Experimental Demonstration of Commutation Relations Using Intensity Correlations

Technical Summary: Experimental Demonstration of Commutation Relations Using Intensity Correlations

Problem Statement
The canonical commutation relation is a foundational element of quantum theory, serving as the basis for the Heisenberg uncertainty principle. While uncertainty relations have been rigorously tested across various physical systems, direct experimental verification of the underlying commutation relation itself has remained elusive. This difficulty stems from the fact that non-commuting observables lack a joint probability distribution, and their product is generally non-Hermitian, precluding straightforward measurement. Previous experimental demonstrations have been largely restricted to finite-dimensional qubit systems or have relied on indirect methods in the continuous-variable domain, such as sequential photon addition and subtraction.

Methodology
The authors propose and demonstrate a method to extract the expectation value of the field-operator commutator by quantitatively comparing two distinct intensity correlation functions: cross-correlation and auto-correlation.

1. Theoretical Framework:

   • Cross-Correlation ($g^{(2)}_{cross}$): Utilizing the standard Hanbury Brown and Twiss (HBT) setup, where an optical field is split by a 50:50 beam splitter and detected by two independent detectors. This measurement yields a normally ordered correlation function: $g^{(2)}_{cross}(\tau) \propto \langle \hat{a}^\dagger(t) \hat{a}^\dagger(t+\tau) \hat{a}(t) \hat{a}(t+\tau) \rangle$.
   • Auto-Correlation ($g^{(2)}_{auto}$): Obtained by recording a time-resolved detection record from a single detector (or merging two streams) and correlating the signal with itself. This yields: $g^{(2)}_{auto}(\tau) \propto \langle \hat{a}^\dagger(t) \hat{a}(t) \hat{a}^\dagger(t+\tau) \hat{a}(t+\tau) \rangle$.
   • The Commutator Connection: Theoretical analysis shows that for a pure quantum state, the difference between these functions at zero time delay ($\tau=0$) is directly proportional to the expectation value of the bosonic commutator $[\hat{a}, \hat{a}^\dagger]$. Specifically, $g^{(2)}_{auto}(0) - g^{(2)}_{cross}(0) = \langle [\hat{a}, \hat{a}^\dagger] \rangle / \langle \hat{a}^\dagger \hat{a} \rangle$. The zero-delay peak in auto-correlation, often viewed as an artifact, is identified as the direct experimental manifestation of the non-commutativity of field operators.

2. Experimental Implementation:
   The authors validated this framework using two distinct light sources:

   • Single-Photon Source: A single $^{174}\text{Yb}^+$ ion trapped at the focus of a deep parabolic mirror. The ion was driven by a 369.5 nm laser, and emitted fluorescence was collected, split by a beam splitter, and detected by two photomultiplier tubes (PMTs). Detection efficiency was rigorously characterized via a state-sensitive method ($\eta \approx 4.61\%$).
   • Coherent Source: A frequency- and power-stabilized 369.5 nm laser (linewidth $\approx 10$ Hz) scattered off the ion-trap assembly into the detection path.
   • Data Processing: Time-tagged detection events were recorded. For $g^{(2)}_{cross}$, the streams from the two detectors were correlated. For $g^{(2)}_{auto}$, the streams were merged and correlated with themselves. Corrections were applied for optical losses, detector dead times, and finite measurement time windows.

Key Results

• Single-Photon State: The experiment yielded $g^{(2)}_{auto}(0) = 714.93 \pm 0.19$ and $g^{(2)}_{cross}(0) = 0.016 \pm 0.002$. Using the measured mean photon number and detection efficiency, the extracted commutator expectation value was $\langle [\hat{a}, \hat{a}^\dagger] \rangle = 1.01 \pm 0.04$, consistent with the theoretical value of unity.
• Coherent State: For the laser source, $g^{(2)}_{auto}(0) = 841.68 \pm 0.24$ (consistent with $1 + 1/\langle N \rangle$) and $g^{(2)}_{cross}(0) = 0.97 \pm 0.02$. The calculated commutator expectation value was $\langle [\hat{a}, \hat{a}^\dagger] \rangle = 1.00 \pm 0.02$, again in quantitative agreement with quantum theory.
• Temporal Behavior: For $\tau \neq 0$, the auto- and cross-correlation functions were found to be identical, confirming that field operators commute at different times, i.e., $[\hat{a}(t), \hat{a}^\dagger(t')] = \delta(t-t')$.

Significance and Claims
The paper claims to provide the first direct experimental demonstration of the bosonic commutation relation for optical field operators based on intensity correlation measurements. The authors emphasize that the distinction between auto- and cross-correlation is not merely technical but reflects a fundamental consequence of operator non-commutativity.

Key contributions highlighted by the authors include:

1. Direct Access: Establishing a framework to determine commutation relations through intensity correlations without requiring sequential operations or finite-dimensional qubit systems.
2. Universality: The method applies to arbitrary optical fields, operating independently of whether the statistics are classical (coherent) or quantum (single-photon).
3. Reinterpretation of Artifacts: Reframing the zero-delay auto-correlation peak from a technical artifact to a direct experimental manifestation of the bosonic commutation relation.
4. Experimental Robustness: Demonstrating that this fundamental quantum property can be verified using standard intensity-correlation techniques (HBT and auto-correlation) accessible in typical optical laboratories.

The authors conclude that the difference between these correlation functions at zero delay contains fundamental information often overlooked, providing a robust and accessible method to verify the non-commutativity of quantum field operators.