- Email: [email protected]
Quantum theory of an atomic parametric oscillator
1 December 1997
PHYSICS
LETTERS
A
Physics Letters A 236 ( 1997) 84-88
ELSEYIER
Quantum theory of an atomic parametric oscillator N.S. Ananikian ‘, KG. Petrosyan 2 Department ufTheoretical Physics. Yerevan Physics Institute, Alikhanian Br 2, 375036 Yerevan, Armenia Received 27 June 1997; revised manuscript received IS September 1997; accepted for publication 18 September 1997 Communicated by L.J. Sham
Abstract We present a quantum theory of an atomic parametric oscillator which describes a gas of weakly interacting trapped bosons. The particles are assumed to be confined in a damped atomic cavity pumped by a coherent matter beam. We consider the three trapped levels which are the pumped and two damped modes. The system is described by a FokkerPlanck equation (FPE) for the quasi-distribution function in the complex P-representation for the density operator. Using a known steady-state solution of the FPE we derive the boson number distribution function. On the basis of an exact treatment we investigate the quantum behavior of the device, especially in the critical region. As a result, a quantum noise-induced effect is revealed. We suggest to use the device as a correlated matter waves generator. @ 1997 Published by Elsevier Science B.V.
1. Introduction Recent achievements in manipulating atoms by the use of magnetic and optical fields have led to experimental realizations of the Bose-Einstein condensation by several research groups [ 11. Very recently, one of them has produced the first atom laser [ 21, which has become possible by the realization of an output coupler for Bose-Einstein condensed atoms. There are several theories for atom lasers, including the cases of noninteracting [3] and interacting [4,5] samples of bosons. A theory for excitonpolariton lasers was developed in Ref. [6]. The above-mentioned theoretical works concern both the atom and the exciton-polariton lasers, and are related to schemes which produce coherent matter beams;
’ E-mail: ananikajerewan I .yerphi.am. ? Also at: The Institute for Physical Research, National Academy of Sciences, Ashtarak 2, 378410 Armenia.
hence they can be regarded as coherent matter wave generators. In our Letter we look at a further development of the idea having in mind the history of nonlinear optics. We propose a device which produces correlated atomic beams in very close analogy to optical parametric oscillators. In this Letter we introduce an exactly solvable model of an atomic parametric oscillator, which describes the quantum behavior of the proposed correlated matter waves generator. The treatment of the interacting Bose gas in this Letter resembles that of Refs. [ 451. Note that in Ref. [ 41 the interactions are assumed to be caused by photon exchanges between the particles. In Ref. [5] the interactions are due to collisions. The main feature of these treatments is the introduction of two-body interparticle interactions. We also introduce the two-body interactions and consider a simplified model in order to be able to provide an exact analytical treatment of the process under consideration. Let us also mention Ref. [ 71, where another scheme of atomic field am-
0375-9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PI/ SO375-9601(97)00770-6
N.S. Ananikian, KG. PetrosyadPhysics
plification was considered, in which an initial matter field was split into two other waves, reproducing a parametric interaction. Our model is quite different due to the interaction mechanism considered here. The layout of this Letter is as follows. In Section 2 we present the model Hamiltonian and write down a master equation for the density operator. We transform the master equation into the Fokker-Planck equation (FPE) for a quasi-distribution in the complex P-representation for the density matrix. The exact steady-state solution to the FPE is then presented and the behavior of the second-order correlators is discussed. In Section 3 we derive the boson number distribution function and investigate the stationary quantum behavior, including the critical region. Also in this section, special attention is paid to the quantum noiseinduced phenomenon that is revealed in our model.
2. Model of an atomic parametric
oscillator
Let us consider an atomic cavity [ 8,4] which provides the discrete levels labeled as +, 0, and -. We assume that the 0 level is pumped by a coherent matter wave. The injected particles undergo two-body interactions, which leads to transitions of one of the particles to the level + and its interacting partner to the level -. This is a process that creates a pair of bosons on the + and - levels simultaneously, annihilating two particles at the 0 level, and the process has an obvious resemblance to the nondegenerate parametric processes well-known in nonlinear optics. In addition, the present two-body interactions involve an exchange of particles from the + and - levels, i.e. a boson from level + can interact with another one at the level -, resulting in their exchange. This interaction is an atomic analogue of the nonlinear optical process of cross-phase modulation [ 91. The Hamiltonian that describes the above interactions in the atomic cavity can be written as follows, H = fi~~,&q
+ Ziw+u;u+ + liw-a+_~-
+ hG( u;u:u;
+ uof2u+u_)
+ A*@lo+‘u; + li/y+&2~
+ n&u’2u2
+fig+uo+u@l~u++~g-u~u&_+fixu$z+u+u_, (1)
Letters A 236 (1997) 84-88
85
where ul and ak (k = 0, -, +) are the usual boson creation and annihilationoperators for the trapped levels; wk (k = 0, -, +) are the trap frequencies; and the phenomenological coefficients G, g*, x, ~0, x+ are nonlinear couplings responsible for two-body interactions [ 4,5]. We assume next that the pumped 0 level is macroscopically populated and undepleted, and treat it as a coherent wave, which allows us to replace the 0 level operator by a c-number amplitude aa -+ fi with N being the number of bosons on the 0 level. Another assumption we will use is that the intralevel pair couplings are much smaller than the interlevel one (,y* < x), i.e. the f- interactions dominate the -t+ and -ones. This can be achieved by a specific choice of selection rules [4] which can be well engineered in a trap by applying laser fields. The laser-matter interactions in the trap will select dominant interactions and correspond to resonantly enhanced processes satisfying momentum conservation conditions [ 41. Then the model Hamiltonian takes the following form in the interaction picture, H = fiGN( a++
+ a+~_ ) + ligN( &a’
+ hXu~u+u’u_,
+ cz+a_ ) (2)
where we put g, = g- = g. We allow for the damping of the levels + and via an output coupling [2] or tunneling through the trap potential barrier. Thus we arrive at a model of a dissipative process which can be described by the following master equation for the density operator p ofthesystem [lO,ll],
dP _ I,H,p]
X -
ifi
+~~k(?Uk/Xl+Z:oip~“;-uk) k=-
(3) with yk (k = +, -) being the atomic cavity damping constants. Note that the master equation (3) with the Hamiltonian (2) has been considered in Ref. [ 121 in the context of quantum optics for the case of equal damping constants y+ = y-. There second-order correlators were calculated and analyzed for the case when the system possesses two critical regions simultaneously. In our atomic parametric oscillator case we will mainly restrict ourselves to the consideration of the
N.S. Ananikian, K.G. PetrosyadPhysics
86
boson number distribution functions for the same symmetric case of equal damping constants. However, here there will be just one critical point for the atomic trap case under consideration. To analyze the system we transform the master equation (3) into a FPE for a quasi-distribution function in the complex P-representation of the density operator [ lo,13 1. In the complex P-representation the density operator is expressed by the quasi-probability distribution function P (a+, a:, a_. , a? >,
Letters A 236 (1997) 84-88 8’
”
““’
”
”
“’
”
“’
6-
I-
>/’ *A
dMW+,a;,a-,a’)
P=
c
C’
where the complex c-numbers (Y+ and (Y- correspond to the operators a+ and a_, respectively; dp = da,dazdcr_daf is a measure and CC’ are path contours in the complex planes. Using Eqs. (3) and (4) we derive the following FF’E for the P-function,
(-&I-(7+ + -igN)
- &[-(r_
+igN)cr_
-i(GN+xa+a-)a:]
- -&[-(y._
-igN)cu?
+i(GN+xoT~~?)a+l
I,
,,1
,
m(GN+~a+(~-)~(GN+xcr=cr+)~*
x exp( 2a+cu + 2fl+P), where fi is a normalization troduced the notation
(6) constant
and we have in-
_,=
The steady-state solution for the P-function, which is equivalent to the density operator, enables us to exactly calculate all the normally ordered operator moments in the steady-state regime. The semiclassical steady-state solution can be obtained from the deterministic part of Eq. (5) and reads
cy: +i(GN+xffTa?)(Y-1
- &]-(y+ +
I,
iX
- i(GN+xff+a-)a?1
+
Pst =
I
1000
h=2(y+kN)
igN)a+
- i&(GN
,,
0
(4)
$P=
--n--r7
-c 4000 2000 3000 5000 Fig. I. Classical steady-state (full curve), quantum mechanical mean (dashed line) boson numbers and the boson number distribution’s most probable values (locations of the maxima, squares) of the f levels versus the number of injected Bose condensed particles for G/y = 5 x 10w4, g/y = 3 X 10P4, x/y = 0. I. o
+ p+a-)
+ip aa!;a+ (GN + x+Y+) + -
P.
(5)
As was shown [ 121 the FFE (4) has a steady-state solution, which can be obtained using the method of potential conditions [ 131. Note that the conditions are satisfied only for the case of equal damping constants y+ = y_ = y, so that in the rest of this Letter we will restrict ourselves to this case. The steady-state solution has the following form,
There is a phase transition when the system approaches the critical threshold value N, = r/d-. The phase transition exists only if the pair interaction at level 0, which produces pairs at the levels + and -, is stronger than the +- interlevel pair exchanges (G > g). We plot the mean boson number (azak), which can be derived using Eq. (6)) and the semiclassical steady-state solution no versus the number of injected particles N in Fig. 1. Second-order auto- and cross-correlators, which are delined as
N.S. Ananikian.
KG.
Petrr~svan/Phvsics
Letters A 236 (1997)
XL%?
87
For the single-level boson number distribution one has to sum over the second-level variable, so that we obtain P(n)
= II
GP(a+,a;,a-,a’)
(8)
4..
1..
1
1
,.,..,,,..,.,,.,.
Nl_/
‘3100
.I(: Ii’
4003
Fig. 2. Auto-correlator (full curve) and cross-correlator line) for the same parameters as in Fig. I. g~c2; =
W4)
5Cli:O
(dashed
(a$2+&_) (u=u+) (u’u-) ’
We then substitute the steady-state solution (6) into the Eq. (8) and perform calculations similar to those of Ref. [ 141. In our case under consideration which is symmetric with respect to the levels + and -- we obtain the following result for the boson number distribution P(n) of a single (say +) level. P(n)
= (a::n+)* - g:‘l
=
00 respectively, are shown in Fig. 2. We see a smooth transition from the below threshold thermal (or chaotic) statistics to the above threshold coherent one in the auto-correlator, while the cross-correlator demonstrates strong below threshold correlations. This behavior is very similar to that of the well-known model of optical parametric oscillators [ 14,151, and we conclude that our system is capable of producing strongly correlated matter waves. More details about the interplay between the cross- and auto-correlations in the mode1 are presented in Ref. [ 121. In the present Letter we turn to the investigation of the system’s behavior using a more complete particle number distribution description.
3. Boson number distribution behavior
and the steady-state
The particle numbers joint probability distribution function P( n+, FZ-) can be expressed via the complex P-representation by the following relation, P(n+,rz_) =
= (rz+,n_Ipln_,n+) dpP(a,,a;.a_,a’)
c C’ X
(a+a~)“’ I1!
(a-a?)“m!
exp( -cy+cy~ - a_ (Yi ) (7)
X
c,=o
(12+ 1)/ (IA
fV)/+m
IGNJz/#“‘“’
I!
(9)
The distribution (9) has a single peak at zero in the below threshold region. Then the maximum of the function changes from zero to a nonzero value as the system passes through a threshold. This has an obvious interpretation; however, it is of more interest to analyze the behavior of the number distribution’s most probable values (the locations of its maxima) and compare this with the classical results, as the maxima of the probability distribution function can be identified with the states where the system spends most time. We calculate the boson number probabilities for the whole relevant domain of the injected particle number N and then determine the corresponding most probable values of the P(n)-function. Fig. I (squares) shows the behavior of the locations of the most probable values depending on N. As compared with the classical (full) curve we observe a quantum noise-induced shift of the threshold. The phenomenon of noise-induced shifts is well known in classical physics (see, e.g., Ref. [ 161 ) and similar effects were obtained in some quantum systems, including the one-atom optical bistability [ 171 and parametrically driven dissipative anharmonic oscillator [ 181. Our results show that the difference between the quantum and classical results becomes essential when the nonlinearity increases. This is due to the increase of quantum fluctuations which arise from the nonlinear
88
N.S. Ananikian. KG. PerrosyadPhysics
couplings; it appears, actually, that the quantum diffusion terms in the FEE and the atomic pair interactions are described by the same parameters. In conclusion, we have presented a quantum theory of an atomic parametric oscillator which describes a weakly interacting Bose gas trapped in a damped atomic cavity which contains three discrete energy levels. We have calculated the boson number distribution function and presented a detailed description of the system’s behavior. Special attention was paid to the revealed quantum noise-induced effect of the shift of the threshold. The model is very close to the well-known optical systems [ 14,15,12] and thus it can be proposed as a correlated matter waves generator, which is a further extension of ideas related to the coherent atomic beam generator [4] and atom lasers [ 351.
Acknowledgement The authors are grateful to K.V. Kheruntsyan for valuable discussions. This work was partly supported by the grant 2 1 l-529 1 YPI of the German Bundesministerium fur Forschung und Technologie.
References 1 I ] M. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269 ( 1995) 198; CC. Bradley, CA. Sackett, J.J. Tollet, R. Hulet, Phys. Rev. Lett. 75 (1995) 1687;
Letters A 236 (I 997) 84-88
K.B. Davis, M.-O. Mewes, M.R. Andrews. N.J. van Druten. D.S. Durfee, D.M. Kum. W. Ketterle. Phys. Rev. Lett. 75 (1995) 3969. [2] M.-O. Mewes. M.R. Andrews, D.M. Kum, D.S. Durfee, C.G. Townsend, W. Ketterle, Phys. Rev. Lett. 78 (1997) 582. [3] H.M. Wiseman. M.J. Collett, Phys. Lett. A 202 (1995) 246; R.J.C. Spreeuw, T. Pfau. U. Janicke, M. Wilkens, Europhys. L&t. 32 (1995) 469; M. Olshanii, Y. Castin, J. Dalibard, in: Proc. of the 12th Int. Conf. on Laser Spectroscopy, eds. M. Inguscio, M. Allegrini, A. Lasso (World Scientific, Singapore, 1995). 141 A.M. Guzman, M. Moore, P. Meystre, Phys. Rev. A 53 ( 1996) 977. 151 M. Holland, K. Burnett, C. Gardiner, J.I. Cirac, P. Zoller. Phys. Rev. A 54 ( 1996) R1757. 161 A. Imamoglu. R.J. Ram, S. Pau, Y. Yamamoto, Phys. Rev. A 53 (1996) 4250. [71 Ch.J. Borde, Phys. Lett. A 204 (1995) 217. 181 W. Zhang, P Meystre. E.M. Wright, Phys. Rev. A S2 ( 1995) 498. f91 N. Imoto. H.A. Haus, Y. Yamamoto. Phys. Rev. A 52 ( 1985) 2287. 1101C.W. Gardiner, Quantum Noise (Springer, Berlin. 199 I ). 1111 D.F. Walls, G.J. Milbum, Quantum Optics (Springer, Berlin. 1994). 1121 K.G. Petrossian, Opt. Commun. 134 (1997) 99. [I31 C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin. 1986). 1141 K.J. McNeil, C.W. Gardiner, Phys. Rev. A 28 ( 1983) 1560. 1151 JOSA BIO ( 1993) Nos. 9, I I, Special issues on Optical Parametric Oscillation and Amplification. R. Lefever. Noise-Induced Transitions [I61 W. Horsthemke, (Springer, Berlin 1984). 1171 C. Savage, H.J. Carmichael, IEEE J. Quantum Electron. QE24 (1988) 1495. 1181 K.V. Kheruntsyan, Opt. Commun. 127 (1996) 230.
PHYSICS
LETTERS
A
Physics Letters A 236 ( 1997) 84-88
ELSEYIER
Quantum theory of an atomic parametric oscillator N.S. Ananikian ‘, KG. Petrosyan 2 Department ufTheoretical Physics. Yerevan Physics Institute, Alikhanian Br 2, 375036 Yerevan, Armenia Received 27 June 1997; revised manuscript received IS September 1997; accepted for publication 18 September 1997 Communicated by L.J. Sham
Abstract We present a quantum theory of an atomic parametric oscillator which describes a gas of weakly interacting trapped bosons. The particles are assumed to be confined in a damped atomic cavity pumped by a coherent matter beam. We consider the three trapped levels which are the pumped and two damped modes. The system is described by a FokkerPlanck equation (FPE) for the quasi-distribution function in the complex P-representation for the density operator. Using a known steady-state solution of the FPE we derive the boson number distribution function. On the basis of an exact treatment we investigate the quantum behavior of the device, especially in the critical region. As a result, a quantum noise-induced effect is revealed. We suggest to use the device as a correlated matter waves generator. @ 1997 Published by Elsevier Science B.V.
1. Introduction Recent achievements in manipulating atoms by the use of magnetic and optical fields have led to experimental realizations of the Bose-Einstein condensation by several research groups [ 11. Very recently, one of them has produced the first atom laser [ 21, which has become possible by the realization of an output coupler for Bose-Einstein condensed atoms. There are several theories for atom lasers, including the cases of noninteracting [3] and interacting [4,5] samples of bosons. A theory for excitonpolariton lasers was developed in Ref. [6]. The above-mentioned theoretical works concern both the atom and the exciton-polariton lasers, and are related to schemes which produce coherent matter beams;
’ E-mail: ananikajerewan I .yerphi.am. ? Also at: The Institute for Physical Research, National Academy of Sciences, Ashtarak 2, 378410 Armenia.
hence they can be regarded as coherent matter wave generators. In our Letter we look at a further development of the idea having in mind the history of nonlinear optics. We propose a device which produces correlated atomic beams in very close analogy to optical parametric oscillators. In this Letter we introduce an exactly solvable model of an atomic parametric oscillator, which describes the quantum behavior of the proposed correlated matter waves generator. The treatment of the interacting Bose gas in this Letter resembles that of Refs. [ 451. Note that in Ref. [ 41 the interactions are assumed to be caused by photon exchanges between the particles. In Ref. [5] the interactions are due to collisions. The main feature of these treatments is the introduction of two-body interparticle interactions. We also introduce the two-body interactions and consider a simplified model in order to be able to provide an exact analytical treatment of the process under consideration. Let us also mention Ref. [ 71, where another scheme of atomic field am-
0375-9601/97/$17.00 @ 1997 Published by Elsevier Science B.V. All rights reserved. PI/ SO375-9601(97)00770-6
N.S. Ananikian, KG. PetrosyadPhysics
plification was considered, in which an initial matter field was split into two other waves, reproducing a parametric interaction. Our model is quite different due to the interaction mechanism considered here. The layout of this Letter is as follows. In Section 2 we present the model Hamiltonian and write down a master equation for the density operator. We transform the master equation into the Fokker-Planck equation (FPE) for a quasi-distribution in the complex P-representation for the density matrix. The exact steady-state solution to the FPE is then presented and the behavior of the second-order correlators is discussed. In Section 3 we derive the boson number distribution function and investigate the stationary quantum behavior, including the critical region. Also in this section, special attention is paid to the quantum noiseinduced phenomenon that is revealed in our model.
2. Model of an atomic parametric
oscillator
Let us consider an atomic cavity [ 8,4] which provides the discrete levels labeled as +, 0, and -. We assume that the 0 level is pumped by a coherent matter wave. The injected particles undergo two-body interactions, which leads to transitions of one of the particles to the level + and its interacting partner to the level -. This is a process that creates a pair of bosons on the + and - levels simultaneously, annihilating two particles at the 0 level, and the process has an obvious resemblance to the nondegenerate parametric processes well-known in nonlinear optics. In addition, the present two-body interactions involve an exchange of particles from the + and - levels, i.e. a boson from level + can interact with another one at the level -, resulting in their exchange. This interaction is an atomic analogue of the nonlinear optical process of cross-phase modulation [ 91. The Hamiltonian that describes the above interactions in the atomic cavity can be written as follows, H = fi~~,&q
+ Ziw+u;u+ + liw-a+_~-
+ hG( u;u:u;
+ uof2u+u_)
+ A*@lo+‘u; + li/y+&2~
+ n&u’2u2
+fig+uo+u@l~u++~g-u~u&_+fixu$z+u+u_, (1)
Letters A 236 (1997) 84-88
85
where ul and ak (k = 0, -, +) are the usual boson creation and annihilationoperators for the trapped levels; wk (k = 0, -, +) are the trap frequencies; and the phenomenological coefficients G, g*, x, ~0, x+ are nonlinear couplings responsible for two-body interactions [ 4,5]. We assume next that the pumped 0 level is macroscopically populated and undepleted, and treat it as a coherent wave, which allows us to replace the 0 level operator by a c-number amplitude aa -+ fi with N being the number of bosons on the 0 level. Another assumption we will use is that the intralevel pair couplings are much smaller than the interlevel one (,y* < x), i.e. the f- interactions dominate the -t+ and -ones. This can be achieved by a specific choice of selection rules [4] which can be well engineered in a trap by applying laser fields. The laser-matter interactions in the trap will select dominant interactions and correspond to resonantly enhanced processes satisfying momentum conservation conditions [ 41. Then the model Hamiltonian takes the following form in the interaction picture, H = fiGN( a++
+ a+~_ ) + ligN( &a’
+ hXu~u+u’u_,
+ cz+a_ ) (2)
where we put g, = g- = g. We allow for the damping of the levels + and via an output coupling [2] or tunneling through the trap potential barrier. Thus we arrive at a model of a dissipative process which can be described by the following master equation for the density operator p ofthesystem [lO,ll],
dP _ I,H,p]
X -
ifi
+~~k(?Uk/Xl+Z:oip~“;-uk) k=-
(3) with yk (k = +, -) being the atomic cavity damping constants. Note that the master equation (3) with the Hamiltonian (2) has been considered in Ref. [ 121 in the context of quantum optics for the case of equal damping constants y+ = y-. There second-order correlators were calculated and analyzed for the case when the system possesses two critical regions simultaneously. In our atomic parametric oscillator case we will mainly restrict ourselves to the consideration of the
N.S. Ananikian, K.G. PetrosyadPhysics
86
boson number distribution functions for the same symmetric case of equal damping constants. However, here there will be just one critical point for the atomic trap case under consideration. To analyze the system we transform the master equation (3) into a FPE for a quasi-distribution function in the complex P-representation of the density operator [ lo,13 1. In the complex P-representation the density operator is expressed by the quasi-probability distribution function P (a+, a:, a_. , a? >,
Letters A 236 (1997) 84-88 8’
”
““’
”
”
“’
”
“’
6-
I-
>/’ *A
dMW+,a;,a-,a’)
P=
c
C’
where the complex c-numbers (Y+ and (Y- correspond to the operators a+ and a_, respectively; dp = da,dazdcr_daf is a measure and CC’ are path contours in the complex planes. Using Eqs. (3) and (4) we derive the following FF’E for the P-function,
(-&I-(7+ + -igN)
- &[-(r_
+igN)cr_
-i(GN+xa+a-)a:]
- -&[-(y._
-igN)cu?
+i(GN+xoT~~?)a+l
I,
,,1
,
m(GN+~a+(~-)~(GN+xcr=cr+)~*
x exp( 2a+cu + 2fl+P), where fi is a normalization troduced the notation
(6) constant
and we have in-
_,=
The steady-state solution for the P-function, which is equivalent to the density operator, enables us to exactly calculate all the normally ordered operator moments in the steady-state regime. The semiclassical steady-state solution can be obtained from the deterministic part of Eq. (5) and reads
cy: +i(GN+xffTa?)(Y-1
- &]-(y+ +
I,
iX
- i(GN+xff+a-)a?1
+
Pst =
I
1000
h=2(y+kN)
igN)a+
- i&(GN
,,
0
(4)
$P=
--n--r7
-c 4000 2000 3000 5000 Fig. I. Classical steady-state (full curve), quantum mechanical mean (dashed line) boson numbers and the boson number distribution’s most probable values (locations of the maxima, squares) of the f levels versus the number of injected Bose condensed particles for G/y = 5 x 10w4, g/y = 3 X 10P4, x/y = 0. I. o
+ p+a-)
+ip aa!;a+ (GN + x+Y+) + -
P.
(5)
As was shown [ 121 the FFE (4) has a steady-state solution, which can be obtained using the method of potential conditions [ 131. Note that the conditions are satisfied only for the case of equal damping constants y+ = y_ = y, so that in the rest of this Letter we will restrict ourselves to this case. The steady-state solution has the following form,
There is a phase transition when the system approaches the critical threshold value N, = r/d-. The phase transition exists only if the pair interaction at level 0, which produces pairs at the levels + and -, is stronger than the +- interlevel pair exchanges (G > g). We plot the mean boson number (azak), which can be derived using Eq. (6)) and the semiclassical steady-state solution no versus the number of injected particles N in Fig. 1. Second-order auto- and cross-correlators, which are delined as
N.S. Ananikian.
KG.
Petrr~svan/Phvsics
Letters A 236 (1997)
XL%?
87
For the single-level boson number distribution one has to sum over the second-level variable, so that we obtain P(n)
= II
GP(a+,a;,a-,a’)
(8)
4..
1..
1
1
,.,..,,,..,.,,.,.
Nl_/
‘3100
.I(: Ii’
4003
Fig. 2. Auto-correlator (full curve) and cross-correlator line) for the same parameters as in Fig. I. g~c2; =
W4)
5Cli:O
(dashed
(a$2+&_) (u=u+) (u’u-) ’
We then substitute the steady-state solution (6) into the Eq. (8) and perform calculations similar to those of Ref. [ 141. In our case under consideration which is symmetric with respect to the levels + and -- we obtain the following result for the boson number distribution P(n) of a single (say +) level. P(n)
= (a::n+)* - g:‘l
=
00 respectively, are shown in Fig. 2. We see a smooth transition from the below threshold thermal (or chaotic) statistics to the above threshold coherent one in the auto-correlator, while the cross-correlator demonstrates strong below threshold correlations. This behavior is very similar to that of the well-known model of optical parametric oscillators [ 14,151, and we conclude that our system is capable of producing strongly correlated matter waves. More details about the interplay between the cross- and auto-correlations in the mode1 are presented in Ref. [ 121. In the present Letter we turn to the investigation of the system’s behavior using a more complete particle number distribution description.
3. Boson number distribution behavior
and the steady-state
The particle numbers joint probability distribution function P( n+, FZ-) can be expressed via the complex P-representation by the following relation, P(n+,rz_) =
= (rz+,n_Ipln_,n+) dpP(a,,a;.a_,a’)
c C’ X
(a+a~)“’ I1!
(a-a?)“m!
exp( -cy+cy~ - a_ (Yi ) (7)
X
c,=o
(12+ 1)/ (IA
fV)/+m
IGNJz/#“‘“’
I!
(9)
The distribution (9) has a single peak at zero in the below threshold region. Then the maximum of the function changes from zero to a nonzero value as the system passes through a threshold. This has an obvious interpretation; however, it is of more interest to analyze the behavior of the number distribution’s most probable values (the locations of its maxima) and compare this with the classical results, as the maxima of the probability distribution function can be identified with the states where the system spends most time. We calculate the boson number probabilities for the whole relevant domain of the injected particle number N and then determine the corresponding most probable values of the P(n)-function. Fig. I (squares) shows the behavior of the locations of the most probable values depending on N. As compared with the classical (full) curve we observe a quantum noise-induced shift of the threshold. The phenomenon of noise-induced shifts is well known in classical physics (see, e.g., Ref. [ 161 ) and similar effects were obtained in some quantum systems, including the one-atom optical bistability [ 171 and parametrically driven dissipative anharmonic oscillator [ 181. Our results show that the difference between the quantum and classical results becomes essential when the nonlinearity increases. This is due to the increase of quantum fluctuations which arise from the nonlinear
88
N.S. Ananikian. KG. PerrosyadPhysics
couplings; it appears, actually, that the quantum diffusion terms in the FEE and the atomic pair interactions are described by the same parameters. In conclusion, we have presented a quantum theory of an atomic parametric oscillator which describes a weakly interacting Bose gas trapped in a damped atomic cavity which contains three discrete energy levels. We have calculated the boson number distribution function and presented a detailed description of the system’s behavior. Special attention was paid to the revealed quantum noise-induced effect of the shift of the threshold. The model is very close to the well-known optical systems [ 14,15,12] and thus it can be proposed as a correlated matter waves generator, which is a further extension of ideas related to the coherent atomic beam generator [4] and atom lasers [ 351.
Acknowledgement The authors are grateful to K.V. Kheruntsyan for valuable discussions. This work was partly supported by the grant 2 1 l-529 1 YPI of the German Bundesministerium fur Forschung und Technologie.
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