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Preventing Disentanglement by Symmetry Manipulations

Preventing Disentanglement by Symmetry Manipulations. G. Gordon, A. Kofman, G. Kurizki Weizmann Institute of Science, Rehovot 76100, Israel Sponsors: EU, ISF. Outline. Decoherence mechanisms General formalism Modulation schemes Numerical example Conclusions.

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Preventing Disentanglement by Symmetry Manipulations

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  1. Preventing Disentanglement by Symmetry Manipulations G. Gordon, A. Kofman, G. Kurizki Weizmann Institute of Science, Rehovot 76100, Israel Sponsors: EU, ISF

  2. Outline • Decoherence mechanisms • General formalism • Modulation schemes • Numerical example • Conclusions

  3. Decoherence Scenarios: Single Particle Ion trap Cold atom in (imperfect) optical lattice Keller et al. Nature 431, 1075 (2004) Häffner et al. Nature 438 643 (2005) Jaksch et al. PRL 82, 1975 (1999) Mandel et al. Nature 425, 937 (2003) Ion in cavity Kreuter et al. PRL 92 203002 (2004)

  4. Ft() G() Single Particle Solution A. G. Kofman and G. Kurizki, Nature 405, 546 (2000), PRL 87, 270405 (2001), PRL 93,130406(2004) |(t)i=kk(t)|ki|gi+(t)|vaci|ei (t)=e-J(t)(0) J(t)=2s-11 dG()Ft(+a) Reservoir coupling spectrum G()!()|()|2 / Impulsive phase modulation (Caused by Repetitive Weak Pulses) (t)=ei[t/], ¿ J=2G(a+/) Spectral intensity of modulation Ft()=|t()|2 Fidelity: Dynamic decoupling. Viola & Lloyd PRA 58 2733 (1998) Shiokawa & Lidar PRA 69 030302(R) (2004) Vitali & Tombesi PRA 65 012305 (2001) F(t)=|h(0)|(t)i|2=e-2< J(t)

  5. Decoherence Scenarios: Many Particles Lisi & Mølmer, PRA 66, 052303 (2002); Sherson & Mølmer, PRA 71, 033813 (2005) Coupled atoms’ vibrations in imperfect optical lattice Ions’ vibrations in trap Ions in cavity “Sudden Death”, Yu & Eberly PRL 93 140404 (2004)

  6. k,b,1 |ki |2i |1i |gi |2i |1i |gi k,a,1 (b) (a) a,1(t) b,2(t) The System • Particles: • Ions • Cold atoms • Bath: • Cavity modes • Vibrational modes • Bath-particle coupling • Modulations: • AC Stark Shifts: RF fields, Lasers • Coupling modulation: On-off switch

  7. |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi |2i |1i |gi The Multipartite Wavefunction b a |(t)i = kk(t)|ki­|gia|gib + a,1(t)|vaci­|1ia|gib + a,2(t)|vaci­|2ia|gib + b,1(t)|vaci­|gia|1ib + b,2(t)|vaci­|gia|2ib

  8. The Multipartite Solution Gordon, Kurizki, Kofman J. Opt. B. 7 283, (2005); Opt. Comm. (in press) (t)=e-J(t)(0) Decoherence Matrix Jjj',nn'(t) = s0tdt's0t'dt''jj',nn'(t'-t'')Kjj',nn'(t',t'')eij,nt'-ij',n't'' Bath Matrix Modulation Matrix Kjj',nn'(t,t')=*j,n(t)j',n'(t') j,n(t)=eis0 dj,n() jj',nn'(t)=s dGjj',nn'()e-i t Gjj',nn‘(w)=~-2kk,j,n*k,j',n'(-wk)

  9. Decoherence Matrix Elements Diagonal elements: Individual particle decoherence Jjj,nn(t)=2s-11 dGjj,nn()Ft,j,n(+j,n) Off-diagonal elements: Cross-decoherence Jjj',nn'(t) = s0tdt's0t'dt''jj',nn'(t'-t'')*j,n(t`)j`n`(t``)eij,nt'-ij',n't''

  10. Fe(t)=|1,1(0)|2 |j=12n=12c*j,n(0)cj,n(t)|2 j=12n=12|cj,n(t)|2 The Fidelity F(t)=|h(0)|(t)i|2 Definitions: Mixing parameters: cj,n(t)=j,n(t)/1,1(t) Decay parameter: A(t)=1,1(t)(j,n|cj,n(t)|2)1/2 F(t)=Fp(t)Fe(t) Population Preservation Fp(t)=|A(t)|2 F(t) of single particle Entanglement Preservation

  11. The Fidelity: Example Initial entangled state: |(0)i=1/√2(|gia|1ib+|1ia|gib) No Cross-decoherence, different decoherence rates: a(t)=1/√2 e-Ja(t) b(t)=1/√2 e-Jb(t) Population preservation: probability of having a particle in an excited state Fp(t)=(e-2Ja(t)+e-2Jb(t))/2 Entanglement preservation: Given that a particle is in an excited state, a measure of entanglement preservation compared to initial state. Fe(t)=1/2+e- J(t)/(1+e-2 J(t)) J(t) = Jb(t)-Ja(t)

  12. Modulation Schemes Tasks N identical independent particles N different independent particles No Modulation ,Global Modulation N decoherence free qubits Decoherence Free Subspace Viola et al. PRL 85, 3520 (2000); Wu & Lidar, PRL 88, 207902 (2002)

  13. Numerical Example: Setup • Two three-level particles • Coupling: Gaussian, Gjj`,nn`()/ exp(-2/j,n2)exp(-2/j`,n`2) • Different for each particle • Cross-decoherence • Impulsive phase modulation j,n(t)=ei[t/j,n]j,n • Global Scheme: Identical modulation to all particles • Local Scheme: Addressability, specific modulations • Initial Entangled State: |(0)i=1/√2(|-ia|gib+|gia|-ib) |-ij=1/√2(|1ij-|2ij) ”dark state”

  14. Global Modulation Condition: j,n(t)=(t)8 j,n Jjj`,nn`(t)=2s-11 dGjj`,nn`()Ft(+j,n) > 0 J(t) • General Decoherence Matrix • Cross-coupling particles, Different coupling to bath • Population loss Entanglement loss

  15. F Fp Fe Numerical Example: Global Modulation  No Symmetry Decoherence Matrix Elements Decoherence Matrix Fidelity

  16. Task 1: Eliminating cross-decoherence J(t) Condition: j,n(t)j`,n`(t)8 j,j`,n,n` • Diagonal Decoherence Matrix • Effectively: N different independent particles • Separated particles, Coupled to different baths • Population loss Entanglement loss Jjj',nn'(t) = s0tdt's0t'dt''jj',nn'(t'-t'')*j,n(t`)j`n`(t``)eij,nt'-ij',n't''=0 8 j j`, n n`

  17. F Fp Fe Numerical Example: Local modulations  Eliminate cross-decoherence Decoherence Matrix Elements Decoherence Matrix Fidelity

  18. Ft() Ft() G() G() Task 2: Equating decoherence rates Condition: j,n(t)j`,n`(t)8 j,j`,n,n` Jjj,nn(t)=2s-11 dGjj,nn()Ft,j,n(+j,n) =J(t) • Decoherence Matrix / Identity Matrix • Imposes permutation symmetry • Effectively: N independent identical particles • Separated particles, identical coupling to baths • Reduce problem to single particle decoherence control • Population loss Entanglement preservation = J(t) Decaying Entangled state |(t)i=e-J(t)|(0)i

  19. F Fp Fe Numerical Example: Equating decoherence rates Decoherence Matrix Elements Decoherence Matrix Fidelity

  20. Task 3: Equating decoherence and cross-decoherence Condition: j,n(t)¼j`,n`(t)8 j,j`,n,n` Jjj`,nn`(t)=J(t) J(t) • All Decoherence Matrix Elements Equal • Imposes permutation symmetry • Cross-coupled particles, identical coupling to the same bath • Anti-symmetric state = Decoherence-Free Subspace Very difficult

  21. Optimal modulation scheme J(t) • For N three-level particles • Equatingintraparticle decoherence and cross-decoherence of each particle • Eliminating interparticle cross-decoherence • Anti-symmetric state of each particle |-ij=1/√2(|1ij-|2ij) = Decoherence Free Subspace Condition: j,n(t)¼j`,n`(t)8 j=j`,n,n` j,n(t)j`,n`(t)8 j j`,n,n’ N decoherence free qubits

  22. F Fp Fe Numerical Example: Optimal scheme Decoherence Matrix Elements Decoherence Matrix Fidelity |(0)i=1/√2(|-ia|gib+|gia|-ib) |-ij=1/√2(|1i2-|2ij)

  23. Numerical Example: Summary

  24. Single particle: No modulation: Lifetime = 1.168 s With modulation: Lifetime ¼ 1.4 s Multiple ions in cavity: Position in cavity: a-b¼ 15% Suggested Experimental Setup:Multiple 40Ca+ Ions in Cavity Kreuter et al. PRL 92 203002 (2004);Barton et al. PRA 62 032503 (2000) Experimental parameters: Finesse¼ 35000 32D5/2 a = 729 nm Cavity mode width¼ 12 GHz Required impulsive phase modulation rate ~  Single ion in cavity: Three level system: |gi = 42S1/2 |1i = 32D3/2 |2i = 32D5/2 1/2 = 1.026 Two ions in each cavity + Local modulations 1/√2(|gia|1ib-|1ia|gib) = DFS

  25. Conclusions • Local modulations can • Impose permutation symmetry • Introduce a Decoherence-Free Subspace • Reduce the task of multipartite disentanglement to that of a single relaxing particle • Universal dynamical decoherence control formalism gives the modulations’ conditions for each task • Optimal modulation scheme for N three-level particles • Can impose many-particle DFS Thank You !!!

  26. Modulation Criteria Ft,j,n()=Ft,j`,n`() Global modulation j,n(t)=j`,n`(t) G() Ft,j,n() Ft,j`,n`() Elimination of cross-decoherence j,n(t)j`,n`(t) G() Ft,j,n() Creation of DFS Ft,j`,n`() j,n(t)¼j`,n`(t) G()

  27. Multilevel Cross-Decoherence No modulation Ft,j`,n(+j`) Ft,j,n(+j) j,n(t)=0 G() Ft,j`,n(+j`) Ft,j,n(+j) Global modulation G() j,n(t)=j`,n`(t) Creation of DFS Ft,j,n(+j) Ft,j`,n(+j`) j,n(t)¼j`,n`(t) G()

  28. Quasi-periodic Modulation j,n(t)=ll e-ilt J(t)=2llG(+l) Ft,j,n() G() l-3 l-2 l-1 l l+1 l+2 l+3

  29. General Bath Formalism Decoherence Matrix Jjj',nn'(t) = s0tdt's0t'dt''jj',nn'(t'-t'')Kjj',nn'(t',t'')eij,nt'-ij',n't'' Bath Matrix jj',nn'(t)=s dGjj',nn'()e-i t Gjj',nn‘(w)=~-2kk,j,n*k,j',n'(-wk) The Same Bath: Separate Baths: (independent particles) k,j,nk,j`,n`=0 8 jj`,n,n`,k Particle j coupled to modes {kj} Particle j` coupled to modes {kj`}  {kj}Å{kj`}=; k,j,n=k8 j,n

  30. |2i V(t) |1i 2 1 |0i "Dark State” No Cross-Decoherence, levels coupled to separate baths |2i 12 |1i 2 1 |0i Exploiting Cross-Decoherence to create “dark state”

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