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Atomic coherence initiated noise free energy transfer in a resonant medium
Optics Communications North-Holland
106 (1994)
OPTICS COMMUNICATIONS
237-241
Atomic coherence initiated noise free energy transfer in a resonant medium G.S. Agarwal
‘, M.O. Scully ’ and H. Walther
Max-Planck Institutfir Received
2 September
Quantenoptik, D-85740 Garching, Germany 1993
Energy transfer from pump to probe in a resonant medium is accompanied by the addition of spontaneous noise of the medium. A scheme is presented where this energy transfer can be made noise free i.e., a coherent signal is transformed into a coherent signal with larger intensity. The scheme uses a A-system with injected coherence between two ground levels -this coherence leads to the transfer of energy from the pump to the probe beam.
Several authors have considered the transfer of energy from pump to probe in a two-level atomic medium [ 11. The gain at the probe frequency has been extensively discussed. However it turns out that the gain process is noisy [ 21 i.e., the medium adds spontaneous noise to the probe field. Multilevel schemes for the energy transfer have also been considered and a very popular scheme is based on the Stokes processes in a A-system. Even this scheme is noisy [ 3 1. A question thus arises if it is possible to envisage a scheme of energy transfer in a resonant medium so that the transfer of energy is free of noise. The noise free transfer is in the sense that a coherent probe field is transformed into a coherent field with larger intensity. We propose a scheme where the energy transfer process is noise free. Our proposal makes use of atomic coherences [ 41 in a A-system. In our model the pump is also treated dynamically. Before we discuss our model and the possibility of noise free transfer of energy let us describe briefly the idea of noise free energy transfer. Let us consider a pump field of frequency w, and probe field of frequency f&. Let a and b be the annihilation operators for pump and probe fields. Let the probe field be ini’ School of Physics,
University of Hyderabad, Hyderabad 500 134, India. * Department of Physics, Texas A and M University, College Station, TX 77843, USA.
0030-4018/94/%07.00
0 1994 Elsevier
SSDI 0030-4018(93)E0598-A
tially in a coherent state I p). We then look for an optical process which would lead to the following transformation on the probe field
(1) The probe field gains energy at the cost of the pump field. The transformation ( 1) implies the following transformation on the P-function of the probe field pb(p,
0)
=~cz’(P-h+~b(8)
=~‘*‘(P-8).
(2)
It should be borne in mind that the transformation (2) does not imply that the operator b transforms like b-b (p”lp). It would also be erroneous to conclude from eq. (2) that any commutation relations are violated. The relation (2) is to be contrasted from the result for standard amplifiers #l S”)(B-j&)+-$exp(
-
*)
,
where CJcorresponds to the noise photons added by the medium. Consider the situation shown in the fig. 1 where a A-system, with two ground states with some initial coherence produced by some external means like a microwave field, interacts with external fields. We RI Quantum
Science B.V. All rights reserved.
noise in linear amplifiers sively. See, for example, refs. [ 51.
has been studied
exten-
237
Volume 106, number
4,5,6
15March 1994
OPTICS COMMUNICATIONS
P=-i[Hl,pl+Lp,
(7)
where the Liouvillian L contains the effects of pumping and relaxation. The matrix elements of Lp in the atomic Hilbert space are (LP)aa = -YP’Paa>tLP)abt tLP)ab
=
-
=
-
(Y/2h%bf,
(r/2)pab,
(Lp)ii=-A(pli_1)+(y/2)p,, (Lp)bb’= #JO)>
=
(lb>+e”lb’>)/j’2
Fig. 1. Schematic illustration of the scheme used for noise free energy transfer. The initial preparation of the atomic system is givenby
IVIA(O
(4)
We consider the interaction of the atomic medium with two fields resonant on the transitions 1a) c* I b ) and I a) +-, lb’). The interaction hamiltonian can be written as Hi =fig(ala)(bI
+bla)
(b’l +h.c.)
,
(5)
where g is the coupling constant #* and a and b are the annihilation operators for the fields on the two atomic transitions )a) * I b ) , I a) *--*I b’) . The mode a (b) will be taken as the pump (probe) mode. We would transfer energy from a mode to b mode. The excited state of the atom can decay to the ground state at the rate y. For simplicity we will assume that the two branching ratios are equal so that the decay rate for each transition is y/2. We can now derive a density matrix equation for the field alone. Let PF be the reduced density matrix defined by PF = TratomsP .
(6)
In order to derive the dynamical equation for the fields, we write the equation for the combined Iieldatom system as w In eq. (4) we assume that the populations of the levels 1b) and 1b’) are equal. Similarly in eq. (5 ) we assume that the two coupling constants are equal. These assumptions are made for simplicity. The general case can be treated similarly without, however, changing the essential physics.
238
- fe-‘“)
.
(8)
All A dependent terms correspond to the injected coherence and populations. In the absence of HI the steady state atomic correlations can be calculated to be
will work in the interaction picture and take the initial state of the atomic medium as I~A(0))=(l/fi)(Ib)+ei@Ib’)).
-A&
i=borb’,
=( Ib’)(al.la)(b’I)=fe-y”2, ((lb)(al),(la)(b’l))=$e-v/*e@.
(9)
The equation for pr can be obtained from eqs. (7 )(9) and by using standard master equation methods #3. The atomic medium is to be treated as a bath for the problem with time correlations given by (9). The master equation requires the knowledge of bath correlations. In our derivation we also assume that bath atoms are independent of each other. The final result of the calculation is the master equation for the fields a and b /& = - r(A +Ap, - 2&A
++&A +A) ,
where A is the boson operator A=(l/fi)(a+be’@), The parameter r=
(2g2/r)N
(10)
defined by
[A,A+]=l.
(11)
r in ( 10) is >
(12)
where N is the number of atoms in the medium. Note that g* is inversely proportional to volume and hence g2N and r are proportional to the density of atoms. Equation ( 10) gives the dynamical evolution of the two fields a and b. We now present the solution of eq. ( 10). For this purpose we introduce an operator B which is also a linear combination of a and b x3 For master equation methods see, for example, refs. [ 61.
B= (l/JZ)(a-be’@)
,
and has the following
commutation
[B,B’]=l,
[B,A]=O,
(13)
[B,A’]=O.
B=F,,
(14)
(15)
ci=-(r/2)(a+be’@)+F,, (16)
Note that the coupling between a and b modes is produced because of the coherence between two ground states of the medium #4. Furthermore the normally ordered correlations of F, and Fb are zero. However the antinormally ordered correlations of F’s are nonzero. This is important to maintain the correctness of quantum mechanical commutation relations of a’s and b’s. The density matrix eq. ( 10) can be solved by introducing the coherent states of A and B
(17) Using ( 12 ) and ( 17 ) it can be shown that the P-distribution satisfies the equation
ap z
=I--& (o!“P)
+c.c.,
as >t)=d(2)(aA-a~oqO)e-rf)
>
(19)
if initially the modes A and B are in coherent 1a,!,O) ) and 1aB”)) respectively
states
xs(‘)(a,-ap)
where the noise sources, F’s have zero mean value and all normally ordered correlations of F’s are zero. So, as long as one is calculating the normally ordered expectation values involving A and B operators, one does not need the noise terms FA, Fsr etc. The Langevin equations for a and b can be written from ( 11) , (13) and (15):
b=-(T/2)(ae-‘@+b)+Fb.
P(aA,
relations
The structure of ( 10) and ( 14) implies that all the moments involving B, Bt operators do not evolve in time. The quantum Langevin equations for A and B can be written as A= -~A+F,,
15 March 1994
OPTICS COMMUNICATIONS
Volume 106, number 4,5,6
(18)
P((yA, %, 0) =6(2’((w,
-aJjO’)
.
6’2’(aB-a~‘)
Note the absence of di’sion terms in ( 18) and this very fact leads to noise free transfer of energy from pump to probe. The result (20) shows that p(t)=la~O)e-”
9a~“))(a~O)e-“,a&o’i
of the P-function
is
x4 The semiclassical equations of the form ( 16) have also been derived by Fill et al. [ 71 in connection with lasers without inversion. The existence of the matched fields also follows from this paper.
,
(21)
if p(0)=la~O’,a~O’)(aa”),aSo)I
.
(22)
Note further that the operators A and B are linearly related to a and b (eqs. ( 11) and ( 13)) with no mixing of annihilation and creation operators. Thus if a system is in a coherent state of A and B, then it is also in a coherent state of a and b. Therefore eqs. (21), (22), (11) and (13) lead to p(t)=
la(t),B”(t)>(~i(t),P”(t)l,
ifA0)=IwB)(a,BI,
(23)
~%=(a/2)(l+e-“)-(P/2)(1-e-~)e’@, B= (p/2) (1 +e-“)
- (a/2)e-‘@(
(24) 1 -eert)
.
(25)
It should be borne in mind that we are working in the interaction picture and that the transformation (a, fi)- (a, 8) is not a unitary transformation. We next concentrate on the properties of b mode. From (23) we can write the density matrix for the b mode as pCb’(t) = Ik)>
(im
I.
(26)
The probe will gain energy if
lib>
I > ISI ,
A
and hence the time dependence simple:
(20)
(27)
which indeed is possible as eq. ( 25 ) shows. The gain process continues till both modes acquire equal intensities. In the limit of large T’t the gain condition (27) can be written as
la12+21/31 Ial ~~~~~-~~,+~~,+~~~31B12,
(28)
where $,, @aare the phases of (Yand /I, respectively. 239
Volume
106, number
4,5,6
OPTICS
15March 1994
COMMUNICATIONS
Clearly this condition can be satisfied over a very wide range of values of I a I, I PI, q3,,q+, @.The phase of the output pis determined by the amplitudes 1a 1, (/3I and the phases &,, efl and @ We have thus shown: “The mode b gains energy and this process is noise free as the input coherent state is transformed into a coherent state - this has been achieved by transferring energy from the mode a to the mode b. This transfer is mediated by the atomic coherence.” We also note #5 that if the coherence between the two ground states were absent then instead of eq. ( 12) one would have obtained
which is as large as I a I ‘z+z-I /?I ‘. It should be noted that the field b is optimized if $-@,+da+ a=O, though it is not necessary to operate our system under such conditions as long as the condition (27 ) is satisfied. We conclude by giving the results for other input states of the probe field b. Let the probe field’s initial state be characterized by the distribution Pb(B) . From (23) we see that the distribution at time t will be
pF = - (r/2)
Pb(P’) = j ~‘2’(8’-&))
+terms
(a+up, -2u~u++p,u+u)
with u+b .
The solution
(29)
4./Gmizcos(@-$a+$fi+~)
(33)
which on using (25 ) reduces to Pb(B> t)=
,
j( ((ye-(~/2)r,pe-‘r/2)tl
xpb
absence of the injected coherence the medium acts like an absorber. The steady state of (24) teresting properties: &= f’z C(t)=a/2-
(l+&’
(30)
ifPF(0)=la,j3)(a,PI,andthusnotransferofenergy from pump to probe occurs. The amplitude of the probe mode decays at the rate r/2. Thus in the
and (25 ) has some in-
p+
(
(a/2)e-'@( 1 -e-“), (1 +e-“)/2
o >.
If the field b is in a thermal state at time t = 0, then on interaction with the A-scheme of fig. 1 it is transformed into a mixture of coherent and incoherent components, i.e., Pb(/T, O)= &exp
nn
(P/2)ei@,
(
-IpIz
n >’
(a/2)e-‘@,
(G/j!) = -e’@.
(31)
Thus in the steady state the amplitudes of the two fields become equuP’.The relative phase is fixed by the atomic coherence in the system. For @=n, even the phases of the fields on the transition I a) H Ib) , I a) c* Ib’) are the same. From the form offlit is also clear that the differential gain Slp12/Slj?12 is given by 115One might think of the present system as a frequency
converter. However, there are differences. The frequency conversion in a non resonant medium is typically described by the hamiltonian rc(a+b+ab+). This hamiltonian leads to periodic transfer of energy from one mode to the other whereas in the scheme discussed in the text we have irreversible transfer of energy from the stronger mode to the weaker mode. This continues till both the modes acquire equal intensities.
240
Pb(B, 0) d2P,
(32)
of eq. (29) is found to be
PF(t)=I(Ye-(~/2)1,~e-‘r/2,r)
&p/2-
>
In a recent paper, Harris [ 81 has considered the following problem - a A-system interacting with fields of the form [l+f(t)]~Lexp(-iwlt){[I+g(t)]~2exp(-iwzt)}resonant on the transition 1a) ++ 1b) [ 1a) - 1b’) 1. He considers the propagation of the tieldsfand g and shows that the medium produces matched fields. In other words the fourier componentsf(z, w) and g(z, o) become identical in the limit z-co irrespective of the input fields. The medium becomes transparent for distances larger compared to many absorption lengths. The model presented in this paper also achieves this as eq. ( 3 1) shows. In Harris’s work the parts tr and tz of the fields ( 1 +f)er, ( 1 +g)tz prepare the atomic system in a state which has the same form as our state (4) with @= II and eI = ~2. Thusone can understand why we obtain similar results though we address to very different physical situations. The steady state ( 3 1) should not be mistaken to be the same as that produced by a beam splitter. First of all the frequencies of the pump and probe are different. Secondly the determinant of the transformation in (3 I ) is zero whereas for a beam splitter it should be one.
OPTICS COMMUNICATIONS
Volume 106, number 4,5,6
15 March 1994
References
ff(t)=fi(1+e-rf)2/4
(35)
Thus, in a sense, our model quenches thermal noise. It should also be noted that state distribution depends on the initial i.e., the system has a memory as can be
the inputthe steady
condition, seen from
(34) by taking the limit t-m pb(B, m)=4Pb(28+ae-“,
0) .
(36)
The memory of the initial state can be traced back to the nonevolution of the mode B [ eqs. ( 13) ] which was a linear combination of a and b modes. In conclusion we have shown how the atomic coherence effects can be used to construct systems which would transfer energy from pump to probe without the addition of noise, i.e., an input probe in a coherent state is transformed into a coherent state with larger mean intensity. We have benefited from discussions with many scientists particularly with C. Benkert, R. Boyd, J.H. Eberly, S.E. Harris and T.W. Mossberg. This work was partially supported by the Office of Naval Research.
[ 1 ] B.R. Mollow, Phys. Rev. A 5 (1972) 2217; F.Y. Wu, S. Ezekiel, M. Ducloy and B.R. Mallow, Pbys. Rev. Lett. 38 (1977) 1077; R.W. Boyd and M. Sargent, J. Opt. Sot. Am. B 5 (1988) 99; M.T. Grtlneisen, K.R. McDonald and R.W. Boyd, J. Opt. Sot. Am. B 5 (1988) 123. [2] A. Gaeta, R.W. Boyd and G.S. Agarwal, Phys. Rev. A 46 (1992) 4271. [3] Y.R. Shen, Phys. Rev. 155 (1967) 921; K.J. McNeil and D.F. Walls, J. Phys. A 7 (1974) 617; D.F. Walls, J. Phys. A 6 ( 1973) 496; see also R.C. Swanson, P.R. Battle and J.L. Carlston, Phys. Rev. Lett. 67 (1991) 38. [4] M.O. Scully, S.Y. Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813; M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. 5 ] S. Frieberg and L. Mandel, Optics Comm. 46 ( 1983) 14 1; R. Loudon and T.J. Shepherd, Optica Acta 3 1 ( 1984) 1243; S. Stenholm, Optics Comm. 58 (1986) 177; R.J. Glauber, in: Laser Spectroscopy, VIII, eds. W. Persson and S. Svanberg (Springer, Berlin, 1987) p. 126; C.M. Caves, Phys. Rev. D 26 (1982) 1817. 1’61 G.S. Agarwal, Quantum optics, Springer Tracts in Modem Physics, Vol. 70 (Springer, Berlin, 1974); W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1974). [7] E.E.Fill,M.O.ScullyandS.Y.Zhu,OpticsComm. 77 (1990) 36. [S] S.E. Harris, Phys. Rev. I&t. 70 (1993) 552.
241
106 (1994)
OPTICS COMMUNICATIONS
237-241
Atomic coherence initiated noise free energy transfer in a resonant medium G.S. Agarwal
‘, M.O. Scully ’ and H. Walther
Max-Planck Institutfir Received
2 September
Quantenoptik, D-85740 Garching, Germany 1993
Energy transfer from pump to probe in a resonant medium is accompanied by the addition of spontaneous noise of the medium. A scheme is presented where this energy transfer can be made noise free i.e., a coherent signal is transformed into a coherent signal with larger intensity. The scheme uses a A-system with injected coherence between two ground levels -this coherence leads to the transfer of energy from the pump to the probe beam.
Several authors have considered the transfer of energy from pump to probe in a two-level atomic medium [ 11. The gain at the probe frequency has been extensively discussed. However it turns out that the gain process is noisy [ 21 i.e., the medium adds spontaneous noise to the probe field. Multilevel schemes for the energy transfer have also been considered and a very popular scheme is based on the Stokes processes in a A-system. Even this scheme is noisy [ 3 1. A question thus arises if it is possible to envisage a scheme of energy transfer in a resonant medium so that the transfer of energy is free of noise. The noise free transfer is in the sense that a coherent probe field is transformed into a coherent field with larger intensity. We propose a scheme where the energy transfer process is noise free. Our proposal makes use of atomic coherences [ 41 in a A-system. In our model the pump is also treated dynamically. Before we discuss our model and the possibility of noise free transfer of energy let us describe briefly the idea of noise free energy transfer. Let us consider a pump field of frequency w, and probe field of frequency f&. Let a and b be the annihilation operators for pump and probe fields. Let the probe field be ini’ School of Physics,
University of Hyderabad, Hyderabad 500 134, India. * Department of Physics, Texas A and M University, College Station, TX 77843, USA.
0030-4018/94/%07.00
0 1994 Elsevier
SSDI 0030-4018(93)E0598-A
tially in a coherent state I p). We then look for an optical process which would lead to the following transformation on the probe field
(1) The probe field gains energy at the cost of the pump field. The transformation ( 1) implies the following transformation on the P-function of the probe field pb(p,
0)
=~cz’(P-h+~b(8)
=~‘*‘(P-8).
(2)
It should be borne in mind that the transformation (2) does not imply that the operator b transforms like b-b (p”lp). It would also be erroneous to conclude from eq. (2) that any commutation relations are violated. The relation (2) is to be contrasted from the result for standard amplifiers #l S”)(B-j&)+-$exp(
-
*)
,
where CJcorresponds to the noise photons added by the medium. Consider the situation shown in the fig. 1 where a A-system, with two ground states with some initial coherence produced by some external means like a microwave field, interacts with external fields. We RI Quantum
Science B.V. All rights reserved.
noise in linear amplifiers sively. See, for example, refs. [ 51.
has been studied
exten-
237
Volume 106, number
4,5,6
15March 1994
OPTICS COMMUNICATIONS
P=-i[Hl,pl+Lp,
(7)
where the Liouvillian L contains the effects of pumping and relaxation. The matrix elements of Lp in the atomic Hilbert space are (LP)aa = -YP’Paa>tLP)abt tLP)ab
=
-
=
-
(Y/2h%bf,
(r/2)pab,
(Lp)ii=-A(pli_1)+(y/2)p,, (Lp)bb’= #JO)>
=
(lb>+e”lb’>)/j’2
Fig. 1. Schematic illustration of the scheme used for noise free energy transfer. The initial preparation of the atomic system is givenby
IVIA(O
(4)
We consider the interaction of the atomic medium with two fields resonant on the transitions 1a) c* I b ) and I a) +-, lb’). The interaction hamiltonian can be written as Hi =fig(ala)(bI
+bla)
(b’l +h.c.)
,
(5)
where g is the coupling constant #* and a and b are the annihilation operators for the fields on the two atomic transitions )a) * I b ) , I a) *--*I b’) . The mode a (b) will be taken as the pump (probe) mode. We would transfer energy from a mode to b mode. The excited state of the atom can decay to the ground state at the rate y. For simplicity we will assume that the two branching ratios are equal so that the decay rate for each transition is y/2. We can now derive a density matrix equation for the field alone. Let PF be the reduced density matrix defined by PF = TratomsP .
(6)
In order to derive the dynamical equation for the fields, we write the equation for the combined Iieldatom system as w In eq. (4) we assume that the populations of the levels 1b) and 1b’) are equal. Similarly in eq. (5 ) we assume that the two coupling constants are equal. These assumptions are made for simplicity. The general case can be treated similarly without, however, changing the essential physics.
238
- fe-‘“)
.
(8)
All A dependent terms correspond to the injected coherence and populations. In the absence of HI the steady state atomic correlations can be calculated to be
will work in the interaction picture and take the initial state of the atomic medium as I~A(0))=(l/fi)(Ib)+ei@Ib’)).
-A&
i=borb’,
=( Ib’)(al.la)(b’I)=fe-y”2, ((lb)(al),(la)(b’l))=$e-v/*e@.
(9)
The equation for pr can be obtained from eqs. (7 )(9) and by using standard master equation methods #3. The atomic medium is to be treated as a bath for the problem with time correlations given by (9). The master equation requires the knowledge of bath correlations. In our derivation we also assume that bath atoms are independent of each other. The final result of the calculation is the master equation for the fields a and b /& = - r(A +Ap, - 2&A
++&A +A) ,
where A is the boson operator A=(l/fi)(a+be’@), The parameter r=
(2g2/r)N
(10)
defined by
[A,A+]=l.
(11)
r in ( 10) is >
(12)
where N is the number of atoms in the medium. Note that g* is inversely proportional to volume and hence g2N and r are proportional to the density of atoms. Equation ( 10) gives the dynamical evolution of the two fields a and b. We now present the solution of eq. ( 10). For this purpose we introduce an operator B which is also a linear combination of a and b x3 For master equation methods see, for example, refs. [ 61.
B= (l/JZ)(a-be’@)
,
and has the following
commutation
[B,B’]=l,
[B,A]=O,
(13)
[B,A’]=O.
B=F,,
(14)
(15)
ci=-(r/2)(a+be’@)+F,, (16)
Note that the coupling between a and b modes is produced because of the coherence between two ground states of the medium #4. Furthermore the normally ordered correlations of F, and Fb are zero. However the antinormally ordered correlations of F’s are nonzero. This is important to maintain the correctness of quantum mechanical commutation relations of a’s and b’s. The density matrix eq. ( 10) can be solved by introducing the coherent states of A and B
(17) Using ( 12 ) and ( 17 ) it can be shown that the P-distribution satisfies the equation
ap z
=I--& (o!“P)
+c.c.,
as >t)=d(2)(aA-a~oqO)e-rf)
>
(19)
if initially the modes A and B are in coherent 1a,!,O) ) and 1aB”)) respectively
states
xs(‘)(a,-ap)
where the noise sources, F’s have zero mean value and all normally ordered correlations of F’s are zero. So, as long as one is calculating the normally ordered expectation values involving A and B operators, one does not need the noise terms FA, Fsr etc. The Langevin equations for a and b can be written from ( 11) , (13) and (15):
b=-(T/2)(ae-‘@+b)+Fb.
P(aA,
relations
The structure of ( 10) and ( 14) implies that all the moments involving B, Bt operators do not evolve in time. The quantum Langevin equations for A and B can be written as A= -~A+F,,
15 March 1994
OPTICS COMMUNICATIONS
Volume 106, number 4,5,6
(18)
P((yA, %, 0) =6(2’((w,
-aJjO’)
.
6’2’(aB-a~‘)
Note the absence of di’sion terms in ( 18) and this very fact leads to noise free transfer of energy from pump to probe. The result (20) shows that p(t)=la~O)e-”
9a~“))(a~O)e-“,a&o’i
of the P-function
is
x4 The semiclassical equations of the form ( 16) have also been derived by Fill et al. [ 71 in connection with lasers without inversion. The existence of the matched fields also follows from this paper.
,
(21)
if p(0)=la~O’,a~O’)(aa”),aSo)I
.
(22)
Note further that the operators A and B are linearly related to a and b (eqs. ( 11) and ( 13)) with no mixing of annihilation and creation operators. Thus if a system is in a coherent state of A and B, then it is also in a coherent state of a and b. Therefore eqs. (21), (22), (11) and (13) lead to p(t)=
la(t),B”(t)>(~i(t),P”(t)l,
ifA0)=IwB)(a,BI,
(23)
~%=(a/2)(l+e-“)-(P/2)(1-e-~)e’@, B= (p/2) (1 +e-“)
- (a/2)e-‘@(
(24) 1 -eert)
.
(25)
It should be borne in mind that we are working in the interaction picture and that the transformation (a, fi)- (a, 8) is not a unitary transformation. We next concentrate on the properties of b mode. From (23) we can write the density matrix for the b mode as pCb’(t) = Ik)>
(im
I.
(26)
The probe will gain energy if
lib>
I > ISI ,
A
and hence the time dependence simple:
(20)
(27)
which indeed is possible as eq. ( 25 ) shows. The gain process continues till both modes acquire equal intensities. In the limit of large T’t the gain condition (27) can be written as
la12+21/31 Ial ~~~~~-~~,+~~,+~~~31B12,
(28)
where $,, @aare the phases of (Yand /I, respectively. 239
Volume
106, number
4,5,6
OPTICS
15March 1994
COMMUNICATIONS
Clearly this condition can be satisfied over a very wide range of values of I a I, I PI, q3,,q+, @.The phase of the output pis determined by the amplitudes 1a 1, (/3I and the phases &,, efl and @ We have thus shown: “The mode b gains energy and this process is noise free as the input coherent state is transformed into a coherent state - this has been achieved by transferring energy from the mode a to the mode b. This transfer is mediated by the atomic coherence.” We also note #5 that if the coherence between the two ground states were absent then instead of eq. ( 12) one would have obtained
which is as large as I a I ‘z+z-I /?I ‘. It should be noted that the field b is optimized if $-@,+da+ a=O, though it is not necessary to operate our system under such conditions as long as the condition (27 ) is satisfied. We conclude by giving the results for other input states of the probe field b. Let the probe field’s initial state be characterized by the distribution Pb(B) . From (23) we see that the distribution at time t will be
pF = - (r/2)
Pb(P’) = j ~‘2’(8’-&))
+terms
(a+up, -2u~u++p,u+u)
with u+b .
The solution
(29)
4./Gmizcos(@-$a+$fi+~)
(33)
which on using (25 ) reduces to Pb(B> t)=
,
j( ((ye-(~/2)r,pe-‘r/2)tl
xpb
absence of the injected coherence the medium acts like an absorber. The steady state of (24) teresting properties: &= f’z C(t)=a/2-
(l+&’
(30)
ifPF(0)=la,j3)(a,PI,andthusnotransferofenergy from pump to probe occurs. The amplitude of the probe mode decays at the rate r/2. Thus in the
and (25 ) has some in-
p+
(
(a/2)e-'@( 1 -e-“), (1 +e-“)/2
o >.
If the field b is in a thermal state at time t = 0, then on interaction with the A-scheme of fig. 1 it is transformed into a mixture of coherent and incoherent components, i.e., Pb(/T, O)= &exp
nn
(P/2)ei@,
(
-IpIz
n >’
(a/2)e-‘@,
(G/j!) = -e’@.
(31)
Thus in the steady state the amplitudes of the two fields become equuP’.The relative phase is fixed by the atomic coherence in the system. For @=n, even the phases of the fields on the transition I a) H Ib) , I a) c* Ib’) are the same. From the form offlit is also clear that the differential gain Slp12/Slj?12 is given by 115One might think of the present system as a frequency
converter. However, there are differences. The frequency conversion in a non resonant medium is typically described by the hamiltonian rc(a+b+ab+). This hamiltonian leads to periodic transfer of energy from one mode to the other whereas in the scheme discussed in the text we have irreversible transfer of energy from the stronger mode to the weaker mode. This continues till both the modes acquire equal intensities.
240
Pb(B, 0) d2P,
(32)
of eq. (29) is found to be
PF(t)=I(Ye-(~/2)1,~e-‘r/2,r)
&p/2-
>
In a recent paper, Harris [ 81 has considered the following problem - a A-system interacting with fields of the form [l+f(t)]~Lexp(-iwlt){[I+g(t)]~2exp(-iwzt)}resonant on the transition 1a) ++ 1b) [ 1a) - 1b’) 1. He considers the propagation of the tieldsfand g and shows that the medium produces matched fields. In other words the fourier componentsf(z, w) and g(z, o) become identical in the limit z-co irrespective of the input fields. The medium becomes transparent for distances larger compared to many absorption lengths. The model presented in this paper also achieves this as eq. ( 3 1) shows. In Harris’s work the parts tr and tz of the fields ( 1 +f)er, ( 1 +g)tz prepare the atomic system in a state which has the same form as our state (4) with @= II and eI = ~2. Thusone can understand why we obtain similar results though we address to very different physical situations. The steady state ( 3 1) should not be mistaken to be the same as that produced by a beam splitter. First of all the frequencies of the pump and probe are different. Secondly the determinant of the transformation in (3 I ) is zero whereas for a beam splitter it should be one.
OPTICS COMMUNICATIONS
Volume 106, number 4,5,6
15 March 1994
References
ff(t)=fi(1+e-rf)2/4
(35)
Thus, in a sense, our model quenches thermal noise. It should also be noted that state distribution depends on the initial i.e., the system has a memory as can be
the inputthe steady
condition, seen from
(34) by taking the limit t-m pb(B, m)=4Pb(28+ae-“,
0) .
(36)
The memory of the initial state can be traced back to the nonevolution of the mode B [ eqs. ( 13) ] which was a linear combination of a and b modes. In conclusion we have shown how the atomic coherence effects can be used to construct systems which would transfer energy from pump to probe without the addition of noise, i.e., an input probe in a coherent state is transformed into a coherent state with larger mean intensity. We have benefited from discussions with many scientists particularly with C. Benkert, R. Boyd, J.H. Eberly, S.E. Harris and T.W. Mossberg. This work was partially supported by the Office of Naval Research.
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