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Engineering correlation and Engineering correlation and entanglement dynamics in spin entanglement dynamics in spin chainschains

T. S. Cubitt

J.I. Cirac

http://www.mpg.de/english/

•Motivation and goals•Time evolution of a chain•Correlation and entanglement wave-packets•Engineering the dynamics: solitons etc.•Fermionic gaussian state formalism•Conclusions


T. S. Cubitt

J.I. Cirac


Conceptual motivation: new Conceptual motivation: new experimentsexperiments

• …motivate new theoretical studies of non-equilibrium behaviour.

• New experiments…

• Many papers on correlations/entanglement of ground states• Fewer on time-dependent behaviour away from equilibrium

• Many papers on correlations/entanglement of ground states.• Fewer on time-dependent behaviour away from equilibrium

Existing resultsExisting results

• In Phys. Rev. A 71, 052308 (2005), we used our entanglement rate equations to bound the time taken to entangle the ends of a length L chain.

• Left open question of whether our pL lower bound is tight.

• In J.Stat.Mech. 0504 (2005) p.010, Calabrese and Cardy investigated the time-evolution of block-entropy in spin chains.

• Bravyi, Hastings and Verstraete (quant-ph/0603121) recently used Lieb-Robinson to prove tighter, linear bound.

Practical motivation: quantum repeatersPractical motivation: quantum repeaters

• “Traditional” solution to entanglement distribution:build a quantum repeater.

• But a real quantum repeater involves particle interactions, e.g. atoms in cavities.

• Alternative (e.g. Popp et al., Phys. Rev. A 71, 042306 (2005)): use entanglement in ground state:

• Getting to the ground state may be unrealistic.• Why not use non-equilibrium dynamics to distribute

entanglement?


T. S. Cubitt


J.I. Cirac


Time evolution of a spin chain (1)Time evolution of a spin chain (1)

• As an exactly-solvable example, we take the XY model…

anisotropy magnetic field

…and start it in some separable state, e.g. all spins +.

• Fourier:


• Solved by a well-known sequence of transformations:

• Bogoliubov:

• Jordan-Wigner:


• Initial state N|+i…

…is vacuum of the cl=zj-

l operators.

• Wick’s theorem: all correlation functions hxm…pni of the ground state of a free-fermion theory can be re-expressed in terms of two-point correlation functions.

• Our initial state is a fermionic Gaussian state: it is fully specified by its covariance matrix:


• Hamiltonian…

…is quadratic in x and p.

• From Heisenberg equations, can show that time evolution under any quadratic Hamiltonian:

corresponds to an orthogonal transformation of the covariance matrix:

Gaussian state stays gaussian under gaussian evolution.


• Initial state is a fermionic gaussian state in xl and pl.

• Time-evolution is a fermionic gaussian operation in xk and p

k.

Connected by Fourier and Bogoliubov transformations

• Fourier and Bogoliubov transformations are gaussian:

Time evolution of a spin chain (phew!)Time evolution of a spin chain (phew!)

• Putting everything together:

xk , pkxk p

kxk p

k xk , pkxk , pk xl , pltime-evolveinitial state



T. S. Cubitt

J.I. Cirac


String correlationsString correlations

• We can get string correlations hanz

lbmi for free…

• Equations are very familiar: wave-packets with envelope S propagating according to dispersion relation .

• Given directly by covariance matrix elements, e.g.:

Two-point correlationsTwo-point correlations

• Two-point connected correlation functions hznz

mi - hznihz

mi can also be obtained from the covariance matrix.

• Again get wave-packets (3 of them) propagating according to dispersion relation k.

• Using Wick’s theorem:

• As with all operationally defined entanglement measures, LE is difficult to calculate in practice.

• Best we can hope for is a lower bound.

What about entanglement?What about entanglement?

• The relevant measure for entanglement distribution (e.g. in quantum repeaters) is the localisable entanglement (LE).

• Definition: maximum entanglement between two sites (spins) attainable by LOCC on all other sites, averaged over measurement outcomes.

• Popp et al., Phys. Rev. A 71, 042306 (2005) : LE is lower-bounded by any two-point connected correlation function.

• In case you missed it: we’ve just calculated this!



T. S. Cubitt

J.I. Cirac


• In particular, around =1.1, =2.0 all three wave-packets in the two-point correlation equations are nearly identical

• ! localised packets propagate at well-defined velocity with minimal dispersion: “soliton-like” behaviour

• In other parameter regimes: narrower wave-packets in nearly linear regions of dispersion relation

• ! packets maintain their coherence as they propagate

• In some parameter regimes: broad wave-packets and a highly non-linear dispersion relation

• ! correlations rapidly disperse and disappear

Correlation wave-packetsCorrelation wave-packets

• In some parameter regimes: broad wave-packets and a highly non-linear dispersion relation

• ! correlations rapidly disperse and disappear: (=10, =2)

Correlation wave-packets (1)Correlation wave-packets (1)

• The system parameters and simultaneously control both the dispersion relation and form of the wave-packets.

• In other parameter regimes, all three wave-packets in the two-point correlation equations are nearly identical

• ! localised packets propagate at well-defined velocity with minimal dispersion: “soliton-like” behaviour: (=1.1, =2)

Correlation/entanglement solitonsCorrelation/entanglement solitons

• In general, time-ordering is essential.• But if parameters change slowly, dropping it gives

reasonable approximation.

• If we stay within “soliton” regime, adjusting the parameters only changes gradient of the dispersion relation, without significantly affecting its curvature.

• ! Can speed up and slow down the “solitons”.

Controlling the soliton velocity (1)Controlling the soliton velocity (1)

• If the parameters are changed with time,

• ! Effective evolution under time-averaged Hamiltonian.

• Starting from =1.1, =2.0 and changing at rate +0.01:

Controlling the soliton velocity (2)Controlling the soliton velocity (2)

• Can calculate this analytically using same machinery as before.• Resulting equations are uglier, but still separate into terms

describing multiple wave-packets propagating and interfering.

““Quenching” correlations (1)Quenching” correlations (1)

• What happens if we do the opposite: rapidly change parameters from one regime to another?

• Choose parameters so that “frozen” packets remain relatively coherent, whilst others rapidly disperse.

Get four types of behaviour for the wave-packets:• Evolution according to 1 for t1, then 2

• Evolution according to 1 for t1, then -2

• Evolution according to 1 until t1, no evolution thereafter• Evolution according to 2 starting at t1

““Quenching” correlations (2)Quenching” correlations (2)

• Choose parameters so that “frozen” packets remain relatively coherent, whilst others rapidly disperse.

• ! can move correlations/entanglement to desired location, then “quench” system to freeze it there.

• E.g. =0.9, =0.5 changed to =0.1, =10.0 at t1=20.0:



T. S. Cubitt

J.I. Cirac


What about entanglement? (2)What about entanglement? (2)

• There is another LE bound we can calculate…

• Recall concurrence:

• Not a covariance matrix element because of |*i.

• Localisable concurrence:

• Operators a, ay commute

• States specified by symmetric covariance matrix

• Gaussian operations $ symplectic transformations of

Bosonic case

Fermionic gaussian formalismFermionic gaussian formalism

• Recap of gaussian state formalism…

• States specified by antisymmetric covariance matrix

• Gaussian operations $ orthogonal transformations of

Fermionic case• Operators c, cy anti-commute

• What’s missing? A fermionic phase-space representation.


• Solution: expand fermionic Fock space algebra to include anti-commuting numbers, or “Grassmann numbers”, n.

• Coherent states and displacement operators now work:cncm|i = -cmcn|i = -n m|i = m n|i = cncm|i

• Grassman algebra:n m = -m n ) n

2=0; for convenience n cm = -cm n

• Grassman calculus:

• Characteristic function for a gaussian state is again gaussian:


• We will need another phase-space representation: the fermionic analogue of the P-representation.

• Essentially, it is a (Grassmannian) Fourier transform of the characteristic function.

• Useful because it allows us to write state in terms of coherent states:

• For gaussian states:

• Finally,

What about entanglement (3)What about entanglement (3)

• Recall bound on localisable entanglement:

• Substituting the P-representation for states and * :

and expanding xn and pn in terms of cn and cn y, the calculation

becomes simple since coherent states are eigenstates of c.• Not very useful since bound 0 in thermodynamic limit N 1.

• However, experimentalists are starting to build atomic analogues of quantum optical setups: e.g. atom lasers, atom beam splitters.

• Fermionic gaussian state formalism may become important as fermionic gaussian states and operations move into the lab.


• Michael Wolf has used fermionic gaussian states to prove an area law for a large class of fermionic systems, in arbitrary dimensions: Phys. Rev. Lett. 96, 010404 (2006)

• Already leading to new theoretical results, e.g.:


Entanglement flowEntanglement flow in multipartite systems in multipartite systems

T. S. Cubitt

J.I. Cirac


ConclusionsConclusions

• Have shown that correlation and entanglement dynamics in a spin chain can be described by simple physics: wave-packets.

Correlation dynamics can be engineered:• Set parameters to produce “soliton-like” behaviour• Control “soliton” velocity by adjusting parameters slowly in time• Freeze correlations at desired location by quenching the system

• Developed fermionic gaussian state formalism, likely to become more important as experimentalists are starting to do gaussian operations on atoms in the lab (atom lasers, atomic beam-splitters…).

The end!

Download - Engineering correlation and entanglement dynamics in spin chains

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