Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical $^{40}\mathrm{Ca}$ nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical $^{40}\mathrm{Ca}$ nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.

arXiv:quant-ph/0509038Physical Review CApr 17, 20067 citations

Residual Quantum Effects beyond Mean‐Field Treatment in Quantum Optics Systems

Yue‐Xun Huang, Ming Li, Zi-Jie Chen +4 more

Abstract Mean‐field treatment (MFT) is frequently applied to approximately predict the dynamics of quantum optics systems. It simplifies the system Hamiltonian by neglecting the quantum statistics of certain modes that are driven strongly by lasers or couple weakly with other modes. However, the neglected quantum correlations between different modes result in unanticipated quantum effects and might lead to significantly distinct system dynamics. Here, a general and systematic theoretical framework based on perturbation theory in company with MFT is provided to capture these quantum effects. The form of nonlinear dissipation and parasitic Hamiltonian as well as their relationship to the nonlinear coupling rate are predicted. Furthermore, the indicator is also proposed as a measure of the accuracy of mean‐field treatment. As an example, this theory is applied to quantum frequency conversion, in which mean‐field treatment is commonly applied, to test its limitation under strong pump and large coupling strength. The analytical results show excellent agreement with the numerical simulations. This work clearly reveals the residual quantum effects neglected by MFT and provides a more precise theoretical framework for nonlinear optics and quantum optics.Abstract Mean‐field treatment (MFT) is frequently applied to approximately predict the dynamics of quantum optics systems. It simplifies the system Hamiltonian by neglecting the quantum statistics of certain modes that are driven strongly by lasers or couple weakly with other modes. However, the neglected quantum correlations between different modes result in unanticipated quantum effects and might lead to significantly distinct system dynamics. Here, a general and systematic theoretical framework based on perturbation theory in company with MFT is provided to capture these quantum effects. The form of nonlinear dissipation and parasitic Hamiltonian as well as their relationship to the nonlinear coupling rate are predicted. Furthermore, the indicator is also proposed as a measure of the accuracy of mean‐field treatment. As an example, this theory is applied to quantum frequency conversion, in which mean‐field treatment is commonly applied, to test its limitation under strong pump and large coupling strength. The analytical results show excellent agreement with the numerical simulations. This work clearly reveals the residual quantum effects neglected by MFT and provides a more precise theoretical framework for nonlinear optics and quantum optics.

arXiv:2111.14636Laser & Photonics ReviewJan 22, 20232 citations

Dynamical Mean-Field Theory

Volodymyr Turkowski

Springer eBooksJan 1, 20210 citations

Dynamical Mean-Field Theory

WORLD SCIENTIFIC eBooksMar 23, 20250 citations

Mean-Field Theory of Phase Transitions

Hidetoshi Nishimori

Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.

Jul 19, 20011 citation

^{40}\mathrm{Ca}

nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.