Statistical and phase properties and number-phase uncertainty relations are systematically investigated for photon states associated with the Holstein-Primakoff realization of the SU(1,1) Lie algebra. Perelomov's SU(1,1) coherent states and the eigenstates of the SU(1,1) lowering generator (the Barut-Girardello states) are discussed. A recently developed formalism, based on the antinormal ordering of exponential phase operators, is used for studying phase properties and number-phase uncertainty relations. This study shows essential differences between properties of the Barut-Girardello states and the SU(1,1) coherent states. The philophase states, defined as states with simple phase-state representations, relate the quantum description of the optical phase to the properties of the SU(1,1) Lie group. A modified Holstein-Primakoff realization is derived, and eigenstates of the corresponding lowering generator are discussed. These states are shown to contract, in a proper limit, to the familiar Glauber coherent states.
A modification of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of a $SU(2)$ algebra. Once it is applied to a quantum Hamiltonian, a subsequent application of the prescription by Lieb to obtain the classical limit for spin operators allows one to write efficiently a classical Hamiltonian for the system. This process is illustrated for the $M$-atom Jaynes-Cummings model.
As an aid to understanding the {\it displacement operator} definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the {\it ladder-operator or minimum-uncertainty} definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying $[A, A^{\dagger}] = 1$, but the $A$'s are not necessarily the Fock space $a$'s; the multiboson system. It has been previously observed that the ground state…
We show that an ensemble of identical d-level atoms can be efficiently described by d-1 collective oscillator degrees of freedom in the vicinity of a product state with all atoms in the same, but otherwise arbitrary single-particle state. We apply our description to two different kinds of spin squeezing: (i) when each spin-F atom is individually squeezed without creating interatomic entanglement and (ii) when a particular collective atomic oscillator mode is squeezed via quantum non-demolition (QND) measurement and feedback. When combined in sequence, the order of the two methods is relevant i…
In this paper we obtained for the Higgs algebra three kinds of single-mode realizations such as the unitary Holstein-Primakoff-like realization, the non-unitary Dyson-like realization and the unitary realization based upon Villain-like realization. The corresponding similarity transformations between the Holstein-Primakoff-like realizations and the Dyson-like realizations are revealed.
We study different representations of a many-particle system. We reveal the stunning beauty of the connections among the nonclassicalities of a many-particle, a single-mode and a two-mode systems. First, we demonstrate that (separable) atomic coherent states~(ACSs), of a many-particle system, mimic the coherent states of single-mode light. This enables the utilization of a many-particle entanglement~(MPE) criterion as a single-mode nonclassicality~(SMNc) condition via Holstein-Primakoff transformation. As an example, we show that spin-squeezing criterion, for MPE, can be transformed into the q…
Operator square-roots are ubiquitous in theoretical physics. They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation. Often the use of a perturbative expansion is the only recourse when dealing with them. In this work we show that under certain conditions differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square-roots. Specifically, for the number operator $\hat n=\hat a^†a$ we show that the square-root $\sqrt{\hat n}$ near $\hat n=0$ can be approximated by a polyno…
A Hamiltonian describing the collective behaviour of N interacting spins can be mapped to a bosonic one employing the Holstein-Primakoff realisation, at the expense of having an infinite series in powers of the boson creation and annihilation operators. Truncating this series up to quadratic terms allows for the obtention of analytic solutions through a Bogoliubov transformation, which becomes exact in the limit N -> infinit. The Hamiltonian exhibits a phase transition from single spin excitations to a collective mode. In a vicinity of this phase transition the truncated solutions predict the …
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 co…
We introduce the negative binomial states with negative binomial distribution as their photon number distribution. They reduce to the ordinary coherent states and Susskind-Glogower phase states in different limits. The ladder and displacement operator formalisms are found and they are essentially the Perelomov's su(1,1) coherent states via its Holstein-Primakoff realisation. These states exhibit strong squeezing effect and they obey the super-Poissonian statistics. A method to generate these states is proposed.
A mean field description of the Dicke model is presented, employing the Holstein-Primakoff realization of the angular momentum algebra. It is shown that, in the thermodynamic limit, when the number of atoms interacting with the photons goes to infinity the energy surface takes a simple form, allowing for a direct description of many observables.
We study the critical properties of the Lipkin-Meshkov-Glick Model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytical case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are non-trivial results, indicating the fidelity susceptibility is not always extensive.
By expressing the Holstein-Primakoff transformation in a symmetric form a modified spin-polaron technique utilized on triangular-lattice antiferromagnet is developed. With the technique, we have treated an extended t-J model, calculated the quasiparticle dispersion, and we also compared the the dispersion with that obtained by other method.
A new solution is proposed to the long-standing problem of describing the quantum phase of a harmonic oscillator. In terms of an'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the oscillator, a regular phase operator is constructed in the Hilbert-Fock space as a strongly convergent power series. It is shown that the eigenstates of the new 'exponential operators are SU(1,1) coherent states in the Holstein-Primakoff realization. In terms of these eigenstates, the diagonal representation of phase densities and a generalized spectal resolution of …
A class of squeezed states for the su(1,1) algebra is found and expressed by the exponential and Laguerre-polynomial operators acting on the vacuum states. As a special case it is proved that the Perelomov's coherent state is a ladder-operator squeezed state and therefore a minimum uncertainty state. The theory is applied to the two-particle Calogero-Sutherland model. We find some new squeezed states and compared them with the classical trajectories. The connection with some su(1,1) quantum optical systems (amplitude-squared realization, Holstein-Primakoff realization, the two mode realization…
We consider a large number of two-level atoms interacting with the mode of a cavity in the rotating-wave approximation (Tavis-Cummings model). We apply the Holstein-Primakoff transformation to study the model in the limit of the number of two-level atoms, all in their ground state, becoming very large. The unitary evolution that we obtain in this approximation is applied to a macroscopic superposition state showing that, when the coherent states forming the superposition are enough distant, then the state collapses on a single coherent state describing a classical radiation mode. This appear a…
A binary mixtures of Bose-Einstein condensate structures exhibit an incredible richness in terms of holding different kinds of phases. Depending on the ratio of the inter- and intra-atomic interactions, the transition from mixed to separated phase, which is also known as the miscibility-immiscibility transition, has been reported in different setups and by different groups. Here, we describe such type of quantum phase transition in an effective Hamiltonian approach, by applying Holstein-Primakoff transformation in the limit of large number of particles. We demonstrate that non-trivial geometri…
Motivated by recent prototypes of engineered atomic spin devices, we study a fully connected system of $N$ spins $1/2$, modeled by the Lipkin-Meshkov-Glick (LMG) model of a collective spin $s=N/2$ in the presence of Markovian dissipation processes. We determine and classify the different phases of the dissipative LMG model with Markovian dissipation, including the properties of the steady-state and the dynamic behavior in the asymptotic long-time regime. Employing variational methods and a systematic approach based on the Holstein-Primakoff mapping, we determine the phase diagram and the spect…
