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arXiv:1812.01842 · Journal of Physics Conference Series

Special functions with mod n symmetry and kaleidoscope of quantum coherent states

Aygül Koçak, Oktay K. Pashaev

Published Apr 1, 2019• 5 citations
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Abstract

The set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.

DOI
10.1088/1742-6596/1194/1/012059

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The set of mod n functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod n exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.
The Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits are generalized to the kaleidoscope of coherent states, related with regular n-polygon symmetry and the roots of unity. This quantum kaleidoscope is motivated by our method of classical hydrodynamics images in a wedge domain, described by qqq-calculus of analytic functions with qqq as a primitive root of unity. The cases of the trinity states and the quartet states are described in details. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry. We show that these states can be generated for an arbitrary nnn by the Quantum Fourier transform and can provide qutrits, ququats and in general, qudit units of quantum information. Relations of our states with quantum groups and quantum calculus are discussed.
The Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits, are generalized to the kaleidoscope of coherent states related with regular n-polygon symmetry and the roots of unity. The cases of the trinity states and the quartet states are described in details. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry. We show that for an arbitrary nnn, these states can be generated by the Quantum Fourier transform and can provide qutrits, ququats and in general, qudit units of quantum information. Relations with quantum groups and quantum calculus are discussed.
The hydrodynamic flow in several bounded domains can be formulated by the image theorems, like the two circle, the wedge and the strip theorems, describing flow by q -periodic functions. Depending on geometry of the domain, parameter q has different geometrical meanings and values. In the special case of the wedge domain, with q as a primitive root of unity, the set of images appears as a regular polygon kaleidoscope. By interpreting the wave function in the Fock-Barman representation as complex potential of a flow, we find mod n projection operators in the space of quantum coherent states, related with operator q -numbers. They determine the units of quantum information as kaleidoscope of quantum states with quantum group symmetry of the q-oscillator. Expansion of Glauber coherent states to these units and corresponding entropy are discussed.
A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ \ensuremath{\surd}2 , where P is the parity operator. Such A arises naturally in the q\ensuremath{\rightarrow}-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-q2{\mathit{q}}^{2}q2${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, ${\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.
We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q -function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n -qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q -functions. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
The properties of the displaced Fock states mod a,n) identical to d(a,a*) mod n), (a complex numbers, D(a,a*) displacement operators, n=0,1,2,. . .) are systematically investigated with emphasis on the connections to the Heisenberg-Weyl group and to its irreducible representations. The displaced Fock states comprise the coherent states mod a) identical to mod a,0) as well as the Fock states mod n) identical to mod 0,n) as particular cases. An orthocompleteness relation for the displaced Fock states in the form of the area integral of the operators mod a,m)(a,n mod over the complex a-plane is derived. It generalizes for m=n the well known completeness relation for the coherent states and leads for m not=n to identities expressing the overcompleteness of the displaced Fock states. A basic formula is obtained for the convolution of the operators mod a,m)(a,n mod with the class of Gaussian functions that have exponents proportional to aa*. The connection of the displaced Fock states to the transition operators from the density operator for a single boson mode to quasiprobabilities is studied in general form and specified to the class of transition operators with the displaced Fock states as their eigenstates and connected by convolutions with Gaussian functions to the coherent-state quasiprobability, the Wigner quasiprobability, and the Glauber-Sudarshan quasiprobability.