Jump to ContentJump to Main Navigation
Lectures on LightNonlinear and Quantum Optics using the Density Matrix$

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

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2016. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 05 December 2016

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

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

Source:
Lectures on Light
Author(s):

Stephen C. Rand

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)
ρij(t)=ρ˜ij(t)exp(iωt),

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)
ρ˙ij=ρ˜˙ijexp(iωt)+iωρ˜ijexp(iωt).

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˜˙ij0, and the equation of motion must be integrated to find a solution, as follows.

Take the interaction to be

(G.3)
Vij(t)=12V˜ij(t)exp(iωt)+V˜ij(t)exp(iωt).

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

(G.4)
ρ˜˙12=iΔρ˜12+V˜12/2i(ρ22ρ11)Γ12ρ˜12,

where Δω0ω. Next, use is made of an integrating factor through the substitution of

(G.5)
ρ˜12=ρ˜12exp[(iΔΓ12)t].

This yields

ρ˜˙12exp[(iΔΓ12)t]+(iΔΓ12)ρ˜12exp[(iΔΓ12)t]==(iΔΓ12)ρ˜12exp[(iΔΓ12)t]+V˜12/2i(ρ22ρ11),

(p.354) or

(G.6)
ρ˜˙12=V˜12/2i(ρ22ρ11)exp[(iΔΓ12)t],

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

(G.7)
ρ˜12=i20tV˜12(t)(ρ22(t)ρ11(t))exp(iΔΓ12)tdt.

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)
tV˜12(t)ρ22(t)ρ11(t)exp(iΔΓ12)tdt=V˜12(t)(ρ22(t)ρ11(t))exp(iΔΓ12)t(iΔΓ12)ttddtV˜12(t)(ρ22(t)ρ11(t))exp(iΔΓ12)t(iΔΓ12)dt.

Under steady-state conditions (ρ˙11=ρ˙22=0), assuming the interaction is turned on adiabatically at early times so that dV˜12(t)/dt0, the second integral on the right is zero. One therefore finds the result

(G.9)
i2tV˜12(t)(ρ22(t)ρ11(t))exp(iΔΓ12)tdt=i2V˜12(t)exp[(iΔΓ12)t](iΔ+Γ12)(ρ11(t)ρ22(t))

and

(G.10)
ρ˜12(t)=iV˜12(t)2(iΔ+Γ12)(ρ11(t)ρ22(t))exp[(iΔΓ12)t].

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

(G.11)
ρ˜12=V˜122(Δ+iΓ12)(ρ11ρ22)=(Ω12/2)(Δ+iΓ12)(ρ11ρ22),

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

(G.12)
ρ12(t)=Ω12/2Δ+iΓ12(ρ11(t)ρ22(t))exp(iωt).

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)/dt0. 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.