Pairwise entanglement and readout of atomic-ensemble and optical wave-packet modes in traveling-wave Raman interactions

We analyze quantum entanglement of Stokes light and atomic electronic polarization excited during single-pass, linear-regime, stimulated Raman scattering in terms of optical wave-packet modes, and atomic-ensemble spatial modes. The output of this process is confirmed to be decomposable into multiple discrete, Bosonic mode pairs, each pair undergoing independent evolution into a two-mode squeezed state. For this we extend the Bloch-Messiah reduction theorem, previously known for discrete linear systems [S. L. Braunstein, Phys. Rev. A 71, 055801 (2005)]. We present typical mode functions in the case of one-dimensional scattering in an atomic vapor. We find that in the absence of dispersion, one mode pair dominates the process, leading to a simple interpretation of entanglement in this continuous-variable system. However, many mode pairs are excited in the presence of dispersion-induced temporal walkoff of the Stokes, as witnessed by the photon-count statistics. We also consider the readout of the stored atomic polarization using the anti-Stokes scattering process. We prove that the readout process can also be decomposed into multiple mode pairs, each pair undergoing independent evolution analogous to a beam-splitter transformation. We show that this process can have unit efficiency under realistic experimental conditions. The shape of the output light wave packet can be predicted. In the case of unit readout efficiency it contains only excitations originating from a specified atomic excitation mode.

Figure 1

Relevant atomic levels in the Stokes and anti-Stokes (upper panel) Raman scattering. Initially atoms reside in state ∣1⟩. During the first scattering process (lower panel) they are driven by strong pump field at a frequency ${\omega }_p$, which creates Stokes optical field $\widehat{a}$ at a frequency ${\omega }_{\mathrm{S}}$ and atomic polarization $\widehat{b}$. Subsequently, the atoms are subjected to a strong field at a frequency ${\omega }_{p^{\prime }}$ which erases atomic polarization we will denote by $\widehat{d}$ shifting the excitation into anti-Stokes optical field $\widehat{c}$ at a frequency ${\omega }_a$ (upper panel).

Figure 2

(Color online) Pictorial representation of the Stokes and subsequent anti-Stokes scattering processes (see Sec. 5) in the laboratory reference frame. The vertical stripe represents the cell in which the nonlinear medium is contained, the shaded parallelograms are the spatiotemporal interaction boundaries, the gray (green in color) stripe represents Stokes pump field envelope $A_p(z,t)$, while the dark gray (red) stripe represents anti-Stokes pump field envelope $A_p^{\prime }(z,t)$. Letters with hats are the annihilation operators of the input and output quantum fields, defined at the respective boundaries of the interaction regions.

Figure 3

(Color online) Pictorial representation of the Stokes scattering process in the Stokes reference frame, defined in Eq. (1). The medium entangles light field $\widehat{a}$ with atoms $\widehat{b}$. The local entanglement amplitude increases with a (fixed) local pump field strength $A_p(z,t)$, represented in the picture by a gray (green in color) stripe. Horizontal stripe corresponds to equal group velocities of pump and Stokes, $\Delta \beta =0$.

Figure 4

Mean number of Stokes photons as a function of coupling strength $g_0$ $\left[{\left(\mathrm{mm}\mathrm{ps}\right)}^{-1∕2}\right]$. Three lines are the results of computations for different $\Delta \beta$ equal 0 (solid line), $-10\mathrm{ps}∕\mathrm{mm}$ (dashed line), and $-30\mathrm{ps}∕\mathrm{mm}$ (dotted line).

Figure 5

Mean number of photons in each of characteristic output modes $⟨{\widehat{n}}_n⟩$ for 15 most-occupied modes. The histograms has been made for coupling yielding average total number of Stokes photons $⟨N_{\mathrm{tot}}⟩={10}^6$ for different $\Delta \beta$ equal 0 (black bar), $-10\mathrm{ps}∕\mathrm{mm}$ (gray bars) and $-30\mathrm{ps}∕\mathrm{mm}$ (empty bars).

Figure 6

(Color online) Pictorial representation of the Stokes scattering for zero group velocity difference $\Delta \beta$ (left) and large negative $\Delta \beta ⪡-{\tau }_p∕L$ (right). The gray box represents the spatiotemporal region of interaction, the dark gray (green in color) stripe is the pumped region, the horizontal (red) arrows are the Stokes emission, while the vertical (blue) arrows represent atomic polarization. The picture has been drawn in the Stokes reference frame.

Figure 7

Optical ${\psi }_n^{\left(\mathit{out}\right)}\left(t\right)$ and atomic ${\varphi }_n^{\left(\mathit{in}\right)}\left(t\right)$ output mode functions for three most excited modes, $n=1$ (solid line), $n=2$ (dashed line), and $n=3$ (dotted line) for various group velocity differences $\Delta \beta$.

Figure 8

Photon count probability distribution $p\left(n\right)$ of Stokes light pulses. Three lines are the results of computations for different $\Delta \beta$ equal 0 (solid line), $-10\mathrm{ps}∕\mathrm{mm}$ (dashed line), and $-30\mathrm{ps}∕\mathrm{mm}$ (dotted line).

Figure 9

Equivalent number of independent thermal modes $\mathcal{M}$ as a function of coupling strength $g_0$ $\left[{\left(\mathrm{mm}\mathrm{ps}\right)}^{-1∕2}\right]$. Three lines are the results of computations for different $\Delta \beta$ equal 0 (solid line), $-10\mathrm{ps}∕\mathrm{mm}$ (dashed line), and $-30\mathrm{ps}∕\mathrm{mm}$ (dotted line).

Figure 10

Readout efficiency ${\eta }_n$ for the most efficiently read modes in anti-Stokes scattering process for $\Delta {\beta }^{\prime }=0$ and coupling $g_0^{\prime }$ equal 0.01, 0.02, 0.05, $0.1{\left(\mathrm{ps}\mathrm{mm}\right)}^{-1∕2}$, for black, gray, and empty bars, respectively.

Figure 11

Fraction of residual atomic polarization $1-\eta$ not transferred to an optical field after the anti-Stokes scattering process as a function of the readout coupling $g_0$ $\left[{\left(\mathrm{ps}\mathrm{mm}\right)}^{-1∕2}\right]$. Results for three group velocity differences $\Delta {\beta }^{\prime }$ equal 0 (solid line), $-10\mathrm{ps}∕\mathrm{mm}$ (dashed line), and $-30\mathrm{ps}∕\mathrm{mm}$ (dotted line) are displayed.

Figure 12

Readout anti-Stokes light modes ${\sigma }_1\left(t\right)$ for fundamental atomic Stokes modes ${\varphi }_1^{\left(\mathit{out}\right)}\left(z\right)$ for various group velocity difference $\Delta \beta =\Delta {\beta }^{\prime }$ and coupling strengths corresponding to readout efficiency peaks for nonzero $\Delta {\beta }^{\prime }$ (compare Fig. 11). Solid, dashed, and dotted lines correspond respectively to subsequent values of $g_0^{\prime }$, listed below each panel.

Figure 13

(Color online) Pictorial representation of the anti-Stokes scattering for zero group velocity difference $\Delta {\beta }^{\prime }$ (a), medium value of $\Delta {\beta }^{\prime }$ (b) and high negative $\Delta {\beta }^{\prime }⪡-{\tau }_p∕L$ (c). The gray box represents the spatiotemporal region of interaction, the dark gray (red in color) stripe is the pumped region, the horizontal (green) arrows are the anti-Stokes emission, while the vertical (blue) arrows represent atomic polarization. The picture is drawn in the anti-Stokes reference frame. Panel (d) gives insight into oscillatory exchange of excitation between light and atoms.