On the derivation of exact eigenstates of the generalized squeezing operator

We construct the states that are invariant under the action of the generalized squeezing operator exp(za†k−z*ak) for arbitrary positive integer k. The states are given explicitly in the number representation. We find that for a given value of k there are k such states. We show that the states behave as n −k/4 when occupation number n → ∞ . This implies that for any k ≥ 3 the states are normalizable. For a given k, the expectation values of operators of the form a†aj are finite for positive integer j < (k/2 − 1) but diverge for integer j ≥ (k/2 − 1). For k = 3 we also give an explicit form of these states in the momentum representation in terms of Bessel functions.


Introduction
The concept of squeezing plays one of the central roles in quantum optics. Squeezed states facilitate measurement and communication in a way not possible with the coherent states which are produced from quantum vacuum. Squeezed states are characterized by the phase-space distribution of the associated momentum-like (P) and position-like (X ) quadrature variables of the field. Their variances obey the Heisenberg principle D Dˆ X P 1 4. Vacuum, coherent, and squeezed states minimally satisfy this inequality and a coherent state is realized when D = DX P. A squeezed state is produced when either of the quadratures is increased at the expense of the other. Under purely harmonic time evolution, squeezed states remain squeezed and, therefore, always minimally satify the Heisenberg relation. However, they will evolve into non-squeezed states if non-harmonic perturbations are introduced to the Hamiltonian. Dodonov [1] and Dell'Anno et al [2] provide extensive lists of references that deal with various aspects of squeezing.
The mathematical realization of a squeezed state in the simplest case is given in terms of the squeezing acting on the vacuum state. Here a † and a are creatrion and annihilation operators and z is a compex-valued parameter. Over time, attempts to generalize this operator to include higher order processes have been made. Different types of generalizations have been investigated. Some of these generalizations involve exponentials of operators that are elements of closed algebras [3][4][5][6]. By contrast, in this work we consider the generalization of the squeezing operator of the form with integer k 3. This kind of generalization turned out to be quite nontrivial. On one hand, it was shown by Fisher et al [7] that the vacuum to vacuum probability amplitude 〈0|U k (z)|0〉 has a zero radius of convergence as a power series with respect to z, for k > 2. On the other hand, Braunstein and McLachlan demonstrated numerically [8,9] that such expressions can still be well defined. Some properties of operator za † k − z * a k were discussed by Nagel [10]. The exact eigenstates of this operator for the case of k = 2 were also constructed by Lo [11]. Elyutin and Klyshko showed [12] that the average occupation number of an arbitrary initial state acted upon by the unitary operator U 3 (z) diverges to infinity for a finite z. All of these challenges lead to various attempts to modify the operators a † and a in U k (z) in such a way that no question of convergence would arise [13][14][15]. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
The k-squeezed states can be physically realized using the following argument. Consider the following unitary transformation of the harmonic This expression can be evaluated explicitly only for k = 1 and 2 [16]. However, for small z we can expand the right-hand side of equation (2) in powers of z and keep only the zeroth and first order terms. Application of the commutation rules for a † and a leads to Thus, a k-th order squeezing would result from a k-th order non-linearity within an optically pumped system.
In this paper we construct k exact degenerate eigenstates of operator U k (z) that have an eigenvalue equal to one and investigate some of their properties.

Invariant states of operator U k (z) in the number representation
Without the loss of generality we can limit ourselves to the case where z is real z = z * = r and consider the eigenstates of the Hermitian operator The eigenvalues of M k are not known except for the cases of k = 1, 2 [10]. In general, the eigenvalue problem of this operator written in the number representation leads to a three term recurrence relation. However, in the case of zero eigenvalue (assuming that it exists) the recurrence relation involves only two terms and as a result explicit eigenstates can be obtained. It is clear that since these eigenstates of M k = ir(a † k − a k ) have zero eigenvalues they are also eigenstates of U k (r) with eigenvalue equal to one, or, in other words, invariant under the action of U k (r).
Although we will be primarily interested in the case of k 3, results of this section also apply to k = 1 and 2. Note that U k (z) commutes with operator where α is the degeneracy index. Acting on 〈n| from the right with the creation and annihilation operators in M k we obtain the following recurrence relation  Here integer α is the degeneracy index that can take values from 0 to (k − 1) and integer m runs from 0 to infinity. The occupation number is given by n = α + 2mk. Number states with occupation numbers that do not satisfy the last equation do not contribute to the eigensates of M k . Γ(x) denotes the gamma function and c(k, α) is the normalization constant given by is the generalized hypergeometric series. The eigenfunctions given by equation (8)  Since the square of the eigenstate behaves as m − k/2 for large m we can conclude that the norm is finite for any k 3. This is because the series of the form å = ¥ m m p 1 converges when p > 1 and diverges when p 1. For k = 1 and 2 the norm (see equation (9)) diverges in agreement with the known exact results for these cases [10,11]. Similarly, we can see that the average for the number operator = † n a a is divergent for k < 5, the averagen 2 diverges for k < 7, etc. In general, for a given k the expectation values of operators of the formn j diverge for the integer j (k/2 − 1).
If we define dimensionless coordinate and momentum operators = then their expectation values for states given by equation (8) vanish. However, if superpositions of the degenerate states (8) are considered then, in general, the average ofX andP will diverge for k 3 but converge for k 4. Expectation values ofX 2 andP 2 behave in the same way as that for a † a, namely, diverge for k = 3 and k = 4, but remain finite for k 5. The divergence of the expectation value of the number operator for k = 3 and k = 4 implies infinite average energy for these states. Figure 1 shows y á ñ a | n k ʼs as functions of occupation number n for k = 3 through k = 7. The inset tables for k = 5 through k = 7 give the computed á ñ n , á ñ n 2 , and second-order intensity correlator g a a n n n n 2 22 2 2 2 , for each allowed value of the degeneracy index α. The g (2) correlator is a particularly useful quantity since it gives the probability of detecting two simultaneous photons normalized by the probability of detecting two photons from a random source. One can generalize this to = á ñ á ñ ( )( ) † g a a n ; k kk k however, such terms will diverge for reasons given above.

Invariant states of U 3 (z) in the momentum representation
Since eigenstates y á ñ a | n k decay slowly as functions of n, it is of interest to consider their behavior in a continuum basis, such as coordinate or momentum representations. In this section we will costruct the invariant states of U k (z) in momentum representation for k = 3. We chose momentum over coordinate representation to demonstrate an interesting mathematical point that will be mentioned below. Rewriting a and a † in terms of dimensionless coordinate and momentum operators as = + (ˆˆ) a X iP 1 2 and = -(ˆˆ) † a X iP 1 2 , inserting them into M 3 , and using momentum representation we obtain the following eigenvalue equation for the zero eigenvalue Here we suppress subscript 3 for k = 3 in the wave function to simplify the notation. This is a second order ordinary differential equation and its two independent solutions are given by [17] j j = = ⎜ ⎟ ⎜ ⎟