The question of finding non-equilibrium stationary states, on which energy and particle transport are established, has been an important open question in Physics since very early stage of Physics. For classical systems, such as heat conductance and electric circuits, one can use phenomenological laws such as Fourier's law and Ohm's law, although even those laws are not derived directly from classical equation of motion. During the past many decades there have been many efforts on trying to deriving those laws from the first principle. When it comes to quantum systems, we do not even have such phenomenological laws. So the ultimate goal of this project is to develop such a theoretical framework to find non-equilibrium stationary states from the first-principle equation of motion of quantum systems.
In equilibrium statistical physics, an equilibrium state of a system (both classical and quantum) is given explicitly by the Boltzmann distribution, which requires only the Hamiltonian (and of course, its' eigenvalues and eigenvectors), bath temperature and chemical potential. For non-equilibrium states, even when we are given all those things, we still don't know how to write down explicitly the distribution function.
Therefore, what we are looking for is a blackbox accepting Hamiltonian, bath temperatures, chemical potentials, and if necessary coupling mechanism and coupling strength to baths, and giving a distribution function. In this project, we consider only quantum systems.
There are already some approaches to this problem, including: the Kubo formula, which treats the central systems as closed systems, the Landauer-Buttiker formula for non-interacting systems and the non-equilibrium Green's function method, which is capable of dealing with interaction. We decided to use another approach based on the open-system scenario: starting from the Schroedinger equation of the composite system of central system+bath, deriving an effective equation of motion of the central system only and then solving the effective equation. The baths are usually assumed to be in thermal equilibrium states all the time (i.e. baths have infinite degree of freedom and coupling strength is small so that back action from the central system to the bath can be neglected). Derivations of the effective equations usually involves additional approximation such as second-order perturbation and Markovian approximation. But those approximations can be relaxed and then more complicated effective equation can be derived. At the current stage of this project, we will not focus on deriving such equations, but rather on solving those equations, developing and applying this framework. One of such equations is called the Redfield equation. It makes the above two approximations but still makes use of the full dynamics of the central systems.
Note the whole picture mentioned above has already been suggested before we started this project, especially in discussion of relaxation towards equilibrium states. However, it is not widely used in describing non-equilibrium stationary states. We believe that it is because of the difficulty in solving the resulting open-system master equation such as the Redfield equation. Using direct methods, one needs to solve an eigenvalue problem of size 4N where N is the size of the system measured in qubits. If efficient methods of solving this equation can be found, then more applications on various physical models of this framework will for sure emerge.
The following lists several projects I have worked on.
However, for interacting systems, unfortunately, those equations of single-particle Green's functions, denoted as G1, are coupled to two-particle Green's function G2. Furthermore, at the next order, equations of G2 are coupled to G3 and so on. This hierarchical structure exists too for equations of equilibrium Green's function. Solving the whole hierarchy is as hard as solve the original equation of density matrices. There has to be some ways to truncate and solve the hierarchy consistently and systematically. In equilibrium Green's function, the BBGKY (Bogoliubov-Born-Green-Kirkwood-Yvon) hierarchy and the cluster expansion provide exactly such a method.
Therefore, here we just need to develop a similar equation hierarchy for the Redfield equation. The only difference is at those additional bath operators in the Redfield equation: they are operators involving many-particle operators. One has to derive too a hierarchical form of those operators together with the interaction term in the Hamiltonian.
In its first-order form, we truncated at terms involving G2, neglected higher Gs and assumed
In its second-order form, we truncated at terms involving G3, neglected higher Gs and assumed
In this work, we use the coherent-state representation approach to convert the 4N-dimensional Redfield equation into a stochastic differential equation with 2N complex variables, so called a generalized Fokker-Planck equation. Analytical expression of the non-equilibrium stationary states are derived for some systems. The accuracy of this method is around 6%, see in Figure Coherent. Manuscript in preparation.
It is well known, and as we used and showed above, that if the bath is ideal that it has infinite degrees of freedom, it is initially in a thermal equilibrium state, and the coupling strength between the bath and the central system is infinitesimal, then the central system will be driven into a stationary state. In its not-so-rigorous derivation, usually second-order perturbation and Markovian approximation are used. So the question is how essential are all the above conditions and approximations for the central system to converge to the stationary states?
This might be answered if we can implement the pure dynamical evolution of the system+bath and add in various approximations and conditions one by one. For example, as in Fig 1st_principle we may take both the bath and the central system to be finite, of course with the bath being much large, and study their pure dynamical evolution with different initial condition of the bath's state. We can then set the initial bath states follows the Boltzmann distribution and check if the central system evolves towards stationary states when the bath size goes to larger and larger. From there we may also be able to understand the difference between Markovian and non-Markovian evolution.
In fact, answers to this question have deeper meaning than what they appears here. These answers could help us establish a framework of studies of general open system, including under what condition infinite-dimension assumption is valid or Markovian approximation is reasonable, or it is fine to assume the bath in equilibrium and so on. This question, how to study open systems, is really essential in quantum computing.
It is so different that they even have different equilibrium limits. Let us first consider the NEGF configuration: lead+system+lead with equilibrium setup where both leads, which can be regarded as semi-infinite non-interacting chains, are in equilibrium states with the same temperatures and chemical potentials. One special case is when the central system is also non-interacting, say a finite non-interacting chain, which has the same natural with the leads. If we ask what is the stationary state of the whole system, one can easily imagine that it will the equilibrium state of the whole composite system of lead+system+lead. In fact, using Dyson's equation for Green's function, one can find exact solution for this case and confirm the above intuitive picture.
Let us now consider the open-system scenario: a central system is coupled to two baths, which may in fact as well be the same semi-infinite non-interacting chains but we start directly from the reduced equation of motion of the central system. From there, if the central system is a non-interacting one we find that the stationary state is the equilibrium state of the central system only.
What is the difference? In the case of NEGF, the equilibrium state is of the whole composite system while in the case of open-system scenario, the equilibrium state is of the central system only. Therefore, although they deal with the same physical question, there is no guarantee that they will have the same quantitative or even qualitative results. Then in what circumstance, will the two be directly comparable? Thinking about a two-spin Ising system for instance with coupling constant Jz. When there is no local magnetic fields on the spins, starting from the equilibrium state of the whole two-spin system,
where p(T) and q(T) depend on T but p(T)+q(T)=1/2, the reduced density matrix of one spin becomes
and
where Jz, inter refers to the coupling strength between the central system and the leads. Under those conditions, ideally the two methods should be comparable. In all of our previous example, coupling to baths are taken to be in the momentum space. We have derived some formulae for coupling in real space, using a configuration similar to Fig 1st_principle(b). Detailed analytical and numerical work has not yet started.
