## Abstract

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

## 1. Introduction

The SU(1,1) algebra has played an important role in the literature on group theory and mathematical physics. Various representations of SU(1,1) have been extensively studied (Barut & Fronsdal 1965; Holman & Biedenharn 1966). In recent years, the SU(1,1) group and its Schwinger representation (Schwinger 1965) have been used in quantum optics in connection with the study of parametric amplifiers (Louisell 1977), interferometers (Yurke *et al*. 1986; Fearn & Loudon 1989) and squeezed states (Bishop & Vourdas 1986). The generators of SU(1,1) may be represented in terms of operators obeying certain commutation relations. In this paper we explore a new way of expressing the generators of SU(1,1).

There is a system of commutation relations which generalizes the standard commutation relations of the simple harmonic oscillator. The generators of SU(1,1) may be expressed in terms of the creation and annihilation operators of the generalized commutation relations. In §2 we study this structure and interpret it as a parity-violating modification, and we investigate its spectral properties. In the course of this analysis, we note that in a special case, the quantum algebra reduces to the algebra introduced by Hodges & Sukumar (2007), which describes the combinatorics of the Bernoulli and Euler numbers. In §3 we identify this algebraic structure as relevant to squeezed states such as the ones considered in quantum optics. In §4 we extend the combinatorial theory developed earlier for the analysis of Bernoulli and Euler numbers, and apply these discrete methods to calculate matrix elements for squeezed states. Section 5 comments briefly on possible extensions of these results.

## 2. Modified commutation relations

Suppose we have operators which satisfy the commutation relations(2.1)

These can be considered as the generators of an SU(1,1) algebra. They can arise from the standard creation and annihilation operators *a*^{†} and *a* of the simple harmonic oscillator when(2.2)

However, there is a more general possibility. Consider a quantum algebra defined by a non-Hermitian operator *A*, its adjoint *A*^{†} and a Hermitian operator *X* satisfying(2.3)

The bilinear operators defined by(2.4)still satisfy the commutation relations (2.1).

The conditions on *X* are stringent. They imply that the operator *X*^{2} commutes with *A* and *A*^{†}, and so with *R*, *L* and *S*. Thus, *X*^{2} is some multiple of the identity. Also, *X* itself commutes with *H*=*A*^{†}*A*. Thus, we can identify the algebra defined by these commutation relations by studying the simultaneous eigenstates of the operators *X* and *H*. We shall then be able to interpret *X* in terms of *parity*.

### (a) Spectral analysis

Let |*n*〉 be an eigenstate of the operators *X* and *H*. The eigenvalue equations(2.5)and the operator relations(2.6)imply that(2.7)showing that the states (*A*|*n*〉) and (*A*^{†}|*n*〉) are also simultaneous eigenstates of the operators *H* and *X*. Iteration of the above equations leads to(2.8)

Thus, a sequence of simultaneous eigenstates of *H* and *X* may be generated from the state |*n*〉. We first consider the case |*α*_{n}|≤1. It will be shown later that all other cases may be related to this case. Then, the above equations show that each operation with *A*^{†} on an eigenstate of *H* produces an eigenstate with a higher eigenvalue or the same eigenvalue for *H* and changes the sign of the eigenvalue of the *X* operator. Similarly, each operation with *A* on an eigenstate of *H* produces another eigenstate with a lower eigenvalue or the same eigenvalue for *H* and changes the sign of the eigenvalue of *X*. *H* is a positive semi-definite operator and so the spectrum of *H* must be positive semi-definite too. This implies that there must be a state such that *A*|0〉=0 with the lowest eigenvalue *λ*_{0}=0, so that a state with an eigenvalue lower than *λ*_{0} is not possible. This must be the groundstate. From the groundstate |0〉, it is then possible to generate a sequence of eigenstates |*n*〉∼*A*^{†n}|0〉. This spectral picture must match the spectrum arising from the consideration of any eigenstate. For example, if we start from the eigenvalue equation for |1〉, then we must have *A*^{2}|1〉=0 to stop the spectrum from going below zero. It may be shown that this is consistent if |1〉∼*A*^{†}|0〉.

The resulting spectral picture is that there is a groundstate |0〉 with eigenvalue 0 for *H* and *α*_{0} for *X* and a set of eigenstates |2*n*〉 with *λ*_{2n}=2*n* and *α*_{2n}=α_{0}, for *n*=0, 1, 2, … There is another sequence of eigenstates |2*n*+1〉 with eigenvalues *λ*_{2n+1}=(2*n*+1+*α*_{0}) and *α*_{2n+1}=−*α*_{0}. The level spacing of the eigenvalues of *H* is 2 for both sequences but the odd sequence is displaced from the even sequence by (1+*α*_{0}).

Now that we have identified all the states and the eigenvalues of *X* in all of them, we may note that we have shown that(2.9)That is, *X* must be a multiple of the *parity operator* .

In what follows, we shall write *X*=*αP*, where the parameter *α* may be freely chosen, so that the eigenvalues of *X* are given by *α*_{n}=(−1)^{n}*α* and *X*^{2}=*α*^{2}. By examining the normalization of the states *A*^{†}|*n*〉 and *A*|*n*〉, it may be shown that(2.10)It follows that(2.11)It is evident from equations (2.11) that *R* and *L* do not connect the states |*n*〉 or even *n* with other states of odd *n* and vice versa. *S* is diagonal in the space of the states |*n*〉 with eigenvalues (2*n*+1+*α*)/2. Thus, the matrices for (*R*, *L*, *S*) always split into even and odd parts.

### (b) Calculating matrix elements

The case *X*=0, *α*=0 corresponds to the standard harmonic oscillator. The spectrum of *H* consists of non-degenerate states *A*^{†n}|0〉 with eigenvalues *n*=0, 1, 2, …, and the matrix elements are well known.

In considering the states and matrix elements for non-zero *α*, it is helpful to study first the special case *α*=1. In this case, the separation between the odd and even sequences of eigenvalues is 2, thus equalling the spacing of the levels in each sequence. The two ladders of eigenvalues overlap. There is a non-degenerate groundstate |0〉 with eigenvalue 0 for *H*. All eigenstates of *H* other than the groundstate are doubly degenerate with even integers 2, 4, 6, … as eigenvalues. Each degenerate excited state consists of two states with opposite parities. The even parity states are *A*^{†2n}|0〉 and the corresponding degenerate odd parity states are *A*^{†(2n−1)}|0〉.

The resulting algebra coincides with the (*R*, *L*, *S*) algebra developed by Hodges & Sukumar (2007). It was shown in that paper how the matrix elements naturally yield the tangent and secant numbers (equivalent to the Bernoulli and Euler numbers). In particular,where *E*_{2m} and *T*_{2m+1} are the secant and tangent numbers, respectively. It was further shown that the operator *X* can be regarded as [*U*^{†},*U*], where *U* is an anti-commuting operator defining a symmetry between the tangent and secant number structures.

For *α*=−1, the situation is essentially the same. The difference is only technical. Equation (2.10) shows that *A*^{†}|0〉=0 and it is not possible to build a spectral ladder based on the state |0〉. Equation (2.10) also shows that *A*|1〉=0 but *A*^{†}|1〉≠0 and now |1〉 is the non-degenerate groundstate with eigenvalue 0 for *H*_{1}.

The theory for general *α* may now be developed by analogy with the case *α*=1. Since the matrices for *R*, *L* and *S* split into even and odd parts that do not connect, we can consider a set of matrices (*R*_{e}, *L*_{e}, *S*_{e}) whose rows and columns are numbered by integers (0, 1, 2, …) but correspond to the operators in the space of even values (*n*=0, 2, 4, …) of |*n*〉, and another set of matrices (*R*_{o}, *L*_{o}, *S*_{o}) whose rows and columns are numbered by integers (0, 1, 2, …) but correspond to the operators in the space of odd values (*n*=1, 3, 5, …) of |*n*〉.

We then write the matrices *M*_{e}=*R*_{e}+*L*_{e} and *M*_{o}=*R*_{o}+*L*_{o} as(2.12)(2.13)It is clear that the even and odd matrices satisfy the relation *M*_{o}(*α*−2)=*M*_{e}(*α*), thus justifying our assertion that it is sufficient to consider |*α*_{n}|≤1. In particular, the matrices for *α*=−1 are just relabellings of the matrices for *α*=1. In the ensuing discussion we shall not pay further attention to the case *α*=−1.

In the case *α*=1, the multiplication of these bidiagonal matrices is equivalent to the use of the recursive triangles set out in §§4–6 of Hodges & Sukumar (2007) for the calculation of the secant and tangent numbers. We can now see these triangles as special cases of the more general algebra obtained by considering these non-standard commutation relations.

For the general case of arbitrary *α*, the spectrum consists of two parity-dependent equispaced ladder of states, which are separated by 1+*α*. A search through the literature in nuclear physics suggests that such parity-dependent spectra can arise for heavy nuclei with octupole deformations. Semi-classically nuclei with octupole deformations have a geometrical arrangement similar to that of the ammonia molecule for which there are two possible mirror symmetric arrangements of the N atom above and below the plane containing the H atoms. The evidence for intrinsic reflection asymmetry in heavy atomic nuclei is discussed by Butler (1998). The level scheme of shown by Cocks *et al*. (1997) suggests that this nucleus may be viewed as a possible candidate for a physical system exhibiting the type of spectra arising from a non-integer value of *α*.

We have shown in this section that a simple change in the standard oscillator algebra leads to a variety of interesting possibilities. We now study some useful applications of the modified algebra.

## 3. Application to squeezed states

We consider an application to the study of squeezed states of the simple harmonic oscillator whose generators obey the same commutation relations as the operators introduced in the study of Bernoulli and Euler numbers. We first summarize the results of §§4 and 5 of Hodges & Sukumar (2007) on the algebra of secant and tangent numbers, putting them into the form of generating functions. The combinatorial property of the sequence of secant numbers *E*_{2n} can be expressed as(3.1)where *R* and *L* are just the operators discussed in §2, in the case *α*=1. This uses the ‘even’ matrices *M*_{e}. In the case of the tangent numbers *T*_{2n+1}, which involve the ‘odd’ matrices *M*_{o}, we need the additional observation that(3.2)to give the analogous generating function as(3.3)

We can also note that and hence(3.4)

We shall now show that these formulae generalize to all *α*. The key idea is that the exponential function appearing in these formulae can be reinterpreted as giving rise to the *squeezed states* of the simple harmonic oscillator such as the ones considered in quantum optics.

### (a) Coherent states and squeezed states

In the context of quantum optics, the mechanisms for the generation of *coherent states* and *Gaussian squeezed states* of the electromagnetic field have been extensively studied (Stoler 1970; Yuen 1976). The general method is to find a unitary operator that acts on the vacuum to generate a state of the field with the required properties.

Coherent states are states of the simple harmonic oscillator for which the variance product Δ*x*Δ*p* is time independent and has the minimum value allowed by the Heisenberg uncertainty principle. It is known that the coherent states are eigenstates of the annihilation operator. Coherent states are Gaussians and may be generated by considering(3.5)where *a* and *a*^{†} are the annihilation and creation operators of the simple harmonic oscillator, respectively, obeying the commutation rule [*a*,*a*^{†}]=1. The operators *x* and *p* are Hermitian linear combinations of *a* and *a*^{†}, satisfying the commutation relation .

Squeezed states are states of the simple harmonic oscillator for which the variances Δ*x* and Δ*p* are time dependent and the variance product exceeds that for the coherent states, but the variance of *x* or *p* can dip below that for the coherent states for part of the time during a cycle whose period is determined by the oscillator frequency. The Gaussian squeezed states, which are eigenstates of a linear combination of the creation and annihilation operators, may be expressed in the form(3.6)

It is well known that the coherent states generated by *U*_{c} can be brought to the normal-ordered form, in which all annihilation operators appear to the right and all the creation operators appear on the left, namely(3.7)thus yielding a useful form for calculating expectation values involving *Ψ*_{c}. There is an analogous normal-ordered form for the Gaussian squeezed states, which arises from the following theorem. If a set of operators satisfy the algebra(3.8)with *b* a complex number, then (*B*, *B*^{†}, *C*) is a closed algebra and the unitary operator(3.9)may be brought to a normal-ordered form(3.10)

This is a new theorem which may be proved by following the same procedure as that used for establishing the normal-ordered form of the squeeze operator (Truax 1985; Schumaker 1986; Sukumar 1989). In fact, the normal-ordered form of *U*_{c} in equation (3.7) is a special case of this general theorem for *C*=1, *b*=0, so that

A corresponding theorem may be given for the operator(3.11)

The normal-ordered form of *V* has a structure similar to that of *U*, given by the replacements tanh→tan, sech→sec. If now we choosethen these general theorems supply the normal-ordered forms for the expressions(3.12)in our generalized (*R*, *L*, *S*) algebra in which *α* may have any value. These forms yield the matrix elements(3.13)and similarly for *V*, with the obvious replacements. Putting *α*=1, *n*=0 in these expressions, we recover the results in equations (3.1)–(3.4).

In the case *α*=0, we obtain the expectation values for the standard harmonic oscillator(3.14)

These are well-known results in the literature on Gaussian squeezed states. We shall derive them by a new combinatorial method in §4.

### (b) An alternative representation of the tangent and secant number algebra

The algebra of (*R*_{e}, *L*_{e}, *S*_{e}) for *α*=1 can also be realized by another algebra defined in the space of standard harmonic oscillator states (|*n*〉, *n*=0, 1, 2, …). If we consider the operators(3.15)which are defined in terms of the oscillator creation and annihilation operators, then(3.16)

The resulting algebra(3.17)defined in the space of the oscillator number states that |*n*〉 is identical to that satisfied by (*R*_{e}, *L*_{e}, *S*_{e}) as given in equation (2.11) but now restricted to the space of even states, so that *S*_{e}=*N*+1 gives *S*_{e}|2*n*〉=(2*n*+1)|2*n*〉. The *U* and *V* operators defined using will therefore lead to the same results as those discussed earlier for the (*R*_{e}, *L*_{e}, *S*_{e}) algebra. The operators *T*_{1} and have earlier been used in a study of squeezed states, which are *not* Gaussians (Sukumar 1989) but are linear superpositions of Gaussian forms.

Similarly, the algebra of (*R*_{o}, *L*_{o}, *S*_{o}) can also be realized by another algebra defined on the harmonic oscillator number states. If we consider(3.18)the resulting algebra(3.19)defined in the space of the oscillator number states is identical to that satisfied by (*R*_{o}, *L*_{o}, *S*_{o}) as given in equation (2.11) but now restricted to the space of odd states, so that *S*_{o}=*N*+1 gives *S*_{o}|2*n*+1〉=(2*n*+2)|2*n*+1〉.

We note that operators involving *N*^{1/2} may appear to be unusual, but such operators have been used to define the phase operator of the electromagnetic field (Susskind & Glogower 1964; Carruthers & Nieto 1965).

## 4. Combinatorial evaluation of the standard oscillator matrix elements

We now return to the standard creation and annihilation operators with [*a*,*a*^{†}]=1, i.e. the case *α*=0. We shall consider the matrix elementswhich, as we have seen, arise in the squeezed states of the simple harmonic oscillator.

These elements can be calculated by the matrices given by the general theory, and the calculation put in the form of a recursive triangle (figure 1).

Here the numbers in the left-hand column give the matrix elements for *n*=0, 1, 2, … It also follows from (3.14) that the generating function for this sequence is given by(4.1)

But we can also obtain these results by discrete combinatorial methods. Hodges & Sukumar (2007) developed such methods for the analysis of the permutation group on *n* elements, thereby finding an operator formalism for the calculation of the number of *alternating* or ‘zig-zag’ permutations. We now apply a similar analysis to the subgroup of permutations consisting of pure *transpositions*, i.e. permutations of 2*m* elements in which every element lies in a 2-cycle.

We shall place a transposition in a *transposition zig-zag class* as follows: the *i*th element of the class is *C* if *i* is in the 2-cycle (*ij*) with *j*>*i*, and it is *D* otherwise. Thus, the transposition (14)(25)(38)(67) is in class *CCCDDCDD*.

We define |*X*| to be the number of transpositions in a class |*X*|, and extend the definition naturally by linearity, so that |*αX*+β*Y*|=*α*|*X*|+*β*|*Y*|. We then find that |*X*| can be identified with the matrix element 〈0|*X*|0〉 in the standard quantum algebra. The proof runs in analogy to the arguments made for the permutation classes. By simple counting arguments, we may establish such properties as(4.2)(4.3)(4.4)(4.5)where (2*m*)!/(2^{m}*m*!) is just the total number of transpositions on 2*m* elements. Thus, the algebra of *C* and *D* is exactly that of *a* and *a*^{†}. The recursive triangle yielding the matrix elements for 〈0|(*a*^{2}+*a*^{†2})^{m}|0〉 can now be interpreted as using combinatorial relations to count the transpositions in the classes of |(*C*^{2}+*D*^{2})^{m}|.

We can also show by combinatorial means that the generating function for the resulting sequence 1, 2, 28, 1112, 87 568, … is given by the square root of the secant. Note first that a typical transposition in the classes of (*C*^{2}+*D*^{2})^{m} will have *m* pairs of consecutive elements associated with *C* and the remaining *m* pairs with *D*, with every *C*-pair member associated with a *D*-pair member by the transposition. For example, with *m*=2, the class *C*^{2}*C*^{2}*D*^{2}*D*^{2} contains (15)(26)(37)(48), and it also contains (15)(28)(36)(47).

The key point in what follows is that there is a more refined characterization of the transpositions, which yields a natural subdivision of the transposition zig-zag classes. It is a simple connectivity property, which can be illustrated by a *chaining graph*. The nodes of the graph are the consecutive pairs, and its edges are defined by the transposition. For the two transpositions we have considered, the chaining graphs are as in figure 2 above.

The chaining graph for any transposition will partition the 2*m* pairs of consecutive elements into a number of disjoint classes, each with its *irreducible* chaining subgraph.

Conversely, given a chaining graph of this form, it defines a class of transpositions all belonging to some class in (*C*^{2}+*D*^{2})^{2m}. It follows that to count the total number of such transpositions, we can enumerate all the possible chaining graphs, and then the number of transpositions associated with each such graph.

This enumeration can be readily performed. We first consider the partitions of the 2*m* pairs effected by the chaining graphs. A typical partition is given bywhere *a*_{i} is the multiplicity of 2*m*_{i} in the partition. There are thendistinct partitions of the pairs.

Given a subset of 2*m*_{i} pairs, the number of *irreducible* chaining graphs of that size is just the same as the number of zig-zag permutations on 2*m*_{i} elements, viz. the tangent number

The number of transpositions associated with an irreducible chaining graph of size 2*m*_{i} is simply

Putting these facts together, we have(4.6)where the sum on the right-hand side is over all partitions, such that , for *m*≥1. This can be rewritten as(4.7)where now the sum is over all *a*_{i}, *m*_{i}, *q*≥1. But, using equation (3.2), this is just(4.8)

As a generalization, where *s* is a positive integer,(4.9)where the sum is taken over all partitions, such that , for *m*≥1. This is readily seen to yield an integer sequence, since

,

the are all integers,

is an integer since it counts the number of distinct partitions

More generally , where *r* and *s* are the positive integers, has an integer sequence. This fact can be interpreted, using equation (3.13), in terms of matrix elements for the squeezed states for the case *α*=2*r*/*s*−1. We anticipate that there are more interesting combinatorial results to be found.

## 5. Discussion

In this paper we have shown that an alteration of the fundamental commutation relation between the creation and annihilation operators of the simple harmonic oscillator by the inclusion of a parity-violating term leads to spectral features that have several physical applications. We have shown that the algebra governing the secant and tangent numbers is a special case of a more general algebra. The expectation values involving Gaussian squeezed states lead to secant and tangent functions and their higher integer powers. However, the generalized algebra enables the evaluation of the expectation values involving a class of non-Gaussian squeezed states that lead to non-integer powers of secant and tangent functions. The establishment of a connection between the operator methods used to calculate expectation values in coherent states and squeezed states and simple triangle schemes arising from a bidiagonal structure of underlying matrices also suggests that further generalizations are possible.

An algebraic scheme for producing parity-dependent spectra may be a useful tool in interpreting the rotational and vibrational states of some nuclei, such as , and possibly some molecules. There are non-trivial consequences which follow if a physical system is indeed governed by an algebra such as the ones considered in this paper. For example, if the even and odd parity rotational bands built on different vibrational states are described by the same parameter *α*, then the separation of the even and odd parity rotational bands must be the same for different vibrational states of the same nucleus or molecule. The preparation and study of octupole-deformed nuclei is a difficult subject experimentally and the published data on nuclei such as only involve nuclei in their vibrational groundstate. Further research towards identifying physical systems exhibiting parity-dependent spectra is in progress.

## Footnotes

- Received April 25, 2007.
- Accepted May 23, 2007.

- © 2007 The Royal Society