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

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

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

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

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

The Multipartite Wavefunction |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |2 i |1 i |g i |  (t) i =  k  k (t)|k i­ |g i a |g i b +  a,1 (t)|vac i­ |1 i a |g i b +  a,2 (t) |vac i­ |2 i a |g i b +  b,1 (t) |vac i­ |g i a |1 i b +  b,2 (t) |vac i­ |g i a |2 i b a b

 t)=e -J(t)  (0) J jj',nn' (t) = s 0 t dt' s 0 t' dt''  jj',nn' (t'-t'')K jj',nn' (t',t'')e i  j,n t'-i  j',n' t''  jj',nn' (t)= s d  G jj',nn' (  )e -i  t G jj',nn‘  ~ -2  k  k,j,n  * k,j',n'  k ) Bath MatrixModulation Matrix K jj',nn' (t,t')=  * j,n (t)  j',n' (t')  j,n (t)=e i s 0  d   j,n (  ) Decoherence Matrix The Multipartite Solution Gordon, Kurizki, Kofman J. Opt. B , (2005); Opt. Comm. (in press)

Decoherence Matrix Elements Diagonal elements: Individual particle decoherence J jj,nn (t)=2  s d  G jj,nn (  )F t,j,n (  +  j,n ) Off-diagonal elements: Cross-decoherence J jj',nn' (t) = s 0 t dt' s 0 t' dt''  jj',nn' (t'-t'')  * j,n (t`)  j`n` (t``)e i  j,n t'-i  j',n' t''

F(t)=| h  (0)|  (t) i | 2 Definitions: Mixing parameters: c j,n (t)=  j,n (t)/  1,1 (t) Decay parameter: A(t)=  1,1 (t)(  j,n |c j,n (t)| 2 ) 1/2 F(t)=F p (t)F e (t) F p (t)=|A(t)| 2 F e (t)=|  1,1 (0)| 2 |  j=1 2  n=1 2 c * j,n (0)c j,n (t)| 2  j=1 2  n=1 2 |c j,n (t)| 2 Population Preservation Entanglement Preservation The Fidelity F(t) of single particle

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

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

Two three-level particles Coupling: Gaussian, G jj`,nn` (  ) / exp(-  2 /  j,n 2 )exp(-  2 /  j`,n` 2 ) –Different for each particle –Cross-decoherence Impulsive phase modulation  j,n (t)=e i[t/  j,n ]  j,n –Global Scheme: Identical modulation to all particles –Local Scheme: Addressability, specific modulations Initial Entangled State: |  (0) i =1/√2(|- i a |g i b +|g i a |- i b ) |- i j =1/√2(|1 i j -|2 i j ) ”dark state” Numerical Example: Setup

J(t) Global Modulation General Decoherence Matrix Cross-coupling particles, Different coupling to bath Population loss Entanglement loss Condition :  j,n (t)=  (t) 8 j,n J jj`,nn` (t)=2  s d  G jj`,nn` (  )F t (  +  j,n )  0

Decoherence Matrix Decoherence Matrix Elements Fidelity FFpFeFFpFe Numerical Example: Global Modulation  No Symmetry

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

Fidelity Decoherence Matrix Elements Decoherence Matrix FFpFeFFpFe Numerical Example: Local modulations  Eliminate cross-decoherence

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) Task 2: Equating decoherence rates Decaying Entangled state |  (t) i =e -J(t) |  (0) i Condition :  j,n (t)  j`,n` (t) 8 j,j`,n,n` J jj,nn (t)=2  s d  G jj,nn (  )F t,j,n (  +  j,n ) =J(t) G(  ) Ft()Ft() Ft()Ft() =

Fidelity Decoherence Matrix Elements Decoherence Matrix FFpFeFFpFe Numerical Example: Equating decoherence rates

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

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

Fidelity Decoherence Matrix Elements Decoherence Matrix FFpFeFFpFe Numerical Example: Optimal scheme |  (0) i =1/√2(|- i a |g i b +|g i a |- i b ) |- i j =1/√2(|1 i 2 -|2 i j )

Numerical Example: Summary

Suggested Experimental Setup: Multiple 40 Ca + Ions in Cavity Experimental parameters: Finesse ¼ D 5/2 a = 729 nm Cavity mode width  ¼ 12 GHz Single particle: No modulation: Lifetime = s With modulation:Lifetime ¼ 1.4 s Required impulsive phase modulation rate ~  Multiple ions in cavity: Position in cavity:  a -  b ¼ 15% Three level system: |g i = 4 2 S 1/2 |1 i = 3 2 D 3/2 |2 i = 3 2 D 5/2  1 /  2 = Kreuter et al. PRL (2004);Barton et al. PRA (2000) Single ion in cavity: Two ions in each cavity + Local modulations 1/√2(|g i a |1 i b -|1 i a |g i b ) = DFS

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 Conclusions Thank You !!!

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

Multilevel Cross-Decoherence No modulation Global modulation Creation of DFS G(  ) F t,j,n (  +  j ) G(  )  j,n (t)=   j,n (t)  j`,n` (t)  j,n (t) ¼  j`,n` (t) F t,j`,n (  +  j` ) G(  ) F t,j,n (  +  j ) F t,j`,n (  +  j` ) F t,j,n (  +  j )F t,j`,n (  +  j` )

Quasi-periodic Modulation  j,n (t)=  l  l e -i l t G(  ) F t,j,n (  ) l l+1 l+2 l+3 l-1 l-2 l-3 J(t)=2  l  l G(  + l )

General Bath Formalism J jj',nn' (t) = s 0 t dt' s 0 t' dt''  jj',nn' (t'-t'')K jj',nn' (t',t'')e i  j,n t'-i  j',n' t''  jj',nn' (t)= s d  G jj',nn' (  )e -i  t G jj',nn‘  ~ -2  k  k,j,n  * k,j',n'  k ) Bath Matrix Decoherence Matrix The Same Bath:  k,j,n =  k 8 j,n Separate Baths: (independent particles)  k,j,n  k,j`,n` =0 8 j  j`,n,n`,k Particle j coupled to modes {k j } Particle j` coupled to modes {k j` }  {k j } Å {k j` }= ;

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