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arXiv:2307.00875 · Physical Review A

Multipartite spin coherent states and spinor states

Tim Byrnes

Published Feb 27, 2024• 7 citations
arXivHTMLPDF

Abstract

Multipartite generalizations of spin coherent states are introduced and analyzed. These are the spin analogs of multimode optical coherent states as used in continuous variable quantum information, but generalized to possess full spin symmetry. Two possible generalizations are given, one of which is a simple tensor product of a given multipartite quantum state. The second generalization uses the bosonic formulation in the Jordan-Schwinger map, which we call spinor states. In the unipartite case, spinor states are equivalent to spin coherent states; however, in the multipartite case, they are no longer equivalent. Some fundamental properties of these states are discussed, such as their observables and covariances with respect to symmetric operators, form-preserving transformations, and entanglement. We discuss the utility of such multipartite spin coherent and spinor states as a way of storing quantum information.

DOI
10.1103/physreva.109.022438

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Multipartite generalizations of spin coherent states are introduced and analyzed. These are the spin analogs of multimode optical coherent states as used in continuous variable quantum information, but generalized to possess full spin symmetry. Two possible generalizations are given, one of which is a simple tensor product of a given multipartite quantum state. The second generalization uses the bosonic formulation in the Jordan-Schwinger map, which we call spinor states. In the unipartite case, spinor states are equivalent to spin coherent states; however, in the multipartite case, they are no longer equivalent. Some fundamental properties of these states are discussed, such as their observables and covariances with respect to symmetric operators, form-preserving transformations, and entanglement. We discuss the utility of such multipartite spin coherent and spinor states as a way of storing quantum information.
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This chapter discusses spinor Bose--Einstein condensates and the various common states that are encountered with such systems. Many analogous concepts to optical systems are discussed, such as spin coherent states, spin squeezed states, uncertainty relations, and quasiprobability distributions such as the Q- and Wigner functions. A gallery of different spin states is shown for both the Q- and Wigner functions is shown, including that for highly non-classical states such as the Schrodinger cat state. Due to the different operators involved for spin systems, we describe the similarities and differences to optical squeezing, and introduce the one-axis and two-axis countertwisting spin squeezed states. The notion of entanglement in such systems, along with and ways of detecting this is are discussed. Several mappings, such as the Holstein--Primakoff transformation between spins and bosonic operators, as well asand the equivalence between condensed and uncondensed systems, is are discussed. We also introduce some key mathematical results involving important states, such as formulas to perform basis transformations between Fock states.
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.
The Jordan-Schwinger map is widely employed to switch between bosonic or fermionic mode operators and spin observables, with numerous applications ranging from quantum field theories of magnetism and ultracold quantum gases to quantum optics. While the construction of observables obeying the algebra of spin operators across multiple modes is straightforward, a mapping between bosonic or fermionic Fock states and spin states has remained elusive beyond the two-mode case. Here, we generalize the Jordan-Schwinger map by algorithmically constructing complete sets of spin states over several bosonic or fermionic modes, allowing one to describe arbitrary multi-mode systems faithfully in terms of spins. As a byproduct, we uncover a deep link between the degeneracy of multi-mode spin states in the bosonic case and Gaussian polynomials. We demonstrate the feasibility of our approach by deriving explicit relations between arbitrary three-mode Fock and spin states, which provide novel interpretations of the genuinely tripartite entangled GHZ and W state classes.
A new method of treating problems of spin systems is developed, in which spin operators are transformed into two kinds of boson operators according to Schwinger's method. Usual treatment of coherent state representation is generalized to spin systems with a fixed magnitude S . Some basic properties and the relations to the atomic coherent state are investigated rather in details. The method is applied to spin relaxation process and a corresponding Fokker-Planck type equation is obtained and solved to give a multiple relaxation process due to nonlinearities. It is shown that our formulation has several advantages over other ones in some respects.