May 17, 2006 - ... P. O. Box 2455, Riyadh 11451, Saudi Arabia. 3Mathematics Department, College of Science, University of Bahrain, P. O. Box 32038, Bahrain.
PHYSICAL REVIEW A 73, 053817 共2006兲
Spin squeezing and entanglement in a dispersive cavity R. N. Deb,1 M. Sebawe Abdalla,2 S. S. Hassan,3 and N. Nayak4
1
Physics Department, Darjeeling Government College, Darjeeling, India Mathematics Department, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia 3 Mathematics Department, College of Science, University of Bahrain, P. O. Box 32038, Bahrain 4 S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-3, Salt Lake City, Kolkata-700098, India 共Received 2 March 2006; published 17 May 2006兲
2
We consider a system of N two-level atoms 共spins兲 interacting with the radiation field in a dispersive but high-Q cavity. Under an adiabatic condition, the interaction Hamiltonian reduces to a function of spin operators which is capable of producing spin squeezing. For a bipartite system 共N = 2兲, the expressions for spin squeezing get very simple, giving a clear indication of close to 100% noise reduction. We analyse this squeezing as a measure of bipartite entanglement. DOI: 10.1103/PhysRevA.73.053817
PACS number共s兲: 42.50.Dv, 03.67.Mn, 03.65.Ud
Spin squeezing has been a subject of interesting studies in the past few years 关1–7兴. Recently, its importance has increased further due to its relation to quantum entanglement 关8–12兴, a basic ingradient of quantum information 关13兴. In addition, spin squeezing has been found to be a useful parameter to quantify entanglement. Various proposals have been made to measure entanglement 关14兴, but spin squeezing is being increasingly preferred since it is a physically measurable quantity defined by simple spin operators. We know that two-level atoms can be described by the Pauli spin algebra with the exception that the z component of the spin represents the population difference between the two levels of the atoms. Thus we take the liberty of referring to a two level system as atoms or spins. The spin squeezing is defined in terms of these operators as follows. Let Sx, Sy, and Sz be the usual collective spin operators for a system of N twolevel atoms. We define a mean spin vector which has the magnitude 兩具S典 兩 = 冑具Sx典2 + 具Sy典2 + 具Sz典2. Then, in a plane normal to 具S典, the uncertainty relation between the pair of mutually orthogonal spin components Sx⬘ and Sy⬘, say, satisfy 关6兴 1 ⌬Sx⬘⌬Sy⬘ 艌 兩具S典兩 2
Hef f = ⌬0S+S− + ⌬1Sz = ⌬0共S2 − Sz2兲 + 共⌬0 + ⌬1兲Sz
共2兲
where
共1兲
with 具Sx⬘典 = 具Sy⬘典 = 0. For a coherent state 关15兴, the equality sign is satisfied with ⌬Sx⬘ = ⌬Sy⬘ = 冑兩具S典 兩 / 2. It may be noted here that 具S典 is in the z⬘ direction and the frames of reference 兵x , y , z其 and 兵x⬘ , y ⬘ , z⬘其 are related by the usual polar and azimuthal angles. In this situation, if one of the components is less than 冑兩具S典 兩 / 2 at the expense of the other such that Eq. 共1兲 is satisfied, then we say that the system of N two-level atoms is spin squeezed. Cavity QED and cavity-related systems 关16,17兴 have been widely studied for obtaining correlated particles, photons or two-level atoms or even both. Agarwal et al. 关17兴 have studied mesoscopic superposition of spin states, the so-called Schrödinger cat states, in a dispersive cavity. These states are a result of existence of nondiagonal elements of the atomic density matrix. Hence there are possibilities that the correlations among the spin components 具SiS j + S jSi典, i , j ⬅ x , y , z, 1050-2947/2006/73共5兲/053817共4兲
are nonzero for i ⫽ j. These correlations have been found to be responsible for spin squeezing. It is our endeavor here to study spin squeezing and its interesting quantum entanglement properties in such a system consisting of N two-level atoms 共spins兲 interacting with the single mode of a cavity having high quality factor and at thermal equilibrium. The cavity temperature is represented by the average number of thermal photons ¯nth present in the cavity. The photon lifetime is represented by 共2兲−1 where is the cavity bandwidth. We consider a highly detuned cavity, that is, ␦ = c − a, c and a being the cavity mode and atomic transition frequencies, respectively, is very large such that the condition 兩i␦ + 兩 Ⰷ g冑N 共where g is the atom-photon coupling constant兲 is satisfied. We further assume that the quality factor Q of the cavity is very high making very small, but we treat the cavity as a dispersive type. Such a system easily satisfies ␦ Ⰷ . The effective Hamiltonian describing the dynamics of the atoms in the cavity takes the form 关17兴
⌬0 =
g 2␦ 2 + ␦2
共3兲
¯ th⌬0. We notice that the cavity damping influand ⌬1 = 2n ences the atomic dynamics through the scaled coupling constant ⌬0. The cavity temperature appears in the terms linear in Sz which produce only a rotation in the atomic system. Hence, the atomic dynamics is independent of cavity temperatures. This is reminiscent of the so-called secular approximation in cavity QED, such as g Ⰷ , where the thermal photons are usually neglected 关16兴. Hence, the Hamiltonian takes the form Hef f = ⌬0共S2 − Sz2兲.
共4兲
It is interesting to note that Hef f is a special case of the so-called Lipkin-Meshkov-Glick 共LMG兲 Hamiltonian for a many-body fermionic system 关18兴. The LMG interaction in various other forms has been utilized to study entanglement 关8兴 and has also been successfully engaged in experiments
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PHYSICAL REVIEW A 73, 053817 共2006兲
DEB et al.
for its generation 关9兴. Kitagawa and Ueda 关1兴 have also considered a similar Hamiltonian to establish the phenomenon of spin squeezing. In this paper, in addition to probing deeper into the spin squeezing dynamics of Hef f , we report that, when the atomic system is in a bipartite state 共N = 2兲, the time evolution of the interaction has quite interesting properties. We show below that the bipartite system is almost 100% spin squeezed. Further, the interaction is also capable of producing almost perfectly entangled bipartite spin states. We find that these properties are strikingly different compared to their multipartite counterparts. We assume that the atoms, before they enter the cavity, are put in a coherent state 关15兴 s
兩, 典 =
1 兺 冑2sCs+ms+m兩s,m典 共1 + 兩兩2兲s m=−s
共5兲
具S␣2 共t兲典 =
⫻cos 2关共2s − 1兲⌬0t − 共s − 1兲⌰2共t兲兴其
1 具SxSy + SySx典 = 共2s − 1兲s sin2 关⌳2共t兲兴共s−1兲 2 ⫻sin 2关共2s − 1兲⌬0t − 共s − 1兲⌰2共t兲兴, 共13兲
具SxSz + SzSx典 = 共2s − 1兲s sin 兵⌳1共t兲其共s−1兲
共6兲
s sin2 . 2
具Sx共t兲典 = 关⌳1共t兲兴
− cos2
cos 2关s⌬0t − 共s − 1兲⌰1共t兲兴 , 2
冊
共14兲 and
s cos关2s⌬0t − 共2s − 1兲⌰1共t兲兴sin , 共8兲
具SySz + SzSy典 = 共2s − 1兲s sin 兵⌳1共t兲其共s−1兲
共9兲 where ⌳1共t兲 and ⌰1共t兲 are given by
1 + sin4 + sin2 cos 2j⌬0t 2 2 2
sin兵2共s − 1兲关⌬0t − ⌰1共t兲兴其 2
− cos2
sin 2关s⌬0t − 共s − 1兲⌰1共t兲兴 . 2
sin2
共11兲
respectively, with j = 1. The averages of their squares are given by
冊
共15兲
共10兲
and
sin 2j⌬0t 2 , tan ⌰ j共t兲 = 2 2 cos 2j⌬0t cos + sin 2 2
冉
⫻ sin2
具Sy共t兲典 = 关⌳1共t兲兴共s−1/2兲s sin关2s⌬0t − 共2s − 1兲⌰1共t兲兴sin ,
⌳ j共t兲 = cos4
cos兵2共s − 1兲关⌬0t − ⌰1共t兲兴其 2
共7兲
The other components of the spin system are 共s−1/2兲
冉
⫻ sin2
and 具Sz2典 = s2cos2 +
共12兲
where the upper and the lower signs are for ␣ ⬅ x and y, respectively. ⌳2共t兲 and ⌰2共t兲 appearing in the above equations are given by, again, Eqs. 共10兲 and 共11兲, respectively, but with j = 2. To evaluate spin squeezing we now rotate the frame 兵x , y , z其 to a new frame 兵x⬘ , y ⬘ , z⬘其 so that one of the new axes is aligned along the mean spin vector 具S典. Our analysis involves z⬘ along 具S典. In this frame, 具Sx⬘典 = 具Sy⬘典 = 0, but 具Sx2⬘典 and 具S2y 典 involve, in addition to 具S2x 典 and 具S2y 典, the following ⬘ correlation functions:
where = tan共 / 2兲ei and s = N / 2. = 0 represents all the atoms in the lower state 兩s , −s典. In the following we take = 0 as it has no effect on the problem we study here. The state 兩 , 0典 can be achieved by, first, sending the atoms in their upper states through an auxiliary cavity. The angle is decided by the duration of atom-field interaction in this cavity. These atoms then enter the cavity where their evolution is governed by the Hamiltonian in Eq. 共4兲. The structure of the Hamiltonian makes Sz and Sz2 constants of motion which, after averaging over the initial condition, take values given by 具Sz典 = − s cos
s 1 + s共2s − 1兲sin2 兵1 ± 关⌳2共t兲兴共s−1兲 2 4
These correlation functions are at the root of spin squeezing 关1–6兴. We notice that all the correlation functions are 0 for a single two-level atom, that is, s = 1 / 2. Thus spin squeezing cannot be obtained from a single atom. We now look for squeezing for s 艌 1. Before we do so, we notice that the dynamics for the duration ⌬0t = 0 to repeats itself during the time to 2 and so on. Hence, it is sufficient to limit the dynamics to 0 ⬍ ⌬0t ⬍ . It can be shown easily that there is no spin squeezing exactly at ⌬0t = . The variance in the y ⬘ quadrature has the form
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SPIN SQUEEZING AND ENTANGLEMENT IN A¼
creases. For = / 2, that is, equal population in the upper and lower levels, we have 具Sz典 = 0. This reduces the variance in the x⬘ quadrature to 2
具⌬Sx⬘典 = 具Sz2典 = s/2
共19兲
where we have used Eq. 共7兲. Since 兩具S典兩 = s cos共2s−1兲 ⌬0t
共20兲
we have 2
2具⌬Sx⬘典 兩具S典兩
冑
2
FIG. 1. Degree of squeezing X = 关2具⌬Sx 典兴 / 兩具S兩 as a function of ⬘ s 艌 2 for = / 16. The dimensionless interaction time ⌬0t = / 2, 9 / 16, and 5 / 16 for the full, broken, and dotted curves, respectively. Note that X ⬍ 1 indicates squeezing. 2
具⌬Sy⬘典 =
s s + 共2s − 1兲sin2 2 4
冋 冉
⫻ 1− 1−
1 sin2 共1 − cos 4⌬0t兲 2
冊
s−1
册
⫻cos 2关共2s − 1兲⌰1共t兲 − 共s − 1兲⌰2共t兲 − ⌬0t兴 . 共16兲 We notice in the above expression that 具⌬S2y 典 ⬎ s / 2 and, ⬘ since the length of the mean spin vector 兩具S典兩 = s冑关⌳1共t兲兴共2s−1兲sin2 + cos2
共17兲
兩具S典兩
⬎1
共2s−1兲
cos
⌬ 0t
⬎ 1.
共21兲
Hence, the system does not exhibit any squeezing for = / 2. This is in contrast to the results in Ref. 关1兴 as squeezing in the y ⬘z⬘ plane was studied there, whereas our study is in the x⬘y ⬘ plane. It may be noted here that for = ⌬0t = / 2, squeezing is not defined since 兩具S典 兩 = 0. We study the bipartite system 共s = 1兲 in detail below as it exhibits interesting properties. In this case, Eqs. 共12兲–共15兲 take a simpler form because of the appearance of s − 1 in the exponents and give a closer view of the noise quenching process. We set ⌬0t = / 2 since the lowest noise level is obtained at this interaction time for all values of s. The spin components are further simplified to 具Sx共t兲典 = −共sin 2兲 / 2, 具Sy共t兲典 = 0, and 具Sz典 = −cos . The averages of their squares reduce to 具S2x 共t兲典 = 1 / 2, 具S2y 共t兲典 = 共1 + sin2 兲 / 2, and 具Sz2共t兲典 = 1 − 共sin2 兲 / 2. But the differences between this system and the system with s ⬎ 1 stem from the correlation function, at ⌬0t = / 2, 具SxSz + SzSx典 = 共− 1兲s−1s共2s − 1兲sin cos2共s−1兲
共18兲
is always satisfied, that is, the y ⬘ component is never squeezed. However, we find noise reduction in the other quadrature. Let us consider small excitation to the upper level, that is, small. We find that for = / 16, representing about 2% of the atoms in their upper states, there is about 21% squeezing when the interaction time is set at ⌬0t = / 2. We notice in Fig. 1 that an atomic system with about 140 atoms can be squeezed at this interaction time. The maximum squeezing takes place at N = 26. The degree of squeezing and the number of atoms that can be squeezed are reduced as we deviate from this interaction time. Figure 1 shows that for ⌬0t = 9 / 16, the squeezing reduces to 15% and for ⌬0t = 5 / 16, it reduces to as low as 10%. The squeezing further reduces as we deviate further from ⌬0t = / 2 and it ultimately vanishes at ⌬0t = as mentioned earlier. The degree of squeezing also deteriorates as we increase the number of atoms in their upper states, that is, as in-
共22兲
which for s = 1 reduces to 具SxSz + SzSx典 = sin .
has a maximum values of s, the condition 2 2具⌬Sy⬘典
1
=
共23兲
The other two correlation functions are 0 anyway at ⌬0t = / 2. Equation 共22兲 clearly indicates the squeezing characteristics for s 艌 2 in Fig. 1. It also points to reduction in squeezing as we increase to / 2. But the variation in squeezing with is quite different for a system with s = 1. Equation 共23兲 shows that spin squeezing should increase as increases to / 2. In fact, we have
冑
2
X = 关2具⌬Sx⬘典兴/兩具S典兩 =
冉冑
cos
1 + sin2
冊
1/2
共24兲
for 0 ⬍ ⬍ / 2. We note that squeezing increases with and it approaches 100% as approaches / 2. Thus, this variation is just in the opposite direction to its behavior in multipartite systems 共s ⬎ 1兲. However, it may be noted from Eq. 共17兲 that 兩具S典 兩 = 0 at = ⌬0t = / 2, that is, squeezing is not defined at these values. In the next quadrant, / 2 ⬍ ⬍ , there is no squeezing since X=
冉冑
− cos
1 + sin2
冊
1/2
.
共25兲
We now use the spin squeezing condition in Eq. 共24兲 to show that the system possesses bipartite entanglement
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Thus we observe that our spin squeezed bipartite system is also entangled when the above conditions for the interaction time and atomic excitation are satisfied. We further notice that states satisfying Eq. 共25兲 also satisfy the entanglement condition in Eq. 共26兲. This means that the bipartite entangled system need not be spin squeezed. However, this situation
arises only for values of close to / 2 such as = 9 / 16 beyond which the entanglement condition is not satisfied. In summary, we have shown that atoms initially in a coherent state get squeezed if they are allowed to interact with a cavity field in thermal equilibrium. A comparison with the results in Ref. 关1兴 shows that it all depends in which plane or component the squeezing of noise is required. But our study has shown that a bipartite system can have almost 100% squeezing. This is a key result in our paper. We have also shown that the system is capable of producing almost perfect bipartite entangled states and so it can be applied to quantum-informatic problems 关19兴. One of us 共M.S.A. is grateful for the financial support from the project Math 2005/32 of the Research Center, College of Science, King Saud University.
关1兴 M. Kitagawa and M. Ueda, Phys. Rev. Lett. 67, 1852 共1991兲; Phys. Rev. A 47, 5138 共1993兲. 关2兴 D. J. Wineland, J. J. Bollinger, W. M. Itano, F. L. Moore, and D. J. Heinzen, Phys. Rev. A 46, R6797 共1992兲; D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, ibid. 50, 67 共1994兲. 关3兴 G. S. Agarwal and R. R. Puri, Phys. Rev. A 49, 4968 共1994兲. 关4兴 L. Vernac, M. Pinard, and E. Giacobino, Phys. Rev. A 62, 063812 共2000兲; A. Dantan, M. Pinard, V. Josse, N. Nayak, and P. R. Berman, ibid. 67, 045801 共2003兲; A. Dantan and M. Pinard, ibid. 69, 043810 共2004兲. 关5兴 C. Genes, P. R. Berman, and A. G. Rojo, Phys. Rev. A 68, 043809 共2003兲. 关6兴 R. N. Deb, N. Nayak, and B. Dutta-Roy, Eur. Phys. J. D 33, 149 共2005兲; N. Nayak, R. N. Deb, and B. Dutta-Roy, J. Opt. B: Quantum Semiclassical Opt. 7, S761 共2005兲. 关7兴 See, for example, J. Hald, J. L. Sørensen, C. Schori, and E. S. Polzik, Phys. Rev. Lett. 83, 1319 共1999兲; A. Kuzmich, L. Mandel, and N. P. Bigelow, ibid. 85, 1594 共2000兲; V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, ibid. 86, 5870 共2001兲. 关8兴 K. Mølmer and A. Sørensen, Phys. Rev. Lett. 82, 1835 共1999兲; A. Sørensen and K. Mølmer, ibid. 82, 1971 共1999兲; R. G. Unanyan and M. Fleischhauer, ibid. 90, 133601 共2003兲.
关9兴 C. A. Sackett, Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, Nature 共London兲 404, 256 共2000兲. 关10兴 A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 共London兲 409, 63 共2001兲. 关11兴 X. Wang and B. C. Sanders, Phys. Rev. A 68, 012101 共2003兲. 关12兴 J. K. Korbicz, J. I. Cirac, and M. Lewenstein, Phys. Rev. Lett. 95, 120502 共2005兲. 关13兴 See, for example,M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information 共Cambridge University Press, Cambridge, U.K., 2000兲. 关14兴 For a review, see M. Horodecki, P. Horodecki, and R. Horodecki, e-print quant-ph/0110032. 关15兴 J. M. Radcliffe, J. Phys. A 4, 313 共1971兲; F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 共1972兲. 关16兴 For a review, see J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73, 565 共2001兲; also see E. Solano, G. S. Agarwal, and H. Walther, Phys. Rev. Lett. 90, 027903 共2003兲. 关17兴 G. S. Agarwal, R. R. Puri, and R. P. Singh, Phys. Rev. A 56, 2249 共1997兲. 关18兴 H. J. Lipkin, N. Meshkov, and A. J. Glick, Nucl. Phys. 62, 188 共1965兲. 关19兴 A. Gábris and G. S. Agarwal, Phys. Rev. A 71, 052316 共2005兲.
关11,12兴. Following Korbicz et al. 关12兴, states with s = 1 are entangled if the condition 2
2具⌬Sx⬘典 = X2兩具S典兩 ⬍ 1
共26兲
is satisfied since we have 具Sx⬘典 = 0. We have seen in the above that X ⬍ 1 is satisfied for ⌬0t = / 2 and 0 ⬍ ⬍ / 2. In this situation we have from Eq. 共17兲 兩具S典兩 = 冑1 − sin4 ⬍ 1.
共27兲
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