Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum superpositions

We discuss a device capable of filtering out two-mode states of light with mode populations differing by more than a certain threshold, while not revealing which mode is more populated. It would allow engineering of macroscopic quantum states of light in a way which is preserving specific superpositions. As a result, it would enhance optical phase estimation with these states as well as distinguishability of “macroscopic” qubits. We propose an optical scheme, which is a relatively simple, albeit nonideal, operational implementation of such a filter. It uses tapping of the original polarization two-mode field, with a polarization-neutral beam splitter of low reflectivity. Next, the reflected beams are suitably interfered on a polarizing beam splitter. It is oriented such that it selects unbiased polarization modes with respect to the original ones. The more an incoming two-mode Fock state is unequally populated, the more the polarizing beam-splitter output modes are equally populated. This effect is especially pronounced for highly populated states. Additionally, for such states we expect strong population correlations between the original fields and the tapped one. Thus, after a photon-number measurement of the polarizing beam-splitter outputs, a feed-forward loop can be used to let through a shutter the field, which was transmitted by the tapping beam splitter. This happens only if the counts at the outputs are roughly equal. In such a case, the transmitted field differs strongly in occupation number of the two modes, while information on which mode is more populated is nonexistent (a necessary condition for preserving superpositions).

Figure 1

Comparison of two filtering techniques: absolute difference (MDF) (a) and orthogonality filter (OF) (b). The dots in (b) symbolize specific possible measurement results of photon numbers. The state of the field filtered by an OF is represented by one of the dots. In the case of MDF, the state is projected onto the whole YES region, which preserves quantum coherence of components occupying both regions. $k$ and $l$ denote numbers of photons in two orthogonal polarization modes.

Figure 2

Photon distribution $p_{\Phi }$ for the macroscopic state $|\Phi \rangle$ computed for $g=1.87$ and filtering threshold ${\delta }_{\mathrm{th}}=0$ (a) and ${\delta }_{\mathrm{th}}=200$ (b). $k$ and $l$ denote numbers of photons in two orthogonal polarization modes. The one-dimensional plots show values of $p_{\Phi }$ for $k=0$ (the left one) and $l=0$ (the bottom one), respectively.

Figure 3

Photon distribution $p_{\Phi }$ for the macroscopic state $|\Phi \rangle$ computed for $g=1.87$, filtering threshold ${\delta }_{\mathrm{th}}=200$, and $50\%$ (a) and $90\%$ (b) of losses. $k$ and $l$ denote numbers of photons in two orthogonal polarization modes. The one-dimensional plots show values of $p_{\Phi }$ for $k=0$ (the left one) and $l=0$ (the bottom one), respectively.

Figure 4

Distinguishability $v$ of macroqubits [Eq. (4)] evaluated for gain $g=1.87$ and several threshold values ${\delta }_{\mathrm{th}}$ as function of losses $R$.

Figure 5

An approximate operational scheme of an MDF. The box MDF in (b) is the setup given in (a). The details are in the main text.

Figure 6

Distribution of the population difference $p^{S,\Delta }\left({\Delta }_r\right)$ in a superposition Fock input state $|{\Psi }_{\mathrm{in}}{\rangle =|n,m\rangle }_r$ conditioned on the measurement of $S=200$ photons and $\Delta =0$ (a), $\Delta =80$ (b), $\Delta =200$ (c) at the PBS output. The vertical dashed lines show the threshold ${\delta }_{\mathrm{th}}=30$. The probability that $|{\Delta }_r|\ge 30$ is given by $p\left(\right|{\Delta }_r|\ge 30)$.

Figure 7

Physical implementation of the MDF with the notation indicating the state evolution in different parts of the setup.

Figure 8

Distribution of the population difference $p^{S,\Delta }\left({\Delta }_t\right)$ in the transmitted beam $t$ after the shutter for the state in Eq. (B9) with $S_0=200$ assuming that $S=20$ photons were registered in the reflected beam and the difference measured by detectors was $\Delta =0$ (a), $\Delta =10$ (b), $\Delta =20$ (c). The vertical dashed lines show the threshold ${\delta }_{\mathrm{th}}=150$. The probability that $|{\Delta }_t|\ge 150$ is given by $p\left(\right|{\Delta }_t|\ge 150)$.

Figure 9

Distribution of the population difference $p^{S,\Delta }\left({\Delta }_t\right)$ [Eq. (B8)] in the transmitted beam $t$ after the shutter for ${\rho }_{\mathrm{in}}=1/2\left(\right|\Phi \rangle \langle \Phi |+|{\Phi }_{\perp }\rangle \langle {\Phi }_{\perp }\left|\right)$ for $g=1.87$ assuming that $S=20$ photons were registered in the reflected beam and the difference measured by detectors was $\Delta =0$ (a), $\Delta =10$ (b), $\Delta =20$ (c). The vertical dashed lines show the threshold ${\delta }_{\mathrm{th}}=40$. The probability that $|{\Delta }_t|\ge 40$ is given by $p\left(\right|{\Delta }_t|\ge 40)$.

Figure 10

Photon-number distribution $p_{\Phi }$ [Eq. (C2)] and distinguishability $v$ [Eq. (4)] of the macroscopic state $|\Phi \rangle$ processed by the setup from Fig. 5b, computed for $g=1.87$, the level of trust $90\%$, and ${\delta }_{\mathrm{th}}=0$ (a), ${\delta }_{\mathrm{th}}=5$ (b), ${\delta }_{\mathrm{th}}=10$ (c), ${\delta }_{\mathrm{th}}=15$ (d). $k$ and $l$ denote numbers of photons in two orthogonal polarization modes.

Figure 11

Distribution of the population difference $p_2^{S,\Delta }\left({\Delta }_t\right)$ [Eq. (C3)] in the transmitted beam $t$ after the shutter for the two-mode state in Eq. (C1) for $g=1.87$ assuming that $S=20$ photons were registered in the reflected beam and the difference measured by detectors was $\Delta =0$ (a), $\Delta =10$ (b), $\Delta =20$ (c).

Figure 12

Distribution of the population difference $p_R^{S,\Delta }\left({\Delta }_t\right)$ [Eq. (D1)] in the transmitted beam $t$ after the shutter for the state in Eq. (C1) subjected to $20\%$ of losses for $g=1.87$ assuming that $S=20$ photons were registered in the reflected beam and the difference measured by detectors was $\Delta =0$ (a), $\Delta =10$ (b), $\Delta =20$ (c).