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On enhancing spectral resolution via correlated spontaneous emission
cm .__
1 March 1997
gi I!/7
OPTICS COMMUNICATIONS
ELSEVIER
Optics Communications 136 ( 1997) 39-43
On enhancing spectral resolution via correlated spontaneous emission Marlan 0. Scully a3b*c, Ulrich W. Rathe apc,d,Chang Su e, Girish S. Agarwal f a Department of Physics, Texas A&M University, College Station, TX 77843, USA ’ Texas Laser Laboratory, HARC, The Woodlands, TX 77381, USA ’ Max-Plan&Institut fcr Quantenoptik, 85748 Garching, Germany ’ Sektion Physik, Ludwig-Mwtimilians-Uniuersit~t, 80333 M&hen, Germany e Department of Physicsand Astronomy, Hunter College of the CUNY, NY 10021, USA ’ Physical Research Laboratory, Navrangpura, Ahmedabad 380009, In&u
Received 19 March 1996; revised version received 20 September
1996; accepted 23 September
1996
Abstract Correlated spontaneous emission from a single three-level atom in a cascade configuration can yield subnatural line narrowing beyond the usual radiative broadening. The effect is observed in the second-order coherence properties of the radiated field.
In a recent Letter [I], we have demonstrated the possibility for a line narrowing in spontaneously emitted radiation via two-photon correlation measurements on atomic cascade radiation. The interferometer used there involved two atoms held in a trap. The purpose of the present article is to demonstrate the effect in an experimentally more accessible setup involving only one atom. Moreover, we here supply a more detailed discussion of the physics involved. Let us start out by noting that the width of an optical transition between two atomic states 1a> and 1b), as measured by conventional spectroscopic means, is gov-
’ Considerable effort has been invested in measurement schemes that potentially resolve spectral lines beyond the “natural” width via “time delay spectroscopy.” For work along these lines, see: Ref. 121. ’ Another approach to sharpening radiative lines using strong dressing fields was presented or demonstrated in Ref. [3]. 3 Lundeen and Pipkin used the separated oscillatory field technique for obtaining subnatural linewidths in measurements of the Lamb shift in H, see Ref. 141. 003s4018/97/$17.00 Copyright PII SOO30-4018(96)00626-S
0 1997 Published
erned by the sum of the radiative decay rates, yU + y,,, out of the two levels in question (see, however, footnotes ‘-3 and references contained therein). It is to be noted that in such conventional measurement schemes, we are dealing with ordinary single-photon events, i.e., single-photon emission and detection. However, as we show in the following, the situation is radically changed when we instead consider two-photon correlated spontaneous emission radiation together with a two-photon (second-order correlation) detection scheme as in Fig. 1. In such a case we find that the second-order correlation function displays spectral features of width y, independent of yhr where y, is the decay rate out of state 1a> into 1b) and yb is the decay rate out of 1b) into 1c). Specifically, we find that the joint probability for registering a count in both detectors D, and D, of Fig. 1 is given by
+ 2e- 2W!&!?]
.
(1)
n
This is the main result of the present paper, and is further elucidated in Fig. 2. In Eq. (l), K is an uninteresting
by Elsevier Science B.V. All rights reserved.
40
M.O. Scully et al./ Optics Communications 136 (1997) 39-43
x II,, l,>,
(2)
where the I a) + 16) (1 b) + I c>) transition involves wave vector k(q), and g,,, (gb,J is the atom-field coupling constant for the I a> -+ I b) ( I b) -+ I c>> transition whose frequency is given by w,~ (w&. We proceed to sketch the calculation for the joint count probability that both detectors at R, and R, (with I R, I = 1R, 1) register a count. This is given by
Fig. 1. An excited three-level atom decays, producing “ + “- and “ - “-polarized photons. Radiation travels through birefringent material to detectors, producing an interference pattern in the two-photon coincidence probability.
P(2)=/om
d$
dt, G(*‘(l,
2),
where the Glauber intensity correlation function G(*)( 1, 2) = G(*)(R,, rl; R,, t2) [14] may be written as GC2’(1, 2) = ‘PC2’(1, 2)*‘PC2’(1,
2)
(4)
with !P(l,
with the inconstant, A = oab - wbc, 7= r(n+- n->/c, dices of refraction n, for radiation from the upper and lower transition, respectively, r is the distance from the atom to the detectors, and ce is the vacuum speed of light; fir, A) is an interference function whose form need not concern us now but is given by Eq. (10). It is the presence of the sharp Lorentzian, of width my,, which interests us here since one normally (i.e., in uncorrelated spontaneous emission) expects a width x + yb. In Fig. 2, we plot the signal AT, A)/( A2 + 7,‘) for different r to demonstrate the line narrowing. For example, the dashed curve corresponding to r = 2 y; ’ shows a width of approximately yay,, independent of yb. A motivation for the present studies derives from the recent beautiful experiments of the NIST group investigating the interference pattern produced by two single atoms [S] along the lines of our earlier “quantum eraser” studies [6]. Another motivation is closely connected with recent work on two-photon down conversion [7-l 11. We now turn to the analysis behind Eq. (1). The type of experiment we have in mind involves the excitation of the atom located at the origin at time t = 0, and the field is in the vacuum state 10) (see Fig. 1). The initial atom-field state may thus be written as 1lu( t = 0)) = 1a) @ IO). For times t 2+ y; ‘, yb ’ the atom decays to 1c) through the intermediate level I b), and the system state vector evolves into I?P(t -+ m)) = I c) @ I I), 4). Here, I +, 4) is the resulting two-photon field state, where $(+) is associated with the 1a) --) I b) (I b) -+ I c)) transition. For the time being, we consider the birefringent material between the atom and the detectors absent. Within the framework of Weisskopf-Wigner theory, we then obtain the expansion of the field state in terms of plane wave states [12]
2) = (0 I3+‘(1)~‘+‘(2)]w).
(9
z(‘)(j) is the usual positive frequency field operator CkZ?& exp[i(k . R, - cktj)] for j = 1, 2. Inserting the state IV) = I $, $> from Eq. (2) into (5) we find q(*)(l,
2) = Ke-iwOb(rl-r/c@(tr
_ r/c>
X eiob=(‘~-r/c)O [ ( t2 - r/c)
,-r,(tl-r/cf - (t, - r/c)]
Xe-Yb[(‘2~‘/C)-(fi-T/C)1 +(1++
2).
(6)
In Eq. (6), I= jRll = ]R,]isthedistancefromtheatom to the detectors, K is an uninteresting constant, and 8(x> is the usual step function. 1
30.5 El ‘B
0
Fig. 2. Signal part fir, A)/(A’ + y,‘> in the joint count probability for different T. The different curves have been normalized for simplicity.
4 If one chooses to formulate the problem in terms of circularly polarized photons, one would have to take into account the fact that a transition emitting right circularly radiation to detector D, emits left circularly radiation to detector D,.
41
M.O. Scully et al./ Optics Communicutions 136 (1997) 39-43 We now take the presence of the birefringent medium into account, assuming a refractive index n, for “ + “polarized $-radiation and n_ for CC- “-polarized 4 (pradiation (n+> n_>. This leads to a modification of the respective retardations in Eq. (6), !PC2)( 1, 2) = Ke- iw,b(rl-r/c+)fj(tl Xei%(‘2-‘/c-@[([z -(t,
-m+>l
+(1
*2>,
c C-
2
(a)
_ I/c+)e-~J’~-r/~+) - T/c_)
W(2,l) (7)
where c+= co/n+ and c_ = co/n_. At this point it is useful to note that the times t; r/c+= TV* (i= 1, 2) are the times when a “+“or “ - “-polarized photon, that is to be detected later by the ith detector, is emitted by the atom. For example, the first term in Eq. (7) corresponds to the emission of a w,~ photon at 7: and subsequent detection of this photon at t, = 5-T+ r/c+ by photodetector D,. Thus the @(S-T> function in Eq. (7) ensures that the o,,, photon is emitted only for positive Q-T, i.e., only after the atom has been excited, and the term exp(- -y,r:> ensures that the emission event will take place in a time T: N l/y, after the excitation of the atom. Likewise, the @(T; - 7:) in Eq. (7) says that the o ,,= photon triggers a count in detector D, only when T; > Q-T, thus requiring that the second (obC> photon is emitted after emission of the w,~ photon, and the term exp[ - yb(r; - r:)] ensures that the whC emission event will take place within a time 7; - 7: l/ yb. The notation (1 @ 2) in Eq. (7) denotes that the w,~ photon now goes to detector D, and the obC photon to detector D ]. Now Eq. (7) can be rewritten as @)(l,
%c -----
1 -3
Fig. 3. Key contributions to coincidence event of detector D, at time t, and detector D, at time t2. Photons are emitted by tbe atom at the retarded times TV*. Lines from atom to detectors represent optical patblengths.
We insert Eq. (7) into Eq. (4) and carry out the integrations of Eq. (3). After a somewhat lengthy calculation, we find PC2)(1, 2) as given by Eq. (1) with f(r,
A) = 1 -e-Y”’
cos AT-
$-sin
AT
u
(10)
In order to understand the line narrowing physics behind the 1/(A2 + -y,‘> term, we explicitly write out the first key term from Eq. (9)
-m
z
L 1 dt,
dt, (-__+_-
0
2
:
x
:---”
1
2) = Ke-i0~,~~;e(~:)e-Yo7;e--iWbc7;
= W)(
=2”---i
1) 2) + @‘2’(2, 1) c----c--., + 2 1 )
(8)
where we introduce the notation @@)(l , 2) to represent amplitude for the CO,,, photon being detected by D, the wbC photon by D,, whereas @*)(2, 1) stands for w,~ photon going to D, and the whr photon going to The diagrammatic representation of these processes established in Fig. 3, the solid (dashed) line denoting w,~ (cIJ~~>photon propagator. The interesting interference terms are included in pC2)(1, 2) =/,I
d’,jo*
dt, Qt2’(1, 2)=@(*)(2,
the and the D,. is the
1) +c.c. (9)
NP
1
A*+ y,”
(1’)
The first factor is the process in Fig. 3(a). In Fig. 4, we depict this case in a space-time diagram with the two solid lines. There, the retarded times T: and T; on the t = 0 axis are the emission times, whose difference 7; - T: is given by r; - 7: = t2 - t, - r(n_ - n+>/c,; but for the second process indicated with dashed lines in Fig. 4, corresponding to Fig. 3(b), 7; - T: = tl - t2 - r(n_n+>/co. From the preceding discussion we see that the time dependence involving y,,t, and y,, 1, cancels out in processes such as Eq. (11). That is, the yh factors in Eq. exp[yb(r2 - 7: > - yb(r; - r: >I = (1 l), exp[ - 2 yhr( n, - n_ j/c,] are time independent. This is
M.O. Scully et al./Optics Communications 136 (1997) 39-43
42
tt
r + Fig. 4. Space-time diagram of the two interfering processes. The two solid lines depict the case of Fig. 3(a), the dashed lines the one of Fig. 3(b).
the key point. It may be restated by noting that the two interfering processes of Fig. 4 spend complementary times in state 1b) and the product of their amplitudes is independent of the difference between detection times t, and t,. Therefore, an integral over tl and t,, yielding the total count probability, will lead to Eq. (11). We can also understand the absence of yb from the Lorentzian denominator from an intuitive “quantum jump’ ’ picture of the cascade decay and subsequent detection. Consider the situation depicted with solid lines in Fig. 4, also shown in Fig. 3(a). By arguing as before that the count of detector D, at time I, corresponds to the emission of an w,,-photon by the atom at the retarded time r: = I, - r/c+, and that the count of D, at tp corresponds to emission of a w,,-photon at a time r; = t, - r/c_, we deduce that the atom spent the time 7; - 7: in the intermediate state 1b). In the spirit of the quantum jump approach the probability amplitude of this process is therefore weighted by a damping term of the form exp[ - ~~(7; - r:)]. In the process with dashed lines in Fig. 4, corresponding to Fig. 3(b), the atom emits the w,,-photon at time rl = tT - r/c+ and the o,,-photon at time r; = t, - r/c_, so that the total time this atom spent in state 1b) is given by 7; - T:. Therefore, the probability amplitude of this process is weighted by exp[ - ~~(7; - T:)]. We see that the product of the two probability amplitudes will be influenced by y,-damping through exp[ - y&r; - 7:) ~~(7; - ?-:>I which is obviously independent of t, and tz. Again, the integral over t, and t2 yielding the total count probability will not have denominators depending on yb but instead produce the Lorentzian in Eq. (1). It is the purpose of the present paper to point out this two-photon line narrowing effect. A possible experiment making use of the sharp Lorentzian in Eq. (1) in order to measure the transition frequencies o,~ and obc can be conceived as follows. We assume 1a> and 1c) to be atomic Sstates, and 1b) to be the m = - 1 sublevel of a P-state with energy obo in the absence of a magnetic field. First we note that the atomic frequency w,, = w,~, + wboc can be measured by, for example, Raman excitation of the
state 1a), in which case ya determines the measurement precision. To get a similarly precise measurement of w,~, - wboc, we could map out the sharp Lorentzian 1/(A2 + y,‘) by varying A = wab - w,,=, e.g., with the help of a variable external magnetic field, and thus provide a good measurement of the magnetic field strength B, for which A = 0 ‘. From Be we obtain o,~, - oboe, and from the Raman spectroscopy w,~, + oboe is known; therefore we have, in principle, determined both w,,,, and o+ to a precision governed by ‘y,. In this context, we stress that the present line narrowing provides a high spectral resolution in the sense that we can separate closely spaced lines according to the Rayleigh criterion. However, the extent to which it leads to a higher experimental accuracy in the final result for the atomic transition frequencies requires further discussion as will be given elsewhere. The reason for this is the exponential damping of the signal with the time delay T by means of the prefactor exp( - 2 ~~7) in Eq. (1) which decreases the visibility of the signal. This trade-off of signal width against signal amplitude is well-known from other methods for subnatural spectroscopy, notably from time delay spectroscopy [2]. Note, however, that although the present scheme is reminiscent of time delay spectroscopy, there exists a major difference between the two approaches. Whereas the signal width in time delay spectroscopy is ultimately limited by the difference of the decay rates, ya - yb, (see Meystre et al. in Ref. [2]) correlated emission spectroscopy is limited by ya only. Therefore, it allows for subnatural spectroscopy in atomic systems where time delay spectroscopy does not produce subnatural linewidths. In conclusion, we emphasize that the underlying physical reason for the interference in Eq. (1) is the correlation involved in the two-photon state produced by the cascade transition as given in Eq. (2). If we had chosen to consider an interrupted cascade transition in the spirit of Ref. [6], the resulting Tc2)(1, 2) for one atom would have had the form qc2’(1, 2) = $,(1)!5(2) + Y$(l)W,(2), and there would have been no subnatural resolution. Finally, we note that the potential application of these considerations to nuclear and molecular spectroscopy, where there is an abundance of oscillator-like states with w ab = Obc’ is also interesting. This work was search, the Welch Research Program. Fry, J. Franson, H.
supported by the Office of Naval ReFoundation, and the Texas Advanced One of us (MOS) wishes to thank E. Pilloff, H. Van Driel, and A. Zeilinger
5 The interference function AT, A) modulating the Lorentzian + y,‘) in Eq. (1) has to be taken into account for this measurement. In this context, the choice of the time delay T is 1/(A*
crucial as seen in Fig. 2.
M.O. Scully et al./Optics
Communications 136 (1997) 39-43
for helpful and stimulating discussions. UR would like to thank the “Studienstiftung des deutschen Volkes” for support.
References [l] U.W. Rathe and M.O. Scully, Len. Math. Phys. 34 (1995) 297. [2] C.S. Wu, Y.K. Lee, N. Benczer-Keller and P. Simms, Phys. Rev. Lett. 5 (1960) 432; I.-J. Ma, J. Mertens, G. zu Pntlitz and G. Schutte, 2. Phys. 208 (1968) 276; G. zu Putlitz, Comm. Atom. Mol. Phys. 1 (1969) 74; G. Copley, B.P. Kibble and G.W. Series, J. Phys. B: Proc. Phys. Sot., London 1 (1968) 724; H. Figger and H. Walther, Z. Phys. 267 (1974) 1; R.C. Hilbom and R.L. deZafra, J. Opt. Sot. Am. 62 (1972) 1492; P. Schenck, R.C. Hilbom and H. Metcalf, Phys. Rev. Lett. 31 (1973) 189; T.S. Luk, L. DiMauro, M. Feldman and H. Metcalf, Phys. Rev. A 24(1981) 864; P. Meystre, M.O. Scully and H. Walther, Optics Comm. 33 (1980) 153. [3] L.M. Narducci, G.-L. Oppo and M.O. Scully, Optics Comm. 75 (1990) 111; L.M. Narducci, M.O. Scully, G.-L. Oppo, P. Ru and J.R. Tredicce, Phys. Rev. A 42 (1990) 1630; D.J. Gauthier, Y. Zhu and T.W. Mossberg, Phys. Rev. Lett. 66 (1991) 2460.
43
[4] S.R. Lundeen and F.M. Pipkin, Phys. Rev. Lett. 34 (1975) 1368; 46 (1981) 232. [S] U. Eichmann, J.C. Bergquist, J.J. Bollmger, J.M. Gilligan, W.M. Itano, D.J. Wineland and M.G. Raizen, Phys. Rev. Lett. 70 (1993) 2359. [6] M.O. Scully and K. Drtihl, Phys. Rev. A 25 (1982) 2208; see also P.G. Kwiat, A.M. Steinberg and R.Y. Chiao, Phys. Rev. A 45 (1992) 7729; P.G. Kwiat, A.M. Steinberg and R.Y. Chiao, Phys. Rev. A 49 (1994) 61. [7] R. Gosh and L. Mandel, Phys. Rev. L&t. 59 (1987) 1903; Z.Y. Ou and L. Mandel, Phys. Rev. Lett. 61 (1988) 50; 54; Z.Y. Ou, L.J. Wang and L. Mandel, Phys. Rev. A 40 (1989) 1428; Z.Y. Ou, X.Y. Zou, L.J. Wang and L. Mandel, Phys. Rev. Lett. 65 (1990) 321; X.Y. Zou, T.P. Grayson and L. Mandel, Phys. Rev. Lctt. 69 (1992) 3041. [8] Y.H. Shih and C.O. Alley, Phys. Rev. Lett. 61 (1988) 2921. [9] M.A. Home, A. Shimony and A. Zeilinger, Phys. Rev. Lett. 62 (1989) 2209. [lo] J.D. Franson, Phys. Rev. Lett. 62 (1989) 2205; Phys. Rev. A 44 (1991) 4552. 1111 P.G. Kwiat, W.A. Vareka, C.K. Hong, H. Nathel and R.Y. Chiao, Phys. Rev. A 41 (1990) 2910; P.G. Kwiat, A.M. Steinberg and R.Y. Chiao, Phys. Rev. A 47 (1993) R2472. [12] H.J. Kimble, A. Mezzacappa and P.W. Milonni, Phys. Rev. A 31 (1985) 3686; H. Huang and J.H. Eberly, J. Mod. Optics 40 (1993) 915; M.O. Scully, in: Proc. 1994 NATO Advanced Study Institute on Advances in Quantum Phenomena (to be published). 1131 R.J. Glauber, in: Quantunm Optics and Electronics, Les Houches 1964 (Gordon and Breach, New York, 1965).
1 March 1997
gi I!/7
OPTICS COMMUNICATIONS
ELSEVIER
Optics Communications 136 ( 1997) 39-43
On enhancing spectral resolution via correlated spontaneous emission Marlan 0. Scully a3b*c, Ulrich W. Rathe apc,d,Chang Su e, Girish S. Agarwal f a Department of Physics, Texas A&M University, College Station, TX 77843, USA ’ Texas Laser Laboratory, HARC, The Woodlands, TX 77381, USA ’ Max-Plan&Institut fcr Quantenoptik, 85748 Garching, Germany ’ Sektion Physik, Ludwig-Mwtimilians-Uniuersit~t, 80333 M&hen, Germany e Department of Physicsand Astronomy, Hunter College of the CUNY, NY 10021, USA ’ Physical Research Laboratory, Navrangpura, Ahmedabad 380009, In&u
Received 19 March 1996; revised version received 20 September
1996; accepted 23 September
1996
Abstract Correlated spontaneous emission from a single three-level atom in a cascade configuration can yield subnatural line narrowing beyond the usual radiative broadening. The effect is observed in the second-order coherence properties of the radiated field.
In a recent Letter [I], we have demonstrated the possibility for a line narrowing in spontaneously emitted radiation via two-photon correlation measurements on atomic cascade radiation. The interferometer used there involved two atoms held in a trap. The purpose of the present article is to demonstrate the effect in an experimentally more accessible setup involving only one atom. Moreover, we here supply a more detailed discussion of the physics involved. Let us start out by noting that the width of an optical transition between two atomic states 1a> and 1b), as measured by conventional spectroscopic means, is gov-
’ Considerable effort has been invested in measurement schemes that potentially resolve spectral lines beyond the “natural” width via “time delay spectroscopy.” For work along these lines, see: Ref. 121. ’ Another approach to sharpening radiative lines using strong dressing fields was presented or demonstrated in Ref. [3]. 3 Lundeen and Pipkin used the separated oscillatory field technique for obtaining subnatural linewidths in measurements of the Lamb shift in H, see Ref. 141. 003s4018/97/$17.00 Copyright PII SOO30-4018(96)00626-S
0 1997 Published
erned by the sum of the radiative decay rates, yU + y,,, out of the two levels in question (see, however, footnotes ‘-3 and references contained therein). It is to be noted that in such conventional measurement schemes, we are dealing with ordinary single-photon events, i.e., single-photon emission and detection. However, as we show in the following, the situation is radically changed when we instead consider two-photon correlated spontaneous emission radiation together with a two-photon (second-order correlation) detection scheme as in Fig. 1. In such a case we find that the second-order correlation function displays spectral features of width y, independent of yhr where y, is the decay rate out of state 1a> into 1b) and yb is the decay rate out of 1b) into 1c). Specifically, we find that the joint probability for registering a count in both detectors D, and D, of Fig. 1 is given by
+ 2e- 2W!&!?]
.
(1)
n
This is the main result of the present paper, and is further elucidated in Fig. 2. In Eq. (l), K is an uninteresting
by Elsevier Science B.V. All rights reserved.
40
M.O. Scully et al./ Optics Communications 136 (1997) 39-43
x II,, l,>,
(2)
where the I a) + 16) (1 b) + I c>) transition involves wave vector k(q), and g,,, (gb,J is the atom-field coupling constant for the I a> -+ I b) ( I b) -+ I c>> transition whose frequency is given by w,~ (w&. We proceed to sketch the calculation for the joint count probability that both detectors at R, and R, (with I R, I = 1R, 1) register a count. This is given by
Fig. 1. An excited three-level atom decays, producing “ + “- and “ - “-polarized photons. Radiation travels through birefringent material to detectors, producing an interference pattern in the two-photon coincidence probability.
P(2)=/om
d$
dt, G(*‘(l,
2),
where the Glauber intensity correlation function G(*)( 1, 2) = G(*)(R,, rl; R,, t2) [14] may be written as GC2’(1, 2) = ‘PC2’(1, 2)*‘PC2’(1,
2)
(4)
with !P(l,
with the inconstant, A = oab - wbc, 7= r(n+- n->/c, dices of refraction n, for radiation from the upper and lower transition, respectively, r is the distance from the atom to the detectors, and ce is the vacuum speed of light; fir, A) is an interference function whose form need not concern us now but is given by Eq. (10). It is the presence of the sharp Lorentzian, of width my,, which interests us here since one normally (i.e., in uncorrelated spontaneous emission) expects a width x + yb. In Fig. 2, we plot the signal AT, A)/( A2 + 7,‘) for different r to demonstrate the line narrowing. For example, the dashed curve corresponding to r = 2 y; ’ shows a width of approximately yay,, independent of yb. A motivation for the present studies derives from the recent beautiful experiments of the NIST group investigating the interference pattern produced by two single atoms [S] along the lines of our earlier “quantum eraser” studies [6]. Another motivation is closely connected with recent work on two-photon down conversion [7-l 11. We now turn to the analysis behind Eq. (1). The type of experiment we have in mind involves the excitation of the atom located at the origin at time t = 0, and the field is in the vacuum state 10) (see Fig. 1). The initial atom-field state may thus be written as 1lu( t = 0)) = 1a) @ IO). For times t 2+ y; ‘, yb ’ the atom decays to 1c) through the intermediate level I b), and the system state vector evolves into I?P(t -+ m)) = I c) @ I I), 4). Here, I +, 4) is the resulting two-photon field state, where $(+) is associated with the 1a) --) I b) (I b) -+ I c)) transition. For the time being, we consider the birefringent material between the atom and the detectors absent. Within the framework of Weisskopf-Wigner theory, we then obtain the expansion of the field state in terms of plane wave states [12]
2) = (0 I3+‘(1)~‘+‘(2)]w).
(9
z(‘)(j) is the usual positive frequency field operator CkZ?& exp[i(k . R, - cktj)] for j = 1, 2. Inserting the state IV) = I $, $> from Eq. (2) into (5) we find q(*)(l,
2) = Ke-iwOb(rl-r/c@(tr
_ r/c>
X eiob=(‘~-r/c)O [ ( t2 - r/c)
,-r,(tl-r/cf - (t, - r/c)]
Xe-Yb[(‘2~‘/C)-(fi-T/C)1 +(1++
2).
(6)
In Eq. (6), I= jRll = ]R,]isthedistancefromtheatom to the detectors, K is an uninteresting constant, and 8(x> is the usual step function. 1
30.5 El ‘B
0
Fig. 2. Signal part fir, A)/(A’ + y,‘> in the joint count probability for different T. The different curves have been normalized for simplicity.
4 If one chooses to formulate the problem in terms of circularly polarized photons, one would have to take into account the fact that a transition emitting right circularly radiation to detector D, emits left circularly radiation to detector D,.
41
M.O. Scully et al./ Optics Communicutions 136 (1997) 39-43 We now take the presence of the birefringent medium into account, assuming a refractive index n, for “ + “polarized $-radiation and n_ for CC- “-polarized 4 (pradiation (n+> n_>. This leads to a modification of the respective retardations in Eq. (6), !PC2)( 1, 2) = Ke- iw,b(rl-r/c+)fj(tl Xei%(‘2-‘/c-@[([z -(t,
-m+>l
+(1
*2>,
c C-
2
(a)
_ I/c+)e-~J’~-r/~+) - T/c_)
W(2,l) (7)
where c+= co/n+ and c_ = co/n_. At this point it is useful to note that the times t; r/c+= TV* (i= 1, 2) are the times when a “+“or “ - “-polarized photon, that is to be detected later by the ith detector, is emitted by the atom. For example, the first term in Eq. (7) corresponds to the emission of a w,~ photon at 7: and subsequent detection of this photon at t, = 5-T+ r/c+ by photodetector D,. Thus the @(S-T> function in Eq. (7) ensures that the o,,, photon is emitted only for positive Q-T, i.e., only after the atom has been excited, and the term exp(- -y,r:> ensures that the emission event will take place in a time T: N l/y, after the excitation of the atom. Likewise, the @(T; - 7:) in Eq. (7) says that the o ,,= photon triggers a count in detector D, only when T; > Q-T, thus requiring that the second (obC> photon is emitted after emission of the w,~ photon, and the term exp[ - yb(r; - r:)] ensures that the whC emission event will take place within a time 7; - 7: l/ yb. The notation (1 @ 2) in Eq. (7) denotes that the w,~ photon now goes to detector D, and the obC photon to detector D ]. Now Eq. (7) can be rewritten as @)(l,
%c -----
1 -3
Fig. 3. Key contributions to coincidence event of detector D, at time t, and detector D, at time t2. Photons are emitted by tbe atom at the retarded times TV*. Lines from atom to detectors represent optical patblengths.
We insert Eq. (7) into Eq. (4) and carry out the integrations of Eq. (3). After a somewhat lengthy calculation, we find PC2)(1, 2) as given by Eq. (1) with f(r,
A) = 1 -e-Y”’
cos AT-
$-sin
AT
u
(10)
In order to understand the line narrowing physics behind the 1/(A2 + -y,‘> term, we explicitly write out the first key term from Eq. (9)
-m
z
L 1 dt,
dt, (-__+_-
0
2
:
x
:---”
1
2) = Ke-i0~,~~;e(~:)e-Yo7;e--iWbc7;
= W)(
=2”---i
1) 2) + @‘2’(2, 1) c----c--., + 2 1 )
(8)
where we introduce the notation @@)(l , 2) to represent amplitude for the CO,,, photon being detected by D, the wbC photon by D,, whereas @*)(2, 1) stands for w,~ photon going to D, and the whr photon going to The diagrammatic representation of these processes established in Fig. 3, the solid (dashed) line denoting w,~ (cIJ~~>photon propagator. The interesting interference terms are included in pC2)(1, 2) =/,I
d’,jo*
dt, Qt2’(1, 2)=@(*)(2,
the and the D,. is the
1) +c.c. (9)
NP
1
A*+ y,”
(1’)
The first factor is the process in Fig. 3(a). In Fig. 4, we depict this case in a space-time diagram with the two solid lines. There, the retarded times T: and T; on the t = 0 axis are the emission times, whose difference 7; - T: is given by r; - 7: = t2 - t, - r(n_ - n+>/c,; but for the second process indicated with dashed lines in Fig. 4, corresponding to Fig. 3(b), 7; - T: = tl - t2 - r(n_n+>/co. From the preceding discussion we see that the time dependence involving y,,t, and y,, 1, cancels out in processes such as Eq. (11). That is, the yh factors in Eq. exp[yb(r2 - 7: > - yb(r; - r: >I = (1 l), exp[ - 2 yhr( n, - n_ j/c,] are time independent. This is
M.O. Scully et al./Optics Communications 136 (1997) 39-43
42
tt
r + Fig. 4. Space-time diagram of the two interfering processes. The two solid lines depict the case of Fig. 3(a), the dashed lines the one of Fig. 3(b).
the key point. It may be restated by noting that the two interfering processes of Fig. 4 spend complementary times in state 1b) and the product of their amplitudes is independent of the difference between detection times t, and t,. Therefore, an integral over tl and t,, yielding the total count probability, will lead to Eq. (11). We can also understand the absence of yb from the Lorentzian denominator from an intuitive “quantum jump’ ’ picture of the cascade decay and subsequent detection. Consider the situation depicted with solid lines in Fig. 4, also shown in Fig. 3(a). By arguing as before that the count of detector D, at time I, corresponds to the emission of an w,,-photon by the atom at the retarded time r: = I, - r/c+, and that the count of D, at tp corresponds to emission of a w,,-photon at a time r; = t, - r/c_, we deduce that the atom spent the time 7; - 7: in the intermediate state 1b). In the spirit of the quantum jump approach the probability amplitude of this process is therefore weighted by a damping term of the form exp[ - ~~(7; - r:)]. In the process with dashed lines in Fig. 4, corresponding to Fig. 3(b), the atom emits the w,,-photon at time rl = tT - r/c+ and the o,,-photon at time r; = t, - r/c_, so that the total time this atom spent in state 1b) is given by 7; - T:. Therefore, the probability amplitude of this process is weighted by exp[ - ~~(7; - T:)]. We see that the product of the two probability amplitudes will be influenced by y,-damping through exp[ - y&r; - 7:) ~~(7; - ?-:>I which is obviously independent of t, and tz. Again, the integral over t, and t2 yielding the total count probability will not have denominators depending on yb but instead produce the Lorentzian in Eq. (1). It is the purpose of the present paper to point out this two-photon line narrowing effect. A possible experiment making use of the sharp Lorentzian in Eq. (1) in order to measure the transition frequencies o,~ and obc can be conceived as follows. We assume 1a> and 1c) to be atomic Sstates, and 1b) to be the m = - 1 sublevel of a P-state with energy obo in the absence of a magnetic field. First we note that the atomic frequency w,, = w,~, + wboc can be measured by, for example, Raman excitation of the
state 1a), in which case ya determines the measurement precision. To get a similarly precise measurement of w,~, - wboc, we could map out the sharp Lorentzian 1/(A2 + y,‘) by varying A = wab - w,,=, e.g., with the help of a variable external magnetic field, and thus provide a good measurement of the magnetic field strength B, for which A = 0 ‘. From Be we obtain o,~, - oboe, and from the Raman spectroscopy w,~, + oboe is known; therefore we have, in principle, determined both w,,,, and o+ to a precision governed by ‘y,. In this context, we stress that the present line narrowing provides a high spectral resolution in the sense that we can separate closely spaced lines according to the Rayleigh criterion. However, the extent to which it leads to a higher experimental accuracy in the final result for the atomic transition frequencies requires further discussion as will be given elsewhere. The reason for this is the exponential damping of the signal with the time delay T by means of the prefactor exp( - 2 ~~7) in Eq. (1) which decreases the visibility of the signal. This trade-off of signal width against signal amplitude is well-known from other methods for subnatural spectroscopy, notably from time delay spectroscopy [2]. Note, however, that although the present scheme is reminiscent of time delay spectroscopy, there exists a major difference between the two approaches. Whereas the signal width in time delay spectroscopy is ultimately limited by the difference of the decay rates, ya - yb, (see Meystre et al. in Ref. [2]) correlated emission spectroscopy is limited by ya only. Therefore, it allows for subnatural spectroscopy in atomic systems where time delay spectroscopy does not produce subnatural linewidths. In conclusion, we emphasize that the underlying physical reason for the interference in Eq. (1) is the correlation involved in the two-photon state produced by the cascade transition as given in Eq. (2). If we had chosen to consider an interrupted cascade transition in the spirit of Ref. [6], the resulting Tc2)(1, 2) for one atom would have had the form qc2’(1, 2) = $,(1)!5(2) + Y$(l)W,(2), and there would have been no subnatural resolution. Finally, we note that the potential application of these considerations to nuclear and molecular spectroscopy, where there is an abundance of oscillator-like states with w ab = Obc’ is also interesting. This work was search, the Welch Research Program. Fry, J. Franson, H.
supported by the Office of Naval ReFoundation, and the Texas Advanced One of us (MOS) wishes to thank E. Pilloff, H. Van Driel, and A. Zeilinger
5 The interference function AT, A) modulating the Lorentzian + y,‘) in Eq. (1) has to be taken into account for this measurement. In this context, the choice of the time delay T is 1/(A*
crucial as seen in Fig. 2.
M.O. Scully et al./Optics
Communications 136 (1997) 39-43
for helpful and stimulating discussions. UR would like to thank the “Studienstiftung des deutschen Volkes” for support.
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