How does one time-evolve a quantum state with its kernel function?

Sep 9, 2023

Sciencemaster

TL;DR Summary: I'd like to model the evolution of a squeezed state by representing it as a kernel function and applying a unitary transformation, but I'm having trouble doing this.

I'd like to model the evolution of a squeezed state and its properties (such as phase at different spatiotemporal coordinates). I know one can represent them using kernel functions (and I have found a paper that gives a kernel function for a squeezed state: https://arxiv.org/pdf/2105.05990.pdf). I've been told one can diagonalize the kernel function in terms of some eigenbasis and then represent the state in terms of a matrix representation with a truncated set of these eigenbasis functions, or alternatively just represent the kernel function in terms of a higher dimensional grid. Apparently you also need to represent the squeezing parameter in terms of a kernel function ($ξa^2→a\hat(k1)ξ(k1,k2)a\hat(k2)$) Once this is done, one can use unitary transformations on the matrix to actually simulate the system.
However, I'm having trouble doing this. Specifically, I'm stuck on trying to make a kernel function matrix from the paper and/or setting up an initial system (for example, let's say we have a Gaussian beam of squeezed light being emitted, and this simulation aims to time evolve this beam). Does anyone have any insights on how I can do this? Any help would be appreciated.

What is the kernel function in the context of quantum mechanics?

The kernel function in quantum mechanics, often referred to as the propagator, is a fundamental concept that describes how a quantum state evolves over time. It essentially provides a way to calculate the probability amplitude for a particle to transition from one state to another over a given time period.

How is the kernel function used to time-evolve a quantum state?

The kernel function, or propagator, is used in the integral form of the Schrödinger equation to evolve a quantum state. Given an initial state $\psi(x,0)$, the state at a later time $t$ can be found using the kernel $K(x, t; x', 0)$ by the integral: $\psi(x,t) = \int K(x, t; x', 0) \psi(x',0) dx'$.

What is the relationship between the kernel function and the Hamiltonian of a system?

The kernel function is closely related to the Hamiltonian $H$ of the quantum system. For a time-independent Hamiltonian, the kernel function can be expressed as $K(x, t; x', 0) = \langle x | e^{-iHt/\hbar} | x' \rangle$, where $e^{-iHt/\hbar}$ is the time-evolution operator. This shows that the kernel function encapsulates the dynamics dictated by the Hamiltonian.

Can the kernel function be determined for any quantum system?

In principle, the kernel function can be determined for any quantum system, but the complexity of finding it depends on the system. For simple systems, such as a free particle or a harmonic oscillator, the kernel function can be derived analytically. For more complex systems, numerical methods or approximations may be required to determine the kernel function.

What are some common methods to compute the kernel function?

Common methods to compute the kernel function include path integral formulations, where the propagator is expressed as a sum over all possible paths a particle can take, and operator methods, which involve using the spectral decomposition of the Hamiltonian. For specific potentials, exact solutions or perturbative approaches can also be employed to find the kernel function.

How does one time-evolve a quantum state with its kernel function?

FAQ: How does one time-evolve a quantum state with its kernel function?

What is the kernel function in the context of quantum mechanics?

How is the kernel function used to time-evolve a quantum state?

What is the relationship between the kernel function and the Hamiltonian of a system?

Can the kernel function be determined for any quantum system?

What are some common methods to compute the kernel function?

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