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Physical Review

Dissipation in Quantum Mechanics. Two-Level System. III

I. R. Senitzky

Published Mar 15, 1965• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.137.a1635

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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.
The study of a two-level system (TLS) coupled to a loss mechanism (LM), the fluctuations of which are taken into account, is motivated by relating the problem to spin-lattice relaxation, radiation damping, and Brownian motion. A TLS of the electric dipole type (coupled to the LM through one variable only) is discussed first. The problem is formulated in terms of the Pauli spin matrices, and the Langevin equations for Brownian motion of a TLS are derived. Their solution---made possible by taking expectation values in LM space---contains the Weisskopf-Wigner exponential decay formula of radiation theory, as well as a second-order shift in frequency of the TLS produced by the (nondispersive) LM. A driving force is added to the problem and expressions are obtained for a driven lossy TLS of the electric dipole type. The same analysis is applied to a TLS of the magnetic dipole type and differential equations for the spin matrices---for which the expectation values in LM space has been taken---are obtained. Taking expectation values in TLS space converts these equations into the Bloch equations. Unlike the electric dipole case, no frequency shift is present.
Previously derived equations of motion---in terms of the expectation values of the Pauli spin matrices---for a two-level system (TLS) coupled to a loss mechanism (LM) are generalized to include higher order interactions which describe the reaction of the LM to the effect of the driving field on the TLS. This generalization results in the addition of a driving term to the equations of motion which is approximately independent of the state of the TLS and involves parameters of the LM that characterize its response at the difference between the TLS frequency $\ensuremath{\omega}$ and the driving-field frequency $\ensuremath{\nu}$. The new driving term is negligible unless the TLS possesses a permanent dipole moment, and, in the case of a magnetic TLS, unless the driving field has a transverse component. A solution of the equations of motion for a magnetic TLS being driven by a transverse rotating field shows that amplification of the driving field may be obtained without population inversion of the TLS provided $\ensuremath{\nu}<\ensuremath{\omega}$, provided the loss constant $\ensuremath{\eta}$ (which is a measure of the response of the LM) has a frequency dependence such that $\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})$ is significant compared to $\ensuremath{\eta}(\ensuremath{\nu})$, and provided the TLS temperature exceeds the LM temperature by an appropriate amount that is minimized when $\frac{\ensuremath{\eta}(\ensuremath{\nu})}{\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$ is a minimum. If $\frac{\ensuremath{\eta}(\ensuremath{\nu})}{\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$ is small compared with unity, amplification is obtained when the TLS temperature is larger than the LM temperature by a factor $\frac{\ensuremath{\omega}}{(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$. The induced emission power is discussed.
The theory of a previous article, dealing with two types of two-level systems coupled to a loss mechanism (LM), is extended. The first extension consists of the consideration of the most general type of two-level system (TLS), in which the dipole moment is expanded in terms of the three Pauli spin matrices and unit matrix, the expansion coefficients being vectors (dipole vectors). The second extension consists of the addition to the thermal-reservoir type of LM of a large number of systems identical to the TLS under consideration. The TLS is described in terms of the time development of the Pauli matrices and differential equations are obtained for their expectation value in the presence of arbitrary driving fields. The Bloch equations for a magnetic dipole of spin \textonehalf{} are exhibited as a special case of these equations, corresponding to a particular combination of the dipole vectors. All other combinations describe electric dipole systems. Equations for two simple special cases of such systems are presented, one treated in the previous article and the other having permanent dipole moment. The frequency of oscillation of a freely decaying TLS is derived and shown to be shifted by an amount that depends on the relationship between the dipole vectors. It is pointed out that the commonly held belief that any TLS can be represented as a magnetic dipole of spin \textonehalf{} is only approximately correct in the presence of dissipation. The conditions under which the differential equations for the expectation values of the dynamical variables of the TLS can be converted into differential equations for macroscopic variables are discussed.
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 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.
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.