Abstract

Electron spins in semiconductor nanostructures are considered as one of the most promising candidates of the building blocks for quantum information processing. Initialization, coherent manipulation, and measurement of a single electron spin have been recently established. Here we explore the possibility to manipulate and measure by optical means the states of two electrons as a single qubit in nanostructures and propose methods of spin-state tomography based on Faraday or Kerr rotation for cases of both a single electron and two electrons, as well as a method of the Bell-state measurement for two electrons.

© 2010 Optical Society of America

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2009 (3)

H. Kosaka, T. Inagaki, Y. Rikitake, H. Imamura, Y. Mitsumori, and K. Edamatsu, “Spin state tomography of optically injected electrons in a semiconductor,” Nature (London) 457, 702-705 (2009).
[CrossRef]

A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev, D. Reuter, A. D. Wieck, T. L. Reinecke, and M. Bayer, “Ultrafast optical rotations of electron spins in quantum dots,” Nat. Phys. 5, 262-266 (2009).
[CrossRef]

O. Cakir and T. Takagahara, “Proposals of nuclear spin quantum memory in group-IV elemental and II-VI semiconductors,” Phys. Rev. B 80, 155323 (2009).
[CrossRef]

2008 (5)

O. Cakir and T. Takagahara, “Quantum dynamics in electron-nuclei coupled spin system in quantum dots: Bunching, revival, and quantum correlation in electron-spin measurements,” Phys. Rev. B 77, 115304 (2008).
[CrossRef]

J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, and S. A. Lyon, “Solid-state quantum memory using the P31 nuclear spin,” Nature (London) 455, 1085-1088 (2008).
[CrossRef]

H. Kosaka, H. Shigyou, Y. Mitsumori, Y. Rikitake, H. Imamura, T. Kutsuwa, K. Arai, and K. Edamatsu, “Coherent transfer of light polarization to electron spins in a semiconductor,” Phys. Rev. Lett. 100, 096602 (2008).
[CrossRef] [PubMed]

J. Berezovsky, M. H. Mikkelsen, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Picosecond coherent optical manipulation of a single electron spin in a quantum dot,” Science 320, 349-352 (2008).
[CrossRef] [PubMed]

D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature (London) 456, 218-221 (2008).
[CrossRef]

2007 (10)

Y. Wu, E. D. Kim, X. Xu, J. Cheng, D. G. Steel, A. S. Bracker, D. Gammon, S. E. Economou, and L. J. Sham, “Selective optical control of electron spin coherence in singly charged GaAs-Al0.3Ga0.7As quantum dots,” Phys. Rev. Lett. 99, 097402 (2007).
[CrossRef] [PubMed]

S. E. Economou and T. L. Reinecke, “Theory of fast optical spin rotation in a quantum dot based on geometric phases and trapped states,” Phys. Rev. Lett. 99, 217401 (2007).
[CrossRef]

M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312-1316 (2007).
[CrossRef]

R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79, 1217-1265 (2007).
[CrossRef]

T. Meunier, I. T. Vink, L. H. Willems van Beveren, K. J. Tielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz, M. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Experimental signature of phonon-mediated spin relaxation in a two-electron quantum dot,” Phys. Rev. Lett. 98, 126601 (2007).
[CrossRef] [PubMed]

C. Emary, X. Xu, D. G. Steel, S. Saikin, and L. J. Sham, “Fast initialization of the spin state of an electron in a quantum dot in the Voigt configuration,” Phys. Rev. Lett. 98, 047401 (2007).
[CrossRef] [PubMed]

X. Xu, Y. Wu, B. Sun, Q. Huang, J. Cheng, D. G. Steel, A. S. Bracker, D. Gammon, C. Emary, and L. J. Sham, “Fast spin state initialization in a singly charged InAs-GaAs quantum dot by optical cooling,” Phys. Rev. Lett. 99, 097401 (2007).
[CrossRef] [PubMed]

K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L. M. K. Vandersypen, “Coherent control of a single electron spin with electric fields,” Science 318, 1430-1433 (2007).
[CrossRef] [PubMed]

T. Takagahara and O. Cakir, “Theoretical aspects of quantum state transfer, correlation measurement and electron-nuclei coupled dynamics in quantum dots,” J. Nanophotonics 1, 011593 (2007).
[CrossRef]

M. Atature, J. Dreiser, A. Badolato, and A. Imamoglu, “Observation of Faraday rotation from a single confined spin,” Nat. Phys. 3, 101-106 (2007).
[CrossRef]

2006 (4)

F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Driven coherent oscillations of a single electron spin in a quantum dot,” Science 442, 766-771 (2006).

J. Berezovsky, M. H. Mikkelsen, O. Gywat, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Nondestructive optical measurements of a single electron spin in a quantum dot,” Science 314, 1916-1920 (2006).
[CrossRef] [PubMed]

M. Atatüre, J. Dreiser, A. Badolato, A. Hogele, K. Karrai, and A. Imamoglu, “Quantum-dot spin-state preparation with near-unity fidelity,” Science 312, 551-553 (2006).
[CrossRef] [PubMed]

S. E. Economou and L. J. Sham, Y. Wu, and D. G. Steel, “Proposal for optical U(1) rotations of electron spin trapped in a quantum dot,” Phys. Rev. B 74, 205415 (2006).
[CrossRef]

2005 (2)

M. V. G. Dutt, J. Cheng, B. Li, X. Xu, Xiaoqin Li, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, S. E. Economou, R. B. Liu, and L. J. Sham, “Stimulated and spontaneous optical generation of electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef] [PubMed]

J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180-2184 (2005).
[CrossRef] [PubMed]

2004 (2)

P. Chen, C. Piermarocchi, L. J. Sham, D. Gammon, and D. G. Steel, “Theory of quantum optical control of a single spin in a quantum dot,” Phys. Rev. B 69, 075320 (2004).
[CrossRef]

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature (London) 432, 81-84 (2004).
[CrossRef]

2003 (5)

X. Li, Y. Wu, D. G. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809-811 (2003).
[CrossRef] [PubMed]

R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, “Zeeman energy and spin relaxation in a one-electron quantum dot,” Phys. Rev. Lett. 91, 196802 (2003).
[CrossRef] [PubMed]

E. Yablonovitch, H. W. Jiang, H. Kosaka, H. D. Robinson, D. S. Rao, and T. Szkopek, “Optoelectronic quantum telecommunications based on spins in semiconductors,” Proc. IEEE 91, 761-780 (2003).
[CrossRef]

A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, “Electron spin relaxation times of phosphorus donors in silicon,” Phys. Rev. B 68, 193207 (2003).
[CrossRef]

J. M. Taylor, C. M. Marcus, and M. D. Lukin, “Long-lived memory for mesoscopic quantum bits,” Phys. Rev. Lett. 90, 206803 (2003).
[CrossRef] [PubMed]

2001 (4)

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64, 052312 (2001).
[CrossRef]

A. A. Kiselev, K. W. Kim, and E. Yablonovitch, “In-plane light-hole g factor in strained cubic heterostructures,” Phys. Rev. B 64, 125303 (2001).
[CrossRef]

H. Kosaka, A. A. Kiselev, F. A. Baron, K. W. Kim, and E. Yablonovitch, “Electron g factor engineering in III-V semiconductors for quantum communications,” Electron. Lett. 37, 464-465 (2001).
[CrossRef]

R. Vrijen and E. Yablonovitch, “A spin-coherent semiconductor photo-detector for quantum communication,” Physica E (Amsterdam) 10, 569-575 (2001).
[CrossRef]

2000 (1)

C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) 404, 247-255 (2000).
[CrossRef]

1999 (1)

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204-4207 (1999).
[CrossRef]

1998 (1)

D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

1995 (1)

A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, “Conditional quantum dynamics and logic gates,” Phys. Rev. Lett. 74, 4083-4086 (1995).
[CrossRef] [PubMed]

1981 (1)

A. Bambini and P. R. Berman, “Analytic solutions to the two-state problem for a class of coupling potentials,” Phys. Rev. A 23, 2496-2501 (1981).
[CrossRef]

1952 (2)

R. B. Dingle, “Some magnetic properties of metals. I. general introduction, and properties of large systems of electrons,” Proc. R. Soc. London 211, 500-516 (1952).
[CrossRef]

R. B. Dingle, “Some magnetic properties of metals. III. diamagnetic resonance,” Proc. R. Soc. London 212, 38-47 (1952).
[CrossRef]

1932 (1)

N. Rosen and C. Zener, “Double Stern-Gerlach experiment and related collision phenomena,” Phys. Rev. 40, 502-507 (1932).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, 1972).

Abstreiter, G.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, “Optically programmable electron spin memory using semiconductor quantum dots,” Nature (London) 432, 81-84 (2004).
[CrossRef]

Ager, J. W.

J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, and S. A. Lyon, “Solid-state quantum memory using the P31 nuclear spin,” Nature (London) 455, 1085-1088 (2008).
[CrossRef]

Arai, K.

H. Kosaka, H. Shigyou, Y. Mitsumori, Y. Rikitake, H. Imamura, T. Kutsuwa, K. Arai, and K. Edamatsu, “Coherent transfer of light polarization to electron spins in a semiconductor,” Phys. Rev. Lett. 100, 096602 (2008).
[CrossRef] [PubMed]

Ardavan, A.

J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, and S. A. Lyon, “Solid-state quantum memory using the P31 nuclear spin,” Nature (London) 455, 1085-1088 (2008).
[CrossRef]

Astashkin, A. V.

A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, “Electron spin relaxation times of phosphorus donors in silicon,” Phys. Rev. B 68, 193207 (2003).
[CrossRef]

Atature, M.

M. Atature, J. Dreiser, A. Badolato, and A. Imamoglu, “Observation of Faraday rotation from a single confined spin,” Nat. Phys. 3, 101-106 (2007).
[CrossRef]

Atatüre, M.

M. Atatüre, J. Dreiser, A. Badolato, A. Hogele, K. Karrai, and A. Imamoglu, “Quantum-dot spin-state preparation with near-unity fidelity,” Science 312, 551-553 (2006).
[CrossRef] [PubMed]

Awschalom, D. D.

J. Berezovsky, M. H. Mikkelsen, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Picosecond coherent optical manipulation of a single electron spin in a quantum dot,” Science 320, 349-352 (2008).
[CrossRef] [PubMed]

J. Berezovsky, M. H. Mikkelsen, O. Gywat, N. G. Stoltz, L. A. Coldren, and D. D. Awschalom, “Nondestructive optical measurements of a single electron spin in a quantum dot,” Science 314, 1916-1920 (2006).
[CrossRef] [PubMed]

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204-4207 (1999).
[CrossRef]

D. D. Awschalom, D. Loss, and N. Samarth, Semiconductor Spintronics and Quantum Computation (Springer-Verlag, 2002).

Badolato, A.

M. Atature, J. Dreiser, A. Badolato, and A. Imamoglu, “Observation of Faraday rotation from a single confined spin,” Nat. Phys. 3, 101-106 (2007).
[CrossRef]

M. Atatüre, J. Dreiser, A. Badolato, A. Hogele, K. Karrai, and A. Imamoglu, “Quantum-dot spin-state preparation with near-unity fidelity,” Science 312, 551-553 (2006).
[CrossRef] [PubMed]

Bambini, A.

A. Bambini and P. R. Berman, “Analytic solutions to the two-state problem for a class of coupling potentials,” Phys. Rev. A 23, 2496-2501 (1981).
[CrossRef]

Barenco, A.

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J. J. L. Morton, A. M. Tyryshkin, R. M. Brown, S. Shankar, B. W. Lovett, A. Ardavan, T. Schenkel, E. E. Haller, J. W. Ager, and S. A. Lyon, “Solid-state quantum memory using the P31 nuclear spin,” Nature (London) 455, 1085-1088 (2008).
[CrossRef]

A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, “Electron spin relaxation times of phosphorus donors in silicon,” Phys. Rev. B 68, 193207 (2003).
[CrossRef]

Vandersypen, L. M.

R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79, 1217-1265 (2007).
[CrossRef]

Vandersypen, L. M. K.

T. Meunier, I. T. Vink, L. H. Willems van Beveren, K. J. Tielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz, M. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Experimental signature of phonon-mediated spin relaxation in a two-electron quantum dot,” Phys. Rev. Lett. 98, 126601 (2007).
[CrossRef] [PubMed]

K. C. Nowack, F. H. L. Koppens, Yu. V. Nazarov, and L. M. K. Vandersypen, “Coherent control of a single electron spin with electric fields,” Science 318, 1430-1433 (2007).
[CrossRef] [PubMed]

F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Driven coherent oscillations of a single electron spin in a quantum dot,” Science 442, 766-771 (2006).

R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, “Zeeman energy and spin relaxation in a one-electron quantum dot,” Phys. Rev. Lett. 91, 196802 (2003).
[CrossRef] [PubMed]

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T. Meunier, I. T. Vink, L. H. Willems van Beveren, K. J. Tielrooij, R. Hanson, F. H. L. Koppens, H. P. Tranitz, M. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Experimental signature of phonon-mediated spin relaxation in a two-electron quantum dot,” Phys. Rev. Lett. 98, 126601 (2007).
[CrossRef] [PubMed]

F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Driven coherent oscillations of a single electron spin in a quantum dot,” Science 442, 766-771 (2006).

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R. Hanson, B. Witkamp, L. M. K. Vandersypen, L. H. Willems van Beveren, J. M. Elzerman, and L. P. Kouwenhoven, “Zeeman energy and spin relaxation in a one-electron quantum dot,” Phys. Rev. Lett. 91, 196802 (2003).
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C. Emary, X. Xu, D. G. Steel, S. Saikin, and L. J. Sham, “Fast initialization of the spin state of an electron in a quantum dot in the Voigt configuration,” Phys. Rev. Lett. 98, 047401 (2007).
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R. Vrijen and E. Yablonovitch, “A spin-coherent semiconductor photo-detector for quantum communication,” Physica E (Amsterdam) 10, 569-575 (2001).
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Phys. Rev. Lett. (11)

Y. Wu, E. D. Kim, X. Xu, J. Cheng, D. G. Steel, A. S. Bracker, D. Gammon, S. E. Economou, and L. J. Sham, “Selective optical control of electron spin coherence in singly charged GaAs-Al0.3Ga0.7As quantum dots,” Phys. Rev. Lett. 99, 097402 (2007).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

X. Xu, Y. Wu, B. Sun, Q. Huang, J. Cheng, D. G. Steel, A. S. Bracker, D. Gammon, C. Emary, and L. J. Sham, “Fast spin state initialization in a singly charged InAs-GaAs quantum dot by optical cooling,” Phys. Rev. Lett. 99, 097401 (2007).
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R. Vrijen and E. Yablonovitch, “A spin-coherent semiconductor photo-detector for quantum communication,” Physica E (Amsterdam) 10, 569-575 (2001).
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Proc. IEEE (1)

E. Yablonovitch, H. W. Jiang, H. Kosaka, H. D. Robinson, D. S. Rao, and T. Szkopek, “Optoelectronic quantum telecommunications based on spins in semiconductors,” Proc. IEEE 91, 761-780 (2003).
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Science (8)

J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180-2184 (2005).
[CrossRef] [PubMed]

F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink, K. C. Nowack, T. Meunier, L. P. Kouwenhoven, and L. M. K. Vandersypen, “Driven coherent oscillations of a single electron spin in a quantum dot,” Science 442, 766-771 (2006).

X. Li, Y. Wu, D. G. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809-811 (2003).
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M. V. Gurudev Dutt, L. Childress, L. Jiang, E. Togan, J. Maze, F. Jelezko, A. S. Zibrov, P. R. Hemmer, and M. D. Lukin, “Quantum register based on individual electronic and nuclear spin qubits in diamond,” Science 316, 1312-1316 (2007).
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Figures (27)

Fig. 1
Fig. 1

Schematic energy level structure for the coherent Raman process. Allowed optical transitions are depicted by x and y, which represent the mutually orthogonal polarizations. Δ denotes the off-resonance energy of the excitation lights relative to the transition energies. The Raman condition for the excitation lights is imposed.

Fig. 2
Fig. 2

Rotation angles of the spin are plotted in units of π as a function of the normalized off-resonance energy Δ σ = Δ t p for the pulse areas of (a) 2 π , (b) 4 π , (c) 6 π , (d) 8 π , and (e) 10 π . The inset exhibits schematically the trajectory of the complex number in Eq. (29), starting from Δ σ = Δ t p = 0 . The clockwise winding angle is measured from the point at Δ σ = 0 and is plotted as the rotation angle in the main figure. The trajectory starts from 1 (1) on the real axis for the cases of (a), (c) and (e) [(b) and (d)] and ends at 1 in the limit of Δ σ = for all cases. The complex number is always on the unit circle, but the trajectories are shifted to show the movement clearly.

Fig. 3
Fig. 3

Fidelity of the spin rotation of a single electron is plotted as a function of the normalized off-resonance. Curves (a), (b), and (c) correspond to the pulse area 2 π , 4 π , and 6 π , respectively.

Fig. 4
Fig. 4

Residual population in the excited state | T after the spin rotation of a single electron is plotted as a function of the normalized off-resonance. Curves (a), (b), and (c) correspond to the pulse area 2 π , 4 π , and 6 π , respectively.

Fig. 5
Fig. 5

Fidelity of the spin rotation of a single electron is compared between the cases of a Gaussian pulse and a sech pulse for the 2 π pulse area.

Fig. 6
Fig. 6

Residual population in the excited state | T after the spin rotation of a single electron is compared between the cases of a Gaussian pulse and a sech pulse for the 2 π pulse area.

Fig. 7
Fig. 7

Four-level system composed of two spin states of an electron (lower levels) and two trion states with different hole spin states (upper levels). Allowed optical transitions are indicated by the x and y polarizations.

Fig. 8
Fig. 8

Fidelity of the spin rotation of a single electron is plotted as a function of the normalized off-resonance in the four-level model. Curves (a), (b), and (c) correspond to the pulse area 2 π , 4 π , and 6 π , respectively.

Fig. 9
Fig. 9

Allowed optical transitions in the Faraday configuration for two electrons. The lower levels represent the four spin states of two electrons: the singlet ( S ) and three triplet ( T 1 , T 0 , T 1 ) states, whereas the upper levels exhibit the negatively doubly charged exciton states ( X 2 ) with indexes indicating the spin state of the electron in the excited orbital and the spin state of the heavy hole.

Fig. 10
Fig. 10

Allowed optical transitions in the Voigt configuration for two electrons. The lower levels represent the four spin states of two electrons: the singlet ( S ) and three triplet ( T 1 , T 0 , T 1 ) states, whereas the upper levels exhibit the negatively doubly charged exciton states ( X 2 ) with indexes indicating the spin state of the electron in the excited orbital and the spin state of the light hole.

Fig. 11
Fig. 11

Examples of elementary optical processes in the Voigt configuration for the triplet states T 1 and T 1 , in which two electrons are spin-aligned in the same direction and occupy the ground and excited orbital states. The upper horizontal lines indicate the ground and excited orbital states of the electron, while the lower horizontal line represents the hole level. A thin (thick empty) arrow represents an electron (a hole) with the spin direction along the arrow. An x ( y ) -polarized light creates an electron with the x ( x ) -directed spin and a light hole state | lh + .

Fig. 12
Fig. 12

Five-level system composed of three lower levels and two upper levels. This is a simplest idealized model for studying the effect of overlapping Λ-type transitions.

Fig. 13
Fig. 13

Fidelity of the spin rotation of two electrons is plotted as a function of the normalized off-resonance in the five-level model. Curves (a), (b), and (c) correspond to the pulse area 2 π , 4 π , and 6 π , respectively.

Fig. 14
Fig. 14

Fidelity of the spin rotation of two electrons is plotted as a function of the normalized off-resonance in the five-level model for two cases, namely, one case where initially the population is prepared within the states |0⟩ and |2⟩ with 90% weight and in the state |4⟩ with 10% weight and the other case where the population is prepared within the subspace spanned only by |0⟩ and |2⟩. The pulse area is 2 π .

Fig. 15
Fig. 15

Λ-type transitions for a single electron in the Voigt configuration. The lower levels indicate the two spin states of the electron, whereas the upper levels represent the trion states associated with the light hole or the heavy hole states. The polarization selection rules are given in terms of the x and y bases, where the in-plane magnetic field is applied in the x direction.

Fig. 16
Fig. 16

A Λ-type transition is chosen from the left-hand side of Fig. 15 and the levels are numbered to simplify theoretical expressions.

Fig. 17
Fig. 17

A Λ-type transition is chosen from the right-hand side of Fig. 15 and the levels are numbered to simplify theoretical expressions.

Fig. 18
Fig. 18

Elementary processes of the Faraday rotation for the case of a single resident electron. σ + ( ) denotes the right-(left-) circularly polarized light. The upper (lower) horizontal line indicates the electron (hole) level. A thin (thick empty) arrow represents an electron (a hole) with the spin direction along the arrow.

Fig. 19
Fig. 19

Elementary processes of the Faraday rotation for the triplet T 1 and T 1 states of two resident electrons.

Fig. 20
Fig. 20

Elementary processes of the Faraday rotation for the singlet S state of two resident electrons.

Fig. 21
Fig. 21

Elementary processes of the Faraday rotation for the triplet T 0 state of two resident electrons. X 2 * and X 2 * denote excited states of the doubly negatively charged exciton.

Fig. 22
Fig. 22

Dependence on the probe photon energy ( ω ) of the Faraday rotation angle for the triplet T 0 state and the singlet S state of two resident electrons. It depends on the sign of the energy difference; namely, (a) | E ( X 2 ) E ( X 2 ) | > 0 , (b) | E ( X 2 ) E ( X 2 ) | < 0 .

Fig. 23
Fig. 23

Dependence on the probe photon energy ( ω ) of the Faraday rotation angle for the three triplet states T 1 , T 0 , T 1 and the singlet state S of two resident electrons. Those for T 0 and S are exhibited for the case of | E ( X 2 ) E ( X 2 ) | > 0 .

Fig. 24
Fig. 24

Apparatus to select a particular polarization component of light. It is composed of a HWP, a QWP, a polarizer which is assumed to transmit only a vertically polarized photon, and a photon detector.

Fig. 25
Fig. 25

Parallel combination of two apparatus to select a particular polarization component of photons.

Fig. 26
Fig. 26

Apparatus to select a particular spin component of an electron. It is composed of a spin HWP, a spin QWP, a spin polarizer which is assumed to transmit only the down spin electron, and an electron detector.

Fig. 27
Fig. 27

Parallel combination of two apparatus to select a particular spin component of two electrons.

Tables (2)

Tables Icon

Table 1 Combinations of the Angles h and q of the Fast Axes of the HWP and QWP to Select a Particular Polarization Component of Light

Tables Icon

Table 2 Combinations of the Angles h and q of the Fast Axes of the Spin HWP and Spin QWP to Select a Particular Spin Component of an Electron

Equations (146)

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H = H 0 + V ,
H 0 = E x | x x | + E x ¯ | x ¯ x ¯ | + E T | T T | ,
V = Ω x ( t ) [ e i ω x t | T x | + e i ω x t | x T | ] Ω y ( t ) [ e i ω y t i δ | T x ¯ | + e i ω y t + i δ | x ¯ T | ] ,
i ψ ̇ = H ψ , ψ = c x | x + c x ¯ | x ¯ + c T | T ,
d d t ( c x c x ¯ c T ) = ( i Ω x ( t ) e i ω x t c T i E x ¯ c x ¯ + i Ω y ( t ) e i ω y t + i δ c T i E T c T + i Ω x ( t ) e i ω x t c x + i Ω y ( t ) e i ω y t i δ c x ¯ ) .
c x ¯ ( t ) = e i E x ¯ t c ̃ x ¯ ( t ) , c T ( t ) = e i E T t c ̃ T ( t ) ,
d d t ( c x c ̃ x ¯ c ̃ T ) = ( i Ω x ( t ) e i ( ω x E T ) t c ̃ T i Ω y ( t ) e i ( ω y E T + E x ¯ ) t + i δ c ̃ T i Ω x ( t ) e i ( ω x E T ) t c x + i Ω y ( t ) e i ( ω y E T + E x ¯ ) t i δ c ̃ x ¯ ) .
E T ω x = E T E x ¯ ω y = Δ
Ω x ( t ) = Ω 0 ( t ) cos θ , Ω y ( t ) = Ω 0 ( t ) sin θ .
( c B c D ) = ( cos θ e i δ sin θ e i δ sin θ cos θ ) ( c x c ̃ x ¯ ) = U ( c x c ̃ x ¯ ) ,
d d t ( c B c ̃ T ) = ( i Ω 0 ( t ) e i Δ t c ̃ T i Ω 0 ( t ) e i Δ t c B ) , d d t c D = 0 .
d 2 d t 2 c B = ( Ω ̇ 0 ( t ) Ω 0 ( t ) i Δ ) d d t c B Ω 0 2 ( t ) c B ,
d 2 d t 2 c ̃ T = ( Ω ̇ 0 ( t ) Ω 0 ( t ) + i Δ ) d d t c ̃ T Ω 0 2 ( t ) c ̃ T .
Ω 0 ( t ) = Ω sech ( σ t ) .
ζ = 1 2 ( 1 + tanh ( σ t ) ) ,
ζ ( 1 ζ ) d 2 d ζ 2 c B + ( 1 2 ( 1 + i Δ σ ) ζ ) d d ζ c B + Ω 2 σ 2 c B = 0 ,
ζ ( 1 ζ ) d 2 d ζ 2 c ̃ T + ( 1 2 ( 1 i Δ σ ) ζ ) d d ζ c ̃ T + Ω 2 σ 2 c ̃ T = 0 .
( c B ( t ) c ̃ T ( t ) ) = ( F ( α , α , γ | ζ ) e i Δ t i α γ * ζ γ * F ( α + γ * , α + γ * , 1 + γ * | ζ ) e i Δ t i α γ ζ γ F ( α + γ , α + γ , 1 + γ | ζ ) F ( α , α , γ * | ζ ) ) × ( c B ( ) c ̃ T ( ) )
α = Ω σ , γ = 1 2 ( 1 + i Δ σ ) ,
c B ( ) = finite , c D ( ) = finite , c ̃ T ( ) = 0 ,
c ̃ T ( ) = 0
α γ F ( α + γ , α + γ , 1 + γ | 1 ) = 0 ,
F ( a , b , c | 1 ) = Γ ( c ) Γ ( c a b ) Γ ( c a ) Γ ( c b ) ,
α γ F ( α + γ , α + γ , 1 + γ | 1 ) = α γ Γ ( 1 + γ ) Γ ( 1 γ ) Γ ( 1 α ) Γ ( 1 + α ) .
x Γ ( x ) = Γ ( x + 1 ) , Γ ( x ) Γ ( 1 x ) = π sin π x ,
α γ Γ ( 1 + γ ) Γ ( 1 γ ) Γ ( 1 α ) Γ ( 1 + α ) = α γ γ Γ ( γ ) Γ ( 1 γ ) Γ ( 1 α ) α Γ ( α ) = sin π α sin π γ .
α γ F ( α + γ , α + γ , 1 + γ | 1 ) = sech ( π Δ 2 σ ) sin π α .
F ( α , α , γ | 1 ) = γ * γ , γ * ( γ * + 1 ) γ ( γ + 1 ) , γ * ( γ * + 1 ) ( γ * + 2 ) γ ( γ + 1 ) ( γ + 2 ) , ,
F ( α , α , γ | 1 ) = e i ϕ .
( c B ( ) c D ( ) ) = ( e i ϕ 0 0 1 ) ( c B ( ) c D ( ) ) .
( c x ( ) c ̃ x ¯ ( ) ) = U ( e i ϕ 0 0 1 ) U ( c x ( ) c ̃ x ¯ ( ) ) ,
U ( e i ϕ 0 0 1 ) U = e i ϕ 2 [ cos ϕ 2 1 i sin ϕ 2 ( n σ ) ] = e i ϕ 2 exp [ i ϕ 2 n σ ] = e i ϕ 2 exp [ i ϕ n S ] ,
n = ( sin 2 θ cos δ , sin 2 θ sin δ , cos 2 θ ) , tan θ = Ω y Ω x .
ρ ( ) = | ψ ( ) ψ ( ) | ,
ψ ( ) = c x ( ) | x + c x ¯ ( ) | x ¯ ,
( c x ( ) c x ¯ ( ) ) = ( cos θ i 2 e i φ i 2 sin θ i 2 e i φ i 2 ) ,
ρ ̇ ( t ) = i [ H 0 + V , ρ ] + Γ ρ ,
F = Tr | x , | x ¯ ρ ideal ( ) ρ ( ) c x , c x ¯ ,
c x , c x ¯ = 1 4 π 0 π d θ i sin θ i 0 2 π d φ i .
Γ T x = Γ T x ¯ = 0.01 meV , γ T x = γ T x ¯ = 0.05 meV ,
ω T ω T ¯ = 0.05 meV , ω x ¯ ω x = 0.05 meV ,
ω x ω y = 0.05 meV ,
Γ T x = Γ T x ¯ = Γ T ¯ x = Γ T ¯ x ¯ = 0.01 meV ,
γ T x = γ T x ¯ = γ T ¯ x = γ T ¯ x ¯ = 0.05 meV ,
| = | 3 2 3 2 , | = | 3 2 3 2 ,
| lh + = 1 2 ( | 3 2 1 2 + | 3 2 1 2 ) or | lh = 1 2 ( | 3 2 1 2 + | 3 2 1 2 ) ,
ω 1 ω 3 = 0.05 meV , ω 2 ω 0 = 0.05 meV ,
ω 4 ω 0 = 0.07 meV ,
Γ 1 0 = Γ 1 2 = Γ 1 4 = Γ 3 0 = Γ 3 2 = Γ 3 4 = 0.01 meV ,
γ 10 = γ 12 = γ 14 = γ 30 = γ 32 = γ 34 = 0.05 meV .
ρ = ( ρ 0 ρ 0 ρ 0 ρ 0 ) = 1 2 ( 1 + s σ ) with s = ( s x , s y , s z ) ,
s x = Tr ρ σ x = ρ 0 + ρ 0 , s y = Tr ρ σ y = i ( ρ 0 ρ 0 ) ,
s z = Tr ρ σ z = ρ 0 ρ 0 ,
P = Tr ρ 2 = 1 2 ( 1 + ( s ) 2 ) .
E test ( t ) = ( E x e x + E y e y ) e i ω t + c.c. ,
ρ ( t = 0 ) = ( ρ 00 0 ρ 02 0 0 ρ 20 0 ρ 22 0 0 0 0 0 ) ,
ρ ̇ = i [ H 0 + V , ρ ] + Γ ρ ,
ρ ̇ 00 = i Ω x e i ω t ρ 10 i Ω x * e i ω t ρ 01 + Γ 1 0 ρ 11 ,
ρ ̇ 11 = i Ω x e i ω t ρ 10 i Ω y e i ω t ρ 12 + c.c. ( Γ 1 0 + Γ 1 2 ) ρ 11 ,
ρ 22 = 1 ρ 00 ρ 11 ,
ρ ̇ 01 = i Ω x e i ω t ( ρ 11 ρ 00 ) i Ω y e i ω t ρ 02 + ( i ω 10 γ 01 ) ρ 01 ,
ρ ̇ 02 = i Ω x e i ω t ρ 12 i Ω y * e i ω t ρ 01 + ( i ω 20 γ 02 ) ρ 02 ,
ρ ̇ 12 = i Ω x * e i ω t ρ 02 + i Ω y * e i ω t ( ρ 22 ρ 11 ) + ( i ω 21 γ 12 ) ρ 12 ,
Ω x = μ 01 x E x , Ω y = μ 21 y E y , ω i j = ( E i E j ) ,
ρ 01 ( t ) = ρ ¯ 01 ( t ) e i ω t , ρ 12 ( t ) = ρ ¯ 12 ( t ) e i ω t ,
ρ ¯ ̇ 01 = i Ω x ( ρ 11 ρ 00 ) i Ω y ρ 02 + ( i ( ω 10 ω ) γ 01 ) ρ ¯ 01 ,
ρ ̇ 02 = i Ω x ρ ¯ 12 i Ω y * ρ ¯ 01 + ( i ω 20 γ 02 ) ρ 02 ,
ρ ¯ ̇ 12 = i Ω x * ρ 02 + i Ω y * ( ρ 22 ρ 11 ) + ( i ( ω ω 12 ) γ 12 ) ρ ¯ 12 .
ρ 01 st = i Ω x ρ 00 0 + i Ω y ρ 02 0 i Δ γ 01 , ρ 21 st = i Ω x ρ 20 0 + i Ω y ρ 22 0 i ( Δ ω 20 ) γ 21 with Δ = ω 10 ω .
P = Tr μ ρ st = ( μ 10 x ρ 01 st + μ 12 y ρ 21 st ) e i ω t + c.c.
= χ E e i ω t + c.c. = ( χ x x χ x y χ y x χ y y ) ( E x E y ) e i ω t + c.c.
χ A = A ( ρ 00 0 i ρ 02 0 i ρ 20 0 ρ 22 0 ) A = | μ 01 x | 2 v 0 Δ ,
ϵ = ϵ 0 + 4 π χ A ,
χ B = B ( ρ 22 0 i ρ 20 0 i ρ 02 0 ρ 00 0 ) B = | μ 2 1 ̃ x | 2 ν 0 Δ ̃ Δ ̃ = ω 1 ̃ 2 ω .
χ tot. = χ A + χ B A ( 1 i ( ρ 20 0 + ρ 02 0 ) i ( ρ 20 0 + ρ 02 0 ) 1 ) .
χ A = A 2 ( 1 + s z i s x + s y i s x + s y 1 s z ) ,
ϵ = ϵ 0 + 4 π χ A = ( ϵ 0 + 2 π A ) 1 + 4 π χ A ,
χ A = A 2 ( s z i s x + s y i s x + s y s z ) = A 2 [ s t n n t σ + ( σ × s ) n ] ,
χ A u 1 = χ 1 u 1 , χ A u 2 = χ 2 u 2 .
E probe ( t , z = 0 ) = ( a u 1 + b u 2 ) e i ω t + c.c. = E probe ( + ) e i ω t + c.c.
E probe ( t , z ) = [ a u 1 e i k 0 ϵ 1 z + b u 2 e i k 0 ϵ 2 z ] e i ω t + c.c. ,
k 0 = ω c and ϵ 1 ( 2 ) = ϵ 0 + 2 π A + 4 π χ 1 ( 2 ) = ϵ ̃ 0 + 4 π χ 1 ( 2 ) .
E probe ( t , z ) = e i ω t i k 0 ϵ ̃ 0 z [ a u 1 + b u 2 i 2 π k 0 z ϵ ̃ 0 ( a χ 1 u 1 + b χ 2 u 2 ) + ] + c.c.
= e i ω t i k 0 ϵ ̃ 0 z [ 1 i η χ A + ] E probe ( + ) + c.c.
η = 2 π k 0 z ϵ ̃ 0 .
E trans. = [ 1 i η χ A ] E probe = [ 1 i η χ A ] ( 1 0 ) = ( 1 i η A s z 2 η A ( s x + i s y ) 2 ) .
| D = 1 2 ( 1 1 ) , | D ¯ = 1 2 ( 1 1 ) ,
| D | E trans. | 2 | D ¯ | E trans. | 2 η A s x + O ( ( η A ) 2 ) .
| R = 1 2 ( 1 i ) , | L = 1 2 ( 1 i ) .
| R | E trans. | 2 | L | E trans. | 2 η A s y + O ( ( η A ) 2 ) ,
E trans. = [ 1 i η χ A ] E probe with E probe = 1 2 ( 1 1 )
= 1 2 ( 1 + η A s x 2 i η A ( s y + s z ) 2 1 η A s x 2 i η A ( s y s z ) 2 ) .
| R | E trans. | 2 | L | E trans. | 2 η A s z + O ( ( η A ) 2 ) ,
E ref. = 1 ϵ 1 + ϵ E probe = 1 ϵ ̃ 0 ( 1 + 2 π χ A ϵ ̃ 0 + ) 1 + ϵ ̃ 0 ( 1 + 2 π χ A ϵ ̃ 0 + ) E probe
= 1 ϵ ̃ 0 1 + ϵ ̃ 0 [ 1 + 4 π χ A ( ϵ ̃ 0 1 ) ϵ ̃ 0 + ] E probe
= 1 ϵ ̃ 0 1 + ϵ ̃ 0 [ 1 + η χ A + ] E probe η = 4 π ( ϵ ̃ 0 1 ) ϵ ̃ 0 .
E Kerr = [ 1 + η χ A + ] E probe .
E Kerr = [ 1 + η χ A + ] E probe E probe = ( 1 0 )
= ( 1 + η A s z 2 η A ( i s x + s y ) 2 ) .
| R | E Kerr | 2 | L | E Kerr | 2 η A s x + O ( ( η A ) 2 ) .
| D | E Kerr | 2 | D ¯ | E Kerr | 2 η A s y + O ( ( η A ) 2 ) ,
E probe = | D = 1 2 ( 1 1 ) ,
| H | E Kerr | 2 | V | E Kerr | 2 η A s z + O ( ( η A ) 2 ) ,
| x = 1 2 ( | σ + + | σ ) ,
φ j ( σ + ) | j | P σ + | i | 2 ( E j , i ω ) ( 2 γ j , i 2 + ( E j , i ω ) 2 ) k ( σ ) | k | P σ | i | 2 ( E k , i ω ) ( 2 γ k , i 2 + ( E k , i ω ) 2 ) ,
j ( σ + ) = X 2 , k ( σ ) = X 2 .
E ( X 2 ) 1 2 ( g c μ B B g v μ B B ) + E 0 ,
E ( X 2 ) 1 2 ( g c μ B B g v μ B B ) + E 0 ,
E ν , n = ( | n | + 1 + 2 ν ) Ω + n 2 ω c
with Ω = ω 0 2 + ω c 2 4 , ω c = e B m * c ,
| H = ( 1 0 ) , | V = ( 0 1 ) .
ρ = 1 2 ( 1 + s σ )
s i = Tr ρ σ i ( i = x , y , z ) ,
( a b ) = a | H + b | V ,
( a ¯ b ¯ ) = U HWP ( h ) ( a b )
with U HWP ( h ) = ( cos 2 h sin 2 h sin 2 h cos 2 h ) ,
( a ¯ b ¯ ) = U QWP ( q ) ( a b )
with U QWP ( q ) = e i π 4 1 2 ( i cos 2 q sin 2 q sin 2 q i + cos 2 q ) ,
( a b ) = U QWP ( q ) U HWP ( h ) ( 0 1 ) = 1 2 ( i sin 2 h + sin 2 ( h q ) i cos 2 h cos 2 ( h q ) ) .
| H = ( 1 0 ) , | V = ( 0 1 ) , | D = 1 2 ( 1 1 ) , | R = 1 2 ( 1 i ) ,
N H Tr ρ | H H | = 1 2 ( 1 + s z ) , N V Tr ρ | V V | = 1 2 ( 1 s z ) ,
N D Tr ρ | D D | = 1 2 ( 1 + s x ) , N R Tr ρ | R R | = 1 2 ( 1 s y ) ,
s x = 2 N D N 0 1 , s y = 1 2 N R N 0 , s z = N H N V N 0
N 0 = N H + N V .
ρ = i , j = 0 3 r i j σ i 1 σ j 2 ,
| H | , | V | ,
( a b ) ( e i ω z t a e i ω z t b ) = e i ω z t ( a e i 2 ω z t b ) ,
| h = ( cos h 2 sin h 2 ) , | h = ( sin h 2 cos h 2 ) .
( a ¯ b ¯ ) = U HWP s ( h ) ( a b )
with U HWP s ( h ) = ( cos h sin h sin h cos h ) .
( a ¯ b ¯ ) = U QWP s ( q ) ( a b )
U QWP s ( q ) = e i π 4 1 2 ( i + cos q sin q sin q i cos q ) ,
( a b ) = U QWP s ( q ) U HWP s ( h ) ( 0 1 ) = 1 2 ( i sin h + sin ( h q ) i cosh + cos ( h q ) ) .
ρ = i , j = 0 3 r i j σ i 1 σ j 2 ,
| c = | s | , | c = | s | ,
| v 1 = | 3 2 , 3 2 = | 1 , 1 | ,
| v 2 = | 3 2 , 1 2 = 2 3 | 1 , 0 | + 1 3 | 1 , 1 | ,
| v 3 = | 3 2 , 1 2 = 1 3 | 1 , 1 | + 2 3 | 1 , 0 | ,
| v 4 = | 3 2 , 3 2 = | 1 , 1 | ,
p σ + = p c v a c r a v 4 r + 1 3 p c v a c r a v 3 r + h.c. ,
p σ = p c v a c r a v 1 r 1 3 p c v a c r a v 2 r + h.c. ,
| c + = 1 2 ( | c + | c ) , | c = 1 2 ( | c + | c ) ,
| v h + = 1 2 ( | v 1 + | v 4 ) , | v h = 1 2 ( | v 1 + | v 4 ) ,
| v l + = 1 2 ( | v 2 + | v 3 ) , | v l = 1 2 ( | v 2 + | v 3 ) ,
p x = p c v ( a c + r a v h r + a c r a v h + r ) + 1 3 p c v ( a c + r a v l r a c r a v l + r ) + h.c. ,
p y = i p c v ( a c + r a v h + r + a c r a v h r ) i 1 3 p c v ( a c + r a v l + r a c r a v l r ) + h.c. .

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