We would like to thank Professor Peter Knight and Dr Barry Garraway for helpful and stimulating discussions and for reading the manuscript. This work was.
JOURNAL OF MODERN OPTICS,
1992,
VOL .
39,
NO .
12, 2481-2499
Interaction of squeezed light with two-level atoms H . MOYA-CESSAt and A . VIDIELLA-BARRANCO Optics Section, The Blackett Laboratory, Imperial College, London SW7 2BZ, England (Received 3 June 1992; revision received 24 August 1992)
Abstract . We investigate some of the fundamental features of the interaction of squeezed light with two-level atoms in the framework of the Jaynes-Cummings model . We start our analysis by calculating the collapses and revivals of the atomic inversion . We discuss the degree of purity of the field (given by the entropy) and its disentanglement from the atomic source . The connection with the evolution of the Q-function is also made . We notice that contrary to the coherent state case, the field turns into a nearly pure (squeezed) state at the revival time as if the field was prepared in a coherent state . The field also becomes a superposition of squeezed states at half of the revival time, and this is confirmed by investigating the photon number distribution . The phase properties of the field are discussed using the Pegg-Barnett formalism .
1.
Introduction
The squeezed states of light [1] exhibit remarkable statistical properties, e .g ., reduced quadrature (or amplitude) noise, oscillations in photon number distribution and sub-Poissonian (or super-Poissonian) statistics, characteristics that are very different from the ones presented by coherent light . These non-classical properties will obviously have consequences for the interaction of such light with matter . The problem of the interaction of a single mode of the electromagnetic field prepared in a squeezed state with a two-level atom in the Jaynes-Cummings model [2] has been already addressed . It was first noticed by Milburn [3] that the `collapse time' of the atomic inversion undergoes a change depending on the degree and orientation of the squeezing, and moreover, the atomic response to the field in a squeezed vacuum state and in a thermal state are very similar . For the field initially in a strongly squeezed state, Satyanarayana et al . [4] found what they called ringing revivals of the atomic inversion, which are secondary revivals related to oscillations in the photon number distribution [5] . As we are going to show, this problem still presents new and interesting features, particularly regarding the properties of the cavity field . It is well known that the field evolution in the Jaynes-Cummings model is quite complicated [6] . If the field is initially prepared in a coherent state (pure state), as time goes on the atom and field become entangled and the field state will become increasingly mixed, returning, however, to a nearly pure state in the middle of the so-called `collapse region', which is when the atom does not respond to the field [6] . The atom is assumed to be prepared either in the excited or in the ground state . When the atom nearly returns to its initial configuration ('revival region') the field does not return to a pure state, as t Permanent address : INAOE, Apdo Postal 51 y 216, Puebla, Pue ., Mexico . 0950-0340/92 $3 .00 © 1992 Taylor & Francis Ltd .
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H. Moya-Cessa and A . Vidiella-Barranco
one could expect . As we are going to see, the situation is rather different if the field is initially prepared in a squeezed coherent state . At first we will present the calculation of the atomic inversion for different squeezing parameters of the initial field . Then we will concentrate on the cavity field evolution . One advantage of the JaynesCummings model is that we can calculate exactly (in the rotating wave approximation) the time-dependent field density operator, which can be expressed in a relatively compact way as : Pew= Cfl+ 1Pr( 0)C.+ 1 + atS„ + 1Pe(0)S,t,+ 1 a,
(1)
where Cn+1 =COs [/~t(aat)1I2],
(1 a)
sin [),t(aat)1/2] S„+1=
(aat ) 1/2
The atom is initially prepared in the excited state, and the field in the state described by Pf (O) . The density operator (1) represents the maximum information we can retrieve from the quantized cavity field . However, it is convenient to calculate particular representations of the density operator . In this paper we are going to use the Q-function [7,8] and the photon number distribution . These are two important quantities to characterize the field : the Q-function (phase space approach) was first applied to the Jaynes-Cummings model by Eiselt and Risken [8], and gives an interesting picture of the field evolution . However, we need some more information about the off-diagonal elements of Pf(t), and a quantity that will play a central role in our analysis is the von Neumann (or total) entropy [9] . The entropy provides the most complete information about the purity of the cavity field [6], and is defined as :
S= -Tr {of (t) In [of(t)]} .
(2)
Another useful definition of entropy is the so-called Shannon entropy, but this only involves diagonal elements of the density operator with respect to some observable basis, and consequently contains less information than the total entropy . The Shannon entropy in the number state basis, defined as . = - > x In [], Ss' n=O
is calculated in reference [4], but their discussion is connected to the appearance of ringing revivals only . We will also investigate some of the phase properties of the cavity field, in connection with the previous analysis (Q-function, entropy) . The time evolution of the phase operator variance as well as the phase probability distribution will be calculated using the Pegg-Barnett formalism [10, 11] . This paper is organized as follows : in section 1 we discuss the atomic excitation comparing the behaviour of the atomic inversion for the field initially prepared in a squeezed coherent state with different squeezing parameters r, i.e . r=0 (coherent state), and r=1 and 2 . The coherent intensity of the initial field employed throughout this paper will be IaI 2 =25 unless otherwise specified . In section 2 we present a study of some aspects of the evolution of the cavity field, again for different
Squeezed light with two-level atoms
2483
squeezing parameters, and starting with the calculation of the time-dependent von Neumann entropy . In the remaining of the subsections, we will discuss the results obtained for the entropy using the evolution of the Q-function for the cavity field as well as its photon number distribution . We analyse in some detail the field behaviour in order to explain the pattern of field purity . We conclude the paper by discussing the phase properties of the cavity field using the Pegg-Barnett formalism .
2.
Atomic dynamics
2 .1 . Atomic inversion The Jaynes-Cummings model [2] consists of a two-level atom with ground state and and excited state je> placed inside a lossless cavity and interacting with a quantized single mode of the electromagnetic field . Its Hamiltonian (in the rotating wave approximation) may be written as : Ig>
H=zhco 0 o 3 +hw(aa+?+) h .(a + +u_at),
(3)
where the atomic operators are a 3 =Ie>, i .e . P„ =exp (_ 10C12)JOC12" In!, the inversion exhibits non-trivial features : the Rabi oscillations collapse and revive after a time that depends on the intensity of the initial field [13] . The atomic inversion in this case is plotted against what we will call scaled time or (At) in figure 1 (a) (for ILI 2 =25), and we clearly see the collapses and revivals . The revival occurs when the different terms in the sum (5) are re-phased, and this happens at a time :
tc R=
2n n1/2 . A
(6)
We see that for each example of field statistics, the atomic response is qualitatively very different . Now we want to consider the initial field prepared in a
H. Moya-Cessa and A . Vidiella-Barranco
2484
1 .00 1 0 .76 0 .50 0 .25 0 .00 -0 .25 -0 .50 -0 .75-1 .00 0
I
10 20
30 40
50 60 70 80 90 100 (a)
10 20
30 40
50 60 70 80 90 100
1 .00 0.75 0.50 0 .25 0 .00 -0 .26 -0 .60 -0 .75 -1 .00 0
(b) 1 .00 0 .75 0 .50 0 .25 ~r 0.00 -0 .26 -0 .50 -0 .76 -1 .00 0
!~
I
I
10 20 30 40 60 60
I
I
70 80 90 100
ocaled time (c) Figure 1 . Atomic inversion as a function of time for the atom initially excited and the field prepared in a squeezed coherent state a 2 =25 with different squeezing parameters (a) r=0, (b) r=1, (c) r=2 .
squeezed coherent state 1~, a>, where ~=r exp (iO) is the (complex) squeezing parameter, and a the coherent amplitude . The properties of the squeezed states are extensively described in reference [1], and here we will just write down its photon number distribution [14] :
PS= (n! cosh r) - 1 [2 tanh r]" x
exp { - JaI 2 - 1 tank r[a* 2 exp (iO)
a + a* exp (i6) tanh r 2 2 +a exp(-iO)]}H" [2 exp (W)tanhr] 1 /2 ] (7)
Squeezed light with two-level atoms
a., 0.04
2485
-
0.02 0.00
•
•
10
20
30 (a)
10
20
30
0.00 0
40
, 40
50
60
, 50
60
(b) 0 .12
0 .04
0.00 •
•
10
20 30
40
60
60 70
80
90
100
n (c) Figure 2 . Photon number distribution of the initial field for different squeezing parameters (a) r=0, (b) r=1, (c) r=2 . In all cases a 2 =25 .
The expression (7) is quite complicated in comparison with the simple Poisson distribution for the coherent state, but the changes in the atomic response might be relatively subtle, depending on the values of the parameters involved . We are going to study the case in which the initial field is a squeezed state with 0 = 0, i .e . both ~ = r and a real . This corresponds to a contour in phase space which is an ellipse with its minor axis parallel to the (x) `real axis', as shown in figure 8 at t = 0 . For this particular choice, the atomic inversion is plotted in figure 1 (b) and 1 (c), (for r=1, 2 respectively) . We immediately notice two features : the increase of the collapse time as r increases [3] and the appearance of ringing revivals [4] . In figure 2 we show the photon number distribution of the initial field for different squeezing parameters (r=0, 1, 2) . The familiar oscillations [5] in the photon number distribution of a squeezed state are present for not so small squeezing parameters . As discussed in reference [4], the additional peaks in the PS., each one with `local' mean photon
2486
H. 1 .00 0 .76 0 .500 .25 r y 0 .00 -0 .25.60 -0 .76 -1 .00 0 1 .00 0.75 0.500.25 0.00 -0.25-0 .50-0.75-1 .00 0
Moya-Cessa and A . Vidiella-Barranco
6
, 10 (a)
, 16
20
P
, 6
10
15
20
10 16 scaled time (c)
20
(b) 1 .00 0 .75 0.60 0.26 0 .00 -0 .25-0 .50-0 .75-1 .00 0
0
Figure 3 . Atomic inversion as a function of time for the atom initially excited and the field prepared in a squeezed vacuum state a=0 with (a) r=0, (b) r=1, (c) r=2 .
numbers, will be responsible for the ringing revivals of the atomic inversion . It is worth mentioning that the revival time also changes for an initial squeezed field, and in fact is given by : tR=(2tc/A)(Ial 2 +sinh 2 r) 112 [3] . For r = 0 we get the revival time in the coherent state case tR [13] . Another interesting case is when the field is prepared in the squeezed vacuum state (a=0) . In figure 3 we plot the atomic inversion for different squeezing parameters (r = 0, 1,2) . For r = 0 we see the ordinary vacuum Rabi oscillations for an initially excited atom . For increasing r, however, the oscillations not only become more irregular, but they also start to resemble the inversion in the case when the field is prepared in a thermal state [3,12] . It is interesting to see that by varying the parameters in the photon number distribution (7) we get atomic responses as diverse as the ones found if the field is prepared in a number state, a thermal state and a coherent state .
Squeezed light with two-level atoms
2487
3. Field dynamics 3 .1 . Evolution of the field entropy We start analysing the time evolution of the field von Neumann entropy [9]. Because in our problem both atom and field are initially in a pure state, the total entropy is going to be zero . The atom and field can be treated as separate subsystems, with reduced density operators given by the tracing operation : Pf(a)(t) = Tra(f) [P(t)],
(8)
where 5(r) is the total density operator (4) . The von Neumann entropy for each one of the subsystems is defined as Sf(a)
=
- Tr (&(.) (t) In [Pf(a) (t)] 1 .
(9)
According to the Araki-Lieb theorem [15], provided the total initial entropy is zero (in a closed system), the entropy of the sub-systems will be equal, i .e . Sf =Sa . So, in the present case it is convenient to calculate the atomic entropy, rather than the field entropy [6] . From the total density operator in (4), Pa(t) = Trf [P(t)] =
CAl 2
X121 2 22 '
with 00 Ai,i = n=0 Y_ < nl p1 n> . In the case of an initial squeezed state jr, a> (r, a real), the corresponding density operator will be P,(0)=lr,a) 1 . 34) . When the field is sub-Poissonian there is an increase in the sharpness of the revivals of the atomic inversion, and the optimum value of r for which the entropy is a minimum lies in the sub-Poissonian interval . In the same graph we also notice the dependence on a . Nevertheless, the minimum always occurs at t=tc, i.e . it does not depend on the squeezing parameter . The entropy exhibits relatively rapid oscillations in the region around the minimum, or more precisely, in the `revival region' . This is where the atomic inversion also shows oscillations . As we can see in figure 6, this correspondence is also true in the case of the ringing revivals investigated in reference [4]. The additional peaks in the photon number distribution for a squeezed state (see figure
0.7 0 .6 0.5 0.4 0.3 0.2 0.1 0.0
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r
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i I
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V
A
J
i
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V
1'0
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10
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20
(b) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 .0
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scaled time Figure 7 .
(c) Field entropy as a function of time for initial fields prepared in the squeezed vacuum state (a=0) with (a) r=0, (b) r=1, (c) r=2 .
Squeezed light with two-level atoms
2491
2 (b), (c)) will give rise to ringing revivals as well as modulated oscillations in the entropy . However, the Shannon entropy in the number state basis does not show a well-defined ringing structure [4], and this demonstrates that more subtle information can retrieved by examining the von Neumann (total) entropy rather than the Shannon entropy, because the latter involves only diagonal elements of p . In figure 7 we can see that, when the initial field is prepared in a squeezed vacuum state, if we increase the squeezing parameter r, the field entropy becomes increasingly irregular and with values characteristic of a mixed state, S z 0 . 7 . This is in qualitative agreement with the fact that the atomic response for the field initially prepared in a squeezed vacuum state is very similar to when it is prepared in a thermal (mixed) state [3,12] .
3 .2 . Evolution of the Q-function The evolution of the Q-function provides valuable information about the cavity field dynamics in the Jaynes-Cummings model [8] . The Q-function is a quasiprobability distribution [7] associated with anti-normally ordered products of annihilation and creation (bosonic) operators, and can be written in a convenient form as :
Q(x,y ; t) =n
« Iif(t)IQ>,
where I#> is a coherent state with amplitude fi=x+iy, and p f (t) is the reduced density operator for the field (equation (1)), obtained from the total density operator (4) . Suppose the field is initially prepared in a coherent state . The splitting and recombination of branches of the Q-function in phase space is clearly associated with the collapses and revivals of the atomic inversion : when a splitting of the Q-function occurs, we have the collapse of the Rabi oscillations, accompanied by a complicated evolution of the field . At the time when the two branches are most `far apart', i .e . at t=tR/2, the interference effects are such that the field is nearly in a pure state . At t = tR, a recombination of the two branches in the Q-function occurs, but in this case the field is no longer in a pure state . If the initial field is prepared in a squeezed coherent state, i .e . Pf (O) = Ia, r>< Oml'
2 49 6
H. Moya-Cessa and A . Vidiella-Barranco
where IOm > are the states of well defined phase which can be expanded as s
1 Iem >
(S+1)1(2
n=0 exp (in0 n,)In>,
(19)
with 0,„=Oo+2nm/(s+l), m=0,I, . . .,s and where 0 0 is an arbitrary reference phase . After having calculated the relevant quantities, the limit s-+oc should be taken, in order to be consistent with the quantum mechanical formalism, in which an infinite state space is used . The formalism allows us to define a continuous phase probability distribution for a general field state described by P as : (20)
dP(0)= d0 .
The phase properties of the cavity field in the Raman-coupled JaynesCummings model as well as in the ordinary Jaynes-Cummings model have already been investigated for the field initially prepared in a coherent state [11] . One of the conclusions in [11] is that the variance of the phase operator in (18), , and the result is : c
)n-k
IE2 =
3
+4
Z D„Dk cos[( .Jn- .Jk)At] (_1
(n-k) 2
(22)
In figure 13 we have I plotted as a function of time for different squeezing parameters r=0, 1, 2 . We notice that for the `optimum' squeezing parameter r = 1, there is a slower degradation of the phase variance oscillations than for other values of r . This means that the field not only returns approximately to a pure state (at t= tc , for instance), but it also (nearly) recovers its fluctuations in phase for a particular value of the squeezing parameter (in this case, for r= 1) . a=
/2
Conclusion The interaction between squeezed light and matter (in the Jaynes-Cummings model) shows some interesting features concerning not only the atomic dynamics [3, 4] but the cavity field as well . The von Neumann (total) field entropy has shown
2497
Squeezed light with two-level atoms
3 .00
2 .00
P 1 .00
0 .00
-n
71
6
Figure 12 . Phase probability distribution as a function of the phase angle 0 at (1) t=0, (2) t=tR /2, (3) r=tR . In this case r=1 .
0.00
1
0
1
I
I
,
I
I
10 20 30 40 60 60 70 80 90 100
igcaled time Figure 13 . Standard deviation a of the phase operator as a function of time for the field initially prepared in a squeezed coherent state with different squeezing parameters (1) r=0, (2) r=1, (3) r=2 .
H . Moya-Cessa and A . Vidiella-Barranco
2498
itself to be a very useful tool to investigate the field dynamics . We found that the field (initially prepared in a squeezed state) evolves approximately into a pure state at half of the revival time as well as at the revival time . The periodic secondary (ringing) revivals discussed in [4], which are actually due to the oscillations in the photon number distribution [5], are also evident if one looks at the entropy evolution as we see in figure 6 . It is worth mentioning that the tendency of the field to turn into a pure state in the revival region has been already noticed [21] when the initial field is prepared in a superposition of coherent states [18] . Although that particular superposition state exhibits squeezing, we cannot fully compare the results in [21] with those obtained here, where ideal squeezed states have been used . An interesting characteristic feature is that the field is in its purest state at tc=(2n/2)(IaJ 2) I/2 rather than at tR=(2n/).)(Ial 2 +sinh 2 r) 112 apparently for any value of the squeezing parameter except for r = 0, i .e . the coherent state case . This means that the coherent amplitude dictates the time at which the field gets less mixed, despite the fact that this mechanism is related to the squeezing nature of the field (remember that in the Jaynes-Cummings model the field becomes squeezed as time goes on [20]) . Further analysis using the evolution of the field Q-function as well as its photon number distribution showed us that the field is approximately in a squeezed state at the revival time and in a coherent superposition of squeezed states at half of the revival time . Given a field coherent intensity IaI 2 , there is an optimum value of the squeezing parameter for which the entropy is a minimum . We see that the quantum interference between states formed during the field evolution in the JaynesCummings model are responsible for subtle and sometimes surprisingly different effects, depending on the initial conditions .
Acknowledgments We would like to thank Professor Peter Knight and Dr Barry Garraway for helpful and stimulating discussions and for reading the manuscript . This work was supported in part by CAPES (Coordenacao de Aperfeic oamento de Pessoal de Nivel Superior), the Mexican Consejo Nacional de Ciencia y Tecnologia (CONACyT), and the Overseas Research Student Awards Scheme . We would also like to thank one of the referees for valuable suggestions .
References [1] LOUDON, R., and KNIGHT, P . L ., 1987, J . mod . Optics, 34,709 ; ZAHEER, K ., and ZUBAIRY, M . S ., in Advances in Molecular and Optical Physics, vol . 28, edited by D . Bates and
[2] [3] [4]
B . Bederson (New York : Academic Press), p . 143 . JAYNES, E . T ., and CUMMINGS, F . W., 1963, Proc . IEEE, 51, 89 ; BARNETT, S . M ., FILIPOWICZ, P ., JAVANAINEN, J ., KNIGHT, P . L ., and MEYSTRE, P ., 1986, in Frontiers in Quantum Optics, edited by E . R . Pike and S . Sarkar (Bristol : Adam Hilger), p .485 . MILBURN, G . J ., 1984, Optica Acta, 31, 671 . SATYANARAYANA, M . V ., RICE, P ., VYAS, R ., and CARMICHAEL, H . J ., 1989, J . opt. Soc.
Am . B,
[5] [6]
6, 228 .
SCHLEICH, W . P ., WALLS, D . F ., and WHEELER, J . A ., 1988, Phys . Rev . A, 38,1177 ; SCHLEICH, W. P ., and WHEELER, J . A ., 1987, J . opt . Soc . Am ., B4, 1715 . PHOENIX, S . J . D ., and KNIGHT, P. L ., 1988, Ann . Phys ., 186, 381 ; GEA-BANACLOCHE, J .,
1990,
Phys . Rev . Lett ., Rev . A, 44, 6203 .
65, 3385 ;
PHOENIX,
S . J . D .,
and KNIGHT,
P . L ., 1991,
Phys.
Squeezed light with two-level atoms
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HILLERY, M ., O'CONNELL, R . F ., SCULLY, M . 0 ., and WIGNER, E . P ., 1984, Phys . Rep . 106, 121 . EISELT, J ., and RISKEN, H ., 1991, Phys . Rev . A, 43, 346 . vON NEUMANN, J ., 1955, Mathematical Foundations of Quantum Mechanics (Princeton : Princeton University Press) . PEGG, D . T ., and BARNETT, S . M ., 1989, J . mod. Optics, 36, 7 . PHOENIX, S . J . D ., and KNIGHT, P . L ., 1990, J . opt . Soc. Am ., B7,116 ; DUNG, H . T ., TANAS, R ., and SHUMOVSKY, A . S ., 1990, Optics Commun ., 79, 462 . KNIGHT, P . L ., and RADMORE, P . M ., 1982, Phys. Lett . A, 90, 342 . NAROZHNY, N . B ., SANCHEZ-MONDRAGON, J . J ., and EBERLY, J . H ., 1981, Phys . Rev . A, 23, 236 . YUEN, H . P ., 1976, Phys . Rev . A, 13, 2226 . ARAKI, M ., and LIEB, E ., 1970, Commun . Math . Phys ., 18,160 . GEA-BANACLOCHE, J ., 1991, Phys . Rev . A, 44, 5913 ; GEA-BANACLOCHE, J ., 1991, Phys . Rev . A, 44, 5913 ; GEA-BANACLOCHE, J ., 1992, Optics Commun ., 88, 531 . BUZEK, V ., MOYA-CESSA, H ., KNIGHT, P . L ., and PHOENIX, S . J . D ., 1992, Phys . Rev . A, 45, 8190. BUZEK, V ., VIDIELLA-BARRANCO, A ., and KNIGHT, P . L ., 1992, Phys . Rev . A, 45, 6570 . TOMBESI, P ., and MECOZZI, A ., 1987, J . opt . Soc . Am ., B4,1700 . MEYSTRE, P ., and ZUBAIRY, M . S ., 1982, Phys. Lett ., 89A, 390 ; KUKLIIVSKI, J . R ., and MADAJCZYK, J . L ., 1988, Phys . Rev . A, 37, 3175 . VIDIELLA-BARRANCO, A ., MOYA-CESSA, H ., and BUZEK, V ., 1982, J . mod. Optics, 39, 1441 .
