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arXiv:quant-ph/0505206 · Physical Review A

Multiquantum eigenstates of a linear chain of coupled qubits

C. J. Mewton, Z. Ficek

Published Jun 29, 2005• 6 citations
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Abstract

We present a technique to identify exact analytic expressions for the multiquantum eigenstates of a linear chain of coupled qubits. A choice of Hilbert subspaces is described that allows an exact solution of the stationary Schr\"odinger equation without imposing periodic boundary conditions and without neglecting end effects, fully including the dipole-dipole nearest-neighbor interaction between the atoms. The treatment is valid for an arbitrary coherent excitation in the atomic system, any number of atoms, any size of the chain relative to the resonant wavelength and arbitrary initial conditions of the atomic system. The procedure we develop is general enough to be adopted for the study of excitation in an arbitrary array of atoms including spin chains and one-dimensional Bose-Einstein condensates.

DOI
10.1103/physreva.71.060311

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We present a technique to identify exact analytic expressions for the multiquantum eigenstates of a linear chain of coupled qubits. A choice of Hilbert subspaces is described that allows an exact solution of the stationary Schr"odinger equation without imposing periodic boundary conditions and without neglecting end effects, fully including the dipole-dipole nearest-neighbor interaction between the atoms. The treatment is valid for an arbitrary coherent excitation in the atomic system, any number of atoms, any size of the chain relative to the resonant wavelength and arbitrary initial conditions of the atomic system. The procedure we develop is general enough to be adopted for the study of excitation in an arbitrary array of atoms including spin chains and one-dimensional Bose-Einstein condensates.
The characteristic polynomial for a chain of dipole-dipole coupled two-level atoms with nearest-neighbour and next-nearest-neighbour interactions is developed for the study of eigenvalues and eigenvectors for single-photon excitations. We find the exact form of the polynomial in terms of the Chebyshev polynomials of the second kind that is valid for an arbitrary number of atoms and coupling strengths. We then propose a technique for expressing the roots of the polynomial as a power series in the coupling constants. The general properties of the solutions are also explored, to shed some light on the general properties that the exact, analytic form of the energy eigenvalues should have. A method for deriving the eigenvectors of the Hamiltonian is also outlined.
In this study, we develop a semi-analytical framework to solve generalized Jaynes-Tavis-Cummings Hamiltonians describing multi-qudit systems coupled via EM resonators. Besides the multi-level generalization we allow for an arbitrary number of resonators and/or modes, with nonidentical couplings to the qudits. Our method is based on generic excitation-number operators which commute with the respective Hamiltonians in the rotating wave approximation (RWA). The validity of the RWA is assessed explicitly. The formalism enables the study of eigenstates, eigenenergies and corresponding time evolutions of such coupled multi-qudit systems. The technique can be applied in cavity quantum electrodynamics and circuit quantum electrodynamics. It is also applicable to atomic physics, describing the coupling of a single-mode photon to an atom. As an example, we solve the Schrödinger equation for a two-qubit-one-resonator system, in principle to arbitrary high excitations. We also solve the Tavis-Cummings Hamiltonian in the one-excitation subspace for an arbitrary number of identical qubits resonantly coupled to one resonator. As a final example, we calculate of the low-excitation spectrum of a coupled two-transmon system.
We investigate various properties of a linear chain of qubits coupled throughthe dipole-dipole interaction. Most of the analysis is performed in the nearest-neighbor approximation. The Hamiltonian is ¯rst diagonalized, and we ex-plore some of the mathematical properties of the eigenstates. The patternof radiation emission from the single-excitation states is given, along withthe demonstration that most of these single-excitation states are subradiant,making them useful in applications such as quantum information storage andprocessing. The entanglement dynamics of the chain is also investigated,mainly showing how very long chains lose initial concurrence, as opposed toshort chains which do not.We conclude with an investigation into trying to move beyond the nearest-neighbor approximation, and derive the characteristic polynomial of theHamiltonian eigenvalue equation in analytic form, and use it to derive apower series approximation for the energy eigenvalues.
We study a matrix product state algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [see S. \"Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537 (1995);S. Rommer and S. \"Ostlund, Phys. Rev. B 55, 2164 (1997)], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions and previous numerical results for the quantum Ising, the antiferromagnetic Heisenberg spin-1/21/21/2, and the bilinear-biquadratic spin-1 models.
In this study, we develop a semi-analytical framework to solve generalized Jaynes-Tavis-Cummings Hamiltonians describing multi-qudit systems coupled via EM resonators. Besides the multi-level generalization we allow for an arbitrary number of resonators and/or modes, with nonidentical couplings to the qudits. Our method is based on generic excitation-number operators which commute with the respective Hamiltonians in the rotating wave approximation (RWA). The validity of the RWA is assessed explicitly. The formalism enables the study of eigenstates, eigenenergies and corresponding time evolutions of such coupled multi-qudit systems. The technique can be applied in cavity quantum electrodynamics and circuit quantum electrodynamics. It is also applicable to atomic physics, describing the coupling of a single-mode photon to an atom. As an example, we solve the Schrodinger equation for a two-qubit-one-resonator system, in principle to arbitrary high excitations. We also solve the Tavis-Cummings Hamiltonian in the one-excitation subspace for an arbitrary number of identical qubits resonantly coupled to one resonator. As a final example, we calculate of the low-excitation spectrum of a coupled two-transmon system.
We describe a quantum system consisting of a one-dimensional linear chain of n identical harmonic oscillators coupled by a nearest neighbor interaction. Two boundary conditions are taken into account: periodic boundary conditions (where the nth oscillator is coupled back to the first oscillator) and fixed wall boundary conditions (where the first oscillator and the nth oscillator are coupled to a fixed wall). The two systems are characterized by their Hamiltonian. For their quantization, we treat these systems as Wigner quantum systems (WQSs), allowing more solutions than just the canonical quantization solution. In this WQS approach, one is led to certain algebraic relations for operators (which are linear combinations of position and momentum operators) that should satisfy triple relations involving commutators and anti-commutators. These triple relations have a solution in terms of the Lie superalgebra gl(1|n). We study a particular class of gl(1|n) representations V(p), the so-called ladder representations. For these representations, we determine the spectrum of the Hamiltonian and of the position operators (for both types of boundary conditions). Furthermore, we compute the eigenvectors of the position operators in terms of stationary states. This leads to explicit expressions for position probabilities of the n oscillators in the chain. An analysis of the plots of such position probability distributions gives rise to some interesting observations. In particular, the physical behavior of the system as a WQS is very much in agreement with what one would expect from the classical case, except that all physical quantities (energy, position, and momentum of each oscillator) have a finite spectrum.