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The Journal of Chemical Physics

On the relaxation of a two-level system driven by a strong electromagnetic field

Eitan Geva, Ronnie Kosloff, James L. Skinner

Published Jun 1, 1995• 132 citations
Publisher

Abstract

A detailed and unified account of the theory of the generalized Bloch equations is presented. The equations apply to a two-level system weakly coupled to a heat bath and subject to a monochromatic rotating field of arbitrary intensity. The relaxation tensor obtained is explicitly field-dependent. The derivation is valid for general coupling to a quantum heat bath. The generalized Bloch equations are shown to be thermodynamically consistent, as opposed to the standard Bloch equations. Different limits of the generalized Bloch equations are examined and related to previous studies. The potential use of the generalized Bloch equations as a probe of the bath spectral density is demonstrated for the case of a two-level system embedded in a Debye solid.

DOI
10.1063/1.468844

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A detailed and unified account of the theory of the generalized Bloch equations is presented. The equations apply to a two-level system weakly coupled to a heat bath and subject to a monochromatic rotating field of arbitrary intensity. The relaxation tensor obtained is explicitly field-dependent. The derivation is valid for general coupling to a quantum heat bath. The generalized Bloch equations are shown to be thermodynamically consistent, as opposed to the standard Bloch equations. Different limits of the generalized Bloch equations are examined and related to previous studies. The potential use of the generalized Bloch equations as a probe of the bath spectral density is demonstrated for the case of a two-level system embedded in a Debye solid.
The Bloch equation and its variants constitute the fundamental dynamical model for arbitrary two-level systems. Many important processes, including those in more complicated systems, can be modeled and understood through the two-level approximation. It is therefore of widespread relevance, especially as it relates to understanding dissipative processes in current cutting-edge applications of quantum mechanics. Although the Bloch equation has been the subject of considerable analysis in the 70 years since its inception, there is still, perhaps surprisingly, significant work that can be done. This paper extends the scope of previous analyses. It provides a framework for more fully understanding the dynamics of dissipative two-level systems. A solution is derived that is compact, tractable, and completely general, in contrast to previous results. Any solution of the Bloch equation depends on three roots of a cubic polynomial that are crucial to the time dependence of the system. The roots are typically only sketched out qualitatively, with no indication of their dependence on the physical parameters of the problem. Degenerate roots, which modify the solutions, have been ignored altogether. Here the roots are obtained explicitly in terms of a single real-valued root that is expressed as a simple function of the system parameters. For the conventional Bloch equation, a simple graphical representation of this root is presented that makes evident the explicit time dependence of the system for each point in the parameter space. Several intuitive, visual models of system dynamics are developed. A Euclidean coordinate system is identified in which any generalized Bloch equation is separable, i.e., the sum of commuting rotation and relaxation operators. The time evolution in this frame is simply a rotation followed by relaxation at modified rates that play a role similar to the standard longitudinal and transverse rates. These rates are functions of the applied field, which provides information towards control of the dissipative process. The Bloch equation also describes a system of three coupled harmonic oscillators, providing additional perspective on dissipative systems.
Previous theory describing the behavior of a general two-level system (TLS) coupled to a loss mechanism and driven by an oscillating field near resonance is generalized to include an arbitrary driving field. Integro-differential equations of motion for the expectation values of the Pauli spin matrices, which describe completely the behavior of a TLS, are derived. The relaxation constants involved in these equations prove to be frequency-dependent, and reactive effects produced by the loss mechanism appear. It is shown that for an approximately monochromatic driving field near resonance, the general equations of motion can be approximated by differential equations. In the case of a magnetic TLS, these differential equations reduce to the Bloch equations if reactive effects are neglected. An exact solution of the general equations of motion for the special, simple case of a magnetic TLS being driven by a transverse rotating field of arbitrary frequency is obtained, and deviation from a Lorentzian resonance shape, caused by the frequency dependence of the relaxation and reactive constants, is found to exist.
Previously derived equations for the expectation values of the dynamical variables of a general two-level dipole system (TLS) coupled to a relaxation mechanism---of which the Bloch equations for a spin-\textonehalf{} magneticdipole system are a special case---are reformulated in the density-matrix formalism in order to point out the explicit difference in the relaxation terms between the present description, which includes electric-dipole systems, and that of conventional treatments that are based on the assumption that any TLS is equivalent to a spin-\textonehalf{} magnetic dipole. The original equations are then solved for a weak monochromatic driving field, and the frequency of maximum power absorption is shown to depend on the type of TLS under consideration. The case of a stronger monochromatic driving field and the resulting saturation effects are studied for a special TLS, a spatially linear electric dipole which may have permanent dipole moment. This case exhibits a strong-field resonance shift that depends on the magnitude of the permanent dipole moment. A driving field consisting of a strong component at resonance and a weak component near resonance is considered in connection with the same TLS. The polarization is shown to contain, in lowest order, a component at the difference frequency with an amplitude determined by the magnitude of the permanent dipole moment.
The recently developed technique combining the weak-coupling limit with the Floquet formalism is applied to a model of a two-level atom driven by a strong laser field and weakly coupled to heat baths. First, the case of a single electromagnetic bath at zero temperature is discussed and the formula for resonance fluorescence is derived. The expression describes the well-known Mollow triplet, but its details differ from the standard ones based on additional simplifying assumptions. The second example describes the case of two thermal reservoirs: an electromagnetic one at finite temperature and the second dephasing one, which can be realized as a phononic or buffer gas reservoir. It is shown using the developed thermodynamical approach that the latter system can work in two regimes depending on the detuning sign: a heat pump transporting heat from the dephasing reservoir to an electromagnetic bath or heating both, always at the expense of work supplied by the laser field.