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Chinese Physics B

A method to calculate effective Hamiltonians in quantum information*

Jun-Hang Ren, Ming-Yong Ye, Xiu‐Min Lin

Published Oct 14, 2019• 1 citation
Publisher

Abstract

Effective Hamiltonian method is widely used in quantum information. We introduce a method to calculate effective Hamiltonians and give two examples in quantum information to demonstrate the method. We also give a relation between the effective Hamiltonian in the Shrödinger picture and the corresponding effective Hamiltonian in the interaction picture. Finally, we present a relation between our effective Hamiltonian method and the James–Jerke method which is currently used by many authors to calculate effective Hamiltonians in quantum information science.

DOI
10.1088/1674-1056/ab4d43

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Effective Hamiltonian method is widely used in quantum information. We introduce a method to calculate effective Hamiltonians and give two examples in quantum information to demonstrate the method. We also give a relation between the effective Hamiltonian in the Shrödinger picture and the corresponding effective Hamiltonian in the interaction picture. Finally, we present a relation between our effective Hamiltonian method and the James–Jerke method which is currently used by many authors to calculate effective Hamiltonians in quantum information science.
James' effective Hamiltonian method has been extensively adopted to investigate largely detuned interacting quantum systems. This method only corresponds to the second-order perturbation theory and cannot be exploited to treat problems which should be solved by using the third- or higher-order perturbation theory. In this paper, we generalize James' effective Hamiltonian method to the higher-order case. Using the method developed here, we reexamine two recently published examples [L. Garziano et al., Phys. Rev. Lett. 117, 043601 (2016); Ken K. W. Ma and C. K. Law, Phys. Rev. A 92, 023842 (2015)]; our results turn out to be the same as the original ones derived from the third-order perturbation theory and adiabatic elimination method, respectively. For some specific problems, this method can simplify the calculating procedure and the resultant effective Hamiltonian is more general.
One of the useful and practical methods for solving quantum-mechanical many- body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called box introduced by Kuo et. al. It has been one of the central problems how to calculate the box accurately' and efficiently. Introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With the transformed Hamiltonian we derive a recursion method of calculating the box non-perturbatively and analytically. The box given in this study corresponds to a non-perturbative solution for the energy-dependent. Hamiltonian interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. The present approach has a possibility of resolving many of the theoretical and practical difficulties encountered in the calculation of effective-interaction and/or Hamiltonian.
Various approaches have been used in the literature for eliminating nonresonant levels in atomic systems and deriving effective Hamiltonians. Important among these are elimination techniques at the level of probability amplitudes, operator techniques to project the dynamics on to the subspace of resonant levels, Green’s function techniques, the James’ effective Hamiltonian approach, etc. None of the previous approaches is suitable for deriving effective Hamiltonians in intracavity situations. However, the James’ approach does work in the case of only two-photon transitions in a cavity. A generalization of the James’ approach works in the case of three-photon transitions in a cavity, but only under Raman-like resonant conditions. Another important approach for adiabatic elimination is based on an adaptation of the Markov approximation well-known in the theory of system–bath interactions. However, this approach has not been shown to work in intracavity situations. In this paper, we present a method of adiabatic elimination for atoms inside cavities in the presence of multiphoton transitions. We work in the Heisenberg picture, and our approach has the advantage that it allows one to derive effective Hamiltonians even when Raman-like resonance conditions do not hold.
We often hear at seminars and presentations ‘Show me the Hamiltonian of your problem and I will tell you what problem you are talking about’. Therefore, to understand what quantum optics is about, we start from the description of a standard Hamiltonian of the problems considered in quantum optics. We illustrate the method of derivation of the explicit form of the Hamiltonian for a simple system that is composed of two subsystems that interact (communicate) with each other. As we shall see, the explicit form of the Hamiltonian is essential to explicitly calculate energy levels of a given combined system and the temporal evolution of an arbitrary operator representing the system.
Abstract The strong interaction between individual Rydberg atoms provides a powerful tool exploited in an ever-growing range of applications in quantum information science, quantum simulation and ultracold chemistry. One hallmark of the Rydberg interaction is that both its strength and angular dependence can be fine-tuned with great flexibility by choosing appropriate Rydberg states and applying external electric and magnetic fields. More and more experiments are probing this interaction at short atomic distances or with such high precision that perturbative calculations as well as restrictions to the leading dipole–dipole interaction term are no longer sufficient. In this tutorial, we review all relevant aspects of the full calculation of Rydberg interaction potentials. We discuss the derivation of the interaction Hamiltonian from the electrostatic multipole expansion, numerical and analytical methods for calculating the required electric multipole moments and the inclusion of electromagnetic fields with arbitrary direction. We focus specifically on symmetry arguments and selection rules, which greatly reduce the size of the Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up to higher multipole orders on a desktop computer. Finally, we present example calculations showing the relevance of the full interaction calculation to current experiments. Our software for calculating Rydberg potentials including all features discussed in this tutorial is available as open source.