1550 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 G. Yang and K. Wang

Quantum-nondemolition measurementby a two-photon transition

in the presence of the Stark shift

Guojan Yang and Kaige Wang

Department of Physics, Beijing Normal University, Beijing 100875, China

Received August 16, 1996; revised manuscript received January 28, 1997

We evaluate the effectiveness of a quantum-nondemolition measurement scheme in an effective two-levelatomic configuration in the presence of the Stark shift. We compare the prediction of this model with that ofits full cascade three-level counterpart and analyze the relationship between the quantum-nondemolition gainand the dynamic instability of the system. © 1997 Optical Society of America [S0740-3224(97)02207-8]

PACS number(s): 42.50.Dv, 03.65.Bz

1. INTRODUCTIONThe scheme for the measurement of quantum nondemoli-tion (QND) in optics by coupling two electromagneticfields through a two-photon cascade transition in a three-level atom is a practical model extensively studied in re-cent years.1–5 Grangier et al.2 and Blockley and Walls3

considered an effective two-level model because the fieldswere highly detuned from the intermediate level. Theadvantage of this detuning lies the ability of the field toavoid absorption and spontaneous-emission noise.1

However, the dynamic Stark shift,6 an intensity-dependent shift in the atomic two-photon detuning thatresults from elimination of the intermediate level, hasbeen neglected. In this paper we take this effect into ac-count to make the model closer to practice. We follow themethod developed in Ref. 7, that is, we use the input–output formalism8 and the Fokker–Planckapproximation9 to determine the explicit parametric ex-pression in the frequency domain of three criteria forQND measurement.10 We verify the validity of this ef-fective two-level model as a simple description of the fullcascade three-level model and find that the QND gain atboth zero and nonzero frequencies can be strengthenedunder the influence of soft-mode instability.

2. PRINCIPLE OF THE CALCULATIONThe QND measurement scheme considered here is illus-trated by the following Hamiltonian3:

H 5 (j51

5

Hj ,

H1 5 \v0Sz 1 (j51

2

\v jaj1aj ,

H2 5 b1S2S1a1

1a1 1 b2S1S2a2

1a2

1 ig~S2a11a2

1 2 S1a1a2!,

0740-3224/97/0701550-06$10.00 ©

H3 5 i\(j51

2

@e jaj1 exp~2iV jt ! 2 e j*aj exp~iV jt !#,

H4 5 (m51

N

~GAsm 1 GA1sm

1!,

H5 5 (j51

2

~GFaj1 1 GF

1aj!, (1)

where the atomic operators are

S6 5 (m51

N

sm6 exp~6ikrm!, Sz 5 (

m51

N

smz; (2)

H1 is the free Hamiltonian for the atoms and the fields;v0 is the atomic transition frequency and v j is the jthmode frequency; H2 is the effective two-level interaction

6;g is the two-photon coupling coefficient, which is of the or-der of g } g1g2 /d, where g1 and g2 are the one-photoncouplings to the intermediate level and d is its detuning;b j 5 gj

2/d ( j 5 1, 2) is a parameter describing the Starkshift of the jth atomic level that is due to one of twoatomic transitions to the intermediate level; N is thenumber of atoms; H3 is the driving term, where e j andV j are, respectively, the amplitude and the frequency ofthe jth driving field; H4 and H5 refer to the dissipativeprocess; GF is a bath operator for the fields, and it resultsin cavity damping at a rate k j ; and GA is a bath operatorfor the atoms, and it leads to spontaneous emission char-acterized by the damping rate of the atomic polarizationg' and the decay rate of the population difference g11 .A Fokker–Planck equation (FPE) is an alternative way

to describe the system of Eqs. (1) approximately. Forthis we scale the operators of the cavity modes, the atoms,and the driving fields in the following way9:

1997 Optical Society of America

G. Yang and K. Wang Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1551

xj* 5 ^aj1&/Ns

1/2, xj 5 ^aj&/Ns1/2,

n6 5 ~21/2NAg11 /g'!21^S6&,

m 5 ~21/2N !21^Sz&, Yj 5 e j /ANs (3)

for j 5 1, 2, where Ns 5 Ag11g'/2g is the saturationphoton number. Following the method introduced in Ref.9, we transform the operator description of the system ofEqs. (1) into a FPE:

]

]tP~x1 , x1* , x2 , x2* , n, n* , m, t !

5 LP~x1 , x1* , x2 , x2* , n, n* , m, t !,

L 5 (j51

2

k jF ]

]xj(~1 1 iu j!xj 2 Yj 1 2Cj

3 $x32j* n 1 id21g12322jxj@1 2 ~21 ! jm#%) 1 c.c.G

1 g'H ]

]nF ~1 1 iD!n 2 mx1x2 1

id

2

3 ~2g12x1* x1 1 g1221x2* x2!nG 1 c.c.J 1 g11

]

]m

3 @m 2 1 1 1/2~n* x1x2 1 nx1* x2* !#

1 (j51

2 2k jnj

Ns

] 2

]xj]xj*

22k1C1

NsS ] 2

]x1]x2n 1 c.c.D 1

g'd

2Ns

3 S ] 2

]x1]nig12x1n 2

] 2

]x2]nig12

21x2n 1 c.c.D1

g'

k1

1

4C1NsH ] 2

]n]n*@~2g' 2 g11!~1 2 m !#

2g11

2 S ] 2

]n 2 nx1x2 1 c.c.D 1g11

2

2g'

] 2

] 2m

3 @1 2 m 1 1/2~n* x1x2 1 nx1* x2* !#J , (4)

where P is the quasi-probability-distribution functionthat generalizes the Glauber P function for the fieldmodes, D 5 (v0 2 V1 2 V2)/g' is the scaled atomic two-photon detuning, u j 5 (v j 2 V j)/k j is the scaled cavitydetuning, Cj 5 Ngd/4k j is the two-photon cooperation co-efficient for the jth mode, n̄ j is the average thermal pho-ton number in the jth mode, and d and g12 are defined,respectively, as d 5 Ag11 /g' and g12 5 g1 /g2 .The semiclassical steady-state solutions of the dynamic

system that can easily be derived from drift matrix of theFPE (4) read as

m 51 1 D82

1 1 D82 1 X1X2, n 5

~1 2 iD8!x1x2

1 1 D82 1 X1X2,

n* 5~1 1 iD8!x1* x2*

1 1 D82 1 X1X2,

Yj 5 xjF S 1 12CjX32j

1 1 D82 1 X1X2D

1 iS u j8 22D8CjX32j

1 1 D82 1 X1X2D G , (5)

where Xj 5 xjxj* is the scaled internal intensity of jthmode and

D8 5 D 2d2

~g12X1 2 g1221X2!,

u j8 5 u j 1 2Cjd21g12

322j@1 2 ~21 ! jm#,(6)

where D8 is the intensity-dependent two-photon detuningcorrected by the Stark shift of the two atomic levels andu j8 is the atomic-dependent cavity detuning. In Eqs. (6),even if the intensity-dependent shift in the two-photondetuning disappears as g12X1 5 g12

21X2 , the atomic-dependent shift in the cavity detuning always exists.Because the fluctuations of the system are much

smaller than the stationary intensities, we linearize FPE(4) around the stationary state [Eqs. (5)] and obtain

]

]tP~a, t ! 5 (

i, j51

7 FAij

]

]a ia j 1

1

2Dij

]2

]a i]a jGP~a, t !,

(7)

where a j ( j 5 1, ..., 7) denote the deviations of the vari-ables x1 , x1* , x2 , x2* , n, n* , and m from their stationaryvalues, A is the drift matrix, and D is the diffusion ma-trix.The effectiveness of the QND measurement scheme is

evaluated by the three criteria developed by Hollandet al.,10 which, in fact, are the normalized quantum corre-lations between two particular quadratures of the differ-ent input and output fields. If p(t) and q(t) are the twoHermitian operators in which we are interested, the cri-teria in the frequency domain can be described by4

Cl2~v!

51

4

U E2`

`

exp~2ivt!^p~t!q 1 qp~t!&dtU2

E2`

`

exp~2ivtp!^p~tp!p&dtpE2`

`

exp~2ivtq!^q~tq!q&dtq

,

(8)

for l 5 1, 2, 3, whose simplified notation is Cl2

5 u^pq&u2/^p2&^q2&. Denoting Qj as one of the system’sinternal operators corresponding to xj , xj* , n, n* , andm, we can derive the correlation coefficients of the opera-tors Qj from the quantum-regression theorem11,12:

1552 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 G. Yang and K. Wang

E0

`

^@Qj~t !, Ql#&exp~2ivt !dt

5 (k

~A 1 ivI !jk21^~Qk , Ql!&,

E0

`

^@Qj~t !, Ql#&exp~ ivt !dt

5 (k

~A 2 ivI !jk21^~Qk , Ql!&, (9)

whereas the correlation coefficients of the cavity opera-tors in the normal order *`

`^:Qj(t)Ql :&exp(2ivt)dt are ob-tained from the covariance matrices13

S 5 ~A 1 ivI !21D~AT 2 ivI !21. (10)

Assuming that a1 refers to the probe field and a2 refersto the signal field, we have the fluctuation operators forthe quadrature components of the input and the outputfields:

Ejin~a, t ! 5 daj

in~t !exp~2ia jin! 1 daj

in1~t !exp~ia j

in!,

Ejout~a, t ! 5 daj

out~t !exp~2ia jout!

1 dajout1~t !exp~ia j

out!, j 5 1, 2.

(11)

Using the boundary condition for the input–output for-malism in a single-ended cavity configuration,8

ajout~t ! 2 aj

in~t ! 5 A2k jaj~t !, (12)

we finally derive from Eqs. (5)–(12) the criteria expressedas

C12~v! 5

u^E2inE1

out&u2

^E2in2&^E1

out2&, C2

2~v! 5u^E2

inE2out&u2

^E2in2&^E2

out2&,

C32~v! 5

u^E2outE1

out&u2

^E2out2&^E1

out2&, (13)

where

^Ejin2& 5 1/4,

^Ejout2& 5

14

1k j

2@S2j21,2j21 exp~22ia j

out!

1 S2j,2j21 exp~2ia jout! 1 S2j21,2j

1 S2j,2j21#, j 5 1, 2 (14)

^E2inE1

out& 5 2Ak1k2

2 $~A 1 iv!1,421

3 exp@2i~a2in 1 a1

out!# 1 ~A 1 iv!2,321

3 exp@i~a2in 1 a1

out!# 1 ~A 1 iv!1,321

3 exp@i~a2in 2 a1

out!# 1 ~A 1 iv!2,421

3 exp@2i~a2in 2 ia1

out!#%,

^E2inE2

out& 5 1/2 cos~a2out 2 a2

in! 2k2

2

3 $~A 1 iv!3,421 exp@2i~a2

in 1 a2out!#

1 ~A 1 iv!4,321 exp@i~a2

in 1 a2out!#

1 ~A 1 iv!3,321 exp@i~a2

in 2 a2out!#

1 ~A 1 iv!4,421 exp@2i~a2

in 2 a2out!#%,

^E2outE1

out& 5 A2k1k2$S1,3 exp@2i~a2out 1 a1

out!#

1 S2,4 exp@i~a2out 1 a1

out!#

1 S1,4 exp@i~a2out 2 a1

out!#

1 S2,3 exp@2i~a2out 2 a1

out!#%. (15)

For the linearized system [Eq. (7)], the third criterion canbe described by the conditional variance

Vc~E2out/E1

out! 5 ^E2out2&~1 2 C3

2!. (16)

The effectiveness of the QND measurement scheme forthe given parameters is determined by the values ofC1

2, C22, and Vc , which are, respectively, the efficiency

of the measurement, the nondemolition property, and ef-ficiency for quantum-state preparation. For the idealscheme we have C1

2 5 C22 5 1 and Vc 5 0. In addi-

tion, it should be emphasized here that when we choosethe output fluctuations as the amplitude or the phasequadrature, the phase shift between the input (output)field and the internal field must be taken into account.For the single-ended cavity, the phase shift between theinput field and the output field is given by4,14

^ajin&

u^ajin&u5

^ajout&

u^ajout&uexp~22iF j!, j 5 1, 2. (17)

In the present analysis, F j is

F j 5 arctanF2u j8~1 1 D82! 2 2CjD8X32jm

1 1 D82 1 2CjX32jmG , (18)

which is derived from Eqs. (5) and (6). For example, ifone measures the QND by reading the phase quadratureof the probe output to pick up the information on the am-plitude quadrature of the signal input, Eqs. (11)–(16)should include the following values: a2

in 5 0, a2out

5 2F2 , and a1out 5 p/2 1 2F1 . In the remainder of

this paper we shall consider that these values have beenused.

3. RESULTSWe exploited the effectiveness of our model numericallyand found some ranges of parameters in which the QNDperformance is good. For comparison we first presenthere (Fig. 1) the results obtained with the parametersused by Poizat et al.4 in analyzing a full cascade three-level model with the assumption that the one-photon de-tuning d is ten times as large as the two-photon detuningD. In addition, we set n̄1 5 n̄2 5 0 because the thermalfluctuation tends to destroy the quantum effects. We de-termine u1 and u2 by taking F j 5 0 ( j 5 1, 2), so twofields are on resonance with the nonlinear cavity. Figure

G. Yang and K. Wang Vol. 14, No. 7 /July 1997 /J. Opt. Soc. Am. B 1553

1 shows that at zero frequency the QND criterion C12

1 C22 is greater than 1.87, so the measurement is actu-

ally performed and the signal is not excessively degraded.The conditional variance Vc falls to 0.017, which indicatesthat the state-preparation capability is quite good. Thequalitative agreement between the effective two-levelmodel and the full three-level model can be determined bya comparison of Figs. 2 and 4 of Ref. 4, which shows thatwhen d is large enough the three-level model can be effec-tively simplified by the two-level model. On the otherhand, there is a small discrepancy between these twomodels. The performance of the two-level model isslightly better than that predicted by the three-levelmodel. (In the latter case, C1

2 1 C22 ' 1.80 and Vc

' 0.12.) The physical explanation of this difference isthat the noise arising from spontaneous decay from theintermediate level of the three-level atom has a negativeinfluence on the quantum behavior of the QND measure-ment, whereas in the effective two-level model it has notbeen taken into account.By scanning numerically in parameter space we find

that the model’s QND nature is closely linked to its dy-namic instability. Figure 2 plots the steady-state inten-sity of mode 1 against the mode’s detuning u1 by adiabati-

Fig. 1. QND criteria versus noise frequency v/k1 for F j 5 0,D 5 2120, C1 5 9.409, C2 5 37.635, g11 /k1 5 2g' /k15 0.821, k2 /k1 5 0.25, Y1 5 1.670, Y2 5 2.092, and g12 5 1.

Fig. 2. Steady-state intensity of mode 1 versus its detuningu1 . The parameters are D 5 60, C1 5 C2 5 200, g11 /k15 2g' /k1 5 1, k2 /k1 5 1, Y1 5 80, Y2 5 5, and g12 5 0.05.u2 is kept equal to u1 .

cally sweeping u1 forward and backward in integration ofthe dynamic equations of the system, keeping u2 5 u1 .The bistability loop induced by the soft-mode instabilitycan be clearly distinguished in the interval 7.93 , u1, 15.10. The undamping oscillation of the system oc-curs between 20.90 and 2.50, and the stationary solu-tions of the rest of the curve are stable. Figure 3 pre-sents the QND criteria spectra as u1 is fixed at 7.93adjacent to the tuning point (u1

th 5 7.92) on the upperbranch of the bistable curve. Two sets of QND gainpeaks can be seen at two frequencies that are related tothe intrinsic frequencies (v int /k1) of the dynamic system.One set of peaks is located at nonzero frequency v int /k15 13.54 with maximum correlation C1

2 1 C22 5 1.51

and conditional variance Vc 5 0.10, and another set is lo-cated at zero frequency, v int /k1 5 0, with C1

2 1 C22

5 1.51 and Vc 5 0.11. Obviously, v int /k1 5 0 is relatedto the soft-mode instability. Figure 4 shows the improve-ment in the QND nature of the measurement as u1 ap-proaches u1

th on the upper branch in the interval 7.93< u1 < 15.00. In Fig. 4 two sets of QND gain peaks ap-pearing at two different frequencies are selected for eachgiven u1 . Figures 4(a) and 4(b) refer to the cases ofv int /k1 approaching 0 and v int /k1 approaching 13.54, re-spectively. [Note from Fig. 4(a) that before u1 reachesu1

th, C22 has gone down slightly while C1

2 is still increas-ing, so the two curves of C1

2 and C22 overlap at u1

th (C1

2 5 0.78 and C22 5 0.72, which are derived from Fig.

3).] The relationship between v int /k1 and u1 in Figs. 4(a)and 4(b) is shown in Fig. 4(c). If the system jumps to thelower branch, three criteria take the approximate valuesC1

2 ' 0, C22 ' 1, and Vc ' 1, and the QND measure-

ment fails. In addition, there was no observable QNDgain either in the neighborhood of the hard-mode insta-bility (u1 ' 20.90 and u1 ' 2.50) or in the other parts ofthe stationary solution curve. These figures show thegreat improvement in the effectiveness of our model notonly at zero frequency but also at nonzero frequency asthe system is driven to the bistability transition point,and the soft-mode instability is indeed helpful for realiz-ing a better QND measurement. As a matter of fact, weobserved similar behavior in analyzing the two-level

Fig. 3. QND criteria versus noise frequency for u1 5 7.93 on theupper branch of the bistable curve. Lines 1, 2, and 3 correspondto C1

2, C22, and Vc , respectively. All other parameters are the

same as for Fig. 2.

1554 J. Opt. Soc. Am. B/Vol. 14, No. 7 /July 1997 G. Yang and K. Wang

Fig. 4. Optimal QND nature at v int /k1 versus u1 on the upperbranch as v int /k1 approaches (a) 0 and (b) 13.54. Curves 1–3 andall other parameters are the same as for Fig. 3. (c) Relationshipbetween v int /k1 and u1 illustrated in (a) and (b). Curves 1 and 2correspond to (a) and (b), respectively.

model in the absence of the Stark shift.7 However, theparameter regions for this behavior in the two models arewidely separated. The reason for the difference is thatthe Stark shift of the atomic levels induces an additionalphase shift between the input field and the output field,which influences the quantum effect of the light fields.

4. CONCLUSIONWe have studied the effectiveness of the QND measure-ment in an effective two-level atomic configuration in thepresence of the Stark shift. Our main results are sum-marized as follows:

(1) We compared the prediction of the full cascadethree-level atomic model with that of the effective two-level atomic model in the resonant case and found themodels in qualitative agreement.(2) We pointed out that, as the system is driven to the

bistability transition point, the QND performance im-proves. This observation is similar to that made for thesqueezing effect, another pure quantum effect, of lightfields15–18 and reveals again that the singularity of thedynamic behavior of the system plays an important rolein governing the system’s quantum behavior.(3) We observed that, at the bistability transition

point, QND gain becomes large at both zero and nonzero

frequencies. The advantage of a nonzero frequency mea-surement is that it may eliminate experimental noisenear zero frequency.4

ACKNOWLEDGMENTSWe thank L. A. Lugiato and Hu Gang for helpful discus-sions. This project was supported by National NaturalScience Foundation of China. K. Wang is grateful for thesupport of the International Center for Theoretical Phys-ics, Training and Research Programme, Trieste, Italy.

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