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## Stephen C. Rand

Print publication date: 2016

Print ISBN-13: 9780198757450

Published to Oxford Scholarship Online: August 2016

DOI: 10.1093/acprof:oso/9780198757450.001.0001

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# (p.353) Appendix G Solving for Off-Diagonal Density Matrix Elements

Source:
Lectures on Light
Publisher:
Oxford University Press

Off-diagonal elements of the density matrix describe charge oscillations initiated by applied fields. Hence the full temporal evolution of their amplitudes reflects transient build-up (or decay) of an oscillation prior to the establishment of any steady-state amplitude. The frequency of the oscillation is $ω$, as determined by the driving field. Hence the general form of the solution is

(G.1)
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where i, j specify the initial and final states of the transition closest in frequency to $ω$ and $ρ˜ij$ is the slowly varying amplitude of the polarization.

The time derivative of $ρij$ is

(G.2)
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If the assumption is made that $ρ˜˙ij=0$, then substitution of Eq. (G.2) into the equation of motion yields an algebraic equation that may be readily solved to find the steady-state solution. More generally $ρ˜˙ij≠0$, and the equation of motion must be integrated to find a solution, as follows.

Take the interaction to be

(G.3)
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Substituting Eqs. (G.3) and (G.2) into the equation of motion for a two-level system, as an example, one finds

(G.4)
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where $Δ≡ω0−ω$. Next, use is made of an integrating factor through the substitution of

(G.5)
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This yields

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(p.354) or

(G.6)
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The next step is to integrate over time between appropriate limits, for example, 0 and t. One finds

(G.7)
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If the interaction starts at time zero, as assumed in Eq. (G.7), then the integral must be performed by taking the transient buildup of the envelope function $V˜12(t′)$ into account. Since $V˜12(t′)$ itself cannot be approximated by a constant over the interval of integration, it cannot be removed from the integral. The steady-state solution, identical to that obtained by solving Eq. (G.4) algebraically with the assumption $ρ˜˙12=0$, may be found directly after setting the lower limit of integration equal to $t′=−∞$ and integrating by parts:

(G.8)
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Under steady-state conditions ($ρ˙11=ρ˙22=0)$, assuming the interaction is turned on adiabatically at early times so that $dV˜12(t′)/dt′≅0$, the second integral on the right is zero. One therefore finds the result

(G.9)
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and

(G.10)
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Using Eq. (G.10) in Eq. (G.5), the steady-state value for the slowly varying polarization amplitude is found to be

(G.11)
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(p.355) and the final result for the off-diagonal coherence (polarization) is therefore

(G.12)
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This result is the same as that obtained by substituting Eq. (G.2) into the equation of motion for $ρ12$ after setting $ρ˜˙12=0$. Determinations of $ρ12$ that include transient behavior associated with a rapid onset of the interaction must include an evaluation of the second term on the right of Eq. (G.8), since then $dV˜12(t′)/dt′≠0$. Similarly, population oscillations or rapid population dynamics require a full evaluation of Eq. (G.8). A useful alternative is to assume the interaction appears instantaneously at time $t=0$, and to evaluate the integral using Eq. (G.7). However, for steady-state conditions we note that the result in Eq. (G.12) agrees with the result obtained in Chapter 5 using the simple substitutions $ρ12(t)=ρ˜12exp(iωt)$ and $ρ˜˙12=0$.