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arXiv:2301.00195 · Physical Review A

Sub-Planck structures and sensitivity of the superposed photon-added or photon-subtracted squeezed-vacuum states

Naeem Akhtar, Jizhou Wu, Jia‐Xin Peng, Wu‐Ming Liu, Gao Xianlong

Published May 25, 2023• 24 citations
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Abstract

The Wigner function of the compass state (a superposition of four coherent states) develops phase-space structures of dimension much less than the Planck scale $\ensuremath{\hbar}$, which are crucial in determining the sensitivity of these states to phase-space displacements. In the present work we introduce compasslike states that may have connection to the contemporary experiments, which are obtained by either adding photons to or subtracting photons from the superposition of two squeezed-vacuum states. We show that when a significant quantity of photons is added (or subtracted), the Wigner functions of these states are shown to have phase-space structures of an area that is substantially smaller than the Planck scale. In addition, these states exhibit sensitivity to displacements that is much higher than the standard quantum limit. Finally, we show that both the size of the sub-Planck structures and the sensitivity of our states are strongly influenced by the average photon number, with the photon-addition case having a higher average photon number leading to the smaller sub-Planck structures and, consequently, being more sensitive to displacement than the photon-subtraction case. Our states offer unprecedented resolution to the external perturbations, making them suitable for quantum sensing applications.

DOI
10.1103/physreva.107.052614

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The Wigner function of the compass state (a superposition of four coherent states) develops phase-space structures of dimension much less than the Planck scale \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ, which are crucial in determining the sensitivity of these states to phase-space displacements. In the present work we introduce compasslike states that may have connection to the contemporary experiments, which are obtained by either adding photons to or subtracting photons from the superposition of two squeezed-vacuum states. We show that when a significant quantity of photons is added (or subtracted), the Wigner functions of these states are shown to have phase-space structures of an area that is substantially smaller than the Planck scale. In addition, these states exhibit sensitivity to displacements that is much higher than the standard quantum limit. Finally, we show that both the size of the sub-Planck structures and the sensitivity of our states are strongly influenced by the average photon number, with the photon-addition case having a higher average photon number leading to the smaller sub-Planck structures and, consequently, being more sensitive to displacement than the photon-subtraction case. Our states offer unprecedented resolution to the external perturbations, making them suitable for quantum sensing applications.
Superposed photon-added and photon-subtracted squeezed-vacuum states exhibit sub-Planck phase-space structures and metrological potential similar to the original compass states (superposition of four coherent states), but are more closely tied to modern experiments. Here, we observe that these compasslike states are highly susceptible to loss of quantum coherence when placed in contact with a thermal reservoir; that is, the interaction with the thermal reservoir causes decoherence, which progressively suppresses the capacity of these states to exhibit interference traits. We focus on the sub-Planck structures of these states and find that decoherence effects on these features are stronger with increasing the average thermal photon number of the reservoir, the squeezing parameter, or the quantity of added (or subtracted) photons to the squeezed-vacuum states. Furthermore, we observe that the sub-Planck structures of the photon-subtracted case survive comparatively longer in the thermal reservoir than their counterparts in the photon-added case, and prolonged contact with the thermal reservoir converts these compasslike states into a classical state.
We investigate the sub-Planck-scale structures associated with the SU(1,1) group by establishing that the Planck scale on the hyperbolic plane can be considered as the inverse of the Bargmann index kkk. Our discussion involves SU(1,1) versions of Wigner functions, and the quantum-interference effect is easily visualized through plots of these Wigner functions. Specifically, the superpositions of four Perelomov SU(1,1) coherent states (compass state) yield nearly isotropic sub-Planck structures in phase space scaling as 1/k1/k1/k compared with 1/k1/\sqrt{k}1/k​ scaling for individual SU(1,1) coherent states and anisotropic quadratically improved scaling for superpositions of two SU(1,1) coherent states (cat state). We show that displacement sensitivity exhibits the same quadratic improvement to scaling.
Coherent-state superpositions are of great importance for many quantum subjects, ranging from foundational to technological, e.g., from tests of collapse models to quantum metrology. Here we explore various aspects of these states, related to the connection between sub-Planck structures present in their Wigner function and their sensitivity to displacements (ultimately determining their metrological potential). We review this for the usual Heisenberg-Weyl algebra associated with a harmonic oscillator and extend it to find analogous results for the su(2)\mathrm{su}(2)su(2) algebra, typically associated with angular momentum. In particular, in the Heisenberg-Weyl case, we identify phase-space structures with support smaller than the Planck action in both Schr\"odinger-cat-state mixtures and superpositions, the latter known as compass states. However, as compared to coherent states, compass states are shown to have N\sqrt{N}N​-enhanced sensitivity against displacements in all phase-space directions (NNN is the average number of quanta), whereas cat states and cat mixtures show such enhanced sensitivity only for displacements in specific directions. We then show that these same properties apply for analogous SU(2) states provided (i) coherent states are restricted to the equator of the sphere that plays the role of phase space for this group, (ii) we associate the role of the Planck action with the size of SU(2) coherent states in such a sphere, and (iii) we associate the role of NNN with the total angular momentum.
We discuss the nonclassical properties of photon subtracted compass states (PSCS). Nonclassical behavior is studied using various parameters like the Wigner function, squeezing, and photon statistical parameters like Mandels Q function, secondorder correlation function, Agarwal Tara A3A_3A3​ criterion, and photon number distribution. Further analysis is being done to investigate the subPlanck structures in the Wigner functions of these PSCS. We also show that the photon subtraction causes the loss of sensitivity due to displacement of the states in phase space.
Environmental decoherence occurs when a quantum system interacts with its surroundings, progressively reducing quantum interference and coherence, complicating the preservation of critical quantum features over time, especially during experimental implementation. The quantum features of a state can be represented in phase space via the Wigner function, which manifests across multiple scales, with decoherence potentially influencing each scale differently, as examined in this work. We consider the compass state and its photon-added and photon-subtracted variants (optimized compass states) as our representative examples, each of which exhibits phase-space features with dimensions beyond the Planck scale, making them suitable for quantum sensing applications. We investigate the interaction of these states with a heat reservoir by employing a range of well-established theoretical tools. We observe that compass states with finer-scale phase-space features are more fragile to decoherence, with parameters favoring greater sub-Planckness in phase space concomitantly increasing the fragility of these compass states to decoherence. Our findings are then validated for generic quantum states interacting with the heat reservoir, for which we provide analytical and numerical investigations, exploring the relationship between quantum state robustness to decoherence and the sizes of their phase-space features; that is, phase-space features at smaller scales decay faster under decoherence, and vice versa.
Compass states deliver a sub-Planck phase-space structure in the sense that sensitivity to phase-space displacement is superior to the sensitivity of displacing the vacuum state in any direction, but this sensitivity is anisotropic: better sensitivity for some directions of phase-space displacement versus others. Here we introduce generalized compass states as superpositions of nnn compass states, with each oriented by $\ensuremath{\pi}/2n$ with respect to its predecessor. Specifically, we derive Wigner functions for these generalized compass states and approximate closed-form expressions for overlaps between generalized compass states and their displaced counterparts. Furthermore, we show that generalized compass states, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, display isotropic sensitivity to phase-space displacement in any direction.