Abstract

We show that a configuration of four birefringent crystals and wave-plates can emulate almost any arbitrary unital channel for polarization qubits encoded in single photons, where the channel settings are controlled by the wave-plate angles. The scheme is applied to a single spatial mode and its operation is independent of the wavelength and the fine temporal properties of the input light. We implemented the scheme and demonstrated its operation by applying a dephasing environment to classical and quantum single-photon states with different temporal properties. The applied process was characterized by a quantum process tomography procedure, and a high fidelity to the theory was observed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
  13. M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
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    [Crossref]
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  19. C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory 47(1), 192–209 (2001).
    [Crossref]
  20. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
    [Crossref]
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    [Crossref]
  22. J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005).
    [Crossref]
  23. A. Shaham and H. S. Eisenberg, “Quantum process tomography of single-photon quantum channels with controllable decoherence,” Phys. Scr. T147, 014029 (2012).
    [Crossref]
  24. A. Shaham and H. S. Eisenberg, “Realizing controllable depolarization in photonic quantum-information channels,” Phys. Rev. A 83(2), 022303 (2011).
    [Crossref]
  25. R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41(12), 2315–2323 (1994).
    [Crossref]

2018 (1)

2012 (2)

A. Shaham and H. S. Eisenberg, “Realizing a variable isotropic depolarizer,” Opt. Lett. 37(13), 2643–2645 (2012).
[Crossref]

A. Shaham and H. S. Eisenberg, “Quantum process tomography of single-photon quantum channels with controllable decoherence,” Phys. Scr. T147, 014029 (2012).
[Crossref]

2011 (3)

A. Shaham and H. S. Eisenberg, “Realizing controllable depolarization in photonic quantum-information channels,” Phys. Rev. A 83(2), 022303 (2011).
[Crossref]

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

2009 (3)

E. Amselem and M. Bourennane, “Experimental four-qubit bound entanglement,” Nat. Phys. 5(10), 748–752 (2009).
[Crossref]

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

J.-S. Xu, C.-F. Li, X.-Y. Xu, C.-H. Shi, X.-B. Zou, and G.-C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103(24), 240502 (2009).
[Crossref]

2008 (1)

2007 (2)

R. B. A. Adamson, L. K. Shalm, and A. M. Steinberg, “Preparation of pure and mixed polarization qubits and the direct measurement of figures of merit,” Phys. Rev. A 75(1), 012104 (2007).
[Crossref]

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

2005 (2)

G. Puentes, D. Voigt, A. Aiello, and J. P. Woerdman, “Experimental observation of universality in depolarized light scattering,” Opt. Lett. 30(23), 3216–3218 (2005).
[Crossref]

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005).
[Crossref]

2004 (1)

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

2001 (2)

C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory 47(1), 192–209 (2001).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

2000 (2)

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62(6), 063808 (2000).
[Crossref]

P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
[Crossref]

1999 (1)

J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A 59(6), 4249–4254 (1999).
[Crossref]

1997 (1)

I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt. 44(11-12), 2455–2467 (1997).
[Crossref]

1996 (1)

M. Horodecki and R. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54(3), 1838–1843 (1996).
[Crossref]

1994 (1)

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41(12), 2315–2323 (1994).
[Crossref]

1972 (1)

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3(4), 275–278 (1972).
[Crossref]

Adamson, R. B. A.

R. B. A. Adamson, L. K. Shalm, and A. M. Steinberg, “Preparation of pure and mixed polarization qubits and the direct measurement of figures of merit,” Phys. Rev. A 75(1), 012104 (2007).
[Crossref]

Aiello, A.

Almeida, M. P.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

Altepeter, J. B.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005).
[Crossref]

P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
[Crossref]

Álvarez, J.-R.

Amselem, E.

E. Amselem and M. Bourennane, “Experimental four-qubit bound entanglement,” Nat. Phys. 5(10), 748–752 (2009).
[Crossref]

Banaszek, K.

Berglund, A. J.

P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
[Crossref]

Bourennane, M.

E. Amselem and M. Bourennane, “Experimental four-qubit bound entanglement,” Nat. Phys. 5(10), 748–752 (2009).
[Crossref]

Branning, D.

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62(6), 063808 (2000).
[Crossref]

Breuer, H.-P.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Calderón-Losada, O.

Cerf, N. J.

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

Chiuri, A.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

Chuang, I. L.

I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt. 44(11-12), 2455–2467 (1997).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, U.K., 2000).

Cirac, J. I.

J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A 59(6), 4249–4254 (1999).
[Crossref]

Davidovich, L.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

De Martini, F.

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

de Melo, F.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

Eisenberg, H. S.

A. Shaham and H. S. Eisenberg, “Realizing a variable isotropic depolarizer,” Opt. Lett. 37(13), 2643–2645 (2012).
[Crossref]

A. Shaham and H. S. Eisenberg, “Quantum process tomography of single-photon quantum channels with controllable decoherence,” Phys. Scr. T147, 014029 (2012).
[Crossref]

A. Shaham and H. S. Eisenberg, “Realizing controllable depolarization in photonic quantum-information channels,” Phys. Rev. A 83(2), 022303 (2011).
[Crossref]

Ekert, A. K.

J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A 59(6), 4249–4254 (1999).
[Crossref]

Filip, R.

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

Fiurášek, J.

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

Giacomini, S.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

Guo, G.-C.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

J.-S. Xu, C.-F. Li, X.-Y. Xu, C.-H. Shi, X.-B. Zou, and G.-C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103(24), 240502 (2009).
[Crossref]

Hor-Meyll, M.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

Horodecki, M.

M. Horodecki and R. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54(3), 1838–1843 (1996).
[Crossref]

Horodecki, R.

M. Horodecki and R. Horodecki, “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54(3), 1838–1843 (1996).
[Crossref]

Huang, Y.-F.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Huelga, S. F.

J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A 59(6), 4249–4254 (1999).
[Crossref]

Imai, H.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

James, D. F. V.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

Jamiolkowski, A.

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys. 3(4), 275–278 (1972).
[Crossref]

Jeffrey, E. R.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005).
[Crossref]

Jozsa, R.

R. Jozsa, “Fidelity for mixed quantum states,” J. Mod. Opt. 41(12), 2315–2323 (1994).
[Crossref]

Karpinski, M.

King, C.

C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory 47(1), 192–209 (2001).
[Crossref]

Kwiat, P. G.

J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat, “Photonic state tomography,” Adv. At., Mol., Opt. Phys. 52, 105–159 (2005).
[Crossref]

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, “Experimental verification of decoherence-free subspaces,” Science 290(5491), 498–501 (2000).
[Crossref]

Laine, E.-M.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Li, C.-F.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

J.-S. Xu, C.-F. Li, X.-Y. Xu, C.-H. Shi, X.-B. Zou, and G.-C. Guo, “Experimental characterization of entanglement dynamics in noisy channels,” Phys. Rev. Lett. 103(24), 240502 (2009).
[Crossref]

Li, L.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Liu, B.-H.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Macchiavello, C.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, “Distributed quantum computation over noisy channels,” Phys. Rev. A 59(6), 4249–4254 (1999).
[Crossref]

Mataloni, P.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

Migdall, A. L.

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62(6), 063808 (2000).
[Crossref]

Munro, W. J.

D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A 64(5), 052312 (2001).
[Crossref]

Nielsen, M. A.

I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the dynamics of a quantum black box,” J. Mod. Opt. 44(11-12), 2455–2467 (1997).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, Cambridge, U.K., 2000).

Nuñez, M.

O’Brien, J. L.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

Pádua, S.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

Piilo, J.

B.-H. Liu, L. Li, Y.-F. Huang, C.-F. Li, G.-C. Guo, E.-M. Laine, H.-P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7(12), 931–934 (2011).
[Crossref]

Puentes, G.

Radzewicz, C.

Ribeiro, P. H. S.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

Ricci, M.

M. Ricci, F. De Martini, N. J. Cerf, R. Filip, J. Fiurášek, and C. Macchiavello, “Experimental purification of single qubits,” Phys. Rev. Lett. 93(17), 170501 (2004).
[Crossref]

Rosati, V.

A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchiavello, and P. Mataloni, “Experimental realization of optimal noise estimation for a general Pauli channel,” Phys. Rev. Lett. 107(25), 253602 (2011).
[Crossref]

Ruskai, M. B.

C. King and M. B. Ruskai, “Minimal entropy of states emerging from noisy quantum channels,” IEEE Trans. Inf. Theory 47(1), 192–209 (2001).
[Crossref]

Salles, A.

M. P. Almeida, F. de Melo, M. Hor-Meyll, A. Salles, S. P. Walborn, P. H. S. Ribeiro, and L. Davidovich, “Environment-induced sudden death of entanglement,” Science 316(5824), 579–582 (2007).
[Crossref]

Sergienko, A. V.

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62(6), 063808 (2000).
[Crossref]

Shaham, A.

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Figures (4)

Fig. 1.
Fig. 1. (a) The experimental setup for the characterization of single-qubit channels. Photon pairs are generated in the BBO crystal, which is located after a lens (L1). The down-converted signal is filtered by passing through a dichroic mirror (DM), a single-mode fiber (SM) and an interference bandpass filter (IF). The photon pairs are split into two ports using a BS, where single-photon detectors (DET$_1$, DET$_2$) are located at the end of each port. The investigated channel ($\mathcal {E}$) is applied only on photons that emerge from one of the ports. The initial polarization state of these photons is determined using a polarizer (POL), a HWP ($\lambda /2$) and a QWP ($\lambda /4$), and their final polarization state is measured using a HWP, a QWP and a polarizer that are placed after the channel. (b) The four-crystal scheme, composed of four perpendicularly-oriented calcite crystals and three HWPs. The thickness of the two outer (inner) crystals is 1 mm (2 mm). (c) A Soleil-Babinet compensator, composed of two translatable quartz wedges and another rectangular fixed quartz crystal (see details in the appendix).
Fig. 2.
Fig. 2. Numerical investigation of the possible channels of the four-crystal scheme in the tetrahedron representation. (a) All possible channels using different orientations of the three HWPs of the four-crystal scheme. (b) All possible channels using the different orientations of the three HWPs of the four-crystal scheme, with the addition of all allowed polarization rotations. These rotations can be implemented using more wave-plates that are placed after the scheme. Points on the dashed yellow line along the edge of the tetrahedron in (a) correspond to a dephasing channel that preserves the length of $D_3$.
Fig. 3.
Fig. 3. Four-crystal scheme as a dephasing channel. (a) Calculated eigenvalues of the $\chi$ matrix as a function of the first HWP angle. (b) Measured eigenvalues of the $\chi$ matrix as a function of the first HWP angle for the dephasing range ($0\leq \theta _1\leq 10^\circ$, gray area of (a)). Eigenvalues that were reconstructed from counts of a single detector (coincidence measurements) are presented as upright triangles (upside-down triangles). Solid lines are the theoretical prediction.
Fig. 4.
Fig. 4. (a) The eigenvalues of the $\chi$ matrix as a function of the time delay $t$ between the two polarization modes. Eigenvalues of process matrices that were reconstructed from counts of single detector (coincidence measurements) are presented as red upright triangles (blue upside-down triangles). Solid lines are Gaussian fits to lead the eye. (b) The oscillations of the Stokes parameter $S_2$ as a function of the time delay between the two polarization modes. Solid line represents a sinusoidal fit.

Equations (13)

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E ( ρ ^ ) = m , n χ m n σ m ρ ^ σ n ,
E ( ρ ^ ) = ( 1 P ) ρ ^ + P σ 3 ρ ^ σ 3 .
P = 1 χ 0 .
χ = 1 4 ( I I + 3 m , n = 1 D m n σ m σ n ) .
| D i ± D j | | 1 ± D k | ,
D i = χ 0 + χ i χ j χ k ,
L Δ n c τ .
D 1 = sin ( 4 θ 1 ) sin ( 4 θ 3 ) cos 2 ( 2 θ 2 ) + cos ( 4 θ 1 ) cos ( 4 θ 2 ) cos ( 4 θ 3 ) ,
D 2 = sin ( 4 θ 2 ) sin ( 2 θ 1 + 2 θ 3 ) cos ( 2 θ 1 ) cos ( 2 θ 3 ) cos 2 ( 2 θ 2 ) cos 2 ( 2 θ 1 + 2 θ 3 ) 1 2 sin ( 4 θ 1 ) sin ( 4 θ 3 ) sin 2 ( 2 θ 2 ) ,
D 3 = sin ( 4 θ 2 ) sin ( 2 θ 1 + 2 θ 3 ) cos ( 2 θ 1 ) cos ( 2 θ 3 ) cos 2 ( 2 θ 2 ) cos 2 ( 2 θ 1 + 2 θ 3 ) 1 2 sin ( 4 θ 1 ) sin ( 4 θ 3 ) sin 2 ( 2 θ 2 ) .
θ 1 = θ 3 = θ 2 2 .
P = 1 χ m = 3 cos 4 ( 4 θ 1 ) + 2 cos 2 ( 4 θ 1 ) + 1 2 .
F ( χ ^ 1 , χ ^ 2 ) = ( Tr ( χ ^ 1 χ ^ 2 χ ^ 1 ) ) 2 .

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