Abstract

Bichromatic laser pumping is an effective tool to control (e.g., to drive into an entangled state) solid-state quantum bits of different nature. For clusters of resonantly interacting ions under bichromatic laser pumping, we present a theoretical approach and approximate analytical solution for quantum states dynamics. The solution provides an optimal ratio of laser pulse intensities needed for creating the maximally entangled states and performing quantum gates. Numerical simulation corroborates the analytical results.

© 2013 Optical Society of America

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    [CrossRef]
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2009

Y. Khodorkovsky, G. Kurizki, and A. Vardi, “Decoherence and entanglement in a bosonic Josephson junction: Bose-enhanced quantum Zeno control of phase diffusion,” Phys. Rev. A 80, 023609 (2009).
[CrossRef]

2008

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

2006

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

2005

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

2004

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

2003

M. Teodorescu-Frumosu and G. Jaeger, “Quantum Lorentz-group invariants of n-qubit systems,” Phys. Rev. A 67, 052305 (2003).
[CrossRef]

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

S. K. Sekatskii, M. Chergui, and G. Dietler, “Coherent fluorescence resonance energy transfer: construction of nonlocal multiparticle entangled states and quantum computing,” Europhys. Lett. 63, 21–27 (2003).
[CrossRef]

2002

D. Petrosyan and G. Kurizki, “Scalable solid-state quantum processor using subradiant two-atom states,” Phys. Rev. Lett. 89, 207902 (2002).
[CrossRef]

2000

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

S. Swain, P. Zhou, and Z. Ficek, “Intensity-intensity correlations and quantum interference in a driven three-level atom,” Phys. Rev. A 61,043410 (2000).
[CrossRef]

1999

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

L. Quiroga and N. F. Johnson, “Entangled Bell and Greenberger-Horne-Zeilinger states of excitons in coupled quantum dots,” Phys. Rev. Lett. 83, 2270–2273 (1999).
[CrossRef]

1998

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

A. Ekert and R. Jozsa, “Quantum algorithms: entanglement-enhanced information processing,” Phil. Trans. R. Soc. A 356, 1769–1782 (1998).
[CrossRef]

1995

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

1994

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

1992

1971

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

1965

V. V. Osiko, “Thermodynamics of optical centers in crystals CaF2-TR3+,” Sov. Phys. Solid State 7, 1294–1302 (1965).

1954

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Basiev, T. T.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

Basieva, I. T.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

Beck, W.

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

Bell, J.

J. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

Benhelm, J.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

Blatt, R.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

Bonadeo, N. H.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Brennen, G. K.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

Caves, C. M.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

Chen, G.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Chergui, M.

S. K. Sekatskii, M. Chergui, and G. Dietler, “Coherent fluorescence resonance energy transfer: construction of nonlocal multiparticle entangled states and quantum computing,” Europhys. Lett. 63, 21–27 (2003).
[CrossRef]

Deutsch, I. H.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Dietler, G.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

S. K. Sekatskii, M. Chergui, and G. Dietler, “Coherent fluorescence resonance energy transfer: construction of nonlocal multiparticle entangled states and quantum computing,” Europhys. Lett. 63, 21–27 (2003).
[CrossRef]

Ekert, A.

A. Ekert and R. Jozsa, “Quantum algorithms: entanglement-enhanced information processing,” Phil. Trans. R. Soc. A 356, 1769–1782 (1998).
[CrossRef]

Elsaesser, T.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

Erugin, N. P.

N. P. Erugin, Linear Systems of Ordinary Differential Equations with Periodic and Quasi-periodic Coefficients (USSR Academy of Sciences, 1963).

Fedorov, V. V.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

Ficek, Z.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

S. Swain, P. Zhou, and Z. Ficek, “Intensity-intensity correlations and quantum interference in a driven three-level atom,” Phys. Rev. A 61,043410 (2000).
[CrossRef]

Flytzanis, C.

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

Gammon, D.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Georgescu, S.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Greenberger, D. M.

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond Bell’s theorem,” in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos, ed. (Kluwer, 1989), pp. 73–76.

Horne, M. A.

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond Bell’s theorem,” in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos, ed. (Kluwer, 1989), pp. 73–76.

Jaeger, G.

M. Teodorescu-Frumosu and G. Jaeger, “Quantum Lorentz-group invariants of n-qubit systems,” Phys. Rev. A 67, 052305 (2003).
[CrossRef]

G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics (Springer, 2009).

Jessen, P. S.

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

Johnson, N. F.

L. Quiroga and N. F. Johnson, “Entangled Bell and Greenberger-Horne-Zeilinger states of excitons in coupled quantum dots,” Phys. Rev. Lett. 83, 2270–2273 (1999).
[CrossRef]

Jozsa, R.

A. Ekert and R. Jozsa, “Quantum algorithms: entanglement-enhanced information processing,” Phil. Trans. R. Soc. A 356, 1769–1782 (1998).
[CrossRef]

Karasik, A. Y.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

Karasik, A. Ya.

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

Katzer, D. S.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Khodorkovsky, Y.

Y. Khodorkovsky, G. Kurizki, and A. Vardi, “Decoherence and entanglement in a bosonic Josephson junction: Bose-enhanced quantum Zeno control of phase diffusion,” Phys. Rev. A 80, 023609 (2009).
[CrossRef]

Kirchmair, G.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

Kornienko, A. A.

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

Kurizki, G.

Y. Khodorkovsky, G. Kurizki, and A. Vardi, “Decoherence and entanglement in a bosonic Josephson junction: Bose-enhanced quantum Zeno control of phase diffusion,” Phys. Rev. A 80, 023609 (2009).
[CrossRef]

D. Petrosyan and G. Kurizki, “Scalable solid-state quantum processor using subradiant two-atom states,” Phys. Rev. Lett. 89, 207902 (2002).
[CrossRef]

Lienau, C.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

Lupei, A.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Lupei, V.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Merzbacher, E.

E. Merzbacher, Quantum Mechanics (Wiley, 1998).

Mueller, K.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

Nanau, P. M.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Orlovskii, Y. V.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

Osiko, V. V.

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

V. V. Osiko, “Thermodynamics of optical centers in crystals CaF2-TR3+,” Sov. Phys. Solid State 7, 1294–1302 (1965).

Papashvili, A. G.

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

Park, D.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Petrosyan, D.

D. Petrosyan and G. Kurizki, “Scalable solid-state quantum processor using subradiant two-atom states,” Phys. Rev. Lett. 89, 207902 (2002).
[CrossRef]

Prokhorov, A. M.

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

Pukhov, K. K.

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

Quiroga, L.

L. Quiroga and N. F. Johnson, “Entangled Bell and Greenberger-Horne-Zeilinger states of excitons in coupled quantum dots,” Phys. Rev. Lett. 83, 2270–2273 (1999).
[CrossRef]

Reeves, R.

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

Roos, C. F.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

Sekatskii, S. K.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

S. K. Sekatskii, M. Chergui, and G. Dietler, “Coherent fluorescence resonance energy transfer: construction of nonlocal multiparticle entangled states and quantum computing,” Europhys. Lett. 63, 21–27 (2003).
[CrossRef]

Sham, L. J.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Smirnov, M. Z.

Steel, D. G.

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Stoicescu, C.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Swain, S.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

S. Swain, P. Zhou, and Z. Ficek, “Intensity-intensity correlations and quantum interference in a driven three-level atom,” Phys. Rev. A 61,043410 (2000).
[CrossRef]

Teodorescu-Frumosu, M.

M. Teodorescu-Frumosu and G. Jaeger, “Quantum Lorentz-group invariants of n-qubit systems,” Phys. Rev. A 67, 052305 (2003).
[CrossRef]

Tiseanu, C.

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

Tscherbakov, I. A.

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

Unold, T.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

Vardi, A.

Y. Khodorkovsky, G. Kurizki, and A. Vardi, “Decoherence and entanglement in a bosonic Josephson junction: Bose-enhanced quantum Zeno control of phase diffusion,” Phys. Rev. A 80, 023609 (2009).
[CrossRef]

Ver Steeg, K.

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

Ver Steeg, K. W.

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

Voronko, Y. K.

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

Wieck, A. D.

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

Zeilinger, A.

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond Bell’s theorem,” in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos, ed. (Kluwer, 1989), pp. 73–76.

Zhou, P.

S. Swain, P. Zhou, and Z. Ficek, “Intensity-intensity correlations and quantum interference in a driven three-level atom,” Phys. Rev. A 61,043410 (2000).
[CrossRef]

Chem. Phys.

V. V. Fedorov, W. Beck, T. T. Basiev, A. Y. Karasik, and C. Flytzanis, “Fine level splitting of aggregate neodymium centers in CaF2 crystals,” Chem. Phys. 257, 275–281 (2000).
[CrossRef]

Europhys. Lett.

S. K. Sekatskii, M. Chergui, and G. Dietler, “Coherent fluorescence resonance energy transfer: construction of nonlocal multiparticle entangled states and quantum computing,” Europhys. Lett. 63, 21–27 (2003).
[CrossRef]

J. Lumin.

T. T. Basiev, V. V. Fedorov, A. Ya. Karasik, and K. K. Pukhov, “Strong coherent interaction of Nd3+-Nd3+ pair ions in CaF2 crystal,” J. Lumin. 81, 189–197 (1999).
[CrossRef]

K. Ver Steeg, A. Y. Karasik, R. Reeves, and T. T. Basiev, “Accumulated photon echo in CaF2–YF3:Nd3+ crystals,” J. Lumin. 60/61, 742–744 (1994).
[CrossRef]

J. Opt. Soc. Am. B

JETP Lett.

T. T. Basiev, A. Y. Karasik, A. A. Kornienko, A. G. Papashvili, and K. K. Pukhov, “Supersensitive electronic transition in impurity Nd-Nd nanoclusters in CaF2 crystal,” JETP Lett. 78, 319–321 (2003).
[CrossRef]

Nat. Phys.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[CrossRef]

Opt. Commun.

S. K. Sekatskii, T. T. Basiev, I. T. Basieva, G. Dietler, V. V. Fedorov, A. Y. Karasik, Y. V. Orlovskii, and K. K. Pukhov, “Experimental preparation of entangled Bells vacuum–single exciton and vacuum–biexciton states for pair centers of neodymium ions in a crystal,” Opt. Commun. 259, 298–303 (2006).
[CrossRef]

Phil. Trans. R. Soc. A

A. Ekert and R. Jozsa, “Quantum algorithms: entanglement-enhanced information processing,” Phil. Trans. R. Soc. A 356, 1769–1782 (1998).
[CrossRef]

Phys. Rev.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[CrossRef]

Phys. Rev. A

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

S. Swain, P. Zhou, and Z. Ficek, “Intensity-intensity correlations and quantum interference in a driven three-level atom,” Phys. Rev. A 61,043410 (2000).
[CrossRef]

M. Teodorescu-Frumosu and G. Jaeger, “Quantum Lorentz-group invariants of n-qubit systems,” Phys. Rev. A 67, 052305 (2003).
[CrossRef]

Y. Khodorkovsky, G. Kurizki, and A. Vardi, “Decoherence and entanglement in a bosonic Josephson junction: Bose-enhanced quantum Zeno control of phase diffusion,” Phys. Rev. A 80, 023609 (2009).
[CrossRef]

Phys. Rev. B

V. Lupei, A. Lupei, C. Tiseanu, S. Georgescu, C. Stoicescu, and P. M. Nanau, “High-resolution optical spectroscopy of YAG:Nd: a test for structural and distribution models,” Phys. Rev. B 51, 8–17 (1995).
[CrossRef]

I. T. Basieva, T. T. Basiev, G. Dietler, K. K. Pukhov, and S. K. Sekatskii, “Quantum control of exciton states in clusters of resonantly interacting fluorescent particles using biharmonic laser pumping,” Phys. Rev. B 74, 165329 (2006).
[CrossRef]

Phys. Rev. Lett.

L. Quiroga and N. F. Johnson, “Entangled Bell and Greenberger-Horne-Zeilinger states of excitons in coupled quantum dots,” Phys. Rev. Lett. 83, 2270–2273 (1999).
[CrossRef]

D. Petrosyan and G. Kurizki, “Scalable solid-state quantum processor using subradiant two-atom states,” Phys. Rev. Lett. 89, 207902 (2002).
[CrossRef]

T. Unold, K. Mueller, C. Lienau, T. Elsaesser, and A. D. Wieck, “Optical control of excitons in a pair of quantum dots coupled by the dipole-dipole interaction,” Phys. Rev. Lett. 94, 137404 (2005).
[CrossRef]

G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum logic gates in optical lattices,” Phys. Rev. Lett. 82, 1060–1063 (1999).
[CrossRef]

Science

G. Chen, N. H. Bonadeo, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, and L. J. Sham, “Optically induced entanglement of excitons in a single quantum dot,” Science 289, 1906–1909 (2000).
[CrossRef]

Sov. Phys. JETP

T. T. Basiev, A. Y. Karasik, V. V. Fedorov, and K. W. Ver Steeg, “Optical echo spectroscopy and phase relaxation of Nd3+ ions in CaF2 crystals,” Sov. Phys. JETP 86, 156–163 (1998).
[CrossRef]

V. V. Osiko, Y. K. Voronko, A. M. Prokhorov, and I. A. Tscherbakov, “Investigation of the mechanism of the elementary act of excitation-energy transfer between rare-earth ions in crystals,” Sov. Phys. JETP 33, 510–515 (1971).

Sov. Phys. Solid State

V. V. Osiko, “Thermodynamics of optical centers in crystals CaF2-TR3+,” Sov. Phys. Solid State 7, 1294–1302 (1965).

Other

G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics (Springer, 2009).

J. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University, 1987).

E. Merzbacher, Quantum Mechanics (Wiley, 1998).

D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond Bell’s theorem,” in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, M. Kafatos, ed. (Kluwer, 1989), pp. 73–76.

G. Jaeger, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entanglement, mixedness, and spin-flip symmetry in multiple-qubit systems,” http://arxiv.org/pdf/quant-ph/0307124.pdf .

N. P. Erugin, Linear Systems of Ordinary Differential Equations with Periodic and Quasi-periodic Coefficients (USSR Academy of Sciences, 1963).

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Figures (7)

Fig. 1.
Fig. 1.

Equal-intensity bichromatic pumping on resonant frequencies ω 0 V and ω 0 + V . (a) Real parts of the wave-function coefficients. Gray lines, zeroth-order approximation Eq. (27); dashes, first-order of λ expression Eq. (26), λ = 0.3 . (b) Density matrix elements, λ = 0.01 ( A 1 = A 2 = 1.4 V / 100 ).

Fig. 2.
Fig. 2.

Ground-state dynamics (absolute value of the coefficient before | 1 in psi function) under bichromatic pumping on resonant frequencies ω 0 V and ω 0 + V , A 2 = 2 A 1 = 0.42 V ( k = 2 , λ = 0.15 ). (a) Gray line, zeroth-order PT approximation; dashes, first-order PT approximation Eq. (28). (b) Dashed line, zeroth-order solution Eq. (24); thin dotted line, first-order PT approximation Eq. (28).

Fig. 3.
Fig. 3.

Dynamics of density matrix elements in a dimer under bichromatic pumping at the resonant frequencies ω 0 V and ω 0 + V with properly chosen intensity ratios: (a)  A 1 = 1.4 V / 100 and A 2 = 0.41 A 1 ( λ = 0.01 , k = 2 1 , and I 2 / I 1 = 3 2 2 ); (b)  A 2 = 1.4 V / 100 and A 1 = A 2 / 1.41 ( λ = 0.007 , k = 2 + 1 , and I 2 / I 1 = 3 + 2 2 ). Arrows point at the moments of the maximally entangled Bell states.

Fig. 4.
Fig. 4.

Dynamics of the dimer density matrix elements under bichromatic pumping at the resonant frequencies ω 0 V and ω 0 + V , with the intensity ratios selected to create the uniform three-state superposition I 2 / I 1 = 1 3 / 2 . (a)  A 1 = 1.4 V / 100 and A 2 = 0.366 A 1 ( λ = 0.01 , k = ( 3 1 ) / 2 ); (b)  A 2 = 1.4 V / 100 , A 2 = 0.732 A 1 , and A 1 = 1.4 V / 100 ( λ = 0.00732 , k = ( 3 + 1 ) / 2 ). Arrows point at the moments when the populations of the three states are equal ( ρ 1 1 = ρ 00 = ρ 11 = 1 / 3 ).

Fig. 5.
Fig. 5.

Evolution of density matrix elements ρ 2 2 , ρ 1 1 , ρ 00 in tetramer under bichromatic laser radiation on resonant frequencies ω 0 3 V and ω 0 V with the specific intensity ratio (a)  A 1 = 0.014 V ( λ = 0.014 ), I 2 / I 1 = ( A 2 / A 1 ) 2 = 2 4 2 / 3 ( k = ( 2 2 ) / 3 ), (b)  A 2 = 0.014 V , I 2 / I 1 = ( A 2 / A 1 ) 2 = 2 + 4 2 / 3 ( k = ( 2 + 2 ) / 3 ). The first-order approximation, Eq. (39), is shown by dots. Dashed lines on the lower parts of the figures correspond to the phases of the state function coefficients at | 0 and | 1 as related to the phase of the ground state | 2 .

Fig. 6.
Fig. 6.

Dynamics of tetramer density matrix elements under bichromatic pumping. (a) Inversion of population from the ground to the biexciton state under laser pumping at resonant frequencies ω 0 3 V and ω 0 V with the intensity ratio k = 1 / 1.5 , A 1 = 0.014 V ( λ = 0.014 ), and I 2 / I 1 = ( A 2 / A 1 ) 2 = 0.67 . (b) Two-photon transitions under laser pumping with frequencies ω 0 2 V and ω 0 + 2 V with equal intensity: A 2 = A 1 = V / 10 ( λ = 0.1 , k = 1 ).

Fig. 7.
Fig. 7.

Preparation of entangled states in tetramer by bichromatic pumping at frequencies ω 0 2 V and ω 0 + 2 V . (a)  A 1 = V / 10 and A 2 = 1.5 A 1 ( λ = 0.1 , k = 1.5 ); (b)  A 2 = V / 10 = A 1 / 1.5 ( λ = 0.15 , k = 0.67 ).

Equations (45)

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( | 1 2 n ± | 1 2 n ) / 2 ,
H ^ int = 2 d 1 E 1 J ^ x cos ( ω 1 t + φ 1 ) 2 d 2 E 2 J x cos ( ω 2 t + φ 2 ) ,
H ^ = ( ω 0 ω ˜ ) J ^ z + V ( J ^ z 2 J ^ 2 ) ( d 1 E 1 cos ( Δ 1 t ) + d 2 E 2 cos ( Δ 2 t ) ) J ^ x + ( d 1 E 1 sin ( Δ 1 t ) + d 2 E 2 sin ( Δ 2 t ) ) J y .
S ( n ) 2 = Tr ( ρ ρ ˜ ) ,
ρ ˜ = σ y n ρ * σ y n
α = 0 ; α = 1 ; 2 θ 0 θ 1 = π .
ψ ˙ = i H ψ
Y ˙ = Y P ( t ) λ .
Y = exp ( t B ( λ ) ) · ( E + Z ( λ , t ) ) ,
B ( λ ) = λ B 1 + λ 2 B 2 + , Z ( λ , t ) = λ Z 1 ( t ) + λ 2 Z 2 ( t ) + ,
B 1 = 1 2 π 0 2 π P ( t ) d t , Z 1 = 0 t P ( t ) d t t B 1 ,
B 2 = 1 2 π 0 2 π ( Z 1 ( t ) P ( t ) B 1 Z 1 ( t ) ) d t , Z 2 = 0 t ( Z 1 ( t ) P ( t ) B 1 Z 1 ( t ) ) d t t B 2 ,
H = [ V A 1 e i V t + A 2 e i V t 2 0 A 1 e i V t + A 2 e i V t 2 0 A 1 e i V t + A 2 e i V t 2 0 A 1 e i V t + A 2 e i V t 2 V ] .
ψ ˙ = i λ ψ · [ 0 e 2 i τ + k 0 e 2 i τ + k 0 k e 2 i τ + 1 0 k e 2 i τ + 1 0 ] .
B 1 = [ 0 i k 0 i k 0 i 0 i 0 ] ,
Z 1 = 1 2 [ 0 1 e 2 i τ 0 1 + e 2 i τ 0 k + k e 2 i τ 0 k k e 2 i τ 0 ] ,
B 2 = i 2 [ 2 k 1 0 1 k + k 2 0 1 4 k + k 2 0 1 k + k 2 0 ( 2 k ) k ] ,
Z 2 = 1 4 [ e 2 i τ 1 k e 2 i τ + k e 2 i τ 0 k 2 1 k + e 2 i τ + k e 2 i τ k 2 e 2 i τ 0 e 2 i τ 1 k 2 + 2 k e 2 i τ 2 k e 2 i τ + k 2 e 2 i τ 0 1 k k 2 e 2 i τ + k e 2 i τ + k 2 e 2 i τ 0 k e 2 i τ + k 2 e 2 i τ k 2 k e 2 i τ ] .
Y 2 = exp ( τ ( λ B 1 + λ 2 B 2 ) ) · ( E + λ Z 1 ( τ ) + λ 2 Z 2 ( τ ) ) exp ( τ ( λ B 1 + λ 2 B 2 ) ) · ( E + λ Z 1 ( τ ) ) .
F [ k 2 e x 1 τ + k 2 e x 2 τ + 2 e x 3 τ 2 ( 1 + k 2 ) k e x 1 τ k e x 2 τ 2 1 + k 2 k ( e x 1 τ + e x 2 τ 2 e x 3 τ ) 2 ( 1 + k 2 ) k e x 1 τ k e x 2 τ 2 1 + k 2 e x 1 τ + e x 2 τ 2 e x 1 τ e x 2 τ 2 1 + k 2 k ( e x 1 τ + e x 2 τ 2 e x 3 τ ) 2 ( 1 + k 2 ) e x 1 τ e x 2 τ 2 1 + k 2 e x 1 τ + e x 2 τ + 2 k 2 e x 3 τ 2 ( 1 + k 2 ) ] .
Y 2 = F · ( E + λ Z 1 ( τ ) ) .
ψ ( 0 ) = ( 0 0 1 ) ,
ψ 1 = k ( e x 1 τ + e x 2 τ 2 e x 3 τ ) 2 ( 1 + k 2 ) λ ( e x 1 τ e x 2 τ ) ( 1 e 2 i τ ) 4 1 + k 2 , ψ 0 = e x 1 τ e x 2 τ 2 1 + k 2 + λ k ( e x 1 τ + e x 2 τ + ( k 2 1 ) e x 3 τ ) ( 1 e 2 i τ ) 2 ( 1 + k 2 ) , ψ 1 = e x 1 τ + e x 2 τ + 2 k 2 e x 3 τ 2 ( 1 + k 2 ) λ k ( e x 1 τ e x 2 τ ) ( 1 e 2 i τ ) 4 1 + k 2 .
| ψ k ( e x 1 τ + e x 2 τ 2 e x 3 τ ) 2 ( 1 + k 2 ) | 1 + e x 1 τ e x 2 τ 2 1 + k 2 | 0 + e x 1 τ + e x 2 τ + 2 k 2 e x 3 τ 2 ( 1 + k 2 ) | 1 .
| ψ ( τ ) = 1 2 ( cos θ 1 + i λ 2 sin θ ( 1 e 2 i τ ) ) | 1 + ( i 2 sin θ λ 2 cos θ ( 1 e 2 i τ ) ) | 0 + 1 2 ( cos θ + 1 + i λ 2 sin θ ( 1 e 2 i τ ) ) | 1 ,
θ = 2 λ τ = A 1 V τ = A 2 V τ .
| ψ ( t ) 1 2 ( cos θ 1 ) | 1 + i 2 sin θ | 0 + 1 2 ( cos θ + 1 ) | 1 .
ψ 1 cos ( μ τ ) + k 2 k 2 + 1 λ k ( k 2 + 1 ) 3 / 2 ( sin ( 2 τ ) sin ( μ τ ) + i sin ( μ τ 2 τ ) + sin ( μ τ ) 2 ( μ 1 ) i sin ( μ τ + 2 τ ) + sin ( μ τ ) 2 ( μ + 1 ) ) + λ 2 ( 1 k 2 ) 2 ( k 2 + 1 ) 3 / 2 ( k 2 ( e 2 i τ e i μ τ ) + ( 1 e 2 i τ i μ τ ) / 2 μ 2 + k 2 ( e 2 i τ e i μ τ ) + ( 1 e 2 i τ + i μ τ ) / 2 μ + 2 ) .
ρ 11 = k 2 2 ( 1 + k 2 ) 2 ( 3 + cos [ ( x 1 x 2 ) τ / i ] 2 cos [ ( x 1 x 3 ) τ / i ] 2 cos [ ( x 2 x 3 ) τ / i ] ) k λ 2 ( 1 + k 2 ) 3 / 2 Re [ e 2 i τ sin ( 2 μ τ ) e ( x 1 x 3 ) τ + e ( x 1 x 3 2 i ) τ + e ( x 2 x 3 ) τ e ( x 2 x 3 2 i ) τ ] + O ( λ 2 ) .
ρ 11 = k 2 2 ( 1 + k 2 ) 2 ( 3 + cos ( 2 μ τ ) 2 cos [ ( μ 3 Δ / 2 ) τ ] 2 cos [ ( μ + 3 Δ / 2 ) τ ] ) ,
ρ 00 = 1 cos [ ( x 1 x 2 ) τ / i ] 2 ( 1 + k 2 ) = 1 cos [ 2 μ τ ] 2 ( 1 + k 2 ) ,
ρ 1 1 = 1 + 2 k 4 + cos ( 2 μ τ ) + 2 k 2 ( cos [ ( μ + 3 Δ / 2 ) τ ] + cos [ ( μ 3 Δ / 2 ) τ ] ) 2 ( 1 + k 2 ) 2 ,
ρ 10 = k 2 ( 1 + k 2 ) 2 ( cos ( 2 μ τ ) e i τ ( μ + 3 Δ / 2 ) + e i τ ( μ 3 Δ / 2 ) ) ,
ρ 1 1 = k 2 ( 1 + k 2 ) 2 ( 1 2 k 2 + cos ( 2 μ τ ) e i τ ( μ + 3 Δ / 2 ) e i τ ( μ 3 Δ / 2 ) + k 2 e i τ ( μ + 3 Δ / 2 ) + k 2 e i τ ( μ 3 Δ / 2 ) ) ,
ρ 0 1 = i sin ( 2 μ τ ) + k 2 e i τ ( μ + 3 Δ / 2 ) k 2 e i τ ( μ 3 Δ / 2 ) 2 ( 1 + k 2 ) 2 .
S ( 2 ) 2 = | 2 ψ 1 ψ 1 ψ 0 2 | 2 ,
S ( 2 ) 2 = | k ( e x 1 τ + e x 2 τ 2 e x 3 τ ) ( e x 1 τ + e x 2 τ + 2 k 2 e x 3 τ ) ( 1 + k 2 ) ( e x 1 τ e x 2 τ ) 2 2 ( 1 + k 2 ) 2 | 2 .
H ( 1 , 2 ) = H * ( 2 , 1 ) = λ ( e 6 i τ + k e 4 i τ ) , H ( 2 , 3 ) = H * ( 3 , 2 ) = λ ( e 4 i τ + k e 2 i τ ) 1.5 , H ( 3 , 4 ) = H * ( 4 , 3 ) = λ ( e 2 i τ + k ) 1.5 , H ( 4 , 5 ) = H * ( 5 , 4 ) = λ ( 1 + k e 2 i τ ) .
| ψ ( τ ) = ( k e i λ α τ ( e i λ α τ 1 ) 2 2 ( 1 + 1.5 k 2 ) λ ( 1 e 2 i τ ) sin ( λ α τ ) 2 1 + 1.5 k 2 ) 1.5 | 0 + ( sin ( λ α τ ) 1 + 1.5 k 2 + λ k ( e 2 i τ 1 ) ( 5 ( cos ( λ α τ ) + 1 ) + 3 ( k 2 1 ) ) 4 ( 1 + 1.5 k 2 ) ) | 1 + ( cos ( λ α τ ) + 1.5 k 2 ( 1 + 1.5 k 2 ) λ k ( e 2 i τ 1 ) sin ( λ α τ ) 2 1 + 1.5 k 2 ) | 2 .
S ( 4 ) 2 = | ψ 0 2 + 6 ψ 2 ψ 2 | 2 / 9 .
B 1 = 0 ; B 2 = i [ k 2 + 1 / 5 0 k 2 1.5 0 0 0 0.3 2.5 k 2 0 k 0 k 2 1.5 0 1 + k 2 0 1.5 0 k 0 0.3 k 2 2.5 0 0 0 1.5 0 1 + k 2 / 5 ] ,
Z 1 ( 1 , 2 ) = Z 1 ( 2 , 1 ) * = e 5 i τ 1 5 k + k e i τ , Z 1 ( 2 , 3 ) = Z 1 ( 3 , 2 ) * = ( k k e i τ + e 3 i τ 1 3 ) 1.5 , Z 1 ( 3 , 4 ) = Z 1 ( 4 , 3 ) * = ( 1 + e i τ + 1 e 3 i τ 3 k ) 1.5 , Z 1 ( 4 , 5 ) = Z 1 ( 5 , 4 ) * = 1 e i τ + 1 e 5 i τ 5 k .
| ψ ( τ ) 0.25 ( 2 e 1.2 i τ λ 2 + 1.23 e i τ λ 2 / 5 + 0.77 e 3.4 i τ λ 2 ) | 2 0.478 e 3.4 i τ λ 2 ( 1 + e 3.6 i τ λ 2 ) | 0 + 0.25 ( 2 e 1.2 i τ λ 2 + 1.23 e i τ λ 2 / 5 + 0.77 e 3.4 i τ λ 2 ) | 2 .
| ψ ( τ ) ( 0.13 e 5.8 i τ λ 2 + 0.28 e 0.33 i τ λ 2 0.41 e 1.6 i τ λ 2 ) | 2 + ( 0.16 e 5.8 i τ λ 2 0.28 e 0.33 i τ λ 2 + 0.12 e 1.6 i τ λ 2 ) | 0 + ( 0.04 e 5.8 i τ λ 2 + 0.19 e 0.33 i τ λ 2 + 0.76 e 1.6 i τ λ 2 ) | 2 .
| ψ ( τ ) ( 0.13 e 2.6 i τ λ 2 0.41 e 0.73 i τ λ 2 + 0.28 e 0.14 i τ λ 2 ) | 2 + ( 0.47 e 2.6 i τ λ 2 0.06 e 0.73 i τ λ 2 0.4 e 0.14 i τ λ 2 ) | 0 + ( 0.38 e 2.6 i τ λ 2 + 0.22 e 0.73 i τ λ 2 + 0.4 e 0.14 i τ λ 2 ) | 2 .

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