Large Phase-by-Phase Modulations in Atomic Interfaces

M. Artoni and A. Zavatta

Phys. Rev. Lett. 115, 113005 – Published 10 September 2015

Abstract

Phase-resonant closed-loop optical transitions can be engineered to achieve broadly tunable light phase shifts. Such a novel phase-by-phase control mechanism does not require a cavity and is illustrated here for an atomic interface where a classical light pulse undergoes radian level phase modulations all-optically controllable over a few micron scale. It works even at low intensities and hence may be relevant to new applications of all-optical weak-light signal processing.

Received 16 March 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.113005

Authors & Affiliations

M. Artoni^1,2,3 and A. Zavatta²

¹European Laboratory for Nonlinear Spectroscopy (LENS), I-50019 Sesto Fiorentino, Firenze, Italy
²Istituto Nazionale di Ottica (INO-CNR), I-50125 Firenze, Italy
³Department of Engineering and Information Technology, Brescia University, I-25133 Brescia, Italy

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Issue

Vol. 115, Iss. 11 — 11 September 2015

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Images

Figure 1
(a) Phase Control. Two weak coherent wave packets $ω_{m}$ and $ω_{p}$ , with equal initial intensities ( $n_{m} = n_{p}$ ) and length ( $L$ ), and two coupling beams $ω_{c m}$ and $ω_{c p}$ couple to an atomic interface of length $z$ [30]. The couplings relative phase $Δ ϕ_{c}$ , the phase $ϕ_{p}$ and detuning $δ_{p}$ of the control wave packet are used to steer the phase $ϕ_{m}$ of the signal wave packet. Here $ϕ_{m} (ϕ_{p})$ and $n_{m} (n_{p})$ are the initial mean phase angle and mean photon number of the coherent state excitation $α_{m}$ ( $α_{p}$ ) in the mode $m (p)$ . The two couplings Rabi frequency are fixed at $Ω_{c} = 2 γ_{2}$ and the Raman detuning at $Δ = 20 γ_{2}$ . (b) The interface. The level scheme corresponds to the transition $5 {{}^{2}S}_{1 / 2} \to 5 {{}^{2}P}_{3 / 2}$ ( ${}^{87}{Rb}$ $D_{2}$ line) at $λ_{21} = 780.24 nm$ with decay rates $γ_{2} = 2 π \times 6 MHz$ and $γ_{3} = 2 π \times 1 kHz$ . The wave packets with a bandwidth $c / L \sim 1 / τ ≲ γ_{2} ≪ Δ$ are double resonant with the couplings ( $ω_{m} - ω_{c m} ≃ ω_{p} - ω_{c p}$ ) through a nearly resonant ( $δ_{p} ≃ δ_{c p} ≲ γ_{2}$ ) transition and a far off-resonant ( $δ_{m} ≃ δ_{c m} = Δ$ ) Raman transition between the two hyperfine grounds $| 1 ⟩$ and $| 3 ⟩$ . (c) Measurement. Mach-Zehnder setup used to measure $ϕ_{m} (z, t)$ at the photodetector (PD). The interferometer may also be used to implement a low-light optical switch [33].
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Figure 2
Signal phase modulations accumulated at time $t_{0} = 0.1 τ$ by a narrow band signal Gaussian wave packet of duration $τ = 5.2 μ s$ for arbitrary interfaces of lengths $z$ and phases $Φ$ . Each curve represents values ${z, Φ}$ for which $ϕ_{m} (z, t_{0})$ has the same value. Phases are in rad with an atomic density $N / V ≃ 10^{13} {cm}^{- 3}$ and a length scale $L = 1 mm$ . All other parameters are as in Fig. 1.
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Figure 3
(a) Signal phase $ϕ_{m} (z, t_{0})$ vs $z$ ( $Φ = π / 4$ ) when $δ_{p} = 0$ and $N / V ≃ 10^{13} {cm}^{- 3}$ (green), $δ_{p} = 0$ and $N / V ≃ 5 \times 10^{12} {cm}^{- 3}$ (blue), $δ_{p} = 0.03 γ_{2}$ and $N / V ≃ 10^{13} {cm}^{- 3}$ (brown). Inset: Inverse tangent term (green) in Eq. (1) around $z ≃ 0.0054 L$ (box). Large shifts arise from a swift change in sign of the numerator (dash) over $μ m$ length scale where the denominator (solid) has only smooth variations. (b) Signal phase $ϕ_{m} (z_{0}, t_{0})$ vs $Φ$ when $z_{0} = 0.006 L$ (blue), $z_{0} = 0.01 L$ (violet), and $z_{0} = 0.14 L$ (green). For $z_{0} = 0.01 L$ the phase undergoes a $π$ shift for $Φ ≃ 2.9$ (blue-dot). Inset: The phase plotted $π$ jumps are due to a quadrant swap [33].
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Figure 4
Signal phase $ϕ_{m} (z_{0}, t)$ (green line) vs time for an interface of length $z_{0} = 5.4 μ m$ and intensity of the signal (purple fill) and control (blue fill) pulses. The signal drops to zero at $t_{0} ≃ - 0.6 τ$ . Right, inset: Loci where the denominator (continuous) and numerator (dashed) of the inverse tangent term in Eq. (1) are zero. At the intersections (dots) the inverse tangent is undefined and the field intensity is zero. Left, inset: Enlargement of the box in the main frame showing the signal phase singularity corresponding to the lower intersection point inside the green square box shown in the right inset. The singularity appears around the point $t_{0} = - 0.6 τ \pm δ t$ ( $δ t = 5 \times 10^{- 11} τ$ ).
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