Optical Bloch equations: Informal motivation without the Schrödinger equation

Richard P.

Feynman

,

Frank L.

Vernon

, Jr., and

Robert W.

Hellwarth

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Pierre Meystre and Murray Sargent III, Elements of Quantum Optics (Springer, New York, 1999), 3rd ed.

Peter W. Milonni and Joseph H. Eberly, Lasers (Wiley, New York, 1988), especially Chaps. 6 and 8.

Bruce W. Shore, The Theory of Coherent Atomic Excitation, Simple Atoms and Fields (Wiley, New York, 1990), Vol. 1, especially Chap. 8.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).

Marc D. Levenson and Satoru S. Kano, Introduction to Nonlinear Laser Spectroscopy, Revised edition (Academic, New York, 1988), pp. 29–47.

Discussed in many optics books. See, for example, Eugene Hecht, Optics (Addison-Wesley, New York, 1998), 3rd ed., Chap. 3.

Reference 3, pp. 27–39.

David J. Griffiths, Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, NJ, 1999), 3rd ed., pp. 398–405.

S. G. Lipson, H. Lipson, and D. S. Tannhauser, Optical Physics (Cambridge University Press, New York, 1995), 3rd ed., pp. 383–394.

A spring might seem like a contrived visual image; it’s just a way to picture oscillations about a stable equilibrium.

Jerry B. Marion and Stephen T. Thornton, Classical Dynamics of Particles and Systems (Saunders/Harcourt Brace, New York, 1995), 4th ed., Chap. 3.

Keith R. Symon, Mechanics, 3rd ed. (Addison-Wesley, Reading, MA, 1971), Chap. 2.

S. C. Bloch, Introduction to Classical and Quantum Harmonic Oscillators (Wiley, New York, 1997).

See, e.g., Refs. 12,13 14.

Reference 2, pp. 68–69.

Reference 4, p. 423.

H. Haken, Light, Vol. 1, Waves, Photons, Atoms (North-Holland, New York, 1981), pp. 14–15, 133–135.

Reference 10, pp. 419–422.

For a discussion of phasors see, for example, Ref. 7 or Ferrel G. Stremler, Introduction to Communication Systems (Addison-Wesley, New York, 1990), 3rd ed.

Mary B.

James

and

David J.

Griffiths

, “,” (), – ().

Joshua B.

Diamond

, “,” (), – ().

Bruce A.

Sherwood

, “,” (), – ().

P. W.

Milonni

, “,” (), ().

Bart G.

de Grooth

, “,” (), – ().

Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1963), Vol. I, Chap. 31.

Reference 7, pp. 91–95.

Reference 10, pp. 389–393.

Reference 5, pp. 5–7. We use different normalization for U and $V.$

One check, for example, is that differentiating $A=(U2+V2)1/2$ and then using Eqs. (10) and (11) with $Fo=0$ give $dA/dt=−βA$ as required for exponential decay.

Over any whole number of cycles, $cos2(ωt)¯=1/2$ and $cos(ωt)×sin(ωt)¯=0.$

Too much should not be read into the fact that we use the same symbols (e.g., β) in both the classical and semiclassical cases. There are no claims here about assigning a numerical value to β. By extension, there are no claims about comparing values of parameters between the two theories. We keep the same symbols only to facilitate comparing the forms of various equations.

The length squared of the Bloch vector is $L2=u2+v2+w2.$ The time rate of change of $L2$ is $(d/dt)L2=2(uu̇+vv̇+wẇ).$ Using Eqs. (25), this becomes $(d/dt)L2=−2β(L2+w2+2w).$ This last form shows that if β=0, an atom starting in the ground state with $L=1$ will always have $L=1.$ Hence, when $w=±1,$ putting the atom completely in the upper or lower eigenstate, the atom has no dipole moment, as required. (Additionally, $L=1$ means that $u2+v2$ can never exceed 1, showing that the dipole moment for this quantum transition has an upper limit.) Now, for cases with β≠0, $(d/dt)L2=−2β(L2+w2+2w)$ can be either positive or negative. But it cannot be positive if $L=1,$ because $w2+2w$ can never be less than −1. This means that L can never grow beyond 1. So whether β is zero or not, the length of the Bloch vector cannot exceed 1, assuring that there is no dipole moment whenever $w=±1.$ (This endnote addresses only the form of the equations, without bringing to bear additional information that the physical interpretation of the equations would allow.)

They are the optical Bloch equations in the dipole approximation and rotating wave approximation.