Open Quantum systems

No quantum system can be completely isolated from its environment. As a direct result a quantum system is never pure \(Tr\{ \bf \rho^2 \} < 1\). A pure state is unitary equivalent to a zero temperrature grounds state frobidden by the third law of thermodynamics. A complete description of a quantum system requires to include the environment. The ourcome of this process is that oltimately we end with the state of whole universe describned by a wavefunction\(\Psi\). 

Even if the combined system is pure and can be described by a wavefunction \(\Psi\), a subsystem in general cannot be described by a wave function. This observation motivated the formalism of density operators and matrices introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau  in 1927 and Felix Bloch in 1946. In general the state of a subsystem is described by the density operator \(\rho_S\)and an observable by the scalar product\(< {\bf A}>= (\rho \cdot {\bf A })= Tr\{ \rho {\bf A} \}\). There is no way to know if the combined system is pure from the knowledge of the observales of the sybsystem. In particular if the combined system is entangled the system state is not pure.

Open system dynamics

The theory of open quantum systems seeks an economical treatment of the dynamics of observables that can be associated with the system. Typical observables are energy and coherence. Loss of energy to the environment  is termed quantum dissipation. Loss of coherence is termed quantum decoherence.  The reduction problem is difficult resulting in a diversity of approaches that have been attempted. A common objective is to derive a redced descrption where the system's dynamics is considered explicitly and the bath is described implicitly.

When the interaction between the system and the environment is weak a time dependent perturbation theory seems appropriate. The typical assumption is that the system and bath are initially uncorrolated \(\rho(0) =\rho_S \otimes \rho_B\).The idea has been originated by F. Bloch and followed by Redfield known as Redfield equation. The drawbak of the Redfield eqution is that it does not conserve the positivity of the density operator.

An alternative derivation employing projection operator techniques is know as Nakajima–Zwanzig equation. The derivation highlight the problem that the reduced dynamics is non-local in time: \(\partial_t{\rho }_\mathrm{S}=\mathcal{P}L{{\rho}_\mathrm{rel}}+\int_{0}^{t}{dt'\mathcal{K}({t}'){{\rho }_\mathrm{S}}(t-{t}')}.\)The effect of the bath is hidden in the momrey kernel \(\kappa(\tau)\). Additional assumptions of a fast bath are required to lead to a time local equation \(\partial_t \rho_S = {\cal L} \rho_s\). 

Another approach emeres as a analogue of classical dissipation theory develped by R. Kubo and Y. Tanimura . This approach is connected to Hierarchical equations of motion which embeds the density operator in a larger space of auxillary operaors such that a time local equation is obtained for the whole set.

A formal construction of the Markovian local equation of motion is an alternative to a reduced derivation. The theory is based on an axiomatic appproach. The basic starting point is a completely positive map. The assumption is that the inital sysem-environmnt state is uncorrelated \(\rho(0) =\rho_S \otimes \rho_B\)and the combined dynamics is unitary. Such a map falls under the category of Kraus map. The most general type of Markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix ρ that is trace-preserving and completely positive for any initial condition is the  Gorini–Kossakowski–Sudarshan–Lindblad equation or GKSL equation. For observales it becomes:\( \frac{d}{dt}A=+{\frac {i}{\hbar }}[H,A]+{\frac {1}{\hbar }}\sum _{k=1}^{\infty }{\big (}L_{k}^{\dagger }AL_{k}-{\frac {1}{2}}\left(AL_{k}^{\dagger }L_{k}+L_{k}^{\dagger }L_{k}A\right){\big )} \). The family of maps generated by the GKSL equation forms a quantum dynamical semi-group. In some fields such as quantum optics the term Lindblad superoperator is often used to express the quantum master equation for a dissipative system. EB Davis, derived the GKSL Markovian master equations using perturbation theory thus fixing the flaws of  the Redfield equation. Davises construction leads the stationary solution to thermal equilibrium.

In 1981, Amir Caldeira and Anthony J. Leggett proposed a simplifying assumption in which the bath is decomposed to normal modes representd as harmonic oscillators linearly coupled to the system. As a result the influence of the bath can be summerized by the  bath spectral function. This method is known as the Caldeira–Leggett or harmonic bath model. To proceed typically, the path integral description of quantum mechanics is employed.

The harmonic normal-mode bath leads to a physically consistent picture of quantum dissipation. Nevertheless its ergodic properties are too weak. The dynamcs does not generate wide scale entangelment between the bath modes.

An alternativ bath model is a spin bath. At low temperature and weak system-bath coupling therse two bath models are equivalent. But for higher excitation the spin bath has strong ergodic properties. Once the system is coupled significant entagelment is generated between all modes. A spin bath can simulate a harmonic bath but the oposite is not true.

Basic Construction of the surrogate Hamiltonian

In the stochastic surrogate Hamiltonian approach, the  system device is subject to dissipative forces due to coupling to  primary baths. In turn, the primary baths are subject to interactions with a secondary bath:

H_T=H_S+H_B+H_B''+H_SB+H_BB''

where H_S represents the system, H_B represent the primary baths, H_B''  the secondary baths, and H_SB the system–bath interaction. H_BB''  the primary/secondary baths’ interactions. The system Hamiltonian H_S describes molecular nuclear modes: The molecular Hamiltonian has the form:

\({\bf H_S} = \frac{{\bf P}^2}{2m} +V({\bf R})\)

The reduction in computational complexity is obtained by splitting the representation to a system that requires a full quantum description and a bath described implicitly. The final outcome is equations of motion for relevant dynamics, which are computationally tractable.

The bath is described by a fully quantum formulation. The method employed is the stochastic surrogate Hamiltonian  \cite{k238,k250}. Briefly, the bath is divided into a primary part interactingwith the system directly and a secondary bath which eliminates recurrence. The primary bath Hamiltonian is composed of a collection of two-level-systems.

\({\bf H_B} ~~=~~ \hbar \sum_j \omega_j {\bf \sigma^+_j} {\bf  \sigma_j } \)

The energies \(\omega_j\) represent the spectrum of the bath and \({\bf \sigma_{\pm}}\) are bath excitation/de-excitation operators.  The system-bath interaction \(\bf H_{SB}\) can be chosen to represent different physical processes. Specifically  we choose an interaction leading to vibrational relaxation:
\({\bf{H}_{SB} }= \hbar f({\bf R_s}) \otimes \sum_{j}^{N}  \lambda_{j} ({\bf{\sigma}_{j}^{\dagger}}+\bf{\sigma}_{j})\) ,
where \(f({\bf R_S})\)) is a dimensionless function of the system nuclear coordinates \({\bf R_S}\) and \(\lambda_j\)is the system-bath coupling frequency of bath mode j. when the system-bath coupling is characterized by a spectral density \(J(\omega)=\sum_{j} |\lambda^2|\delta (\omega-\omega_j) \)then \(\lambda_j =\sqrt{J(\omega_j)/\rho_j} ~~and ~~\rho_j=(\omega_{j+1}-\omega_j)^{-1} \) is the density of bath modes.The secondary bath is also composed of noninteracting two-level-systems (TLS) at temperature T with the same frequency spectrum as the primary bath. At random times the states of primary and secondary bath modes of the same frequency are swapped at a rate
\(\Gamma_j \) .  
The swapping procedure permits description of both dephasing and energy relaxation. The final results are obtained by averaging over the stochastic realizations. The swap makes the bath effectively infinite. Each swap operation eliminates the quantum correlation between the bath mode swapped and system and other bath modes. 
This loss of system-bath correlation leads to dephasing.

The stochastic surrogate Hamiltonian approach is a fully quantum treatment of system-bath dynamics. The method is not Markov constrained, and is based on a  wavefunction  construction. The system and bath are initially correlated, since the initial state is the combined thermal state  generated from the coupled system-bath Hamiltonian by propagating in imaginary time 
until the correlated ground state is obtained. The transport properties of the surrogate Hamiltonian are consistent with the second law of thermodynamics. Additional entanglement is generated by the dynamics. Convergence of the model is obtained by increasing the number of bath modes and the number of stochastic realizations. 

A central operation in the stochastic surrogate Hamiltonian is the swap: replacing one spin component of the bath with another. An important  technical issue is how to perform this operation within a global wavefunction description of the system and primary bath.

Decoherence is a fundamental process where a quantum system loses its wave properties enabling interference. As a result classical behavior emerges. Decoherence is a clear concept studied at least for 50 years nevertheless, it is still ill defined, it cannot be associated with a unique observable. New quantum technologies require control on the decoherence and knowledge on the transition from quantum to classical behavior. In the implementation of possible future quantum technologies and quantum information processing  fast decoherence is destructive. A signature of decoherence is purity loss where purity is defined by .

Gil Katz, David Gelman, Mark A. Ratner, and Ronnie Kosloff Stochastic surrogate Hamiltonian, J. Chem.Phys. 129 034108 (2008). 

Erik Torrontegui, and Ronnie Kosloff Activated and non-activated dephasing in a spin bath New Journal of Physics, 18, 093001 (2016).