## Stochastic decoherence of qubits

Optics Express, Vol. 8, Issue 2, pp. 145-152 (2001)

http://dx.doi.org/10.1364/OE.8.000145

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### Abstract

We study the stochastic decoherence of qubits using the Bloch equations and the Bloch sphere description of a two-level atom. We show that it is possible to describe a general decoherence process of a qubit by a stochastic map that is dependent on 12 independent parameters. Such a stochastic map is constructed with the help of the damping basis associated with a Master equation that describes the decoherence process of a qubit.

© Optical Society of America

## 1 Introduction

*b*⃗=(

*u*,

*v*,

*w*) gives a simple, yet powerful, geometrical description of the coherent and the incoherent dynamics of a two-level atom. The classical by now textbook of Allen and Eberly [1] provides a detailed description of the two-level dynamics. In this description the coherent dynamics is represented by a rotation of the Bloch vector on a Bloch sphere defined as

*u*

^{2}+

*v*

^{2}+

*w*

^{2}=const, while the incoherent dynamics leads to a spiraling of the Bloch vector into a steady state that is not necessarily on the Bloch sphere. The physical source of damping characterizing the incoherent part of the dynamics has been attributed to quantum or stochastic fluctuations of various properties of the reservoir coupled to the two-level atom. Perhaps the best known source of decoherence is the spontaneous emission damping due to quantum vacuum fluctuations of the electromagnetic field [2]. Other possible sources of noise can involve stochastic phase, frequency or amplitude fluctuations. There is vast literature devoted to stochastic models of collisions, phase diffusion or frequency fluctuations [3]. In the framework of a theory involving a driven two-level atom by a multiplicative Gaussian white noise, the Bloch vector description is still valid, with damping rates given by the diffusion coefficients of the corresponding fluctuations.

5. C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory **44**, 2724–2748 (1998). [CrossRef]

## 2 Decoherence of two level atoms

*T*

_{2}gives a decay rate of the atomic dipole moment characterized by

*u*and

*v*, while the longitudinal lifetime

*T*

_{1}gives the decay rate of the atomic inversion

*w*into an equilibrium state

*w*

_{eq}. The Bloch equations have been applied as the most successful tool used in quantum optics to describe and to study the impact of incoherent effects on the two-level atom dynamics.

## 3 Decoherence of qubits

*B*matrix is transformed as follows:

*i.e.*, it has to satisfy the condition:

*ρ*)≥0. Such positive maps play a central role in quantum information theory [6

6. M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on *M*_{2},” preprint quantum-ph/0005004, http://xxx.lanl.gov/

*ρ*) directly from the Bloch equations.

*Stochastic map for spontaneous emission noise*

*w*

_{eq}=-1. The damping terms are characterized by the Einstein A coefficient of spontaneous emission. In geometrical terms, the dynamics of a coherently driven qubit with spontaneous noise can be seen as the following three operations on the Bloch sphere. Due to coherent interaction, the first operation amounts to a rotation of the Bloch vector. The spontaneous emission noise leads to two additional operations: a damping and a linear shift of the Bloch vector. In this case we have Φ

_{A}:

*B*↦

*B*

^{′}with

**M**=

**RN**

_{A}has been written as a product of a coherent rotation

**R**and the spontaneous emission noise damping

**N**

_{A}. This damping noise matrix is diagonal

*Stochastic map for frequency diffusion noise*

7. K. Wódkiewicz and J. H. Eberly, “Random telegraph theory of effective Bloch equations with applications to free induction decay,” Phys. Rev. A **32**, 992–1001 (1985). [CrossRef] [PubMed]

*w*

_{eq}=0,

*i.e.*, the steady inversion becomes thermal. The damping rate

_{Γ}leads to no translation of the Bloch vector, i.e.,

*b*⃗

_{0}=0 and as in the case of the spontaneous emission noise

**M**=

**RN**

_{Γ}with a diagonal damping matrix

**N**

_{Γ}with eigenvalues

## 4 Arbitrary stochastic noise

*x*3 matrix M and by 3 real shift parameters

*b*⃗

_{0}. This means that an arbitrary linear stochastic map is fully characterized by an affine transformation labeled by 12 real independent parameters.

**M**=

**RN**, where the damping matrix

**N**is symmetric and is characterized by 6 real parameters. The damping matrix

**N**can be diagonalized by another orthogonal transformation

**T**, leading to a diagonal matrix

**Λ**with eigenvalues Λ

_{1}, Λ

_{2}, Λ

_{3}. Each of the two orthogonal transformations

**R**and

**T**corresponds geometrically to a rotation (with inversion) of the Bloch vector on the Bloch sphere. The resulting matrix

**Λ**describes the three fundamental decoherence damping eigenvalues of the qubit in some selected principal axes of the Bloch sphere.

*b*⃗·

*b*⃗=1, while mixed states correspond to

*b*⃗·

*b*⃗≤1. Under the stochastic map (5) the density operator becomes

*h*

_{R}and

*h*

_{T}correspond to the two orthogonal transformations (rotations) of the Bloch vector and

*S*

_{L,R}are some superoperators acting on the density operator. The form of these superoperators remains to be established. In the next section we provide an explicit construction of such a transformation using a Master equation.

8. K. Kraus, *States, Effects and Operations: Fundamental Notions of Quantum Theory* (Springer-Verlag, Berlin Heidelberg, 1983). [CrossRef]

## 5 General stochastic transformation

*F*

_{i}and

10. H. J. Briegel and B. -G. Englert, “Quantum optical master equations: The use of damping bases,” Phys. Rev. A **47**, 3311–3328 (1993). [CrossRef] [PubMed]

*ρ*(0). Such a time evolution is given by

_{λ}=

*e*

^{λt}and give the dynamical evolution of the density operator. From the properties of the damping basis (24) we obtain

## 6 Stochastic noise of bloch equations

10. H. J. Briegel and B. -G. Englert, “Quantum optical master equations: The use of damping bases,” Phys. Rev. A **47**, 3311–3328 (1993). [CrossRef] [PubMed]

*u*

^{2}+

*v*

^{2}+

*w*

^{2}=1 is mapped into the interior of the Bloch ball. This is obtained if:

6. M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on *M*_{2},” preprint quantum-ph/0005004, http://xxx.lanl.gov/

11. C. King and M. B. Ruskai, “Minimal Entropy of States Emerging from Noisy Channels,” preprint quantum-phy/9911079, http://xxx.lanl.gov/

## 7 Conclusions

## Acknowledgments

## References and links

1. | L. Allen and J. H. Eberly, |

2. | G. S. Agarwal, |

3. | C. W. Gardiner, |

4. | M. B. Plenio and P. L. Knight, “Realistic lower bounds for the factorisation time of large numbers on a quantum computer,” Phys. Rev. A |

5. | C. H. Benett and P. W. Shore, “Quantum Information Theory,” IEEE Trans. Info. Theory |

6. | M. B. Ruskai, S. Szarek, and E. Werner, “A Characterisation of Completely-Positive Trace Preserving Maps on |

7. | K. Wódkiewicz and J. H. Eberly, “Random telegraph theory of effective Bloch equations with applications to free induction decay,” Phys. Rev. A |

8. | K. Kraus, |

9. | C. W. Gardiner, |

10. | H. J. Briegel and B. -G. Englert, “Quantum optical master equations: The use of damping bases,” Phys. Rev. A |

11. | C. King and M. B. Ruskai, “Minimal Entropy of States Emerging from Noisy Channels,” preprint quantum-phy/9911079, http://xxx.lanl.gov/ |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 17, 2000

Published: January 15, 2001

**Citation**

Krzysztof Wodkiewicz, "Stochastic decoherence of qubits," Opt. Express **8**, 145-152 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-145

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### References

- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
- G. S. Agarwal, Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Springer, Berlin, Heidelberg, 1974), Vol. 70.
- C. W. Gardiner, Handbook of Stochastic Processes (Springer, Berlin, Heidelberg, 1984).
- M. B. Plenio and P. L. Knight, "Realistic lower bounds for the factorisation time of large numbers on a quantum computer," Phys. Rev. A 53, 2986-2990 (1996). [CrossRef] [PubMed]
- C. H. Benett and P. W. Shore, "Quantum Information Theory," IEEE Trans. Info. Theory 44, 2724-2748 (1998). [CrossRef]
- M. B. Ruskai, S. Szarek and E. Werner, "A Characterisation of Completely-Positive Trace Pre-serving Maps on M 2," preprint quantum-ph/0005004, http://xxx.lanl.gov/
- K. W�dkiewicz and J. H. Eberly, "Random telegraph theory of effective Bloch equations with applications to free induction decay," Phys. Rev. A 32, 992-1001 (1985). [CrossRef] [PubMed]
- K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory (Springer-Verlag, Berlin Heidelberg, 1983). [CrossRef]
- C. W. Gardiner, Quantum Noise (Springer, Berlin, Heidelberg,1991).
- H. J. Briegel and B. -G. Englert, "Quantum optical master equations: The use of damping bases," Phys. Rev. A 47, 3311-3328 (1993). [CrossRef] [PubMed]
- C. King and M. B. Ruskai, "Minimal Entropy of States Emerging from Noisy Channels," preprint quantum-phy/9911079, http://xxx.lanl.gov/

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