Revealing geometric phases in modular and weak values with a quantum eraser

Mirko Cormann, Mathilde Remy, Branko Kolaric, and Yves Caudano

Phys. Rev. A 93, 042124 – Published 28 April 2016

Abstract

We present a procedure to completely determine the complex modular values of arbitrary observables of pre- and postselected ensembles, which works experimentally for all measurement strengths and all postselected states. This procedure allows us to discuss the physics of modular and weak values in interferometric experiments involving a qubit meter. We determine both the modulus and the argument of the modular value for any measurement strength in a single step, by simultaneously controlling the visibility and the phase in a quantum eraser interference experiment. Modular and weak values are closely related. Using entangled qubits for the probed and meter systems, we show that the phase of the modular and weak values has a topological origin. This phase is completely defined by the intrinsic physical properties of the probed system and its time evolution. The physical significance of this phase can thus be used to evaluate the quantumness of weak values.

Received 20 August 2015
Revised 9 February 2016

DOI:https://doi.org/10.1103/PhysRevA.93.042124

Physics Subject Headings (PhySH)

Geometric & topological phases Weak values & weak measurements

General Physics

Authors & Affiliations

Mirko Cormann^1,2,*, Mathilde Remy¹, Branko Kolaric^1,2, and Yves Caudano^1,2,†

¹Research Centre in Physics of Matter and Radiation (PMR), Department of Physics, University of Namur, 61 Rue de Bruxelles, B-5000 Namur, Belgium
²Namur Center for Complex Systems (naXys), 61 Rue de Bruxelles, B-5000 Namur, Belgium

^*mirko.cormann@unamur.be
^†yves.caudano@unamur.be

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Issue

Vol. 93, Iss. 4 — April 2016

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Images

Figure 1
Representation in the Bloch sphere of the relevant (a)–(c) meter states. (a) The red plane is perpendicular to the control state $\vec{r}$ . It contains all final meter states $\vec{q}$ and $- \vec{q}$ implementing the quantum eraser condition. (b),(c) The blue plane contains the initial meter state $\vec{m}$ and the control state $\vec{r}$ . The final meter states (b) ${\vec{q}}_{Re}$ in the blue plane and (c) ${\vec{q}}_{Im}$ perpendicular to it measure the real and imaginary parts of the modular value, respectively.
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Figure 2
Quantum controlled evolution: (a) protocol, (b) representation in the Bloch sphere of the relevant probe states, and (c) experimental setup. (b) Probe Bloch sphere with initial $\vec{i}$ and final $\vec{f}$ states, ${\hat{σ}}_{n}$ observable rotation axis $\vec{n}$ , and the mirror image ${\vec{i}}^{'}$ of $\vec{i}$ with respect to the $\vec{n}$ axis. The solid angle $Ω_{i n i^{'} f}$ associated to the geometric phase in Eq. (13) is obtained by consecutively following the three great circle arcs $i \to n \to i^{'}$ (blue), $i^{'} \to f$ (red), and $f \to i$ (red). (c) The setup comprises three areas: the state preparation with the two qubit generation (I), the meter measurement by detectors $D_{1}$ and $D_{2}$ (II), and the final probe postselection by $D_{3}$ (III). The coincidence counts $N_{13}$ and $N_{23}$ are acquired by four single-photon counting modules (SPCMs) placed in the meter and probe paths.
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Figure 3
(a) Visibility and (b) argument as a function of the postselected polarization $| ψ_{f} 〉 = cos (α) | H 〉 + sin (α) | V 〉$ with preselected $| H 〉$ polarization for three initial meter states: $θ_{1} = 0.499 π$ and $P_{1 m} = 0.882 \pm 0.002$ (red squares), $θ_{2} = 0.297 π$ and $P_{2 m} = 0.836 \pm 0.002$ (blue circles), $θ_{3} = 0.092 π$ and $P_{3 m} = 0.956 \pm 0.001$ (black triangles). (c) Positive or negative solution criterion. Values larger or smaller than unity (violet solid horizontal line) admit the positive or negative solution ${|σ_{x, w}|}_{\pm}$ , respectively. Final meter states are (a),(b) $| D 〉$ and $| A 〉$ and (c) $| H 〉$ and $| V 〉$ . Gray solid lines represent theoretical curves.
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Figure 4
Weak values determined from phase and visibility measurement for the three strengths $θ_{1}$ (red squares), $θ_{2}$ (blue circles), and $θ_{3}$ (black triangles), and from the standard weak-measurement technique (violet diamonds) using relation (5) with the weak strength $θ_{4} = 0.025 π$ and purity $P_{4 m} = 0.982 \pm 0.001$ . All setups use a $| D 〉, | A 〉$ basis for the final meter measurement (the additional measurement in the $| L 〉, | R 〉$ meter basis required by standard weak measurements was not performed since $Im σ_{x, w} (α) = 0$ here).
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Figure 5
The solid angle $Ω_{a b b_{e} a_{e}}$ of the four vertices $\vec{a}, \vec{b}, {\vec{b}}_{e}$ , and ${\vec{a}}_{e}$ represented in the Bloch sphere (the red surface). The states $| a_{e} 〉$ and $| b_{e} 〉$ lying on the equator are horizontal lifts of the associated states $| a 〉$ and $| b 〉$ , respectively. The solid angle of the counterclockwise sequence of states $| a 〉 \to | b 〉 \to | b_{e} 〉 \to | a_{e} 〉 \to | a 〉$ is in direct relationship to the Pancharatnam connection $arg 〈 b | a 〉$ .
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