Generation of steady state squeezing in micromaser

Volume 78, number 5,6

OPTICS COMMUNICATIONS

15 September 1990

Generation of steady state squeezing in micromaser S. Qamar, K. Zaheer and M.S. Zubairy Department of Electronics, Quaid-i-Azam University, lslamabad. Pakistan Received 16 January 1990; revised manuscript received 18 June 1990

It is shown that squeezing can be generated in steady state, in a micromaser pumped with coherently excited atoms. We consider initial vacuum, number and thermal state of the cavity field and show that the field evolves to a squeezed state with a proper choice of the interaction time. In all the cases, and in particular the latter two, the initial phase-noise is "cleaned" during the evolution and the resultant field is almost ideally squeezed. It is also shown that the steady state photon statistics is sub-poissonian.

I. Introduction The single-atom m i c r o m a s e r has been a system o f f u n d a m e n t a l interest in q u a n t u m optics. The present interest stems from a possibility o f studying the discrete nature o f light a n d its interaction with a t o m s [1 ]. In a micromaser, a b e a m o f atoms, in excited state, is injected in a cavity where they interact for a time rim with the field and then leave the cavity. The injection rate is kept low so that at one time, at most only one a t o m is inside the interaction region. Recent advances in high-Q cavity techniques (Q ~ 10 ~1), a n d the use o f R y d b e r g a t o m s has m a d e it possible to achieve the idealized situation o f a single-mode coupling [2,3]. Because o f their large dipole m o m e n t , R y d b e r g a t o m s couple strongly to the c a v i t y - m o d e and the single-atom m a s e r oscillation has been d e m o n s t r a t e d experimentally [4,5 ]. W i t h the continuous a t o m i c injection, the m a s e r oscillation reaches a steady state due to the finite loss o f the electromagnetic field from the cavity. In such a situation, the field statistics essentially d e p e n d s on the d u r a t i o n o f interaction zint, and can be sub-poissonian for certain values o f this p a r a m e t e r [6,7 ]. If the interaction time is chosen properly, the emission p r o b a b i l i t y at a particular p h o t o n n u m b e r can be m a d e arbitrarily small so that the p h o t o n distribu-

tion function is truncated at that p h o t o n number. Such a state is called a t r a p p i n g state [8]. Schemes based on the m a n i p u l a t i o n o f interaction times, i.e., t i m e o f flight o f the a t o m s through the cavity, to generate a pure n u m b e r state have been discussed [ 8,9 ]. In a recent p a p e r [ 10 ], a new set o f nonclassical states, n a m e l y the 'tangent' and 'cotangent' states were shown to evolve in a micromaser, if the initial state o f the cavity field is between two trapping states a n d the a t o m s are injected in a coherent superposition o f states. In the present c o m m u n i c a t i o n , we show that a squeezed state can be generated in a mic r o m a s e r in steady state. We consider three different initial states for the cavity field, i.e., v a c u u m state, n u m b e r state and thermal state. In all these cases, the cavity field evolves to a squeezed state. By considering the q u a d r a t u r e uncertainty product, it is also seen that the field p r o d u c e d is in a m i n i m u m uncertainty state, i.e., it is an ideal squeezed state. These results are quite surprising since the initial phase noise associated with the n u m b e r state and the thermal state is 'cleaned' during the field evolution. In sec. 2, we discuss the physical system a n d obtain an equation for the field density matrix. Section 3 contains numerical results for the t e m p o r a l q u a d r a t u r e variances a n d the n o r m a l l y o r d e r e d n u m b e r fluctuations for different initial states o f the cavity field.

Research supported, in part, by the Pakistan Science Foundation and the World Laboratory Centre for High Energy Physics and Cosmology, Islamabad, Pakistan. 0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

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OPTICS COMMUNICATIONS

2. The system and equation of motion for the field density matrix

sin(at)

-- igpba COS(fir)pv - -

o¢

'

We consider a situation in which a monokinetic beam of two-level Rydberg atoms, prepared in a coherent superposition I~/(O) )ATOM = C a la) +Cb I b ) ,

( 1)

of the upper ( [ a ) ) and lower ( [ b ) ) atomic levels is injected in a high-Q maser cavity. The atoms are injected at a low rate so that at most one atom is present in the cavity at a time. Assuming that the atoms couple to a single-mode of the cavity field, through a coupling constant g, the interaction is governed by the Jaynes-Cummings hamiltonian [ 11 ],

H=h~oa~+hg2ata+hg(atcr_ +a+a) .

(2)

Here a+ ( a _ ) is the raising (lowering) operator for the atom, at(a) is the creation (annihilation) operator for the photon, 09 is the atomic transition frequency and (2 is the frequency of the cavity eigen mode. At resonance 09 = 12, the interaction hamiltonian is

HI =hg(atcr_ +a+a) .

(3)

N o w assuming that at time ti an atom enters the cavity and spends a time r inside during which it interacts with the cavity field, the field density operator p after a time r can be written as

p(ti + r ) = eft(r) p ( t i ) .

(4)

The form of the operator ~ depends on the model under consideration and can be obtained from the time evolution operator as c~( T)p( ti ) =Paa COS(OLT) PF COS(ogr)

+ igpbaa ~

Pv COS(O~r)

- ipabg COS(O~r) PV

+g2pbba ~ P v q-g2paaa*

342

sin(at)

oL

sin(fir) at

sin(fir)fi at

PF

sin(at)

og

a

15 September 1990

lgpaba

t sin(at) OL

a

Pv cos (fir)

+Pbb COS(fir) PF cos(fir) .

(5)

In eq. (5), a - g ( a a t ) 1/2, fi-g(ata) 1/2 and

po=C, CT,

i,j=a, b,

(6)

give the elements of the atomic density matrix at the injection time t~. If the atoms are injected at a rate R, the field density operator at time t is given by

p(t)=~Kp(O),

(7)

where K=Rt is the number of atoms which interact with the cavity field in time t. Here we assume that no dissipation takes place during the experiment i.e., the cavity life time rc >> t, and that the atomic beam is monochromatic so that all the atoms spend equal time in the cavity. Effects of the field dissipation and the spread in velocities will be discussed elsewhere. In a lossless case, the field density matrix at a particular time depends on two parameters, namely, the number o f atoms passed through the cavity and the interaction time. Generally the atoms are not injected at regular intervals; rather there is a distribution for the injection times. The formalism developed in ref. [12 ] to incorporate the p u m p statistics uses a parameter p which denotes the probability that an atom is excited to its upper level before entering the cavity. The formalism actually deals with the random injection times and not the random excitation of the atoms. In the notation of that paper, if p is the probability that an atom enters the cavity after a time R - ~, then p ( t ) = ~=o~(K) (I-p)K-kpkf~kp(O)'

(8)

K is the number of atoms corresponding to a regular injection, and k is the number of atoms actually entering the cavity. For p = 1, eq. (8) reduces to eq. (7) which describes the kicked JCM. For p - , 0 , R - - , ~ , such that pR = r (finite), one obtains the well known Scully-Lamb master equation [13] which has been shown to correspond to Poisson injection statistics [ 14]. In this limit, the equation of motion for an element of the density matrix is given by

Volume 78, number 5,6

OPTICS COMMUNICATIONS

Pnm(t) =rpaa[p,m(t) c o s ( g r ~

) cos(grx//m+ 1 )

+ P , - l , m - , (t) sin(grx/~) s i n ( g r x / ~ ) ]

+ rpbb[pn + l,m+, (t) sin ( g z ~ )

sin ( g z ~ )

+p,m(t) cos(grv/n) cos(gr,,fm) ] +rip,b[p,,m+~ (t) c o s ( g r ~

) sin (gzx//m+ 1 )

--p,_Lm(t) sin(gzv/n) c o s ( g r v / m ) ] --irpb,[p,+ Lm(t) sin ( g z x / n + 1 )cos ( g r ~

1)

--P . . . . l(t) c o s ( g r x / n ) s i n ( g r x / m ) ] --rp,,m(t).

(9) In the next section, we discuss the role of p u m p statistics and the interaction times in the generation of squeezing.

3. Trapping states and generation of squeezing The idea of trapping states can be originally attributed to ref. [ 7 ]. If the interaction time of the atom is chosen such that g~q+

lr=qn,

q = 1, 2, 3, ....

gx/~qr=qn,

q = 1, 2, 3, ....

(11)

an atom entering the cavity in its ground state will undergo complete q Rabi oscillation to leave the cavity in the same state. The maser is then said to be in a downward trapping state. Clearly, the number state immediately following a downward trapping state is an upward trapping state. In both the cases, the atoms entering the cavity see the field as a 2qn pulse and a "steady state" is achieved even in the absence of losses, so that

p(ti+~)=p(ti),

t i + l = t i + R -1

criterion for a micromaser [ 6 ] despite the fact that the field changes during the interaction time r. We are presently investigating a possibility of generating a squeezed state in the micromaser. To this end, the coherence terms in eq. (9), i.e., the terms proportional to Pab and Pba play an important role and lead to the build up of off-diagonal elements from an initial diagonal field state. With the preceding discussion in mind, which was in the context of purely excited or deexcited atoms, a careful look at eq. (9) reveals that the contribution from the coherence terms becomes zero as an upward trapping state is encountered. In the absence of losses, if a trapping state exists, the steady state field distribution is independent of the noise associated with the random injection times and depends entirely on the interaction time. We can therefore, use eq. (7) instead of eq. (9) to study the field statistics as a function of the number of atoms passed through the cavity. In particular, we calculate the variances in the two quadrature defined by Xl = ½ ( a + a ~) ,

X2=(i/2)(a-a*)

(12)

Eq. (12) represents the conventional steady state

(13a)

,

(13b)

and the, normally ordered photon-number variance

(10)

an excited atom will undergo q Rabi oscillations and will leave the cavity in an excited state. The maser is said to be in upward trapping state since no probability flows from state nq to state nq-t- 1 with the injection of successive atoms. Consequently, with the injection of inverted atoms, the maser field evolves to a number state Inq). On the other hand if

15 September 1990

:a: =

(a*ataa) - ( a ' a ) 2 (a'a)

(14)

A quadrature of the field is said to be squeezed if AXi2 < 1 / 4 ,

i= 1,2.

(15)

The photon statistics is poissonian, sub- and superpoissonian depending whether :a: is equal to, less than or greater than zero respectively. In the following we choose Ca and Cb to be out of phase by n/2. Under this condition, Pab =P~,a and the density matrix elements Pn,, remain real so that the quadrature XI is maximally squeezed. Figs. 1, 2 and 3 show the quadrature variances (z~kXl)2, (z~¥2)2 and the uncertainty product AXj AX2 for an initial vacuum, number and thermal state of the cavity field respectively. In all these cases, the temporal behaviour of quadrature variance is qualitatively similar i.e., it initially increases and then decreases to a steady state value. The maximum squeezing obtained for initial vacuum and number state is about 50% and that for the initial thermal 343

Volume 78, number 5,6

OPTICS COMMUNICATIONS

1-50 ,

1.00, .875

1-25 ,," ',, \

•7 5

6 25 {J

.5ooi!

15 September 1990

u~

\ x

\

k-

yi ""

u

"75 ! '<,"', '',

E

50i

C

g •375

1-00 ~r,,' ",

",~

# .25o ' 25

125

.........

0 [

25

50

Number

of

125 ._m

1-00 i 75F .50 ~

50

-

75

Number

Atoms

Fig. 1. Plot of quadrature variances (a) ~t~l, (b) AX~ and (c) the uncertainty product AX~AX2, against the number of atoms passed through the cavity for initial vacuum state. Scaled interaction time gr = n/xI-N, and upper level probability p~ = 0.7.

2 "50 2"25 2' 00 I -75 150

25

75

-

100 of

125

] 150

Atoms

Fig. 3. Plot of quadrature variances (a) AXe, (b) AX~ and (c) the uncertainty product AX~AX2, against the number of atoms passed through the cavity for initial thermal state with n= 1. Scaled interaction time gr= n/x~, and upper level probability p~,=0.7.

-40 .35 )'

.3o

,, x C O

~

~0

-.~ b

C -25

Q

250[

/

• 20

.15

75

Number

25

0

of Atoms

50

75

Number of Atoms Fig. 2. Plot of quadrature variances (a) AXe, (b) AX~ and (c) the uncertainty product AX~AX2, against the number of atoms passed through the cavity for initial number state with n = 4. All other parameters are same as in fig. 1. state is a b o u t 52%. A larger s q u e e z i n g in t h e latter case can be a t t r i b u t e d to a h i g h e r t r a p p i n g state, n q = 2 0 , as o p p o s e d to r t q : 10 for earlier cases. T h i s was p r i m a r i l y d o n e t o i n c l u d e the initial p r o b a b i l i t y d i s t r i b u t i o n , w i t h i n the n u m e r i c a l limits. In all these cases the u n c e r t a i n t y p r o d u c t stabilizes at close to the m i n i m u m u n c e r t a i n t y level. Figs. 4 a n d 5 also s h o w the q u a d r a t u r e v a r i a n c e s a n d the u n c e r t a i n t y p r o d u c t for an initial v a c u u m a n d n u m b e r state r e s p e c t i v e l y o f the c a v i t y field, b u t for d i f f e r e n t initial a t o m i c c o n d i t i o n s . T h e steady state s q u e e z i n g in t h e s e cases is a b o u t 20%. H o w ever, c o m p a r i s o n w i t h figs. I a n d 2 shows t h a t the 344

Fig. 4. Same as fig. 1 except paa=0.3.

2.25 i

2-00: 1.75 1.5o

~o125 •~ l-P0 >

75

(

.5oi 25 0

i

25 50 Number of Atoms Fig. 5. Same as fig. 2 except p~=0.3.

75

Volume 78, number 5,6

OPTICS COMMUNICATIONS

We have s h o w n that in this cavity system, u p to 52% s q u e e z i n g is possible a n d to o u r knowledge, n o other f u n d a m e n t a l l i m i t o n squeezing exists. These are presently b e i n g investigated. In fig. 6, n o r m a l l y ordered v a r i a n c e : a: i n the p h o t o n n u m b e r for i n i t i a l v a c u u m , n u m b e r a n d t h e r m a l states is p l o t t e d as a f u n c t i o n o f n u m b e r o f atoms. I n all these cases phot o n d i s t r i b u t i o n finally shows s u b - p o i s s o n i a n statistics.

0 a -10

2'5 .50 "15 Number of Atoms

15 September 1990

100

Fig. 6. Plot of photon number variance against the number of atoms passed through the cavity for (a) initial vacuum state (b) initial number state and (c) initial thermal state. All parameters corresponding to these curves are same as the corresponding parameters in figs. 1, 2, 3.

a p p r o a c h to the steady state is q u a l i t a t i v e l y different a n d d e p e n d s o n the i n i t i a l a t o m i c as well as the field c o n d i t i o n . I n a d d i t i o n the t r a n s i e n t s q u e e z i n g is larger t h a n the steady state squeezing. I n conclusion, we have shown that i n a lossless case, s q u e e z i n g c a n be g e n e r a t e d in steady state i n a m i c r o m a s e r b y i n j e c t i n g a t o m i c c o h e r e n c e a n d b y adj u s t i n g the velocity o f the a t o m i c b e a m . As m e n t i o n e d earlier this system has b e e n c o n s i d e r e d b y Slosser et al. [ lO] where they have s h o w n t h a t the cavity field is a p u r e state w h i c h they called c o t a n gent state. Values o f steady state s q u e e z i n g c a n b e o b t a i n e d u s i n g their results as well. T h e m a x i m u m s q u e e z i n g d e p e n d s o n the t r a p p i n g state nq a n d the a t o m i c a m p l i t u d e s Ca a n d Cb. We h a v e s h o w n t h a t the steady state s q u e e z i n g is i n d e p e n d e n t o f the initial cavity field w h i c h is in a g r e e m e n t with the results o f ref. [ 1 0 ] , b u t the a p p r o a c h to steady state d e p e n d s o n the i n i t i a l a t o m i c as well as field state.

References [ 1 ] See for example, F. Diedrich, J. Krause, G. Rempe, M.O. Scully and H. Walther, IEEE J. Quantum Elect. QE-24 (1988) 1314. [ 2 ] S. Haroche and J.M. Raymond, in: Advances in Atomic and Molecular Physics, Vol. 20, eds. D.R. Bates and B. Bederson (Academic Press, 1985) p. 350. [3] J.A.C. Gallas, G. Leuchs, H. Walther and H. Figger, in: Advances in Atomic and Molecular Physics, Vol. 20, eds. D.R. Bates and B. Bederson (Academic Press, 1985) p. 414. [4] D. Meschede, H. Walther and G. Miiller, Phys. Rev. Lett., 54 (1985) 551. [5] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. 58 (1987) 353. [ 6 ] P. Filipowicz, J. Javanainen and P. Meystre, Phys. Rev. A34 (1986) 3077. [ 7 ] P. Filipowicz, J. Javanainen and P. Meystre, J. Opt. Soc. Am. B 3 (1986) 906. [8] P. Meystre, G. Rempe and H. Walther, Optics Lett. 13 (1988) 1078. [9] J. Krause, M.O. Scully, T. Walther and H. Walther, Phys. Rev. A 39 (1989) 1915. [ 10] J. Slosser, P. Meystre and S. Braunstein, Phys. Rev. Len. 63 (1989) 934. [ 11 ] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 (1963) 89. [ 12 ] J. Bergou, L. Davidovich, M. Orszag, C. Benkert, M. Hillery and M.O. Scully, Optics Comm. 72 (1989) 82. [ 13 ] L. Davidovich, J.M. Raimond, M. Brune and S. Haroche, Phys. Rev. A 36 (1987) 3771. [ 14] L.A. Lugiato, M.O. Scully and H. Walther, Phys. Rev. A 36 (1987) 740.

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