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Physical Review Letters

Two-Level Systems with Relaxation

David E. Rourke, Larisa A. Khodarinova, Alexander A. Karabanov

Published Apr 22, 2004• 18 citations
Publisher

Abstract

A method is presented for the coherent control of two-level systems when T2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, Gamma, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T2. In general, the effect of T2 on Gamma cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.

DOI
10.1103/physrevlett.92.163003

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A method is presented for the coherent control of two-level systems when T2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, Gamma, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T2. In general, the effect of T2 on Gamma cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.
Pulsed radio-frequency transient phenomena of resonant two-level spins are often described by the Bloch equations where the phenomenological dipole (T2{T}_{2}T2​) and population (T1{T}_{1}T1​) decay times appear. A similar description has been carried over recently into the microwave and infrared regions as well. At visible-ultraviolet wavelengths, however, a more general solution is required because the transition levels do not constitute an isolated system and also spontaneous emission can no longer be disregarded. As a consequence, the lower (1) and upper (2) transition levels are characterized by vastly different decay rates (${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$) and the lifetime T1{T}_{1}T1​ is no longer a defined quantity. In view of laser-induced coherent transient phenomena now being observed, we present general solutions of the density-matrix equations of motion for an optically resonant two-level system in contact with a multilevel reservoir where ${\ensuremath{\gamma}}_{1}\ensuremath{\ne}{\ensuremath{\gamma}}_{2}$. Particular and perturbative solutions involving intensity-dependent damping and frequency shifts are included with applications to echo and nutation phenomena.
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 effects of incoherent feeding and decay on a coherently driven two-level system are investigated using density-matrix techniques. A closed-form solution that includes the effects of relaxation and spontaneous emission between the two levels is obtained for cases in which the sample size is both small and large compared to the wavelength of the coherent field. The formalism developed herein is somewhat more general than the original approach of Bloch, Bloembergen, Purcell, Pound, and Redfield, insofar as it is applicable to cases where the trace of density matrix is not constant in time. As such, several new phenomena are predicted. Specifically, the steady-state solutions reveal that under appropriate conditions a significant off-resonance coherent component can be maintained indefinitely by the combination of driving field and incoherent feeding. We term this effect "kinetic coherence" and develop its formalism for both optical and radio-frequency fields.
Density matrix equations for the relaxation of a two-level system are studied. Standard dogma, Γ ik ≥ (γi + γk)/2, about the relaxation constant for the off-diagonal Γ ik ) and diagonal (γi) matrix elements is verified for the case of up- and down-transitions with high rates w 21 and w 12. Non-negativity of the density matrix imposes much weaker limitation, Γ21 ≥ (½)[(w 21)½ - (w 12)½]2. An almost ‘physical’ system is suggested, which describes to a certain approximation such decrease of transverse relaxation constant Γ.
Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes, and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled-harmonic-oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times (T1 and T2) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories.
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.