Engineering multipartite coupling in doubly pumped parametric down-conversion processes

We investigate the quantum state generated by optical parametric down-conversion in a ${\chi }^{\left(2\right)}$ medium driven by two noncollinear light modes. The analysis shows the emergence of multimode, namely, three- or four-mode, entangled states in a subset of the spatiotemporal modes generated by the process. These appear as bright spots against the background fluorescence, providing an interesting analogy with the phenomenology recently observed in two-dimensional nonlinear photonic crystals. We study two realistic setups. (1) Noncritical phase matching in a periodically poled lithium tantalate slab, characterized by a three-mode entangled state. (2) A type I setup in a beta-barium borate crystal, where the spatial walk-off between the two pumps can be exploited to make a transition to a four-mode entangled state. In both cases, we show that the properties of the state can be controlled by modulating the relative intensity of two pump waves, making the device a versatile tool for quantum state engineering.

Figure 1

(a) Schematic of the dual pump source, where two noncollinear modes of amplitudes ${\alpha }_1$ and ${\alpha }_2$ pump a ${\chi }^{\left(2\right)}$ medium, and generate three-mode entangled states in specific sets of spatiotemporal modes (Sec. 3b). The two pumps can be seen as deriving from a single pump ${\overline{\alpha }}_p=\sqrt{|{\alpha }_1{|}^2+{\left|{\alpha }_2\right|}^2}$. We will show that the source is formally equivalent to the scheme (b), in which the medium is pumped by a single beam of amplitude ${\overline{\alpha }}_p$, and one of the two parties of the EPR state thereby generated is mixed with an arbitrary input mode ${\widehat{d}}_{i-}$ on a beam-splitter with reflection and transmission coefficients $\frac{R}{T}=\frac{|{\alpha }_2{|}^2}{|{\alpha }_1{|}^2}$. ${\varphi }_{-}$ are local phase shift by half of the pump phase difference.

Figure 2

Decomposition of the four-mode entangled state generated by a doubly pumped BBO in the resonance conditions demonstrated in Sec. 3. This is equivalent to a pair of independent EPR states mixed on a beam-splitter BS and followed by phase rotations (gray boxes). The squeeze parameters ${\mathrm{\Lambda }}_{\sigma }, {\mathrm{\Lambda }}_{\delta }$ of the EPR states, and the BS transmission and reflection coefficients are controlled by the intensity ratio of the two pump modes [see Eqs. (34) and (35)]. The description holds for specific quadruplets of spatiotemporal modes which will be characterized in Sec. 3a.

Figure 3

(a) Geometry of the scheme for the $\mathrm{e}\to \mathrm{e},\mathrm{e}$ process in a periodically poled $\mathrm{LiTa}{0}_3$ slab. $O_1,O_2,O_3$ are the crystal principal axes. All the fields are polarized along the optical $O_3$ axis, and propagate at small angles with the $O_2$ axis. The pumps are slightly tilted along $x=O_1$. (b) Far-field distribution of down-converted light from numerical simulations (see text). The plot shows a $20\mathrm{nm}$ bandwidth around $1064\mathrm{nm}$.

Figure 4

PPLT pumped at $532\mathrm{nm}$, by two pump modes tilted at ${\theta }_{p1,p2}=\mp 1.2^{\circ }$ (internal angle). (a) Plot in the 3D space $({\theta }_x,{\theta }_y,\lambda )$ of the surfaces ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$ [Eq. (5)] defining phase matching for each pump, calculated with the Sellmeier formulas in Ref. [28], for $\mathrm{T}={75}^{\circ }$ and ${\mathrm{\Lambda }}_{\mathrm{pol}}=7.79\mu \mathrm{m}$. (b) Section at ${\lambda }_s={\lambda }_i=1064\mathrm{nm}$, showing the triplet of entangled modes. (c) and (d) Sections at two conjugate wavelengths ${\lambda }_s=900\mathrm{nm}$ and ${\lambda }_i=1300\mathrm{nm}$, showing two independent triplets of modes (stars and dots).

Figure 5

Numerical simulations of Eq. (1) for a doubly pumped PPLT, showing the Fourier $(q_x,q_y)$ and angular $({\theta }_x,{\theta }_y)$ intensity distributions of light down-converted in the bandwidth $950-1210$ nm, from two Gaussian pump pulses of peak amplitudes ${\alpha }_1$ and ${\alpha }_2, 1\mathrm{ps}$ duration and $400\mu \mathrm{m}$ waist, tilted at ${\theta }_{p1,p2}=\mp 1.2^{\circ }$. In rows (a)–(c) ${\alpha }_2={\alpha }_1$, with the three hot-spot branches becoming progressively brighter for increasing propagation length. In rows (d)–(f), ${\alpha }_2={\alpha }_1/4$, corresponding to $R\simeq 6\%$ in the equivalent scheme of Fig. 1, and the left branch is much less intense. The overall peak gain is $\overline{g}=\sqrt{g_1^2+g_2^2}=1.2{\text{mm}}^{-1}$, other parameters as in Fig. 4.

Figure 6

Geometry of the scheme, for the $\mathrm{e}\to \mathrm{oo}$ process in a BBO crystal, cut at ${\gamma }_0=33.{44}^{\circ }$. The two pumps propagate mainly along $z$, with small transverse tilts ${\vec{Q}}_1$ and ${\vec{Q}}_2$. The orientation of ${\vec{Q}}_2-{\vec{Q}}_1$ with respect to the optical axis $O_3$ [lying in the plane $(y^{\prime },z)]$ determines different configurations: in the example (A) the pumps form roughly the same angle with the optical axis, while in (B) and (C) they propagate at different angles with $O_3$, and have different wave numbers.

Figure 7

BBO doubly pumped at $352\mathrm{nm}$. ${\theta }_{p1}=0;{\theta }_{p2}=1.2^{\circ }$. [(a)–(c)] 3D plots in the $(\lambda ,{\theta }_x,{\theta }_y)$ space of the phase-matching surfaces ${\mathrm{\Sigma }}_1$ and ${\mathrm{\Sigma }}_2$ in Eq. (5) calculated with the Sellmeier formulas from [35]. [(d)–(i)] Angular phase-matching curves at the two conjugate wavelengths ${\lambda }_s=0.6\mu \mathrm{m}$ [(d)–(f)] and ${\lambda }_i=0.85\mu \mathrm{m}$ [(g)–(i)], illustrating the transition to resonance. The symbols show the position of shared and coupled modes. For $\beta =0$ (left column), two independent triplets of entangled modes coexist (dots and stars). At ${\beta }_1^{\mathrm{res}}=-7.{16}^{\circ }$ (central column) and ${\beta }_2^{\mathrm{res}}=9.{98}^{\circ }$ (right column), resonance is achieved. The two triplets of modes merge into four entangled modes.

Figure 8

(a) Squeeze eigenvalues ${\mathrm{\Lambda }}_{\sigma }$ and ${\mathrm{\Lambda }}_{\delta }$ in Eq. (34), normalized to the reference squeeze parameter $\overline{g}$ of a single pump carrying all the energy. The inset shows the maximum of ${\mathrm{\Lambda }}_{\sigma }$, occurring at $|g_2|=\sqrt{2}\left|g_1\right|$. (b) Mixing coefficients of the unitary transformation in Eq. (30).

Figure 9

Illustration of the resonance in the case $\beta =\frac{\pi }{2}$. The blue thick line represents the ellipsoid of the refraction indexes, ${\vec{p}}_1, \overrightarrow{p_2}$, and $\overline{p}$ show the directions of propagation of pump 1, pump 2, and of the pump carrier respectively, while $\vec{S}$ is the propagation direction of its Poynting vector. At resonance, it superimposes to the direction of propagation of pump 1.