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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.


Measured data, best fit and residuals.Spin population as a function of detuning and switch-off time of the laser beam. a is for a laser beam centred in zone B, while for b the beam was displaced towards zone C by 64 μm. From left to right are plots of the experimental data, the populations generated from the best fit Hamiltonian, and the residuals. Each data point results from 100 repetitions of the experimental sequence. The data in a consist of an array of 100 × 101 experimental settings, while that shown in b consists of an array of 201 × 201 settings. This leads to smaller error bars in the reconstructed Hamiltonian for the latter. For the Hamiltonian estimation, the data was weighted according to quantum projection noise.
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f2: Measured data, best fit and residuals.Spin population as a function of detuning and switch-off time of the laser beam. a is for a laser beam centred in zone B, while for b the beam was displaced towards zone C by 64 μm. From left to right are plots of the experimental data, the populations generated from the best fit Hamiltonian, and the residuals. Each data point results from 100 repetitions of the experimental sequence. The data in a consist of an array of 100 × 101 experimental settings, while that shown in b consists of an array of 201 × 201 settings. This leads to smaller error bars in the reconstructed Hamiltonian for the latter. For the Hamiltonian estimation, the data was weighted according to quantum projection noise.

Mentions: We first perform a comparison in which the ion velocity is the same but the beam position is changed. Thus we expect to obtain two different profiles for Ω(t) but the same velocity profile, which is closely related to δ(t). Experimental data is shown in Fig. 2 alongside the results of fitting performed using our iterative method. The beam positions used for each data set differ by ∼64 μm along the transport axis, but the transport waveform used was identical. It can be seen from the residuals that the estimation is able to find a Hamiltonian, which results in a close match to the data.


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)

Measured data, best fit and residuals.Spin population as a function of detuning and switch-off time of the laser beam. a is for a laser beam centred in zone B, while for b the beam was displaced towards zone C by 64 μm. From left to right are plots of the experimental data, the populations generated from the best fit Hamiltonian, and the residuals. Each data point results from 100 repetitions of the experimental sequence. The data in a consist of an array of 100 × 101 experimental settings, while that shown in b consists of an array of 201 × 201 settings. This leads to smaller error bars in the reconstructed Hamiltonian for the latter. For the Hamiltonian estimation, the data was weighted according to quantum projection noise.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f2: Measured data, best fit and residuals.Spin population as a function of detuning and switch-off time of the laser beam. a is for a laser beam centred in zone B, while for b the beam was displaced towards zone C by 64 μm. From left to right are plots of the experimental data, the populations generated from the best fit Hamiltonian, and the residuals. Each data point results from 100 repetitions of the experimental sequence. The data in a consist of an array of 100 × 101 experimental settings, while that shown in b consists of an array of 201 × 201 settings. This leads to smaller error bars in the reconstructed Hamiltonian for the latter. For the Hamiltonian estimation, the data was weighted according to quantum projection noise.
Mentions: We first perform a comparison in which the ion velocity is the same but the beam position is changed. Thus we expect to obtain two different profiles for Ω(t) but the same velocity profile, which is closely related to δ(t). Experimental data is shown in Fig. 2 alongside the results of fitting performed using our iterative method. The beam positions used for each data set differ by ∼64 μm along the transport axis, but the transport waveform used was identical. It can be seen from the residuals that the estimation is able to find a Hamiltonian, which results in a close match to the data.

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.