Higher-order two-mode and multimode entanglement in Raman processes

Sandip Kumar Giri, Biswajit Sen, Anirban Pathak, and Paresh Chandra Jana

Phys. Rev. A 93, 012340 – Published 25 January 2016

Abstract

The existence of higher-order entanglement in the stimulated and spontaneous Raman processes is established using the perturbative solutions of the Heisenberg equations of motion for various field modes that are obtained using the Sen-Mandal technique and a fully quantum mechanical Hamiltonian that describes the stimulated and spontaneous Raman processes. Specifically, the perturbative Sen-Mandal solutions are exploited here to show the signature of the higher-order two-mode and multimode entanglement. In some special cases, we have also observed higher-order entanglement in the partially spontaneous Raman processes. Further, it is shown that the depth of the nonclassicality indicators (parameters) can be manipulated by the specific choice of coupling constants, and it is observed that the depth of nonclassicality parameters increases with the order.

Received 22 September 2015

DOI:https://doi.org/10.1103/PhysRevA.93.012340

Physics Subject Headings (PhySH)

Quantum entanglement

Quantum Information, Science & Technology

Authors & Affiliations

Sandip Kumar Giri^1,2, Biswajit Sen³, Anirban Pathak⁴, and Paresh Chandra Jana²

¹Department of Physics, Panskura Banamali College, Panskura-721152, India
²Department of Physics, Vidyasagar University, Midnapore-721102, India
³Department of Physics, Vidyasagar Teachers' Training College, Midnapore-721101, India
⁴Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India

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Vol. 93, Iss. 1 — January 2016

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Images

Figure 1
Two-photon stimulated Raman scheme. The pump photon is converted into a Stokes photon and a phonon. The pump photon can also mix with a phonon to produce an anti-Stokes photon.
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Figure 2
Higher-order intermodal entanglement in stimulated Raman process with $|α_{1}| = 10, |α_{2}| = 8, |α_{3}| = 0.01$ and $|α_{4}| = 1$ using HZ-1 criteron is shown for different values of phase angle ( $ϕ = 0, \frac{π}{2}$ , and $π$ ) in pump mode. Higher-order intermodal entanglement is observed in (a) pump-Stokes mode for phase angle $0,$ (b) Stokes-vibration phonon mode for phase angle $ϕ = \frac{π}{2},$ (d) pump-anti-Stokes mode for phase angle $0,$ (f) Stokes-anti-Stokes mode for phase angle 0; and not observed in (c) pump-vibration phonon mode and (e) vibration phonon-anti-Stokes mode. In all the plots, the smooth line, dotted line, and dash-dotted line are used for the $m = 1$ and $n = 1, 2$ , and 3, respectively. In (e), $n = 2$ and 3 are multiplied by $10^{3}$ and $10^{6}$ , respectively. While in all the remaining cases, $n = 1$ and 2 are shown 1500 and 50 times, respectively. All the quantities plotted here and in the following plots are dimensionless.
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Figure 3
Higher-order intermodal entanglement in stimulated Raman process is illustrated using HZ-2 criterion with different phase angle ( $ϕ = 0, \frac{π}{2}$ , and $π$ ) in pump mode for $|α_{1}| = 10, |α_{2}| = 8, |α_{3}| = 0.01$ , and $|α_{4}| = 1$ . Specifically, higher-order intermodal entanglement is observed in (b) for Stokes-vibration phonon mode with phase angle $ϕ = \frac{π}{2},$ (c) for pump-vibration phonon mode with phase angle $ϕ = \frac{π}{2},$ and (f) for Stokes-anti-Stokes mode with phase angle $\frac{π}{2}$ . However, in (a), (d), and (e) higher-order intermodal entanglement is not observed for pump-Stokes, pump-anti-Stokes and vibration phonon-anti-Stokes modes, respectively. The smooth, dotted and dash-dotted lines are used for $m = 1$ and $n = 1, 2$ , and 3, respectively. Here, for (e) $n = 2$ and 3 are multiplied by $10^{3}$ and $10^{6}$ , respectively. For all the remaining cases, $n = 1$ and 2 are shown 1500 and 50 times, respectively.
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Figure 4
The variation of three-mode entanglement among pump, Stokes and vibration phonon modes (a) for stimulated Raman process using $|α_{1}| = 10, |α_{2}| = 8, |α_{3}| = 0.01, |α_{4}| = 1$ (b) for spontaneous Raman process using $|α_{1}| = 10, |α_{2}| = |α_{3}| = |α_{4}| = 0$ with the smooth, dashed, and dash-dotted lines corresponding to $ϕ = 0, \frac{π}{2}$ , and $π,$ respectively.
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Figure 5
Four-mode entanglement among pump, Stokes, vibration phonon, and anti-Stokes modes is depicted in (a) stimulated Raman process using $|α_{1}| = 10, |α_{2}| = 8, |α_{3}| = 0.01, |α_{4}| = 1$ (b) partial spontaneous Raman process using $|α_{1}| = 10, |α_{2}| = |α_{3}| = 0$ . Here, the smooth line, dashed line and dash-dotted line corresponds to $ϕ = 0, \frac{π}{2}$ , and $π,$ respectively.
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