Engineering the Dynamics

Engineering Entanglement andEngineering Entanglement andCorrelation Dynamics in Spin ChainsCorrelation Dynamics in Spin Chains[1]

T. S. Cubitt1,2 and J. I. Cirac1

•How are long-range correlations created? This question has been studied extensively for ground states of quantum systems, both for correlation functions and, more recently, for entanglement[2]. But there are far fewer studies of what happens away from equilibrium.

Motivation

•Whereas, previously, precise control of many-body quantum systems was experimentally infeasible, with recent experimental developments directed towards quantum information processing it is becoming increasingly realistic. And whereas, motivated by experimental viability, theoretical studies of many-body quantum systems concentrated on equilibrium states, studies of non-equilibrium behaviour are correspondingly increasingly well-motivated.

New theoretical

studies

New experiments

•They are also of practical interest, e.g. for quantum repeaters[3]. Physically, quantum repeaters consist of a chain of quantum systems interacting with their neighbours. Recently[4], it has been suggested that the correlations inherent in the ground state of this interacting system could be used as an alternative way to distribute long-range entanglement. However, reaching the ground state is unrealistic in many systems. Why not instead use the non-equilibrium dynamics to distribute entanglement?

Correlation Wave Packets

•This has the form of two wave packets with envelope S/2 propagating in opposite directions along the chain according to a dispersion relation given by the system spectrum k. Thus the evolution of the string correlations can be described by very simple physics.•What about two-point connected correlation functions? Using Wick’s theorem:

•The easiest correlation functions to calculate are string correlation functions:

Wave packetsagain!

(six of them)

•What about entanglement? Localisable entanglement (the relevant measure for entanglement distribution) is bounded from below by any two-point connected correlation function[4].

•The system spectrum, and therefore the form of the dispersion relation, depends only on global properties of the system, as do the shapes of the correlation wave packets. Thus the dynamics can be controlled by simple, global system parameters.

•In some parameter regimes, broad (in frequency-space) wave packets and a highly non-linear dispersion relation cause the correlations to rapidly disperse and disappear:

•However, in other regimes, narrower wave packets located in nearly linear regions of the dispersion relation remain well-localised as they propagate:

Controlling Evolving Correlations

•How possible is it to control these correlation wave packets? If the system parameters are changed slowly with time, the dispersion relation can be altered without significantly changing the wave-packet envelopes, allowing us to speed up (or slow down) the correlation packets:

•We can also consider doing the opposite: abruptly changing the system parameters at time t1. The time-evolved covariance matrix in this scenario can be calculated analytically, and splits into various wave-packet terms.

•From the Heisenberg evolution equations, any evolution governed by a quadratic Hamiltonian corresponds to an orthogonal transformation of the covariance matrix, defined bywhere .

The System

•Consider a chain of spins, initially in a separable (uncorrelated) state. We are interested in investigating the dynamics of correlations and entanglement when interactions between neighbouring spins are switched on.

•As a simple, exactly solvable model, we take the XY-model for a chain of spin-1/2 particles:

•Similarly, the Bogoliubov and Fourier transformations are orthogonal:

•The initial state |is the vacuum of the Majorana operators obtained after the Jordan-Wigner transformation. By Wick’s theorem, it can be defined by it’s covariance matrix:

•Putting all this together, the time-evolved state is given by:

•This can be diagonalised by applying a Jordan-Wigner, Fourier, and Bogoliubov transformations:Majorana operators

x = j - i

i j

Gaussian states stay gaussian under gaussian operations

[1] T. S. Cubitt and J. I. cirac, quant-ph/0701053 (2007)

[2] A. Osterloh et al., Nature 416, 608 (2002);T. Osborne and M. A. Nielsen, Phys. Rev. A 66, 032110 (2002)

[3] H. J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998); W. Dür et al., Phys. Rev. A 59, 169 (1999)

[4] M. Popp et al., Phys. Rev. A 71, 042306 (2005)

References

Conclusions

•We have shown for a simple spin model that long-range correlations are created by propagation of correlation wave-packets.

•By changing easily accessible, global parameters of the system, the propagation velocity and dispersion of these packets can be engineered.

•If the parameters are changed during the evolution, we can speed up, slow down, or freeze the propagating correlation packets.

•This has obvious applications to quantum repeater setups.•Some start propagating according to the initial dispersion relation and subsequently evolve according to the new one. Some subsequently propagate backwards. Some only start evolving after the parameters are changed. And some undergo no further evolution after the change:

•We can use the latter to allow correlations to propagate to a desired location, then “quench” the system to freeze them there.

1Max-Planck-Institut für Quantenoptik, Garching, Germany 2Department of Mathematics, University of Bristol, UK