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Estimation of a general time-dependent Hamiltonian for a single qubit.

de Clercq LE, Oswald R, Flühmann C, Keitch B, Kienzler D, Lo HY, Marinelli M, Nadlinger D, Negnevitsky V, Home JP - Nat Commun (2016)

Bottom Line: The initially unknown Hamiltonian arises from transporting an ion through a static laser beam.Hamiltonian estimation allows us to estimate the spatial beam intensity profile and the ion velocity as a function of time.The estimation technique is general enough that it can be applied to other quantum systems, aiding the pursuit of high-operational fidelities in quantum control.

View Article: PubMed Central - PubMed

Affiliation: Institute for Quantum Electronics, ETH Zürich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland.

ABSTRACT
The Hamiltonian of a closed quantum system governs its complete time evolution. While Hamiltonians with time-variation in a single basis can be recovered using a variety of methods, for more general Hamiltonians the presence of non-commuting terms complicates the reconstruction. Here using a single trapped ion, we propose and experimentally demonstrate a method for estimating a time-dependent Hamiltonian of a single qubit. We measure the time evolution of the qubit in a fixed basis as a function of a time-independent offset term added to the Hamiltonian. The initially unknown Hamiltonian arises from transporting an ion through a static laser beam. Hamiltonian estimation allows us to estimate the spatial beam intensity profile and the ion velocity as a function of time. The estimation technique is general enough that it can be applied to other quantum systems, aiding the pursuit of high-operational fidelities in quantum control.

No MeSH data available.


Estimates of time-dependent co-efficients.(a) The effective detuning δ(t) and (b) Rabi frequency Ω(t) obtained from the two data sets. Blue and red solid lines show data obtained having the beam centred in zone B and with the beam displaced by a few tens of microns. Dashed lines indicate the s.e. on the mean of these estimates, which are obtained using resampling. For a the inset shows a close-up of the estimated δ(t) in the regions where the estimates overlap, showing that these do not give the same value. (c) The estimated velocity  of the ion obtained after applying wavefront correction. The inset shows that this produces consistent results.
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f3: Estimates of time-dependent co-efficients.(a) The effective detuning δ(t) and (b) Rabi frequency Ω(t) obtained from the two data sets. Blue and red solid lines show data obtained having the beam centred in zone B and with the beam displaced by a few tens of microns. Dashed lines indicate the s.e. on the mean of these estimates, which are obtained using resampling. For a the inset shows a close-up of the estimated δ(t) in the regions where the estimates overlap, showing that these do not give the same value. (c) The estimated velocity of the ion obtained after applying wavefront correction. The inset shows that this produces consistent results.

Mentions: The estimated coefficients of the Hamiltonian extracted from the two data sets are shown in Fig. 3a,b. To estimate the relevant errors of our reconstruction, we have performed non-parametric resampling with replacement, optimizing for the solution using the same set of B-spline functions as was used for the experimental data to provide a new estimate for the Hamiltonian. This is repeated for a large number of samples, resulting in a distribution for the estimated values of δ(t) and Ω(t) from which we extract statistical properties such as the s.e. The error bounds shown in Fig. 3 correspond to the s.e. on the mean obtained from these distributions (see Methods for further details). It can be seen that the values of δ(t) for the two different beam positions have a similar form but a fixed offset for the region where the reconstructions overlap. We believe that this effect arises from the non-planar wavefronts of the laser beam. Inverting the expression for δ(t) to obtain the velocity of the ion, we find . Using this correction, we find that the two velocity profiles agree if we assume that the ion passes through the centre of the beam at a distance of 2.27 mm before the minimum beam waist, a value which is consistent with experimental uncertainties due to beam propagation and possible mis-positioning of the ion trap with respect to the fixed final focusing lens. The velocity estimates taking account of this effect are shown in Fig. 3c.


Estimation of a general time-dependent Hamiltonian for a single qubit.

de Clercq LE, Oswald R, Flühmann C, Keitch B, Kienzler D, Lo HY, Marinelli M, Nadlinger D, Negnevitsky V, Home JP - Nat Commun (2016)

Estimates of time-dependent co-efficients.(a) The effective detuning δ(t) and (b) Rabi frequency Ω(t) obtained from the two data sets. Blue and red solid lines show data obtained having the beam centred in zone B and with the beam displaced by a few tens of microns. Dashed lines indicate the s.e. on the mean of these estimates, which are obtained using resampling. For a the inset shows a close-up of the estimated δ(t) in the regions where the estimates overlap, showing that these do not give the same value. (c) The estimated velocity  of the ion obtained after applying wavefront correction. The inset shows that this produces consistent results.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4834628&req=5

f3: Estimates of time-dependent co-efficients.(a) The effective detuning δ(t) and (b) Rabi frequency Ω(t) obtained from the two data sets. Blue and red solid lines show data obtained having the beam centred in zone B and with the beam displaced by a few tens of microns. Dashed lines indicate the s.e. on the mean of these estimates, which are obtained using resampling. For a the inset shows a close-up of the estimated δ(t) in the regions where the estimates overlap, showing that these do not give the same value. (c) The estimated velocity of the ion obtained after applying wavefront correction. The inset shows that this produces consistent results.
Mentions: The estimated coefficients of the Hamiltonian extracted from the two data sets are shown in Fig. 3a,b. To estimate the relevant errors of our reconstruction, we have performed non-parametric resampling with replacement, optimizing for the solution using the same set of B-spline functions as was used for the experimental data to provide a new estimate for the Hamiltonian. This is repeated for a large number of samples, resulting in a distribution for the estimated values of δ(t) and Ω(t) from which we extract statistical properties such as the s.e. The error bounds shown in Fig. 3 correspond to the s.e. on the mean obtained from these distributions (see Methods for further details). It can be seen that the values of δ(t) for the two different beam positions have a similar form but a fixed offset for the region where the reconstructions overlap. We believe that this effect arises from the non-planar wavefronts of the laser beam. Inverting the expression for δ(t) to obtain the velocity of the ion, we find . Using this correction, we find that the two velocity profiles agree if we assume that the ion passes through the centre of the beam at a distance of 2.27 mm before the minimum beam waist, a value which is consistent with experimental uncertainties due to beam propagation and possible mis-positioning of the ion trap with respect to the fixed final focusing lens. The velocity estimates taking account of this effect are shown in Fig. 3c.

Bottom Line: The initially unknown Hamiltonian arises from transporting an ion through a static laser beam.Hamiltonian estimation allows us to estimate the spatial beam intensity profile and the ion velocity as a function of time.The estimation technique is general enough that it can be applied to other quantum systems, aiding the pursuit of high-operational fidelities in quantum control.

View Article: PubMed Central - PubMed

Affiliation: Institute for Quantum Electronics, ETH Zürich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland.

ABSTRACT
The Hamiltonian of a closed quantum system governs its complete time evolution. While Hamiltonians with time-variation in a single basis can be recovered using a variety of methods, for more general Hamiltonians the presence of non-commuting terms complicates the reconstruction. Here using a single trapped ion, we propose and experimentally demonstrate a method for estimating a time-dependent Hamiltonian of a single qubit. We measure the time evolution of the qubit in a fixed basis as a function of a time-independent offset term added to the Hamiltonian. The initially unknown Hamiltonian arises from transporting an ion through a static laser beam. Hamiltonian estimation allows us to estimate the spatial beam intensity profile and the ion velocity as a function of time. The estimation technique is general enough that it can be applied to other quantum systems, aiding the pursuit of high-operational fidelities in quantum control.

No MeSH data available.