quantum mechanics

(a)
e


?p ? g


(b)



?



R(t)

g


?p
s g

Fig. 5.4. Single-photon source via intracavity STIRAP. (a) A three-level atom is pumped by a classical ?eld of Rabi frequency ?p(t) and is coupled to the cavity ?eld with vacuum Rabi frequency g. (b) The compound system atom + cavity ?eld initially in state |s, 0) undergoes STIRAP transfer into the state |g, 1) with a single photon in the cavity mode which escapes the cavity through a partially transmitting mirror with the rate ?, forming a single-photon wavepacket with temporal shape R(t).


We need, however, to consider the e?ects of atomic and cavity ?eld relax- ation in more detail. Assuming that the lower atomic levels |s) and |g) are metastable (long-lived), the state |e, 0) of the compound system decays via the atomic spontaneous emission with the rate ? , while the state |g, 1) decays due to the cavity-?eld relaxation with the rate ?. Thus, the eigenstates |D)
and |B±), whose energy separation |?± ? ?0| is characterized by ?¯, acquire
certain widths determined by ? and ?. As a result, the nonadiabatic coupling between the dark |D) and bright |B±) eigenstates is small if they do not over- lap, which requires that the condition ?¯ > ? +? be satis?ed at all times. Since
the vacuum Rabi frequency g is constant, while ?p(t) is time-dependent, so that ?p(ti) g and ?p(tf ) > g, at resonance ? = 0, the above requirement translates to the strong coupling condition g > ?, ?. In addition, the adiabatic following condition (3.135) requires that ?p(t) changes su?ciently slowly for the condition max(?p) ?p 1, with ?p being a characteristic raising time of ?p(t), to be satis?ed.
To verify the possibility of intracavity STIRAP, we thus have to solve the
Liouville equation for the density matrix ? of the compound system,

?
?t ? = ?

i
k [H, ?]+ Lat? + Lcav?, (5.94)

where H is the Hamiltonian of (5.89), while Lat? and Lcav? describe, respec- tively, the atomic and cavity mode relaxations. The cavity Liouvillian Lcav? is that of (5.15), and the atomic Liouvillian Lat? is given by
Lat? = . 1 ?el.2 ?le? ?el ? ?ee? ? ? ?ee.
l
= . ?el ?le? ?el ? 1 ? .?ee? + ? ?ee. , (5.95)
l


where the index l = s, g,... runs over all the lower levels of the atom to which the upper level |e) can decay. Hence, ?es and ?eg represent the spontaneous
decay rates from |e) to levels |s) and |g), respectively, while ? = .l ?el is the
total decay rate of |e). For the initial state |s, 0), the Hamiltonian H acts in the Hilbert space H = { |s, 0), |e, 0), |g, 1)}. Then the decay of the cavity ?eld takes the system outside of space H, to state |g, 0) ?/ H. On the other hand, the decay of state |e, 0) due to the atomic relaxation can take the system to one of the states |s, 0), |g, 0) or |l, 0), where |l) denotes any lower atomic state other than |g) or |s). The states |g, 0) and |l, 0) are not in space H, while |s, 0) ? H. This necessitates our use of the density matrix approach, since with the amplitude equations we could not properly take into account the decay channel |e, 0)? |s, 0).


8



4



0
1



0.5



0
1


(a)







(b)







(c)




g






?s,0;s,0


?g,1;g,1
?e,0;e,0


?p(t)















Pemit(t)



0.5

R(t)



0
?5 0 5
Time

Fig. 5.5. Dynamics of the intracavity STIRAP with g = ? = 4? . (a) Rabi fre- quencies g and ?p(t); (b) populations ?s,0;s,0(t), ?e,0;e,0 (t) and ?g,1;g,1(t); and (c) emission rate R(t) and probability Pemit (t). The light-gray curves in (c) correspond to R(t) and Pemit (t) for an initially excited two-level atom with the parameters of Fig. 5.2, i.e., g = ? and ? = 4? . Time is measured in units of ? ?1.


In Fig. 5.5, we plot the results of the numerical solution of the equations for all the relevant elements of the density matrix ? (see Prob. 5.6). We ?nd that, when the strong coupling condition is not quite satis?ed, during the


evolution the intermediate excited state |e, 0) acquires small but ?nite pop- ulation ?e,0;e,0, which is due to nonadiabatic transitions. On the other hand, assuming the absorption in the cavity mirrors to be negligible, ?abs ?tr c ?, the probability ?g,1;g,1 of the single-photon state |g, 1) decays due to the leak- age of the cavity ?eld through the partially transparent mirror, the outgoing photon pulse having the temporal shape R(t) = ?tr?g,1;g,1(t). Yet, for the parameters of Fig. 5.5, at the end of the process the photon emission prob- ability Pemit(tf ) attains the value 0.93, which is close to the ideal. When we take g = 8? , with the other parameters unchanged, so as to better satisfy the strong coupling condition, we obtain completely adiabatic evolution of the system with negligible population of the excited state |e, 0), achieving Pemit(tf ) ” 0.98.
Under these conditions, using simple arguments, we can derive an analytic
expression for the shape of the outgoing single-photon pulse R(t). As the system adiabatically follows the dark state |D(t)) of (5.92), we expect that at any time t ? [ti, tf ] the ratio of populations of states |s, 0) and |g, 1) is given by
?s,0;s,0(t) = cot2 ?(t) , (5.96)
?g,1;g,1(t)
where the mixing angle ?(t) is de?ned through cot ?(t) = g/?p(t). Next, since under adiabatic evolution, the excited state |e, 0) is never signi?cantly populated, the sum of populations of the initial |s, 0) and ?nal |g, 1) states decays only via the cavity ?eld relaxation,

? .?


(t)+ ?

(t). = ???


(t) . (5.97)

?t s,0;s,0

g,1;g,1

g,1;g,1

Using (5.96) with ?p(t) ƒ= 0 for t ? [ti, tf ], one can now derive the rate equation for the population of state |g, 1) (see Prob. 5.7), whose solution is


?g,1;g,1(t)= ?g,1;g,1(ti) exp

. ¸ t
?
ti


dtr

? + ?tr cot2 ?(tr) .
1+ cot2 ?(tr)


. (5.98)


At the initial time ti, we assume that g ?p(ti) ƒ= 0, and ?s,0;s,0(ti) = 1. From (5.96) we then have ?g,1;g,1(ti) = tan2 ?(ti), which should be used in the above solution. It turns out that under the strong coupling condition g > ?, ? , the analytic solution (5.98) is practically indistinguishable from the exact numerical solution of the full set of density matrix equations. This con?rms the validity of the adiabatic approximation that led to our starting equations (5.96) and (5.97).
We can now write the expression for the pulse shape in the explicit form
?tr ?p (tr )

?2(ti)

?¸ t

2g2 r

2 r ?

R(t)= ?tr p
g2

exp ?

dtr
ti

?p (t ) ? ??p (t )
g2 + ?2(tr) ?

, (5.99)


which shows that, by carefully changing the pump ?eld Rabi frequency ?p(t), and thereby the mixing angle ?(t), we can manipulate at will the temporal characteristics of the outgoing pulse. In particular, we can fully control the timing and the temporal shape, or the bandwidth, of the single-photon pulse. Once the photon has left the cavity, we may recycle the system by switching o? the pump ?eld and preparing the atom in the initial state |s). We could then repeat the process to generate another photon with precise timing and pulse shape. This system can thus serve as a deterministic and e?cient source of tailored single-photon pulses.