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Laser cooling force in noisy quadrature of squeezed light
Optics Communications 267 (2006) 124–127 www.elsevier.com/locate/optcom
Laser cooling force in noisy quadrature of squeezed light G.M. Saxena *, Ashish Agarwal Time and Frequency Section, National Physical Laboratory, Dr. K.S.Krishnan Road, New Delhi 110012, India Received 9 December 2005; received in revised form 21 April 2006; accepted 5 June 2006
Abstract The laser cooling of atoms is a result of the combined effect of Doppler shift, light shift and polarization gradient. These are the phenomena which generally introduce frequency shift and uncertainty. However, they combine gainfully in realizing laser cooling and trapping of the atoms. In this paper we discuss the laser cooling of atoms in the presence of the squeezed light with the decay of atomic dipole moment into noisy quadrature. We show that the higher decay rate of the atomic dipole moment into the noisy quadrature, which leads to decrease in the signal to noise ratio, may contribute in realizing larger cooling force vis-a`-vis with coherent laser light. Ó 2006 Elsevier B.V. All rights reserved. PACS: 32.80.Pj; 42.50.Vk; 42.50.Dv; 42.50.Lc
1. Introduction The laser cooling of the atoms and ions has been a subject of extensive studies in recent years [1]. The laser cooling of atoms has number of applications i.e., in the Cs fountain clock, Bose–Einstein condensation, test of principles of Quantum mechanics and the study of time dependence of the fundamental constants. In the laser cooling experiment, the polarization gradient [1] and dark state [2] techniques are used for cooling the multi-level atoms to the temperatures much lower than the ‘Doppler limit’. However, very low temperature may also be reached by using the squeezed vacuum [3] even for the two-level atoms. It has been shown [4] that the temperatures below the Doppler limit can be obtained for the two-level atoms in a near-resonant standing wave squeezed field. We establish in this paper that by using the noisy quadrature of the squeezed vacuum under certain phase matching conditions, the laser cooling force may be increased vis-a`-vis the cooling with laser light. We shall see that the decay of the atomic dipole moment into the noisy quadrature of the squeezed light results in higher cooling force in spite of *
Corresponding author. Tel.: +91 11 25734617. E-mail address: [email protected] (G.M. Saxena).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.06.005
the fact that the signal to noise ratio in noisy quadrature is smaller than that in vacuum state. It is quite understandable as the laser cooling is after all the result of the combined effect of light shift, Doppler shift and polarization gradient. These effects, in general, introduce shifts or uncertainties in the frequency. However, it is a well established observation that these effects have helped in cooling the atoms to a very low temperature. And now we have one more similar factor, the noisy quadrature of the squeezed vacuum. The atoms in the noisy quadrature decay at a rate faster than the spontaneous emission rate. This may increase the momentum diffusion leading to the increase in the temperature. But the squeezing at higher intensities increases the frictional coefficient which lowers the temperature and more than offsets the rise in the temperature due to momentum diffusion for a standing wave field near atomic resonance [4]. We report in this paper that the noisy quadrature of the squeezed light can be gainfully used in realizing the higher cooling force vis-a`-vis normal laser light. In this paper we study the cooling force on the Cs atoms in the optical molasses formed by the coherent squeezed state and the squeezed vacuum with the certain phase matching conditions. The squeezed vacuum with proper phase interacts with those atoms, which have enhanced
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
dipole decay rate in the noisy quadrature. We show that for the combination of the coherent squeezed state and the squeezed vacuum, the atoms can be cooled much below the Doppler limit even for a resonant driving field. It occurs when the phase of the coherent squeezed state is different from 0 or p and the phase of the squeezed vacuum is 0 or p relative to the driving field, respectively. 2. Squeezed light cooling force In this section we shall discuss the cooling force on the atoms with the squeezed light. We first, briefly, describe one of the methods of generating the squeezed light using an optical parametric oscillator (OPO) [5] before considering the cooling of atoms in the presence of the cw source of the squeezed light. A brief description of the squeezed light generation using OPO is included for defining the parameters that characterize the squeezed states. In the set-up for the squeezed light an external-cavity diode laser locked to Cs transition at 852 nm serves as a primary source of light or pump for the input of OPO. The laser light at 852 nm is first sent to a frequency-doubling cavity. The cavity output produces a pump light at 426 nm for the OPO and it is frequency locked to the incident laser light. The doubling cavity contains a Potassium Niobate (KNbO3) nonlinear crystal. The OPO cavity is a four-mirror folded cavity with two curved and two plain mirrors. The non-linear medium at the waist between curved mirrors is a-cut KNbO3 crystal. The spontaneous parametric fluorescence, produced by degenerate down conversion into a sub-harmonic mode of the OPO, generates the squeezed vacuum. In a degenerate OPO, the pump field at 2x is split by a non-linear crystal into two photons of frequencies x each. The average values of the photon annihilation and creation operators in the squeezed vacuum are hak i ¼ haþ k i ¼ 0 [6] and that of second order photon annihilation and creation operai tors are haþ k ak i ¼ N ðxÞ; hak ak i ¼ MðxÞe /, here N(x) corresponds to the photon number, M(x) is the two-photon correlation function and / is the squeezed field phase. The photon number and two-photon correlation functions are not independent of each other. They satisfy the inequality for the quantum squeezed field [7], N 2 < M 2 ðxÞ 6 N ðxÞ½N ðxÞ þ 1:
ð1aÞ
Whereas in the case of the classically squeezed field N and M satisfy the inequality M 6 N. The factor (N + 1) instead of N in the above equation arises due to the quantum nature of the field. In the broadband case M(x) = gN(x)[N(x) + 1] with 0 6 g 6 1. Here g is the degree of correlation. In this paper we shall use the quantum squeezed field which satisfies the inequality (1a). For an ideal degenerate OPO [8], we have N ðxÞ ¼ ðk2 l2 Þ½1=ðx2 þ l2 Þ 1=ðx2 k2 Þ=4; 2
2
2
2
2
2
MðxÞ ¼ ðk l Þ½1=ðx þ l Þ þ 1=ðx k Þ=4; and
ð1bÞ ð1cÞ
2
MðxÞ ¼ N ðxÞ½N ðxÞ þ 1;
125
ð1dÞ
where k = j/2 + e, l = j/2 e, with j and e being the cavity decay constant and the amplification factor, respectively, for OPO. The values of N and M functions may be selected by the OPO parameters. The OPO generates an ideal squeezed vacuum with N = sinh2 r, and jMj = cosh r sinh r, here r is the squeeze factor. The signal and idler outputs of the OPO may be combined at a 50/50 beam-splitter to produce a squeezed-vacuum state and a squeezed-coherent state [9]. In the conventional laser cooling set-up for the fountain atomic clock, the Cs atoms are cooled and trapped by an optical molasses formed by three mutually perpendicular standing-wave laser beams. The cooling force due to the optical molasses has been calculated by several groups and is well documented [10]. The cooling force on the slowly moving atoms with the density matrix q, in a standing-wave squeezed-coherent field [11,12] is given by F ¼ ihqr ðX hq12 i Xhq21 iÞ ¼ hqr XhrY isc ;
ð2Þ
here qr = k tan(kx), k = 2p/k, k = 852 nm, and X is the Rabi frequency of the optical field. hrYisc, the y-component of the atomic Block vector in the presence of the squeezedcoherent light is given by hrY isc ¼ Xðd þ cjMj sin /Þ=2Dsc ;
ð3aÞ
where d is the laser frequency detuning from the resonance, c is the normal atomic decay rate, / is the relative phase of the driving laser field with respect to the squeezed-coherent light, and 1 Dsc ¼ n c2 n2 þ d2 c2 jMj2 þ X2 ðn=2 þ jMj cos /Þ; 4 ð3bÞ and n = 1 + 2N. The force given by Eq. (2) is calculated assuming kv < c and d < c, where v is the velocity of the atoms. In the above calculations for the cooling force, the squeezed light may couple in free space to atoms in a small fraction of 4p solid angle. To overcome this problem Gardiner [6] suggested coupling squeezed light modes to the atom in an optical cavity, which is basically small in dimensions. We may consider bad cavity scenario in which the Bloch equations are effectively similar to those in the free space with reduced value of strength of two-photon correlation. In the bad cavity scenario j c, g with g > c. Here g, j and c are the measure of the atom – cavity coupling, cavity and atomic decay rates, respectively. The bad cavity environment modifies the free space values of N and M to Nc and Mc, respectively with Nc = BN and Mc = BM [13]. Here B is generally referred to as the beta factor of the cavity and it may be defined as ratio of spontaneous decay rate into cavity mode to the total spontaneous decay rate. B is related to C = g2/jc, the single atom cooperative parameter for an atom. We have B = C/(1 + C) and as B < 1 for positive finite values of C, the free space values of N and M in the bad cavity environment
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
are reduced. The lowering of N and M values results from the fact that in the bad cavity environment the atom is always subjected from the open side of the cavity to the unsqueezed field modes. This exposure to unsqueezed field leads to smaller value of strength of two-photon correlations. The decay rate c is also modified to cC = c(C + 1) in the cavity environment. For very large values of C we have B = 1 and in this case the bad cavity environment may be approximated by the free space. For experimental purposes modified N, M and c values may be used depending on the bad cavity parameters. However, for the sake of generality we will consider the free space atom-squeezed light interaction. This treatment may be adapted for any laser cooling experimental setup in the bad cavity environment by choosing the modified values of N, M and c. We observe from Eqs. (2) and (3a) that the magnitude of the cooling force basically depends on (d + cjMj sin /) and there is non-vanishing cooling force even at zero detuning (d = 0). The term cjMj sin / in the expression for cooling force arises due to the interaction of atoms with squeezedcoherent light – a quantum field. From this term it follows that the squeezed light enhances the cooling force. The twophoton correlation factor M depends on the degree of squeezing. The decay rate c is also an important parameter in determining the cooling force. Several authors have shown that the squeezed states might be used for manipulating the atomic spontaneous decay rate [8,9]. As discussed in the paper, for increased decay rate a larger cooling force may be realized. On interacting with the squeezed vacuum, the atoms de-excite into either of the two quadratures with different decay rates. The atoms in the squeezed (in-phase) quadrature relax at a slower rate while those in the noisy (out-of-phase) quadrature decay at faster relaxation rate compared to normal vacuum. Specifically, when the relative phase U between the coherent driving field and the squeezed vacuum is 0, the atoms decay into the out-of-phase or the noisy quadrature with the larger decay rate,
the faster decay rate from Eq. (4a) in (2), the cooling force may be expressed as, F sv ¼ qr XhrY isvsc :
hrY isvsc ¼ 1=2b½D þ 2jMjðn=2 þ jMjÞ sin /=D; 2
ð6Þ
2
ð7Þ
In the above expressions, the normalized detuning D and Rabi frequency b are defined as D d/c and b X/c, respectively. It may be readily shown that with the modified y-component of the atomic Block vector hrYisvsc, the cooling force is increased depending on the phase / and squeeze parameter r. The cooling force is calculated for the different values of these parameters. The relevant parameter quantifying the degree of squeezing is given by the factor 2(jMj N). Georgiades et al. [14] have developed a source of squeezed light exhibiting approximately 75% squeezing. This corresponds to N = 9/16, (a) 6 5 4 3 2 1 0 3
2
1
0
6
5
4
9
8
7
10
β 6
ð4aÞ
ð4bÞ
In general, the squeezed states are used for reducing the quantum noise or the decay rate of the atoms. However, in this paper, we demonstrate that even the noisy quadrature could be gainfully utilized in realizing higher cooling force than that with either laser light or quieter quadrature of the squeezed light. Let us consider that the atoms in the optical molasses, formed by the squeezed-coherent light of phase /, decay with the faster rate c0 into the noisy quadrature (U = 0) of the squeezed vacuum. We calculate the cooling force on the atoms in the combined presence of the coherent squeezed state and squeezed vacuum. On incorporating
C oo lin g For ce ( γ k / 2)
5
When the relative phase U between the coherent driving field and the squeezed vacuum is p atoms decay at slower rate, cp ¼ cðN þ 1=2 MÞ:
2
2
D ¼ n=2 ½b þ 2D þ ðn=2 þ jMjÞ þ b jMj cos /:
(b)
c0 ¼ cðN þ 1=2 þ MÞ:
ð5Þ
Here hrYisvsc, the y-component of the atomic Bloch vector in the combined presence of the coherent squeezed state and the squeezed vacuum, with some mathematical simplifications, is given by the expression,
Cooling Force ( γ k / 2)
126
4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
β
Fig. 1. The dependence of spatially averaged force on the normalized Rabi frequency b of the driving field in the presence (solid and dotted line) and absence (dashed line) of squeezed vacuum. The solid line is for U = 0 or p (out-of-the phase) and dotted line is for U = p/2 (in-phase quadrature). Here degree of squeezing is chosen to be 75% and / = 0.8p. Plot (a) is for the resonant case D = 0, and (b) for the off-resonant case D = 0.1. Cooling force hck/2 = 1 is the Doppler limit.
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
jMj = 15/16 and r 0.7. With this kind of source of squeezed light one can efficiently cool and trap atoms. We have calculated the cooling force for the squeezed light and for the combination of squeezed vacuum and squeezed coherent light, and denoted them as F and Fsv, respectively. We compare the cooling force F and Fsv in Figs. 1 and 2 for different cases. As shown in Fig. 1, for small values of normalized Rabi frequency b, F (shown by dashed line) is greater than Fsv (shown by solid curve). For b > 3, this trend reverses. There is a steep rise in Fsv compared to F for the increasing value of b. Fig. 1(a) is a plot for the resonant case. We find that the squeezed light can give rise to a cooling force even at zero detuning. In Fig. 1(b), we plot the off-resonance case. This shows similar behavior as in Fig. 1(a) except that the cooling force increases slightly. For squeezed (in-phase) quadrature instead of the noisy quadrature, i.e., (a) 8 7
5
( γ k / 2)
Cooling force
6
4 3 2 1.0 0.8 0.5
1 0 0
φ/π
0.3 20
40
60
Degree of Squeezing (%)
0.0 80
for the phase U = p/2, the cooling force (shown by dotted line in Fig. 1) is much smaller than F (shown by dashed line). This shows the utility of the noisy quadrature over the inphase quadrature of the squeezed light for efficient cooling of the atoms. In Fig. 2, we show that the relative phase / and the degree of squeezing play significant role in enhancing the laser cooling force in the noisy quadrature of the squeezed light. Figs. 2(a) and (b) are for the resonant and off-resonant squeezed light, respectively. These figures show that the cooling force is sensitive to small changes in the relative phase between fields, the detuning from the resonance, value of Rabi frequency and degree of squeezing. 3. Conclusion We have established in this paper that the noisy quadrature of the squeezed vacuum in the presence of coherent squeezed state could be gainfully utilized in realizing higher cooling force than that with either laser light or with the squeezed vacuum. Besides, for some values of the phase of the coherent squeezed field with respect to the laser field and for the appropriate value of two-photon correlation M, the atoms may experience large cooling force even for the resonant squeezed light. It is very unexpected result because the noisy quadrature is generally unused part of the squeezed light. However, there may be increase in the momentum diffusion in noisy quadrature. This enhancement in momentum diffusion [4] may be more than offset and counteracted by squeezing at higher laser intensities. As the squeezed field at higher intensities increases the cooling force near atomic resonance for standing wave field due to accelerated cooling process without changing the atom – squeezed field interaction. References
(b) 8 7 6
Cooling force ( γ k / 2)
127
5 4 3 2 0.9
1
0.6
0 0
0.3 20
40
Degree of Squeezing (%)
60
φ/π
0.0 80
Fig. 2. A three-dimensional plot of spatially averaged cooling force Fsv as a function of the degree of squeezing and relative phase / for (a) the resonant case D = 0, and (b) the off-resonant case D = 0.1. Normalized Rabi frequency b = 10.
[1] D. Wineland, W. Itano, Phys. Rev. A 20 (1979) 1521; S. Chu, C. Wieman, J. Opt. Soc. Am. B 6 (1989) 1961; J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 (1989) 2023. [2] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. CohenTannoudji, J. Opt. Soc. Am. B 6 (1989) 2112. [3] Y. Shevy, Phys. Rev. Lett. 64 (1990) 2905. [4] R. Graham, D.F. Walls, W.P. Zhang, Phys. Rev. A 44 (1991) 7777. [5] E.S. Polzik, J. Carri, H.J. Kimble, Appl. Phys. B 55 (1992) 279. [6] C.W. Gardiner, Phys. Rev. Lett. 56 (1986) 1917; A.S. Parkins, C.W. Gardiner, Phys. Rev. A 40 (1990) 3796. [7] H.J. Carmichael, A.S. Lane, D.F. Walls, Phys. Rev. Lett. 58 (1997) 2539; H.J. Carmichael, A.S. Lane, D.F. Walls, J. Mod. Opt. 34 (1987) 821. [8] H. Ritsh, P. Zoller, Opt. Commun. 64 (1987) 523; H. Ritsh, P. Zoller, Phys. Rev. A 38 (1988) 4657. [9] C. Kim, P. Kumar, Phys. Rev. Lett. 73 (1994) 1605. [10] H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer Verlag, New York, 1999. [11] B.J. Dalton, Z. Ficek, S. Swain, J. Mod. Opt. 46 (1999) 379. [12] Z. Ficek, W.S. Smyth, S. Swain, Phys. Rev. A 52 (1995) 4126. [13] W.S. Smyth, S. Swain, Phys. Rev. A 53 (1996) 2846; P. Zhou, S. Swain, Opt. Commun. 131 (1996) 153. [14] N.P. Georgiades, E.S. Polzik, K. Edamatsu, H.J. Kimble, A.S. Parkins, Phys. Rev. Lett. 75 (1995) 3426.
Laser cooling force in noisy quadrature of squeezed light G.M. Saxena *, Ashish Agarwal Time and Frequency Section, National Physical Laboratory, Dr. K.S.Krishnan Road, New Delhi 110012, India Received 9 December 2005; received in revised form 21 April 2006; accepted 5 June 2006
Abstract The laser cooling of atoms is a result of the combined effect of Doppler shift, light shift and polarization gradient. These are the phenomena which generally introduce frequency shift and uncertainty. However, they combine gainfully in realizing laser cooling and trapping of the atoms. In this paper we discuss the laser cooling of atoms in the presence of the squeezed light with the decay of atomic dipole moment into noisy quadrature. We show that the higher decay rate of the atomic dipole moment into the noisy quadrature, which leads to decrease in the signal to noise ratio, may contribute in realizing larger cooling force vis-a`-vis with coherent laser light. Ó 2006 Elsevier B.V. All rights reserved. PACS: 32.80.Pj; 42.50.Vk; 42.50.Dv; 42.50.Lc
1. Introduction The laser cooling of the atoms and ions has been a subject of extensive studies in recent years [1]. The laser cooling of atoms has number of applications i.e., in the Cs fountain clock, Bose–Einstein condensation, test of principles of Quantum mechanics and the study of time dependence of the fundamental constants. In the laser cooling experiment, the polarization gradient [1] and dark state [2] techniques are used for cooling the multi-level atoms to the temperatures much lower than the ‘Doppler limit’. However, very low temperature may also be reached by using the squeezed vacuum [3] even for the two-level atoms. It has been shown [4] that the temperatures below the Doppler limit can be obtained for the two-level atoms in a near-resonant standing wave squeezed field. We establish in this paper that by using the noisy quadrature of the squeezed vacuum under certain phase matching conditions, the laser cooling force may be increased vis-a`-vis the cooling with laser light. We shall see that the decay of the atomic dipole moment into the noisy quadrature of the squeezed light results in higher cooling force in spite of *
Corresponding author. Tel.: +91 11 25734617. E-mail address: [email protected] (G.M. Saxena).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.06.005
the fact that the signal to noise ratio in noisy quadrature is smaller than that in vacuum state. It is quite understandable as the laser cooling is after all the result of the combined effect of light shift, Doppler shift and polarization gradient. These effects, in general, introduce shifts or uncertainties in the frequency. However, it is a well established observation that these effects have helped in cooling the atoms to a very low temperature. And now we have one more similar factor, the noisy quadrature of the squeezed vacuum. The atoms in the noisy quadrature decay at a rate faster than the spontaneous emission rate. This may increase the momentum diffusion leading to the increase in the temperature. But the squeezing at higher intensities increases the frictional coefficient which lowers the temperature and more than offsets the rise in the temperature due to momentum diffusion for a standing wave field near atomic resonance [4]. We report in this paper that the noisy quadrature of the squeezed light can be gainfully used in realizing the higher cooling force vis-a`-vis normal laser light. In this paper we study the cooling force on the Cs atoms in the optical molasses formed by the coherent squeezed state and the squeezed vacuum with the certain phase matching conditions. The squeezed vacuum with proper phase interacts with those atoms, which have enhanced
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
dipole decay rate in the noisy quadrature. We show that for the combination of the coherent squeezed state and the squeezed vacuum, the atoms can be cooled much below the Doppler limit even for a resonant driving field. It occurs when the phase of the coherent squeezed state is different from 0 or p and the phase of the squeezed vacuum is 0 or p relative to the driving field, respectively. 2. Squeezed light cooling force In this section we shall discuss the cooling force on the atoms with the squeezed light. We first, briefly, describe one of the methods of generating the squeezed light using an optical parametric oscillator (OPO) [5] before considering the cooling of atoms in the presence of the cw source of the squeezed light. A brief description of the squeezed light generation using OPO is included for defining the parameters that characterize the squeezed states. In the set-up for the squeezed light an external-cavity diode laser locked to Cs transition at 852 nm serves as a primary source of light or pump for the input of OPO. The laser light at 852 nm is first sent to a frequency-doubling cavity. The cavity output produces a pump light at 426 nm for the OPO and it is frequency locked to the incident laser light. The doubling cavity contains a Potassium Niobate (KNbO3) nonlinear crystal. The OPO cavity is a four-mirror folded cavity with two curved and two plain mirrors. The non-linear medium at the waist between curved mirrors is a-cut KNbO3 crystal. The spontaneous parametric fluorescence, produced by degenerate down conversion into a sub-harmonic mode of the OPO, generates the squeezed vacuum. In a degenerate OPO, the pump field at 2x is split by a non-linear crystal into two photons of frequencies x each. The average values of the photon annihilation and creation operators in the squeezed vacuum are hak i ¼ haþ k i ¼ 0 [6] and that of second order photon annihilation and creation operai tors are haþ k ak i ¼ N ðxÞ; hak ak i ¼ MðxÞe /, here N(x) corresponds to the photon number, M(x) is the two-photon correlation function and / is the squeezed field phase. The photon number and two-photon correlation functions are not independent of each other. They satisfy the inequality for the quantum squeezed field [7], N 2 < M 2 ðxÞ 6 N ðxÞ½N ðxÞ þ 1:
ð1aÞ
Whereas in the case of the classically squeezed field N and M satisfy the inequality M 6 N. The factor (N + 1) instead of N in the above equation arises due to the quantum nature of the field. In the broadband case M(x) = gN(x)[N(x) + 1] with 0 6 g 6 1. Here g is the degree of correlation. In this paper we shall use the quantum squeezed field which satisfies the inequality (1a). For an ideal degenerate OPO [8], we have N ðxÞ ¼ ðk2 l2 Þ½1=ðx2 þ l2 Þ 1=ðx2 k2 Þ=4; 2
2
2
2
2
2
MðxÞ ¼ ðk l Þ½1=ðx þ l Þ þ 1=ðx k Þ=4; and
ð1bÞ ð1cÞ
2
MðxÞ ¼ N ðxÞ½N ðxÞ þ 1;
125
ð1dÞ
where k = j/2 + e, l = j/2 e, with j and e being the cavity decay constant and the amplification factor, respectively, for OPO. The values of N and M functions may be selected by the OPO parameters. The OPO generates an ideal squeezed vacuum with N = sinh2 r, and jMj = cosh r sinh r, here r is the squeeze factor. The signal and idler outputs of the OPO may be combined at a 50/50 beam-splitter to produce a squeezed-vacuum state and a squeezed-coherent state [9]. In the conventional laser cooling set-up for the fountain atomic clock, the Cs atoms are cooled and trapped by an optical molasses formed by three mutually perpendicular standing-wave laser beams. The cooling force due to the optical molasses has been calculated by several groups and is well documented [10]. The cooling force on the slowly moving atoms with the density matrix q, in a standing-wave squeezed-coherent field [11,12] is given by F ¼ ihqr ðX hq12 i Xhq21 iÞ ¼ hqr XhrY isc ;
ð2Þ
here qr = k tan(kx), k = 2p/k, k = 852 nm, and X is the Rabi frequency of the optical field. hrYisc, the y-component of the atomic Block vector in the presence of the squeezedcoherent light is given by hrY isc ¼ Xðd þ cjMj sin /Þ=2Dsc ;
ð3aÞ
where d is the laser frequency detuning from the resonance, c is the normal atomic decay rate, / is the relative phase of the driving laser field with respect to the squeezed-coherent light, and 1 Dsc ¼ n c2 n2 þ d2 c2 jMj2 þ X2 ðn=2 þ jMj cos /Þ; 4 ð3bÞ and n = 1 + 2N. The force given by Eq. (2) is calculated assuming kv < c and d < c, where v is the velocity of the atoms. In the above calculations for the cooling force, the squeezed light may couple in free space to atoms in a small fraction of 4p solid angle. To overcome this problem Gardiner [6] suggested coupling squeezed light modes to the atom in an optical cavity, which is basically small in dimensions. We may consider bad cavity scenario in which the Bloch equations are effectively similar to those in the free space with reduced value of strength of two-photon correlation. In the bad cavity scenario j c, g with g > c. Here g, j and c are the measure of the atom – cavity coupling, cavity and atomic decay rates, respectively. The bad cavity environment modifies the free space values of N and M to Nc and Mc, respectively with Nc = BN and Mc = BM [13]. Here B is generally referred to as the beta factor of the cavity and it may be defined as ratio of spontaneous decay rate into cavity mode to the total spontaneous decay rate. B is related to C = g2/jc, the single atom cooperative parameter for an atom. We have B = C/(1 + C) and as B < 1 for positive finite values of C, the free space values of N and M in the bad cavity environment
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
are reduced. The lowering of N and M values results from the fact that in the bad cavity environment the atom is always subjected from the open side of the cavity to the unsqueezed field modes. This exposure to unsqueezed field leads to smaller value of strength of two-photon correlations. The decay rate c is also modified to cC = c(C + 1) in the cavity environment. For very large values of C we have B = 1 and in this case the bad cavity environment may be approximated by the free space. For experimental purposes modified N, M and c values may be used depending on the bad cavity parameters. However, for the sake of generality we will consider the free space atom-squeezed light interaction. This treatment may be adapted for any laser cooling experimental setup in the bad cavity environment by choosing the modified values of N, M and c. We observe from Eqs. (2) and (3a) that the magnitude of the cooling force basically depends on (d + cjMj sin /) and there is non-vanishing cooling force even at zero detuning (d = 0). The term cjMj sin / in the expression for cooling force arises due to the interaction of atoms with squeezedcoherent light – a quantum field. From this term it follows that the squeezed light enhances the cooling force. The twophoton correlation factor M depends on the degree of squeezing. The decay rate c is also an important parameter in determining the cooling force. Several authors have shown that the squeezed states might be used for manipulating the atomic spontaneous decay rate [8,9]. As discussed in the paper, for increased decay rate a larger cooling force may be realized. On interacting with the squeezed vacuum, the atoms de-excite into either of the two quadratures with different decay rates. The atoms in the squeezed (in-phase) quadrature relax at a slower rate while those in the noisy (out-of-phase) quadrature decay at faster relaxation rate compared to normal vacuum. Specifically, when the relative phase U between the coherent driving field and the squeezed vacuum is 0, the atoms decay into the out-of-phase or the noisy quadrature with the larger decay rate,
the faster decay rate from Eq. (4a) in (2), the cooling force may be expressed as, F sv ¼ qr XhrY isvsc :
hrY isvsc ¼ 1=2b½D þ 2jMjðn=2 þ jMjÞ sin /=D; 2
ð6Þ
2
ð7Þ
In the above expressions, the normalized detuning D and Rabi frequency b are defined as D d/c and b X/c, respectively. It may be readily shown that with the modified y-component of the atomic Block vector hrYisvsc, the cooling force is increased depending on the phase / and squeeze parameter r. The cooling force is calculated for the different values of these parameters. The relevant parameter quantifying the degree of squeezing is given by the factor 2(jMj N). Georgiades et al. [14] have developed a source of squeezed light exhibiting approximately 75% squeezing. This corresponds to N = 9/16, (a) 6 5 4 3 2 1 0 3
2
1
0
6
5
4
9
8
7
10
β 6
ð4aÞ
ð4bÞ
In general, the squeezed states are used for reducing the quantum noise or the decay rate of the atoms. However, in this paper, we demonstrate that even the noisy quadrature could be gainfully utilized in realizing higher cooling force than that with either laser light or quieter quadrature of the squeezed light. Let us consider that the atoms in the optical molasses, formed by the squeezed-coherent light of phase /, decay with the faster rate c0 into the noisy quadrature (U = 0) of the squeezed vacuum. We calculate the cooling force on the atoms in the combined presence of the coherent squeezed state and squeezed vacuum. On incorporating
C oo lin g For ce ( γ k / 2)
5
When the relative phase U between the coherent driving field and the squeezed vacuum is p atoms decay at slower rate, cp ¼ cðN þ 1=2 MÞ:
2
2
D ¼ n=2 ½b þ 2D þ ðn=2 þ jMjÞ þ b jMj cos /:
(b)
c0 ¼ cðN þ 1=2 þ MÞ:
ð5Þ
Here hrYisvsc, the y-component of the atomic Bloch vector in the combined presence of the coherent squeezed state and the squeezed vacuum, with some mathematical simplifications, is given by the expression,
Cooling Force ( γ k / 2)
126
4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
β
Fig. 1. The dependence of spatially averaged force on the normalized Rabi frequency b of the driving field in the presence (solid and dotted line) and absence (dashed line) of squeezed vacuum. The solid line is for U = 0 or p (out-of-the phase) and dotted line is for U = p/2 (in-phase quadrature). Here degree of squeezing is chosen to be 75% and / = 0.8p. Plot (a) is for the resonant case D = 0, and (b) for the off-resonant case D = 0.1. Cooling force hck/2 = 1 is the Doppler limit.
G.M. Saxena, A. Agarwal / Optics Communications 267 (2006) 124–127
jMj = 15/16 and r 0.7. With this kind of source of squeezed light one can efficiently cool and trap atoms. We have calculated the cooling force for the squeezed light and for the combination of squeezed vacuum and squeezed coherent light, and denoted them as F and Fsv, respectively. We compare the cooling force F and Fsv in Figs. 1 and 2 for different cases. As shown in Fig. 1, for small values of normalized Rabi frequency b, F (shown by dashed line) is greater than Fsv (shown by solid curve). For b > 3, this trend reverses. There is a steep rise in Fsv compared to F for the increasing value of b. Fig. 1(a) is a plot for the resonant case. We find that the squeezed light can give rise to a cooling force even at zero detuning. In Fig. 1(b), we plot the off-resonance case. This shows similar behavior as in Fig. 1(a) except that the cooling force increases slightly. For squeezed (in-phase) quadrature instead of the noisy quadrature, i.e., (a) 8 7
5
( γ k / 2)
Cooling force
6
4 3 2 1.0 0.8 0.5
1 0 0
φ/π
0.3 20
40
60
Degree of Squeezing (%)
0.0 80
for the phase U = p/2, the cooling force (shown by dotted line in Fig. 1) is much smaller than F (shown by dashed line). This shows the utility of the noisy quadrature over the inphase quadrature of the squeezed light for efficient cooling of the atoms. In Fig. 2, we show that the relative phase / and the degree of squeezing play significant role in enhancing the laser cooling force in the noisy quadrature of the squeezed light. Figs. 2(a) and (b) are for the resonant and off-resonant squeezed light, respectively. These figures show that the cooling force is sensitive to small changes in the relative phase between fields, the detuning from the resonance, value of Rabi frequency and degree of squeezing. 3. Conclusion We have established in this paper that the noisy quadrature of the squeezed vacuum in the presence of coherent squeezed state could be gainfully utilized in realizing higher cooling force than that with either laser light or with the squeezed vacuum. Besides, for some values of the phase of the coherent squeezed field with respect to the laser field and for the appropriate value of two-photon correlation M, the atoms may experience large cooling force even for the resonant squeezed light. It is very unexpected result because the noisy quadrature is generally unused part of the squeezed light. However, there may be increase in the momentum diffusion in noisy quadrature. This enhancement in momentum diffusion [4] may be more than offset and counteracted by squeezing at higher laser intensities. As the squeezed field at higher intensities increases the cooling force near atomic resonance for standing wave field due to accelerated cooling process without changing the atom – squeezed field interaction. References
(b) 8 7 6
Cooling force ( γ k / 2)
127
5 4 3 2 0.9
1
0.6
0 0
0.3 20
40
Degree of Squeezing (%)
60
φ/π
0.0 80
Fig. 2. A three-dimensional plot of spatially averaged cooling force Fsv as a function of the degree of squeezing and relative phase / for (a) the resonant case D = 0, and (b) the off-resonant case D = 0.1. Normalized Rabi frequency b = 10.
[1] D. Wineland, W. Itano, Phys. Rev. A 20 (1979) 1521; S. Chu, C. Wieman, J. Opt. Soc. Am. B 6 (1989) 1961; J. Dalibard, C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6 (1989) 2023. [2] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, C. CohenTannoudji, J. Opt. Soc. Am. B 6 (1989) 2112. [3] Y. Shevy, Phys. Rev. Lett. 64 (1990) 2905. [4] R. Graham, D.F. Walls, W.P. Zhang, Phys. Rev. A 44 (1991) 7777. [5] E.S. Polzik, J. Carri, H.J. Kimble, Appl. Phys. B 55 (1992) 279. [6] C.W. Gardiner, Phys. Rev. Lett. 56 (1986) 1917; A.S. Parkins, C.W. Gardiner, Phys. Rev. A 40 (1990) 3796. [7] H.J. Carmichael, A.S. Lane, D.F. Walls, Phys. Rev. Lett. 58 (1997) 2539; H.J. Carmichael, A.S. Lane, D.F. Walls, J. Mod. Opt. 34 (1987) 821. [8] H. Ritsh, P. Zoller, Opt. Commun. 64 (1987) 523; H. Ritsh, P. Zoller, Phys. Rev. A 38 (1988) 4657. [9] C. Kim, P. Kumar, Phys. Rev. Lett. 73 (1994) 1605. [10] H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer Verlag, New York, 1999. [11] B.J. Dalton, Z. Ficek, S. Swain, J. Mod. Opt. 46 (1999) 379. [12] Z. Ficek, W.S. Smyth, S. Swain, Phys. Rev. A 52 (1995) 4126. [13] W.S. Smyth, S. Swain, Phys. Rev. A 53 (1996) 2846; P. Zhou, S. Swain, Opt. Commun. 131 (1996) 153. [14] N.P. Georgiades, E.S. Polzik, K. Edamatsu, H.J. Kimble, A.S. Parkins, Phys. Rev. Lett. 75 (1995) 3426.