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A quantum trajectory unraveling of the superradiance master equation1
25 May 2000
Optics Communications 179 Ž2000. 417–427 www.elsevier.comrlocateroptcom
A quantum trajectory unraveling of the superradiance master equation 1 H.J. Carmichael a b
a,),2
, Kisik Kim
b
Abteilung fur ¨ Quantenphysik, UniÕersitat ¨ Ulm, D-89069 Ulm, Germany Department of Physics, Inha UniÕersity, Inchon 402-751, South Korea
Received 23 September 1999; received in revised form 12 November 1999; accepted 16 November 1999
Abstract We unravel the superradiance master equation into quantum trajectories whose stochastic wavefunction is conditioned on photon counting records resolved in photon emission time and direction. Numerical results are reported which illustrate the wealth of statistical information made available by simulating the counting records. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Superradiance and related collective radiation phenomena have been topics of interest in atomic physics and quantum optics since the work of Dicke w1x. Although that work dealt with a somewhat idealized model, a great deal of attention has been given to more accessible versions of superradiance in the intervening years w2x. Today, in view of a series experimental developments, something much closer to Dicke’s original conception can be realized. With the use of ion traps and laser cooling methods, atomic radiators can be localized within an optical
) Corresponding author. E-mail: [email protected] 1 We dedicate this paper to Marlan Scully on the occasion of his 60th birthday. 2 On leave from the Department of Physics, University of Oregon, Eugene, Oregon 97403-1274.
wavelength. While localizing a number of ions simultaneously is problematic due to Coulomb repulsion, an experiment with a pair of trapped Baq 138 ions has reported observations of superradiant and subradiant emission at approximately the 1% level w3x. In a second, more extended system of two trapped 198 Hgq ions, Young’s interference, but not superradiance, has been observed w4x. Experimental work with cold neutral atoms and Bose–Einsten condensates opens yet another path to interesting new studies of collective radiation phenomena w5x. Theoretical developments since Dicke’s work have also been very broad. In particular, a master equation for an extended spatial arrangement of atoms has been derived w6–10x. Static dipole–dipole interactions are included in this equation, but retardation and propagation effects are not taken into account. Our work is directed towards this superradiance master equation. We reformulate the equation to make the rich photon statistics of superradiance accessible – possibly to analysis, but at least to the power of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 9 4 - X
418
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
numerical simulations. We have in mind not only the statistics of the photon emission times, but also the statistics of the photon emission directions; thus, the stochastic initiation of directed emission is something, in particular, we would wish eventually to study. The present paper is merely a preparation for anything quite so ambitious. Here, we execute an unraveling of the master equation on the basis of quantum trajectory ideas w11x. The density operator is replaced by an ensemble of pure states, each of which undergoes a stochastic evolution. The evolution follows the individual photon emissions through a sequence of quantum jumps, the state being labelled at any time by the accumulated record of photon emissions – the photon counting record. There has been considerable interest in such unravelings of master equations in recent years w11–15x. A notable feature of the current example is that in the case of the superradiance master equation the jump operators are generally collective atomic lowering operators, not the lowering operators for the individual atoms. Each collective operator is labelled by a direction of photon emission and this provides access to a detailed statistical description of the angular distribution of the radiation. We review the model and the superradiance master equation in Section 2. In Section 3 we define what we mean by photon counting records and develop the unraveling of the time-dependent density operator as an ensemble of pure states labelled by these records. We describe the stochastic, quantum trajectory implementation of the unraveling in Section 4, and also report in this section the results of some numerical simulations. As an example, we consider the radiative decay of a string of five initially excited atoms; results illustrating basic features of both the temporal and directional photon statistics are reported. A summary and some brief conclusions are presented in Section 5.
nance frequency v 0 and with aligned dipole moments d. For each atom there are pseudo-spin operators sˆiy, sˆiq, and sˆi z , with commutation relations
sˆiy , sˆjq s yd i , j sˆi z ,
sˆi " , sˆj z s .2 d i , j sˆi " . Ž 2.1 .
The master equation describing spontaneous emission from such a collection is standard. In the electric-dipole, rotating-wave, and Born–Markov approximations, the reduced density operator for the internal state of the atoms satisfies w6–10x dr dt
s Lr ,
Ž 2.2 .
where L is the generalized Liouvillian Žsuperoperator. L ' yi v 0
1 2
N
Ý sˆi z ,P is1
N
yi
Ý
Di j sˆiq sˆjy ,P q
i/js1
N
1 2
Ý
gi j
i , js1
= 2 sˆjyP sˆiqy sˆiq sˆjyPyP sˆiq sˆjy ,
ž
Ž 2.3 .
/
with
Di j ' g
3 4
½
y 1 y dˆP rˆi j
ž
q 1 y 3 dˆP rˆi j
ž
2
ž
/
2
/
cos j i j
ji j sin j i j
j i2j
q
cos j i j
j i3j
/5 Ž 2.4 .
and
gi j ' g
3 2
½
1 y dˆP rˆi j
ž
q 1 y 3 dˆP rˆi j
ž
sin j i j
2
/
2
/
ji j
ž
cos j i j
j i2j
y
sin j i j
j i3j
/5
,
Ž 2.5 .
2. The source master equation where We consider a collection if N two-state atoms located at positions r i , i s 1, . . . , N, all with reso-
j i j ' k 0 ri j s 2p ri jrl0 ,
ri j ' ri y rj
Ž 2.6 .
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
and
gs
1
4v 03 d 2 3
4pe 0 3"c
Ž 2.7 .
is the Einstein A coefficient; dˆ and rˆi j are unit vectors in the directions of d and r i j . The Liouvillian Ž2.3. is comprised of three terms: one proportional to v 0 which evolves the atoms freely, one with coefficients Di j to account for the static dipole–dipole interactions, and the third term with coefficients g i j describing the actual process of photon emission. Solving the master equation is a difficult task. Its solution depends in a sensitive way on the spatial arrangement of the atoms, and even a moderate number of atoms leads to a very large number of equations. For systems of macroscopic size with the atoms distributed randomly in space, approximations are justified that can simplify the problem w2x. In the opposite extreme, the two-atom case can of course be solved exactly with no particular difficulty w8x. Our approach, using quantum trajectory theory, is suited best to systems of intermediate size. Implemented numerically, it can potentially solve the master equation for a few to a few tens of atoms. It is likely to prove useful, though, even for macroscopic numbers of atoms, since it accounts, quite generally, for the stochastic initiation of superradiant emission. It is indeed the stochastic nature of the emission that interests us the most. Ultimately, it is not just the solution to the master equation that we seek; we would like to make a full characterization of the statistics of the emitted light; and not only is the distribution of photon emission times of interest but also the distribution of the photon emission directions. Traditionally, one obtains information on such things by calculating correlation functions of the electromagnetic field. Of course, formally, through the quantum regression theorem, the solution to the master equation is needed here as well. An explicit calculation of correlation functions sets out, however, on an indirect path to the mentioned distributions. The quantum trajectory formalism goes directly to this raw statistical information. It does so by unraveling the master equation evolution into an ensemble of photon emission sequences, each sequence labelled by specific emission times and directions. A numerical algorithm exists to implement the
419
unraveling and generate realizations of the sequences of emissions. The unraveling and the numerical algorithm are the subjects of the next two sections. 3. Unraveling the master equation with photon counting records We aim to write the density operator r Ž t . as an explicit mixture of pure states and replace its evolution under Ž2.2. by a stochastic evolution of those states. It is well-known that this can be done for any master equation in Lindblad form and an extensive literature exists describing the various procedures that might be followed w11–15x. We wish to achieve more, however, than a formal pure-state decomposition of r Ž t . 3 ; we require each pure state to be labelled by a specific photon emission sequence. To achieve this end, we follow Carmichael’s procedure for unraveling a master equation by working backwards from photon counting records to an inferred dynamics of the source w11x. Setting aside the atoms and their master equation for the moment, we begin by considering the electromagnetic field that is radiated by the atoms. Solving Heisenberg equations of motion in the Markov approximation, the electric field operator may be written as a sum of a free field and a source field. Its positive frequency component is w7,10,11x EˆŽq. Ž R ,t . s Eˆf Žq. Ž R ,t . q EˆsŽq. Ž R ,t . ,
Ž 3.1 .
with EˆsŽq. Ž R ,t R . s y =
1
v 02 d
4pe 0 c 2
N
Ý is1
ž dˆ= Rˆ / = Rˆ e i
Ri
i
i k 0 Ž R iyR.
sˆiy Ž t . ,
Ž 3.2 . 3
A Lindblad master equation adds to the Liouville equation ˆ r xr i" a series of terms in the form ÝmŽ2 Lˆm r Lˆ†m y r˙ s w H, Lˆ†m Lˆm r y r Lˆ†m Lˆm . w16x. The most direct way to recast the superradiance master equation in this form would be to diagonalize the matrix of coefficients g i j . The operators Lˆm arrived at in this way are merely formal, however. It would appear that in general, when the atoms are sufficiently close to one another that they are not resolvable as individual sources, these operators are not associated with any accessible Žobservable. physical process in the manner of the operators Sˆp q in Eq. Ž4.4..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
420
where t R ' t q Rrc, t ) 0, and R i ' R y ri .
Ž 3.3 .
Assuming ri j < R i , for all i and j, we may replace Eq. Ž3.2. by EˆsŽq. Ž R ,t R . s yeˆRˆ EˆsŽq. Ž R ,t R . ,
Ž 3.4 .
where EˆsŽq. Ž R ,t R . s
1
v 02 d
1 y Ž dˆP Rˆ .
4pe 0 c 2 R N
2 1r2
™
emission from N atoms, every record asymptotically Ž t `. contains exactly N events Ž Cm ,Tm .. An ensemble of repeated experiments realizes an ensemble of different counting records. A given record appears with a particular probability in the ensemble and the photon statistics of the emitted light is characterized by these record probabilities 5. They are determined by the source-field operator Ž3.5. and the initial state N
r Ž 0. s < c Ž 0. : ² c Ž 0. < , ˆ
= Ý eyi k 0 RP r isˆiy Ž t .
< c Ž 0 . : s Ł < q :i is1
Ž 3.5 .
Ž 3.8 .
is1
and eˆRˆ is a unit vector in the direction of Ž dˆ= Rˆ. = ˆ The origin of coordinates is chosen to be at the R. N center of the distribution of atoms ŽÝ is1 r i s 0.. We base the unraveling of the master equation on photon counting records for the electromagnetic field Ž3.5.. Consider an array of Žimagined 4 . ideal photodetectors covering the surface of a sphere of radius R. Individual detectors are located at Ž up , f q ., p s 1, . . . , pmax , q s 1, . . . ,qmax and subtend solid angles DV p q at the center of the sphere. The detection events registered by this array define the photon detection records. We denote a specific record, up to time t, by
REC4 ' Ž Cn ,Tn . ; Ž Cny1 ,Tny1 . ; . . . ; Ž C1 ,T1 . 4 , Ž 3.6 . where Cm indicates the location of a photon count Ždirection of a photon emission., Cm g Ž p,q, DV p q . ;1 F p F pmax ,1 F q F qmax 4 ,
Ž 3.7a . and Tm ' w tm ,tm q d tm .
Ž 3.7b .
indicates the time of its occurrence, with t G t n ) t ny 1 ) PPP ) t 1 ) 0. In the case of spontaneous
of the atoms. Thus, we introduce superoperators Sp q '
2 e0 c " v0
Ž R 2DV p q . EˆsŽq.
up , f q , Ž Rrc . q
P EˆsŽy. up , f q , Ž Rrc . q ,
Ž 3.9 .
p s 1, . . . , pm ax , q s 1, . . . , q m ax , such that trw Sp q r Ž t .x is the average photon flux, at time t, registered in the solid angle DV p q by the detector located at Ž up , f q .. Substituting from Eq. Ž3.5., Sp q s g
3 8p
DV p q 1 y dˆP Rˆ p q
N
=
Ý
ž
2
/
ˆ
eyi k 0 R p qP r i j sˆjyP sˆiq ,
Ž 3.10 .
i , js1
where Rˆ p q ' RˆŽ up , f q . is a unit vector in the direction of Ž up , f q .. Here the source-field superoperators are expressed explicitly as sums of superoperators that act on the atoms. Hence, we may use photon detection theory and the Born–Markov treatment of open systems to arrive at an expression for the record probabilities in the convenient form w18,19,11x PREC Ž t . s tr B Ž t y t n . Jn B Ž t n y t ny1 . = Jny 1 PPP J1 B Ž t 1 . r Ž 0 . =dt n d t ny1 PPP d t 1 ,
Ž 3.11 .
4
The photodetectors exist as a purely conceptual device. There is no suggestion that photon counting records must in actual fact be made. Assuming the atoms constitute a genuine photoemissive source, there is no difference between the absorption of a photon by an actual detector and its continued propagation into the reaches of free space.
5 Photon counting records cannot account for every statistical property of the radiated field. Its spectrum, for example, is not determined by photon counting records. The spectrum is accounted for in an independent set of records, which correspond to a complementary unraveling of the same master equation w17x.
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
where Jm g Sp q ;1 F p F pmax ,1 F q F qmax 4 ,
Ž 3.12 .
and BŽ x . ' e L B x , with
Ž 3.13 .
LB ' L y Ý Sp q .
Ž 3.14 .
Every emitted photon is accounted for in the records. Hence, because eÕery photon is account for, the record probabilities may be rewritten in terms of a pure state unraveling of the evolution of r Ž t .. For this purpose, we define the scaled source-field operators Sˆp q '
s
(
2 e0 c
Ž R 2DV p q .
" v0
(
g
3 8p
=
EˆsŽy. up , f q , Ž Rrc . q
DV p q 1 y dˆP Rˆ p q
ž
N
Ý eyi k
ˆ
0 R p qP r i
The essential formal requirement to reach Eq. Ž3.17. from Eq. Ž3.11. is that LB be a commutator. It is for this reason that every photon must appear in the detection record – otherwise there is an efficiency h - 1 multiplying the sum in Eq. Ž3.14.. In the present example it must also be true that
Ý Sˆp q P Sˆp† q s Ý g i j sˆjyP sˆiq , p, q
p, q
Ý Sˆp† q Sˆp q s Ý g i j sˆiq sˆjy ; p, q
sˆiy ,
Ž 3.15 .
such that Sp q s Sˆp q P Sˆp† q .
Ž 3.16 .
The rewriting of the record probabilities Ž3.11. is then PREC Ž t . s ² c REC Ž t . < c REC Ž t . :d t n d t ny1 PPP d t 1 , Ž 3.17 . with < c REC Ž t . : s Bˆ Ž t y t n . Jˆn Bˆ Ž t n y t ny1 . = Jˆny 1 PPP Jˆ1 Bˆ Ž t 1 . < c Ž 0 . : ,
hence HˆB has the alternative form N 1 N HˆB ' " v 0 Ý sˆi z q " Ý Di j sˆiq sˆjy 2 is1 i/js1
½
5
Ý Sˆp† q Sˆp q .
Ž 3.19 .
Ž 3.24 .
pq
ž
/
and developing the formal solution for r Ž t . as a superoperator Dyson expansion with the source-field superoperators Sp q s Sˆp q P Sˆp† q in the role of the interaction terms w18,11x. The expansion takes the form
rŽ t. s
Ý
t
Ý
ns0 C 1 , . . . C n
and ˆ Bˆ Ž x . ' eŽ H B r i " . x ,
2
At this stage the principal pieces of the developed are the records Ž3.6. and their probabilities Ž3.17.. Thus, we have a theory of photon counting, though no explicit connection to an unraveling of r Ž t .. The unraveling itself follows by writing the master equation as dr s Lr s LB q Ý Sˆp q P Sˆp† q r Ž 3.25 . dt p, q
where Jˆm g Sˆp q ;1 F p F pmax ,1 F q F qmax ,
1
`
Ž 3.18 .
Ž 3.23 .
i, j
yi"
is1
Ž 3.22 .
i, j
which is demonstrated in Appendix A. Similarly, it is true that
2 1r2
/
421
tn
H0 d tH0 d t
ny 1
PPP
t2
H0 d t
=< c REC Ž t . : ² c REC Ž t . < , and we arrive in this way at the unraveling r Ž t . s Ý PREC Ž t . < c REC Ž t . : ² c REC Ž t . < ,
1
Ž 3.26 . Ž 3.27 .
REC
Ž 3.20 . where
with HˆB ' " v 0
1 2
yi"
1 2
N
N
Ý sˆi z q " Ý is1
< c REC Ž t . : '
Di j sˆiq sˆjy
i/js1
N
Ý i , js1
g i j sˆiq sˆjy .
Ž 3.21 .
< c REC Ž t . :
(² c
REC
Ž t . < c REC Ž t . :
Ž 3.28 .
is the desired pure state labelled by the photon emission sequence Ž3.6.. The time evolution of the unraveling is implicit in Eq. Ž3.18..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
422
4. A stochastic implementation: quantum trajectories
or in the most likely instance that there is no photon count, a continuous evolution generated by the propagator Ž3.20.; with HˆB in the form Ž3.21., Ž Di i ' 0.
The formal unraveling helps very little with making calculations. Finding a closed form expression for < c REC Ž t .: presents an even more difficult task than solving the master equation. Fortunately, there exists a stochastic implementation of the formalism which opens a path ahead. In this section we describe the implementation of the unraveling as an ensemble of quantum trajectories and illustrate the wealth of statistical information that is made available through simulation of the photon counting records. As an example, we study the temporal and angular distributions of photon emissions from a string of five initially excited atoms.
d < c REC :
4.1. The quantum trajectory algorithm Ideally, we would calculate < c REC Ž t .:, with the record arbitrary, from which the record probabilities and the photon counting distributions follow. The quantum trajectory algorithm generates a solution for < c REC Ž t .:. Not, though, for an arbitrary record, but for a particular record generated stochastically as the trajectory proceeds; repeated trajectories automatically realize an ensemble of records in accord with the probabilities Ž3.17.. We may thus move our attention away from < c REC Ž t .: to the realized records viewed as simulated data. The quantum trajectory algorithm advances from < c REC Ž t .: to < c REC Ž t q d t .: by deciding randomly between all possibilities for the next entry in the record – a photon count in some DV p q or an interval d t without a photon count. The decision is made based on conditional probabilities, probabilities conditioned on the record so far realized and deduced by Baysian inference w11,17x. In the present case, in the time step d t, there is either a quantum jump < c REC Ž t . :
™ Jˆ < c m
REC
Ž t . :,
Ž 4.1 .
with Jˆm selected from amongst the Sˆp q in accord with independent photon counting probabilities
Gp q Ž t . d t s ² c REC Ž t . < Sˆp† q Sˆp q < c REC Ž t . :d t ,
Ž 4.2 .
N
sy
dt
Ý i , js1
ž
gi j 2
q i Di j sˆiq sˆjy < c REC : ,
/
Ž 4.3a . or alternatively, in view of Eq. Ž3.23., d < c REC : dt N
ž
s yi
Ý i/js1
Di j sˆiq sˆjyy
1 2
Ý Sˆp† q Sˆp q pq
/
< c REC : .
Ž 4.3b . It is clear here that Eqs. Ž4.2. and Ž4.3b. constitute the quantum jump unraveling of the Lindblad superoperator N 1 N L s yi v 0 Ý sˆi z ,P y i Ý Di j sˆiq sˆjy ,P 2 is1 i/js1 1 q 2
Ý ž 2 Sˆp q P Sˆp† q y Sˆp† q Sˆp q P yP Sˆp† q Sˆp q / , pq
Ž 4.4 . which is equivalent to the generalized Liouvillian Ž2.3. in view of Eqs. Ž3.22. and Ž3.23.. Thus, the latter equations relating the near-field coefficients g i j to the far-field coefficients d i j ŽAppendix A. are the central results permitting the unraveling. Its main feature when comparing N independent atoms is that the operators Jˆm g Sˆp q 4 are collectiÕe jump operators. This is of course dictated by the physics of the situation; a photon emitted by N closely spaced atoms Ž ri j ; l0 . cannot, in principle, be labelled as an emission by any particular atom 6 . As an example, we consider the string of five atoms illustrated in Fig. 1. The string is laid out along the z axis with a spacing s l 0 between adjacent atoms; we write r i s Ž i y 3 . s l0 zˆ , i s 1,2, . . . ,5. Ž 4.5 . The dipole moment is in the Ž x, z .-plane and dˆP rˆi j s cos a .
6
See footnote 3.
Ž 4.6 .
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
Fig. 1. Coordinate system for the string of five atoms.
The initial state of the atoms is that given in Eq. Ž3.8.. The photon counting is simulated for an array of 5,000 detectors, with
up s Ž p y 0.5 . = 0.02p ,
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latter case, the distributions are essentially exponential with decay times wŽ N y m q 1.g xy1 , m s 1,2, . . . ,5, as is expected for spontaneous emission from independent atoms. In the former, although photon 1 is still emitted with average waiting time Ž5g .y1 , the rate of every subsequent emission is enhanced by the superradiant effect w1–3x. In actual fact, there are more longer as well as shorter waiting times, and the distributions are not exponential any more. Thus, the cascade of photon emissions passes through both superradiant and subradiant states, with the details in a particular instance depending on the directions of the emissions. Subradiant emission is particularly noticeable in the waiting time for photon 5. The long tail in the distribution D n 5 Žt . is illustrated clearly in Fig. 3.
p s 1,2, . . . ,50, Ž 4.7a .
and
f q s Ž q y 0.5 . = 0.02p ,
q s 1,2, . . . ,100.
Ž 4.7b . The detector solid angles vary with the polar angle,
DV p q s 0.02p = 0.02p sin up ,
Ž 4.8 .
and we define the solid angle integrated over azimuth
DV p s 2p = 0.02p sin up .
Ž 4.9 .
Within each photon counting record, m s 1,2, . . . ,5 labels the sequence of photon emissions according to their order in time. 4.2. Example: temporal photon statistics We present first some results for the temporal photon statistics, looking for evidence of a collective enhancement of the spontaneous emission rate. Define D nmŽt ., m s 1,2, . . . ,5, to be the number of times in an ensemble of quantum trajectories that photon m is emitted in the time interval wt ,t q Dt ., t ' t for m s 1, else t ' t y tmy1 , irrespective of the direction of emission. The D nmŽt ., so defined, are waiting-time distributions. Fig. 2 compares the waiting-time distributions for two atomic separations, 0.25 l0 and 2.5 l 0 . In the
Fig. 2. Distributions of photon waiting times for an atomic dipole orientation a s 0 and separation Ži. ss 0.25 and Žii. ss 2.5 wavelengths; from an ensemble of 20,000 trajectories with bin size gDt s 0.01.
424
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
Fig. 3. Distribution of waiting times for photon 5 for the parameters of Fig. 2; from an ensemble of 100,000 trajectories with bin size gDt s 0.01.
The model studied by Dicke w1x shows the strongest superradiant effect. This is obtained for a
Fig. 4. Distributions of photon waiting times for an atomic dipole orientation a s 0 and separation ss 0.025 wavelengths: Ži. without dipole–dipole interactions Ž Di j s 0., Žii. with dipole–dipole interactions included; from an ensemble of 20,000 trajectories with bin size gDt s 0.01.
™´ ™
small system Ž2p ri jrl0 0 g i j g . with the dipole–dipole interactions neglected. The mean waiting times are predicted to be w m Ž N y m q 1.g xy1 – i.e., for m s 1,2, . . . ,5 the sequence Ž5g .y1 , Ž8g .y1 , Ž9g .y1 , Ž8g .y1 , Ž5g .y1 . We recover this result with an atomic separation 0.025 l 0 and the dipole–dipole terms in Ž4.3a. set to zero. The waiting-time distributions are plotted as curves Ži. in Fig. 4. It is known, though, that in a small system, the dipole–dipole terms are important and can dramatically upset the predictions of the Dicke model w2x. This is what we find for the string of five atoms at the atomic separation 0.025 l 0 . The true distributions of waiting times are curves Žii. in Fig. 4. Thus, when dipole–dipole
Fig. 5. Polar photon counting distributions for an atomic dipole orientation a s 0 and separation ss 2.5 wavelengths: Ža. summing photons 1–5 Ž20,000 trajectories., Žb. summing photons 2–4 when photon 1 is emitted into the solid angle DV s 2p =0.04p around u s 0.5p Ž100,000 trajectories., Žc. summing photons 2–4 when photon 1 is emitted into the solid angle DV s 2p sinŽ0.44p .=0.04p around u s 0.44p Ž100,000 trajectories..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
425
interactions are included the waiting times return to those for independent atoms. 4.3. Example: directional photon statistics A second notable feature of superradiance is the directionality of the emission: the strong dependence of the direction of emission on the spatial configuration of the atoms. Five atoms is too few to make a serious study of these effects; although photon counting records for even five atoms show a rich angular dependence. We illustrate the angular dependence with some conditional distributions. The distributions show how the direction of photon 1 biases the directions of subsequent photon emissions. Define D nmŽ up ., m s 1,2, . . . ,5, p s 1,2, . . . , pmax , to be the number of times in an ensemble of quantum trajectories that photon m is emitted into the
Fig. 7. As in Fig. 5 with dipole orientation a sp r2.
solid angle DV p around up , irrespective of the azimuth and time of the emission. Then 5
Ý s Ž up . '
D nm Ž up .
ms1
Ž 4.10 .
DV p
is the unconditional distribution over polar angle, summed for photon 1 to photon 5. Define D nmŽ up ; u , DV ., m s 2, . . . ,5, p s 1,2, . . . , pmax , to be the number of times in an ensemble of quantum trajectories that photon m is emitted into the solid angle DV p around up , irrespective of the azimuth and time of the emission, and photon 1 is emitted into the solid angle DV around u . Then 5
Ý s 1 Ž up ;u , DV . ' Fig. 6. As in Fig. 5 with separation ss 0.5 wavelengths.
D nm Ž up ;u , DV .
ms2
DV p
Ž 4.11 .
426
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
is the conditional distribution over polar angle, summed for photon 2 to photon 4. Distributions Ž4.10. and Ž4.11., constructed from simulated photon counting records, are plotted in Figs. 5–8 for two atomic separations, 2.5 l0 and 0.5 l0 , and with the atomic dipoles parallel to the axis of the string of atoms, and perpendicular to that axis. In all cases the unconditional distribution s Ž up . shows the sin2-dependence expected of dipole radiators. The conditional distributions, on the other hand, show additional structure correlated with the direction, u , selected for the emission of photon 1; also the locations of the additional peaks track the angle u . The correlation is the seed around which directional emission in a larger sample will develop. It is accounted for by the collective character of the jump operator Jˆ1 , which prepares a phased state following the first photon emission. The acquired phases are
such that the angular dependence of the emission rates Gp q Ž t . acquire, in turn, an interference maximum in the direction of photon 1. Thus, the expectation that the first photon would appear at a maximum, not a minimum, of an interference fringe is, by Baysian inference, written into the subsequence evolution of the quantum trajectory 7. A similar mode of description is employed for interfering Bose condensates with conserved atom number w20–22x and interfering Fock states of the electromagnetic field w23,24x. In fact, collective spontaneous emission is essentially equivalent to the latter example.
5. Summary and conclusions The superradiance master equation formulates the difficult problem of spontaneous emission from N two-state atoms with arbitrary locations in space. Such a system exhibits complex photon statistics, both with respect to the times of the photon emissions and the photon emission directions. We have unraveled the master equation as an ensemble of quantum trajectories, introducing a conditional dynamics inferred from photon counting records. We are thus able to simulate the photon emission sequences. The unraveling follows directly from a rewriting of the master equation in Lindblad form, where one must use collective jump operators that resolve the photon emissions in direction. Explicit results were presented for a string of five atoms, considering different atomic separations and orientations of the atomic dipoles. At intermediate atomic separations superradiant and subradiant emission was obtained; dipole–dipole interactions were found to be detrimental at separations much smaller than a wavelength. Correlations in the directions of successive photon emissions were also demonstrated.
7
Fig. 8. As in Fig. 6 with dipole orientation a sp r2.
The logic of the quantum jumps is retrospective and inferential, not causative. There is no suggestion that the photon detection causes the string of radiators to acquire a phase. We would differentiate measurement Ždetection. scenarios that change the unconditional dynamics of the source – the master equation itself – from those, as here, that do not Žsee also footnote 4..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
A straightforward extension to homodyne and heterodyne unravelings follows from the reported Lindblad form of the master equation, and all the unravelings hold equally with a coherent driving field included. We expect the current work, with such extensions, to be useful in the study of a wide range of collective radiation phenomena.
427
polar coordinates with the polar angle, u , taken with respect to an axis in the direction of d,ˆ we find
di j s g i j s g
3 8p
2
Hd V sin u e
i k 0 nŽ ˆ u , f .P r i j
,
Ž A.5 .
which establishes our result.
References Acknowledgements This work was supported by the National Science Foundation under Grant No. PHY-9531218 and by a Research Award of the Alexander von HumboldtStiftung. H.J.C. thanks Professor W. Schleich for his support and hospitality during his stay at the University of Ulm.
w1x w2x w3x w4x
w5x w6x w7x w8x w9x w10x
Appendix A We show that
w11x
N
Ý Sˆp q P Sˆp† q s Ý g i j sˆjyP sˆiq . pq
Ž A.1 .
i , js1
We make the demonstration for the limit of a continuum of point detectors, in which case
™ Ý d sˆ
w12x w13x
N
Ý Sˆp q P Sˆp† q p, q
ij
jyP
sˆiq ,
Ž A.2 .
i , js1
with
di j s g
3 8p
Hd V ˆ 1 y Ž dˆP Rˆ . R
2
ˆ
e i k 0 RP r i j ;
w16x w17x
Ž A.3 .
we must show that the g i j are equal to the d i j . The g i j arise from a standard calculation in perturbation theory w7,9,10x. Formally, they are given by an integral similar to ŽA.3.:
gi j s g
3
2
Hd V ˆ Ž eˆ ˆ P dˆ. e 8p k
k
ˆ rij i k 0 kP
,
w14x w15x
Ž A.4 .
where the unit vector eˆkˆ is perpendicular to kˆ and lies in the plane containing kˆ and d.ˆ In fact, the integral ŽA.4. is precisely ŽA.3.; evaluating both in
w18x w19x w20x w21x w22x w23x w24x
R.H. Dicke, Phys. Rev. 93 Ž1954. 99. M. Gross, S. Haroche, Phys. Rep. 93 Ž1982. 301. R.G. DeVoe, R.G. Brewer, Phys. Rev. Lett. 76 Ž1996. 2049. U. Eichmann, J.C. Berquist, J.J. Bollinger, J.M. Gilligan, W.M. Itano, D.J. Wineland, M.G. Raizen, Phys. Rev. Lett. 70 Ž1993. 2359. S. Inouye et al., Science 285 Ž1999. 571. A.A. Belavkin, B.Ya. Zeldovich, A.M. Perelomov, V.S. Popov, Sov. Phys. JETP 56 Ž1969. 264. R.H. Lehmberg, Phys. Rev. A 2 Ž1970. 883. R.H. Lehmberg, Phys. Rev. A 2 Ž1970. 889. G.S. Agarwal, Phys. Rev. A 2 Ž1970. 2038. G.S. Agarwal, Springer Tracts in Modern Physics, vol. 70, Springer, Berlin, 1974. H.J. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, New Series: Monographs, vol. m18, Springer, Berlin, 1993. K. Mølmer, Y. Castin, Quantum Semiclass. Opt. 8 Ž1996. 1. P. Zoller, C.W. Gardiner, Quantum Noise in Quantum Optics: The Stochastic Schrodinger Equation, in: Quantum Fluc¨ tuations, Les Houches, Session Lxiii, 27 June – 28 July, 1995, Elsevier, New York, 1997. M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70 Ž1998. 101. I. Percival, Quantum State Diffusion, Cambridge University Press, Cambridge, 1998. G. Lindblad, Commun. Math. Phys. 48 Ž1976. 119. H.J. Carmichael, Quantum Jumps Revisited: An Overview of Quantum Trajectory Theory, in: A. Jadczyk, Ph. Blanchard ŽEds.., Quantum Future, Proc. Xth Max Born Symposium, Wrocław, September 24–27, 1997, Springer, Berlin, 1999. H.J. Carmichael, S. Singh, R. Vyas, P.R. Rice, Phys. Rev. A 39 Ž1989. 1200. H.J. Carmichael, J. Opt. Soc. Am. B 4 Ž1987. 1588, Appendix A. Y. Javanainen, S.M. Yoo, Phys. Rev. Lett. 76 Ž1996. 161. J.I. Cirac, C.W. Gardiner, M. Narashewski, P. Zoller, Phys. Rev. A 54 Ž1996. R3714. J. Ruostekoski, M.J. Collet, R. Graham, D.F. Walls, Phys. Rev. A 57 Ž1998. 511. Y.-T. Chough, Phys. Rev. A 55 Ž1997. 3143. K. Mølmer, Phys. Rev. A 55 Ž1997. 3195.
Optics Communications 179 Ž2000. 417–427 www.elsevier.comrlocateroptcom
A quantum trajectory unraveling of the superradiance master equation 1 H.J. Carmichael a b
a,),2
, Kisik Kim
b
Abteilung fur ¨ Quantenphysik, UniÕersitat ¨ Ulm, D-89069 Ulm, Germany Department of Physics, Inha UniÕersity, Inchon 402-751, South Korea
Received 23 September 1999; received in revised form 12 November 1999; accepted 16 November 1999
Abstract We unravel the superradiance master equation into quantum trajectories whose stochastic wavefunction is conditioned on photon counting records resolved in photon emission time and direction. Numerical results are reported which illustrate the wealth of statistical information made available by simulating the counting records. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Superradiance and related collective radiation phenomena have been topics of interest in atomic physics and quantum optics since the work of Dicke w1x. Although that work dealt with a somewhat idealized model, a great deal of attention has been given to more accessible versions of superradiance in the intervening years w2x. Today, in view of a series experimental developments, something much closer to Dicke’s original conception can be realized. With the use of ion traps and laser cooling methods, atomic radiators can be localized within an optical
) Corresponding author. E-mail: [email protected] 1 We dedicate this paper to Marlan Scully on the occasion of his 60th birthday. 2 On leave from the Department of Physics, University of Oregon, Eugene, Oregon 97403-1274.
wavelength. While localizing a number of ions simultaneously is problematic due to Coulomb repulsion, an experiment with a pair of trapped Baq 138 ions has reported observations of superradiant and subradiant emission at approximately the 1% level w3x. In a second, more extended system of two trapped 198 Hgq ions, Young’s interference, but not superradiance, has been observed w4x. Experimental work with cold neutral atoms and Bose–Einsten condensates opens yet another path to interesting new studies of collective radiation phenomena w5x. Theoretical developments since Dicke’s work have also been very broad. In particular, a master equation for an extended spatial arrangement of atoms has been derived w6–10x. Static dipole–dipole interactions are included in this equation, but retardation and propagation effects are not taken into account. Our work is directed towards this superradiance master equation. We reformulate the equation to make the rich photon statistics of superradiance accessible – possibly to analysis, but at least to the power of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 9 4 - X
418
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
numerical simulations. We have in mind not only the statistics of the photon emission times, but also the statistics of the photon emission directions; thus, the stochastic initiation of directed emission is something, in particular, we would wish eventually to study. The present paper is merely a preparation for anything quite so ambitious. Here, we execute an unraveling of the master equation on the basis of quantum trajectory ideas w11x. The density operator is replaced by an ensemble of pure states, each of which undergoes a stochastic evolution. The evolution follows the individual photon emissions through a sequence of quantum jumps, the state being labelled at any time by the accumulated record of photon emissions – the photon counting record. There has been considerable interest in such unravelings of master equations in recent years w11–15x. A notable feature of the current example is that in the case of the superradiance master equation the jump operators are generally collective atomic lowering operators, not the lowering operators for the individual atoms. Each collective operator is labelled by a direction of photon emission and this provides access to a detailed statistical description of the angular distribution of the radiation. We review the model and the superradiance master equation in Section 2. In Section 3 we define what we mean by photon counting records and develop the unraveling of the time-dependent density operator as an ensemble of pure states labelled by these records. We describe the stochastic, quantum trajectory implementation of the unraveling in Section 4, and also report in this section the results of some numerical simulations. As an example, we consider the radiative decay of a string of five initially excited atoms; results illustrating basic features of both the temporal and directional photon statistics are reported. A summary and some brief conclusions are presented in Section 5.
nance frequency v 0 and with aligned dipole moments d. For each atom there are pseudo-spin operators sˆiy, sˆiq, and sˆi z , with commutation relations
sˆiy , sˆjq s yd i , j sˆi z ,
sˆi " , sˆj z s .2 d i , j sˆi " . Ž 2.1 .
The master equation describing spontaneous emission from such a collection is standard. In the electric-dipole, rotating-wave, and Born–Markov approximations, the reduced density operator for the internal state of the atoms satisfies w6–10x dr dt
s Lr ,
Ž 2.2 .
where L is the generalized Liouvillian Žsuperoperator. L ' yi v 0
1 2
N
Ý sˆi z ,P is1
N
yi
Ý
Di j sˆiq sˆjy ,P q
i/js1
N
1 2
Ý
gi j
i , js1
= 2 sˆjyP sˆiqy sˆiq sˆjyPyP sˆiq sˆjy ,
ž
Ž 2.3 .
/
with
Di j ' g
3 4
½
y 1 y dˆP rˆi j
ž
q 1 y 3 dˆP rˆi j
ž
2
ž
/
2
/
cos j i j
ji j sin j i j
j i2j
q
cos j i j
j i3j
/5 Ž 2.4 .
and
gi j ' g
3 2
½
1 y dˆP rˆi j
ž
q 1 y 3 dˆP rˆi j
ž
sin j i j
2
/
2
/
ji j
ž
cos j i j
j i2j
y
sin j i j
j i3j
/5
,
Ž 2.5 .
2. The source master equation where We consider a collection if N two-state atoms located at positions r i , i s 1, . . . , N, all with reso-
j i j ' k 0 ri j s 2p ri jrl0 ,
ri j ' ri y rj
Ž 2.6 .
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
and
gs
1
4v 03 d 2 3
4pe 0 3"c
Ž 2.7 .
is the Einstein A coefficient; dˆ and rˆi j are unit vectors in the directions of d and r i j . The Liouvillian Ž2.3. is comprised of three terms: one proportional to v 0 which evolves the atoms freely, one with coefficients Di j to account for the static dipole–dipole interactions, and the third term with coefficients g i j describing the actual process of photon emission. Solving the master equation is a difficult task. Its solution depends in a sensitive way on the spatial arrangement of the atoms, and even a moderate number of atoms leads to a very large number of equations. For systems of macroscopic size with the atoms distributed randomly in space, approximations are justified that can simplify the problem w2x. In the opposite extreme, the two-atom case can of course be solved exactly with no particular difficulty w8x. Our approach, using quantum trajectory theory, is suited best to systems of intermediate size. Implemented numerically, it can potentially solve the master equation for a few to a few tens of atoms. It is likely to prove useful, though, even for macroscopic numbers of atoms, since it accounts, quite generally, for the stochastic initiation of superradiant emission. It is indeed the stochastic nature of the emission that interests us the most. Ultimately, it is not just the solution to the master equation that we seek; we would like to make a full characterization of the statistics of the emitted light; and not only is the distribution of photon emission times of interest but also the distribution of the photon emission directions. Traditionally, one obtains information on such things by calculating correlation functions of the electromagnetic field. Of course, formally, through the quantum regression theorem, the solution to the master equation is needed here as well. An explicit calculation of correlation functions sets out, however, on an indirect path to the mentioned distributions. The quantum trajectory formalism goes directly to this raw statistical information. It does so by unraveling the master equation evolution into an ensemble of photon emission sequences, each sequence labelled by specific emission times and directions. A numerical algorithm exists to implement the
419
unraveling and generate realizations of the sequences of emissions. The unraveling and the numerical algorithm are the subjects of the next two sections. 3. Unraveling the master equation with photon counting records We aim to write the density operator r Ž t . as an explicit mixture of pure states and replace its evolution under Ž2.2. by a stochastic evolution of those states. It is well-known that this can be done for any master equation in Lindblad form and an extensive literature exists describing the various procedures that might be followed w11–15x. We wish to achieve more, however, than a formal pure-state decomposition of r Ž t . 3 ; we require each pure state to be labelled by a specific photon emission sequence. To achieve this end, we follow Carmichael’s procedure for unraveling a master equation by working backwards from photon counting records to an inferred dynamics of the source w11x. Setting aside the atoms and their master equation for the moment, we begin by considering the electromagnetic field that is radiated by the atoms. Solving Heisenberg equations of motion in the Markov approximation, the electric field operator may be written as a sum of a free field and a source field. Its positive frequency component is w7,10,11x EˆŽq. Ž R ,t . s Eˆf Žq. Ž R ,t . q EˆsŽq. Ž R ,t . ,
Ž 3.1 .
with EˆsŽq. Ž R ,t R . s y =
1
v 02 d
4pe 0 c 2
N
Ý is1
ž dˆ= Rˆ / = Rˆ e i
Ri
i
i k 0 Ž R iyR.
sˆiy Ž t . ,
Ž 3.2 . 3
A Lindblad master equation adds to the Liouville equation ˆ r xr i" a series of terms in the form ÝmŽ2 Lˆm r Lˆ†m y r˙ s w H, Lˆ†m Lˆm r y r Lˆ†m Lˆm . w16x. The most direct way to recast the superradiance master equation in this form would be to diagonalize the matrix of coefficients g i j . The operators Lˆm arrived at in this way are merely formal, however. It would appear that in general, when the atoms are sufficiently close to one another that they are not resolvable as individual sources, these operators are not associated with any accessible Žobservable. physical process in the manner of the operators Sˆp q in Eq. Ž4.4..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
420
where t R ' t q Rrc, t ) 0, and R i ' R y ri .
Ž 3.3 .
Assuming ri j < R i , for all i and j, we may replace Eq. Ž3.2. by EˆsŽq. Ž R ,t R . s yeˆRˆ EˆsŽq. Ž R ,t R . ,
Ž 3.4 .
where EˆsŽq. Ž R ,t R . s
1
v 02 d
1 y Ž dˆP Rˆ .
4pe 0 c 2 R N
2 1r2
™
emission from N atoms, every record asymptotically Ž t `. contains exactly N events Ž Cm ,Tm .. An ensemble of repeated experiments realizes an ensemble of different counting records. A given record appears with a particular probability in the ensemble and the photon statistics of the emitted light is characterized by these record probabilities 5. They are determined by the source-field operator Ž3.5. and the initial state N
r Ž 0. s < c Ž 0. : ² c Ž 0. < , ˆ
= Ý eyi k 0 RP r isˆiy Ž t .
< c Ž 0 . : s Ł < q :i is1
Ž 3.5 .
Ž 3.8 .
is1
and eˆRˆ is a unit vector in the direction of Ž dˆ= Rˆ. = ˆ The origin of coordinates is chosen to be at the R. N center of the distribution of atoms ŽÝ is1 r i s 0.. We base the unraveling of the master equation on photon counting records for the electromagnetic field Ž3.5.. Consider an array of Žimagined 4 . ideal photodetectors covering the surface of a sphere of radius R. Individual detectors are located at Ž up , f q ., p s 1, . . . , pmax , q s 1, . . . ,qmax and subtend solid angles DV p q at the center of the sphere. The detection events registered by this array define the photon detection records. We denote a specific record, up to time t, by
REC4 ' Ž Cn ,Tn . ; Ž Cny1 ,Tny1 . ; . . . ; Ž C1 ,T1 . 4 , Ž 3.6 . where Cm indicates the location of a photon count Ždirection of a photon emission., Cm g Ž p,q, DV p q . ;1 F p F pmax ,1 F q F qmax 4 ,
Ž 3.7a . and Tm ' w tm ,tm q d tm .
Ž 3.7b .
indicates the time of its occurrence, with t G t n ) t ny 1 ) PPP ) t 1 ) 0. In the case of spontaneous
of the atoms. Thus, we introduce superoperators Sp q '
2 e0 c " v0
Ž R 2DV p q . EˆsŽq.
up , f q , Ž Rrc . q
P EˆsŽy. up , f q , Ž Rrc . q ,
Ž 3.9 .
p s 1, . . . , pm ax , q s 1, . . . , q m ax , such that trw Sp q r Ž t .x is the average photon flux, at time t, registered in the solid angle DV p q by the detector located at Ž up , f q .. Substituting from Eq. Ž3.5., Sp q s g
3 8p
DV p q 1 y dˆP Rˆ p q
N
=
Ý
ž
2
/
ˆ
eyi k 0 R p qP r i j sˆjyP sˆiq ,
Ž 3.10 .
i , js1
where Rˆ p q ' RˆŽ up , f q . is a unit vector in the direction of Ž up , f q .. Here the source-field superoperators are expressed explicitly as sums of superoperators that act on the atoms. Hence, we may use photon detection theory and the Born–Markov treatment of open systems to arrive at an expression for the record probabilities in the convenient form w18,19,11x PREC Ž t . s tr B Ž t y t n . Jn B Ž t n y t ny1 . = Jny 1 PPP J1 B Ž t 1 . r Ž 0 . =dt n d t ny1 PPP d t 1 ,
Ž 3.11 .
4
The photodetectors exist as a purely conceptual device. There is no suggestion that photon counting records must in actual fact be made. Assuming the atoms constitute a genuine photoemissive source, there is no difference between the absorption of a photon by an actual detector and its continued propagation into the reaches of free space.
5 Photon counting records cannot account for every statistical property of the radiated field. Its spectrum, for example, is not determined by photon counting records. The spectrum is accounted for in an independent set of records, which correspond to a complementary unraveling of the same master equation w17x.
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
where Jm g Sp q ;1 F p F pmax ,1 F q F qmax 4 ,
Ž 3.12 .
and BŽ x . ' e L B x , with
Ž 3.13 .
LB ' L y Ý Sp q .
Ž 3.14 .
Every emitted photon is accounted for in the records. Hence, because eÕery photon is account for, the record probabilities may be rewritten in terms of a pure state unraveling of the evolution of r Ž t .. For this purpose, we define the scaled source-field operators Sˆp q '
s
(
2 e0 c
Ž R 2DV p q .
" v0
(
g
3 8p
=
EˆsŽy. up , f q , Ž Rrc . q
DV p q 1 y dˆP Rˆ p q
ž
N
Ý eyi k
ˆ
0 R p qP r i
The essential formal requirement to reach Eq. Ž3.17. from Eq. Ž3.11. is that LB be a commutator. It is for this reason that every photon must appear in the detection record – otherwise there is an efficiency h - 1 multiplying the sum in Eq. Ž3.14.. In the present example it must also be true that
Ý Sˆp q P Sˆp† q s Ý g i j sˆjyP sˆiq , p, q
p, q
Ý Sˆp† q Sˆp q s Ý g i j sˆiq sˆjy ; p, q
sˆiy ,
Ž 3.15 .
such that Sp q s Sˆp q P Sˆp† q .
Ž 3.16 .
The rewriting of the record probabilities Ž3.11. is then PREC Ž t . s ² c REC Ž t . < c REC Ž t . :d t n d t ny1 PPP d t 1 , Ž 3.17 . with < c REC Ž t . : s Bˆ Ž t y t n . Jˆn Bˆ Ž t n y t ny1 . = Jˆny 1 PPP Jˆ1 Bˆ Ž t 1 . < c Ž 0 . : ,
hence HˆB has the alternative form N 1 N HˆB ' " v 0 Ý sˆi z q " Ý Di j sˆiq sˆjy 2 is1 i/js1
½
5
Ý Sˆp† q Sˆp q .
Ž 3.19 .
Ž 3.24 .
pq
ž
/
and developing the formal solution for r Ž t . as a superoperator Dyson expansion with the source-field superoperators Sp q s Sˆp q P Sˆp† q in the role of the interaction terms w18,11x. The expansion takes the form
rŽ t. s
Ý
t
Ý
ns0 C 1 , . . . C n
and ˆ Bˆ Ž x . ' eŽ H B r i " . x ,
2
At this stage the principal pieces of the developed are the records Ž3.6. and their probabilities Ž3.17.. Thus, we have a theory of photon counting, though no explicit connection to an unraveling of r Ž t .. The unraveling itself follows by writing the master equation as dr s Lr s LB q Ý Sˆp q P Sˆp† q r Ž 3.25 . dt p, q
where Jˆm g Sˆp q ;1 F p F pmax ,1 F q F qmax ,
1
`
Ž 3.18 .
Ž 3.23 .
i, j
yi"
is1
Ž 3.22 .
i, j
which is demonstrated in Appendix A. Similarly, it is true that
2 1r2
/
421
tn
H0 d tH0 d t
ny 1
PPP
t2
H0 d t
=< c REC Ž t . : ² c REC Ž t . < , and we arrive in this way at the unraveling r Ž t . s Ý PREC Ž t . < c REC Ž t . : ² c REC Ž t . < ,
1
Ž 3.26 . Ž 3.27 .
REC
Ž 3.20 . where
with HˆB ' " v 0
1 2
yi"
1 2
N
N
Ý sˆi z q " Ý is1
< c REC Ž t . : '
Di j sˆiq sˆjy
i/js1
N
Ý i , js1
g i j sˆiq sˆjy .
Ž 3.21 .
< c REC Ž t . :
(² c
REC
Ž t . < c REC Ž t . :
Ž 3.28 .
is the desired pure state labelled by the photon emission sequence Ž3.6.. The time evolution of the unraveling is implicit in Eq. Ž3.18..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
422
4. A stochastic implementation: quantum trajectories
or in the most likely instance that there is no photon count, a continuous evolution generated by the propagator Ž3.20.; with HˆB in the form Ž3.21., Ž Di i ' 0.
The formal unraveling helps very little with making calculations. Finding a closed form expression for < c REC Ž t .: presents an even more difficult task than solving the master equation. Fortunately, there exists a stochastic implementation of the formalism which opens a path ahead. In this section we describe the implementation of the unraveling as an ensemble of quantum trajectories and illustrate the wealth of statistical information that is made available through simulation of the photon counting records. As an example, we study the temporal and angular distributions of photon emissions from a string of five initially excited atoms.
d < c REC :
4.1. The quantum trajectory algorithm Ideally, we would calculate < c REC Ž t .:, with the record arbitrary, from which the record probabilities and the photon counting distributions follow. The quantum trajectory algorithm generates a solution for < c REC Ž t .:. Not, though, for an arbitrary record, but for a particular record generated stochastically as the trajectory proceeds; repeated trajectories automatically realize an ensemble of records in accord with the probabilities Ž3.17.. We may thus move our attention away from < c REC Ž t .: to the realized records viewed as simulated data. The quantum trajectory algorithm advances from < c REC Ž t .: to < c REC Ž t q d t .: by deciding randomly between all possibilities for the next entry in the record – a photon count in some DV p q or an interval d t without a photon count. The decision is made based on conditional probabilities, probabilities conditioned on the record so far realized and deduced by Baysian inference w11,17x. In the present case, in the time step d t, there is either a quantum jump < c REC Ž t . :
™ Jˆ < c m
REC
Ž t . :,
Ž 4.1 .
with Jˆm selected from amongst the Sˆp q in accord with independent photon counting probabilities
Gp q Ž t . d t s ² c REC Ž t . < Sˆp† q Sˆp q < c REC Ž t . :d t ,
Ž 4.2 .
N
sy
dt
Ý i , js1
ž
gi j 2
q i Di j sˆiq sˆjy < c REC : ,
/
Ž 4.3a . or alternatively, in view of Eq. Ž3.23., d < c REC : dt N
ž
s yi
Ý i/js1
Di j sˆiq sˆjyy
1 2
Ý Sˆp† q Sˆp q pq
/
< c REC : .
Ž 4.3b . It is clear here that Eqs. Ž4.2. and Ž4.3b. constitute the quantum jump unraveling of the Lindblad superoperator N 1 N L s yi v 0 Ý sˆi z ,P y i Ý Di j sˆiq sˆjy ,P 2 is1 i/js1 1 q 2
Ý ž 2 Sˆp q P Sˆp† q y Sˆp† q Sˆp q P yP Sˆp† q Sˆp q / , pq
Ž 4.4 . which is equivalent to the generalized Liouvillian Ž2.3. in view of Eqs. Ž3.22. and Ž3.23.. Thus, the latter equations relating the near-field coefficients g i j to the far-field coefficients d i j ŽAppendix A. are the central results permitting the unraveling. Its main feature when comparing N independent atoms is that the operators Jˆm g Sˆp q 4 are collectiÕe jump operators. This is of course dictated by the physics of the situation; a photon emitted by N closely spaced atoms Ž ri j ; l0 . cannot, in principle, be labelled as an emission by any particular atom 6 . As an example, we consider the string of five atoms illustrated in Fig. 1. The string is laid out along the z axis with a spacing s l 0 between adjacent atoms; we write r i s Ž i y 3 . s l0 zˆ , i s 1,2, . . . ,5. Ž 4.5 . The dipole moment is in the Ž x, z .-plane and dˆP rˆi j s cos a .
6
See footnote 3.
Ž 4.6 .
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
Fig. 1. Coordinate system for the string of five atoms.
The initial state of the atoms is that given in Eq. Ž3.8.. The photon counting is simulated for an array of 5,000 detectors, with
up s Ž p y 0.5 . = 0.02p ,
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latter case, the distributions are essentially exponential with decay times wŽ N y m q 1.g xy1 , m s 1,2, . . . ,5, as is expected for spontaneous emission from independent atoms. In the former, although photon 1 is still emitted with average waiting time Ž5g .y1 , the rate of every subsequent emission is enhanced by the superradiant effect w1–3x. In actual fact, there are more longer as well as shorter waiting times, and the distributions are not exponential any more. Thus, the cascade of photon emissions passes through both superradiant and subradiant states, with the details in a particular instance depending on the directions of the emissions. Subradiant emission is particularly noticeable in the waiting time for photon 5. The long tail in the distribution D n 5 Žt . is illustrated clearly in Fig. 3.
p s 1,2, . . . ,50, Ž 4.7a .
and
f q s Ž q y 0.5 . = 0.02p ,
q s 1,2, . . . ,100.
Ž 4.7b . The detector solid angles vary with the polar angle,
DV p q s 0.02p = 0.02p sin up ,
Ž 4.8 .
and we define the solid angle integrated over azimuth
DV p s 2p = 0.02p sin up .
Ž 4.9 .
Within each photon counting record, m s 1,2, . . . ,5 labels the sequence of photon emissions according to their order in time. 4.2. Example: temporal photon statistics We present first some results for the temporal photon statistics, looking for evidence of a collective enhancement of the spontaneous emission rate. Define D nmŽt ., m s 1,2, . . . ,5, to be the number of times in an ensemble of quantum trajectories that photon m is emitted in the time interval wt ,t q Dt ., t ' t for m s 1, else t ' t y tmy1 , irrespective of the direction of emission. The D nmŽt ., so defined, are waiting-time distributions. Fig. 2 compares the waiting-time distributions for two atomic separations, 0.25 l0 and 2.5 l 0 . In the
Fig. 2. Distributions of photon waiting times for an atomic dipole orientation a s 0 and separation Ži. ss 0.25 and Žii. ss 2.5 wavelengths; from an ensemble of 20,000 trajectories with bin size gDt s 0.01.
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H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
Fig. 3. Distribution of waiting times for photon 5 for the parameters of Fig. 2; from an ensemble of 100,000 trajectories with bin size gDt s 0.01.
The model studied by Dicke w1x shows the strongest superradiant effect. This is obtained for a
Fig. 4. Distributions of photon waiting times for an atomic dipole orientation a s 0 and separation ss 0.025 wavelengths: Ži. without dipole–dipole interactions Ž Di j s 0., Žii. with dipole–dipole interactions included; from an ensemble of 20,000 trajectories with bin size gDt s 0.01.
™´ ™
small system Ž2p ri jrl0 0 g i j g . with the dipole–dipole interactions neglected. The mean waiting times are predicted to be w m Ž N y m q 1.g xy1 – i.e., for m s 1,2, . . . ,5 the sequence Ž5g .y1 , Ž8g .y1 , Ž9g .y1 , Ž8g .y1 , Ž5g .y1 . We recover this result with an atomic separation 0.025 l 0 and the dipole–dipole terms in Ž4.3a. set to zero. The waiting-time distributions are plotted as curves Ži. in Fig. 4. It is known, though, that in a small system, the dipole–dipole terms are important and can dramatically upset the predictions of the Dicke model w2x. This is what we find for the string of five atoms at the atomic separation 0.025 l 0 . The true distributions of waiting times are curves Žii. in Fig. 4. Thus, when dipole–dipole
Fig. 5. Polar photon counting distributions for an atomic dipole orientation a s 0 and separation ss 2.5 wavelengths: Ža. summing photons 1–5 Ž20,000 trajectories., Žb. summing photons 2–4 when photon 1 is emitted into the solid angle DV s 2p =0.04p around u s 0.5p Ž100,000 trajectories., Žc. summing photons 2–4 when photon 1 is emitted into the solid angle DV s 2p sinŽ0.44p .=0.04p around u s 0.44p Ž100,000 trajectories..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
425
interactions are included the waiting times return to those for independent atoms. 4.3. Example: directional photon statistics A second notable feature of superradiance is the directionality of the emission: the strong dependence of the direction of emission on the spatial configuration of the atoms. Five atoms is too few to make a serious study of these effects; although photon counting records for even five atoms show a rich angular dependence. We illustrate the angular dependence with some conditional distributions. The distributions show how the direction of photon 1 biases the directions of subsequent photon emissions. Define D nmŽ up ., m s 1,2, . . . ,5, p s 1,2, . . . , pmax , to be the number of times in an ensemble of quantum trajectories that photon m is emitted into the
Fig. 7. As in Fig. 5 with dipole orientation a sp r2.
solid angle DV p around up , irrespective of the azimuth and time of the emission. Then 5
Ý s Ž up . '
D nm Ž up .
ms1
Ž 4.10 .
DV p
is the unconditional distribution over polar angle, summed for photon 1 to photon 5. Define D nmŽ up ; u , DV ., m s 2, . . . ,5, p s 1,2, . . . , pmax , to be the number of times in an ensemble of quantum trajectories that photon m is emitted into the solid angle DV p around up , irrespective of the azimuth and time of the emission, and photon 1 is emitted into the solid angle DV around u . Then 5
Ý s 1 Ž up ;u , DV . ' Fig. 6. As in Fig. 5 with separation ss 0.5 wavelengths.
D nm Ž up ;u , DV .
ms2
DV p
Ž 4.11 .
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H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
is the conditional distribution over polar angle, summed for photon 2 to photon 4. Distributions Ž4.10. and Ž4.11., constructed from simulated photon counting records, are plotted in Figs. 5–8 for two atomic separations, 2.5 l0 and 0.5 l0 , and with the atomic dipoles parallel to the axis of the string of atoms, and perpendicular to that axis. In all cases the unconditional distribution s Ž up . shows the sin2-dependence expected of dipole radiators. The conditional distributions, on the other hand, show additional structure correlated with the direction, u , selected for the emission of photon 1; also the locations of the additional peaks track the angle u . The correlation is the seed around which directional emission in a larger sample will develop. It is accounted for by the collective character of the jump operator Jˆ1 , which prepares a phased state following the first photon emission. The acquired phases are
such that the angular dependence of the emission rates Gp q Ž t . acquire, in turn, an interference maximum in the direction of photon 1. Thus, the expectation that the first photon would appear at a maximum, not a minimum, of an interference fringe is, by Baysian inference, written into the subsequence evolution of the quantum trajectory 7. A similar mode of description is employed for interfering Bose condensates with conserved atom number w20–22x and interfering Fock states of the electromagnetic field w23,24x. In fact, collective spontaneous emission is essentially equivalent to the latter example.
5. Summary and conclusions The superradiance master equation formulates the difficult problem of spontaneous emission from N two-state atoms with arbitrary locations in space. Such a system exhibits complex photon statistics, both with respect to the times of the photon emissions and the photon emission directions. We have unraveled the master equation as an ensemble of quantum trajectories, introducing a conditional dynamics inferred from photon counting records. We are thus able to simulate the photon emission sequences. The unraveling follows directly from a rewriting of the master equation in Lindblad form, where one must use collective jump operators that resolve the photon emissions in direction. Explicit results were presented for a string of five atoms, considering different atomic separations and orientations of the atomic dipoles. At intermediate atomic separations superradiant and subradiant emission was obtained; dipole–dipole interactions were found to be detrimental at separations much smaller than a wavelength. Correlations in the directions of successive photon emissions were also demonstrated.
7
Fig. 8. As in Fig. 6 with dipole orientation a sp r2.
The logic of the quantum jumps is retrospective and inferential, not causative. There is no suggestion that the photon detection causes the string of radiators to acquire a phase. We would differentiate measurement Ždetection. scenarios that change the unconditional dynamics of the source – the master equation itself – from those, as here, that do not Žsee also footnote 4..
H.J. Carmichael, K. Kim r Optics Communications 179 (2000) 417–427
A straightforward extension to homodyne and heterodyne unravelings follows from the reported Lindblad form of the master equation, and all the unravelings hold equally with a coherent driving field included. We expect the current work, with such extensions, to be useful in the study of a wide range of collective radiation phenomena.
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polar coordinates with the polar angle, u , taken with respect to an axis in the direction of d,ˆ we find
di j s g i j s g
3 8p
2
Hd V sin u e
i k 0 nŽ ˆ u , f .P r i j
,
Ž A.5 .
which establishes our result.
References Acknowledgements This work was supported by the National Science Foundation under Grant No. PHY-9531218 and by a Research Award of the Alexander von HumboldtStiftung. H.J.C. thanks Professor W. Schleich for his support and hospitality during his stay at the University of Ulm.
w1x w2x w3x w4x
w5x w6x w7x w8x w9x w10x
Appendix A We show that
w11x
N
Ý Sˆp q P Sˆp† q s Ý g i j sˆjyP sˆiq . pq
Ž A.1 .
i , js1
We make the demonstration for the limit of a continuum of point detectors, in which case
™ Ý d sˆ
w12x w13x
N
Ý Sˆp q P Sˆp† q p, q
ij
jyP
sˆiq ,
Ž A.2 .
i , js1
with
di j s g
3 8p
Hd V ˆ 1 y Ž dˆP Rˆ . R
2
ˆ
e i k 0 RP r i j ;
w16x w17x
Ž A.3 .
we must show that the g i j are equal to the d i j . The g i j arise from a standard calculation in perturbation theory w7,9,10x. Formally, they are given by an integral similar to ŽA.3.:
gi j s g
3
2
Hd V ˆ Ž eˆ ˆ P dˆ. e 8p k
k
ˆ rij i k 0 kP
,
w14x w15x
Ž A.4 .
where the unit vector eˆkˆ is perpendicular to kˆ and lies in the plane containing kˆ and d.ˆ In fact, the integral ŽA.4. is precisely ŽA.3.; evaluating both in
w18x w19x w20x w21x w22x w23x w24x
R.H. Dicke, Phys. Rev. 93 Ž1954. 99. M. Gross, S. Haroche, Phys. Rep. 93 Ž1982. 301. R.G. DeVoe, R.G. Brewer, Phys. Rev. Lett. 76 Ž1996. 2049. U. Eichmann, J.C. Berquist, J.J. Bollinger, J.M. Gilligan, W.M. Itano, D.J. Wineland, M.G. Raizen, Phys. Rev. Lett. 70 Ž1993. 2359. S. Inouye et al., Science 285 Ž1999. 571. A.A. Belavkin, B.Ya. Zeldovich, A.M. Perelomov, V.S. Popov, Sov. Phys. JETP 56 Ž1969. 264. R.H. Lehmberg, Phys. Rev. A 2 Ž1970. 883. R.H. Lehmberg, Phys. Rev. A 2 Ž1970. 889. G.S. Agarwal, Phys. Rev. A 2 Ž1970. 2038. G.S. Agarwal, Springer Tracts in Modern Physics, vol. 70, Springer, Berlin, 1974. H.J. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics, New Series: Monographs, vol. m18, Springer, Berlin, 1993. K. Mølmer, Y. Castin, Quantum Semiclass. Opt. 8 Ž1996. 1. P. Zoller, C.W. Gardiner, Quantum Noise in Quantum Optics: The Stochastic Schrodinger Equation, in: Quantum Fluc¨ tuations, Les Houches, Session Lxiii, 27 June – 28 July, 1995, Elsevier, New York, 1997. M.B. Plenio, P.L. Knight, Rev. Mod. Phys. 70 Ž1998. 101. I. Percival, Quantum State Diffusion, Cambridge University Press, Cambridge, 1998. G. Lindblad, Commun. Math. Phys. 48 Ž1976. 119. H.J. Carmichael, Quantum Jumps Revisited: An Overview of Quantum Trajectory Theory, in: A. Jadczyk, Ph. Blanchard ŽEds.., Quantum Future, Proc. Xth Max Born Symposium, Wrocław, September 24–27, 1997, Springer, Berlin, 1999. H.J. Carmichael, S. Singh, R. Vyas, P.R. Rice, Phys. Rev. A 39 Ž1989. 1200. H.J. Carmichael, J. Opt. Soc. Am. B 4 Ž1987. 1588, Appendix A. Y. Javanainen, S.M. Yoo, Phys. Rev. Lett. 76 Ž1996. 161. J.I. Cirac, C.W. Gardiner, M. Narashewski, P. Zoller, Phys. Rev. A 54 Ž1996. R3714. J. Ruostekoski, M.J. Collet, R. Graham, D.F. Walls, Phys. Rev. A 57 Ž1998. 511. Y.-T. Chough, Phys. Rev. A 55 Ž1997. 3143. K. Mølmer, Phys. Rev. A 55 Ž1997. 3195.