Measurement of the topological Chern number by continuous probing of a qubit subject to a slowly varying Hamiltonian

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Measurement of the topological Chern number by continuous probing of a qubit subject to a slowly varying Hamiltonian

Peng Xu, Alexander Holm Kiilerich, Ralf Blattmann, Yang Yu, Shi-Liang Zhu, and Klaus Mølmer

Phys. Rev. A 96, 010101(R) – Published 5 July 2017

Abstract

We analyze a measurement scheme that allows determination of the Berry curvature and the topological Chern number of a Hamiltonian with parameters exploring a two-dimensional closed manifold. Our method uses continuous monitoring of the gradient of the Hamiltonian with respect to one parameter during a quasiadiabatic quench of the other. Measurement backaction leads to disturbance of the system dynamics, but we show that this can be compensated by a feedback Hamiltonian. As an example, we analyze the implementation with a superconducting qubit subject to time-varying, near-resonant microwave fields, equivalent to a spin-1/2 particle in a magnetic field.

Received 31 March 2017

DOI:https://doi.org/10.1103/PhysRevA.96.010101

Physics Subject Headings (PhySH)

Geometric & topological phases Quantum feedback Quantum fluctuations & noise Quantum measurements Quantum metrology Weak values & weak measurements

Stochastic differential equations

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Peng Xu^1,2,*, Alexander Holm Kiilerich², Ralf Blattmann², Yang Yu^1,3, Shi-Liang Zhu^1,3, and Klaus Mølmer^2,†

¹National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 230039, China
²Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
³Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China

^*xupengqh201461@163.com
^†moelmer@phys.au.dk

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Issue

Vol. 96, Iss. 1 — July 2017

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Images

Figure 1
The plotted curves show the dependence of the integral (3) or (8) on $Δ_{2} / Δ_{1}$ for different quench times $t_{q} = 1 μ s$ (green, lower curve), $2 μ s$ (blue, middle curve), and $15 μ s$ (red, upper curve). We assume $ϕ = 0$ , $Ω_{1} / 2 π = \frac{1}{3} Δ_{1} / 2 π = 10 MHz$ . The Bloch sphere on the left (right) shows the evolution of the quantum state during the quench for $Δ_{2} = \frac{1}{3} Δ_{1} (Δ_{2} = 1.1 Δ_{1})$ .
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Figure 2
(a) A two-level system driven by a time-dependent Hamiltonian is monitored by homodyne detection of a transmitted microwave field. The stochastic homodyne current can be used to control a feedback microwave field applied to the qubit. (b) The dynamical evolution of the quantum state is calculated with $Ω_{1} = Δ_{2} = \frac{1}{3} Δ_{1} = 10 M Hz$ . Results are shown for adiabatic evolution with $t_{q} = 15 μ s$ (red, in $x z$ plane) and for quasiadiabatic evolution with $t_{q} = 1 μ s$ (green, blue, and dotted magenta). The dotted magenta curve depicts the evolution absent probing, the blue curve shows the evolution with high measurement strength ( $κ / 2 π = 0.37 M Hz$ ), and the green curve (matching the dotted magenta curve) indicates the evolution under a strong measurement employing the feedback described in the text. (c) The deviation in the Chern number (integral in Eq. (8) calculated from the unconditional evolution) from the expected integer value, introduced by measurement backaction. Curves are displayed as a function of the probing strength $κ$ for the cases of no probing (green, lower line), probing (blue, upper line), and probing with feedback (magenta, middle line). Results are shown for $Δ_{2} = 0$ , where in the absence of probing, $C_{y} = 1$ , and the inset refers to $Δ_{2} = 2 Δ_{1}$ , where we expect $C_{y} = 0$ . (d) The value of $C_{y}$ for different detuning ratios $Δ_{2} / Δ_{1}$ and different values of the probing strength. We use $Ω_{1} = \frac{1}{3} Δ_{1}$ and the remaining parameters are set to optimize the measurement proposal (see text); $Δ_{1} / 2 π = 16.1 M Hz$ and $t_{q} = 0.96 μ s$ . Lower, dashed curves (cool colors) depict the case of probing without feedback and the upper, full curves (warm colors) show the case with feedback. The dotted black curve shows $C_{y}$ without probing. (e) The Chern number for different detuning ratios $Δ_{2} / Δ_{1}$ with the optimized measurement strength $κ / 2 π = 0.37 M Hz$ (left panel) and with a much larger strength $κ / 2 π = 3.0 M Hz$ (right panel). Shaded areas illustrate error bars with $N = 383$ experimental repetitions. Noisy, magenta (upper) and green (lower) curves, consistent with the error bars and ensemble average signals $C_{y}$ , illustrate the results for $C_{V}$ from simulations of the measurement and feedback procedure.
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