Identifying a two-state Hamiltonian in the presence of decoherence

Jared H. Cole, Andrew D. Greentree, Daniel K. L. Oi, Sonia G. Schirmer, Cameron J. Wellard, and Lloyd C. L. Hollenberg

Phys. Rev. A 73, 062333 – Published 27 June 2006

Abstract

Mapping the system evolution of a two-state system allows the determination of the effective system Hamiltonian directly. We show how this can be achieved even if the system is decohering appreciably over the observation time. A method to include various decoherence models is given and the limits of this technique are explored. This technique is applicable both to the problem of calibrating a control Hamiltonian for quantum computing applications and for precision experiments in two-state quantum systems. The accuracy of the results obtained with this technique are ultimately limited by the validity of the decoherence model used.

Received 21 September 2005

DOI:https://doi.org/10.1103/PhysRevA.73.062333

Authors & Affiliations

Jared H. Cole^1,*, Andrew D. Greentree¹, Daniel K. L. Oi², Sonia G. Schirmer², Cameron J. Wellard¹, and Lloyd C. L. Hollenberg¹

¹Centre for Quantum Computer Technology, School of Physics, The University of Melbourne, Melbourne, Victoria 3010, Australia
²Department of Applied Mathematics and Theoretical Physics, The University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*Electronic address: j.cole@physics.unimelb.edu.au

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 73, Iss. 6 — June 2006

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) Example double sided frequency spectrum showing the labeling of peaks for the case of no decoherence. The zero-frequency peak $F (0) = F_{0}$ and the oscillation frequency peak $F (d) = F_{p}$ while the effect of only taking a finite number of measurements is to produce a noise floor. The distribution of the noise floor gives an uncertainty estimate for the peak heights and therefore the system parameters.Reuse & Permissions
Figure 2
(Color online) The $z$ projection of the time evolution of a two-state system after preparation in the $z = 1$ state, for several different dephasing rates. The magnitude of the Hamiltonian $(d = 1)$ is kept constant while the angle between the components is varied. (a) For $θ = π ∕ 2$ , the oscillations decay exponentially. (b) With $θ = π ∕ 4$ , the system undergoes a different evolution, but still decays to the mixed state $z (\infty) = 0$ . Note the difference in scales on the vertical axes in (a) and (b).Reuse & Permissions
Figure 3
(Color online) The Fourier transform of the system evolution given in Fig. 2, in the presence of dephasing, plotted for $ω \geq 0$ . The magnitude of the Hamiltonian, $d$ , is kept constant and results displayed for (a) $θ = π ∕ 2$ , and (b) $θ = π ∕ 4$ . The peak is reduced when $θ \neq π ∕ 2$ , as expected, and the zero-frequency component increases. As the decoherence rate increases, the peaks broaden and the frequency shifts slightly.Reuse & Permissions
Figure 4
(Color online) The evolution of a two-state system under the influence of both spontaneous absorption and emission, in the limit of large detuning. For this example, the emission rate is five times that of spontaneous absorption $(Γ_{-} ∕ Γ_{+} = 5)$ . The path taken by the system depends on which initial state is used, though the asymptotic behavior is the same. The two paths are labeled $z_{- 1}$ and $z_{1}$ depending on whether the system is initialized in the ground $[z (0) = - 1]$ or excited state $[z (0) = 1]$ , respectively.Reuse & Permissions
Figure 5
(Color online) Simulated data for evolution due to the example Hamiltonian, $H = 0.93 σ_{x} + 0.38 σ_{z}$ . (a) The time series data for $N_{e} = 50$ (points) which approximates the ensemble average of the evolution (solid line). (b) The Fourier transform of the time domain data (points) plotted with the fitted function (solid line) using the estimated parameters in Table .Reuse & Permissions
Figure 6
(Color online) The uncertainty estimate as a function of number measurements for an example Hamiltonian $H = 0.93 σ_{x} + 0.38 σ_{z}$ undergoing pure dephasing in the bare qubit basis. The fractional uncertainty for the $1 σ$ confidence interval is plotted for both the $σ_{z}$ component of the Hamiltonian and the dephasing rate $Γ_{z}$ . The scaling is approximately proportional to $1 ∕ \sqrt{N_{T}}$ and the absolute fractional uncertainty is independent of the decoherence rate.Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information