On the exponential form of the displacement operator for different systems

Abstract

The family of displacement operators D x p ( , ), a central concept in the theory of coherent states of a quantum mechanical harmonic oscillator, has been successfully generalized to systems of quantized, cyclic or finite position coordinates. However, out of the plethora of mutually equivalent expressions for the displacement operators valid in the continuous case, only few are directly applicable in the other systems of interest. The aim of this paper is to strengthen the analogy between the different cases by identifying the root cause of the issues accompanying the straightforward generalization of certain important expressions and, more importantly, offering alternative ones of general validity. Ultimately we arrive at an algorithm allowing one to express any displacement operator as an exponential of a pure imaginary multiple of a generalized 'quadrature' observable that is not obtained by a linear combination of position and momentum observables but rather by a shear transform of one of them in the systemʼs phase space.

Wigner distribution on a double cylinder phase space for studying quantum error correction protocol

“…It leads to the construction of a double cylinder phase space. We then generalize the results obtained on the rotational Wigner distribution on a single cylinder phase space, as introduced in [31][32][33][34], or for the phase-number Wigner distribution one [35,36]. The representation of a quantum state in a double cylinder phase space is totally equivalent to the rectangular phase space.…”

Producing delocalized frequency-time Schrödinger cat-like states with HOM interferometry

“…In order to complete the mathematical analogy between the frequency basis and the quadrature position one, we can also notice that the frequency-time phase space of a single photon is non-commutative similarly to the quadrature position-momentum phase space as shown in [18]. As it happens one can define non-commuting frequency-time displacement operators [18] which obeys the Weyl algebra [19] and hence have a complete mathematical correspondance with the position-momentum quadrature displacement operators. As a consequence, the frequency-time phase space of a single photon exhibits a paving structure due to the non-commutativity of these displacement operators as is the case for the (x, p) phase space.…”

Quantum Harmonic Analysis of the Density Matrix

“…commonly used in quantum optics (a and a † are the annihilation and creation operators; see Potoček [84] for a discussion of these notational issues).…”