[en] We derive a Markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and applies equally well to distinguishable and indistinguishable atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find closed-form formulas for a number of relevant states (Gaussian states, Fock states, and thermal states). In particular, we show that dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion.
Disciplines :
Physics
Author, co-author :
Damanet, François ; Université de Liège > Département de physique > Optique quantique
Braun, Daniel; Institut für theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany
Martin, John ; Université de Liège > Département de physique > Optique quantique
Language :
English
Title :
Master equation for collective spontaneous emission with quantized atomic motion
Publication date :
February 2016
Journal title :
Physical Review. A, Atomic, molecular, and optical physics
A. Einstein, Verhandlungen der Deutschen Physikalischen Gesellschaft 18, 318 (1916).
V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930). EPJAFV 1434-6001 10.1007/BF01336768
E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69, 37 (1946). PHRVAO 0031-899X 10.1103/PhysRev.69.37
V. P. Bykov, Sov. Phys. JETP 35, 269 (1972).
E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). PRLTAO 0031-9007 10.1103/PhysRevLett.58.2059
S. John, Phys. Rev. Lett. 58, 2486 (1987). PRLTAO 0031-9007 10.1103/PhysRevLett.58.2486
R. H. Dicke, Phys. Rev. 93, 99 (1954). PHRVAO 0031-899X 10.1103/PhysRev.93.99
R. Bonifacio, P. Schwendiman, and F. Haake, Phys. Rev. A 4, 302 (1971). 0556-2791 10.1103/PhysRevA.4.302
R. Bonifacio, P. Schwendiman, and F. Haake, Phys. Rev. A 4, 854 (1971). 0556-2791 10.1103/PhysRevA.4.854
L. M. Narducci, C. A. Coulter, and C. M. Bowden, Phys. Rev. A 9, 829 (1974). 0556-2791 10.1103/PhysRevA.9.829
R. J. Glauber and F. Haake, Phys. Rev. A 13, 357 (1976). 0556-2791 10.1103/PhysRevA.13.357
M. Gross and S. Haroche, Phys. Rep. 93, 301 (1982). PRPLCM 0370-1573 10.1016/0370-1573(82)90102-8
G. Nienhuis and F. Schuller, J. Phys. B: At. Mol. Phys. 20, 23 (1987). JPAMA4 0022-3700 10.1088/0022-3700/20/1/008
R. G. DeVoe and R. G. Brewer, Phys. Rev. Lett. 76, 2049 (1996). PRLTAO 0031-9007 10.1103/PhysRevLett.76.2049
P. A. Braun, D. Braun, F. Haake, and J. Weber, Eur. Phys. J. D 2, 165 (1998). EPJDF6 1434-6060 10.1007/s100530050126
P. A. Braun, D. Braun, and F. Haake, Eur. Phys. J. D 3, 1 (1998). EPJDF6 1434-6060 10.1007/s100530050142
Y. N. Chen, C. M. Li, D. S. Chuu, and T. Brandes, New J. Phys. 7, 172 (2005). NJOPFM 1367-2630 10.1088/1367-2630/7/1/172
E. Akkermans, A. Gero, and R. Kaiser, Phys. Rev. Lett. 101, 103602 (2008). PRLTAO 0031-9007 10.1103/PhysRevLett.101.103602
D. Braun, J. Hoffman, and E. Tiesinga, Phys. Rev. A 83, 062305 (2011). PLRAAN 1050-2947 10.1103/PhysRevA.83.062305
R. Wiegner, J. von Zanthier, and G. S. Agarwal, Phys. Rev. A 84, 023805 (2011). PLRAAN 1050-2947 10.1103/PhysRevA.84.023805
S. Oppel, R. Wiegner, G. S. Agarwal, and J. von Zanthier, Phys. Rev. Lett. 113, 263606 (2014). PRLTAO 0031-9007 10.1103/PhysRevLett.113.263606
R. Wiegner, S. Oppel, D. Bhatti, J. von Zanthier, and G. S. Agarwal, Phys. Rev. A 92, 033832 (2015). PLRAAN 1050-2947 10.1103/PhysRevA.92.033832
D. Bhatti, J. von Zanthier, and G. S. Agarwal, Sci. Rep. 5, 17335 (2015). 2045-2322 10.1038/srep17335
S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, Science 285, 571 (1999). SCIEAS 0036-8075 10.1126/science.285.5427.571
B. Coffey and R. Friedberg, Phys. Rev. A 17, 1033 (1978). 0556-2791 10.1103/PhysRevA.17.1033
F. Friedberg and S. R. Hartmann, Phys. Rev. A 10, 1728 (1974). 0556-2791 10.1103/PhysRevA.10.1728
H. S. Freedhoff, J. Phys. B: At. Mol. Phys. 19, 3035 (1986). JPAMA4 0022-3700 10.1088/0022-3700/19/19/017
H. S. Freedhoff, J. Phys. B: At. Mol. Phys. 20, 285 (1987). JPAMA4 0022-3700 10.1088/0022-3700/20/2/012
W. Feng, Y. Li, and S.-Y. Zhu, Phys. Rev. A 88, 033856 (2013). PLRAAN 1050-2947 10.1103/PhysRevA.88.033856
R. Friedberg, S. R. Hartmann, and J. T. Manassah, Phys. Lett. A 40, 365 (1972). PYLAAG 0375-9601 10.1016/0375-9601(72)90533-6
T. Bienaimé, N. Piovella, and R. Kaiser, Phys. Rev. Lett. 108, 123602 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.108.123602
M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 037901 (2001). PRLTAO 0031-9007 10.1103/PhysRevLett.87.037901
R. M. Sandner, M. Müller, A. J. Daley, and P. Zoller, Phys. Rev. A 84, 043825 (2011). PLRAAN 1050-2947 10.1103/PhysRevA.84.043825
P. Debye, Ann. Phys. 348, 49 (1913). ANPYA2 0003-3804 10.1002/andp.19133480105
I. Waller, Z. Phys. A 17, 398 (1923). 1434-6001 10.1007/BF01328696
J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). PRLTAO 0031-9007 10.1103/PhysRevLett.74.4091
K. Mølmer and A. Sørensen, Phys. Rev. Lett. 82, 1835 (1999). PRLTAO 0031-9007 10.1103/PhysRevLett.82.1835
P. Lodahl, A. Floris van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, Nature (London) 430, 654 (2004). NATUAS 0028-0836 10.1038/nature02772
J. Javanainen, J. Opt. Soc. Am. B 5, 73 (1988). JOBPDE 0740-3224 10.1364/JOSAB.5.000073
R. G. Brewer, Phys. Rev. A 52, 2965 (1995). PLRAAN 1050-2947 10.1103/PhysRevA.52.2965
A. W. Vogt, J. I. Cirac, and P. Zoller, Phys. Rev. A 53, 950 (1996). PLRAAN 1050-2947 10.1103/PhysRevA.53.950
D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003). RMPHAT 0034-6861 10.1103/RevModPhys.75.281
D. Braun and J. Martin, Phys. Rev. A 77, 032102 (2008). PLRAAN 1050-2947 10.1103/PhysRevA.77.032102
H. Pichler, A. J. Daley, and P. Zoller, Phys. Rev. A 82, 063605 (2010). PLRAAN 1050-2947 10.1103/PhysRevA.82.063605
J. Cerrillo, A. Retzker, and M. B. Plenio, Phys. Rev. Lett. 104, 043003 (2010). PRLTAO 0031-9007 10.1103/PhysRevLett.104.043003
J. Yin and J. Javanainen, Phys. Rev. A 51, 3959 (1995). PLRAAN 1050-2947 10.1103/PhysRevA.51.3959
B. Dubetsky and P. R. Berman, Phys. Rev. A 53, 390 (1996). PLRAAN 1050-2947 10.1103/PhysRevA.53.390
P. R. Berman, Phys. Rev. A 55, 4466 (1997). PLRAAN 1050-2947 10.1103/PhysRevA.55.4466
G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, Phys. Rev. A 59, 3797 (1999). PLRAAN 1050-2947 10.1103/PhysRevA.59.3797
M. J. McDonnell, J. P. Home, D. M. Lucas, G. Imreh, B. C. Keitch, D. J. Szwer, N. R. Thomas, S. C. Webster, D. N. Stacey, and A. M. Steane, Phys. Rev. Lett. 98, 063603 (2007). PRLTAO 0031-9007 10.1103/PhysRevLett.98.063603
M. Roghani, and H. Helm, Phys. Rev. A 77, 043418 (2008). PLRAAN 1050-2947 10.1103/PhysRevA.77.043418
K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. D. Martin, Phys. Rev. Lett. 93, 233001 (2004). PRLTAO 0031-9007 10.1103/PhysRevLett.93.233001
F. Robicheaux, J. V. Hernández, T. Topçu, and L. D. Noordam, Phys. Rev. A 70, 042703 (2004). PLRAAN 1050-2947 10.1103/PhysRevA.70.042703
E. Altiere, D. P. Fahey, M. W. Noel, R. J. Smith, and T. J. Carroll, Phys. Rev. A 84, 053431 (2011). PLRAAN 1050-2947 10.1103/PhysRevA.84.053431
J. Pellegrino, R. Bourgain, S. Jennewein, Y. R. P. Sortais, A. Browaeys, S. D. Jenkins, and J. Ruostekoski, Phys. Rev. Lett. 113, 133602 (2014). PRLTAO 0031-9007 10.1103/PhysRevLett.113.133602
S. Y. Buhmann, Dispersion Forces, Springer Tracts In Modern Physics Vol. 248 (Springer, New York, 2012).
E. A. Power and S. Zienau, Philos. Trans. R. Soc. 251, 427 (1959). PTRMAD 1364-503X 10.1098/rsta.1959.0008
C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et atomes, Introduction à l'électrodynamique quantique (EDP Sciences, Paris, 1987).
We emphasize the quantum treatment of the atomic motion by writing the center-of-mass position and momentum operators (Equation presented) and (Equation presented), in contrast to the classical center-of-mass position and momentum which are denoted by r and p, respectively.
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2006).
D. A. Lidar, Z. Bihary, and K. B. Whaley, Chem. Phys. 268, 35 (2001). CMPHC2 0301-0104 10.1016/S0301-0104(01)00330-5
See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevA.93.022124 for technical details about the derivation of our master equation.
This choice is perfectly justified in the optical domain since optical modes of the electromagnetic field in thermal equilibrium at a temperature below 300 K are essentially in the ground state.
C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, Berlin, Heidelberg, New York, 2004).
The RWA is valid as long as the relaxation time of the system, (Equation presented), is much larger than the typical time scale (Equation presented) of its intrinsic evolution. Here the intrinsic evolution corresponds to the internal dynamics of the atoms; hence (Equation presented). We thus have (Equation presented) with (Equation presented) the fine-structure constant, (Equation presented) the Bohr radius, and (Equation presented) the wavelength of the emitted radiation. In the optical domain, this ratio is much smaller than one and the dipole approximation and RWA are entirely justified.
M. J. Stephen, J. Chem. Phys. 40, 669 (1964). JCPSA6 0021-9606 10.1063/1.1725188
G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches, Springer Tracts In Modern Physics Vol. 70 (Springer, New York, 1974), p. 1.
In this work, we use for the Fourier transform (Equation presented) and its inverse (Equation presented) the convention F k f = ∫ R3e-i k· rf(r)d r, F r-1 g =∫ R3ei k· rg(k) d k(2π)3.
The principal value arising here is a mere consequence of second-order perturbation theory since resonant intermediate transitions must be discarded [71].
M. Donaire, R. Guérout, and A. Lambrecht, Phys. Rev. Lett. 115, 033201 (2015). PRLTAO 0031-9007 10.1103/PhysRevLett.115.033201
Note that this is a consequence of the second-order perturbative treatment for the atom-field interaction. As van der Waals dipole-dipole interactions between two atoms in the same internal state result from a fourth-order perturbative development, they are not considered in this work.
Z. Heping and L. Fucheng, Chin. Phys. Lett. 12, 203 (1995). CPLEEU 0256-307X 10.1088/0256-307X/12/4/004
D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979). 0556-2791 10.1103/PhysRevA.20.1521
J. Gillis and G. Weiss, Math. Comp. 14, 60 (1960). MCMPAF 0025-5718 10.1090/S0025-5718-1960-0109902-X
M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., New York, 1970).
The generalized hypergeometric series is defined by Fq p (a; b;z)= k=0+ (a1)k...(ap)k(b1)k...(bq)k zkk! with (Equation presented) the Pochhammer symbol defined as (Equation presented) and (a)k=a(a+1)(a+2)(a+k-1)= (a+k-1)!(a-1)!.
There is no contradiction with the fact that we consider the e.m. field in vacuum. Indeed, since the motional frequency (Equation presented) is much smaller than the field frequencies (Equation presented) which contribute significantly to the dynamics, the motional and the e.m. field states can be taken respectively thermally excited and in vacuum at the same time.
K. Huang, Introduction to Statistical Physics (Taylor and Francis, London, 2001).
