New proof of closed formulas for Bogoljubov boson transformations

Abstract

Abstract. The fact that the algebra of the operators b +a, b 2 and 2b+b+l is closed is utilized for the derivation of closed formulas for Bogoljubov transformations by means of the previously developed technique for treating the angular momentum algebra.Recently, there appeared several papers [1, 2] in which closed formulas for the Bogoljubov transformations were derived. The closed formulas find application e.g. in considering the RPA and other transformations (in which two sets of boson creation and annihilation operators are connected through linear relations) if it is required to determine matrix elements of the new (old) operators in the old (new) basis. An example of such a situation is treated in Ref. 4. We shall return to this problem and rederive these relations from the point of view of the method presented in a previous note [3] (to be referred to as I). As we shall see our derivation is at least as direct as that given in Ref. 2 enabling us to present one of the closed formulas in a more compact form. Our treatment exhibits a great resemblance to the angular momentum discussion dealt with in the previous note The operators {., a= + 1, enable us a more general, and simplified, treatment of the operator algebra. First, obviously,and we have to evaluatewhere, in the latter version, the annihilation operator stands to the right of the creation operator. We, further, needwhich proves that the algebra of operators ~-2 ~o,~2o and 2~_1~1+1 is closed. Now, we try, in analogy to Eq. (I.23), to represent the operator B in the following manner B=exp{F1 ~2} exp{F2(2~_ 1 ~lq_ 1)} exp{F 3 ~2

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

“…Of course, the complete Hilbert space is the direct sum of these sub-spaces, H = H 1/4 ⊕ H 3/4 . In the present and in similar contexts, the properties of the SU(1, 1) group have been used by , Malkin et al (1970), Gilmore (1974), Perelomov (1977), Kelemen (1975), Witschel (1975), Fisher et al (1984, Schumaker and Caves (1985), Varró (2015), Wünsche (1999Wünsche ( ), (2003Wünsche ( ), (2017aWünsche ( ), (2017b.…”

“…We note that the anti-normal form can also be obtained similarly, (A.8) These expressions (A.7) and (A.8) are equivalent to the ones derived long ago by Husimi (1953), who used two-variable generating functions of S mn , and the L 2 representation (Hermite functions) for the oscillator basis states (see also . For a real ζ equation (A.7) reduces to the equation ( 27) in Kelemen's (1975) paper. See also Tanabe (1973), Rashid (1975), Satyanarayana (1985) and Mendaš and Popović (1995).…”

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

“…. In the present and in similar contexts, the properties of the ) 1 , 1 ( SU group have been used by Popov and Perelomov (1969), Malkin, Man'ko and Trifonov (1970), Gilmore (1974), Perelomov (1977), Kelemen (1975), Witschel (1975), Fisher, Nieto and Sandberg (1984 or Schumaker and Caves (1985), Wünsche (1999Wünsche ( , 2003Wünsche ( , 2017a.…”

“…For a real ] Eq. (A.7) reduces to the equation ( 27) in Kelemen's (1975) paper. See also Tanabe (1973), Rashid (1975), Satyanarayana (1985) and Mendaš and Popoviü (1995).…”