Quantum trajectories and open many-body quantum systems

Abstract

The study of open quantum systems – microscopic systems exhibiting quantum coherence that are coupled to their environment – has become increasingly important in the past years, as the ability to control quantum coherence on a single particle level has been developed in a wide variety of physical systems. In quantum optics, the study of open systems goes well beyond understanding the breakdown of quantum coherence. There, the coupling to the environment is sufficiently well understood that it can be manipulated to drive the system into desired quantum states, or to project the system onto known states via feedback in quantum measurements. Many mathematical frameworks have been developed to describe such systems, which for atomic, molecular, and optical (AMO) systems generally provide a very accurate description of the open quantum system on a microscopic level. In recent years, AMO systems including cold atomic and molecular gases and trapped ions have been applied heavily to the study of many-body physics, and it has become important to extend previous understanding of open system dynamics in single- and few-body systems to this many-body context. A key formalism that has already proven very useful in this context is the quantum trajectories technique. This method was developed in quantum optics as a numerical tool for studying dynamics in open quantum systems, and falls within a broader framework of continuous measurement theory as a way to understand the dynamics of large classes of open quantum systems. In this article, we review the progress that has been made in studying open many-body systems in the AMO context, focussing on the application of ideas from quantum optics, and on the implementation and applications of quantum trajectories methods in these systems. Control over dissipative processes promises many further tools to prepare interesting and important states in strongly interacting systems, including the realisation of parameter regimes in quantum simulators that are inaccessible via current techniques.

PACS:

• 03.65.Yz Decoherence
• open systems
• quantum statistical methods
• 42.50.-p Quantum optics
• 37.10.-x Atom
• molecule
• and ion cooling methods
• 67.85.-d Ultracold gases
• trapped gases

Keywords:

• open quantum systems
• quantum trajectories
• quantum optics
• time-dependent dynamics
• time-dependent density matrix renormalisation group
• decoherence

Notes

1. In a many-body context, a “small” quantum system might be relatively large and complex – it should just be very substantially smaller, e.g. in the scale of total energy, or total number of degrees of freedom, when compared with the environment to which it is coupling

2. See, e.g. Ref. [202] for a general discussion of random number generators. For typical calculations, only a few tens of thousands of random numbers must be generated, and good quality pseudorandom number generators suffice. See Ref. [203] for a recent discussion of improvements to that are possible using quantum random number generators.

3. We can't completely take the limit physically, as we have made important approximations regarding the fact that we consider dynamics over long timescales compared with . However, we can take the limit of vanishing δ t from the point of view of numerical implementations after the other approximations have been implemented. See Appendix A for a more detailed discussion.

4. For a more detailed review of these methods, see, e.g. Ref. [226].

5. Although atoms consist of protons, neutrons, and electrons, a tightly bound atom at low energy can be assumed to maintain bosonic or fermionic character provided short-distance physics can be eliminated, and the typical separation of atoms is much larger than their size.

6. Note that this transfer can also be interpreted as measurement of the position of the atom within the lattice site.