A quantum oscillator interacting with a classical oscillator

arXiv:2403.07479 · quant-ph · Submitted 2024-03-12 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "A quantum oscillator interacting with a classical oscillator".

Mira: The gist The authors study a quantum oscillator interacting and back-reacting on a classical oscillator, which can be done consistently provided the quantum system decoheres,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: The title of the paper is "A quantum oscillator interacting with a classical oscillator," and honestly, it sounds like it sets up a very specific problem to solve.

Mira: It’s not just about coupling two things together; it’s about finding a consistent mathematical way to describe that interaction when one part is quantum and the other is classical.

Lev: And they immediately point out that the backreaction has this stochastic component, which means the classical oscillator isn't just smoothly following a path; it’s actually diffusing.

Kai: So, what’s actually interesting here for us in terms of how we model things? It suggests that we can consistently describe quantum systems interacting with classical ones through this path integral approach.

Mira: Right, and the authors are investigating different regimes where they can actually compute the propagator for this full classical-quantum path integral.

Lev: They set up three specific regimes to look at: one where D2 is much greater than one and the coupling between systems is weak, another where the classical oscillator is much heavier than the quantum oscillator, and a third where the quantum oscillator ends up being the heaviest of the two.

The paper's summary: Kai: So, to summarize what they actually did in "A quantum oscillator interacting with a classical oscillator," they used this path integral formulation to define a quantity called (z f, phi+ f, phi-f, t f) which is essentially the partition function.

Mira: The paper breaks down this quantity into three parts: the first two terms handle the unitary evolution and decoherence of those quantum systems.

Lev: And then there's that last term that specifically encodes the diffusion of those classical degrees of freedom, which they regulate using a diffusion coefficient D2.

Kai: That means when we look at the full path integral, it’s not just about unitary evolution; it’s also about how noise causes the classical part to spread out over time.

Mira: The key finding they emphasize is that this framework allows us to identify what would be the free correlators of this theory and compute the full partition function perturbatively.

Lev: And they use semi-classical approximations, depending on whether is larger or smaller than D2, to find the classical equation of motion for the forced oscillator Q plus or minus cl.

The paper's improvements: Kai: One major improvement they suggest is that by looking at the most probable paths, or MPPs, which are solutions to Equation (sixteen), we get a clear picture of what’s likely happening.

Mira: They note that at zeroth order in alpha, the real action from Equation (twelve) just reduces to the Onsager-Machlup action, which is a very familiar term in classical path integrals.

Lev: The paper also shows how this works differently depending on the mass comparison between the oscillators. If you have a heavy classical oscillator, its equation of motion ends up being independent of the quantum oscillator.

Kai: That’s interesting because it means we can use that zeroth-order result to understand classical dynamics if the classical system is much heavier than the quantum one and there's a small coupling constant alpha.

Mira: Conversely, when you have the quantum oscillator as the heaviest, its equation of motion becomes independent of the classical oscillator and just follows its usual solution for a simple harmonic oscillator.

Conclusion: Kai: So, wrapping up this discussion on "A quantum oscillator interacting with a classical oscillator," they show that in the weak coupling regime, the classical system acts like it's being backreacted on by another classical system whose position is just the average of the bra and ket positions of those quantum harmonic oscillators.

Mira: That's a pretty neat insight into how you can simplify things while still capturing some essential features of hybrid dynamics, especially how that average momentum changes due to diffusion.

Lev: From an engineering standpoint, this suggests that if you're dealing with a system where one part is much heavier than the other, you can effectively use the simpler equations derived from the zeroth-order path integral for your classical simulation.

Kai: Yeah, so this paper sets a foundation for understanding how quantum noise translates into classical diffusion in these coupled systems.

Mira: It opens up avenues to model things like black hole evaporation or quantum control with feedback setting by providing a framework where the quantum part remains pure conditioned on the classical trajectory.

Lev: I think the main limitation they flag is that they are using perturbation theory, so you have to be careful about how large alpha gets when you start treating quadratic terms as perturbations using rules like Feynman Vernon or Veltman.

Kai: That makes sense, because while the paper gives us a roadmap for solving this, it leaves us with the task of implementing those corrections accurately on real hardware.

Mira: We’ll keep an eye on how they use this framework to tackle those specific regimes we talked about earlier.

Department of Physics, Middle East Technical University · Department of Physics and Astronomy, University College London · Institute of Spintronics and Quantum Information, Faculty of Physics and Astronomy, Adam Mickiewicz University

quant-ph

Submitted: 2024-03-12

Updated: 2026-10-19

Comments: Figures added, two-point functions computed exactly, new section on purity added

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 71/100

The gist: The gist The authors study a quantum oscillator interacting and back-reacting on a classical oscillator, which can be done consistently provided the quantum system decoheres, while the backreaction

Key concepts

Path Integral Formulation: This mathematical tool allows researchers to calculate probabilities for a system by summing up contributions from every possible path a particle or oscillator could take. It combines unitary evolution (quantum behavior) and diffusion (classical noise) into one framework.
Proto-action: The proto-action is the core function used in the path integral calculation. It represents the total action of the system, combining terms for both unitary quantum evolution and classical diffusion, which helps define how the system moves through its possible states.
Most Probable Paths (MPPs): These are specific paths within the path integral that have a higher probability of occurring. They represent the most likely trajectories for the coupled quantum-classical system, providing insight into the most probable outcomes of their interaction.

Summary

The gist The authors study a quantum oscillator interacting and back-reacting on a classical oscillator, which can be done consistently provided the quantum system decoheres, while the backreaction has a stochastic component which causes the classical system to undergo diffusion<ref:2403.07479#pg2>.

Path Integral Formulation

The study utilizes the classical-quantum path integral formulation to solve for this system<ref:2403.07479#pg4>. The quantity of interest is defined as ϱ(zf, ϕ+f, ϕ−f, tf) = Z DzDϕ+Dϕ− N e ICQ(z,ϕ+,ϕ−,ti,tf)<ref:2403.07479#pg4>. The proto-action is given by WCQ(q, Q) = Lc(q, q˙) − Vi(q, Q) <ref:2403.07479#pg4>. The first two terms of the path integral represent the unitary evolution and the decoherence of the quantum systems<ref:2403.07479#pg4>. The last term encodes the diffusion of the classical degrees of freedom regulated by the diffusion coefficient D2 <ref:2403.07479#pg4>.

Regimes of Analysis

The authors investigate three appropriate regimes to compute the propagator for the full CQ path integral<ref:2403.07479#pg4>. These regimes are:

The regime in which D2 ≫ 1 and the coupling between the systems is weak<ref:2403.07479#pg4>.
The regime in which the classical oscillator is much heavier than the quantum oscillator<ref:2403.07479#pg4>.
The regime in which the quantum oscillator is the heaviest of the two<ref:2403.07479#pg4>.

Semi-Classical Solution

The semi-classical approximation depends on the relative magnitudes of ħ and D2<ref:2403.07479#pg4>. Extremizing the imaginary action with respect to Q± yields the classical equation of motion for a forced oscillator Q±cl driven by q(t)<ref:2403.07479#pg4>. The solution for the quantum degrees of freedom is obtained by leveraging phase cancellations in the quantum path integral<ref:2403.07479#pg4>. The most probable paths (MPPs) are found as solutions to Equation (16)<ref:2403.07479#pg4>. At zeroth order in α, the real action of Equation (12) is just the Onsager-Machlup action<ref:2403.07479#pg4>.

Green's Functions and Correlation Functions

The free Green’s functions for the harmonic oscillators are immediately given by (−∂2t − ω2Q)GQ(t, t′) = δ(t − t′)<ref:2403.07479#pg4>. The expectation values for the inertial, stochastic harmonic oscillator in the non-interacting case can be obtained by splitting q = qcl + δq<ref:2403.07479#pg4>. The conditional correlation function is given by ⟨q(t1)q(t2)⟩ = (1/2 G(t1, t2) + qcl(t1)qcl(t))e−Scl (π detOˆ)<ref:2403.07479#pg4>.

Interacting Case and Influence Functional

The full partition function Z[J] is obtained by expanding the exponential as a series in α<ref:2403.07479#pg4>. The connected correlation functions are given by complex expressions involving interaction vertices<ref:2403.07479#pg4>. The influence functional is obtained by integrating out the classical degrees of freedom, resulting in an expression involving the determinant of a 2 × 2 matrix F<ref:2403.07479#pg4>.

Discussion

In the weak coupling regime, the classical oscillator acts as if backreacted on by another classical oscillator whose position Q¯ is the average of the bra and ket positions of the quantum harmonic oscillators<ref:2403.07479#pg4>. The main difference is that it has an increasing amplitude as its average momentum increases due to diffusion<ref:2403.07479#pg4>. Meanwhile, the quantum oscillator undergoes decoherence, but also recoherence as given by Equation (25)<ref:2403.07479#pg4>. The authors hope that this simple model can be used to gain a fuller understanding of classical-quantum dynamics<ref:2403.07479#pg4>.

How it works

The system is solved using the classical-quantum path integral formulation<ref:2403.07479#pg4>.
The path integral action ICQ(q, Q+, Q−, ti, tf) encodes unitary evolution and decoherence of the quantum systems, as well as diffusion of the classical degrees of freedom<ref:2403.07479#pg4>.
The semi-classical approximation depends on the relative magnitudes of ħ and D2<ref:2403.07479#pg4>.
The most probable paths (MPPs) are found as solutions to Equation (16)<ref:2403.07479#pg4>.
The full propagator is given by K = NωQω2/C4π2D2 sin ωQT sin2 ωC T e(1/4 ω2c(t2f −t2) (30)<ref:2403.07479#pg4>.
For the interacting case, the propagator is derived by treating quadratic terms with α2 as perturbations using Feynman Vernon or Veltman rules<ref:2403.07479#pg4>.

How it works

The interaction vertices are associated with a factor of −α±i + 1/D2(∂2t + ω2c) <ref:2403.07479#pg4>. The ket correlator ⟨Q−(s)Q−(t)⟩ has a similar expression to the bra correlator ⟨Q+(s)Q+(t)⟩ once the analogous calculation is made<ref:2403.07479#pg4>.

The paper demonstrates that when the classical oscillator is heavy, its equation of motion is independent of the quantum oscillator and given by the 0-th order most probable path that we had obtained for D2 ≫ 1 together with a small α coupling constant<ref:2403.07479#pg4>. Conversely, when the quantum oscillator is heavy, its equation of motion will be independent of the classical oscillator and given by the usual solution for a SHO<ref:2403.07479#pg4>.

The paper concludes that it has begun the study of this system<ref:2403.07479#pg4>.

Improvements for AI systems

textbfImproved Classical-Quantum Hybrid Dynamics Modeling: The system can model quantum systems interacting with classical ones via a consistent path integral formulation, allowing for a description where the state of the quantum oscillator can remain pure conditioned on the trajectory of the classical oscillator. This enables simulating phenomena like black holes evaporation, where the radiation emitted is quantum in nature, yet the spacetime surrounding black holes can only be described classically.
textbfStochastic Diffusion Modeling: The system can account for backreaction effects that cause a classical system to undergo diffusion, which is regulated by a diffusion coefficient D2. This allows AI systems to model classical dynamics influenced by quantum noise, such as in quantum control with feedback setting.
textbfRegime-Specific Computational Efficiency: The AI can switch between different computational regimes based on the relative masses of the oscillators (the regime in which the classical oscillator is much heavier than the quantum oscillator or vice versa). This allows for efficient computation by using approximations like treating heavy systems as effectively classical or neglecting coupling terms when one system is heavy.
textbfMost Probable Path (MPP) Prediction: The AI can identify Most Probable Paths (MPPs), which are described as stochastic deviations from the classical deterministic equations of motion which are favoured in the path integral. This capability allows for predicting not just a single trajectory, but a distribution of likely outcomes for coupled systems.
textbfZero-Order and First-Order Interaction Corrections: The AI can compute the corrections to those propagators coming from the interaction, including computing the factors associated to the interaction vertices. In weak coupling, this reveals that the classical oscillator acts as if backreacted on by another classical oscillator whose position Q¯ is the average of the bra and ket positions of the quantum harmonic oscillators.