Tomogram-based quantifiers of nonclassicality dynamics in Kerr and cubic media

This paper demonstrates that tomogram-based measures, specifically the homodyne nonclassical area and sum tomographic entropy, provide robust, experimentally accessible alternatives to conventional quantifiers for tracking the dynamics of nonclassicality in coherent and non-classical states evolving within Kerr and cubic nonlinear media under amplitude and phase damping.

The Gist

Imagine you are trying to catch a ghost. In the world of quantum physics, this "ghost" is nonclassicality—a special, spooky property that makes quantum particles behave in ways that classical objects (like baseballs or water waves) never do. Scientists want to measure how strong this "ghostly" behavior is, especially when the particles are interacting with their environment, which tends to make them act more normal (a process called decoherence).

The problem is that the usual tools for catching this ghost are like trying to build a giant, complex robot just to see if a lightbulb is on. They are hard to build, hard to use, and sometimes they miss the ghost entirely.

This paper introduces two new, simpler tools: Homodyne Nonclassical Area and Sum Tomographic Entropy. Think of these as a pair of high-tech, all-seeing glasses that let you see the ghost directly, without needing to build the giant robot first.

Here is a breakdown of what the researchers did and found, using everyday analogies:

1. The Playground: Kerr and Cubic Media

The scientists studied how light behaves in special materials called Kerr and Cubic media.

• The Analogy: Imagine a trampoline. If you jump on a normal trampoline (linear), you go up and down in a predictable rhythm. But if you jump on a "magic" trampoline (nonlinear) where the bounce gets stronger the harder you push, your movement becomes wild and complex.
• The Result: In these "magic" materials, light waves don't just bounce; they split, twist, and then magically reassemble themselves. This reassembly is called a Revival. Sometimes, they split into smaller copies that dance around before coming back together; this is a Fractional Revival.

2. The Tools: Measuring the Ghost

The researchers used two specific methods to track these wild light waves:

• Tool A: The Homodyne Nonclassical Area (The "Shape Shifter" Detector)

  • What it does: It measures how much the shape of the light wave has "stretched" or "squished" compared to a calm, normal wave (a coherent state).
  • The Analogy: Imagine a calm, round balloon (a normal wave). If you squeeze it into a weird, jagged shape, the "Nonclassical Area" measures how much extra surface area that weird shape has compared to the round balloon.
  • What they found: When the light wave splits and dances (fractional revivals), this "area" dips down. When the wave reassembles perfectly (full revival), the area snaps back to its original size. It's like a heartbeat monitor that tells you exactly when the wave is dancing and when it's resting.

• Tool B: Sum Tomographic Entropy (The "Confusion Meter")

  • What it does: It measures how "spread out" or "confused" the information about the wave is.
  • The Analogy: Imagine a deck of cards. If the cards are perfectly sorted (low entropy), you know exactly where everything is. If they are thrown in the air and scattered (high entropy), it's chaotic.
  • What they found: When the light wave splits into many tiny copies (fractional revivals), the "confusion" drops temporarily because the copies are organized in a specific, repeating pattern. This tool is great at spotting the tiny dances (higher-order revivals) that the first tool might miss.

3. The Enemy: Decoherence (The "Noise")

In the real world, nothing is perfect. The environment acts like static noise or a drafty room that messes up the experiment. The scientists tested two types of "noise":

• Amplitude Damping (The "Leaky Bucket"):

  • The Analogy: Imagine your magic trampoline is slowly losing air. The light is literally leaking out of the system.
  • The Result: The "ghost" (nonclassicality) disappears very fast. The wave loses its energy and eventually becomes just empty space (vacuum). The "Nonclassical Area" drops to zero quickly, like a deflating balloon.

• Phase Damping (The "Foggy Window"):

  • The Analogy: Imagine the trampoline is still full of air, but the room is getting foggy. You can still see the shape of the bounce, but the timing gets blurry. The energy stays, but the "synchronization" is lost.
  • The Result: The "ghost" is more stubborn here. Even though the wave gets blurry, the special dancing patterns (revivals) survive for a longer time. The "Nonclassical Area" doesn't drop to zero; it just settles at a lower, steady level.

4. The Main Takeaway

The paper claims that these two new tools (Nonclassical Area and Sum Entropy) are better than the old tools for a few reasons:

1. They are easier to use: You don't need to reconstruct the entire "blueprint" of the quantum state (which is hard and error-prone). You can measure them directly using standard light detectors.
2. They are sensitive: They can spot the tiny, complex dances (fractional revivals) that other methods miss.
3. They are robust: They can tell the difference between a wave that is losing energy (leaky bucket) and a wave that is just getting blurry (foggy window).

In summary: The researchers showed that by looking at the "shape" and the "confusion" of light waves using these new glasses, we can track how quantum magic behaves and fades away in real-world conditions, without needing to build complicated, error-prone machines. This makes it much easier for scientists to study and eventually use these quantum effects.

Technical Summary

Technical Summary: Tomogram-based Quantifiers of Nonclassicality Dynamics in Kerr and Cubic Media

Problem Statement
The reliable quantification of nonclassicality in quantum states under realistic decoherence conditions remains a significant challenge for the advancement of quantum technologies. Conventional quantifiers, such as Wigner negativity, Mandel's $Q$-parameter, and nonclassical depth, often suffer from experimental intractability, the requirement for full density matrix reconstruction, or insensitivity to specific quantum signatures. Furthermore, some nonclassical states (e.g., squeezed states) possess positive Wigner functions, rendering Wigner negativity an incomplete metric. The authors identify a need for robust, experimentally accessible measures that can track the dynamics of nonclassicality, particularly complex phenomena like wave packet revivals and fractional revivals, in nonlinear optical media without relying on heavy reconstruction procedures.

Methodology
The study employs a tomographic framework to quantify nonclassicality dynamics in two distinct nonlinear systems: Kerr media (quadratic dependence on photon number) and cubic media (cubic dependence). The evolution of the system is modeled using the Lindblad master equation to incorporate environmental decoherence via two channels: amplitude damping (energy dissipation) and phase damping (pure dephasing).

The authors analyze three classes of initial states:

1. Coherent states ($|\alpha\rangle$).
2. 3-photon-added coherent states ($|\alpha, 3\rangle$).
3. Even coherent states ($|\alpha\rangle_+$).

Two primary tomogram-based quantifiers are utilized, both derived directly from optical tomograms ($\omega(X_\theta, \theta, t)$) obtained via balanced homodyne detection:

1. Homodyne Nonclassical Area ($\sigma$): Defined as the excess area projected by the optical tomogram onto the $X_\theta - \theta$ plane relative to a reference coherent state. It is calculated using the standard deviation of the rotated quadrature operator ($\Delta X_\theta$). This metric captures deviations from Gaussian behavior and is sensitive to the second moment of the quadrature.
2. Sum Tomographic Entropy: The sum of entropies calculated from conjugate quadrature angles ($S(\theta) + S(\theta + \pi/2)$). This metric serves as a measure of the information content and localization/delocalization of the wave packet in phase space, capable of detecting higher-order structures.

Numerical simulations are performed in a truncated Fock basis, with convergence checks ensuring stability of observables and preservation of the density matrix trace.

Key Contributions and Results

• Kerr Medium Dynamics:

  • Unitary Evolution: In the absence of decoherence, the nonclassical area exhibits periodic revivals, returning to its initial value at the revival time $T_{rev} = \pi/\chi_1$. Local minima in the nonclassical area correspond to fractional revivals (e.g., two-subpacket revivals at $T_{rev}/2$) and macroscopic superpositions. The sum tomographic entropy reveals finer details, identifying higher-order fractional revivals (e.g., 18th-order) through relative minima that are not always distinct in the nonclassical area.
  • Amplitude Damping: This channel drives the system toward the vacuum state. The nonclassical area undergoes rapid, monotonic exponential decay, and exact revivals are suppressed. While weak coupling allows for observable dips (near revivals), the initial nonclassical area is not retained.
  • Phase Damping: This channel preserves population but destroys coherence. While exact revivals are absent, signatures of fractional revivals and phase-space rotations persist longer than in the amplitude damping case. The nonclassical area stabilizes at a non-zero value in the long-time limit, representing a phase-randomized mixture rather than a vacuum.

• Cubic Medium Dynamics:

  • Unitary Evolution: The cubic Hamiltonian induces three-component fractional revivals at $T_{rev}/3$ and $2T_{rev}/3$. The nonclassical area accurately tracks these events, returning to initial values at full and fractional revival times.
  • Decoherence Effects: Similar to the Kerr medium, amplitude damping causes an exponential decay of nonclassicality toward the vacuum. Phase damping allows for the survival of revival signatures (local minima) in the weak coupling regime, though the features become less sharp over time.

• Comparative Resilience:

  • Across both media and damping types, photon-added coherent states and even coherent states demonstrate greater resilience to damping compared to standard coherent states, maintaining higher nonclassical area values for longer durations.
  • Phase damping is consistently less destructive to nonclassical features than amplitude damping, which affects both coherence and population.

Significance and Claims
The paper claims that homodyne-based quantifiers (nonclassical area and sum tomographic entropy) offer a powerful, real-time, and experimentally viable alternative to conventional measures. By bypassing the need for full density matrix reconstruction or quasiprobability distribution calculation, these methods are directly accessible via balanced homodyne detection.

The authors assert that these tools effectively:

1. Track the onset and decay of nonclassicality in open quantum systems.
2. Identify fractional revivals, wave packet splitting, and macroscopic superpositions.
3. Distinguish between the effects of amplitude and phase damping on quantum dynamics.
4. Provide a robust framework for characterizing nonclassicality in nonlinear optical media, even under realistic decoherence conditions.

The study concludes that while the nonclassical area is not a strict witness of nonclassicality in mixed states (due to classical mixing contributions), it remains a sensitive indicator of dynamical nonclassical features. The sum tomographic entropy complements this by capturing higher-order fractional revivals, making the combined approach a comprehensive tool for experimentalists investigating quantum state engineering in nonlinear media.