**A quantum speed limit (QSL) in quantum mechanics denotes the shortest amount of time required for a quantum system to transition between two distinguishable states. QSL and time-energy uncertainty relations are tightly connected.**

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The speed of evolution of a quantum system is constrained by quantum mechanics, a fundamental phenomenon of nature. The research of precise boundaries in quantum metrology, the computational limitations of physical systems, and the best quantum control algorithms all make use of this bound. The QSL is the least amount of time a system requires to transition from its initial state to its orthogonal state.

One of the fundamental concepts in the physical universe is time, which is important in practically every physical theory formulated. Nonetheless, comprehending time has proven to be difficult, and it is frequently used as a metric. The rate at which a quantum system may change in time is naturally constrained by the uncertainty correlation between time and energy fluctuation.

Both closed and open quantum systems have been explored in relation to QSL. Closed quantum systems are those in which the system's energy is preserved across time. This is consistent with the well-known quantum mechanical theories in which a system's development is unitary.

A quantum-mechanical system that interacts with the environment, sometimes referred to as an external quantum system, is referred to as an open quantum system in physics. In real-life applications of quantum mechanics, open systems are at play. This makes the study of these systems very crucial.

## Quantum Speed Limit in a Cavity

In a recent theoretical investigation, the quantum speed limit of a single atom contained in a Fabry-Perot microresonator was studied.

The interaction between a single atom and a single photon in open space is weak. Yet, atom-light interaction may be greatly improved by tightly confining the photons inside a microresonator with a high-quality factor Q and connecting the atoms to the resonator mode. The core of cavity quantum electrodynamics (CQED) is based on this line of exploration.

Ultrahigh finesse Fabry-Perot microresonators have contributed significantly to the development of this field over the past few decades. Single atoms have been trapped inside the resonator mode of these conventional resonators, leading to a number of groundbreaking discoveries.

The Fabry-Perot interferometer makes use of the multiple beam interference phenomena that develop when light enters a chamber enclosed by two parallel highly reflecting mirrors. A portion of the light is transmitted out each time it comes into contact with one of the surfaces, while the remaining component is reflected back.

Experimentally, single atoms have been trapped in the mode of a high-finesse optical cavity in a domain of strong coupling by combining the methods of laser cooling and trapping with those of CQED. This first understanding of trapped atoms in CQED has opened up a variety of quantum physics explorations, including the dynamics of individual quantum objects and theoretical models to advance the field.

## Squeezed Light

Light with noise below the conventional quantum limit is said to have the quantum mechanical property of being squeezed. One can alter a light wave's characteristics to lessen the quantum noise. A light wave can have its phase or amplitude "squeezed." An angle is often used to characterize a wave's phase or any particular point in its cycle. A wave has a zero-degree beginning and rotates 360 degrees around its center to experience its peak and trough. It is possible to optimize the mode by experimenting with squeezed light at various phase angles.

In a cavity, one laser can be set to drive the nonlinear medium, while a powerful laser drives the optical cavity mode. When a driving laser is used on a second-order nonlinear medium, the cavity mode will be squeezed.

The Bogoliubov squeezing transformation can be used to derive the effective Hamiltonian.

The stationary solutions of the associated Schrödinger equation are obtained by diagonalizing Hamiltonians, which is a common application of the Bogoliubov transformation. By changing the two optical driving fields, the effective quadratic optomechanical coupling strength may be increased above the decay rate of the cavity mode. A squeezed vacuum field can be used to completely reduce the thermal noise of the squeezed cavity mode.

## Theoretical Reasoning

In quantum mechanics, frequently, a time-evolved state becomes fully distinguishable or orthogonal to the starting state during the arbitrary dynamical evolution of the quantum system. The anticipated value of a particular observable, however, remains the same or fluctuates more slowly.

An external Hamiltonian drives the evolution of the initial state to its orthogonal state in a closed quantum system with an internal Hamiltonian and an initial state. The system's initial state and end state may be distinguished in this case. The average energy of the quantum system has zero evolution time since the initial and end states are energetically equivalent.

The non-Hermitian Schrödinger equation for the initial excited state may be used to generate the analytical expression of the evolved atom state, and the quantum speed limit time corresponds extremely well for both the analytical expression and the master equation method.

The evolution of the probabilities for Markov processes for systems that switch between states continuously is described by master equations, which are differential equations.

A Markov process is a random process in which the future is unrelated to the past. Hence, Markov processes are the stochastic counterparts of the differential equations-based descriptions of deterministic processes.

## Future Outlook

According to the QSL models, it is advantageous to speed up the evolution of the quantum state when there is a significant amount of detuning, strong driving, and coupling. The initial state's properties have a greater impact on the speed of evolution in the first superposition case. The results show that the QSL is affected by the starting state as well as the system characteristics.

The study of the dynamics and characteristics of open quantum systems is of special interest because of the significance of these systems in real-world applications. The QSL time is one of the characteristics that have implications for developing fruitful applications in a range of contemporary research areas, including quantum computing, quantum thermodynamics, and quantum information theory, among many others.

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**References and Further Reading**

Ya-Jie Ma, Xue-Chen Gao, Shao-Xiong Wu and Chang-shui Yu. The quantum speed limit of a single atom in a squeezed optical cavity mode. Accepted Manuscript online 20 February 2023. Chinese Physical Society and IOP Publishing Ltd. DOI 10.1088/1674-1056/acbd2b

Elisa Will, Luke Masters, Arno Rauschenbeutel, Michael Scheucher, and Jürgen Volz. Coupling a Single Trapped Atom to a Whispering-Gallery-Mode Microresonator. Phys. Rev. Lett. **126**, 233602 – Published 11 June 2021. DOI:https://doi.org/10.1103/PhysRevLett.126.233602

C Campbell et al Quantum control and quantum speed limits in supersymmetric potentials. 2022 New J. Phys. 24 095001 DOI 10.1088/1367-2630/ac89a4

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