PHYSICAL REVIEW A 72, 055803 共2005兲
Controlling subluminal to superluminal behavior of group velocity with squeezed reservoir 1
Amitabh Joshi,1,* Shoukry S. Hassan,2 and Min Xiao1
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA Department of Mathematics, College of Science, University of Bahrain, P. O. Box 32038, Kingdom of Bahrain 共Received 1 June 2005; published 30 November 2005兲
2
We consider a three-level ⌳-type atomic medium in an electromagnetically induced transparency configuration interacting with two independent broadband squeezed baths. We show that absorptive and dispersive properties of the medium can be controlled using squeezed bath parameters and coupling field strength such that the medium can have subluminal to superluminal group velocity for the probe pulse. DOI: 10.1103/PhysRevA.72.055803
PACS number共s兲: 42.50.Gy, 42.50.Lc
Over the past 15 years, a great deal of attention has been devoted to studying the phenomena of electromagnetically induced transparency 共EIT兲 and related topics 关1–4兴. This is because of the potential applications of EIT in lasing without inversion 关5兴, high-efficiency nonlinear optical processes 关6兴, efficient propagation of a laser beam through an optically thick medium 关7兴, high-efficiency population transfer via a coherent adiabatic process 关8兴, controlled optical bistability and multistability 关9兴, slow light 关10兴, and storage of optical pulses 关11兴. The phenomenon of EIT can be considered as an interrelated group of processes 共e.g., the coherent population trapping, coherent adiabatic population transfer兲 that results from quantum mechanical coherence and interference in a multilevel system. EIT has been studied extensively and many significant experimental and other results were obtained for a system with three levels in a ⌳ configuration 关1–4兴. In all such studies of EIT, normal heat baths interacting with two transitions of ⌳ configuration are considered. Here, we investigate absorptive 共dispersive兲 properties of the EIT medium as well as related effects such as subluminal and superluminal propagation of light, in an ensemble of three-level atoms in the ⌳ configuration interacting with two laser beams and two independent broadband squeezed baths 关12兴. The central frequency of one of the squeezed baths is near to one of the atomic transitions; whereas, that of the other bath is close to the other atomic transition. There are pairwise correlations between modes symmetrically placed around the central frequency of each of the two baths 关12兴. However, we assume that there is no correlation between the modes belonging to different baths. A recent work shows that a two-level atomic system, damped by a broadband squeezed vacuum and driven by a weak laser, exhibits a finite refractive index accompanied by zero absorption and can lead to the situation of superluminal group velocity 关13兴. Some other interesting works were also reported recently on the superluminal light propagation of probe pulses 关14兴. We consider a three-level atomic system in the ⌳ configuration 共Fig. 1兲 with levels 兩l典 共l = 1 , 2 , 3兲 having energies E1 ⬎ E2 ⬎ E3. The amplitudes of the external driving fields are E P 共the probe field with frequency P兲 and EC 共the coupling field with frequency C兲 interacting with 兩1典 to 兩3典
and 兩1典 to 兩2典 transitions, respectively. The master equation for the density operator of this system under dipole and rotating-wave approximations is given by
˙ = − i关Hatom + Hint, 兴 − i关B1共t兲A13 + A31B†1共t兲, 兴 − i关B2共t兲A12 + A21B†2共t兲, 兴,
共1兲
where 3
Hatom = 兺 EiAii,
Hint = d1E P共A31 + A13兲 + d2EC共A21 + A12兲,
i=1
共2兲 Amn = 兩m典具n兩, 共m , n = 1 , 2 , 3兲 are the atomic ladder operators and Bi共i = 1 , 2兲 are the bath operators; whereas, d1共d2兲 is the dipole operator corresponding to the transition 兩1典 to 兩3典 共兩1典 to 兩2典兲. The two squeezed baths are assumed to be independent and their modes are ␦ correlated, such that † 具Bl共t兲Bm 共t⬘兲典 = 2␥l共nl + 1兲␦共t − t⬘兲␦lm ,
具B†l 共t兲Bm共t⬘兲典 = 2␥lnl␦共t − t⬘兲␦lm , 具Bl共t兲Bm共t⬘兲典 = 2␥lml␦共t − t⬘兲␦lme−2i⍀lt , † 共t⬘兲典 = 2␥lm*l ␦共t − t⬘兲␦lme2i⍀lt, 具B†l 共t兲Bm
共l,m = 1,2兲,
共3兲
in which 2⍀l is the frequency of the pump driving the lth bath, 2␥l, 共l = 1 , 2兲 are the Einstein A coefficients for the transitions 兩1典 to 兩3典 and 兩1典 to 兩2典, respectively. The ml’s are the squeezing parameters 共ml = 兩ml兩exp共i⌽l兲, ⌽l is the phase of lth squeezed bath兲, such that 兩ml兩2 艋 nl共nl + 1兲, the equality sign holds in case of an ideal bath that yields a maximum degree of squeezing. In this work, we keep the equality sign for simplicity. Because of the ␦ correlation, the bath variables can be eliminated from Eq. 共1兲 关12兴 resulting in the master equation
/t = − i关H0, 兴 + Ltherm − Lsqz ,
共4兲
where *Corresponding author. Email address:
[email protected] 1050-2947/2005/72共5兲/055803共4兲/$23.00
H0 = ␣ P共A13 + A31兲 + ␣C共A12 + A21兲 + ⌬ P共A11 + A22兲 − ⌬CA22 , 055803-1
©2005 The American Physical Society
PHYSICAL REVIEW A 72, 055803 共2005兲
BRIEF REPORTS
a perturbation over ␣ P / ␣C and we can solve Eq. 共6兲 perturbatively. We thus obtain the solution of the atomic operator in the first order for the probe transition as 具A13典共1兲 = i关共2␥1m*1 + F*兲/共兩F兩2 − 4␥21兩m1兩2兲兴␣ P具A33典共0兲 , 共7兲 where, F = − i⌬ P + ␥1共2n1 + 1兲 + ␥2共n2 + 1兲
FIG. 1. 共a兲 Schematics of the three-level system in ⌳-type configuration interacting with two broadband squeezed bath reservoirs.
Ltherm = ␥1共n1 + 1兲共2A31A13 − A11 − A11兲 + ␥1n1共2A13A31 − A33 − A33兲 + ␥2共n2 + 1兲共2A21A12 − A11 − A11兲 + ␥2n2共2A12A21 − A22 − A22兲, 2 2 − A13 兲 Lsqz = ␥1兩m1兩e−i1共2A13A13 − A13
+ ␥2兩m2兩e + ␥2兩m2兩e
i2
共2A12A12 −
共2A21A21 −
2 A12
2 A21
−
共1兲 = i共2␥1m*1 + F*兲/共兩F兩2 − 4␥21兩m1兩2兲.
共9兲
In the following, we assume ␥1 = ␥2 = ␥ and n1 = n2 = n 共which also implies 兩m1兩 = 兩m2兩 = 兩m兩兲 for the sake of simplicity. After some simplifications, it is easy to show that
␥共3n + 2兲 + 2␥n =
共5兲
with Rabi frequencies corresponding to the transitions 兩1典 to 兩3典 and 兩1典 to 兩2典 to be 2␣ P = 2d1E P / ប and 2␣C = 2d2EC / ប, respectively. The corresponding laser detunings are ⌬ P = 共E3 − E1兲 / ប − P and ⌬C = 共E2 − E1兲 / ប − C. Here 1 = 2⌽ P − ⌽1, 2 = 2⌽C − ⌽2 are thus relative phases of squeezed baths, where ⌽ P 共⌽C兲 is the phase of probe 共coupling兲 field. The equations of motion for the atomic averages are given by 具A˙11典 = − i␣ P共具A13典 − 具A31典兲 − i␣C共具A12典 − 具A21典兲 − 2关␥1共n1 + 1兲 + ␥2共n2 + 1兲兴具A11典 + 2␥1n1具A33典 + 2␥2n2具A22典, 具A˙22典 = − i␣C共具A21典 − 具A12典兲 + 2␥2共n2 + 1兲具A11典 − 2␥2n2具A22典, 具A˙12典 = i␣ P具A32典 − i␣C共具A11典 − 具A22典兲 + i⌬C具A12典 − 关␥1共n1 + 1兲 + ␥2共2n2 + 1兲兴具A12典 − 2␥2m*2具A21典, 具A˙23典 = − i␣ P具A21典 + i␣C具A13典 + i共⌬ P − ⌬C兲具A23典 − 关␥1n1 + ␥2n2兴具A23典, 具A˙13典 = − i␣ P共具A11典 − 具A33典兲 + i␣C具A23典 − 关␥1共2n1 + 1兲 + ␥2共n2 + 1兲兴具A13典 − 2␥1m*1具A31典 + i⌬ P具A13典.
Clearly, the linear susceptibility 共1兲 is given by
2 A12 兲
2 − A21 兲,
共8兲
Im关共1兲兴
2 2 − A31 兲 + ␥1兩m1兩ei1共2A31A31 − A31 −i2
+ ␣C2 /关共␥1n1 + ␥2n2兲 − i共⌬ P − ⌬C兲兴.
共6兲
For deriving the equation for the linear susceptibility, we need to solve Eq. 共6兲 in the steady state. For this purpose, we assume the standard EIT conditions 关3兴, e.g., the coupling field is much stronger than the probe field and all the atoms are initially in the ground state 兩3典, i.e., 具A11典共0兲 ⬵ 0, 具A22典共0兲 ⬵ 0, 具A33典共0兲 ⬵ 1, etc. So, the probe field can be considered as
␣C2 + 2␥兩m兩cos共1兲 4␥2n2 + 共⌬ P − ⌬C兲2 , 兩F兩2 − 4␥21兩m兩2 共10兲
⌬P − Re关共1兲兴 =
共⌬ P − ⌬C兲␣C2 − 2␥兩m兩sin共1兲 4␥2n2 + 共⌬ P − ⌬C兲2 兩F兩2 − 4␥21兩m兩2
.
共11兲
To understand how the squeezed bath 共which is characterized by the phase sensitive noise through the parameter 1兲 modifies the EIT characteristics of the three-level ⌳-type system, we plot Im关共1兲兴 and Re关共1兲兴 in Figs. 2共a兲 and 2共b兲 under different settings of phase parameter 1. In Figs. 2共a兲 and 2共b兲, curves A, B, C, and D are for 1 = 0, , / 2, and − / 2, respectively. Note that Im关共1兲兴 is same under the conditions of 1 = ± / 2 关curves C, D, Fig. 2共a兲兴, but Re关共1兲兴 is same under the conditions of 1 = 0, 关curves A, B, Fig. 2共b兲兴. This is also evident from the expressions of these two quantities given above in Eqs. 共10兲 and 共11兲, respectively. From Fig. 2共a兲, it is clear that the EIT dip gets reduced for 1 = 共curve B, background top to dip ratio 1.5兲 when compared to 1 = 0 共curve A, background top to dip ratio 2.2兲. The EIT feature for 1 = / 2, − / 2 共curves C, D background top to dip ratio 1.9兲 is intermediate to 1 = 0 and 1 = cases. On the contrary, from Fig. 2共b兲, one can see that the Re关共1兲兴 for 1 = / 2 共curve C兲 lies above the curves for 1 = 0, 共i.e., curves A, B兲, while the curve for 1 = − / 2 共curve D兲 lies below the curves for 1 = 0, 共curves A, B兲. Most of curve C 关Fig. 2共b兲兴 shows a positive value while curve D 关Fig. 2共b兲兴 shows a negative value. The position of the zero value of Re关共1兲兴 is at ⌬ P / ␥ = ⫿ 1.6 for 1 = ± / 2 关curves C, D of Fig. 2共b兲兴 conditions, which are different from 1 = 0, 关curves A, B of Fig. 2共b兲兴 conditions where this occurs at ⌬ P / ␥ = 0 , ± 0.9. This implies that squeezed bath 共through its phase sensitive noise兲 can control both absorptive and dispersive properties of an EIT medium in an effective manner. One can modify the response of the system in terms of ab-
055803-2
PHYSICAL REVIEW A 72, 055803 共2005兲
BRIEF REPORTS
FIG. 2. Im关共1兲兴 共a兲 and Re关共1兲兴 共b兲 as a function of probe field detuning ⌬ P / ␥ for the parameters ␥1 = ␥2 = ␥ = 1.0, n1 = n2 = n = 0.2, ⌬C / ␥ = 0, and ␣C / ␥ = 1. In 共a兲 and 共b兲, curves A, B, C, and D are for 1 = 0, , / 2, and − / 2, respectively.
sorptive 共dispersive兲 profiles by simply changing the phase of the anisotropic noise distribution of the squeezed bath. We come across an interesting situation by examining Eq. 共11兲 for the Re关共1兲兴. For specific choices of the phase parameter 共1兲 of squeezed bath, i.e., 1 = 0 or 共meaning sin 1 = 0兲 with ⌬C = 0, we get from Eq. 共11兲 Re关共1兲兴 =
⌬ P关1 − ␣C2 /共4␥2n2 + ⌬2P兲兴 兩F兩2 − 4␥21兩m兩2
.
共12兲
If we keep changing ⌬ P and simultaneously adjust the coupling field ␣C by the relation ␣C2 = 4␥2n2 + ⌬2P, for fixed ␥ and n, when m is to be maximally squeezed, then we find that Re关共1兲兴 = 0 always with the variation of ⌬ P. This means that by simultaneously adjusting the coupling field strength while varying probe detuning, it is possible to create a medium in which the probe field has no dispersion or it has a flat refractive index profile for all the frequencies. This is depicted in Fig. 3共b兲. The absorption is given by Im关共1兲兴 is also flat or constant 共⬃1 / 关共5n + 2兲 − 2冑n共n + 1兲兴兲 at all the frequencies under the same conditions. We call this condition of ␣C2 = 4␥2n2 + ⌬2P a “criticality condition” for the coupling field strength. Correspondingly, the subcritical and supercritical conditions can be defined according to ␣C2 ⬍ 4␥2n2 + ⌬2P and ␣C2 ⬎ 4␥2n2 + ⌬2P, respectively. The subcritical condition produces normal absorption 关curve B, Fig. 3共a兲兴 and certain dis
FIG. 3. Im关共1兲兴 共a兲 and Re关共1兲兴 共b兲 as a function of probe field detuning ⌬ P / ␥ for the parameters ␥1 = ␥2 = ␥ = 1.0, n1 = n2 = n = 0.2, ⌬C / ␥ = 0, and 1 = 0. Curves A, B, C, and D are for ␣2C = 4␥2n2 + ⌬2P, ␣2C = 0.01␥2n2 + ⌬2P, ␣2C = 10␥2n2 + ⌬2P, and ␣2C = 20␥2n2 + ⌬2P, respectively.
persion 关curve B, Fig. 3共b兲兴 features for the medium. There is an absorption maximum at ⌬ P = 0 due to the subcriticality condition. Correspondingly, the dispersion is 0 at ⌬ P = 0. On the contrary, if we resort to supercriticality condition of the coupling field, then there is an inverted or reduced absorption 关curves C, D of Fig. 3共a兲兴 in the medium accompanied by opposite dispersion 关curves C, D of Fig. 3共b兲兴. It is now well established that such dispersions can give rise from superluminal to subluminal light propagation 关15兴 and the atomic dispersion from normal to anomalous causes pulse delay to change from positive 共slow light兲 to negative 共superluminal or fast light兲 关16兴. Our main interest is to have light propagation with reduced absorption such that probe pulse does not get attenuated considerably. One can optimize the system parameters to get further reduced absorption and narrower profile beyond curve D of Fig. 3共a兲. So, by changing the coupling field alone around the critical condition, one can control the response of the medium from an inverted absorptive one to an absorptive one 关16兴. This implies that in such a medium, the probe pulse can travel from subluminal to superluminal velocities 共thus producing positive to negative group delays兲 depending upon the value of the coupling field strength 共an experimentally controlled parameter兲 with respect to the criticality condition. Another interesting feature of 共1兲 is revealed by introducing a finite coupling field detuning ⌬C, which is plotted in
055803-3
PHYSICAL REVIEW A 72, 055803 共2005兲
BRIEF REPORTS
FIG. 4. Im关共1兲兴 共a兲 and Re关共1兲兴 共b兲 as a function of probe field detuning ⌬ P / ␥ for the parameters ␥1 = ␥2 = ␥ = 1.0, n1 = n2 = n = 0.2, and 1 = 0. Curves A, B, and C are for ␣2C = 4␥2n2 + ⌬2P共⌬C / ␥ = 0兲, ␣2C = 0.01␥2n2 + ⌬2P共⌬C / ␥ = 1兲, and ␣2C = 10␥2n2 + ⌬2P共⌬C / ␥ = 1兲, respectively; while curve D is for ␣2C = 10␥2n2 + ⌬2P共⌬C / ␥ = −1兲.
note that Im关共1兲兴 now looks like a dispersive profile 关curves B, C, D of Fig. 4共a兲兴 while Re关共1兲兴 looks like an absorptive profile 关curves B, C, D of Fig. 4共b兲兴. This observation is “opposite” to what we usually observe for these quantities 关compare Figs. 4共a兲 and 4共b兲 with Figs. 3共a兲 and 3共b兲兴. The dispersive profile 共now in the “opposite terminology”兲 of Im关共1兲兴 is similar to anomalous dispersion when ⌬C is positive 关B, C in Fig. 4共a兲兴 and it becomes a normal dispersion curve when ⌬C is negative 关D in Fig. 4共a兲兴. The absorptive profile 共in the opposite terminology兲 of Re关共1兲兴 becomes negative 共dip-like兲 when ⌬C is positive 关B, C in Fig. 4共b兲兴 and it becomes a usual positive profile 关D in Fig. 4共b兲兴 under negative ⌬C values. So, we get two distinct features due to finite ⌬C and its sign. Thus, ⌬C alone can synthesize the medium response by its actual value and sign. In conclusion, we have studied the three-level ⌳-type system in EIT preferred conditions while the system is interacting with two independent broadband squeezed baths. The squeezed bath phase alone can control the EIT characteristics as well as the dispersive properties of the medium. Physically, this is because of the anisotropy of the noise distribution seen by the atoms. Also, one can uniquely design the system with custom control absorptive 共dispersive兲 properties just by adjusting the coupling field strength alone 关only for particular choices of squeezed bath phases as mentioned before Eq. 共12兲兴 in accordance with a criticality condition involving probe field detuning and bath parameters. One can thus prepare a dispersionless medium with fixed absorption response or a medium which gives rise to subluminal to superluminal group velocities for the probe laser field.
Fig. 4共a兲 共Im关共1兲兴兲 and Fig. 4共b兲 共Re关共1兲兴兲, respectively. Curve A 共as a reference兲 is for ⌬C / ␥ = 0, while curves B, C are for ⌬C / ␥ = 1, and curve D is for ⌬C / ␥ = −1, with other conditions as mentioned in the caption. It is interesting to
We are thankful to K. Osman for carefully reading the manuscript. S.S.H. thanks P. Toschek for fruitful discussions and drawing attention to the earlier work in Ref. 关1兴. We acknowledge the funding support from the National Science Foundation.
关1兴 For a review, see E. Arimondo, in Progress in Optics, edited by E. Wolf 共North-Holland, Amsterdam, 1996兲, Vol. 35; S. E. Harris, Phys. Today 50共7兲, 36 共1997兲; also see earlier work by T. Hansch et al., Z. Phys. 226, 293 共1969兲; 236, 213 共1970兲. 关2兴 K. J. Boller et al., Phys. Rev. Lett. 66, 2593 共1991兲; J. E. Field et al., ibid. 67, 3062 共1991兲. 关3兴 Y. Li and M. Xiao, Phys. Rev. A 51, R2703 共1995兲; 51, 4959 共1995兲; J. Gea-Banacloche et al., Phys. Rev. A 51, 576 共1995兲; M. Xiao et al., Phys. Rev. Lett. 74, 666 共1995兲. 关4兴 J. P. Marangos, J. Mod. Opt. 45, 471 共1998兲. 关5兴 J. Mompart and R. Corbalan, J. Opt. B: Quantum Semiclassical Opt. 2, R7 共2000兲. 关6兴 S. E. Harris et al., Phys. Rev. Lett. 64, 1107 共1990兲; P. R. Hemmer et al., Opt. Lett. 20, 982 共1995兲; Y. Li and M. Xiao, Opt. Lett. 21, 1064 共1996兲; A. J. Merriam et al., Phys. Rev. Lett. 84, 5308 共2000兲; M. Jain et al., ibid. 77, 4326 共1996兲; H. Wang et al., Phys. Rev. Lett. 87, 073601 共2001兲. 关7兴 A. Kasapi et al., Phys. Rev. Lett. 74, 2447 共1995兲. 关8兴 K. Bergmann et al., Rev. Mod. Phys. 70, 1003 共1998兲. 关9兴 A. Joshi et al., Phys. Rev. A 67, 041801共R兲 共2003兲; A. Joshi and M. Xiao, Phys. Rev. Lett. 91, 143904 共2003兲; A. Joshi et
al., Phys. Rev. A 70, 041802共R兲 共2004兲; A. Joshi et al., Opt. Lett. 30, 905 共2005兲. L. V. Hau et al., Nature 共London兲 397, 594 共1999兲; R. W. Boyd and D. J. Gauthier, in Progress in Optics, edited by E. Wolf 共North-Holland, Amsterdam, 2002兲, Vol. 43, Chap. 6, p. 497. C. Liu et al., Nature 共London兲 409, 490 共2001兲; D. F. Phillips et al., Phys. Rev. Lett. 86, 783 共2001兲; M. Bajcsy et al., Nature 共London兲 426, 638 共2003兲. C. W. Gardiner, Phys. Rev. Lett. 56, 1917 共1986兲; H. J. Carmichael et al., ibid. 58, 2539 共1987兲; B. J. Dalton et al., J. Mod. Opt. 46, 379 共1999兲; and references therein. S. S. Hassan et al., J. Phys. B 30, L777 共1997兲; S. S. Hassan et al., Eur. Phys. J. D 26, 301 共2003兲. G. S. Agarwal et al., Phys. Rev. A 64, 053809 共2001兲; D. Bortman-Arbiv et al., Phys. Rev. A 63, 043818 共2001兲; D. Han et al., Phys. Lett. A 334, 243 共2005兲. L. J. Wang et al., Nature 共London兲 406, 277 共2000兲; P. W. Milonni, J. Phys. B 35, R31 共2002兲. E. E. Mikhailov et al., Phys. Rev. A 69, 063808 共2004兲.
关10兴
关11兴
关12兴
关13兴 关14兴
关15兴 关16兴
055803-4
