An Introduction to the Holstein-Primakoff Transformation, with Applications in Magnetic Resonance

Abstract

We have witnessed an impressive advancement in computer performance in the last couple of decades. One would therefore expect a trickling down of the benefits of this technological advancement to the borough of computational simulation of multispin magnetic resonance spectra, but that has not been quite the case. Though some significant progress has been made, chiefly by Kuprov and collaborators, one cannot help but observe that there is still much to be done. In our view, the difficulties are not to be entirely ascribed to technology, but, rather, may mostly stem from the inadequacy of the conventional theoretical tools commonly used. We introduce in this paper a set of theoretical tools which can be employed in the description and efficient simulation of multispin magnetic resonance spectra. The so-called Holstein-Primakoff transformation lies at the heart of these, and provides a very close connection to discrete mathematics (from graph theory to number theory).The aim of this paper is to provide a reasonably comprehensive and easy-to-understand introduction to the Holstein-Primakoff (HP) transformation (and related bosons) to researchers and students working in the field of magnetic resonance. We also focus on how through the use of the HP transformation, we can reformulate many challenging computing problems encountered in multispin systems as enumerative combinatoric problems. This, one could say, is the HP transformation's primary forte. As a matter of illustration, our main concern here will be on the use of the HP bosons to characterize and eigendecompose a class of multispin Hamiltonians often employed in high-resolution magnetic resonance.

Fundamentals of Quantum Mechanics in Liouville Space

“…In ordinary quantum mechanics in H d , the eigenvalues of observables -like the Hamiltonian or magnetization vector -, which are absolute quantities (at least, up to a constant), are the natural occurrences. Meanwhile, what we experimentally measure in spectroscopic experiments like nuclear magnetic resonance (NMR) are quantities related to the differences between these eigenvalues [45]. We need not look further than the resonance conditions of such experiments to see this is the case.…”

Cavity Quantum Electrodynamics Effects with Nitrogen Vacancy Center Spins in Diamond and Microwave Resonators at Room Temperature

“…3(a,b), that the spinensembles occupy states near the lower boundaries of the Dicke state space. Thus, we may approximate these states as occupation number states of quantized harmonic oscillators (Holstein-Primakoff approximation [17][18][19]), and treat the spin-resonator system as two quan- Hz, as marked, with peaks from zero detuning and outwards. The inset panels show the Dicke states of the spin-ensemble with increasing J for increasing ηs.…”

“…3 of the main text, we show that the collective interaction of the optically cooled spin ensemble with the resonator field leads to Rabi oscillations and cavity mode splittings at room temperature. To gain more insights into the physics, we note that the spin ensemble occupies states near the lower boundary of the Dicke space, which permits us to apply the Holstein-Primakoff approximation [17][18][19] and approximate these states as the occupation number states of quantized harmonic oscillators. As a result, we can approximate the Hamiltonian of the spin-ensemble as Ĥs ≈ J Ĥs,J with Ĥs,J ≈ ω s b+ J bJ , where b+ J , bJ are the bosonic creation and annihilation operator for given J and the spin ensembleresonator coupling Hamiltonian as Ĥa−c ≈ J Ĥa−c,J with Ĥa−c,J = √ 2Jg s â †b J + b+ J â , where the coupling strength √ 2Jg s depends on the numbers J.Since Ĥs is sum of Ĥa−c,J over the different J, the system response can be veiwed as the sum of response for sub-system with given J weighted by the population on the corresponding Dicke ladder.…”

Non-equilibrium quantum thermodynamics of a particle trapped in a controllable time-varying potential

“…Step 1: Discretization of the Hamiltonian can be achieved through the Holstein Primakoff (HP) transformation [33][34][35], which intorduces an effective bosonic system with a fixed dimension N . Such HP bosonic system We are now able to discretize the system by recasting Eqs.…”

“…We refer to Ref. [34] for a detailed discussion on the TSS mapping. With two bosonic modes, one can define the bosonic coherent state as…”