Photon echoes using double-resonance optical pulses

Ning Lu; P. R. Berman

doi:10.1364/JOSAB.2.001883

Journal of the Optical Society of America B
Vol. 2,
Issue 12,
pp. 1883-1893
(1985)
•https://doi.org/10.1364/JOSAB.2.001883

Photon echoes using double-resonance optical pulses

Ning Lu and P. R. Berman

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Ning Lu and P. R. Berman, "Photon echoes using double-resonance optical pulses," J. Opt. Soc. Am. B 2, 1883-1893 (1985)

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Abstract

A theory of photon echoes using double-resonance optical pulses (PEDROP) is presented. Two double-resonance pulses, separated by a time interval T, are used to cause transitions among three states of an atom. Each double-resonance pulse consists of two simultaneously applied laser pulses that drive coupled transitions in the atom. It is shown that for time t > T, six photon echoes are produced (three on each of the coupled transitions) at five distinct times. Moreover, a nonradiating macroscopic coherence is also produced between states of the same parity. With the use of an interrogation pulse to monitor this coherence, three additional photon echoes can be produced on either of the coupled transitions. A Doppler phase diagram is constructed that enables one easily to predict the times and conditions of echo formation. PEDROP combines features of both two-photon and trilevel echoes but offers advantages over these methods for studying relaxation phenomena.

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Term n	${\tilde{ρ}}_{l m}$ (0 → T)	${\tilde{ρ}}_{j k}$ (T → t)	Echo-Formation Time t_n	Echo-Formation Condition
1	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{23}$	2T
2	${\hat{ρ}}_{21}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1}}{k_{2}} \|) T$	k₁k₂ > 0
3	${\tilde{ρ}}_{12}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1}}{k_{2}} \|) T$	k₁k₂ < 0
4	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ < Ω₂
5	${\tilde{ρ}}_{13}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{2}} \|) T$	k₁k₂ < 0, Ω₁ > Ω₂
6	${\tilde{ρ}}_{21}$	${\tilde{ρ}}_{12}$	2T
7	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{2}}{k_{1}} \|) T$	k₁k₂ > 0
8	${\tilde{ρ}}_{23}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{2}}{k_{1}} \|) T$	k₁k₂<0
9	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{1}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ > Ω₂
10	${\tilde{ρ}}_{13}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{1}} \|) T$	k₁k₂ < 0, Ω₁ < Ω₂
11	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{13}$	2T
12	${\tilde{ρ}}_{21}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{1}}{k_{1} + k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ > Ω₂
13	${\tilde{ρ}}_{12}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{1}}{k_{1} + k_{2}} \|) T$	k₁k₂ < 0, Ω₁ < Ω₂
14	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{2}}{k_{1} + k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ < Ω₂
15	${\tilde{ρ}}_{23}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{2}}{k_{1} + k_{2}} \|) T$	k₁k₂ < 0, Ω₁ > Ω₂

Term n	Maximum Value of \| ${\tilde{ρ}}_{l m}$ (0⁺)\|	Choice of Angles Leading to the Maximum Value		Maximum Value of \|a_jla_km*\|	Choice of Angles Leading to the Maximum Value

		Φ₁	θ₁		Φ₂	θ₂
1	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1	2π ± π	$\frac{π}{2}$
2	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/8	$2 π \pm \frac{4 π}{3}$	$\frac{π}{4}$
3	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/2	2π ± π	$\frac{π}{4}$
4	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1/2	$2 π \pm \frac{2 π}{3}$	$arccos (\frac{1}{\sqrt{3}})$
5	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π ± π	0
6	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1	2π ± π	0
7	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/8	$2 π \pm \frac{4 π}{3}$	$\frac{π}{4}$
8	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/2	2π ± π	$\frac{π}{4}$
9	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$
10	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π ± π	$\frac{π}{2}$
11	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π	$\frac{π}{4}$
12	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$
13	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1	2π	$\frac{π}{2}$
14	1/2-	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/2	$2 π \pm \frac{2 π}{3}$	$arccos (\frac{1}{\sqrt{3}})$
15	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1	2π ± π	0

Term n	First Double-Resonance Pulse	Second Double-Resonance Pulse	Sign of k₁k₂ and Ω₁ − Ω₂	Notes
1	$cos \frac{Φ_{1}}{2} = - {tan}^{2} θ_{1}$	$θ_{2} = \frac{π}{2}$	–	^b
2	θ₁ = 0	Φ₂ = 2π	–	–
3	θ₁ = 0	–	–	^b,^c
4	Φ₁ = 2π	$cos \frac{Φ_{2}}{2} = - {tan}^{2} θ_{2}$	k₁k₂ < 0, Ω₁ < Ω₂	–
5	Φ₁ = 2π	θ₂ = 0	–	^b
6	θ₁ = 0	θ₂ = 0	–	^b
7	$cos \frac{Φ_{1}}{2} = - {tan}^{2} θ_{1}$	Φ₂ = 2π	–	–
8	$cos \frac{Φ_{1}}{2} = - {tan}^{2} θ_{1}$	–	–	^b,^d
9	Φ₁ = 2π	$cos \frac{Φ_{2}}{2} = - {cot}^{2} θ_{2}$	k₁k₂ < 0, Ω₁ > Ω₂
10	Φ₁ = 2π	$θ_{2} = \frac{π}{2}$	–	^b
11	Φ₁ = 2π	Φ₂ = 2π;	–	^b
12	θ₁ = 0	Φ₂ = π	k₁k₂ > 0	^c
13	θ₁ = 0	$θ_{2} = \frac{π}{2}$	-	^b
14	$cos \frac{Φ_{1}}{2} = - {tan}^{2} θ_{1}$	Φ₂ = π	k₁k₂ > 0	^d
15	$cos \frac{Φ_{1}}{2} = - {tan}^{2} θ_{1}$	θ₂ = 0	–	^b

Term n	${\tilde{ρ}}_{l m}$ (0 → T)	${\tilde{ρ}}_{j k}$ (T → t)	Echo-Formation Time t_n	Echo-Formation Condition
1	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{23}$	2T
2	${\hat{ρ}}_{21}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1}}{k_{2}} \|) T$	k₁k₂ > 0
3	${\tilde{ρ}}_{12}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1}}{k_{2}} \|) T$	k₁k₂ < 0
4	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ < Ω₂
5	${\tilde{ρ}}_{13}$	${\tilde{ρ}}_{23}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{2}} \|) T$	k₁k₂ < 0, Ω₁ > Ω₂
6	${\tilde{ρ}}_{21}$	${\tilde{ρ}}_{12}$	2T
7	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{2}}{k_{1}} \|) T$	k₁k₂ > 0
8	${\tilde{ρ}}_{23}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{2}}{k_{1}} \|) T$	k₁k₂<0
9	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{1}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ > Ω₂
10	${\tilde{ρ}}_{13}$	${\tilde{ρ}}_{12}$	$(1 + \| \frac{k_{1} + k_{2}}{k_{1}} \|) T$	k₁k₂ < 0, Ω₁ < Ω₂
11	${\tilde{ρ}}_{31}$	${\tilde{ρ}}_{13}$	2T
12	${\tilde{ρ}}_{21}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{1}}{k_{1} + k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ > Ω₂
13	${\tilde{ρ}}_{12}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{1}}{k_{1} + k_{2}} \|) T$	k₁k₂ < 0, Ω₁ < Ω₂
14	${\tilde{ρ}}_{32}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{2}}{k_{1} + k_{2}} \|) T$	k₁k₂ > 0 or k₁k₂ < 0, Ω₁ < Ω₂
15	${\tilde{ρ}}_{23}$	${\tilde{ρ}}_{13}$	$(1 + \| \frac{k_{2}}{k_{1} + k_{2}} \|) T$	k₁k₂ < 0, Ω₁ > Ω₂

Term n	Maximum Value of \| ${\tilde{ρ}}_{l m}$ (0⁺)\|	Choice of Angles Leading to the Maximum Value		Maximum Value of \|a_jla_km*\|	Choice of Angles Leading to the Maximum Value

		Φ₁	θ₁		Φ₂	θ₂
1	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1	2π ± π	$\frac{π}{2}$
2	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/8	$2 π \pm \frac{4 π}{3}$	$\frac{π}{4}$
3	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/2	2π ± π	$\frac{π}{4}$
4	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1/2	$2 π \pm \frac{2 π}{3}$	$arccos (\frac{1}{\sqrt{3}})$
5	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π ± π	0
6	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1	2π ± π	0
7	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/8	$2 π \pm \frac{4 π}{3}$	$\frac{π}{4}$
8	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/2	2π ± π	$\frac{π}{4}$
9	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$
10	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π ± π	$\frac{π}{2}$
11	1/2	2π	$\frac{π}{8}, \frac{3 π}{8}$	1	2π	$\frac{π}{4}$
12	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$
13	1/2	$2 π \pm \frac{π}{2}, 2 π \pm \frac{3 π}{2}$	0	1	2π	$\frac{π}{2}$
14	1/2-	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1/2	$2 π \pm \frac{2 π}{3}$	$arccos (\frac{1}{\sqrt{3}})$
15	1/2	$2 π \pm \frac{2 π}{3}$	$arcsin (\frac{1}{\sqrt{3}})$	1	2π ± π	0

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Journal of the Optical Society of America B