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Localization of a two-level atom via the absorption spectrum
Physics Letters A 364 (2007) 208–213 www.elsevier.com/locate/pla
Localization of a two-level atom via the absorption spectrum Jun Xu ∗ , Xiang-Ming Hu Department of Physics, Huazhong Normal University, Wuhan 430079, China Received 10 October 2006; received in revised form 24 November 2006; accepted 29 November 2006 Available online 11 December 2006 Communicated by P.R. Holland
Abstract We show that it is possible to localize a two-level atom as it passes through a standing-wave field by measuring the probe-field absorption. There is 50% detecting probability of the atom at the nodes of the standing-wave field in the subwavelength domain when the probe field is tuned resonant with the atomic transition. © 2006 Elsevier B.V. All rights reserved. PACS: 42.50.Gy; 32.80.Qk; 32.80.Lg
The subwavelength localization of an atom using optical fields has attracted a great deal of interest in recent years. This is because of the optical methods provide better spatial resolution and have their potential applications to many areas of optical manipulations of atomic degrees of freedom. These examples involve laser cooling [1], Bose–Einstein condensation [2], and atom lithography [3], and measurement of the center-of-mass wave function of moving atoms [4]. Earlier schemes for the localization include the measurement of the phase shift of either the standing wave [5–7] or of the atomic dipole [8] due to the interaction of the atom with the standing wave field, the entanglement between the atom’s position and its internal states [9], and resonance imaging methods [10,11]. More recently, Zubairy and co-workers proposed a scheme based on Autler–Townes spontaneous spectrum [12,13]. In this scheme a three-level atom is used, in which either of two levels that are involved in spontaneous emission is coupled to the auxiliary level by a classical standing field. The Autler–Townes splitting is position dependent and the spontaneously emitted photon carries the information of the atomic center-of-mass motion. The detection of the frequency of the spontaneously emitted photon gives the information of the atom about posi-
* Corresponding author.
E-mail addresses: [email protected] (J. Xu), [email protected] (X.-M. Hu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.11.083
tion along with the standing wave field. This group also suggested the atom localization based on resonance fluorescence in a two-level system [14]. Instead of spontaneous emission, the absorption properties were proposed for atom localization. Paspalakis and Knight [15] presented a scheme based on electromagnetically induced transparency [16]. They considered a three-level system interacting a standing wave field and a weak probe field. The measurement of the probe field absorption determines the position information of the atom. It should be noted that the above two kinds of schemes must operate under off-resonance coupling. Otherwise, the resonant interaction of the atom with the fields leads to a uniform position probability distribution over the wavelength domain of the standing wave. In order to obtain more precise position information, Zubairy and co-workers [17–19] proposed to use four-level systems, in which the excited state of the probe transition is coupled to two auxiliary levels in a loop configuration by three driving fields. Of these three fields, one is the standing wave field while the other two are the travelling fields. The responsible mechanism is that the selective elimination of the spontaneous emission or absorption spectrum reduces the periodicity in the conditional position probability distribution. The measurement of the frequency of the spontaneously emitted photon or the absorption of the weak field leads to sub-half-wavelength localization, i.e., the atom is localized in either of the two half-wavelength regions of the cavity field by the amplitude and phase control. At the same time, there is 50% detecting probability (an improve-
J. Xu, X.-M. Hu / Physics Letters A 364 (2007) 208–213
ment by a factor of 2) of the atom within the subwavelength domain. The improvement is also achieved by employing the interference of double dark resonances [20]. In this Letter, we present the localization of a two-level atom via the probe absorption measurement. It is well known that the strongly driven two-level system displays two absorption peaks and two gain peaks near the Rabi frequency sidebands. On the basis of such spectrum, amplification without population inversion [21] and dressed state two-photon lasers are achievable [22,23]. Here we use a standing-wave field as a driving field. The absorption spectrum of a weak probe field carries the information on the atom position along this standing wave due to the position-dependence Rabi frequency. We find that when the probe field is resonant with the atomic transition, the atom is localized at the nodes of the standing-wave field and the detecting probability of the atom in the subwavelength is 50%, whether or not the standing wave field is resonant with the atomic transition. One of the most remarkable features is the utility of the resonance condition. This is in sharp contrast to the previous schemes based on resonance fluorescence or electromagnetically induced transparency, where the localization is unlikely to occur for the resonant excitation [12,13] or Raman resonance [14]. Consider a two-level atom with the ground state |1 and the excited state |2, which are separated by frequency ω21 . The two-level atom passes through a standing-wave field with frequency ωc and Rabi frequency Ω(x) = G sin(kx), where k = ωc /c is the wavenumber of the classical standing-wave coupling field and G is the amplitude. For simplicity Ω(x) is assumed to be real. The spontaneous emission rate from the excited state to the ground state is γ . The master equation for the atom-field interaction system is derived in the appropriate rotating frame and in the dipole approximation as [24] i ∂ρ = − [H, ρ] + Lρ, ∂t h¯
(1)
where the system Hamiltonian is written as H=
h¯ h¯ Δc σz − Ω(x)(σ21 + σ12 ), 2 2
(2)
L12 ρ describes the atomic decay from level |2 to |1 and takes the form Lρ =
γ (2σ12 ρσ21 − σ22 ρ − ρσ22 ), 2
(3)
where Δc = ω21 − ωc is the detuning between the atomic resonance transition frequency ω21 and the driving field frequency ωc . σij = |ij | (i, j = 1, 2) represents a project operator for i = j and a dipole transition operator for i = j . σz = σ22 − σ11 represents the population difference operator. The equations of motion for the expectation values of the atom operators can be derived from the master equation (1) as i ∂ γ (4) σ12 (t) = − − iΔc σ12 (t) − Ω(x) σz (t) , ∂t 2 2 i ∂ γ (5) σ21 (t) = − + iΔc σ21 (t) + Ω(x) σz (t) , ∂t 2 2
209
∂ σz (t) = −γ 1 + σz (t) ∂t + iΩ(x) σ21 (t) − σ12 (t) .
(6)
The steady-state solutions of Eqs. (4)–(6) are derived as σz (t) s = −
γ 2 + 4Δ2c , + 4Δ2c + 2Ω 2 (x) iΩ(x) σ21 (t) s = σz (t) s . γ − 2iΔc γ2
(7) (8)
Now we assume that the driven system is perturbed by a weak monochromatic probe field with frequency ωp . The linear susceptibility of the probe field at frequency ωp is derived from the Fourier transform of the average value of the two-time commutator of the atomic operator as [21,25] χ(ωp ) = ip 2
∞
σ12 (τ ), σ21
s
eiωp τ dτ,
(9)
0
where p is the transition dipole matrix element of the atom, and the subscript s denotes the steady state of the atomic system. We introduce the average value of the two-time commutator of the atomic operator Y1 (τ ) = [σ12 (τ ), σ21 ]s , Y2 (τ ) = [σ21 (τ ), σ21 ]s , Y3 (τ ) = [σz (τ ), σ21 ]s . From the quantum regression theorem [26], we can obtain their equations of motion from Eqs. (4)–(6) as γ i ∂ (10) Y1 (τ ) = − − iΔc Y1 (τ ) − Ω(x)Y3 (τ ), ∂τ 2 2 γ i ∂ Y2 (τ ) = − + iΔc Y2 (τ ) + Ω(x)Y3 (τ ), (11) ∂τ 2 2 ∂ (12) Y3 (τ ) = −iΩ(x) Y1 (τ ) − Y2 (τ ) − γ Y3 (τ ), ∂τ with the initial conditions Y1 (0) = −σz s , Y2 (0) = 0, Y3 (0) = 2σ21 s . The above equations can be solved exactly after taking the Fourier transform. The linear susceptibility χ(ωp ) is obtained as χ(ωp ) =
p2 2 2 G sin (kx)(Δc − Δp ) − 2iD(ωp ) σz s 2 γ γ × + iΔp D(ωp ) + G2 sin2 (kx) − iΔc 2 2 −1 γ × (13) , − iΔc + iΔp 2
where D(ωp ) = ( γ2 − iΔc )( γ2 − 2iΔc + iΔp )(γ − iΔc + iΔp ), and Δp = ω21 − ωp is the detuning between the atomic resonance transition frequency ω21 and the probe field frequency ωp . Here we are interested in the probe absorption spectrum, which is given by the imaginary part of the linear susceptibility A(ωp ) = Im χ(ωp ).
(14)
It is clear that the probe absorption A(ωp ) depends on the position x through the term sin(kx). It is, in principle, possible to obtain the position information of the atom as it moves through the standing wave field by measuring the probe absorption.
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(a)
(b)
Fig. 1. The 3D probe absorption spectra as functions of kx (−π kx π ) and Δc for G = 10, (a) Δp = 0 and (b) Δp = 5.
Fig. 2. The 2D demonstration of Fig. 1 for chosen atom-driving detunings Δc . Column (a) corresponds to Δp = 0 with (a1 ) Δc = 0, (a2 ) Δc = ±2, (a3 ) Δc = ±5, and column (b) to Δp = 5 with (b1 ) Δc = −0.5, (b2 ) Δc = −4, (b3 ) Δc = −8.
In the following numerical calculation, we scale the parameters (Δp , Δc , G) in units of γ and the absorption spectrum in 2
the unit of pγ . We first fix the parameter G = 10 and present the dependence of the atom localization on atom-probe detuning Δp and the atom-driving detuning Δc . (1) Dependence of the absorption spectrum on the atomdriving detuning for given atom-probe detuning. Plotted in Fig. 1 are the 3D absorption spectra as functions of kx (−π kx π ) and Δc for given atom-probe detunings: (a) Δp = 0 and (b) Δp = 5. For Δp = 0 [Fig. 1(a)], two equally probable peaks appear at the nodes of the standing-wave field and the spectra are symmetric with respect to Δc = 0. That indicates that the atom is localized with the detecting probability
50% in the subwavelength domain when the probe field is resonant with the atomic transition. This is true whether or not the driving field is resonant with the atomic transition. In Fig. 2(a) we show the 2D distribution for Δc = 0 (a1 ), ±2 (a2 ), ±5 (a3 ). It is seen that the spectral width increases with increasing Δc . However, for Δp = 5 [Fig. 1(b)], the localization is confined only in a certain range of the detuning Δc . We plot the corresponding 2D spectra for Δc = −0.5 (b1 ), −4 (b2 ), −8 (b3 ) in Fig. 2(b). For small detuning −Δc Δp , as shown in (b1 ), the spectrum exhibits four absorption peaks and four gain peaks. However, the latter are negligibly small compared to the former. For −6 < Δc < −1, as shown in (b2 ), the spectrum has four absorption peaks. When Δc < −6, as shown in (b3 ), the
J. Xu, X.-M. Hu / Physics Letters A 364 (2007) 208–213
(a)
211
(b)
Fig. 3. The 3D probe absorption spectra as functions of kx (−π kx π ) and Δp for G = 10, (a) Δc = 0 and (b) Δc = 5.
Fig. 4. The 2D demonstration of Fig. 3 for chosen atom-probe detunings Δp . Column (a) corresponds to Δc = 0 with (a1 ) Δp = 0, (a2 ) Δp = ±0.5, (a3 ) Δp = ±1, and column (b) to Δc = 5 with (b1 ) Δp = 0, (b2 ) Δp = −4, (b3 ) Δp = −7.
spectrum displays two absorption peaks, whose widths are significantly increased compared to the resonant case in Fig. 1(a). By comparison, it is more suitable for the atom localization under resonance conditions than under off-resonance conditions. (2) Dependence of the absorption spectrum on the atomprobe detuning for given atom-driving detuning. In Fig. 3 we plot the 3D absorption spectra as functions of kx (−π kx π ) and Δp for given atom-driving detunings (a) Δc = 0 and (b) Δc = 5. The localization is achievable in a very narrow range of the detuning Δp whether or not the driving field is resonant with the atomic transition. For Δc = 0 we plot the corresponding 2D distribution for Δp = 0 (a1 ), ±0.5 (a2 ), ±1 (a3 ) in Fig. 4(a). With increasing probe detuning, there appear
two small gain peaks about the absorption peaks. However, the gain is negligibly small. At the same time, the widths of absorption peaks are slightly increased. For Δc = 5, the 2D spectra are shown for Δp = 0 (b1 ), −4 (b2 ), −7 (b3 ) in Fig. 4(b). Note that there are the same parameters for Fig. 4(b1 ) and Fig. 2(a3 ). In this case the atom is localized at the nodes of the standing-wave field and the detecting probability of the atom in the subwavelength is 50%. When the probe field is slightly detuned from the atomic transition, the spectral peaks begin to split. Increasing the detuning to Δp = −4 (b2 ), one has four well resolved peaks. This means that the atomic detecting probability is 25%. As we further increase the detuning to Δp = −7 (b3 ), four peaks evolve into two peaks. In this case, the two peaks lo-
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Fig. 5. Plot of width (w = kΔx) versus G for Δp = Δc = 0.
cate at the antinodes of the standing-wave field. Comparing the case (b3 ) with the case (b1 ), we find that the positions for localization are shifted by the amount of λ4 and the spectral widths are increased. If we continue to increase the detuning or if the detuning takes positive value, the spectral peaks tend to vanish. The responsible mechanism is easily understood in terms of Stark splitting [27] that is created by the driving field. Due to the Stark splitting, both the upper and lower levels are split into |±, which are shifted from the original level by h¯ δ± ,
2 δ± = Δ2c ± ( Δ2c )2 + ( Ω(x) 2 ) . Then the bare atomic states are expressed as
− s|−, where s =
|1 = s|+ + c|−, |2 = c|+
2 + Ω 2 (x)/4 and c = δ / δ 2 + Ω 2 (x)/4. It is (Ω(x)/2)/ δ+ + + obvious that the dressed state populations and coherences ρij (i, j = +, −) are position-dependent through the term Ω(x). By writing the Hamiltonian for the interaction of the atom with the probe field as
p σ21 Ep e−iωp t + σ12 Ep eiωp t 2 p = − csσ++ − csσ−− + c2 σ+− − s 2 σ−+ Ep e−iωp t 2 + H.c., (15)
Hp = −
we easily see that the probe transition in the bare atomic picture is split into four transitions in the dressed state picture. The resultant probe absorption spectrum is determined by the coherent superposition of the four new transitions. The positiondependent populations and coherences ρij enter the susceptibility expression. At the same time, the absorption spectrum depends on the spacing between the split levels (δ+ − δ− ) through the detunings ω21 − ωp ± (δ+ − δ− ). It is clear that the probe absorption carries the information about the position along the standing wave. By making the measurement of the probe absorption we extract the position information of the atom during its motion in the standing wave field. The conditional probability of finding the atom at position x when the probe field absorption is detected is given by [15, 17–19] W (x) = |N |2 |f (x)|2 A2 , where N is a normalized factor, and f (x) is the center-of-mass wave function of the atom and is assumed to be nearly constant over many wavelengths of the standing wave field. In Fig. 5 we show the dependence
of the width w of the best resolved peaks on the amplitude of position-dependent Rabi frequency under the resonant conditions Δp = Δc = 0. The width becomes small as the amplitude G of the Rabi frequency Ω(x) is increased. The width decreases steeply in the region of 1 G 10, but slowly in the regime of 10 G 50, and then it remains nearly constant for G 50. We obtain a better spatial resolution of λ/100 for G = 10. Such condition is accessible, as in recent experiments on the realization of single atoms in the cavity QED [28,29]. Here two points should be emphasized. Firstly, the remarkable difference between the present scheme and the previous schemes is the utility of the resonance condition in the present scheme. Qamar et al. [14] showed that when resonance fluorescence is used to localize the atom the resonance excitation must be excluded. This is because the resonance fluorescence spectrum exhibits a uniform position distribution and provide no information about the atom localization when the driving field is resonant with the atomic transition in the two-level system. Likewise, several authors showed that the Raman resonances become useless when the absorption spectra are employed in the schemes based on electromagnetically induced transparency [15,20]. In sharp contrast, for the present scheme, it is the resonance conditions that are most suitable for the atom localization. Secondly, the advantage of the present scheme is its accessibility in experimental realization. Zubairy and co-workers [17–19] have shown that using three transitions in a loop configuration can lead to the improvement of the detecting probability and to the phase control. Liu et al. [20] have predicted that using a microwave field to couple the Zeeman levels can raise the detecting probability. It should be noted that when many fields and many levels are involved, the unnecessary cross couplings will occur in realistic systems. For the present scheme we simply use the two-level medium, which interacts with one standing wave field and one weak probe field. In this way one can effectively avoid the unnecessary cross couplings. In conclusion, we have discussed the localization of a twolevel atom in a standing-wave field. When a probe field is tuned resonant with the atomic transition, the absorption spectrum displays its peaks at the nodes of the standing-wave field. The atom is localized with the detecting probability 50% in the subwavelength. This is independent of whether or not the driving field is resonant with the atomic transition. On the other hand, when the probe is detuned from the atomic transition, the localization is achievable in a limited range of the detuning between the standing-wave field and the atomic transition. The essential feature is that the atom localization occurs when both the driving field and the probe field are resonant with the atomic transition. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60378008 and No. 10574052. References [1] S. Chu, C. Wieman, J. Opt. Soc. Am. B 6 (1989) 2020;
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H. Metcalf, P. Van der Straten, Phys. Rep. 244 (1994) 204. [2] G.P. Collins, Phys. Today 49 (3) (1996) 18. [3] K.S. Johnson, J.H. Thywissen, N.H. Dekker, K.K. Berggren, A.P. Chu, R. Younkin, M. Prentiss, Science 280 (1998) 1583; M.O. Scully, K. Druhl, Phys. Rev. A 25 (1982) 2208; A.N. Boto, P. Kok, D.S. Abrams, S.L. Braunstein, C.P. Williams, J.P. Dowling, Phys. Rev. Lett. 85 (2000) 2733; G.S. Agarwal, M.O. Scully, Opt. Lett. 28 (2003) 462. [4] K.T. Kapale, S. Qamar, M.S. Zubairy, Phys. Rev. A 67 (2003) 023805. [5] P. Storey, M. Collett, D.F. Walls, Phys. Rev. Lett. 68 (1992) 472; P. Storey, M. Collett, D.F. Walls, Phys. Rev. A 47 (1993) 405; R. Quadt, M. Collett, D.F. Walls, Phys. Rev. Lett. 74 (1995) 351. [6] F.L. Kien, G. Rempe, W.P. Schleich, M.S. Zubairy, Phys. Rev. A 56 (1997) 2972. [7] M.A.M. Marte, P. Zoller, Appl. Phys. B: Photophys. Laser Chem. 54 (1992) 477. [8] S. Kunze, G. Rempe, M. Wilkens, Europhys. Lett. 27 (1994) 115; M. Brune, M. Harlche, V. Lefevere, J.M. Maimond, N. Zagury, Phys. Rev. Lett. 65 (1990) 976. [9] S. Kunze, K. Dieckmann, G. Rempe, Phys. Rev. Lett. 78 (1997) 2038. [10] J.E. Thomas, L.J. Wang, Phys. Rep. 262 (1995) 311, and references therein. [11] P. Rudy, R. Ejnisman, N.P. Bigelow, Phys. Rev. Lett. 78 (1997) 4906. [12] A.M. Herkommer, W.P. Schleich, M.S. Zubairy, J. Mod. Opt. 44 (1997) 2507. [13] S. Qamar, S.Y. Zhu, M.S. Zubairy, Opt. Commun. 176 (2000) 409. [14] S. Qamar, S.Y. Zhu, M.S. Zubairy, Phys. Rev. A 61 (2000) 063806. [15] E. Paspalakis, P.L. Knight, Phys. Rev. A 63 (2001) 065802. [16] S.E. Harris, Phys. Today 50 (7) (1997) 36; J.P. Marangos, J. Mod. Opt. 45 (1998) 471.
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Localization of a two-level atom via the absorption spectrum Jun Xu ∗ , Xiang-Ming Hu Department of Physics, Huazhong Normal University, Wuhan 430079, China Received 10 October 2006; received in revised form 24 November 2006; accepted 29 November 2006 Available online 11 December 2006 Communicated by P.R. Holland
Abstract We show that it is possible to localize a two-level atom as it passes through a standing-wave field by measuring the probe-field absorption. There is 50% detecting probability of the atom at the nodes of the standing-wave field in the subwavelength domain when the probe field is tuned resonant with the atomic transition. © 2006 Elsevier B.V. All rights reserved. PACS: 42.50.Gy; 32.80.Qk; 32.80.Lg
The subwavelength localization of an atom using optical fields has attracted a great deal of interest in recent years. This is because of the optical methods provide better spatial resolution and have their potential applications to many areas of optical manipulations of atomic degrees of freedom. These examples involve laser cooling [1], Bose–Einstein condensation [2], and atom lithography [3], and measurement of the center-of-mass wave function of moving atoms [4]. Earlier schemes for the localization include the measurement of the phase shift of either the standing wave [5–7] or of the atomic dipole [8] due to the interaction of the atom with the standing wave field, the entanglement between the atom’s position and its internal states [9], and resonance imaging methods [10,11]. More recently, Zubairy and co-workers proposed a scheme based on Autler–Townes spontaneous spectrum [12,13]. In this scheme a three-level atom is used, in which either of two levels that are involved in spontaneous emission is coupled to the auxiliary level by a classical standing field. The Autler–Townes splitting is position dependent and the spontaneously emitted photon carries the information of the atomic center-of-mass motion. The detection of the frequency of the spontaneously emitted photon gives the information of the atom about posi-
* Corresponding author.
E-mail addresses: [email protected] (J. Xu), [email protected] (X.-M. Hu). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.11.083
tion along with the standing wave field. This group also suggested the atom localization based on resonance fluorescence in a two-level system [14]. Instead of spontaneous emission, the absorption properties were proposed for atom localization. Paspalakis and Knight [15] presented a scheme based on electromagnetically induced transparency [16]. They considered a three-level system interacting a standing wave field and a weak probe field. The measurement of the probe field absorption determines the position information of the atom. It should be noted that the above two kinds of schemes must operate under off-resonance coupling. Otherwise, the resonant interaction of the atom with the fields leads to a uniform position probability distribution over the wavelength domain of the standing wave. In order to obtain more precise position information, Zubairy and co-workers [17–19] proposed to use four-level systems, in which the excited state of the probe transition is coupled to two auxiliary levels in a loop configuration by three driving fields. Of these three fields, one is the standing wave field while the other two are the travelling fields. The responsible mechanism is that the selective elimination of the spontaneous emission or absorption spectrum reduces the periodicity in the conditional position probability distribution. The measurement of the frequency of the spontaneously emitted photon or the absorption of the weak field leads to sub-half-wavelength localization, i.e., the atom is localized in either of the two half-wavelength regions of the cavity field by the amplitude and phase control. At the same time, there is 50% detecting probability (an improve-
J. Xu, X.-M. Hu / Physics Letters A 364 (2007) 208–213
ment by a factor of 2) of the atom within the subwavelength domain. The improvement is also achieved by employing the interference of double dark resonances [20]. In this Letter, we present the localization of a two-level atom via the probe absorption measurement. It is well known that the strongly driven two-level system displays two absorption peaks and two gain peaks near the Rabi frequency sidebands. On the basis of such spectrum, amplification without population inversion [21] and dressed state two-photon lasers are achievable [22,23]. Here we use a standing-wave field as a driving field. The absorption spectrum of a weak probe field carries the information on the atom position along this standing wave due to the position-dependence Rabi frequency. We find that when the probe field is resonant with the atomic transition, the atom is localized at the nodes of the standing-wave field and the detecting probability of the atom in the subwavelength is 50%, whether or not the standing wave field is resonant with the atomic transition. One of the most remarkable features is the utility of the resonance condition. This is in sharp contrast to the previous schemes based on resonance fluorescence or electromagnetically induced transparency, where the localization is unlikely to occur for the resonant excitation [12,13] or Raman resonance [14]. Consider a two-level atom with the ground state |1 and the excited state |2, which are separated by frequency ω21 . The two-level atom passes through a standing-wave field with frequency ωc and Rabi frequency Ω(x) = G sin(kx), where k = ωc /c is the wavenumber of the classical standing-wave coupling field and G is the amplitude. For simplicity Ω(x) is assumed to be real. The spontaneous emission rate from the excited state to the ground state is γ . The master equation for the atom-field interaction system is derived in the appropriate rotating frame and in the dipole approximation as [24] i ∂ρ = − [H, ρ] + Lρ, ∂t h¯
(1)
where the system Hamiltonian is written as H=
h¯ h¯ Δc σz − Ω(x)(σ21 + σ12 ), 2 2
(2)
L12 ρ describes the atomic decay from level |2 to |1 and takes the form Lρ =
γ (2σ12 ρσ21 − σ22 ρ − ρσ22 ), 2
(3)
where Δc = ω21 − ωc is the detuning between the atomic resonance transition frequency ω21 and the driving field frequency ωc . σij = |ij | (i, j = 1, 2) represents a project operator for i = j and a dipole transition operator for i = j . σz = σ22 − σ11 represents the population difference operator. The equations of motion for the expectation values of the atom operators can be derived from the master equation (1) as i ∂ γ (4) σ12 (t) = − − iΔc σ12 (t) − Ω(x) σz (t) , ∂t 2 2 i ∂ γ (5) σ21 (t) = − + iΔc σ21 (t) + Ω(x) σz (t) , ∂t 2 2
209
∂ σz (t) = −γ 1 + σz (t) ∂t + iΩ(x) σ21 (t) − σ12 (t) .
(6)
The steady-state solutions of Eqs. (4)–(6) are derived as σz (t) s = −
γ 2 + 4Δ2c , + 4Δ2c + 2Ω 2 (x) iΩ(x) σ21 (t) s = σz (t) s . γ − 2iΔc γ2
(7) (8)
Now we assume that the driven system is perturbed by a weak monochromatic probe field with frequency ωp . The linear susceptibility of the probe field at frequency ωp is derived from the Fourier transform of the average value of the two-time commutator of the atomic operator as [21,25] χ(ωp ) = ip 2
∞
σ12 (τ ), σ21
s
eiωp τ dτ,
(9)
0
where p is the transition dipole matrix element of the atom, and the subscript s denotes the steady state of the atomic system. We introduce the average value of the two-time commutator of the atomic operator Y1 (τ ) = [σ12 (τ ), σ21 ]s , Y2 (τ ) = [σ21 (τ ), σ21 ]s , Y3 (τ ) = [σz (τ ), σ21 ]s . From the quantum regression theorem [26], we can obtain their equations of motion from Eqs. (4)–(6) as γ i ∂ (10) Y1 (τ ) = − − iΔc Y1 (τ ) − Ω(x)Y3 (τ ), ∂τ 2 2 γ i ∂ Y2 (τ ) = − + iΔc Y2 (τ ) + Ω(x)Y3 (τ ), (11) ∂τ 2 2 ∂ (12) Y3 (τ ) = −iΩ(x) Y1 (τ ) − Y2 (τ ) − γ Y3 (τ ), ∂τ with the initial conditions Y1 (0) = −σz s , Y2 (0) = 0, Y3 (0) = 2σ21 s . The above equations can be solved exactly after taking the Fourier transform. The linear susceptibility χ(ωp ) is obtained as χ(ωp ) =
p2 2 2 G sin (kx)(Δc − Δp ) − 2iD(ωp ) σz s 2 γ γ × + iΔp D(ωp ) + G2 sin2 (kx) − iΔc 2 2 −1 γ × (13) , − iΔc + iΔp 2
where D(ωp ) = ( γ2 − iΔc )( γ2 − 2iΔc + iΔp )(γ − iΔc + iΔp ), and Δp = ω21 − ωp is the detuning between the atomic resonance transition frequency ω21 and the probe field frequency ωp . Here we are interested in the probe absorption spectrum, which is given by the imaginary part of the linear susceptibility A(ωp ) = Im χ(ωp ).
(14)
It is clear that the probe absorption A(ωp ) depends on the position x through the term sin(kx). It is, in principle, possible to obtain the position information of the atom as it moves through the standing wave field by measuring the probe absorption.
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(a)
(b)
Fig. 1. The 3D probe absorption spectra as functions of kx (−π kx π ) and Δc for G = 10, (a) Δp = 0 and (b) Δp = 5.
Fig. 2. The 2D demonstration of Fig. 1 for chosen atom-driving detunings Δc . Column (a) corresponds to Δp = 0 with (a1 ) Δc = 0, (a2 ) Δc = ±2, (a3 ) Δc = ±5, and column (b) to Δp = 5 with (b1 ) Δc = −0.5, (b2 ) Δc = −4, (b3 ) Δc = −8.
In the following numerical calculation, we scale the parameters (Δp , Δc , G) in units of γ and the absorption spectrum in 2
the unit of pγ . We first fix the parameter G = 10 and present the dependence of the atom localization on atom-probe detuning Δp and the atom-driving detuning Δc . (1) Dependence of the absorption spectrum on the atomdriving detuning for given atom-probe detuning. Plotted in Fig. 1 are the 3D absorption spectra as functions of kx (−π kx π ) and Δc for given atom-probe detunings: (a) Δp = 0 and (b) Δp = 5. For Δp = 0 [Fig. 1(a)], two equally probable peaks appear at the nodes of the standing-wave field and the spectra are symmetric with respect to Δc = 0. That indicates that the atom is localized with the detecting probability
50% in the subwavelength domain when the probe field is resonant with the atomic transition. This is true whether or not the driving field is resonant with the atomic transition. In Fig. 2(a) we show the 2D distribution for Δc = 0 (a1 ), ±2 (a2 ), ±5 (a3 ). It is seen that the spectral width increases with increasing Δc . However, for Δp = 5 [Fig. 1(b)], the localization is confined only in a certain range of the detuning Δc . We plot the corresponding 2D spectra for Δc = −0.5 (b1 ), −4 (b2 ), −8 (b3 ) in Fig. 2(b). For small detuning −Δc Δp , as shown in (b1 ), the spectrum exhibits four absorption peaks and four gain peaks. However, the latter are negligibly small compared to the former. For −6 < Δc < −1, as shown in (b2 ), the spectrum has four absorption peaks. When Δc < −6, as shown in (b3 ), the
J. Xu, X.-M. Hu / Physics Letters A 364 (2007) 208–213
(a)
211
(b)
Fig. 3. The 3D probe absorption spectra as functions of kx (−π kx π ) and Δp for G = 10, (a) Δc = 0 and (b) Δc = 5.
Fig. 4. The 2D demonstration of Fig. 3 for chosen atom-probe detunings Δp . Column (a) corresponds to Δc = 0 with (a1 ) Δp = 0, (a2 ) Δp = ±0.5, (a3 ) Δp = ±1, and column (b) to Δc = 5 with (b1 ) Δp = 0, (b2 ) Δp = −4, (b3 ) Δp = −7.
spectrum displays two absorption peaks, whose widths are significantly increased compared to the resonant case in Fig. 1(a). By comparison, it is more suitable for the atom localization under resonance conditions than under off-resonance conditions. (2) Dependence of the absorption spectrum on the atomprobe detuning for given atom-driving detuning. In Fig. 3 we plot the 3D absorption spectra as functions of kx (−π kx π ) and Δp for given atom-driving detunings (a) Δc = 0 and (b) Δc = 5. The localization is achievable in a very narrow range of the detuning Δp whether or not the driving field is resonant with the atomic transition. For Δc = 0 we plot the corresponding 2D distribution for Δp = 0 (a1 ), ±0.5 (a2 ), ±1 (a3 ) in Fig. 4(a). With increasing probe detuning, there appear
two small gain peaks about the absorption peaks. However, the gain is negligibly small. At the same time, the widths of absorption peaks are slightly increased. For Δc = 5, the 2D spectra are shown for Δp = 0 (b1 ), −4 (b2 ), −7 (b3 ) in Fig. 4(b). Note that there are the same parameters for Fig. 4(b1 ) and Fig. 2(a3 ). In this case the atom is localized at the nodes of the standing-wave field and the detecting probability of the atom in the subwavelength is 50%. When the probe field is slightly detuned from the atomic transition, the spectral peaks begin to split. Increasing the detuning to Δp = −4 (b2 ), one has four well resolved peaks. This means that the atomic detecting probability is 25%. As we further increase the detuning to Δp = −7 (b3 ), four peaks evolve into two peaks. In this case, the two peaks lo-
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Fig. 5. Plot of width (w = kΔx) versus G for Δp = Δc = 0.
cate at the antinodes of the standing-wave field. Comparing the case (b3 ) with the case (b1 ), we find that the positions for localization are shifted by the amount of λ4 and the spectral widths are increased. If we continue to increase the detuning or if the detuning takes positive value, the spectral peaks tend to vanish. The responsible mechanism is easily understood in terms of Stark splitting [27] that is created by the driving field. Due to the Stark splitting, both the upper and lower levels are split into |±, which are shifted from the original level by h¯ δ± ,
2 δ± = Δ2c ± ( Δ2c )2 + ( Ω(x) 2 ) . Then the bare atomic states are expressed as
− s|−, where s =
|1 = s|+ + c|−, |2 = c|+
2 + Ω 2 (x)/4 and c = δ / δ 2 + Ω 2 (x)/4. It is (Ω(x)/2)/ δ+ + + obvious that the dressed state populations and coherences ρij (i, j = +, −) are position-dependent through the term Ω(x). By writing the Hamiltonian for the interaction of the atom with the probe field as
p σ21 Ep e−iωp t + σ12 Ep eiωp t 2 p = − csσ++ − csσ−− + c2 σ+− − s 2 σ−+ Ep e−iωp t 2 + H.c., (15)
Hp = −
we easily see that the probe transition in the bare atomic picture is split into four transitions in the dressed state picture. The resultant probe absorption spectrum is determined by the coherent superposition of the four new transitions. The positiondependent populations and coherences ρij enter the susceptibility expression. At the same time, the absorption spectrum depends on the spacing between the split levels (δ+ − δ− ) through the detunings ω21 − ωp ± (δ+ − δ− ). It is clear that the probe absorption carries the information about the position along the standing wave. By making the measurement of the probe absorption we extract the position information of the atom during its motion in the standing wave field. The conditional probability of finding the atom at position x when the probe field absorption is detected is given by [15, 17–19] W (x) = |N |2 |f (x)|2 A2 , where N is a normalized factor, and f (x) is the center-of-mass wave function of the atom and is assumed to be nearly constant over many wavelengths of the standing wave field. In Fig. 5 we show the dependence
of the width w of the best resolved peaks on the amplitude of position-dependent Rabi frequency under the resonant conditions Δp = Δc = 0. The width becomes small as the amplitude G of the Rabi frequency Ω(x) is increased. The width decreases steeply in the region of 1 G 10, but slowly in the regime of 10 G 50, and then it remains nearly constant for G 50. We obtain a better spatial resolution of λ/100 for G = 10. Such condition is accessible, as in recent experiments on the realization of single atoms in the cavity QED [28,29]. Here two points should be emphasized. Firstly, the remarkable difference between the present scheme and the previous schemes is the utility of the resonance condition in the present scheme. Qamar et al. [14] showed that when resonance fluorescence is used to localize the atom the resonance excitation must be excluded. This is because the resonance fluorescence spectrum exhibits a uniform position distribution and provide no information about the atom localization when the driving field is resonant with the atomic transition in the two-level system. Likewise, several authors showed that the Raman resonances become useless when the absorption spectra are employed in the schemes based on electromagnetically induced transparency [15,20]. In sharp contrast, for the present scheme, it is the resonance conditions that are most suitable for the atom localization. Secondly, the advantage of the present scheme is its accessibility in experimental realization. Zubairy and co-workers [17–19] have shown that using three transitions in a loop configuration can lead to the improvement of the detecting probability and to the phase control. Liu et al. [20] have predicted that using a microwave field to couple the Zeeman levels can raise the detecting probability. It should be noted that when many fields and many levels are involved, the unnecessary cross couplings will occur in realistic systems. For the present scheme we simply use the two-level medium, which interacts with one standing wave field and one weak probe field. In this way one can effectively avoid the unnecessary cross couplings. In conclusion, we have discussed the localization of a twolevel atom in a standing-wave field. When a probe field is tuned resonant with the atomic transition, the absorption spectrum displays its peaks at the nodes of the standing-wave field. The atom is localized with the detecting probability 50% in the subwavelength. This is independent of whether or not the driving field is resonant with the atomic transition. On the other hand, when the probe is detuned from the atomic transition, the localization is achievable in a limited range of the detuning between the standing-wave field and the atomic transition. The essential feature is that the atom localization occurs when both the driving field and the probe field are resonant with the atomic transition. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grant No. 60378008 and No. 10574052. References [1] S. Chu, C. Wieman, J. Opt. Soc. Am. B 6 (1989) 2020;
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