Now André Carvalho from the physics department at Australian National University in Canberra is talking about “Quantum feedback control for entanglement production”. He’s in a theory group with strong connections to the atom laser experimental group at ANU. This theory group works on measurement and control theory for Bose-Einstein condensates and atom lasers.
The good news: recent advances in real-time monitoring allows the control of quantum systems using feedback.
The big question: can we use feedback to design the system dynamics to produce and stabilize entangled states?
The answer: yes.
Start by considering two atoms in a cavity, interacting with a laser. Think of each atom as a 2-state system — so the Hilbert space of the pair of atoms is
We’ll say what the atoms are doing using not a pure state (a unit vector) but a mixed state (a density matrix). The atoms’ time evolution will be described by Lindbladian mechanics. This is a generalization of Hamiltonian mechanics that allows for dissipative processes — processes that increase entropy! A bit more precisely, we’re talking here about the quantum analogue of a Markov process. Even more precisely, we’re talking about the Lindblad equation: the most general equation describing a time evolution for density matrices that is time-translation-invariant, Markovian, trace preserving and completely positive.
As time passes, an initially entangled 2-atom state will gradually ‘decohere’, losing its entanglement.
But next, introduce feedback. Can we do this in a way that makes the entanglement become large as time passes?
With ‘homodyne monitoring’, you can do pretty well. But with ‘photodetection monitoring’, you can do great! As time passes, every state will evolve to approach the maximally entangled state: the ‘singlet state’. This is the density matrix
corresponding to the pure state
So: the system dynamics can be engineered using feedback to product and stabilize highly entangled state. In fact this is true not just for 2-atom systems, but multi-atom systems! And at least for 2-atom systems, this scheme is robust against imperfections and detection inefficiencies. The question of robustness is still under study for multi-atom systems.
For more details, try:
• A. R. R. Carvalho, A. J. S. Reid, and J. J. Hope, Controlling entanglement by direct quantum feedback.
We discuss the generation of entanglement between electronic states of two atoms in a cavity using direct quantum feedback schemes. We compare the effects of different control Hamiltonians and detection processes in the performance of entanglement production and show that the quantum-jump-based feedback proposed by us in Phys. Rev. A 76 010301(R) (2007) can protect highly entangled states against decoherence. We provide analytical results that explain the robustness of jump feedback, and also analyse the perspectives of experimental implementation by scrutinising the effects of imperfections and approximations in our model.
How do homodyne and photodetection feedback work? I’m not exactly sure, but this quote helps:
In the homodyne-based scheme, the detector registers
a continuous photocurrent, and the feedback Hamiltonian
is constantly applied to the system. Conversely, in
the photocounting-based strategy, the absence of signal
predominates and the control is only triggered after a
detection click, i.e. a quantum jump, occurs.