Abstract
In the framework of nonstationary scattering theory, we study the formation of an entangled state of two identical nonrelativistic spin-1/2 particles as a result of their elastic scattering. The measure of particle entanglement in the final channel is described using pair concurrence. For the indicated quantitative criterion, we obtain general expressions in terms of the direct and exchange scattering amplitudes in the cases of pure and mixed spin states of the pair in the initial channel. We consider the violation of Bell’s inequality in the final channel. We show that as a result of a collision between unpolarized particles, a Werner spin state of the pair forms, which is entangled if the singlet component of the angular differential scattering cross section in the center-of-mass reference frame exceeds the triplet component. We use the process of free electron-electron scattering as an example to illustrate the developed formalism.
Similar content being viewed by others
References
D. Bohm and Y. Aharonov, “Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky,” Phys. Rev., 108, 1070–1076 (1957).
J. Bell, “On the Einstein Podolsky Rosen paradox,” Phys. Phys. Fiz., 1, 195–200 (1964).
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett., 23, 880–884 (1969).
S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Modern Phys., 77, 513–577 (2005); arXiv:quant-ph/0410100v1 (2004).
R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Modern Phys., 81, 865–942 (2009); arXiv:quant-ph/0702225v2 (2007).
J. Stolze and D. Suter, Quantum Computing: A Short Course from Theory to Experiment, Wiley-VCH, Weinheim (2007).
J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, Mineola, N. Y. (2006).
L. Lamata and J. León, “Generation of bipartite spin entanglement via spin-independent scattering,” Phys. Rev. A, 73, 052322 (2006); arXiv:quant-ph/0602090v2 (2006).
R. Feder, F. Giebels, and H. Gollisch, “Entanglement creation in electron-electron collisions at solid surfaces,” Phys. Rev. B, 92, 075420 (2015).
D. Vasilyev, F. O. Schumann, F. Giebels, H. Gollisch, J. Kirschner, and R. Feder, “Spin-entanglement between two freely propagating electrons: Experiment and theory,” Phys. Rev. B, 95, 115134 (2017).
J. Kessler, Polarized Electrons, Springer, Berlin (1985).
J. Schliemann, J. I. Cirac, M. Kus, M. Lewenstein, and D. Loss, “Quantum correlations in two-fermion systems,” Phys. Rev. A, 64, 022303 (2001); arXiv:quant-ph/0012094v2 (2000).
G. C. Ghirardi and L. Marinatto, “General criterion for the entanglement of two indistinguishable particles,” Phys. Rev. A, 70, 012109 (2004); arXiv:quant-ph/0401065v2 (2004).
M. C. Tichy, F. Mintert, and A. Buchleitner, “Essential entanglement for atomic and molecular physics,” J. Phys. B, 44, 192001 (2011); arXiv:1012.3940v2 [quant-ph] (2010).
M. C. Tichy, F. de Melo, M. Kuś, F. Mintert, and A. Buchleitner, “Entanglement of identical particles and the detection process,” Fortschr. Phys., 61, 225–237 (2013).
R. L. Franco and G. Compagno, “Quantum entanglement of identical particles by standard information-theoretic notions,” Sci. Rep., 6, 20603 (2016); arXiv:1511.03445v4 [quant-ph] (2015).
W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett., 80, 2245–2248 (1998); arXiv:quant-ph/9709029v2 (1997).
L. J. B. Goldfarb, “Angular correlations and polarization,” in: Nuclear Reactions (P. M. Endt and M. Demeur, eds.), Vol. 1, North-Holland, Amsterdam (1959), pp. 159–214.
C. J. Joachain, Quantum Collision Theory, North-Holland, Amsterdam (1975).
F. Mintert, A. R. R. Carvalho, M. Kuś, and A. Buchleitner, “Measures and dynamics of entangled states,” Phys. Rep., 415, 207–259 (2005); arXiv:quant-ph/0505162v1 (2005).
P. Rungta, V. Buẑek, C. M. Caves, M. Hillery, and G. J. Milburn, “Universal state inversion and concurrence in arbitrary dimensions,” Phys. Rev. A, 64, 042315 (2001); arXiv:quant-ph/0102040v2 (2001).
R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A, 40, 4277–4281 (1989).
S. Popescu, “Bell’s inequalities versus teleportation: What is nonlocality?” Phys.Rev. Lett., 72, 797–799 (1994).
T. Hiroshima and S. Ishizaka, “Local and nonlocal properties of Werner states,” Phys. Rev. A, 62, 044302 (2000); arXiv:quant-ph/0003058v1 (2000).
B. S. Cirel’son, “Quantum generalizations of Bell’s inequality,” Lett. Math. Phys., 4, 93–100 (1980).
R. H. Dalitz, “On higher Born approximations in potential scattering,” Proc. Roy. Soc. Lond. Ser. A, 206, 509–520 (1951).
W. F. Ford, “Anomalous behavior of the Coulomb T matrix,” Phys. Rev., 133, B1616–B1621 (1964).
S. Weinberg, “Infrared photons and gravitons,” Phys. Rev., 140, B516–B524 (1965).
K. A. Kouzakov, Yu. V. Popov, and V. L. Shablov, “Comment on ‘Exact three-dimensional wave function and the on-shell t matrix for the sharply cut-off Coulomb potential: Failure of the standard renormalization factor’,” Phys. Rev. C, 81, 019801 (2010); arXiv:0908.3137v1 [nucl-th] (2009).
J. D. Dollard, “Asymptotic convergence and the Coulomb interaction,” J. Math. Phys., 5, 729–738 (1964).
S. P. Merkuriev and L. D. Faddeev, Quantum Theory of Scattering for Systems of Several Particles [in Russian], Nauka, Moscow (1985)
English transl: L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particle Systems (Math. Phys. Appl. Math., Vol. 11), Kluwer, Dordrecht (1993).
Acknowledgments
The author is grateful to L. Chotorlishvili for the useful discussions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest. The author declares no conflicts of interest.
Rights and permissions
About this article
Cite this article
Kouzakov, K.A. Quantum Entanglement in the Nonrelativistic Collision Between Two Identical Fermions with Spin 1/2. Theor Math Phys 201, 1664–1679 (2019). https://doi.org/10.1134/S0040577919110102
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577919110102