Open Dynamics under Rapid Repeated Interaction Daniel Grimmer David Layden Eduardo Martin-Martinez Robert Mann University of Waterloo Institute for Quantum Computing June 16, 2016 Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 1 / 19
What happens in a single interaction? Some ancilla is picked (from an ensemble) and engages with the system: Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 2 / 19
What happens in a single interaction? Then, depending on the ancilla chosen, the joint system evolves unitarily: Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 3 / 19
What happens in a single interaction? Following the interaction, the ancilla is discarded: Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 4 / 19
What happens in a single interaction? Finally we average over all ancillas which could have been chosen: Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 5 / 19
What happens in a single interaction? Optional: The ancillas can be reused if they are cleaned. Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 6 / 19
Some Applicable Scenarios Through a Gas: NMR: Nuclear spin interacting with electrons Gravitational Decoherence 1 Atom bombarded by a series of atoms/light pulses: or Entanglement Farming 2 In a Gas: Cavity bombarded by atoms: 1 D. Kafri, J.M. Taylor, G. J. Milburn; New Journal of Physics, Volume 16, June 2014 2 E. Matrin-Martinez, E. Brown, W. Donnelly, A. Kempf; Phys. Rev. A 88, 052310 (2013) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 7 / 19
Results • In fast interaction limit ( δ t → 0), evolution is unitary • Decoherence related to classical/quantum ‘uncertainty’ • Applications - Decoherence in Media - Measurement Problem - Quantum Information Processing - Quantum Thermodynamics Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 8 / 19
Interpolation Scheme Single interaction: ¯ � U δ t , k ( δ t )( · ⊗ ρ A k ) U δ t , k ( δ t ) † � φ ( δ t ) = Σ k p k Tr A k System evolves under repeated interactions, at t = n δ t we have, ρ S ( n δ t ) = ¯ [¯ [ ... ¯ = ¯ � n � � � � � � � � �� � � φ δ t φ δ t φ δ t [ ρ S (0) ... φ δ t ρ S (0) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 9 / 19
Interpolation Scheme Single interaction: ¯ � U δ t , k ( δ t )( · ⊗ ρ A k ) U δ t , k ( δ t ) † � φ ( δ t ) = Σ k p k Tr A k System evolves under repeated interactions, at t = n δ t we have, ρ S ( n δ t ) = ¯ [¯ [ ... ¯ = ¯ � n � � � � � � � � �� � � φ δ t φ δ t φ δ t [ ρ S (0) ... φ δ t ρ S (0) Issue: Only know about discrete time points. Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 9 / 19
Interpolation Scheme Single interaction: ¯ � U δ t , k ( δ t )( · ⊗ ρ A k ) U δ t , k ( δ t ) † � φ ( δ t ) = Σ k p k Tr A k System evolves under repeated interactions, at t = n δ t we have, ρ S ( n δ t ) = ¯ [¯ [ ... ¯ = ¯ � n � � � � � � � � �� � � φ δ t φ δ t φ δ t [ ρ S (0) ... φ δ t ρ S (0) Issue: Only know about discrete time points. Solution: We interpolate the system state as, � � ρ S ( t ) = Ω δ t ( t ) ρ S (0) with exact matching at discrete time points, Ω δ t ( n δ t ) = ¯ φ ( δ t ) n . Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 9 / 19
Interpolation Scheme Single interaction: ¯ � U δ t , k ( δ t )( · ⊗ ρ A k ) U δ t , k ( δ t ) † � φ ( δ t ) = Σ k p k Tr A k System evolves under repeated interactions, at t = n δ t we have, ρ S ( n δ t ) = ¯ [¯ [ ... ¯ = ¯ � n � � � � � � � � �� � � φ δ t φ δ t φ δ t [ ρ S (0) ... φ δ t ρ S (0) Issue: Only know about discrete time points. Solution: We interpolate the system state as, � � ρ S ( t ) = Ω δ t ( t ) ρ S (0) with exact matching at discrete time points, Ω δ t ( n δ t ) = ¯ φ ( δ t ) n . Issue: There are many choices for such a interpolation scheme. Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 9 / 19
Interpolation Scheme Single interaction: ¯ � U δ t , k ( δ t )( · ⊗ ρ A k ) U δ t , k ( δ t ) † � φ ( δ t ) = Σ k p k Tr A k System evolves under repeated interactions, at t = n δ t we have, ρ S ( n δ t ) = ¯ [¯ [ ... ¯ = ¯ � n � � � � � � � � �� � � φ δ t φ δ t φ δ t [ ρ S (0) ... φ δ t ρ S (0) Issue: Only know about discrete time points. Solution: We interpolate the system state as, � � ρ S ( t ) = Ω δ t ( t ) ρ S (0) with exact matching at discrete time points, Ω δ t ( n δ t ) = ¯ φ ( δ t ) n . Issue: There are many choices for such a interpolation scheme. Solution: Restrict to be Markovian, Ω δ t ( t ) = e L δ t t . Yields unique L δ t = 1 � ¯ � δ t log φ ( δ t ) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 9 / 19
Master Equation This effective Liouvillian L δ t can be expanded as a series in δ t generates time evolution for the interpolation scheme, d dt ρ S ( t ) = L δ t [ ρ S ( t )] = L 0 [ ρ S ( t )] + δ t L 1 [ ρ S ( t )] + δ t 2 L 2 [ ρ S ( t )] + . . . For rapid interactions, δ t E / � ≪ 1, we can truncate. Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 10 / 19
Master Equation This effective Liouvillian L δ t can be expanded as a series in δ t generates time evolution for the interpolation scheme, d dt ρ S ( t ) = L δ t [ ρ S ( t )] = L 0 [ ρ S ( t )] + δ t L 1 [ ρ S ( t )] + δ t 2 L 2 [ ρ S ( t )] + . . . For rapid interactions, δ t E / � ≪ 1, we can truncate. We take the general system-ancilla interaction Hamiltonian, H k ( ξ ) = H S ⊗ 1 + 1 ⊗ H A k + H SA k ( ξ ) where ξ = t /δ t and use it to explicitly find the forms of the coefficients L 0 and L 1 . Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 10 / 19
Zeroth Order Liouvillian To zeroth order the evolution is entirely unitary! L 0 [ · ] = − i � [ H eff(0) , · ] 3 D. Layden, E. Matrin-Martinez and A. Kempf; Phys. Rev. A 93, 040301(R) (2016) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 11 / 19
Zeroth Order Liouvillian To zeroth order the evolution is entirely unitary! L 0 [ · ] = − i � [ H eff(0) , · ] where H eff(0) = H S + H (0) . Free evolution plus interaction effects, H (0) = � ρ A k ∫ 1 � � p k Tr A k 0 d ξ H SA k ( ξ ) , k 3 D. Layden, E. Matrin-Martinez and A. Kempf; Phys. Rev. A 93, 040301(R) (2016) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 11 / 19
Zeroth Order Liouvillian To zeroth order the evolution is entirely unitary! L 0 [ · ] = − i � [ H eff(0) , · ] where H eff(0) = H S + H (0) . Free evolution plus interaction effects, H (0) = � ρ A k ∫ 1 � � p k Tr A k 0 d ξ H SA k ( ξ ) , k The system and ancilla do not become entangled at leading order in δ t . Interpretation: Pushing vs. Talking Ancillas push the system but do not have time to talk (entangle) with it. 3 3 D. Layden, E. Matrin-Martinez and A. Kempf; Phys. Rev. A 93, 040301(R) (2016) Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 11 / 19
First Order Liouvillian First subleading dynamics introduce leading order dissipative effect as well as subleading unitary dynamics. L 1 [ · ] = − i � [ H eff(1) , · ] + 1 2 D [ · ] Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 12 / 19
First Order Liouvillian First subleading dynamics introduce leading order dissipative effect as well as subleading unitary dynamics. L 1 [ · ] = − i � [ H eff(1) , · ] + 1 2 D [ · ] The new subleading unitary term is H eff(1) = H (1) + H (1) + H (1) where 1 2 3 � 1 � − i � � � H (1) � = p k G 1 � [ H SA k ( ξ ) , H S ] G 1 ( X ) = ( ξ − 1 / 2) X ( ξ ) d ξ 1 k 0 k � 1 � − i � � � H (1) � = � [ H SA k ( ξ ) , H A k ] p k G 2 G 2 ( X ) = ξ X ( ξ ) d ξ 2 k 0 k � 1 � ξ 1 G 3 ( Y ) = 1 � − i � � � H (1) � = � [ H SA k ( ξ 1 ) , H SA k ( ξ 2 )] d ξ 1 d ξ 2 Y ( ξ 1 , ξ 2 ) p k G 3 3 2 k 0 0 k Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 12 / 19
Leading Order Dissipative Terms The leading order dissipation is D [ · ] = 1 � 2 [ H (0) , [ H (0) , · ]] − 1 � � � p k Tr A k [ G 0 ( H SA k ) , [ G 0 ( H SA k ) , · ⊗ ρ A k ]] � 2 k Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 13 / 19
Leading Order Dissipative Terms The leading order dissipation is D [ · ] = 1 � 2 [ H (0) , [ H (0) , · ]] − 1 � � � p k Tr A k [ G 0 ( H SA k ) , [ G 0 ( H SA k ) , · ⊗ ρ A k ]] � 2 k Dissipation is related to ‘uncertainty’ of interaction, � � � D [ ρ S ] = p k Tr A k Var( C )[ ρ S ⊗ ρ A k ] k where Var( C ) = �� C k 2 �� − �� C k �� 2 Variance: � � Generalized Average: �� C k �� [ ρ SA k ] = ρ A k ⊗ Σ l p l Tr A l C l [ ρ SA l ] C k [ ρ SA k ] = ( i � ) − 1 [ G 0 ( H SA k ) , ρ SA k ] Diff. Evolution Op: Grimmer (UW IQC) Repeated Interactions arXiv:1605.04302 June 16, 2016 13 / 19
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