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American Journal of Physics

Spin and photon echoes

D. C. Haueisen, Thomas D. Bonifield

Published Sep 1, 1975• 2 citations
Publisher

Abstract

The classical interpretation of the spin echo is reviewed, followed by a simple quantum-mechanical derivation of the general echo phenomenon for a two-state system. The photon echo is then examined, and graphs of hydrogenic wave functions for eigenstates and transitional states are shown as examples. As many as possible of the classical notions associated with the spin echo are applied to the case of photon echoes.

DOI
10.1119/1.9694

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The classical interpretation of the spin echo is reviewed, followed by a simple quantum-mechanical derivation of the general echo phenomenon for a two-state system. The photon echo is then examined, and graphs of hydrogenic wave functions for eigenstates and transitional states are shown as examples. As many as possible of the classical notions associated with the spin echo are applied to the case of photon echoes.
The amplitude of the electron-spin or photon echo from an ensemble of isolated atoms (echo atoms) interacting with magnetic-nuclear neighbors is calculated as a function of the magnitude and the orientation of an applied magnetic field $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ and the separation between the two excitation pulses, in the limit that the magnitude of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$ is large compared to the effective local fields at the echo-atom sites due to the nuclear neighbors. The general problem is reduced to the equivalent problem of two-level echo atoms interacting with nuclear neighbors. The latter problem is readily solved and applied to both spin and photon echoes. Experimentally observed spin-echo behavior in ruby is compared to calculated spin-echo behavior using no adjustable parameters, and is found to be in good agreement even when the magnetic field is tilted away from the optic axis thereby causing the echoes to disappear. Calculated photon-echo behavior, in which it is assumed that the interaction constants in the excited state are simply related to those in the ground state, shows the same general features as those that have been observed in experiment. Several results are presented to show the wide range of photon-echo behavior for different values of the interaction constants. A simple theory is presented which explains most observed echo behavior in a straightforward way.
In this chapter we build up the basic concepts of quantum mechanics using two simple examples, following a heuristic approach which is more inductive than deductive. We start with a familiar phenomenon, that of the polarization of light, which will allow us to introduce the necessary mathematical formalism. We show that the description of polarization leads naturally to the need for a two-dimensional complex vector space, and we establish the correspondence between a polarization state and a vector in this space, referred to as the space of polarization states. We then move on to the quantum description of photon polarization and illustrate the construction of probability amplitudes as scalar products in this space. The second example will be that of spin 1/2, where the space of states is again two-dimensional. We construct the most general states of spin 1/2 using rotational invariance. Finally, we introduce dynamics, which allows us to follow the time evolution of a state vector. The analogy with the polarization of light will serve as a guide to constructing the quantum theory of photon polarization, but no such classical analog is available for constructing the quantum theory for spin 1/2. In this case the quantum theory will be constructed without reference to any classical theory, using an assumption about the dimension of the space of states and symmetry principles. The polarization of light and photon polarization The polarization of an electromagnetic wave The polarization of light or, more generally, of an electromagnetic wave, is a familiar phenomenon related to the vector nature of the electromagnetic field. Let us consider a plane wave of monochromatic light of frequency ω propagating in the positive z direction.
As it was emphasized several times above, the phenomenon of the photon echo is an optical analog of the spin echo that has been discovered [ 1 ] about a decade earlier. That is why many ideas from the theory of the spin echo has been transferred to the theory of the photon echo and appeared to be fruitful there. One of these ideas put forward by Fernbach and Proctor [ 2 ] is the idea of the possibility of the use of the phenomenon of the spin echo for writing, storage, and processing of the data. The authors [ 2 ] determined the conditions under which the signals of the spin and stimulated spin echo can reproduce (direct or time-reversed) shape of one of the exciting pulses. The theoretical analysis was proved by the corresponding experiments.
The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.
The mechanism of optical echo (photon echo) formation in solid and gas media is described. The treatment is based on solutions of Schrodinger's equation for the time evolution of atomic amplitudes during an optical excitation pulse sequence, and should be accessible to non-specialists and to students studying undergraduate quantum mechanics. Atomic amplitude evolution diagrams are developed to facilitate the analysis of the echo mechanism and the derivation of the echo time and propagation direction.