Time-varying induced coherence

Abstract

A previously reported interference experiment, involving two parametric downconverters with their idlers aligned, is reanalyzed with a time-dependent modulator introduced between the two downconverters. This change modifies the analysis nontrivially. It is found that the visibility of the interference effect is completely determined by the state of the modulator at one moment on the light cone backward from the detected photon, even though no photon may be passing the modulator at that time. The implication for a delayed-choice experiment is discussed briefly.

Full text

Vol. 9, No. 4/April 1992/J. Opt. Soc. Am. B 605
Time-varying induced coherence
L. J. Wang, X. Y. Zou, and L. Mandel
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627
Received July 23, 1991; revised manuscript received October 18, 1991
A previously reported interference experiment, involving two parametric downconverters with their idlers
aligned, is reanalyzed with a time-dependent modulator introduced between the two downconverters. This
change modifies the analysis nontrivially. It is found that the visibility of the interference effect is completely
determined by the state of the modulator at one moment on the light cone backward from the detected photon,
even though no photon may be passing the modulator at that time. The implication for a delayed-choice
experi-
ment is discussed briefly.
INTRODUCTION
In the early days of quantum mechanics the interpreta-
tion of the Young interference experiment and its electron
equivalent was the subject of considerable discussion.
These discussions led to the general conclusion that any
attempt to determine which slit the particle passes
through, i.e., to identify the path, introduces an uncontrol-
lable disturbance that effectively eliminates the interfer-
ence. The nature of the disturbance was supposed to
depend on the experiment, but it was assumed to be some
causal, physical mechanism.
In recent papers Scully and co-workers have challenged
this conclusion. 4They discussed a two-beam interfer-
ence experiment involving a micromaser; from this experi-
ment it appears that interference can be destroyed without
any need for a significant physical disturbance to come
into play.
4The availability of information about the
photon path is sufficient to eliminate the interference.
Similar conclusions were drawn from recent interfer-
ence experiments involving two parametric downconvert-
ers and a variable-transmissivity filter.5 7 In some ways
the effect is even more striking in this case because of the
evident absence of any direct disturbance acting on the
interfering photons. The experiment discussed below is a
proposed time-dependent modification of an interference
experiment that we previously reported,5 7 which is some-
what reminiscent of so-called delayed-choice experi-
ments.8 9 We conclude that, with the introduction of a
time-dependent switch or modulator, the observed inter-
ference depends on the state of the switch at one critical
moment on the light cone backward from the detected
photon.
REVIEW OF PREVIOUS TIME-INDEPENDENT
INTERFERENCE EXPERIMENTS
We first review the essential features and the theoretical
description of the previous experiments.6'7Consider the
experimental situation illustrated in Fig. 1. Two similar
crystals, NL1 and NL2, centered at positions r, and r2,
have x(2) nonlinear susceptibilities. They are optically
pumped by (classical) mutually coherent light waves of
amplitudes V1
(t) and V
2(t) and center frequency wo
0, with
I1(t)1
2= Ij(t) (j = 1, 2) in units of photons per second. As
a result of the parametric interaction between the fields
and the crystals, downconverted signal (s) and idler (i)
fields are generated from the pump; these propagate in
different directions and have to be treated quantum
mechanically. The signal fields s1and S2 are brought
together and are allowed to interfere, while the two idler
fields il and i2are aligned and made copropagating.
After Fourier decomposition of the fields into discrete
modes of frequency spacing bo, the unitary time evolution
operator U(t, t -t1) representing the interaction between
the light and the crystals from time t - t to time t has
the form7
U(t, t -t) = exp
[[t HI (t')dt']
ih t -t,
(8w )3/2
= exp (27r)1/2 'l
x exp[i(ki - kl'
- ki") r]
(01 (0'1"a
sin(wi' + to," - o9l)tl/2
X 41(° 1SQ)
1
, ; w) (w01'
+ C 1w"
-&)/2
x exp[i(w
1' + wi -wi)(t - t/2)]
x v1(&w
1
)eL
0t(w')a
1il(w)1) + (2)1/2 72
x _ _ _ exp[i(k 2- k2' - k2") r2]
(2 2' 2'
sin(w)2' + w0
2" -(2)tl/2
x 02(2w2".;O
2) (w
2' + w2"-02)/2
x exp[i(w0
2' + &w
2" -2) (t -t4/2)]
x v2(w2)as2(s2')ai2'(0°2") -h.c.} (1)
t, is the interaction time, which may be taken to be much
longer than the coherence time TDC of the downconverted
light but much shorter than the mean time interval be-
tween downconversions. &aj and &ai are photon annihila-
tion operators for signal and idler photons from crystal j
(j = 1, 2). 171j
2is the fraction of incident photons that
0740-3224/92/040605-05$05.00 © 1992 Optical Society of America
Wang et al.