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Working principle of variable beam splitter and phase shifter
This protocol is extracted from research article:
On-demand photonic entanglement synthesizer
Sci Adv, May 17, 2019;

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The EOMs for the variable beam splitter and phase shifter contain a crystal of rubidium titanyl phosphate that is sandwiched between two electrodes. We used commercially available EOM driving circuits that are composed of two fast high-voltage switches (one for each electrode). These switches enabled us to selectively apply 0 or V1 volt to one of these electrodes and 0 or −V2 volt to the other electrode, where V1 > 0 and V2 > 0 can be arbitrarily chosen in advance. The net voltage applied to the crystal can thus be switched among 0, V1, V2, and V1 + V2, and these voltages determine the possible values of T(t) and θ(t). The rise/fall time for the switching is ~10 ns. In this system, it is not possible to switch T(t) and θ(t) among more than three different target values in general because of the design of the EOM driving circuits. As a result, our setup was unable to generate GHZ or cluster states of more than three modes, which require switching of T(t) among four or more different values. This limitation can be overcome by developing another sophisticated EOM driving circuits containing more high-voltage switches to increase the number of selectable voltages. Another solution is to cascade multiple EOMs at the expense of introducing additional transmission loss. Because one EOM can shift T(t) from the initial value (applied voltage, 0 volt) to two different target values (V1 or V2 volt), n cascaded EOMs make it possible to switch T(t) among 2n + 1 different target values. In this way, GHZ or cluster states of arbitrary number of modes can be generated in principle.

In the following, we introduce theoretical description of the action of the variable beam splitter and phase shifter. In Fig. 2C, the k-th beam splitter with transmissivity Tk (k ≥ 2) mixes one mode from a squeezer (annihilation operator $âk′=x̂k′+ip̂k′$) and the other mode coming from the (k − 1)-th beam splitter ($âk−1′′$). After this operation, one of the output modes is measured ($âk−1=x̂k−1+ip̂k−1$), while the other output mode become the input mode of the (k + 1)-th beam splitter after the phase shift of θk ($âk′′$). In Fig. 2D, the same operation is performed with the variable beam splitter and variable phase shifter. In the variable beam splitter, the QWP initially introduces a relative phase offset of 90° between two diagonal polarizations, thereby setting the default transmissivity to 0.5. The polarization-rotation EOM introduces an additional relative phase shift of 2δk ≥ 0, which is proportional to the applied voltage. Under this condition, the function of the k-th beam splitter and phase shifter in Fig. 2C is realized in Fig. 2D as$(âk−1âk′′)=(100eiθk)(sin(δk+45°)−cos(δk+45°)cos(δk+45°)sin(δk+45°))(âk−1′′âk′)$(3)The transmissivity of the variable beam splitter is defined by Tk = sin2k + 45°) in Eq. 3. By gradually increasing the applied voltage and thereby increasing δk from 0° to 45°, Tk can be increased from 0.5 to 1. Thus, any transmissivity between 0.5 and 1 can be chosen in this way. When the transmissivity between 0 and 0.5 is required, the voltage has to be further increased to set δk between 90° and 135°. In this region, however, the sign of the off-diagonal terms in Eq. 3 flips. This sign flip corresponds to the additional phase shift of 180 before and after the beam splitter operation$(Tk1−Tk−1−TkTk)=(100−1)(Tk−1−Tk1−TkTk)(100−1)$(4)

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