Experimental reconstruction of Wigner phase-space current

Yi-Ru Chen, Hsien-Yi Hsieh, Jingyu Ning, Hsun-Chung Wu, Hua Li Chen, You-Lin Chuang, Popo Yang, Ole Steuernagel, Chien-Ming Wu, and Ray-Kuang Lee

Phys. Rev. A 108, 023729 – Published 29 August 2023

Abstract

We experimentally reconstruct Wigner's current of quantum phase-space dynamics. We reveal the “push-and-pull” associated with damping and diffusion due to the coupling of a squeezed vacuum state to its environment. In contrast to classical dynamics, where (at zero temperature) dissipation only “pulls” the system toward the origin of phase space, we also observe an outward “push” because our system has to obey Heisenberg's uncertainty relations. With squeezed vacuum states generated by an optical parametric oscillator at variable pumping levels, we identify the pure squeezing dynamics and its central stagnation point with a topological charge of “ $- 1$ ”. This work demonstrates high resolving power and establishes an experimental paradigm for measuring the quantumness and nonclassicality of the dynamics of open quantum systems.

Received 17 January 2023
Revised 13 March 2023
Accepted 8 August 2023

DOI:https://doi.org/10.1103/PhysRevA.108.023729

Physics Subject Headings (PhySH)

Quantum optics Quantum states of light

Phase space methods

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Yi-Ru Chen ¹, Hsien-Yi Hsieh ¹, Jingyu Ning¹, Hsun-Chung Wu¹, Hua Li Chen², You-Lin Chuang³, Popo Yang¹, Ole Steuernagel^1,4, Chien-Ming Wu¹, and Ray-Kuang Lee ^1,2,3,5,*

¹Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 30013, Taiwan
²Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
³Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
⁴Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, United Kingdom
⁵Center for Quantum Technology, Hsinchu 30013, Taiwan

^*rklee@ee.nthu.edu.tw

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Vol. 108, Iss. 2 — August 2023

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Images

Figure 1
Schematics of phase-space dynamics and the corresponding Wigner currents for (a) a coherent state, (b) a squeezed vacuum state (in a corotating frame), (c) the classical picture of a damped coherent state, (d) the damping current (with inward “pull”), and (e) the diffusing current (with outward “push”). $ω$ is the winding number (9) of $\vec{J}$ 's stagnation point at the origin.
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Figure 2
(a) Measured quantum noise levels (in dB) of squeezing and antisqueezing quadratures at different pump powers. In the ideal case, squeezing and antisqueezing levels should be equal (blue dashed line). Because of losses and phase noise, due to coupling to the environment, we instead observe degraded squeezing performance described by the solid green fit line. (b) The corresponding purity $tr (ρ^{2})$ of our squeezed states is determined using machine-learning enhanced quantum state tomography [10]. We highlight two regions with red boxes where the squeezing is “weak” and “strong.” Specific red markers $A$ and $B$ pick out data points generated for “low” and “high” pump power, which are referred to in Fig. 3.
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Figure 3
Top and middle rows: Snapshots of Wigner current distributions, displayed as blue arrows (in arbitrary units), overlaid over colored contours of the corresponding Wigner distributions of squeezed vacuum states. The top row represents a weakly squeezed state generated at an OPO pump power of 4.75 mW and selected by marker $A$ in Fig. 2. Similarly, the middle row represents a strongly squeezed state at 88 mW pump power, selected by marker $B$ . The Wigner current displays, circumscribed by red boxes in the middle row, are shown magnified in the bottom row. They illustrate how, even for strong coupling to the environment, the topological charge $ω (0, 0)$ of ${\vec{J}}_{sys}$ is preserved. First column (a)–(c) : Wigner current ${\vec{J}}_{exp}$ , reconstructed from experimental data. Second column (d)–(f): the ideal Wigner current ${\vec{J}}_{sys}$ , fitted to pure squeezed states. Third column (g)–(i): Thermal contributions in the Wigner current ${\vec{J}}_{env} = {\vec{J}}_{exp} - {\vec{J}}_{sys}$ , extracted from the experimental data (by subtraction of the second column from the first column data). Fourth column (j)–(l): Dissipative part of Wigner current ${\vec{J}}_{damp}$ in Eq. (10). Fifth column (m, n): Diffusive part of Wigner current ${\vec{J}}_{diff}$ in Eq. (10). Note that adding ${\vec{J}}_{damp}$ and ${\vec{J}}_{diff}$ in the fourth and fifth columns, respectively, yields ${\vec{J}}_{env}$ shown in the third column: ${\vec{J}}_{env} = {\vec{J}}_{damp} + {\vec{J}}_{diff}$ . We emphasize that, to make them clearly comparable with ${\vec{J}}_{exp}$ and ${\vec{J}}_{sys}$ in (a, d), we had to increase the magnitudes of ${\vec{J}}_{env}, {\vec{J}}_{damp}$ , and ${\vec{J}}_{diff}$ in (g), (j), and (m) by scale factors 375, 125, and 125, respectively.
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Figure 4
Snapshots of Wigner current distributions (arrows, displayed in arbitrary units) of squeezed vacuum states at moderate squeezing, with the colored contours of corresponding Wigner distributions shown in the background. From top to bottom, the corresponding OPO pump powers are 0.25, 1.75, 3.25, and 4.75 mW. From left to right shows the experimental Wigner current ${\vec{J}}_{exp}$ , the ideal Wigner Current ${\vec{J}}_{sys}$ , the environmental current ${\vec{J}}_{env} = {\vec{J}}_{exp} - {\vec{J}}_{sys}$ , the damping current ${\vec{J}}_{damp}$ , and the diffusive current ${\vec{J}}_{diff}$ , respectively. Note that we also have ${\vec{J}}_{env} = {\vec{J}}_{damp} + {\vec{J}}_{diff}$ .
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