## Multimode unraveling of master equation and decoherence problem

Optics Express, Vol. 2, Issue 9, pp. 347-354 (1998)

http://dx.doi.org/10.1364/OE.2.000347

Acrobat PDF (278 KB)

### Abstract

An unraveling of master equation for a set of fields interfering with one another is developed and conditions are found under which decoherence can be avoided for conditional and unconditional evolution of one of this fields.

© Optical Society of America

*t*

_{eq}to the state of thermal equilibrium. In quantum physics a system coupled to a reservoir displays yet another fundamental phenomenon: in the timescale much shorter than

*t*

_{eq}the density matrix of quantum system becomes diagonal in some basis, defined by the system-reservoir interaction Hamiltonian, and then tends to thermal equilibrium preserving this diagonality [1–3

1. W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?,” Phys. Rev. D **24**, 1516 (1981). [CrossRef]

4. S. M. Barnett and S. J. D. Phoenix, “The principles of quantum cryptography,” Phil. Trans. R. Soc. Lond. A **354**, 793 (1996). [CrossRef]

5. D. P. DiVincenzo, “Quantum computation,” Science **270**, 255 (1995). [CrossRef]

*a*and

*a*

^{+}respectively. One of the cavity mirrors is semi-transparent, so that the field in the cavity interacts with the modes of external, initially vacuum field. The density operator of such a system obeys a well-known master equation of Lindblad form:

*H*=

*ωa*

^{+}

*a*is a Hamiltonian of internal evolution,

*ω*is the mode frequency,

*γ*

_{a}is the decay constant of the cavity and

*ħ*is set to unity. The solution of Eq. (1) can be represented as integral over so-called quantum trajectories [6–8

6. M. D. Srinivas and E. B. Davies, “Photon counting probabilities in quantum optics,” Opt.Acta **28**, 981 (1981). [CrossRef]

*J*

_{a}and

*S*(

*t*) are defined as follows:

*L*is any Liuvillian superoperator, can be unraveled in the following way:

*J*, choice of which determines the physical sense of such an expansion.

*ρ*(

*t*|

*t*

_{1},…,

*t*

_{n}) represents the conditional (non-normalized) density operator of the cavity field given exactly

*n*photons were registered by the photodetector in the time interval [0,

*t*) at times

*t*

_{1},…,

*t*

_{n}, while its normp

*p*(

*t*|

*t*

_{1},…,

*t*

_{n}) =

*Tr*{

*ρ*(

*t*|

*t*

_{1},…,

*t*

_{n})} gives the probability of the corresponding outcome. In this interpretation Eq. (2) can be considered just as an averaging over all possible detector records. The structure of Eq. (3) shows that the conditional state of the field under continuous measurement consists of jumps at times

*t*

_{1}, …,

*t*

_{n}described by superoperator

*J*

_{a}and continuous non-unitary evolution between the counts, described by superoperator

*S*(

*t*). It is easy to see that if the initial state of the field is a pure state

*ρ*(0) = |

*ψ*

_{a}(0)〉 〈

*ψ*

_{a}(0)| it remains pure under conditional evolution:

*e*

^{-γata+a/2}in Eq. (8) all conditional states tend to vacuum (equilibrium) state, however without any decoherence, which arises only as a result of averaging over all possible trajectories, given by Eq. (2). This provides the following interpretation of the decoherence process: due to coupling of the cavity mode to external modes on the semi-transparent mirror, photons leave cavity at some times

*t*

_{1}, …,

*t*

_{n}and the state of the system decays to vacuum. If we know exactly these times, gathering all photons on the surface of a photodetector, the conditional state of the field preserves its quantum coherence and only the loss of information connected with non-unitary detector efficiency or not perfect photon resolution brings about the decoherence of the system state. Thus, to avoid decoherence caused by interaction with a set of bosonic field modes, it is enough to collect all outgoing radiation on a photodetector and resolve all counts - the conditional state of the system with respect to observed photocount sequence will remain pure.

*t*

_{1},…,

*t*

_{n}, it possesses a fundamental property of interference with other fields and itself. Moreover, in many experiments one is interested in phase properties of the cavity field, which are measured with the use of homodyne techniques, that is the external field undergoes interference with the field of local oscillator before being measured. So we should answer the following question: can we avoid in principle de-coherence of the cavity state if the external field is subject to interference with other auxiliary field, so that the times

*t*

_{1}, …,

*t*

_{n}at which photons left the cavity cannot be more obtained directly by photodetection? The answer on this question is non-trivial and generally depends on the state of the auxiliary field.

*b*is initially exited in the state |

*ψ*

_{b}(0)〉. The external fields of the two cavities are mixed on a symmetric beam-splitter and measured by two photodetectors. The master equation for the joint density operator of modes

*a*and

*b*is

*γ*

_{b}and

*H*

_{b}, are the decay rate and Hamiltonian of the corresponding cavity. Eq. (9) can be unraveled in such a way that it will provide an expression for the conditional state of two modes for given photocount sequences at two detectors. We make this unraveling using the rule that the jump of the conditional wave function after detecting a photon at a photodetector is given by the operator of the field incident on this detector [7].

*S*(

*t*), satisfying the equation

*J*

_{a},

*J*

_{b}and

*S*(

*t*) preserve the purity of the state. So the conditional state of two modes is a pure state at any time provided that it was pure at the beginning.

*t*

_{1}, and one at the second detector at time

*t*

_{2}>

*t*

_{1}, the conditional (non-normalized) state at time

*t*>

*t*

_{1},

*t*

_{2}is

*a*(first term), (ii) first photon came from the cavity

*b*, second came from the cavity

*a*(second term), (iii) vice versa (third term), and (iv) both photons came from the cavity

*b*(forth term). as we can in no way determine the paths of photons the resulting wavefunction is a superposition of four possible wavefunctions.

*a*alone! The reason is that the states of two modes become entangled, that is the joint state does not factorize:

*b*, the state of mode a becomes a mixture.

*S*(

*t*) does not entangle states:

*ψ*

_{a}〉 or |

*ψ*

_{b}〉 is a coherent state. It is easy to see from the implicit form of the superoperators

*J*

_{a},

*J*

_{b}, and

*S*(

*t*) that coherent state of mode

*a*or

*b*remains coherent under conditional evolution of the joint state. So we can state that the joint conditional state of two modes under continuous measurement performed after interference on a beam-splitter is disentangled if and only if one of these modes is in a coherent state initially.

*n*single mode fields of the same frequency

*ω*described by operators

*a*

_{i},

*i*= 1, …,

*n*in

*n*cavities with decay rates

*γ*

_{i},

*i*= 1, …,

*n*. Such a system is described by master equation

*H*

_{i}=

*a*

_{i}is the Hamiltonian of internal evolution of the

*i*th mode. We suppose that the outgoing fields of the cavities are mixed with one another on a set of beam splitters and the resulting

*n*beams are detected by

*n*photodetectors. The field operators on the detectors

*U*

_{ij}=

*i*th detector brings about a reduction

*ρ*→

*S*(

*t*):

*n*-mode field is again a pure and generally entangled state.

*a*and decay rate

*γ*

_{a}. Initially the field inside the cavity is exited in a superposition of coherent states with amplitudes

*α*and -

*α*(a so-called Schrodinger cat state [10

10. V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. **36** 3(1995). [CrossRef]

*φ*is arbitrary real phase and

*N*is a normalization constant. The conditional evolution of the state vector of the field under continuous measurement is given by Eq. (8). Substituting Eq. (28) into Eq. (8) we obtain

*n*is number of counts in the time interval [0,

*t*). Eq. (29) shows that the conditional state of field remains a superposition of two coherent states with altering sign between them [11

11. B. M. Garraway and P. L. Knight, “Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space,” Phys. Rev. A **50**, 2548 (1994). [CrossRef] [PubMed]

*α*〉 〈

*α*| and |-

*α*〉 〈-

*α*|. Decoherence again arises as a result of averaging over all possible detector records, however in this special case the conditional state does not depend on times

*t*

_{1}, …,

*t*

_{n}of photon arrivals, but only on total number of photons in the interval [0,

*t*), or more precisely there are only two possible wave functions |

*t*)〉: one for even and one for odd values of

*n*, which differ only by sign between states |

*α*〉 and |-

*α*〉. When a photon leaves the cavity this sign is switched to the opposite value. This sign switching has a simple physical interpretation only for the case of

*π*, and can be realized by the phase shift operator exp{

*iπa*

^{+}

*a*}:

12. D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. **78**, 840 (1997). [CrossRef]

*π*, which must be performed during a time small compared to cavity photon lifetime

*α*|

^{-2}. In such a scheme the conditional state of the field is given, instead of Eq. (8), by the following expression:

12. D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. **78**, 840 (1997). [CrossRef]

*H*and initial state |

*ψ*

_{a}(0)〉. We may describe the effect of instant feedback by some operator

*U*and write a more general form of Eq. (31):

*t*to be independent of photon arrival times

*t*

_{n}and photocount number

*n*, that is

*c*(

*t*|

*n*,

*t*

_{1}, …

*t*

_{n}) is a c-number defining the probability of the corresponding count sequence and |

*ψ′*

_{a}(

*t*)〉 is some vector depending only on time. Letting

*n*= 0 in Eq. (35) we find that |

*ψ′*

_{a}(

*t*)〉 can be chosen as

*ψ′*

_{a}(

*t*)〉,

*t*∈ [0, +∞) is an eigenstate of operator

*Ua*:

*c*(

*t*) is a c-number. So, if for given

*H*and |

*ψ*

_{a}(0)〉 we find an operator

*U*satisfying Eq. (37) and the corresponding physical process, which can be controlled by feedback loop, then the decoherence of intracavity field induced by decay can be completely removed.

## Appendix

*ψ*

_{a}〉, |

*ψ*

_{b}〉, |

*ψ′*

_{a}〉 |

*ψ′*

_{b}〉 |

*ψ″*

_{a}〉, and |

*ψ″*

_{b}〉. Multiplying both sides of Eq. (A1) by bra vector 〈

*φ*

_{a}| such that 〈

*φ*

_{a}|

*ψ*

_{a}〉 = 0 we obtain

*c*is a non-zero c-number, or (ii)

*ψ*

_{a}| gives

*ψ*

_{b}〉 is an eigenstate of operator

*b*(coherent state).

*a*|

*ψ*

_{a}〉 ⊥ |

*φ*

_{a}〉 for any |

*ψ*

_{a}〉 ⊥ |

*ψ*

_{a}〉, therefore

*a*|

*ψ*

_{a}〉 =

*c′*|

*ψ*

_{a}〉 where

*c′*is a c-number. That is |

*ψ*

_{a}) is an eigenstate of operator

*a*(coherent state).

*ψ*

_{a}〉 or |

*ψ*

_{b}〉 is a coherent state.

## References

1. | W. H. Zurek, “Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?,” Phys. Rev. D |

2. | W. H. Zurek, “Environment-induced superselection rules,” Phys. Rev. D |

3. | W. H. Zurek, “Decoherence and transition from quantum to classical,” Phys. Today |

4. | S. M. Barnett and S. J. D. Phoenix, “The principles of quantum cryptography,” Phil. Trans. R. Soc. Lond. A |

5. | D. P. DiVincenzo, “Quantum computation,” Science |

6. | M. D. Srinivas and E. B. Davies, “Photon counting probabilities in quantum optics,” Opt.Acta |

7. | H. Carmichael, |

8. | S. Ya. Kilin |

9. | S. Ya. Kilin, D. B. Horoshko, and V. N. Shatokhin, “Quantum instabilities and decoherence problem,” Acta Phys. Pol. A |

10. | V. Buzek and P. L. Knight, “Quantum interference, superposition states of light, and nonclassical effects,” Prog. Opt. |

11. | B. M. Garraway and P. L. Knight, “Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space,” Phys. Rev. A |

12. | D. B. Horoshko and S. Ya. Kilin, “Direct detection feedback for preserving quantum coherence in an open cavity,” Phys. Rev. Lett. |

13. | H. Mabuchi and P. Zoller, “Inversion of quantum jumps in quantum optical systems under continuos observation,” Phys. Rev. Lett. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

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

**ToC Category:**

Focus Issue: Control of loss and decoherence in quantum systems

**History**

Original Manuscript: December 1, 1997

Published: April 27, 1998

**Citation**

Dmitri Horoshko and Sergey Kilin, "Multimode unravelling of master equation and decoherence problem," Opt. Express **2**, 347-354 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-347

Sort: Journal | Reset

### References

- W. H. Zurek, "Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?," Phys. Rev. D 24, 1516 (1981). [CrossRef]
- W. H. Zurek, "Environment-induced superselection rules," Phys. Rev. D 26, 1862 (1982). [CrossRef]
- W. H. Zurek, "Decoherence and transition from quantum to classical," Phys. Today 44, 36 (1991). [CrossRef]
- S. M. Barnett and S. J. D. Phoenix, "The principles of quantum cryptography," Phil. Trans. R. Soc. Lond. A 354, 793 (1996). [CrossRef]
- D. P. DiVincenzo, "Quantum computation," Science 270, 255 (1995). [CrossRef]
- M. D. Srinivas and E. B. Davies, "Photon counting probabilities in quantum optics," Opt. Acta 28, 981 (1981). [CrossRef]
- H. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).
- S. Ya. Kilin Quantum Optics. Fields and Their Detection (Minsk, 1990).
- S. Ya. Kilin, D. B. Horoshko, and V. N. Shatokhin, "Quantum instabilities and decoherence problem," Acta Phys. Pol. A93, 97 (1998).
- V. Buzek and P. L. Knight, "Quantum interference, superposition states of light, and nonclassical effects," Prog. Opt. 36 3(1995). [CrossRef]
- B. M. Garraway and P. L. Knight, "Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space," Phys. Rev. A 50, 2548 (1994). [CrossRef] [PubMed]
- D. B. Horoshko and S. Ya. Kilin, "Direct detection feedback for preserving quantum coherence in an open cavity," Phys. Rev. Lett. 78, 840 (1997). [CrossRef]
- H. Mabuchi and P. Zoller, "Inversion of quantum jumps in quantum optical systems under continuos observation," Phys. Rev. Lett. 76, 3108 (1996). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.