K_ corresponding to Equations (7), obtained by substituting the "box" facto-
rials for the conventional ones, give normalizable states [11].
We may alternatively carry over the definition
(a -  at)16 z >= zlLz > (20)
to the q-boson case. For the choice z = 0 we obtain
i_,z >= Af-IZ _i [2i- 1]q!!12 i
i=0 [2i]q!!
with normalization
> (21)
N2 = 1 1 ,[2i - 1]q!! (22)
i=0 [2i]q!!
The symbol [r]q![ has the expected meaning Jr]q!! = [r]q[r- 2]q[r-4]q... and the first term in (22)
is 1. The squeezing properties of states defined in this way, for various values of the parameters
and q were calculated in [11].
A more basic definition of squeezed states in the quantum group case arises from generalizing
the automorphism group property given in the previous section for the conventional case, Equation
(8) and Equation (9).. One may seek by analogy to define q-squeezed states in terms of the
automorphism quantum group of the quantum Heisenberg-Weyl algebra of q-bosons. Consider
the quantum plane d la Manin generated by two elements _ and 7 as defined by Woronowicz [12],
satisfying the following commutation relations:
_7 = pTo
c_7" = /tT*O
3'7* = 7*7
o*o - 7"7 = I
oo*-/_27" 7 = i.
We now introduce a conjugation A _ A defined by its effect on
1. c-numbers c _ c*, (complex conjugation)
2. q-numbers (quantum plane) 6 = o* d* = o -_ = tiT* -r* = 1
(23)
9O
3. operators A =_ q½(N2-N)Atq-½(N_-N) (q real ).
Under this transformation, A = A, A-B =/)A, ; and the boson a satisfying
aa ? _ qata = q-N
maps to _ , with the pair a, fi satisfying
aS - #fia = 1. (24)
with # = q2. The two-dimensional fundamental representation of SU,(1, 1) is given by
a #7* ] (25)U _ C[
and u satisfies uJfi = J where
The transformation
[10]J= 0 -p
[a,a] [a,a]u (26)
is an automorphism which preserves Equation (24). Squeezed states in the quantum group context
may now be defined as the eigenstates of the transformed a, thus generalizing the results of [11].
Finally, we note that one may derive an analogue of Yuen's result [2] cited above on the
optimal signal-to-Quantum Noise ratio q-photons [13]; the corresponding bound for q-photons
may be shown to be
p, = 4[N,]q[N, + 1]q/([Ns + 1],- [Nslq) 2. (27)
that is, for a radiation field in terms of photons satisfying the modified commutation relations
of the quantum group version of the Heisenberg-Weyl Algebra. This ratio is always less than the
value in the conventional case, attained for the SU(1, 1) squeezed states.
References
[1] R. J. Glauber, Phys. Rev. 131, 2766 (1963)
[2] H. P. Yuen, Physics Letters 56A, 105 (1976)
[3] D. Stoler, Phys. Rev. D 1, 3217 (1970)
[4] M. Arik and D. D. Coon, J. Math. Phys. 17, 524 (1976)
[5] A. J. MacFarlane, J. Phys. A: Math. Gen. 22, 4581 (1989)
[6] L. C. Biedenharn, J. Phys. A: Math. Gen. 22,L873(1989)
91
[7] J. KatrieI and A. I. Solomon, J.Phys. A: Math. Gen. 24, 2093 (1991)
[8] P. P. Kulish and E. V. Damaskinsky, J. Phys. A: Math.Gem 23, L415 (1990)
[9] V. Buzek, J. Mod. Opt. 38, 801(1991)
[10] E. Celeghini, M. Rasett], and G. Vitiello, Phys. Rev. Lett. 66, 2056 (1991)
[11] A. I. Solomon and J. Katriel, J. Phys. A: Math.Gen. 23, L1209 (1990)
[12] S. L. Woronowicz,Commun. Math. Phys. 136, 399(1991)
[13] A. I. Solomon , to be published.
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The q-harmonic oscillators, q-coherent states and the q-symplecton

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