## Quantum control by compensation of quantum fluctuations

Optics Express, Vol. 2, Issue 9, pp. 339-346 (1998)

http://dx.doi.org/10.1364/OE.2.000339

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### Abstract

We show that the influence of quantum fluctuations in the electromagnetic field vacuum on a two level atom can be measured and consequently compensated by balanced homodyne detection and a coherent feedback field. This compensation suppresses the decoherence associated with spontaneous emissions for a specific state of the atomic system allowing complete control of the coherent state of the system.

© Optical Society of America

^{11. M. B. Plenio, V. Vedral, and P. L. Knight, “Quantum error correction in the presence of spontaneous emission”, Phys. Rev. A 55, 67 (1997). [CrossRef] ,22. C. W. Gardiner, Quantum Noise, (Springer-Verlag, Berlin1991).,33. H. Carmichael, An Open Systems Approach to Quantum Mechanics, (Springer-Verlag, Berlin1993).}. However, this is by no means the only way of observing the electromagnetic field propagating away from a quantum system. As pointed out by Ueda

^{44. M. Ueda and M. Kitagawa, “Reversibility in quantum measurement processes”, Phys. Rev. Lett. 68, 3424 (1992). [CrossRef] [PubMed] }, a measurement of the emitted field which is sensitive to the vacuum state as well is logically reversible, as opposed to the sudden transition to the ground state connected with a photon detection event. Therefore it seems preferable to apply measurement schemes different from photon detection if quantum coherence is to be controlled.

*τ*which is much smaller than the lifetime 1/Γ of the excited atomic state is much smaller than the vacuum fluctuations observed on this timescale. Thus it is possible to interpret the fields measured as quantum fluctuations of the electromagnetic field impinging on the system. In this sense the measurement is a measurement of the forces acting on the system and not a measurement of the system state itself. It should be possible to compensate the effect of the observed quadrature component of the electromagnetic field by a coherent field of opposite sign. However, the effect of the unobserved quadrature component must also be compensated if decoherence is to be suppressed. To find out, how this can be achieved as well, it is necessary to investigate the back-action of the homodyne detection on the atomic system.

^{55. H. P. Yuen and J. H. Shapiro, in Coherence and Quantum Optics IV, ed. by L. Mandel and E. Wolf (Plenum, New York1978), pp. 719.,66. S. L. Braunstein, “Homodyne Statistics”, Phys. Rev. A 42, 474 (1990). [CrossRef] [PubMed] ,77. W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection”, Phys. Rev. A 47, 4227 (1993). [CrossRef] [PubMed] ,88. A. Luis and J. Perina, “Generalized measurements in 8-port homodyne detection” Quantum Semiclass. Opt. 8, 873 (1996). [CrossRef] }. Since the observed fields are small, we will only consider that part of the measurement base composed of the zero or one photon contributions. The effective non-orthogonal measurement base is given by

*δn*of such a coherent field can be calculated from equation (1) by

*n*

^{2}〉 =

*α*

^{*}

*α*and a mean value of 〈∆

*n*〉 =

*α*

^{*}

*β*+

*β*

^{*}

*α*. If the measured value of ∆

*n*is identified as 2 |

*α*| times the quadrature component of the measured light field in phase with the local oscillator, this result exactly corresponds to the quantum uncertainty of 1/4 and a shift by the component of

*β*in phase with

*α*. This result confirms the interpretation of homodyne detection as a projective measurement of the quadrature component in phase with the local oscillator.

^{1010. H. F. Hofmann and G. Mahler, “Measurement models for time-resolved spectroscopy: A comment”, Quantum Semiclass. Opt. 7, 489 (1995). [CrossRef] }. In the following, however, we will assume fast time-resolved measurements performed on the field long before the emission probability from an excited state approaches unity. During the short time intervals

*τ*with Γ

*τ*≪ 1, the one-photon component of the wavefunction corresponds to a photon in a field mode with a rectangular envelope: zero field amplitude for distances

*r*from the atomic system with

*r*>

*cτ*and a constant probability of finding a photon at distances of 0 <

*r*<

*cτ*. Therefore, the photon possibly emitted during the time interval

*τ*is in a well defined mode. Thus it is possible to write down the wave function which evolves from the light field vacuum and an arbitrary state of the two level atom given by

*G*〉 is the atomic ground state and |

*Ẽ*〉 is the excited state in the interaction picture, i.e. without the phase dynamics at the frequency

*ω*

_{0}of the atomic transition. After the time interval

*τ*, the entangled state of the atomic system and the electromagnetic field is

*n*in the homodyne detection during the time interval

*τ*. The wavefunction |

*ψ*(

*τ*)〉 of the atomic system after the measurement reads

*τ*≪ 1, the squared length of this state vector which corresponds to the probability of measuring ∆

*n*is approximately independent of the system state and is given by the vacuum distribution,

*τ*. The higher order terms do have some effect on timescales of 1/Γ, corresponding to a large number of measurement intervals

*τ*. These effects will be discussed elsewhere

^{99. H. F. Hofmann, G. Mahler, and O. Hess, “Quantum control of atomic systems by time resolved homodyne detection”, Phys. Rev. A 57, in press (1998). [CrossRef] }. In the following we will concentrate on the short time fluctuations effective on a timescale of

*τ*.

*ψ*(0)〉 + |

*δψ*(

*τ*)〉, such that |

*δψ*(

*τ*)〉 is the change of the system state orthogonal to |

*ψ*(0)〉, then this change is approximately given by

*n*is a Gaussian, this equation describes a diffusion process. Statistically, the diffusion steps cancel on average, causing decoherence because the uncertainty of the actual path chosen by the system dynamics increases with each unknown step. In our scenario however, the length and the direction of each step has been measured by homodyne detection.

^{33. H. Carmichael, An Open Systems Approach to Quantum Mechanics, (Springer-Verlag, Berlin1993).,1111. H. Carmichael, “Stochastic Schrödinger equations: What they mean and what they can do.” in Coherence and Quantum Optics VII, ed. by J. Eberly, L. Mandel, and E. Wolf, (Plenum, New York1996), pp. 177.}and applied to problems of continuous feedback scenarios in

^{1212. H. M. Wiseman, “Quantum theory of continuous feedback”, Phys. Rev. A 49, 2133 (1994). [CrossRef] [PubMed] }. It has not been derived from a master equation of the open system, however, and the field-atom interaction is described using the Schroedingers equation of Wigner-Weisskopf theory, retaining the full atom-field entanglement up to the projective measurement.

*s*is the expectation value of the population inversion and

_{z}*s*and

_{x}*s*are the in-phase and the out-of-phase dipole moments of the atomic system, respectively. The change in the Bloch vector of the atomic system

_{y}*δ*s conditioned by a measurement of ∆

*n*within the time interval

*τ*is then given by

*s*. It is exactly equal to the effects of a coherent field with an amplitude of ∆

_{y}*n*/|2

*α*|. The non-linear part is shown in Fig. 2. For positive ∆

*n*, this contribution draws the Bloch vector towards the

*s*= +1 pole of the Bloch sphere. For negative ∆

_{x}*n*, the Bloch vector moves towards the

*s*= - 1 pole.

_{x}*s*gained in the measurement. Positive values of ∆

_{x}*n*make a positive dipole component

*s*more likely and negative values of ∆

_{x}*n*make a negative dipole component

*s*more likely. Although the information obtained in a single measurement is almost negligible, the relative suppression of the amplitude of one dipole eigenstate and the corresponding amplification of the amplitude corresponding to the other dipole eigenstate causes a change in the state of the atomic system unless the system is already in an eigenstate of the dipole component with

_{x}*s*= ±1.

_{x}^{1313. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of a spin-1/2 particle can turn out to be 100”, Phys. Rev. Lett. 60, 1351 (1988). [CrossRef] [PubMed] }.

^{44. M. Ueda and M. Kitagawa, “Reversibility in quantum measurement processes”, Phys. Rev. Lett. 68, 3424 (1992). [CrossRef] [PubMed] }. It can be compensated if the previous state of the system is known with sufficient precision. In the following, we shall focus on atomic system states with

*s*= 0. For such states,

_{y}*δs*is also zero and the whole diffusion process takes place in the

_{y}*s*plane. The diffusion steps may then be identified as rotations around the

_{x}, s_{z}*s*axis. By defining the angle

_{y}*θ*such that cos

*θ*=

*s*and sin

_{z}*θ*=

*s*, the diffusion step in the

_{x}*s*plane may be written as

_{x},s_{z}*n*is equivalent to a Rabi rotation around the

*s*axis proportional to the quadrature component measured in the homodyne detection. Despite the quantum mechanical dependence of this Rabi rotation on

_{y}*θ*, it is possible to compensate the effects of the quantum fluctuations by simply reversing the rotations corresponding to each measurement. The feedback field

*f*necessary to stabilize a state of the atomic system with

*θ*=

*is given by*θ ¯

_{0}is the measurement result associated with the quantum fluctuations which are to be compensated by the feedback term. Each time interval

*τ*is therefore associated with a diffusion step caused by the quantum fluctuations and a time delayed feedback which compensates the diffusion step. The total field interacting with the atomic system is given by a coherent state of field amplitude

*f*(∆

*n*

_{0}). This corresponds to vacuum-state quantum fluctuations shifted by

*f*(∆

*n*

_{0}). Consequently the measurement result ∆

*n*corresponding to the time interval

_{next}*τ*during which the feedback field acts on the system will be composed of a stochastic effect of the quantum fluctuations ∆

_{next}*n*and a shift

_{qf}*δ*caused by the feedback field

_{next}*n*should be applied for the determination of the subsequent feedback field.

_{qf}*s*:

_{x}*Dipole eigenstate*. For cos

*= 0 the system is in an eigenstate of the in-phase dipole component*θ ¯

*s*. No measurements of

_{x}*s*, whether weak or strong, will change this. Therefore, the compensating field necessary to suppress the effects of quantum fluctuations is equal to the compensation of the classically expected Rabi rotation. Also note that a coherent field along the unknown field quadrature would not affect this state, since the Bloch vector is parallel to the axis of Rabi rotations caused by fields ±

_{x}*π*/2 out of phase with the local oscillator.

*Ground state*. For cos

*= -1 no feedback is necessary for stabilization. This means that the effects of the Rabi rotation and the weak measurement associated with a homodyne detection result ∆*θ ¯

*n*automatically compensate each other. This is a result of the fact that the ground state is polarized by the field in such a way that the dipole emissions interfere destructively with the field. At the same time, the observed field makes a dipole more likely which emits radiation interfering constructively with the fluctuations. This effect may also be understood in terms of energy conservation. The ground state atom absorbs the field by the destructive interference of dipole emission and incoming field, but at the same time it emits radiation associated with the quantum fluctuations of the dipole variables. Both effects cancel and energy conservation is preserved.

*s*plane (left) and into the

_{y}, s_{z}*s*plane (right) the continuous change of the effective diffusion step as the feedback signal is increased from no feedback (cos

_{x}, s_{z}*s*= cos

_{z}*s*axis, indicating that the feedback stabilizes this value of the inversion expectation value

_{z}*s*regardless of the phase of the dipole oscillations relative to the local oscillator.

_{z}*s*plane of the Bloch vector representation of the atomic system, randomly fluctuating forces cause rotations around the

_{x},s_{z}*s*axis. This effect corresponds to that of a coherent driving field and can consequently be compensated by Rabi rotations of opposite sign induced by a feedback field. The decoherence caused by quantum fluctuations can be suppressed completely with a precision limited only by the time delay between the emission of the field and the measurement by homodyne detection. Even though one quadrature component of the light field remains unobserved, we have demonstrated that quantum control of an arbitrary state of a two-level atomic system is possible by simply applying a coherent feedback field.

_{y}## References

1. | M. B. Plenio, V. Vedral, and P. L. Knight, “Quantum error correction in the presence of spontaneous emission”, Phys. Rev. A |

2. | C. W. Gardiner, |

3. | H. Carmichael, |

4. | M. Ueda and M. Kitagawa, “Reversibility in quantum measurement processes”, Phys. Rev. Lett. |

5. | H. P. Yuen and J. H. Shapiro, in |

6. | S. L. Braunstein, “Homodyne Statistics”, Phys. Rev. A |

7. | W. Vogel and J. Grabow, “Statistics of difference events in homodyne detection”, Phys. Rev. A |

8. | A. Luis and J. Perina, “Generalized measurements in 8-port homodyne detection” Quantum Semiclass. Opt. |

9. | H. F. Hofmann, G. Mahler, and O. Hess, “Quantum control of atomic systems by time resolved homodyne detection”, Phys. Rev. A |

10. | H. F. Hofmann and G. Mahler, “Measurement models for time-resolved spectroscopy: A comment”, Quantum Semiclass. Opt. |

11. | H. Carmichael, “Stochastic Schrödinger equations: What they mean and what they can do.” in |

12. | H. M. Wiseman, “Quantum theory of continuous feedback”, Phys. Rev. A |

13. | Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of a spin-1/2 particle can turn out to be 100”, Phys. Rev. Lett. |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

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

**ToC Category:**

Focus Issue: Control of loss and decoherence in quantum systems

**History**

Original Manuscript: December 14, 1997

Published: April 27, 1998

**Citation**

Holger Hofmann, Ortwin Hess, and Guenter Mahler, "Quantum control by compensation of
quantum fluctuations," Opt. Express **2**, 339-346 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-9-339

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### References

- M. B. Plenio, V. Vedral, and P. L. Knight, "Quantum error correction in the presence of spontaneous emission ", Phys. Rev. A 55, 67 (1997). [CrossRef]
- C. W. Gardiner, Quantum Noise, (Springer-Verlag, Berlin 1991).
- H. Carmichael, An Open Systems Approach to Quantum Mechanics, (Springer-Verlag, Berlin 1993).
- M. Ueda and M. Kitagawa, "Reversibility in quantum measurement processes", Phys. Rev. Lett. 68, 3424 (1992). [CrossRef] [PubMed]
- H. P. Yuen and J. H. Shapiro, in Coherence and Quantum Optics IV, ed. by L. Mandel and E. Wolf (Plenum, New York 1978), pp. 719.
- S. L. Braunstein, "Homodyne Statistics", Phys. Rev. A 42, 474 (1990). [CrossRef] [PubMed]
- W. Vogel and J. Grabow, "Statistics of di’erence events in homodyne detection", Phys. Rev. A 47, 4227 (1993). [CrossRef] [PubMed]
- A. Luis and J. Perina, "Generalized measurements in 8-port homodyne detection" Quantum Semiclass. Opt. 8, 873 (1996). [CrossRef]
- H. F. Hofmann, G. Mahler, and O. Hess, "Quantum control of atomic systems by time resolved homodyne detection", Phys. Rev. A 57, in press (1998). [CrossRef]
- H. F. Hofmann and G. Mahler, "Measurement models for time-resolved spectroscopy: A comment", Quantum Semiclass. Opt. 7, 489 (1995). [CrossRef]
- H. Carmichael, "Stochastic Schroedinger equations: What they mean and what they can do." in Coherence and Quantum Optics VII, ed. by J. Eberly, L. Mandel, and E. Wolf, (Plenum, New York 1996), pp. 177.
- H. M. Wiseman, "Quantum theory of continuous feedback", Phys. Rev. A 49, 2133 (1994). [CrossRef] [PubMed]
- Y. Aharonov, D. Z. Albert, and L. Vaidman, "How the result of a measurement of a component of a spin-1/2 particle can turn out to be 100", Phys. Rev. Lett. 60, 1351 (1988). [CrossRef] [PubMed]

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