Generating overlap between compass states and squeezed, displaced, or Fock states

Arman and Prasanta K. Panigrahi

Phys. Rev. A 109, 033724 – Published 25 March 2024

Abstract

We investigate a broad class of nonclassical states, composed of superposed squeezed and displaced number states. The phase-space structure is analyzed, keeping in mind the Heisenberg limited sensitivity in parameter estimation. Appropriate squeezing and displacement parameters are identified, wherein the state fidelity in comparison to the metrologically sensitive compass state is more than $99 %$ . Also, the equal variance in small shift measurements, for the proposed and compass state, shows identical behavior of the average photon number. In metrological application, the number variance of the compass state, which is small for low coherent amplitude, suggests its potential to estimate damping parameters. Finally, we present the theoretical models corresponding to their preparation.

Received 26 March 2023
Revised 9 October 2023
Accepted 20 February 2024

DOI:https://doi.org/10.1103/PhysRevA.109.033724

Physics Subject Headings (PhySH)

Quantum states of light

General PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Arman ^1,* and Prasanta K. Panigrahi ^1,2,†

¹Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur-741246, West Bengal, India
²Centre for Quantum Science and Technology (CQST), Siksha ‘O' Anusandhan Deemed to be University, Khandagiri Square, Bhubaneswar-751030, Odisha, India

^*a19rs004@iiserkol.ac.in
^†Corresponding author: pprasanta@iiserkol.ac.in

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Vol. 109, Iss. 3 — March 2024

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Figure 1
Comparative PND analysis for SSDNS vs KS states. (a) Strong overlap in PND plot between SSDNS $ψ_{1}$ and $ψ_{KS}^{0} (-)$ with parameters $(r, α, n, β) = [0.15, 0.61, 1, 0.75 (1 + i)]$ , suggesting potential squeezing and displacement strength for proximity to the KS state. (b) Reduced overlap with $β = 2$ , while other parameters remain constant. (c) PND comparison of SSNS $ψ_{2}$ and KS $ψ_{KS}^{2} (+)$ with parameters $(r, n) = (0.3, 2)$ and $β = 1.5$ reveals substantial similarity. (d) Noticeably decreased overlap as $β$ for the KS state becomes 2.
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Figure 2
The interference appears in the Wigner function of SSDNS for squeezing $r = 0.45$ and displacement $α = 2$ for different Fock number: (a) $n = 1$ , (b) $n = 2$ , (c) $n = 3$ , (d) $n = 4$ . The phase-space distribution for (a)–(d) is compared with two different KS, $ψ_{KS}^{0} (-)$ and $ψ_{KS}^{1} (-)$ . We present the plots of KS $ψ_{KS}^{0} (-)$ with parameters (e) $β = 1.7 e^{3 i π / 4}$ and (f) $β = 2 e^{3 i π / 4}$ . Additionally, we provide the plots of KS $ψ_{KS}^{1} (-)$ for (g) $β = 1.7 e^{3 i π / 4}$ and (h) $β = 2 e^{3 i π / 4}$ .
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Figure 3
The sub-Planck scale structure appears in the phase space of SSNS for squeezing $r = 0.45$ with Fock number (a) $n = 1$ , (b) $n = 2$ , (c) $n = 3$ , (d) $n = 4$ , as well as in the same distribution for different KSs: (e) $ψ_{KS}^{1} (+)$ , (f) $ψ_{KS}^{2} (+)$ with $β = 1.5$ , and (g) $ψ_{KS}^{3} (+)$ , (h) $ψ_{KS}^{4} (+)$ with $β = 2$ . Significant similarity is observed between plots of pairs: (a) and (e), (b) and (f), (c) and (g), and (d) and (h).
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Figure 4
Comparative variance and average photon number $〈 n 〉$ analysis for SSDNS vs KS: (a) Blue line follows $F_{Q} (ψ_{1}) = F_{Q} (ψ_{KS}^{0} (-))$ leading to two constraints $A^{ψ_{1}} = A^{K S}$ and $B^{ψ_{1}} = B^{KS}$ . This curve corresponds to equal variance for SSDNS with $n = 1$ and $ψ_{KS}^{0} (-)$ , with $β$ being the absolute of coherent amplitude $β e^{i π / 4}$ . (b) $〈 n 〉$ and (c) product of variance and $〈 n 〉$ become the same for constraints in (a), at each $θ$ with respect to SSDNS and $ψ_{KS}^{0} (-)$ . (d) Both states have the same variance changing with $θ$ at small $β = 1.01$ and remain constant at $β = 2.02$ .
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Figure 5
Comparative variance and average photon number $〈 n 〉$ analysis for SSNS vs KS: (a) Both curves follow $F_{Q} (ψ_{2}) = F_{Q} (ψ_{KS}^{l} (+))$ , leading to a constraint $A^{ψ_{2}} = A^{KS}$ . Blue curve corresponds to equal variance for SSNS with $n = 1$ and $ψ_{KS}^{1} (+)$ , with $β$ being the coherent amplitude. Similarly, the red curve describes the same variance for SSNS with $n = 2$ and $ψ_{KS}^{2} (+)$ . The following pairs: SSNS $n = 1$ (yellow) and $ψ_{KS}^{1} (+)$ (black), and SSNS $n = 2$ (blue) and $ψ_{KS}^{2} (+)$ (red), have overlapping lines for (b) $〈 n 〉$ and (c) the variance multiplied by $〈 n 〉$ under constraint in (a), showing resemblance for SSNS and KS. (d) The variance remains independent of $θ$ and overlaps along points in (a), for SSNS ( $n = 1$ ) with $ψ_{KS}^{1}$ . For illustration, we use two sets of points with parameters $r$ and $β$ : $(r, β) = (0.15, 1.0)$ represented by the thick line and $= (0.6, 2.07)$ denoted by the dotted line.
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Figure 6
The regions of the pairs (a) SSDNS and $ψ_{KS}^{0} (-)$ and (b) SSNS and $ψ_{KS}^{1} (+)$ show values of $Δ^{2} N / {〈 N 〉}^{2}$ (thick) and ${〈 N 〉}^{- 1}$ (dashed), which closely resemble single-photon states. (a) $β$ for both SSDNS: $S [r] (D [α] + D [- α]) | 1 〉$ (black) and $S [r] (D [α] - D [- α]) | 0 〉$ (red) with squeezing $r = 0.1$ represents $α$ , while coherent amplitude for $ψ_{KS}^{0} (-)$ (blue). (b) $β$ for SSNS (red) with $n = 1$ is squeezing ( $r$ ), and coherent amplitude $ψ_{KS}^{1} (+)$ (blue). Both (a) and (b) have regions for parameter $β$ , where $Δ^{2} N / {〈 N 〉}^{2} ⟶ 0$ and ${〈 N 〉}^{- 1} ⟶ 1$ .
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