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Transferring squeezing and statistics in coupled circuits
Physica A 311 (2002) 188 – 198
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Transferring squeezing and statistics in coupled circuits H. Rodriguesa , D. Portes Jr.a , S.B. Duarteb;∗ , B. Baseiac a Centro
Federal de Educaca˜ o Tecnologica -CEFET=RJ, CEP 20.271-110, Rio de Janeiro (RJ), Brazil Brasileiro de Pesquisas F$sicas =CNPq, CEP 22.290-180, Rio de Janeiro (RJ), Brazil c Instituto de F$sica, Universidade Federal de Goi as, CEP 74.001-970, Goiˆania (GO), Brazil
b Centro
Received 22 February 2002
Abstract In previous papers we have studied stationary [Physica A 197 (1993) 364] and transient [Physica A 268 (1999) 121] solutions of a quantized LC-circuit showing squeezing e7ect generated by a time-dependent parameter. Here we extend this issue studying the transfer of this e7ect in coupled circuits, as well as the time evolution and transfer of statistical properties from one to c 2002 Elsevier Science B.V. All rights reserved. the other. PACS: 03.65.Bz; 42.50.Dv Keywords: LC-circuits; Quantum noise; Statistical properties
1. Introduction Generation of coherent ?elds out of a classical current was ?rstly shown by Glauber [1] (see also Nussenzveig), the quantized ?eld evolving from a vacuum state |0 to a coherent state |(a| = |; a being the annihilation operator). Later on, Louisell [2] extended this result for a quantized LC-circuit. These results have been employed in “quantum state engineering” to generate various kinds of ?eld states when starting from a coherent ?eld previously prepared inside a high-Q microwave cavity (see, e.g., [3]). More recently, Refs. [4,5] showed that when the LC-circuit has a time-dependent parameter (e.g., the inductance L(t)), this system evolves from a vacuum state to a squeezed one, i.e., Fai2 ¡ 14 (with Faj2 ¿ 14 ); i; j =1; 2; i = j and Fai Faj ¿ 14 to attain the Heisenberg relation. Here, a1 = (a + a+ )=2 and a2 = (a − a+ )=2i; a+ (a) being the ∗
Corresponding author. E-mail address: [email protected] (S.B. Duarte).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 8 2 5 - 7
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
189
creation (annihilation) operator, as de?ned below. For each system considered (e.g., light ?eld, oscillator, trapped ion, LC-circuit, etc.) a and a+ (hence a1 and a2 ) have their speci?c physical meaning. For example, for light ?eld a1 and a2 stand for its quadrature components whereas for any kind of oscillator a1 and a2 stand for its position and momentum, respectively. The present work is an extension of those in Refs. [4,5], for the case of two coupled LC-circuits. The issue here is to verify the transference of some nonclassical e7ects between these two sub-systems, the result being specially relevant when one wants to generate such e7ects in one of them, whose direct access is diKcult. In this case, the strategy is producing the e7ect upon one accessible sub-system, as done e.g. in Refs. [4,5], and then transfer the e7ect to the other interacting sub-system. Here we will consider the quantum noise reduction below the vacuum level (squeezing) and statistical properties for each sub-system, their time evolution and transfer from one sub-system to the other. This paper is organized as follows. In Section 2 we present the Hamiltonian describing our complete (coupled) system and obtain the Heisenberg equations of motion and their solutions. In Section 3 we calculate dispersions of the operators qj and pj , de?ned below, allowing us to study squeezing e7ect and its transfer from one sub-system to the other. In Section 4 we present an exact solution of the Schrodinger equation describing our coupled system to calculate statistical properties of the two sub-systems: (A) the Mandel-Q factor, allowing us to verify the occurrence of sub-Poissonian e7ect and its transfer; (B) the statistical distribution Pn(i) , describing the ith oscillator, plus its transfer from one to the other and (C) its mixing properties, via Tr[(i) (t)]2 , where (i) is the (reduced) density operator for the ith oscillator. Section 5 contains the concluding remarks. We have written all equations so compact as possible to abbreviate the presentation.
2. Model Hamiltonian and Heisenberg equations of motion To extend the model employed in Refs. [4,5] we start from the Hamiltonian describing our system composed of two coupled LC-circuits 2 1 2 1 2 1 (1) q j + Lj ij + Lj Lk ij ik ; H= 2Cj 2 2 j; k=1
where q j (ij ) stand for the jth charge (current) operator [2], Cj (Lj ) is the capacitance (inductance) of the jth circuit, j = 1; 2. Setting the conjugate momenta (j; k = 1; 2; j = k), pj = Lj ij + Lj Lk ik ; (2) and mj = (1 − 2 )Lj ;
(3)
190
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
we obtain the Hamiltonian in the modi?ed form pj2 pj pk 1 1 2 2 ; + mj !j q j + √ H= 2 2 mj mk 2mj
(4)
j=k
where !j2 =
1 : mj C j
(5)
From Eq. (4), we obtain the Heisenberg equations of motion. They result in compact forms (with k = j), dq j pj pk = + √ (6) dt mj m j mk and dpj = −mj !j2 q j : dt Next, writing q j and pj in terms of the creation and annihilation operators ˝ (aj + aj+ ) qj = 2mj !j and
pj =
˝mj !j 2
aj − aj+ i
(7)
(8)
:
leads Hamiltonian (4) to the form 1 !j aj+ aj − (aj − aj+ )(ak − ak+ ) ; H=˝ 2
(9)
(10)
j=k
where √ !1 !2 : (11) 2 Now, assuming that the inductance Lj = Lj (t) (mj = mj (t); !j = !j (t); = (t)), we obtain the Heisenberg equations for the operators aj and aj+ (with j = k). We ?nd for aj (t), m˙ j daj 1 !˙ j aj+ − i[!j aj + (ak − ak+ )] ; + (12) = dt mj 2 !j =
the same being valid for aj+ , replacing aj ↔ aj+ and i → −i. We write aj (t) in the form aj (t) = jj (t)aj (0) + ∗jj (t)aj+ (0) + jk (t)ak (0) + ∗jk (t)ak+ (0)
(13)
with jk (0) = jk ;
∗jk (0) = 0
(14)
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
191
and (with j = k) |jj |2 − |jj |2 + |jk |2 − |jk |2 = 1
(15)
to hold the commutation relation [aj ; aj+ ] = 1. Inserting Eq. (13) into Eq. (12) leads to the equation of motion for jk (t); jk (t) in a compact form. We have for jk (t) djk (16) = j jk − i[!j jk + (kk − kk )] dt and for jj (t) djj = j jj − i[!j jj + (kj − kj )] (17) dt ˙ with j = 12 (!˙ j =!j − m˙ j =mj ), where 1 = L=(4L) and 2 = 0 from Eqs. (3) and (5). Similar equations are valid for jk (t) and jj (t), replacing ↔ ∗ in Eqs. (16) and (17). 3. Squeezing e ect and its transfer For initial coherent states we have aj (0)=j , aj+ (0)=j∗ , and obtain (with j = k), 2˝ (Re{(jj + jj )j } + Re{(jk + jk )k }) ; (18) q j = mj !j
(19) pj = 2˝mj !j (Im{(jj − jj )j } + Im{(jk − jk )k }) : The substitution of aj (t) from Eq. (13), and its adjoint aj+ (t); in the expressions n(t) = aj+ (t)aj (t) and aj2 (t), with aj = jj j + ∗jj j∗ + jk k + ∗jk k∗ ), gives aj2 = aj 2 + jj vjj + jk vjk ;
(20)
nj = |aj |2 + |jj |2 + |jk |2 :
(21)
Also, application of Eqs. (8), (9) and (13) leads to ˝2 (|jj + jj |2 + |jk + vjk |2 ) ; 2 mj !j ˝mj !j pj2 = pj 2 + (|jj − jj |2 + |jk − jk |2 ) : 2
q2j = q j 2 +
(22) (23)
2 2 2 Eqs. (18) – (23) allow us to obtain the variances FO = O − O for the dimensionless operators qj = (mj !j =˝)q j and pj = (1=˝mj !j )pj . We have, for j = k; j = 1; 2; k = 1; 2,
Fq2j = 12 (|jj + jj |2 + |jk + jk |2 )
(24)
Fp2j = 12 (|jj − jj |2 + |jk − jk |2 ) :
(25)
and
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 0.36
0.27
TDO TIO 0.32
q-Variance
q-Variance
0.26
0.25
0.28
0.24
0.24 0.2
TDO TIO 0.23
(a)
0.16 0
10
time (ω0t)
20
30
0
10
(b)
time (ω0t)
20
30
0.4
q-Variance
0.3
0.2
0.1
TD O TI O 0
(c)
0
10
time (ω0t)
20
30
Fig. 1. Plots of variances Fq2TIO (solid line) and Fq2TDO (dashed line) for =0:1 (a), =0:5 (b), and =0:9 (c) and ! = 0:5. The transference of squeezing e7ects in the dimensionless charge operator q is shown. For this ?gure and all subsequent ones the initial excitation of each oscillator is given by n1 (0)=4, n2 (0)=0. The vertical long dashed line marks the instant !0 " when linear variation of the inductance is set o7.
Figs. 1a–c show the plots of q-variances versus time for particular values of parameters involved in the system. All plots are for L1 = L1 (t); L2 = L0 with L0 being constant. Hereafter, for brevity, we will call TDO and TIO for time-dependent and time-independent oscillators, respectively. We will assume the time dependence in the TDO coming from L1 (t) = L0 + !t, with
! = 0:5 when 0 ¡ !0 t ¡ !0 "; in all plots we √ have taken !0 " = 6 10, with !0 = L01C1 . As a general behavior, Fig. 1 shows the squeezing e7ect emerging originally in the TDO being transferred to the TIO. (As well known, a TDO (TIO) can (cannot) exhibit the squeezing e7ect when isolated.) The transfer of this e7ect is more eKcient when the coupling parameter increases taking values = 0:1; 0:5 and 0:9 (Figs. 1a–c).
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193
4. Statistical properties and its transfer In this section we study, for each sub-system, the occurrence of sub-Poissonian e7ect (Section 4.1), the statistical distribution of excitations (Section 4.2) and mixing properties (Section 4.3), as follows 4.1. Sub-Poissonian statistics The Mandel factor (see, e.g., [6]) Fnj2 (t) − nj (t) Qj = nj (t)
(26)
allows us to verify the occurrence of sub-Poissonian e7ect for a sub-system: Q ¡ 0 denotes the sub-system as sub-Poissonian, hence nonclassical; it is super-Poissonian, hence classical when Q ¿ 0. The border line (Q = 0) stands for Poissonian statistics characterizing a coherent state. When Q approaches its lower bound (Q = −1) the state becomes maximum sub-Poissonian, coinciding with the statistics exhibited by a system prepared in a number state (Fn2 = 0 → Q = −1, cf. Eq. (26)). Now, using Eqs. (13), (20) and (21) we have nj2 (t) = nj (t) + |aj2 |2 + 4(|jj |2 + |jk |2 )|aj |2 + 2(|jj |2 + |jk |2 )2 from which one obtains the Mandel-Q factor(cf. Eq. (26)), |aj2 |2 + nj (t)2 − 2|aj |4 Qj (t) = : nj (t)
(27) (28)
Figs. 2a–c show plots of the Mandel-Q factor versus time for =0:1; 0:5 and 0:9. As a general behavior we note that the statistics passes from sub- to super-Poissonian when the interaction parameter increases, the e7ect being signi?cant for & 0:9, no matter the rate of the time-variation !. Also, we observe the transfer of sub-Poissonian e7ect from the TDO to the TIO. (As well known, a TDO (TIO) will (will not) exhibit this e7ect when isolated.) The occurrence of a sub-Poissonian statistics is enhanced when the coupling parameter increases but diminishes if becomes very large ( & 0:9). 4.2. Statistical distribution To describe statistical distribution, we must know the wave function |$(t) = cn1 n2 (t)|n1 n2
(29)
n 1 n2
where {|n1 n2 } are eigenstates of operator n1 and n2 at t = 0. Nevertheless, it is possible to obtain |$(t) as the coeKcient function {; }. With this purpose, we must ?rst obtain the fundamental state for the operators aj (t), i.e., aj (t)|0 = 0 ; where |0 =
n1 n 2
cn01 n2 |n1 n2 :
(30) (31)
194
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 0.06
0.6
TDO TIO
TDO TIO
0.04
Mandel-Factor
Mandel-Factor
0.4 0.02
0
0.2
0 -0.02
-0.04
-0.2 0
(a)
10
time (ω 0t)
20
30
0
(b)
10
time (ω 0t)
20
30
5.0
TDO TIO
Mandel-Factor
4.0
3.0
2.0
1.0
0.0
-1.0
0
10
(c)
time (ω 0t)
20
30
Fig. 2. Mandel factor Q as a function of time, for = 0:1 (a), = 0:5 (b), and = 0:9 (c) with ! = 0:5, showing sub-Poissonian statistics and its transfer between the sub-systems.
Using Eqs. (13) and (30), we can obtain 12 ∗21 − 22 ∗11 n1 − 1 12 ∗22 − 22 ∗12 n2 0 0 0 cn1 n2 = c(n1 −1)n2 + cn1 (n2 −1) ∗ − ∗ ∗ − ∗ 11 22 n1 11 22 n1 12 21 12 21
(32)
or, equivalently, cn01 n2 = cn01 (n2 −1)
21 ∗12 − 11 ∗22 ∗ − ∗ 11 22 12 21
n2 − 1 21 ∗11 − 11 ∗21 0 + c(n ∗ − ∗ 1 −1)n2 n2 11 22 12 21
n1 : n2
(33)
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
195
Using this recursive relation we obtain, step by step, all the coeKcients {cn01 n2 } from the 0 ; this later being determined by the normalization constraint. The identity value of c00 ∗ 21 11 − 11 ∗21 = 12 ∗22 − 22 ∗12 assures the occurrence between the two expressions. In fact, this identity is preserved by the equations for the coeKcients {; }. Once we know the state |0 we can obtain the time-dependent base {|n1 (t)n2 (t)} (eigenstates of the operators n1 e n2 ), using the equation |n1 (t)n2 (t) = √
1 √ (a+ (t))n1 (a2+ (t))n1 |0 : n1 ! n2 ! 1
(34)
Finally, we obtain the time-dependent coeKcients cn1 n2 (t) appearing in Eq. (29), cn1 n2 (t) = n1 n2 | $(t) = n1 (t)n2 (t) | $(0) :
(35)
Now, the density operator (t) = |$(t)$(t)| describes the entire (coupled) system and allows us to obtain the reduced density operator describing the ith sub-system (with i; j = 1; 2; j = i), (i) (t) = Trj (|$(t)$(t)|)
(36)
yielding the statistical distribution for the ith oscillator: Pn(i) (t) = ni |(i) (t)|ni . Fig. 3 shows the plots of Pn(i) (t) as a function of n, for tree times: !0 t=0 and !0 t=2:37; !0 t= 5:85 for initial excitations: n1 (0) = 4; n2 (0) = 0, where n1 (0)(n2 (0)) stands for the initial excitation in the TDO (TIO). For all these plots = ! = 0:5. As a general behavior we note that the statistical distribution describing each oscillator changes as time evolves and we note the statistical distribution Pn describing the TDO at !0 t = 0 being transferred to the TIO at a certain time, e.g., at time about !0 t = 2:37 (Fig. 3b). However, when both oscillators have the same initial excitation, i.e.: n1 (0) = n2 (0) we no longer observe this transfer. In this case, the TIO returns to its initial statistical distribution Pn(2) (0), the same not occurring for the TDO (not shown in ?gures). 4.3. Mixing properties For the ith oscillator we calculate Tr[(i) (t)]2 , which measures its degree of mixing. Although the entire system remains in a pure state for all time, namely: [(t)]2 = (t), hence Tr[(t)]2 = Tr[(t)] = 1, each sub-system becomes mixed for t ¿ 0, when Tr[(i) (t)]2 ¡ 1; i = 1; 2, except for isolated times as we see in Fig. 4 showing the plot of Tr[(i) (t)]2 versus time for the same values of parameter used in Fig. 3. In the present case, we note that the state describing each oscillator is mixed for t ¿ 0, becoming near a pure state at isolated points, as for !0 t = 4:43; 27:83; etc. Comparing Figs. 3 and 4 we see that the transference of statistical distribution (see Fig. 3 for !0 t = 2:37) occurs when the sub-systems are near pure states, i.e., Tr[(t)]2 1 (we have Tr[(t)]2 0:98 for !0 t = 2:37, in Fig. 4). The same does not occur for other times where Tr[(t)]2 = 1. Actually, it is shown that the pure states assumed by an evolving sub-system have no obligation to be the same [9].
196
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 1 0.8
ω 0t = 0
0.6 0.4 0.2 0 0.8
TDO
ω0t = 2.37
TIO
Pn
0.6 0.4 0.2 0 0.8
ω0t = 5.85
0.6 0.4 0.2 0 0
4
8
12
n Fig. 3. Time evolution of statistic distribution Pn , for = 0:5; ! = 0:5, showing the statistic transfer between sub-systems.
5. Concluding remarks We have studied quantum noise reduction, statistical properties and transfer of these e7ects from a TDO to a TIO, the ?rst (second) sub-system standing for a time-dependent (independent) LC-circuit. For this purpose, we have employed the Schrodinger picture— more appropriate to the study of statistical properties—and present the exact solution of the Schrodinger equation governing our system, constituted by these two interacting LC-circuits [cf. Eq. (1)]. The occurrence of quantum noise below the vacuum level (squeezing) in the TDO and its transfer to the TIO are shown in Figs. 1a–c. No such e7ect occurs in the TIO when isolated. When the two oscillators are coupled, the e7ect occurs similarly in both oscillators before the time variation of L(t) has initiated (t ¡ 0; not shown in ?gures). The e7ect and its transfer increase when grows (Figs. 1a–c). The enhanced squeezing, originally generated in the TDO and transferred √ to the TIO, remains there after the time variation of L1 (t) terminates at !0 " = 6 10), hence transfer occurs permanently, being more eKcient when the coupling-parameter grows.
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
197
1
Tr (ρ2)
0.96
0.92
0.88
0.84 0
10
20
30
time (ω 0t) Fig. 4. Time evolution of mixing properties (Tr[(i) (t)]2 ) of sub-systems, showing that the sub-systems return to pure states for some isolated times.
Figs. 2– 4 concern with various statistical aspects characterizing our sub-systems. Figs. 2a–c stand for Mandel-Q factor showing the occurrence of sub-Poissonian effect in the TDO and its transfer to the TIO: in these plots we have used ! = 0:5, although this parameter does not a7ect the results signi?cantly. We note in these ?gures that the sub-Poissonian e7ect transferred to the TIO increases when increases (compare Figs. 2a and b), but diminishes when becomes very large ( ¿ 0:9), see Fig. 2c. Hence, contrary to the case of squeezing, there is an interval of -values where sub-Poissonian e7ect and its transfer are enhanced. Now, before the time variation is set on, the statistical aspects of both oscillators are similar, the statistics becoming super-Poissonian if is large ( & 0:9), but changes successively between sub- and super-Poissonian if is not large (not shown in ?gures): for about 0.5, during certain time intervals one oscillator may be sub-Poissonian while the other is super-Poissonian and vice versa (Fig. 2b). Fig. 3 stands for the time evolution of statistical distribution Pn(i) (t) describing the ith oscillator i = 1; 2. A striking result to be stressed is the possibility of transferring a desired statistical distribution from the TDO to the TIO: Note in this ?gure that the statistical distribution of the TDO at t = 0 is transferred to the TIO at !0 t 2:37. Also note in Fig. 4 that Tr[(i) (2:37)]2 1, when the reduced density operator (i) becomes near a pure state. Fig. 4 concerns with mixing properties. It exhibits Tr[(i) (t)]2 versus time showing both oscillators starting from pure states at time !t = 0 and becoming mixed states for !t ¿ 0. When !t ¿ 0 we obtain Tr[(i) (t)]2 ¡ 1, except for isolated times when each oscillator assumes a pure state [Tr[(i) (t)]2 = 1]. As mentioned before, in general, the pure state assumed by an evolving sub-system are not necessarily the same: it can be the initial pure state, or another.
198
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
In resume, transfer of squeezing, sub-Poissonian e7ect and statistical distribution occur from the TDO to the TIO in the present scheme. These results are specially interesting when, e.g., the TDO is of easy direct manipulation while the TIO is of diKcult direct access. In this case, we call the TDO (TIO) as the active (passive) oscillator. One such example is found in Ref. [7] where a trapped ion (see, e.g., [8]), laser cooled onto its quantum ground state level can be used as a very low temperature detector for RF-signal (source oscillator) applied to the trap electrodes, the source oscillator being prepared in a squeezed state through the coupling with the ion. A point deserving future attention in this perspective concerns with the improvement of the present scheme (see Eq. (1)), making the transfer of squeezing and sub-Poissonian e7ects more eKcient, as it is eKcient for the transfer of statistical distribution. Finally, we remark that we have found the exact solution describing our system (cf. Section 4.2), which constitutes an available tool for potential application to other aspects of the system, another point to be explored in a future work. Acknowledgements The authors (SBD, BB) thank the CNPq, PRONEX (BB) and FAPERJ (HR, DP Jr.) for the partial support. References [1] R.J. Glauber, Phys. Rev. 130 (1963) 2529; H.M. Nussenzveig, Introduction to Quantum Optics, Gordon and Breach, New York, 1973, pp. 73. [2] W.H. Louisell, Quantum Statistical Properties of Radiation, Wesley, New York, 1973. [3] C.C. Gerry, P.L. Knight, Am. J. Phys. 65 (1997) 964 and references therein. [4] B. Baseia, A.L. de Brito, Physica A 197 (1993) 364. [5] D. Portes Jr., H. Rodrigues, S.B. Duarte, B. Baseia, Physica A 268 (1999) 121. [6] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994. [7] D.J. Heizein, D.J. Wineland, Phys. Rev. A 42 (1990) 2977. [8] F. Dietrich, J.C. Bergquist, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 62 (1989) 403; D.J. Heizein, et al., Phys. Rev. Lett. 66 (1991) 2080. [9] A.R. Gomes, Collapse and Revival of the wavefunction in coupled oscillators, M.Sc. Thesis, Federal University of Paraiba, Jo˜ao Pessoa, Brazil, 1996.
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Transferring squeezing and statistics in coupled circuits H. Rodriguesa , D. Portes Jr.a , S.B. Duarteb;∗ , B. Baseiac a Centro
Federal de Educaca˜ o Tecnologica -CEFET=RJ, CEP 20.271-110, Rio de Janeiro (RJ), Brazil Brasileiro de Pesquisas F$sicas =CNPq, CEP 22.290-180, Rio de Janeiro (RJ), Brazil c Instituto de F$sica, Universidade Federal de Goi as, CEP 74.001-970, Goiˆania (GO), Brazil
b Centro
Received 22 February 2002
Abstract In previous papers we have studied stationary [Physica A 197 (1993) 364] and transient [Physica A 268 (1999) 121] solutions of a quantized LC-circuit showing squeezing e7ect generated by a time-dependent parameter. Here we extend this issue studying the transfer of this e7ect in coupled circuits, as well as the time evolution and transfer of statistical properties from one to c 2002 Elsevier Science B.V. All rights reserved. the other. PACS: 03.65.Bz; 42.50.Dv Keywords: LC-circuits; Quantum noise; Statistical properties
1. Introduction Generation of coherent ?elds out of a classical current was ?rstly shown by Glauber [1] (see also Nussenzveig), the quantized ?eld evolving from a vacuum state |0 to a coherent state |(a| = |; a being the annihilation operator). Later on, Louisell [2] extended this result for a quantized LC-circuit. These results have been employed in “quantum state engineering” to generate various kinds of ?eld states when starting from a coherent ?eld previously prepared inside a high-Q microwave cavity (see, e.g., [3]). More recently, Refs. [4,5] showed that when the LC-circuit has a time-dependent parameter (e.g., the inductance L(t)), this system evolves from a vacuum state to a squeezed one, i.e., Fai2 ¡ 14 (with Faj2 ¿ 14 ); i; j =1; 2; i = j and Fai Faj ¿ 14 to attain the Heisenberg relation. Here, a1 = (a + a+ )=2 and a2 = (a − a+ )=2i; a+ (a) being the ∗
Corresponding author. E-mail address: [email protected] (S.B. Duarte).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 8 2 5 - 7
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
189
creation (annihilation) operator, as de?ned below. For each system considered (e.g., light ?eld, oscillator, trapped ion, LC-circuit, etc.) a and a+ (hence a1 and a2 ) have their speci?c physical meaning. For example, for light ?eld a1 and a2 stand for its quadrature components whereas for any kind of oscillator a1 and a2 stand for its position and momentum, respectively. The present work is an extension of those in Refs. [4,5], for the case of two coupled LC-circuits. The issue here is to verify the transference of some nonclassical e7ects between these two sub-systems, the result being specially relevant when one wants to generate such e7ects in one of them, whose direct access is diKcult. In this case, the strategy is producing the e7ect upon one accessible sub-system, as done e.g. in Refs. [4,5], and then transfer the e7ect to the other interacting sub-system. Here we will consider the quantum noise reduction below the vacuum level (squeezing) and statistical properties for each sub-system, their time evolution and transfer from one sub-system to the other. This paper is organized as follows. In Section 2 we present the Hamiltonian describing our complete (coupled) system and obtain the Heisenberg equations of motion and their solutions. In Section 3 we calculate dispersions of the operators qj and pj , de?ned below, allowing us to study squeezing e7ect and its transfer from one sub-system to the other. In Section 4 we present an exact solution of the Schrodinger equation describing our coupled system to calculate statistical properties of the two sub-systems: (A) the Mandel-Q factor, allowing us to verify the occurrence of sub-Poissonian e7ect and its transfer; (B) the statistical distribution Pn(i) , describing the ith oscillator, plus its transfer from one to the other and (C) its mixing properties, via Tr[(i) (t)]2 , where (i) is the (reduced) density operator for the ith oscillator. Section 5 contains the concluding remarks. We have written all equations so compact as possible to abbreviate the presentation.
2. Model Hamiltonian and Heisenberg equations of motion To extend the model employed in Refs. [4,5] we start from the Hamiltonian describing our system composed of two coupled LC-circuits 2 1 2 1 2 1 (1) q j + Lj ij + Lj Lk ij ik ; H= 2Cj 2 2 j; k=1
where q j (ij ) stand for the jth charge (current) operator [2], Cj (Lj ) is the capacitance (inductance) of the jth circuit, j = 1; 2. Setting the conjugate momenta (j; k = 1; 2; j = k), pj = Lj ij + Lj Lk ik ; (2) and mj = (1 − 2 )Lj ;
(3)
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
we obtain the Hamiltonian in the modi?ed form pj2 pj pk 1 1 2 2 ; + mj !j q j + √ H= 2 2 mj mk 2mj
(4)
j=k
where !j2 =
1 : mj C j
(5)
From Eq. (4), we obtain the Heisenberg equations of motion. They result in compact forms (with k = j), dq j pj pk = + √ (6) dt mj m j mk and dpj = −mj !j2 q j : dt Next, writing q j and pj in terms of the creation and annihilation operators ˝ (aj + aj+ ) qj = 2mj !j and
pj =
˝mj !j 2
aj − aj+ i
(7)
(8)
:
leads Hamiltonian (4) to the form 1 !j aj+ aj − (aj − aj+ )(ak − ak+ ) ; H=˝ 2
(9)
(10)
j=k
where √ !1 !2 : (11) 2 Now, assuming that the inductance Lj = Lj (t) (mj = mj (t); !j = !j (t); = (t)), we obtain the Heisenberg equations for the operators aj and aj+ (with j = k). We ?nd for aj (t), m˙ j daj 1 !˙ j aj+ − i[!j aj + (ak − ak+ )] ; + (12) = dt mj 2 !j =
the same being valid for aj+ , replacing aj ↔ aj+ and i → −i. We write aj (t) in the form aj (t) = jj (t)aj (0) + ∗jj (t)aj+ (0) + jk (t)ak (0) + ∗jk (t)ak+ (0)
(13)
with jk (0) = jk ;
∗jk (0) = 0
(14)
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
191
and (with j = k) |jj |2 − |jj |2 + |jk |2 − |jk |2 = 1
(15)
to hold the commutation relation [aj ; aj+ ] = 1. Inserting Eq. (13) into Eq. (12) leads to the equation of motion for jk (t); jk (t) in a compact form. We have for jk (t) djk (16) = j jk − i[!j jk + (kk − kk )] dt and for jj (t) djj = j jj − i[!j jj + (kj − kj )] (17) dt ˙ with j = 12 (!˙ j =!j − m˙ j =mj ), where 1 = L=(4L) and 2 = 0 from Eqs. (3) and (5). Similar equations are valid for jk (t) and jj (t), replacing ↔ ∗ in Eqs. (16) and (17). 3. Squeezing e ect and its transfer For initial coherent states we have aj (0)=j , aj+ (0)=j∗ , and obtain (with j = k), 2˝ (Re{(jj + jj )j } + Re{(jk + jk )k }) ; (18) q j = mj !j
(19) pj = 2˝mj !j (Im{(jj − jj )j } + Im{(jk − jk )k }) : The substitution of aj (t) from Eq. (13), and its adjoint aj+ (t); in the expressions n(t) = aj+ (t)aj (t) and aj2 (t), with aj = jj j + ∗jj j∗ + jk k + ∗jk k∗ ), gives aj2 = aj 2 + jj vjj + jk vjk ;
(20)
nj = |aj |2 + |jj |2 + |jk |2 :
(21)
Also, application of Eqs. (8), (9) and (13) leads to ˝2 (|jj + jj |2 + |jk + vjk |2 ) ; 2 mj !j ˝mj !j pj2 = pj 2 + (|jj − jj |2 + |jk − jk |2 ) : 2
q2j = q j 2 +
(22) (23)
2 2 2 Eqs. (18) – (23) allow us to obtain the variances FO = O − O for the dimensionless operators qj = (mj !j =˝)q j and pj = (1=˝mj !j )pj . We have, for j = k; j = 1; 2; k = 1; 2,
Fq2j = 12 (|jj + jj |2 + |jk + jk |2 )
(24)
Fp2j = 12 (|jj − jj |2 + |jk − jk |2 ) :
(25)
and
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 0.36
0.27
TDO TIO 0.32
q-Variance
q-Variance
0.26
0.25
0.28
0.24
0.24 0.2
TDO TIO 0.23
(a)
0.16 0
10
time (ω0t)
20
30
0
10
(b)
time (ω0t)
20
30
0.4
q-Variance
0.3
0.2
0.1
TD O TI O 0
(c)
0
10
time (ω0t)
20
30
Fig. 1. Plots of variances Fq2TIO (solid line) and Fq2TDO (dashed line) for =0:1 (a), =0:5 (b), and =0:9 (c) and ! = 0:5. The transference of squeezing e7ects in the dimensionless charge operator q is shown. For this ?gure and all subsequent ones the initial excitation of each oscillator is given by n1 (0)=4, n2 (0)=0. The vertical long dashed line marks the instant !0 " when linear variation of the inductance is set o7.
Figs. 1a–c show the plots of q-variances versus time for particular values of parameters involved in the system. All plots are for L1 = L1 (t); L2 = L0 with L0 being constant. Hereafter, for brevity, we will call TDO and TIO for time-dependent and time-independent oscillators, respectively. We will assume the time dependence in the TDO coming from L1 (t) = L0 + !t, with
! = 0:5 when 0 ¡ !0 t ¡ !0 "; in all plots we √ have taken !0 " = 6 10, with !0 = L01C1 . As a general behavior, Fig. 1 shows the squeezing e7ect emerging originally in the TDO being transferred to the TIO. (As well known, a TDO (TIO) can (cannot) exhibit the squeezing e7ect when isolated.) The transfer of this e7ect is more eKcient when the coupling parameter increases taking values = 0:1; 0:5 and 0:9 (Figs. 1a–c).
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
193
4. Statistical properties and its transfer In this section we study, for each sub-system, the occurrence of sub-Poissonian e7ect (Section 4.1), the statistical distribution of excitations (Section 4.2) and mixing properties (Section 4.3), as follows 4.1. Sub-Poissonian statistics The Mandel factor (see, e.g., [6]) Fnj2 (t) − nj (t) Qj = nj (t)
(26)
allows us to verify the occurrence of sub-Poissonian e7ect for a sub-system: Q ¡ 0 denotes the sub-system as sub-Poissonian, hence nonclassical; it is super-Poissonian, hence classical when Q ¿ 0. The border line (Q = 0) stands for Poissonian statistics characterizing a coherent state. When Q approaches its lower bound (Q = −1) the state becomes maximum sub-Poissonian, coinciding with the statistics exhibited by a system prepared in a number state (Fn2 = 0 → Q = −1, cf. Eq. (26)). Now, using Eqs. (13), (20) and (21) we have nj2 (t) = nj (t) + |aj2 |2 + 4(|jj |2 + |jk |2 )|aj |2 + 2(|jj |2 + |jk |2 )2 from which one obtains the Mandel-Q factor(cf. Eq. (26)), |aj2 |2 + nj (t)2 − 2|aj |4 Qj (t) = : nj (t)
(27) (28)
Figs. 2a–c show plots of the Mandel-Q factor versus time for =0:1; 0:5 and 0:9. As a general behavior we note that the statistics passes from sub- to super-Poissonian when the interaction parameter increases, the e7ect being signi?cant for & 0:9, no matter the rate of the time-variation !. Also, we observe the transfer of sub-Poissonian e7ect from the TDO to the TIO. (As well known, a TDO (TIO) will (will not) exhibit this e7ect when isolated.) The occurrence of a sub-Poissonian statistics is enhanced when the coupling parameter increases but diminishes if becomes very large ( & 0:9). 4.2. Statistical distribution To describe statistical distribution, we must know the wave function |$(t) = cn1 n2 (t)|n1 n2
(29)
n 1 n2
where {|n1 n2 } are eigenstates of operator n1 and n2 at t = 0. Nevertheless, it is possible to obtain |$(t) as the coeKcient function {; }. With this purpose, we must ?rst obtain the fundamental state for the operators aj (t), i.e., aj (t)|0 = 0 ; where |0 =
n1 n 2
cn01 n2 |n1 n2 :
(30) (31)
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 0.06
0.6
TDO TIO
TDO TIO
0.04
Mandel-Factor
Mandel-Factor
0.4 0.02
0
0.2
0 -0.02
-0.04
-0.2 0
(a)
10
time (ω 0t)
20
30
0
(b)
10
time (ω 0t)
20
30
5.0
TDO TIO
Mandel-Factor
4.0
3.0
2.0
1.0
0.0
-1.0
0
10
(c)
time (ω 0t)
20
30
Fig. 2. Mandel factor Q as a function of time, for = 0:1 (a), = 0:5 (b), and = 0:9 (c) with ! = 0:5, showing sub-Poissonian statistics and its transfer between the sub-systems.
Using Eqs. (13) and (30), we can obtain 12 ∗21 − 22 ∗11 n1 − 1 12 ∗22 − 22 ∗12 n2 0 0 0 cn1 n2 = c(n1 −1)n2 + cn1 (n2 −1) ∗ − ∗ ∗ − ∗ 11 22 n1 11 22 n1 12 21 12 21
(32)
or, equivalently, cn01 n2 = cn01 (n2 −1)
21 ∗12 − 11 ∗22 ∗ − ∗ 11 22 12 21
n2 − 1 21 ∗11 − 11 ∗21 0 + c(n ∗ − ∗ 1 −1)n2 n2 11 22 12 21
n1 : n2
(33)
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
195
Using this recursive relation we obtain, step by step, all the coeKcients {cn01 n2 } from the 0 ; this later being determined by the normalization constraint. The identity value of c00 ∗ 21 11 − 11 ∗21 = 12 ∗22 − 22 ∗12 assures the occurrence between the two expressions. In fact, this identity is preserved by the equations for the coeKcients {; }. Once we know the state |0 we can obtain the time-dependent base {|n1 (t)n2 (t)} (eigenstates of the operators n1 e n2 ), using the equation |n1 (t)n2 (t) = √
1 √ (a+ (t))n1 (a2+ (t))n1 |0 : n1 ! n2 ! 1
(34)
Finally, we obtain the time-dependent coeKcients cn1 n2 (t) appearing in Eq. (29), cn1 n2 (t) = n1 n2 | $(t) = n1 (t)n2 (t) | $(0) :
(35)
Now, the density operator (t) = |$(t)$(t)| describes the entire (coupled) system and allows us to obtain the reduced density operator describing the ith sub-system (with i; j = 1; 2; j = i), (i) (t) = Trj (|$(t)$(t)|)
(36)
yielding the statistical distribution for the ith oscillator: Pn(i) (t) = ni |(i) (t)|ni . Fig. 3 shows the plots of Pn(i) (t) as a function of n, for tree times: !0 t=0 and !0 t=2:37; !0 t= 5:85 for initial excitations: n1 (0) = 4; n2 (0) = 0, where n1 (0)(n2 (0)) stands for the initial excitation in the TDO (TIO). For all these plots = ! = 0:5. As a general behavior we note that the statistical distribution describing each oscillator changes as time evolves and we note the statistical distribution Pn describing the TDO at !0 t = 0 being transferred to the TIO at a certain time, e.g., at time about !0 t = 2:37 (Fig. 3b). However, when both oscillators have the same initial excitation, i.e.: n1 (0) = n2 (0) we no longer observe this transfer. In this case, the TIO returns to its initial statistical distribution Pn(2) (0), the same not occurring for the TDO (not shown in ?gures). 4.3. Mixing properties For the ith oscillator we calculate Tr[(i) (t)]2 , which measures its degree of mixing. Although the entire system remains in a pure state for all time, namely: [(t)]2 = (t), hence Tr[(t)]2 = Tr[(t)] = 1, each sub-system becomes mixed for t ¿ 0, when Tr[(i) (t)]2 ¡ 1; i = 1; 2, except for isolated times as we see in Fig. 4 showing the plot of Tr[(i) (t)]2 versus time for the same values of parameter used in Fig. 3. In the present case, we note that the state describing each oscillator is mixed for t ¿ 0, becoming near a pure state at isolated points, as for !0 t = 4:43; 27:83; etc. Comparing Figs. 3 and 4 we see that the transference of statistical distribution (see Fig. 3 for !0 t = 2:37) occurs when the sub-systems are near pure states, i.e., Tr[(t)]2 1 (we have Tr[(t)]2 0:98 for !0 t = 2:37, in Fig. 4). The same does not occur for other times where Tr[(t)]2 = 1. Actually, it is shown that the pure states assumed by an evolving sub-system have no obligation to be the same [9].
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198 1 0.8
ω 0t = 0
0.6 0.4 0.2 0 0.8
TDO
ω0t = 2.37
TIO
Pn
0.6 0.4 0.2 0 0.8
ω0t = 5.85
0.6 0.4 0.2 0 0
4
8
12
n Fig. 3. Time evolution of statistic distribution Pn , for = 0:5; ! = 0:5, showing the statistic transfer between sub-systems.
5. Concluding remarks We have studied quantum noise reduction, statistical properties and transfer of these e7ects from a TDO to a TIO, the ?rst (second) sub-system standing for a time-dependent (independent) LC-circuit. For this purpose, we have employed the Schrodinger picture— more appropriate to the study of statistical properties—and present the exact solution of the Schrodinger equation governing our system, constituted by these two interacting LC-circuits [cf. Eq. (1)]. The occurrence of quantum noise below the vacuum level (squeezing) in the TDO and its transfer to the TIO are shown in Figs. 1a–c. No such e7ect occurs in the TIO when isolated. When the two oscillators are coupled, the e7ect occurs similarly in both oscillators before the time variation of L(t) has initiated (t ¡ 0; not shown in ?gures). The e7ect and its transfer increase when grows (Figs. 1a–c). The enhanced squeezing, originally generated in the TDO and transferred √ to the TIO, remains there after the time variation of L1 (t) terminates at !0 " = 6 10), hence transfer occurs permanently, being more eKcient when the coupling-parameter grows.
H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
197
1
Tr (ρ2)
0.96
0.92
0.88
0.84 0
10
20
30
time (ω 0t) Fig. 4. Time evolution of mixing properties (Tr[(i) (t)]2 ) of sub-systems, showing that the sub-systems return to pure states for some isolated times.
Figs. 2– 4 concern with various statistical aspects characterizing our sub-systems. Figs. 2a–c stand for Mandel-Q factor showing the occurrence of sub-Poissonian effect in the TDO and its transfer to the TIO: in these plots we have used ! = 0:5, although this parameter does not a7ect the results signi?cantly. We note in these ?gures that the sub-Poissonian e7ect transferred to the TIO increases when increases (compare Figs. 2a and b), but diminishes when becomes very large ( ¿ 0:9), see Fig. 2c. Hence, contrary to the case of squeezing, there is an interval of -values where sub-Poissonian e7ect and its transfer are enhanced. Now, before the time variation is set on, the statistical aspects of both oscillators are similar, the statistics becoming super-Poissonian if is large ( & 0:9), but changes successively between sub- and super-Poissonian if is not large (not shown in ?gures): for about 0.5, during certain time intervals one oscillator may be sub-Poissonian while the other is super-Poissonian and vice versa (Fig. 2b). Fig. 3 stands for the time evolution of statistical distribution Pn(i) (t) describing the ith oscillator i = 1; 2. A striking result to be stressed is the possibility of transferring a desired statistical distribution from the TDO to the TIO: Note in this ?gure that the statistical distribution of the TDO at t = 0 is transferred to the TIO at !0 t 2:37. Also note in Fig. 4 that Tr[(i) (2:37)]2 1, when the reduced density operator (i) becomes near a pure state. Fig. 4 concerns with mixing properties. It exhibits Tr[(i) (t)]2 versus time showing both oscillators starting from pure states at time !t = 0 and becoming mixed states for !t ¿ 0. When !t ¿ 0 we obtain Tr[(i) (t)]2 ¡ 1, except for isolated times when each oscillator assumes a pure state [Tr[(i) (t)]2 = 1]. As mentioned before, in general, the pure state assumed by an evolving sub-system are not necessarily the same: it can be the initial pure state, or another.
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H. Rodrigues et al. / Physica A 311 (2002) 188 – 198
In resume, transfer of squeezing, sub-Poissonian e7ect and statistical distribution occur from the TDO to the TIO in the present scheme. These results are specially interesting when, e.g., the TDO is of easy direct manipulation while the TIO is of diKcult direct access. In this case, we call the TDO (TIO) as the active (passive) oscillator. One such example is found in Ref. [7] where a trapped ion (see, e.g., [8]), laser cooled onto its quantum ground state level can be used as a very low temperature detector for RF-signal (source oscillator) applied to the trap electrodes, the source oscillator being prepared in a squeezed state through the coupling with the ion. A point deserving future attention in this perspective concerns with the improvement of the present scheme (see Eq. (1)), making the transfer of squeezing and sub-Poissonian e7ects more eKcient, as it is eKcient for the transfer of statistical distribution. Finally, we remark that we have found the exact solution describing our system (cf. Section 4.2), which constitutes an available tool for potential application to other aspects of the system, another point to be explored in a future work. Acknowledgements The authors (SBD, BB) thank the CNPq, PRONEX (BB) and FAPERJ (HR, DP Jr.) for the partial support. References [1] R.J. Glauber, Phys. Rev. 130 (1963) 2529; H.M. Nussenzveig, Introduction to Quantum Optics, Gordon and Breach, New York, 1973, pp. 73. [2] W.H. Louisell, Quantum Statistical Properties of Radiation, Wesley, New York, 1973. [3] C.C. Gerry, P.L. Knight, Am. J. Phys. 65 (1997) 964 and references therein. [4] B. Baseia, A.L. de Brito, Physica A 197 (1993) 364. [5] D. Portes Jr., H. Rodrigues, S.B. Duarte, B. Baseia, Physica A 268 (1999) 121. [6] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994. [7] D.J. Heizein, D.J. Wineland, Phys. Rev. A 42 (1990) 2977. [8] F. Dietrich, J.C. Bergquist, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 62 (1989) 403; D.J. Heizein, et al., Phys. Rev. Lett. 66 (1991) 2080. [9] A.R. Gomes, Collapse and Revival of the wavefunction in coupled oscillators, M.Sc. Thesis, Federal University of Paraiba, Jo˜ao Pessoa, Brazil, 1996.