# Understanding Quantum Interference with Coherent and Fock States using Strawberry Fields

I tried to understand the interference patterns observed when coherent or Fock states of light are passed through a beamsplitter.

My expertise lies in astrophysics, and quantum optics is not my specialty. I am unable to explain this difference obtained from the following simulartion. I would appreciate it if someone knowledgeable in this area could provide an explanation. Thank you in advance.

The simulations are designed as follows:

• A pair of modes (representing two paths in a double-slit experiment) are prepared in either coherent or Fock states.
• [Coherent state] the amplitude is 1 (= Poisson distribution with a mean value of 1)
• [Fock state] a single photon state + a vacuum state
• A phase shift is applied to one of the modes to simulate the path difference effect in an interference setup.
• A beamsplitter combines these modes, resulting in interference.
• Photon number measurements are performed to observe the resulting interference pattern.

All the codes and outputs are available from my google colab page.

The outcomes of these simulations are the probability of detecting photons in each mode against the applied phase shift. In the case of coherent states, we expect to see a continuous, wave-like interference pattern. In contrast, Fock state interference should exhibit distinct quantum interference characteristics due to the particle nature of single photons.

The results and the codes are as follows.

import strawberryfields as sf
from strawberryfields import ops
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import poisson

def run_state_simulation(num_phases, state_preparation, filename):
# Initialization
phases = np.linspace(-2 * np.pi, 2 * np.pi, num_phases)
probabilities_00 = np.zeros(num_phases)
probabilities_01 = np.zeros(num_phases)
probabilities_10 = np.zeros(num_phases)
eng = sf.Engine(backend="fock", backend_options={"cutoff_dim": 10})

# Simulate for each phase
for i, phase in enumerate(phases):
prog = sf.Program(2)
with prog.context as q:
state_preparation(q)  # State preparation
# Apply phase shift and beamsplitter
ops.Rgate(phase) | q[1]
ops.BSgate() | (q[0], q[1])

# Run the program and get probabilities of different photon numbers
state = eng.run(prog).state
probabilities_00[i] = state.fock_prob([0, 0])
probabilities_01[i] = state.fock_prob([0, 1])
probabilities_10[i] = state.fock_prob([1, 0])

# Plotting interference patterns
plt.plot(phases, probabilities_00, ".", label="Photons = [0,0]")
plt.plot(phases, probabilities_01, ".", label="Photons = [0,1]")
plt.plot(phases, probabilities_10, ".", label="Photons = [1,0]")
plt.plot(phases, 1/np.e**2 * np.ones(len(phases)), "--", label=r"1/$e^2$")
plt.ylabel("Probability in Modes")
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.grid(alpha=0.4)
plt.savefig(filename)
plt.show()

# Case 1: Coherent State (using state)
def prepare_coherent_state(q):
amplitude = 1.0
ops.Coherent(amplitude) | q[0]
ops.Coherent(amplitude) | q[1]

def apply_phase_shift(q, phase):
ops.Rgate(phase) | q[0]

num_phases = 200  # Number of phase values to simulate
# Run the simulation and plot the results
run_state_simulation(num_phases, prepare_coherent_state, "stw_young_coherent_state.png")


The outputs from the coherent state are trigonometric functions.

# Case 2: Fock State (using state)
def prepare_fock_state(q):
ops.Fock(1) | q[0]
ops.Fock(0) | q[1]

num_phases = 200  # Number of phase values to simulate
# Run the simulation and plot the results
run_state_simulation(num_phases, prepare_fock_state, "stw_young_fock_state.png")


The outputs from the fock state are constant against phase.

The coherent state simulation aligns with our classical understanding of light as a wave, so-called the Young’s double-slit experiment. However, the Fock state simulation did not show any modulation against phase. Would it be a natural consequence from the quantum optics? I would appreciate it if someone could provide an explanation.

This question is on 2023.11.23, using sf.version() = 0.23.0

Hi @SHINYA_YAMADA, welcome to the Forum!

We’re taking a look at your question and will be back with an answer soon.

My colleague Filippo mentioned that Fock states are symmetric under rotations so the rotation gate doesn’t do anything.

I hope this helps you!

Let me know if you have any further questions.

Thank you for your quick feedback. I could understand how the rotations work in this case, but I wish to know more on how to use Strawberry Fields for the interferometer.

I am interested in simulating a demonstration of an interferometer that utilizes vacuum and single-photon states, as proposed in the paper ‘Longer-Baseline Telescopes Using Quantum Repeaters’ by Daniel Gottesman, Thomas Jennewein, and Sarah Croke, published in Phys. Rev. Lett. 109, 070503 on August 16, 2012. https://arxiv.org/pdf/1107.2939.pdf

Depending on the orientation of the “baseline” (the relative position of the telescopes in the interferometer), the light has a relative phase shift \phi between the two telescopes L and R, resulting in the state:
|0\rangle_L|1\rangle_R+e^{i \phi}|1\rangle_L|0\rangle_R ~~~ (1)
with |0\rangle and |1\rangle indicating 0 and 1-photon states. If we
measure \phi with high precision, that tells us the source’s
location very precisely. \phi is proportional to the baseline, so longer baselines produce a more accurate measurement of the source’s position.

However, I am uncertain about the method for such a simulation using Strawberry Fields. This approach is gaining increasing attention in recent research as a novel method for astronomical observation.

I am unsure about the fundamental assumptions regarding photons from space. Is it appropriate to consider these photons as being in a coherent state with an average photon number of one? Or should they be considered as a superposition of vacuum and single-photon states? I am seeking clarity on these basic premises. I am wondering if using Strawberry Fields for simulation could be helpful in understanding the fundamental premises.

The default state in Strawberry Fields is the vacuum state (see here). So if you look for example at the squeezing gate here you will notice that you start with a vacuum state and then squeeze it.

You should also notice that not all states have a mean photon number of one. Thermal states for example can have different mean photon numbers.

In general I think our page on Conventions and Formulas can answer many of your questions.