Phase Shifter and Mach-Zehnder Interferometer Example

This example first demonstrates the use of the PhaseShifter model on a single path.

It then shows how the same phase shifter becomes physically relevant inside a simple Mach-Zehnder interferometer (MZI) constructed from two beam splitters and one phase shifter.

We also make use of NumberStateSource to generate the input state.

from __future__ import annotations

import numpy as np

from symop.devices.models import BeamSplitter, NumberStateSource, PhaseShifter
from symop.modes.envelopes import GaussianEnvelope
from symop.modes.labels import Path, Polarization
from symop.polynomial.state.ket import KetPolyState

# Visualization Package
import symop.viz as VI

Setup

Create one single-photon source, one phase shifter, and two 50/50 beam splitters for the Mach-Zehnder interferometer.

src = NumberStateSource(
    envelope=GaussianEnvelope(omega0=100.0, sigma=50.0, tau=0.0),
    polarization=Polarization.H(),
    n=1,
)

ps = PhaseShifter(phi=np.pi / 2)

bs1 = BeamSplitter(theta=np.pi / 4, phi_r=0.0)
bs2 = BeamSplitter(theta=np.pi / 4, phi_r=0.0)

Generate the input state

Start from vacuum and populate one path with a single photon.

vac = KetPolyState.vacuum()

state_in = src(vac, ports={"out": Path("src_out")}).with_label("input")

2) Standalone phase shifter

Apply the phase shifter directly to the occupied path.

For an isolated single path, this changes only the phase of the state. By itself, this does not change number statistics, but the phase becomes important once the state interferes with another path.

state_shifted = ps(
    state_in,
    ports={"path": Path("src_out")},
).with_label("phase_shifted")

VI.display_many(state_in, state_shifted)

Why does the standalone output look almost the same?

The phase shifter applies the unitary

U(phi) = exp(i phi n)

to the selected mode. At the operator level, this induces

a^dagger -> exp(i phi) a^dagger

so the excitation on that path acquires a phase factor.

For a single isolated path, this does not change the direct photon-number probabilities. The effect becomes observable when the mode later interferes with another path.

3) First beam splitter: create the two MZI arms

Send the single input path through a 50/50 beam splitter. The unused second input is treated as vacuum.

state_after_bs1 = bs1(
    state_in,
    ports={
        "in0": Path("src_out"),
        "in1": Path("unused_in"),
        "out0": Path("arm0"),
        "out1": Path("arm1"),
    },
).with_label("after_bs1")

VI.display_many(state_in, state_after_bs1)

4) Apply the phase shifter to one arm

This creates a relative phase between the two interferometer arms.

state_after_ps = ps(
    state_after_bs1,
    ports={"path": Path("arm0")},
).with_label("after_phase_shifter")

VI.display_many(state_after_bs1, state_after_ps)

5) Second beam splitter: recombine the two arms

The relative phase now affects the interference at the output.

state_out = bs2(
    state_after_ps,
    ports={
        "in0": Path("arm0"),
        "in1": Path("arm1"),
        "out0": Path("out0"),
        "out1": Path("out1"),
    },
).with_label("mzi_output")

VI.display_many(state_after_ps, state_out)

Why does the phase matter in the MZI?

After the first beam splitter, the photon amplitude is distributed over two paths. The phase shifter changes the relative phase of one arm. When both arms are recombined at the second beam splitter, this relative phase changes the interference pattern and therefore redistributes the output amplitudes between the two output ports.

In this sense, the standalone phase shifter prepares a phase that becomes physically visible through interference.

6) Density-state version

The same sequence can also be applied to density states.

state_in_dense = state_in.to_density().with_label("input_density")

state_after_bs1_dense = bs1(
    state_in_dense,
    ports={
        "in0": Path("src_out"),
        "in1": Path("unused_in"),
        "out0": Path("arm0"),
        "out1": Path("arm1"),
    },
).with_label("after_bs1_density")

state_after_ps_dense = ps(
    state_after_bs1_dense,
    ports={"path": Path("arm0")},
).with_label("after_phase_shifter_density")

state_out_dense = bs2(
    state_after_ps_dense,
    ports={
        "in0": Path("arm0"),
        "in1": Path("arm1"),
        "out0": Path("out0"),
        "out1": Path("out1"),
    },
).with_label("mzi_output_density")

VI.display_many(
    state_in_dense,
    state_after_bs1_dense,
    state_after_ps_dense,
    state_out_dense,
)