pump-coupled micromasers: coherent and incoherent coupling

25 May 2000

Ž .Optics Communications 179 2000 289–313www.elsevier.comrlocateroptcom

Pump-coupled micromasers: coherent and incoherent coupling 1

Janos A. Bergou ), Mark Hillery, Pal Bogar´ ´ ´Department of Physics and Astronomy, Hunter College of the City UniÕersity of New York, 695 Park AÕenue, New York, NY 10021, USA

Received 11 July 1999; received in revised form 27 October 1999; accepted 1 November 1999

Abstract

We consider two micromasers coupled via the pumping beam of initially excited two-level atoms traversing the twocavities in a sequence. The atomic populations and coherences pumping the second micromaser are, therefore, prepared bythe first one. To separate the effects of atomic coherence we compare the two cases of incoherent and coherent coupling, i.e.,

Ž .when the states of the atoms are measured between the cavities ‘which path’ and when not. Using an exact solution for theincoherent case, we find that the second micromaser undergoes abrupt transitions between distinct phases of the photonstatistics triggered, via injected absorption, by the first one. In the case of coherent coupling, we consider the exact masterequation of the fields first and, after finding its nonlinear expansion, apply standard Fokker–Planck techniques. Comple-menting the nonlinear treatment, we also employ numerical simulations to investigate a different regime of the system. Inaddition to the incoherent effects, we find that atomic coherence significantly modifies the mean and the variance of the

Ž .photon number of the second field and, at the same time, establishes first order phase locking and second order correlationsbetween the two fields at steady state. The time evolution exhibits abrupt jumps in the locking of the relative phase betweenzero and p accompanied by shut-offs of the second micromaser. The coherence effects are particularly important in theregion of small pumping parameters. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 42.50.Dv; 42.52qx

1. Introduction

Advances in Rydberg-state spectroscopy and inthe construction of high-Q superconducting mi-crowave cavities made it possible to detect the sin-gle-atom single-mode coupling in a series of micro-

) Corresponding author. Fax: q1-212-772-5390; e-mail:[email protected]

1 It is both a pleasure and an honor to dedicate this work to ourteacher, mentor, longtime collaborator and — above all — friend,Professor Marlan O. Scully on the occasion of his 60th birthday.His love, enthusiasm, and dedication to physics inspired many ofus to follow him in this endeavor.

w x Žmaser experiments 1–3 for a recent review, seew x .Ref. 1 . This system realizes the fundamental

w xJaynes–Cummings model 4 of the interaction of atwo-level atom with a single mode of the quantizedradiation field. It exhibits genuine quantum featuresthat are normally masked by macroscopic environ-mental fluctuations. Some of the prominent examplesare the collapse and revival of the Rabi nutationw x w5,6 , generation of nonclassical photon statistics 1–

x w3,7–13 , macroscopic quantum superpositions 14–x17 , and the quantum-nondemolition measurements

w xof the photon number 18,19 . The experimentalproof of the quantum collapse and revival predictedby the Jaynes–Cummings model was reported by

0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 99 00669-0

( )J.A. Bergou et al.rOptics Communications 179 2000 289–313290

w xRef. 6 . For the quantum theory of the micromaser,w xsee Ref. 7 . The semiclassical theory is given by

w xRef. 8 . Quantum island states have been found inw xthe micromaser by Ref. 12 .

Coupling such quantum devices together enablesus to study quantum correlation effects between non-

w x Žlocal radiation fields 20–23 . For the linear theoryw xof pump-coupled micromasers see Ref. 20 ; nonlo-

cal microwave field quantum states have been foundw x .in a similar system by Ref. 21 . Various aspects of

the complementarity principle and other fundamen-tals of quantum mechanics can be investigated in this

w xsystem 24–28 . In the present paper, we considertwo micromasers that are coupled sequentially by thepumping atomic beam according to Fig. 1. The finalstate of atoms is measured in a nonselective way byfield ionization detectors. The usual micromaser con-ditions are assumed: there is at most one atom in thecavities at a time and the cavity lifetime is muchlonger than the interaction time. The first micro-maser affects the behavior of the second one viaaltering the state of the pumping atoms. It changesthe population of the two atomic states and at thesame time creates atomic coherence that, entering thesecond cavity, delivers information about the field inthe first one. The latter effect is responsible for theensuing quantum correlation between the two spa-

w xtially separated fields. It has been shown in Ref. 22that arbitrary entanglement of fields can be producedat steady state in the absence of cavity losses.ŽPump-coupled lossless micromasers with condi-tional measurements of atoms have been studied in

w x .Ref. 22 . In the case when dissipation is alsopresent but does not exceed a certain threshold, pureentanglement can be generated in the transient regimew x23 . Here we show that correlation between the

Fig. 1. Schematic arrangement of two micromasers coupled by abeam of two-level atoms the state of which are measured after theinteraction by field ionization detectors.

fields, though not pure entanglement, can survive atsteady state even in the presence of large dissipation.As a result, the two fields are locked in phase andexhibit interference effects when superposed. Wefind that the magnitude of the steady-state first ordercorrelation is particularly large in the small pumpingregime, switching its sign from positive to negativeat a particular value of the pumping parameter. Thiscorresponds to switching the relative phase of thefields from in-phase to out-of-phase. Similarly, thesecond order correlation exhibits positive and nega-tive values as a function of the pumping parameter,i.e., the photons coming from the two differentsources can be either correlated or anticorrelated.Investigating the behavior of the second micromaseralone we find that atomic coherence contributes tothe field and modifies the photon statistics. Thismanifests itself in significant changes in the expecta-tion value and variance of the photon number. Theeffects disappear as soon as we acquire informationabout the state of the atoms between the two cavities.This is reminiscent of obtaining ‘which-path’ infor-mation and consequently losing interference inYoung’s double-slit experiment.

The paper is organized as follows. In the nextsection we analyze the incoherent problem in detailby presenting a simple exact solution. Section 3introduces the master equation for the coherent prob-lem. The lowest order nonlinear expansion and anensuing Fokker Planck treatment is presented inSection 4. The results of numerical simulations aregiven in Section 5. Section 6 is devoted to discus-sions and summary.

2. Incoherent coupling: exact solution

The first micromaser modifies the behavior of thesecond one via altering the state of the atoms in thepumping beam emerging from the first cavity. Thereare two effects that need to be distinguished. Theincoherent effect, that is due to the change in thepopulation of the two atomic levels and the coherenteffect, that is a result of the coherence generatedbetween the levels. In the present section the formerone is studied alone in order to contrast its character-istics with the latter one in the sections to follow.

( )J.A. Bergou et al.rOptics Communications 179 2000 289–313 291

Therefore, let us assume for now that we doacquire ‘which-path’ information about the state ofthe atoms between the two cavities and, conse-quently, destroy the coherence between them. In thiscase, the second micromaser is driven by a beam ofatoms in an incoherent mixture of their upper andlower states generated by the first micromaser. It is

w x Žshown in Refs. 12,13 the same result from aw x.different derivation can be found in Ref. 29 that

the steady state photon statistics of the field forPoissonian pumping is given by

Y YY 2n r b qn mrNaa m b exY Yp sp . 1Ž .Ł Y Yn 0 Y 2ž /r b q n q1 mrNŽ .ms1 bb m b ex

Here, the double-primes denote quantities pertainingto the second micromaser. It has a cavity decay rate,

Y Yg , average number of thermal photons, n , atom-b

field coupling constant, gY, interaction time, tY and

atom injection rate, r. Thus, the average number ofatoms in the second cavity during the photon lifetimeis given by NY

'rrgY and the pumping parameterex

Y Y Y Y Y Y Yby u ' g t N and b ' sin u mrN . The( (ž /ex m ex

single-primed version of the same parameters willrefer to the first micromaser. We want to note herethat N and u can be varied in the two micromasersex

separately via their free parameters of g, t and g ,although r should be considered the same for both.The steady state populations of the incoming atoms,r and r , are prepared by the first micromaseraa bb

and we want to express them as functions of the firstfield.

It is easy to show via tracing over the field statesX X2 † †² Ž .: Ž .(that r s sin g t a a , where a a is thebb 1 1 1 1

Ž .annihilation creation operator of the first field. Onthe other hand, we also know that the equation ofmotion for the average photon number in the firstmicromaser in the case of Poissonian pumping isgiven by

² :d n1 X X X X2 †² : ² :(sr sin g t a a yg n yn ,Ž .ž /1 1 1 bd t2Ž .

where n 'a† a is the number operator of the first1 1 1w xfield 1–3,7,8 . Thus, the outgoing incoherent atomic

population is very simply coupled to the averagenumber of photons and at steady state it reads as

Fig. 2. Density plot of the photon statistics of the second micro-maser in the pump parameter, u , vs. photon number, k, space for

X Y X YN s300 and N s1000 and n s n s0.5 in the case ofex ex b b

incoherent coupling.

X XŽ² : . Ž .r s n yn rN . Applying this to Eq. 1 , webb 1 b ex

readily find that the steady state photon statistics ofthe second micromaser in the case of incoherentcoupling is given by

Y 2 X X X Y Yn ² :b N y n y n rN q n mrNŽ .Ž .m ex 1 b ex b exY Yp s p .Łn 0 Y 2 X X Y Yž /² : Ž .b n y n rN q n q1 mrNŽ .ms1 m 1 b ex b ex

3Ž .The photon statistics resulting from this expression isshown in Fig. 2, and the average photon numbersand the photon number fluctuations are shown in

Ž . Ž .Fig. 3 a and Fig. 3 b .In order to determine the effect of injecting atoms

which are in an incoherent superposition of theirŽ .upper and lower states it is useful to examine Eq. 1

Yin the case n s0. This gives usb

n Y 2r baa mY Yp sp . 4Ž .Łn 0 Y 2 Yž /r b q mrNŽ .ms1 bb m ex

We can find the peaks in this distribution by notingthat pY will be greater than pY if the last factor inn ny1

the product, i.e. the factor with msn, is greaterthan 1. This implies that a peak will occur when

r bY 2

aa ns1, 5Ž .Y 2 Y

r b q nrNŽ .bb n ex


Ž . ² : ² :Fig. 3. a Average steady state photon numbers, n and n ,1 2

of the first and second micromasers as functions of the pumpparameter, u , depicted by dot-dashed and solid lines, respectively.They are calculated from the photon statistics depicted in Fig. 3.Ž .b Normalized standard deviations of the steady state photon

22² : ² : ² :number distributions, s s n y n r n ,is1,2 of the(Ž .i i i i

first and second micromasers as functions of the pump parameter,u , depicted by dot-dashed and solid lines, respectively.

and if the expression on the left-hand side is adecreasing function of n in the neighborhood of the

Ž .point at which Eq. 5 is satisfied. This equation canbe simplified to give

n2b s . 6Ž .n N r yrŽ .ex aa bb

From this we see that the photon number peaksoccur at the intersection points between a straight

Ž .line starting at the origin the right-hand side andŽthe square of a stretched sine curve the left-hand

.side . The slope of the line is inversely proportionalto the inversion of the injected atoms, so that as theinversion decreases, the slope increases. This has theeffect of moving the peaks to lower photon numbers,

a result which is to be expected. Therefore, bydecreasing the inversion of the atoms going into thesecond cavity, the first maser tends to decrease theaverage number of photons in the second.

Decreasing the inversion of the atoms which enterthe second cavity is not the only effect which thefield in the first cavity has; by its action on the atomsit causes coherence to develop between the twofields, and, in particular, it leads to correlations

w xbetween their phases 20 . Because the phase of thefield in the first cavity is completely random, it isequally likely to be anywhere between 0 and 2p, themean value of the atomic coherence injected into thesecond cavity is zero, so that atomic coherence in theconventional sense is not responsible for the field

w xcorrelations. As was discussed in Ref. 20 , thesecorrelations arise from the fact that the atom canfollow two possible paths between the two cavities,one in which the atom is in its upper state and theother in which it is in its lower state, and these twopaths interfere. Expectation values which measure

² † :the correlations between the fields, such as a a ,1 2

are proportional to the product of the amplitudes foreach path. In the model in which the atoms are in anincoherent superposition of their two states between

Ž .the cavities, which leads to Eq. 3 , this product isalways zero, and the interference effects which pro-duce field coherence are absent. By comparing theresults of the incoherent model to those of the actualsystem we can determine for which quantities theinterference effects matter and how large they are.

3. Coherent coupling: the exact master equation

In Section 2 we have seen how the first micro-maser affected the behavior of the second one viaincoherently altering the population of the atomiclevels. The rest of the paper considers the coherentproblem where we also allow for atomic coherenceto be transferred from the first to the second fieldŽi.e., we do not measure the state of the atoms

.between the cavities and study its effects as com-pared to the incoherent case. In this and the nextsection we deal with the master equation for thetwo-field interaction-picture density operator, r, thatcan be found by finding the coarse-grained timederivative first. The density operator, r Žk ., after the


k th atom passed through the cavities can be calcu-lated as

Y X X† Y†Žk . Žky1. Žky1.r sFr 'Tr U U r r U U ,atom atom

7Ž .

where U X and UY are the time evolution operatorsfor the Jaynes–Cummings model in the first andsecond cavities, respectively, r is the atomicatom

Ž .density operator initially excited atom and we traceover the atomic states to find the reduced operatorfor the fields. For Poissonian pumping the coarse-grained time derivative can be obtained by multiply-ing the change of the density matrix due to one atom,Ž .Fy I r, where I is the identity operator, by theatomic injection rate, r. Adding the Liouvillian terms,LL

Xr and LL

Yr, to account for dissipation in both

cavities we readily obtain the exact master equationgiven by

d rX Ysr Fy1 rqLL rqLL r . 8Ž . Ž .

d t

Here, the Liouvillian, LLXr, for the first cavity reads

as

Xg

X X † † †LL r' n q1 2 a r a ya a ryr a aŽ . Ž .b 1 1 1 1 1 1ž /2

X † † †qn 2 a r a ya a ryr a a , 9Ž .Ž .b 1 1 1 1 1 1

and similarly for the second cavity, LLYr, with g

Y,Y †Ž .n , and a a . In the number representation we canb 2 2

write the master equation in the form given by

d 1n ,m n ,m n ,m n ,m n ,m y11 1 1 1 1 1 1 1 1 1r sy m r qA r

d t 2n ,m n ,m n ,m n ,m n y1,m2 2 2 2 2 2 2 2 2 2

X

m ,n n y1,m n ,m1 1 1 1 1 1qA r qIm ,n n ,m y1 n ,m2 2 2 2 2 2

X Y

n q1,m q1 n ,m1 1 1 1yI qIn ,m n ,m2 2 2 2

Y

n ,m1 1yI , 10Ž .n q1,m q12 2

where n , m , n , m G0, and1 1 2 2

mn1 ,m1 s2 r 1ya

Xa

X CY iŽ n q1 m q1 n q1,m q11 1 2 2n ,m2 2

X X Y X Xyb b C qg n q1Ž ..n q1 m q1 n ,m b1 1 2 2

=Xn qm y2 n m qn n qm q2(ž / ž1 1 1 1 b 1 1

Y Yy2 n q1 m q1 qg n q1(Ž . Ž . Ž ./1 1 b

=Yn qm y2 n m qn n qm q2(ž / ž2 2 2 2 b 2 2

y2 n q1 m q1 , 11(Ž . Ž . Ž ./2 2

and

An1 ,m1 sra

Xb

X SY , 12Ž .n m n ,m1 1 2 2n ,m2 2

together withX

n ,m X X Y X X1 1I s rb b C qg n n m(ž /n m n ,m b 1 11 1 2 2n ,m2 2

=n y1,m y1 n ,mX X1 1 1 1

r yg n q1 n m r ,(Ž .b 1 1n ,m n ,m2 2 2 2

13Ž .

andY

n ,m X X Y Y Y Y1 1I s ra a b b qg n n m(ž /n q1 m q1 n m b 2 21 1 2 2n ,m2 2

=n ,m n ,mY Y1 1 1 1

r yg n q1 n m r(Ž .b 1 2n y1,m y1 n ,m2 2 2 2

yrbX

bX

bY

bY

rn1y1 ,m1y1

n m n m1 1 2 2 n ,m2 2

qraX

bX

aY

bY

rn1 ,m1y1

n q1 m n m1 1 2 2 n y1,m2 2

qrbX

aX

bY

aY

rn1y1 ,m1 , 14Ž .n m q1 n m1 1 2 2 n ,m y12 2

X X X X X XŽ . Ž .with a s cos u nrN ,b s sin u nrN for( (n ex n ex

the first and aY, b

Y for the second micromaser, andn n


Y Y Y Y Y YŽ Ž .. ŽS ssin u nrN y mrN ,C scos u =( (n,m ex ex n,mY YŽ ..nrN y mrN . This master equation reduces( (ex ex

to the detailed balance of the photon statistics givenby

X Xd n ,n n ,n n q1,n q11 1 1 1 1 1

r sI yId t n ,n n ,n n ,n2 2 2 2 2 2

Y Y

n ,n n ,n1 1 1 1qI yI . 15Ž .n ,n n q1,n q12 2 2 2

Summing over n , that is tracing over the second2

micromaser, eliminates the third and fourth terms onthe right hand side resulting in

pX sIX yI , 16Ž .˙n n n q11 1 1

where we use pX'r

X sÝ rn1 ,n1 and IX

'n n ,n n n1 1 1 2 1n ,n2 2

Ý In1 ,n1 . Thus, we reobtain the detailed balancen2 n ,n2 2

equation of the independent first micromaser thatprovides us with its well-known steady-state photon

X X wstatistics after solving I s 0 for p 1–n n1 1

x3,7,8,12,13 .Ž .On the other hand, the sum of Eq. 15 over n1

traces over the first micromaser eliminating the firsttwo terms in the equation and yields

pY sIY yIY , 17Ž .˙n n ,n n ,n q12 1 2 1 2

where pY' r

Y s Ý rn1 , n1 and IY

'n n , n n n , n2 2 2 1 1 2n , n2 2

Ý In1 ,n1

Y

. Hence, we need to solve IY s0,, i.e.,n n ,n1 1 2n ,n2 2

NYb

Y 2 aX 2 r

n1 ,n1 ybX 2 r

n1 ,n1Ýex n n q1 n q12 1 1ž /n y1,n y1 n ,n2 2 2 2n1

q2 NYa

Yb

Ya

Xb

Xr

n1y1 ,n1Ýex n n n q1 n2 2 1 1 n ,n y12 2n1

Y Y Y Yqn n r y n q1 n r s0,Ž .b 2 n y1,n y1 b 2 n ,n2 2 2 2

18Ž .

for the steady state.

Here we used the symmetry of the density matrix,n1 ,m1 m1 ,n1 Ž .r sr . It is clear from Eq. 18 that then ,m m ,n2 2 2 2

off-diagonal terms play an essential role in the pho-Ž .ton statistics see the second term on the l.h.s. . This

is generally true: one can find that the field-densitymatrix contains non-zero off-diagonal terms that arecoupled to the two adjacent diagonals. This makes itdifficult to solve for the photon statistics of thesecond field even numerically. On the other hand,the off-diagonal coupling causing the trouble here isresponsible for the quantum correlation building upbetween the two fields and the inseparability of thedensity matrix — exactly what we are interested in.Tracing over either of the two micromasers wouldresult in a diagonal density matrix and the correla-tion would be lost. It can be shown that we reobtainthe result of Section 2 for the incoherent problemwhen assuming zero off-diagonals and separable

n1 ,m1 X Y Ž .density matrix, r sr r ,, in Eq. 18 .n ,m n ,m1 1 2 2n ,m2 2

Here, rX is the first and r

Y is the secondn ,m n ,m1 1 2 2

field-density matrix. In this case we obtain

Y Y YY X2 2² :N b a qn n pex n n q1 b 2 n y12 1 2

Y Y YY X2 2² :y N b b q n q1 n p s0.Ž .ex n n q1 b 2 n2 1 2

19Ž .X X X 2Ž² : . ² : ŽAfter substituting n yn rN for b and1 b ex n q11

² X 2 : ² X 2 :..similarly a s1y b the incoherentn q1 n q11 1

Ž .photon statistics given by Eq. 3 in Section 2 isreobtained.

In the next two sections we shall attempt to getaround this problem in two different ways. First, weconsider a nonlinear expansion of the master equa-tion and apply standard Fokker–Planck techniques.Then, in Section 5, we present the results of numeri-cal simulations.

4. Nonlinear master equation and the Fokker–Planck equation

4.1. DeriÕation of the Fokker–Planck equation

We obtain the nonlinear master equation by ex-Ž .panding Eq. 8 to fourth order in the coupling


w xconstant, g 20 . This will allow us to gain someanalytical insight into the region around the firstthreshold and the onset of saturation. Assuming thatthe two cavities are identical, i.e., g'gX sgY, t'

tX st

Y, g'gX sg

Y, and therefore N 'N X sNYex ex ex

and u'uX su

Y, the equation in the operator formreads as

1 d r u 2† † †sy r a a qa a q2 a a� 4Ž .1 1 2 2 1 2

g d t 2

4u 2† †q r a a qa aŽ .½ 1 1 2 24!Nex

† † † † †q4a a a a q4 a a qa a a aŽ .1 1 2 2 1 1 2 2 1 2

q3 a a† qa a† q2 a† aŽ .1 1 2 2 1 2

=r a a† qa a† q2 a a† y4 a† qa†Ž . Ž .1 1 2 2 1 2 1 2

= † †r a a a q3a q 3a qa a aŽ . Ž . 51 1 1 2 1 2 2 2

1X † †wq n q1 a r a ya a r.� Ž .b 1 1 1 12

X Y† †qn a r a ya a r q n q1Ž .Ž .b 1 1 1 1 b

= a r a† ya† a rŽ .2 2 2 2

Y † †qn a r a ya a r qH .c., 20Ž .4Ž .b 2 2 2 2

where H.c. stands for Hermitian conjugate. Thisequation could be directly used, e.g., to calculate theequations of motion of quantities such as the photonnumber expectation values in the first and second

² : ² :cavities, n and n , or the first and second1 2² : ² †order field correlation functions, n ' a a q12 1 2

† : ² :a a and n n . Here, we apply another standard1 2 1 2

technique instead that, besides calculating the abovequantities, can also be employed to find the noisecharacteristics of the system. We transform the abovenonlinear master equation into a c-number Fokker–Planck equation for the Glauber–Sudarshan P repre-sentation of the field-density matrix. Finding the driftand diffusion coefficients of the Fokker–Planckequation the first and second moments of the Pdistribution, i.e., the average values of the fieldamplitudes and their fluctuations, can be calculated.

Substituting the Glauber–Sudarshan representationform of the field-density matrix,

r t s d2a d2a P a ,a ) ,a ,a ) ,tŽ . Ž .H 1 2 1 1 2 2

=< :² <a ,a a ,a , 21Ž .1 2 1 2

Ž .into Eq. 20 we arrive at the following equation ofŽ U U .motion for P a ,a ,a ,a ,t1 1 2 2

1 E P2

) )s yE d yE d qE D� a a a a a a a a1 1 1 1 1 1 1 1g E t

qE 2) ) D2

) ) D ) yE d yE ) d )a a a a a a a a a a1 1 1 1 1 1 2 2 2 2

qE 2 D qE 2) ) D ) ) qE 2

) D )a a a a a a a a a a a a2 2 2 2 2 2 2 2 2 2 2 2

qE 2 D qE 2) ) D ) ) qE 2

) D )a a a a a a a a a a a a1 2 1 2 1 2 1 2 1 2 1 2

qE 2) D ) Pq . . . , 22Ž .4a a a a1 2 1 2

Ž U U .where PsP a ,a ,a ,a ,t . We use the notation,1 1 2 2E E 22E s and E s , for the derivatives whilea a aEa Ea Eai i ji i j

the higher order terms, E 3 , i, j, ks1, 2, area a ai j k

omitted. The drift coefficients, lower case d, and thediffusion coefficients, upper case D, are given in theAppendix. Introducing the intensities, I and I , and21

iw1phases, w and w , of the two fields as a ' I e(1 2 1 1i w 2and a ' I e we can transform the Fokker–(2 2

Planck equation into polar form, where P is nowconsidered as a function of I , I , and w , w . In1 2 1 2

this way the Fokker–Planck equation becomes

1 E P2 2s yE d yE d qE D qE D� I I w w I I I I w w w w1 1 1 1 1 1 1 1 1 1 1 1g E t

qE 2 D yE d yE d qE 2 DI w I w I I w w I I I I1 1 1 1 2 2 2 2 2 2 2 2

qE 2 D qE 2 D qE 2 Dw w w w I w I ,w I I I I2 2 2 2 2 2 2 2 1 2 1 2

qE 2 D qE 2 D qE 2 D P ,4w w w w I w I w I w I w1 2 1 2 1 2 1 2 2 1 2 1

23Ž .


Žwhere PsP I ,w , I ,w ,t. The corresponding drift1 1 2 2

and diffusion terms are as follows

u 4X 2 2 2d sn qu q I u y1 y I q3 I q1 ,Ž . Ž .I b 1 1 11 3Nex

24Ž .

u 4Y 2 2d sn qu q I u y1 yŽ .I b 22 3Nex

= 2I q3 I q1q3 I q1 2 I q1Ž . Ž .2 2 1 2

q2cos w yw I IŽ .(1 2 1 2

=

4u 112u y 2 I q2 I q , 25Ž .1 2ž /3N 2ex

d s0, 26Ž .w1

4I u 71 2d ssin w yw u y 2 I q ,Ž .w 1 2 1(2 ž /I 3N 22 ex

27Ž .4u

X 2D s I n qu y 2 I q1 , 28Ž . Ž .I I 1 b 11 1 3Nex

4uY 2D s I n qu y 2 I q1q3 I q3žI I 2 b 2 12 2 3Nex

q4cos w yw I I , 29Ž . Ž .( /1 2 1 2

41 u I1X 2D s n qu y q1 , 30Ž .w w b1 1 ž /4 I 3N 21 ex

41 u I2Y 2D s n qu y q1q3 I q3w w b 12 2 ž4 I 3N 22 ex

q2cos w yw I I , 31Ž . Ž .(1 2 1 2 /43u

D sy I I q2cos w yw I IŽ .(I I 1 2 1 2 1 21 2 Nex

=

4u 52u y 4 I q2 I q , 32Ž .1 2ž /3N 2ex

4cos w yw u 5Ž .1 2 2D s u y I q ,w w 11 2 ž /3N 22 I I( ex1 2

33Ž .

D s0, 34Ž .I w1 1

4uD sysin w yw I I , 35Ž . Ž .(I w 1 2 1 22 2 3Nex

D ssin w ywŽ .I w 1 21 2

=

4I u 51 2u y 4 I q , 36Ž .1( ž /I 3N 22 ex

D sysin w ywŽ .I w 1 22 1

=

4I u 52 2u y I q2 I q .1 2( ž /I 3N 21 ex

37Ž .

It is advantageous to introduce the new variables,w'w yw and m'w qw , since the above co-1 2 1 2

efficients depend only on the relative phase, w, ofthe two fields. This should not be surprising sinceone can intuitively expect that it is the phase differ-ence between the fields that is relevant and not thephases of the fields separately. In this case the driftterms of the newly introduced quantities can bewritten as d sd yd and d sd qd , whilew w w m w w1 2 1 2

their diffusion coefficients are D sD qDww w w w w1 1 2 2

yD , D sD qD qD , and D s2-w w mm w w w w w w wm1 2 1 1 2 2 1 2

Ž .D y D , together with D s D yw w w w I w I w1 1 2 2 1 1 1

D , D sD yD , D sD qD , andI w I w I w I w I m I w I w1 2 2 2 1 2 2 1 1 1 1 2

D s D q D . This is the Fokker–PlanckI m I w I w2 2 1 2 2

equation that we wish to investigate.

4.2. Applying the Fokker–Planck equation

From the Fokker–Planck equation, we now calcu-late the time-dependent and steady-state properties ofthe two fields. It can be shown in general that

I d² : ² :x s d , 38Ž .x

g d t

1 d² :d xd y

g d t

² : ² : ² :s d d y q d d x q 1qd D , 39Ž .Ž .x y x y x y


where x and y are any of the four dynamicalvariables of the equation, I , I , w, and m, d xsxy1 2² :x , and d is either 1 or 0 depending whether thex y

two variables x and y are the same or different,respectively. After expanding the drift coefficients,

² : Ž .d , in terms of d x around the average, x , Eq. 38x

becomes

1 d² : <x (d , 40Ž .²all:x

g d t

where the drift coefficient, d , is taken at the aver-x

age values of all the dynamical variables on the rightŽ .hand side. Similarly, Eq. 39 reads as

1 d² : < ² :d xd y (E d d I d y²all:I x 11g d t

< ² : < ² :qE d d I d y qE d dwd y²all: ²all:I x 2 w x2

< ² : < ² :qE d dmd y qE d d I d x²all: ²all:m x I y 11

< ² : < ² :qE d d I d x qE d dwd x²all: ²all:I y 2 w y2

< ² : <qE d q dmd x q 1qd D .Ž .²all: ²all:m y x y x y

41Ž .

Solving the system of differential equations in Eq.Ž .40 we can find the time dependence of the averagedynamical variables, while the system of the alge-

<braic equations, d s0 yields the steady state.²all:x

The results will be given in Section 4.3.Ž .Applying these solutions, Eq. 41 can be used in

a similar fashion to obtain the fluctuations. In partic-ular, taking the time derivative to be zero and solv-

Ž .ing the system of algebraic equations in Eq. 41 forthe present problem, the steady-state noise propertiesof the fields can be calculated as follows

DI I2 1 1² :d I sy , 42Ž . Ž .1 E dI I1 1 ² :all

² : ² : ² :² :d I d I s I I y I I1 2 1 2 1 2

2² :D qE d d IŽ .I I I I 11 2 1 2sy , 43Ž .E d qE dI I I I1 1 2 2 ² :all

D I E dI 2 I I2 2 1 2² : ² :d sy y d I d I ,Ž .I 1 22 E d E dI I I I2 2 2 2² : ² :all all

44Ž .

Dww2² :dw sy . 45Ž . Ž .E dw w ² :all

Ž .Since in the P representation the stochastic aver-Žage of the intensity is connected to the quantum

.mechanical expectation value of the photon number² : ² :as I s n , is1, 2, and the intensity fluctuationi i

to the normally ordered photon number variance as²Ž .2: ² Ž .2 :d I s : Dn : , the total photon number vari-i i

ance in each cavity field is given by

2 2 2² : ² : ² : ² : ² :Dn s : Dn : q n s d I q I .Ž . Ž . Ž .i i i i i

46Ž .

It is sub-Poissonian if the standard deviation,

222² : ² : ² :n y n d IŽ .i i is s s q1 , 47Ž .))ni ² : ² :n Ii i

²Ž .2:is smaller than 1, i.e., d I is negative. This isi

the procedure that we will follow to treat the photonnumber fluctuations.

We now want to examine the correlations be-tween the quantum fluctuations of the field in thetwo cavities. In order to do so we shall study theobservables

1†a ' a qa , 48Ž .Ž .12 ,q 12 122

i†a ' a ya . 49Ž .Ž .12 ,y 12 122

where a 'a a† . These operators are the ones in12 1 2

terms of which difference squeezing between twofields is defined, and in the semiclassical limit they

w xgive us information about their relative phase 30 . Inthis limit we replace the field amplitudes by mean

Žvalues plus small fluctuations, i.e., a ™ r qi i. Ž .dr exp w qdw , is1, 2. This implies thati i i

a ™ r qd r r qd r cos dw ydw ,Ž . Ž . Ž .12 ,q 1 1 2 2 1 2

50Ž .

a ™y r qd r r qd r sin dw ydw .Ž . Ž . Ž .12 ,y 1 1 2 2 1 2

51Ž .


Keeping the lowest order terms in the fluctuations,we find that in the semiclassical limit

2Da ™ r d r qr d r cos w ywŽ . Ž . Ž .12 ,q 1 2 2 1 1 2

2yr r dw ydw sin w yw ,Ž . Ž .1 2 1 2 1 2

52Ž .2

Da ™ r d r qr d r sin w ywŽ . Ž . Ž .12 ,y 1 2 2 1 1 2

2qr r dw ydw cos w yw .Ž . Ž .1 2 1 2 1 2

53Ž .

As we have seen the phase difference between thetwo cavities locks to either 0 or p, and this implies

Ž .2 Ž .2Ž .2that Da ™ r r dw ydw . Therefore, in12,y 1 2 1 2Ž .the semiclassical limit dr <r and dw <1 andi i i

for wsw yw equal to either 0 or p, the relative1 2

phase fluctuations can be expressed as

2² :DaŽ .12 ,y2² :Dw s . 54Ž . Ž .² :² :n n1 2

The variances of a can be expressed using their12 "

normally ordered forms as

12 2² : ² : ² :Da s : Da : q n qn 55Ž . Ž . Ž .12 ," 12," 1 24

and, therefore,

2² : ² :: Da : 1 n qnŽ .12 ,y 1 22² :Dw s qŽ . ² :² : ² :² :n n 4 n n1 2 1 2

² :1 I q I1 22² :s dw q , 56Ž . Ž .² :² :4 I I1 2

²Ž .2: ² : ² :where dw is the phase noise and I , I are1 2

the average intensities obtained from the P represen-tation. Note that this expression implies that if

² :I I1 22² :Dw - , 57Ž . Ž .² :² :4 I I1 2

then the two-mode state is nonclassical.We can also determine when, and if, the fields are

difference squeezed. The observables a obey the12,"

uncertainty relation

12 2 2² :² : <² : <Da Da G n yn , 58Ž . Ž . Ž .12 ,q 12,y 1 216

and a two-mode state is said to be difference squeezedw xif 30

12² : <² : <Da - n yn . 59Ž . Ž .12 ," 1 24

Such a state is often highly nonclassical, because thecondition for being nonclassical,

12² : <² : <Da - n qn , 60Ž . Ž .12 ," 1 24

is much less stringent than that for difference squeez-ing. In order to check for difference squeezing weshall calculate the quantity

² :² :4 n n1 2 2² :s s DwŽ .d ( <² : <n yn1 2

2² :² :² : ² :4 I I dw q I q IŽ .1 2 1 2s . 61Ž .) <² : <I y I1 2

If s is smaller than 1, then the two-mode field isd

difference squeezed which indicates a high degree ofcorrelation.

A similar procedure can be constructed using theQ-representation of the field-density matrixŽ U U . ² < < :Q a ,a ,a ,a ,t s a ,a r a ,a , by finding1 1 2 2 1 2 1 2

the corresponding Fokker–Planck equation. The es-sential difference between the P- and Q-representa-tions is that they are connected to normally andantinormally ordered operator expressions, respec-tively. Their results for the photon number and phaseexpectation values and the corresponding noises donot differ significantly. We employ the P-representa-tion in this section and the Q-representation in theinvestigation of the time dependence of the system.There we need the Q-function because, in most ofthe cases, the numerical solution for the P-represen-tation is not possible due to singularities.

4.3. Results of the Fokker–Planck treatment

4.3.1. The steady-state behaÕiorIn order to find the steady state of the fields we

need to solve the system of algebraic equations,<d , obtained by replacing the time derivative in²all:xŽ .Eq. 40 with zero. Since the first micromaser is


independent of the second one, d together with dI w1 1

Ž Ž .and d do not depend on I or w see Eq. 24 ,I I 2 21 1

Ž . Ž .. <26 and 28 . Thus, d s0, a second order² I :I 11

² :algebraic equation for I , can be solved in itself1² : ² : Ž .for I s n one solution is physical only and1 1

Ž .then the solution can be used in Eq. 47 to find s .n1

These results recover the well-known exact solutionfor a stand-alone micromaser.

Next, let us look at the drift term, d syd ,w w 2

Ž .given by Eq. 27 . The possible steady states ob-<tained from the equation, d s0, are w s0²all:w ss

Ž .and p satisfying sin w s0. This means that theress

is a steady state phase locking between the two fieldsas a result of the coupling. We find the stablesolutions by applying a small perturbation, Dw<

w , to the steady state phase. In this case, thess

equation of motion for the phase becomes

1 d DwŽ .sA sin w qDw ("ADw , 62Ž . Ž .ss

g d t

Ž .where A is the factor multiplying sin w yw in1 2Ž .Eq. 27 , the plus and minus signs correspond to

w s0 and p, respectively, and we used the linearssŽ .expansion, sin Dw (Dw. In order to have a stable

solution the overall sign of "A on the right-handside must be negative. In this case the perturbationdecays back to the steady state while in the othercase diverges. This means that the solutions, w s0ss

or p, are stable if A is respectively negative or² :positive. Using the above solution for I in the1

equation, As0, we finally arrive at a third orderequation in u . The only physical solution of this

'equation, u ( 2 , gives the pumping parameter at2

which the steady-state locking switches from in-phaseto out-of-phase as the pumping is increased. Thispoint coincides with the ‘second threshold’ of theincoherently coupled system of Section 2 where theintensity of the second field drops to approximatelyzero, and the atoms exit the first cavity predomi-nantly in their lower state. Clearly, the present coher-ently coupled system cannot differ significantly fromthe incoherent one at this particular value of thepumping parameter, u , because the dominance of2

the lower atomic state between the cavities preventsthe transfer of atomic coherence into the secondmicromaser. The switch of the relative phase, i.e.,the ‘decoupling’ of the fields, and the drop of theintensity of the second field at u are both the2

² :Fig. 4. Steady state normalized photon number in the second cavity, n rN , obtained from the Fokker–Planck equation in the2 ex

P-representation as a function of the pumping parameter, u . In the upper-left panel the N -dependence is shown by switching fromexX YŽ . Ž .N s50 dot-dash line to N s500 solid line and, at the same time, keeping n sn s0.0 in both cases. The dependence on theex ex b b

X Y Ž .thermal photon number is illustrated in the other three panels using N s100 and n sn s0.0 dot-dash lines while choosingex b bX Y X Y X Yn s3.0 and n s3.0 in the upper-right, n s3.0 and n s0.0 in the lower-left, n s0.0 and n s3.0 in the lower-right panels, all depictedb b b b b b

by solid lines.


manifestations of the depleted upper atomic state.Later we show that the depletion of the upper stateand other related effects can take place not only as afunction of u but also in the time evolution of thesystem, as a function of time.

Let us now look at the steady state intensity of thesecond micromaser in detail. Knowing the solutions

² : ² : <for I and w one can solve d s0, a fourth²all:1 I2

² :order algebraic equation for I , and then apply( 2

the solution to calculate the normalized variances,Ž . Ž .s and s , using Eq. 47 and 59 , respectively.n w2

² : ² :The steady-state photon number, n s I , is2 2

depicted in Fig. 4 as a function of the pumping'parameter, u . We do find the point, u ( 2 , where2

the phase locking switches from zero to p and thephoton number drops significantly, in agreement withthe argument in the previous paragraph. The photonnumber exhibits a very high peak between zero and

² : ² :u . We find that n grows faster with u than n2 2 1

and the peak exactly coincides with the peak in s .n1

In addition, the peak below u is much higher in the2Ž Ž ..coherently coupled system cf. Figs. 4 and 3 a .

Significant modifications can be found in s , toon2

Ž .see Fig. 5 . The first peak in the incoherent noise ofSection 2 is split into a double-peaked structureexhibiting a decrease of the noise down to the Pois-sonian noise level around u(1. This dip in sn2

² :coincides with the peak of n and s . On the2 n1

other hand, no indications of the second noise-peakŽ Ž ..of the incoherent problem see Fig. 3 b around

u(3.5 are found, the noise in this region of u

slowly decreases instead. The discontinuity aroundu(1.5 is due to the breakdown of the P-representa-

² :tion for small n . It is interesting to see how the2

double-peaked structure depends on the parameters,X YN , n and n . To mention two of the most robustex b b

effects only we want to note that increasing Nex

results in an increase of the first sub-peak only. OnXthe other hand, an increase in n , quite surprisingly,b

results in a decrease of the first and, at the sametime, in an increase of the second sub-peak. It can be

Ž . Ž .seen from Eqs. 43 and 44 that the noise in the²Ž .2:first micromaser, d I , contributes to the correla-1

² :tion, d I d I , and the correlation contributes to the1 2²Ž .2:noise of the second micromaser, d I . This sug-2

gests that the extra noise introduced by the thermalradiation in the first micromaser modifies the gener-ated atomic coherence in such a way that, afterentering the second cavity, it can result both inenlarged and reduced photon number fluctuations inthe second micromaser depending on the pumpingparameter.

The noise in the relative phase, s , is also veryw

sensitive to the above control parameters and, in a

Fig. 5. Steady state normalized standard deviation of the photon number in the second cavity, s , corresponding to Fig. 4.n2


Fig. 6. Steady state noise in the relative phase, s , corresponding to Fig. 4. The boundary between squeezed and non-squeezed states isw

Ž . Ž .represented by the s s1 line, while the dotted line separates classical above line and nonclassical under line behavior using N s100w exX Y Ž Ž ..and n sn s0.0 the boundary does not significantly depend on the parameters, see Eq. 61 .b b

Ž .way, behaves similarly to s see Fig. 6 . Then2

increase of N results in a significant increase in theexXmain peak of the noise while the increase in nb

results in a decrease of the noise in the region ofsmall u . We find no squeezing in the relative phase

Ž .the condition is s -1 and, apart from the lower-w

Žleft panel in Fig. 6, no nonclassical behavior theŽ ..condition is given by Eq. 61 . However, it is

surprising to see that finite temperature in the firstŽ .micromaser and there only makes the system be-

² :Fig. 7. Steady state normalized first order correlation, n r2 N , corresponding to Fig. 4.12 ex


Fig. 8. Steady state second order correlation in the form, R Ž2., corresponding to Fig. 4.

Žhave nonclassically for u-0.5 lower-left panel in.Fig. 6 . This is similar to the case above for sn2

where we found that thermal photons in the firstcavity can result in reducedrenlarged photon num-ber noise in the second micromaser. The lower-rightpanels in Figs. 5 and 6 show that thermal radiation inthe second cavity, however, results in increased sn2

and, at the same time, moves the system into theclassical regime in s . Clearly, thermal noise in thew

first cavity can strengthen the coherence between thetwo fields in some region of the pumping parametervia enhancing the generated atomic coherence, whilethermal noise in the second cavity can only weakenit by scrambling the phase between the injectedatomic coherence and the field of the second micro-maser.

Finally, we want to study the steady-state correla-tion between the two fields. The first order correla-

² : ² † 2:tion, n s a a qa † , is calculated in the pre-12 1 2 1 a² : Ž² :.² :sent formalism as n s2cos w I I , while(12 1 2

² : ² :the second order correlation as n n s I I . In1 2 1 2

order to find the steady states of these quantities wetransform the Fokker—Planck equation by introduc-ing the new variables, I s I I or I s I I , calcu-(x 1 2 x 1 2

late their drift coefficients, and solve the algebraic< ² :equations, d s0, for I . Let us first consider²all:I xx

² :n and, besides keeping I , introduce the new12 1

variable, I s I I . After transforming the(x 1 2

Fokker–Planck equation the new drift term for I isx

given by

21 I I 1 Ix 1 xd s d q d y DI I I I Ix 1 2 1 1ž / ž /2 I I 4 I I1 x x 1

2I1q D yD , 63Ž .I I I I2 2 1 2ž /Ix

while, obviously, d has not changed. KnowingI1

² : ² : <I and w the equation, d s0, becomes a²all:1 Ix

² :fourth order equation for I that can be solvedx² :and, thus, the first order correlation, n s212

Ž² :.² :cos w I , can be calculated at steady state. Itx

can be seen in Fig. 7 that this correlation is zero atu , while its magnitude has two maxima on the two2

sides of u . The change of its sign from positive to2

negative is, obviously, due to the switch of therelative phase from zero to p. The essential messageof this result is that the two fields will exhibitstationary interference effects when superposed.

In order to find the second order correlation,² :n n , let us introduce the new variable as I s I I1 2 x 1 2

and find its drift term in the transformed Fokker–Planck equation,

Ixd s d q I d qD . 64Ž .I I 1 I I Ix 1 2 1 2I1


² :Similarly to the above, using the solutions for I1² : <and w the equation, d s0, is a fourth order²all:Ix

² :equation for I I . We show in Fig. 8 the normal-( 1 2

ized second order correlation given by

² :n n1 2Ž2.R s y1. 65Ž .² :² :n n1 2

It can be seen that the photons coming from the twoŽ Ž2. .separate sources are positively correlated R )0

Ž Ž2. .below u and anticorrelated R -0 above it.2ŽAnticorrelation is sensitive to N the higher N theex ex

Ž2..lower the magnitude of R but insensitive tothermal photons, while the opposite holds for posi-tive correlation.

4.3.2. The time-dependent behaÕiorLet us now investigate the time evolution of the

system toward steady state. A scaled time is intro-duced as Tsg t and we employ the Q-representa-tion to investigate the time-dependent behavior ofthe system because it turns out to be more conve-nient for this purpose. The drift coefficients obtainedfrom the Fokker–Planck equation for the Q-functionare given by

u 4XQ 2d sn q1q I u y1 y I I y1 , 66Ž . Ž . Ž .I b 1 1 11 3Nex

u 4YQ 2d sn q1q I u y1 y I I y1Ž . Ž .I b 2 2 22 3Nex

q3I 2 I y1 q2cos w yw I IŽ . Ž .(1 2 1 2 1 2

=

4u 52u y 2 I q2 I y , 67Ž .1 2ž /3N 2ex

and

I1Q Qd syd sysin w ywŽ .w w 1 2 (2 I2

=

4u 12u y 2 I y , 68Ž .1ž /3N 2ex

since dQ s0. We numerically solve the system of 3w1

coupled differential equations, by using these termsŽ . ² : ² : ² : ² :in Eq. 38 , for x s I , I and w si-Q Q Q1 2

multaneously. Some of our selected results are asfollows.

Ž . Ž .In Fig. 9 a and Fig. 9 b some trajectories aredepicted as functions of time. It can be seen that forus1.1 the phase jumps occur if the fields are startedout-of-phase, opposite to the final equilibrium. How-ever, it is interesting to note that for us2 phasejumps can take place even if the system has been

Žstarted in its final equilibrium phase see time evolu-Ž ..tion in the lower left-hand panel in Fig. 9 b . There

are actually two phase jumps in this case, jumpingout of equilibrium phase and back, resulting in a flipin the intensity of the second micromaser.

The time evolution of the field in the secondmicromaser can be understood as a consequence oftwo coexisting driving mechanisms, the stimulatedemissionrabsorption and the driving atomic coher-ence. We have seen how the former one governedthe incoherently coupled system in Section 2 whilethe new features of the coherent coupling are theconsequences of the latter mechanism. The stimu-lated emissionrabsorption is related to the incoher-ent population of the atoms entering the secondcavity and the intensity of the actual field inside thecavity. The populations of the two atomic statesdetermine the weight of emission as compared toabsorption while the intensity of the actual field isrelated to the magnitude of the energy flowing in thetwo directions. Furthermore, the atoms do not carryany phase information into the second cavity.

In the case of the coherent driving mechanism,however, the main feature is the condition on thephase of the atomic coherence relative to that of thefield. The two coupled oscillators, the atomic dipoleand the field, are locked to a relative phase of pr2and the direction of the flow of energy between thempoints toward the one that is behind. On the otherhand, the populations of the two atomic states limitthe magnitude of the atomic coherence. The smallerthe population difference the stronger the dipole, i.e.,in a definite atomic state the coherence is zero. Thefirst micromaser is independent of the second oneand serves to prepare the atoms for the second fieldby modifying their population coherence. Since ev-ery atom enters the first cavity in its excited stateenergy can flow from the atoms to the field only and,therefore, the phase of the first field is constantly lateby pr2 behind that of the atom. The phase of thesecond field, however, can be both before or behindthe atoms by pr2 depending whether the field gains


Ž . ² : ² :Fig. 9. a : Time development of the system using various initial conditions. The intensities, I and I , and the magnified relativeQ Q1 2X Y² : Ž .phase, 10 w , are depicted by dotted, solid and dot-dashed lines, respectively. The parameters are N s100 and n sn s1.0. b : TimeQ ex b b

² : ² : ² :development of the system using various initial conditions. The intensities, I and I , and the magnified relative phase, 10 w , areQ Q Q1 2X Ydepicted by dotted, solid and dot-dashed lines, respectively. The parameters are N s100 and n sn s1.0.ex b b

energy from the dipole or vice versa. This is whyzero and p are the two possibilities for the lockingof the relative phase between the two fields.

Whether the field in the second cavity is ampli-fied or not depends on the net effect of the twomechanisms, the stimulated emissionrabsorption andthe driving atomic coherence. They can act construc-

tively when pumping or damping the field simultane-ously, and destructively when one of them is pump-ing while the other is damping. Let us illustrate thephysics of the coexistence of the two mechanismsthrough the particular example when the intensity ofthe first field is initially zero. In this case, the stateof the first few atoms reaching the second cavity is


dominantly the excited state and, therefore, the in-version is large but the dipole strength is small in the

Žearly part of the time development. We assume thatthe pumping parameter is small enough so that thechange in the population of the first few atoms whentraversing the first cavity is small as compared to a

.full Rabi cycle. The initial phase of the coherencerelative to the second field depends on the initialrelative phase between the two fields while the

Žstrength of the stimulated emission inversion is.large depends on the initial intensity of the second

field. Suppose we start from a relative phase of p

and the intensity in the second field is very small.The time dependence of this case, using the pumpingparameter us2, is depicted in the lower-left panel

Ž .of Fig. 9 b . Due to the initial phase, p, the atomicdipole is absorbing from the second field in thebeginning and due to the small initial intensity thestimulated emission is weak. Thus, after a very smallincrease in the intensity due to the weak stimulatedemission and zero dipole strength at Ts0 the fieldremains very small for some time. However, thedipole strength is increasing and the weight of thestimulated emission as compared to the absorption isdecreasing in time due to the increasing intensity ofthe first field and the consequent increase of thelower state population. Therefore, the intensity be-comes zero, i.e., the second field depletes com-pletely, when the dipole together with the stimulatedabsorption have used up all the energy provided by

Žthe stimulated emission. At this point around Ts.0.75 driving via the dipole takes over and, after

switching the phase between the atom and the sec-ond field to qpr2 due to the still remaining inver-sion, starts pumping the field. This jump in therelative phase is not related to the depletion of the

Župper atomic state as the one that we have seen in.the steady state behavior but to the depletion of the

field by the dipole instead. This is a coherence effectwhere the atomic coherence turns the field off byopposing the pumping via the weak stimulated emis-sion and then takes over to drive the field at theopposite relative phase. Due to the simultaneouspumping by the dipole driving mechanism and the

Žstimulated emission still strong, because the inver-.sion is positive the intensity of the field starts to

grow fast. At the same time, as a result of theincreasing field in the first cavity the population of

the lower atomic state is constantly increasing. Thismeans that the weight of the stimulated emission inthe pumping is constantly decreasing until finally theatoms become pure absorbers when the upper statepopulation disappears. At the same time, the dipolestrength increases to a maximum when the inversionturns to zero and, similarly to the stimulated emis-sion, becomes zero when the upper state is depleted.Since its phase is independent of the inversion, thedipole constantly pumps the field because its phase,qpr2, relative to the field does not change with thechange of the atomic population during this intervalof time. It is the stimulated absorption only thatlowers the intensity of the field. The result of the twoprocesses is that the intensity, after reaching a maxi-mum, falls back to zero when the upper state be-

Ž .comes empty around Ts2 . This shut-off of thesecond field is also connected to a jump in therelative phase between the fields opposite to the oneabove. The atomic population, as the intensity in thefirst field is increasing in time, starts flowing back tothe upper state. As a consequence, some stimulatedemission appears besides the still dominant absorp-tion together with a very weak atomic coherence.Since the inversion is still negative the phase of thearising atomic coherence relative to the second fieldbecomes ypr2 indicating that energy flows fromthe field to the atom. This implies a jump by p

when comparing the relative phase of the newlyarising field after the shut-off to the one before it.Clearly, this shut-off is an incoherent effect since itis the result of the depletion of the incoherent upperstate of the atoms. The resulting stimulated absorp-tion shuts the field off. Then the driving, albeitweak, dipole mechanism develops a new field at adifferent phase. Finally, the intensity of the first fieldsettles to a steady state, freezing the inversion to aparticular value, and, as a consequence of this, deter-mining the weight of stimulated emission as com-pared to absorption. The atomic coherence alsoreaches its steady state remaining in the same phaseafter the second phase jump.

It is clear from the above that the two intensityshut-offs and the corresponding phase jumps are theresults of two different mechanisms. The first phasejump is a consequence of the particular initial condi-tion for the relative phase and the resulting initialbehavior of the dipole driving mechanism. This can


be seen by starting the same system from anotherinitial relative phase, e.g. zero, as depicted in the

Ž .upper two panels of Fig. 9 b . The first phase jumpis missing and only the second one can be found. Onthe other hand, the two panels in the right column inthis figure show what happens when the second fieldis started from a high initial intensity. It can be seenin the lower-right panel that due to the high intensityand high inversion the stimulated emission is sostrong in the beginning that it dominates over theabsorption effect of the dipole mechanism. It followsthat the first shut-off of the field and the correspond-ing phase jump are missing in this case. However,the figure exhibits a significant drop in the intensitythat, in fact, realizes the ‘second’ shut-off due to thestill effective depletion of the upper atomic state.The corresponding ‘second’ phase jump is obviouslymissing since the dipole is already in the properphase. This means that, contrary to the first intensityshut-off, the second one does not depend on particu-lar initial conditions for the intensity and phase ofthe second field. Since it is connected to the deple-tion of the atoms, not the fields, it depends on thefirst field only. Clearly, the first atoms in the beamare not completely inverted for nonzero initial firstfield but their population between the cavities isturned toward the lower state as the initial intensityof the first field is increased. This makes the first andsecond intensity shut-offs shift toward the first atomsof the pumping beam, i.e., toward earlier times, withdecreasing time separation. Finally, when the initialintensity of the first field becomes so high that itdepletes the upper state population of the very firstatom completely the first intensity shut-off pointdisappears. As a summary, we have seen that therealization of the first intensity shut-off dependsstrongly on the initial conditions regarding the rela-tive phase and the intensities of the two fields. Thesecond shut-off, however, always exists. The instantwhen it occurs depends on the initial conditions forthe intensity of the first field.

Finally, we want to note that the above qualitativecharacteristics of the time dependence are generaland do not depend significantly on the pumpingparameter, provided we remain inside the regionconsidered in the present section. As it is apparent

Ž .from comparing Fig. 9 a , illustrating the case ofŽ .us1.1, to Fig. 9 b , where us2, the system is

governed by the same above discussed mechanismsin both cases except that the second shut-off ismissing in the case of us1.1, the first one can berealized only. The second shut-off point is missingfor us1.1 because, in this case, the fields reachtheir steady states earlier than the upper atomic stateis depleted. In other words, the pumping parameterdetermines where the system settles down whenevolving along a universal trajectory. For smallerpumping, i.e. larger losses, the steady state is locatedon an earlier part of the trajectory while in the caseof larger pumping it gets further ahead. This is whythe steady state behavior of the system as a functionof the pumping parameter, u , is reminiscent of thetime evolution for a given u .

In conclusion to this section we want to remark atthe end that the particular strength of this nonlineartreatment is that it is analytical. Besides the formalunderstanding of the system by considering the formof the equations another advantage of this formalismis that we can consider setups with high photon

Ž .numbers i.e., high N without any difficulties.ex

This cannot be said about the exact numerical ap-proaches because, as we will see in the next section,the size of the density matrix in the photon numberrepresentation is greatly limited by the capacities ofthe computers. The price to pay, however, is that thenonlinear theory is not exact but it provides a goodcomplementary to the exact numerical simulationspresented in the following section.

5. Numerical simulations

We assume that the interaction time an atomspends in the cavities is much shorter than the cavity

Žlifetime. In a typical experimental setup the differ-.ence is three orders of magnitude. In this case we

can ignore the decay of the fields during the time anatom is inside the cavities and separate the evolutionof the system into two parts: atom-field interactionŽ . Ž .pumping and decay of the fields damping . Thus,the field-density matrix at the instant when an atom

Ž .leaves cavity 2 can be calculated from Eq. 7 result-Ž .ing in the matrix r 0 , the decay of which is then

calculated from this initial condition as a function oftime by applying the solution of the standard master


equation for a field mode of an empty cavity dampedto a reservoir of finite temperature given by

n ` mk AŽk . yg t Žk .r t se CŽ . 2 Ý Ýn n ,m , l mqkq1Bls0 msny1

=

X ny1 X lA BŽk .r 0 . 69Ž . Ž .mž / ž /A B

Here, r Žk .'r with ksmyn, g is the cavityn nm

decay rate and t is time. The coefficients are givenby

l mqkq l mŽk .C ' y1Ž .n ,m , l ž / ž /l ny1

=mqk mr . 70Ž .( ž /ž / nnqk

Xyg t yg tŽ .Ž . Ž .A' n q1 1ye , B'1qn 1ye , Ab bX Xyg t yg t yg tŽ .'e yn 1ye and B 'A ye , whereb

n is the average number of thermal photons. Abw xderivation of this solution is given in Refs. 23,31,32

Žfor the generation of entangled trapping states inw xpump-coupled micromasers see Ref. 23 ; the effect

of cavity losses is also discussed here; the problem

has been studied in great detail in, for example, Ref.w x31 ; for a recent solution see also the example in

w x.Ref. 33 . It is easy to show that in the special caseŽ .of zero temperature, i.e., n s0, Eq. 69 reduces tob

`mqk mŽk . yg Žnqk r2. tr t s1Ž . Ý (n ž /ž / nnqk

ms0

=my nyg t Žk .1ye r 0 . 71Ž . Ž . Ž .m

The two fields decay according to this time depen-dent density matrix during the time interval until thenext atom arrives. Hence, the time evolution of thefields for a stream of atoms is calculated by numeri-cally iterating the two cycles of pumping and damp-

Ž . Ž .ing by applying Eqs. 7 and 69 , respectively. Thepump statistics of the micromasers can be taken intoaccount via the distribution of time intervals of thedecay cycles in the procedure. Here, however, weconsider regular pumping statistics only.

In Fig. 10 one can see that, similarly to the resultsof the analytical theory, the average photon number

² :of the second field, n , grows faster with u than2² :that of the first one, n . We also find the dip in the1

² :Fig. 10. Steady state normalized photon number in the second cavity, n rN , and normalized standard deviation, s , obtained from2 ex n2

numerical simulations as functions of the pumping parameter, u . In the left two panels the N -dependence is shown by switching fromexX YŽ . Ž . Ž .N s5 dot-dash line to N s20 solid line and, at the same time, keeping n s0.0 n 'n sn . The dependence on the thermalex ex b b b b

photon number is illustrated in the right two panels dependence on the thermal photon number is illustrated in the right two panels usingŽ . Ž .N s10 with n s0.0 dot-dash lines and n s0.5 solid lines .ex b b


² :Fig. 11. Steady state normalized first order correlation, n r2 N , and normalized standard deviation, s , corresponding to Fig. 10.12 ex n12

photon number noise, s , around us1 which coin-n2

cides with the first peak of s . On the other hand,n1

since the first micromaser makes a transition back toa lower order ‘phase’ of small photon number at the

w xtrapping states 12,13 , the atoms entering the second

cavity are predominantly in their upper states atthese pumping parameters. It follows that atomiccoherence is not significant in these cases and thesystem is very similar to the incoherently coupledscheme. Therefore, similar switching mechanisms

Fig. 12. Steady state first and second order correlations in the forms, R Ž1. and R Ž2., in the upper and lower rows, respectively,corresponding to Fig. 10.


can be found here at the trapping states as those inSection 2. At the same time, the second micromaserbehaves very similarly to the first one at thesepumping parameters, reproducing approximately thesame photon number and noise, because it is pumped

² :by almost fully inverted atoms. The smaller the n1

at the trapping state the more similarly the twomicromasers operate. In the case of finite cavitytemperature, the trapping states go away and bothmicromasers recover their ‘phase-transition’-like op-

Ž .eration right panels in Fig. 10 .Considering the behavior of the steady state first

² :order correlation, n , in Fig. 11 and the corre-12

sponding standard deviation given by

22² : ² :n y n12 12s s 72Ž .)n12 <² : <n12

we first find that the results of the analytical treat-ment are confirmed in the region of small pumpingparameters. However, we can investigate largepumping parameters now, in the case of exact nu-merical simulations, that are beyond the range ofvalidity of the nonlinear expansion. It can be seen in

² :Fig. 11 that n approaches zero with increasing u12

and, on the other hand, it is also approximately zeroat the trapping states. This means that the off-diago-nal elements of the density matrix adjacent to the

Ždiagonal go to zero with increasing u and at the.trapping states at steady state and the first order

correlation disappears. However, the fields do notbecome completely uncorrelated. The lower two pan-els in Fig. 12 show that, apart from the trappingstates, significant second order correlation, R Ž2., canbe found even for large u . This means that, eventhough the matrix becomes diagonal for large pump-ing parameters at steady state, correlation betweenthe fields exists via the diagonal terms. Nevertheless,the first and second order correlations both suggestthat the largest correlation between the two fieldscan be found in the small pumping region. In particu-lar, Fig. 12 shows that in the weak pumping regimethe second order correlation of the fields, R Ž2.,undergoes an oscillation between positive and nega-tive values. The photons coming from the two sepa-rate sources can, therefore, be both positively corre-

Ž Ž2. . Ž Ž2. .lated R )0 and anticorrelated R -0 atsteady state depending on the pumping parameter.

Consider the sum and difference of the steadystate normalized photon numbers in the two cavities,

² :Fig. 13. Left two panels: Steady state normalized photon number in the second cavity, n rN , and normalized standard deviation, s ,2 ex n1Ž . Žobtained from numerical simulations as functions of the pumping parameter, u , for coherent solid lines and incoherent coupling dot-dash

X Y. Ž .lines . The parameters are, N s20 and n s0.0 n 'n sn . Right two panels: Steady state normalized second order correlation,ex b b b b² : 2n n rN , and normalized standard deviation, s , corresponding to the left two panels.1 2 ex n n1 2


² : ² : ² :n s n " n , and the corresponding stan-" 1 2

dard variances, s , given byn"

2 2² : ² : ² : ² :² :s n qs n "2 n n y n nŽ .n 1 n 2 1 2 1 21 2s s . 73Ž .n (" <² : ² : <n " n1 2

s can be employed to determine whether then"

second order correlation between the two fields be-haves nonclassically, using the condition given by

2 2² : ² : ² : ² :n "n y n "n - n q n . 74Ž . Ž .1 2 1 2 1 2

Ž .This condition is obtained, similarly to Eq. 61 , bystudying the regular and pathological behavior of the

Ž .P-function. Using s , for example, Eq. 74 impliesnq

that the system is nonclassical if s -1. In the casenq

of single-mode fields like, e. g., in the two micro-masers separately, such a condition for the standarddeviation of the photon numbers in the modes deter-

Ž .mined the sub-Poissonian super-Poissonian statis-Ž .tics of the photons, i.e., the antibunching bunching

of the photons in the modes. In this case the condi-tion looks formally the same but it is for the photonnumber of the two-field system. We find that, insome regions of u , s , can exhibit nonclassicalnq

behavior even though s and s are both super-n n1 2

Poissonian. This is the result of the nonzero anticor-² : ² :² :relation, n n - n n , the effect of which1 2 1 2

Ž .can be seen in Eq. 73 . Similarly, s is enhancednq

as compared to s and s in those regions wheren n1 2

the two fields exhibit positive correlation, i.e.,² : ² :² :n n ) n n . Considering the difference1 2 1 2

² : ² : ² :photon number, n s n y n , the most sig-y 1 2

nificant effect is that, in some regions of the pump-ing parameter, the photon number in the second

² :cavity, n , grows faster with u than that of the2² :first one, n . The effect of second order correla-1

tion on s is exactly reversed as compared to thatnyŽ .on s as it can be seen in Eq. 73 . That is, s isn nq y

Ž .decreased increased in the regions of positiveŽ .negative correlations.

The consequences of coherences are more appar-² : ² :ent in Fig. 13, when comparing n , s , n n ,2 n 1 22

and s obtained in the case of coherent couplingn n1 2

to the corresponding quantities of the incoherentcoupling. It can be seen that the most significanteffects can be found in the region of small pumpingparameters, as we expect from the analysis above.

² : ² :Here, n and n n are enhanced while s and2 1 2 n2

s exhibit a dip due to the coherences. One cann n1 2

also see that the coherent and incoherent schemesresult in approximately the same behavior at thetrapping states. The above mentioned effects are also

² : ² :present in n and s , viz., n can be negativey n y"

and s can be enhancedrreduced depending onn"

² :positivernegative correlation. In particular, nycannot be negative in the incoherent scheme because² : ² :n ) n for any u . On the other hand, signifi-1 2

cant discrepancies can be seen in s and sn nq y

between the coherent and incoherent cases, espe-cially in the region of small u , arising mainly fromabove correlation effects.

Some of the results of the numerical simulationsfor the time dependence of the system are summa-rized as follows. We apply regular pumping statisticsand the time separation between the atoms is 1rNex

s1r20 in the units of the cavity lifetime. The fieldsare started from vacuum and the cavity temperature

X Yis zero, n 'n sn s0.0. As a reference, in Fig.b b b² :14 we show the time development of n as a1

function of the injected number of atoms for us0.5,1.0, 1.5, 2.0, 2.5 and 3.0. Comparing this figure to

² :Fig. 15 one can see that n , similarly to the2

steady-state behavior as a function of u , grows faster² :than n as a function of the atom number. On the1

other hand, it undergoes a flip similar to the shut-offsfound in the nonlinear theory indicating a depletion

Fig. 14. Time development of the average photon number in the² :first micromaser, n , as a function of the number of atoms.1

YHere, N s20 and n s0.0, and the pumping parameter for theex b

curves are u s0.5, 1.0, 1.5, 2.0, 2.5 and 3.0 from left to right,respectively.


of the upper atomic state due to the first micromaser.The flips become shorter as u is increasing becauseit takes less atoms for larger u to reach a completelydepleted upper atomic state. The corresponding first

² :order correlation, n , depicted in Fig. 16, switches12

the sign of its steady-state value from positive tonegative. In particular, after the second threshold the

² :steady-state value of n is negative, reached via a12² :flip related to that in n . This corresponds to the2

switch of the steady-state relative phase in the semi-classical picture from zero to p and can be ex-plained similarly to the arguments in Section 4.3.2.

In conclusion to this section we can say that theexact numerical simulations verified the analyticalresults of the previous section in the region of smallpumping parameters and, at the same time, providedfurther insight into the behavior of the system bothhere and in the cases of strong pumping. We found,in particular, that the largest correlations between thefields are typical in the region of weak pumpingsuggesting that the atomic coherence generated bythe first micromaser is the largest here. In this regime,the photon statistics of the first micromaser consistof a single wide bell-shaped curve while at largerpumping parameters they consist of several distinctsharp peaks. Therefore, it is suggested that the for-mer structure of a wide single peak is optimum forthe largest atomic coherence to be generated. This issupported by the thermal effect found in the previoussection, viz., thermal radiation in the first cavity canenhance the generation of atomic coherence becauseit widens the peaks of the photon statistics and, at thesame time, establishes a wide background similar to

Fig. 15. Time development of the average photon number in the² :second micromaser, n , as a function of the number of atoms2

YŽ .corresponding to Fig. 14 n s0.0 .b

² :Fig. 16. Time development of the first order correlation, n , as12

a function of the number of atoms correspondings to Fig. 15.

the optimum structure for the largest atomic coher-ence.

6. Summary

In the present paper we considered two micro-masers coupled by sharing a common pumping beamof two-level atoms. This system is reminiscent ofYoung’s two-slit experiment because the atoms havetwo indistinguishable paths to reach the same finalstate, viz., they can be in a superposition state be-tween the two cavities. Acquiring informationŽ .‘which path’ about the state of the atoms betweenthe cavities results in a loss of atomic coherence andthe second micromaser is pumped by atoms in anincoherent mixture of the two atomic states. Compar-ing the case of incoherent coupling to the coherentone, we studied the consequences of atomic coher-ence in this system. We first found that in the case ofincoherent coupling the absorption prepared by thefirst field and injected into the second one switchesthe second micromaser between distinct phases ofthe photon statistics. An exact solution was used toinvestigate this system.

Allowing for atomic coherence to be transferredinto the second cavity the behavior of the system issignificantly modified. We attacked the problem intwo different ways. First, we found the exact masterequation for the field-density matrix and, after anonlinear expansion, transformed it into a Fokker–Planck equation. Following standard techniques weused the drift and diffusion terms of this equation toinvestigate the behavior of the fields. Secondly, wecarried out a numerical simulation of the problem


where the evolution of the fields is separated intodriving cycles applying the exact Jaynes–Cummingsinteraction between the atoms and the fields, anddamping cycles calculating the decay of the fields inthe empty cavities. The two approaches are comple-mentary. The nonlinear expansion can be used for

Žarbitrarily large photon numbers the larger the bet-.ter but works in the region of small pumping param-

eters only. The numerical simulation, however, islimited to small photon numbers due to the finitespeed and storage capacities of computers but can beused for arbitrary values of the pumping parameter.

We found that atomic coherence significantlymodifies the behavior of the system, and the coher-ence effects are crucial in the region of small pump-ing parameters, in particular. The photon number andthe number noise of the field in the second cavitydeviate from the corresponding quantities in the in-coherently coupled system because besides the stim-ulated emission related to the incoherent populationof the atoms, the atomic coherence, as a new drivingmechanism, also contributes to the field. We haveseen the consequences of the interplay between thetwo mechanisms in the nonlinear treatment particu-larly clearly. Atomic coherence also establishes acorrelation between the two fields both in the tran-sient and in the steady-state regimes. The first andsecond order correlations were studied. The firstorder correlation corresponding to a phase locking ofthe fields switches its sign between positive andnegative values resulting in jumps of the relativephases between zero and p both as a function oftime during the time development and as a functionof the pumping parameter in the steady-state behav-ior. On the other hand, the steady-state second ordercorrelation between the photons coming from thetwo separate sources exhibits an oscillation betweenpositive and negative values as a function of thepumping parameter. As we mentioned above, theeffect of atomic coherence is the most significant inthe region of small pumping parameters. In particu-lar, the first order correlation approaches zero withincreasing u and the correlations established by the

Ždiagonal terms of the field-density matrix i.e., sec-.ond order survive only. We want to remark that

atomic coherence injected into the second cavity isthe largest when the photon statistics of the firstmicromaser consist of a single wide peak. In this

way, finite temperature of the first cavity wideningthe photon statistics of the first micromaser canimprove the generated atomic coherence and resultin a larger correlation between the fields. Finitetemperature of the second cavity, on the other hand,can only destruct coherence due to the excessivenoise.

In conclusion, we have shown that correlationbetween the two micromasers is a robust effect bothin the transient and the steady-state regimes in thepresence of significant losses and temperature. Weexpect that these theoretical results could be testedby the presently available experimental facilities.

Acknowledgements

This research was supported by a grant from theOffice of Naval Research, Grant No. N00014-92-J-1233, by a grant from the National Science Founda-tion, Grant No. PHY-9403601, and by a grant fromthe PSC-CUNY Research Award Program.

Appendix A

The drift and diffusion coefficients of theŽ .Fokker–Planck equation, Eq. 22 , read as

)

)d s dŽ .a a1 1

1 u 42 )s u y1 a y a 4a a q7 ,Ž . Ž .1 1 1 12 4!Nex

A.1Ž .)

)d s dŽ .a a2 2

1 u 42 )s u y1 a y a 4a a q7Ž . Ž .2 2 2 22 4!Nex

q16a a a ) q2 q24a a a ) q1Ž . Ž .1 1 1 2 1 1

) )q8a a a qa a , A.2Ž .Ž .2 1 2 1 2

u 4)

) )D s D sy 3a a , A.3Ž .Ž .a a a a 1 11 1 1 1 4!Nex

u 4)

) )D s D sy 3a a q4a a ,Ž .Ž .a a a a 2 2 1 22 2 2 2 4!Nex

A.4Ž .


u 4X 2 )

)D sn qu y 10a a q8 , A.5Ž .Ž .a a b 1 11 1 4!Nex

u 4Y 2 )

)D sn qu y 10a a q8a a b 2 22 2 4!Nex

) ) )q24 a a q1 q12 a a qa a ,Ž . Ž .1 2 1 2 1 2

A.6Ž .)

) )D s DŽ .a a a a1 2 1 2

u 4

w xsy 12a a q8a a q18a a ,1 1 2 2 1 24!Nex

A.7Ž .

u 4) 2 )

) )D s D su y 20 a a q1Ž .Ž .a a a a 1 11 2 1 2 4!Nex

) )q8a a q18a a .2 2 1 2

A.8Ž .

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pump-coupled micromasers: coherent and incoherent coupling

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