Transitionless quantum driving in open quantum systems

Transcription

1 Shortcuts to Adiabaticity, Optimal Quantum Control, and Thermodynamics! Telluride, July 2014 Transitionless quantum driving in open quantum systems G Vacanti! R Fazio! S Montangero! G M Palma! M Paternostro! V Vedral!! New Journal of Physics! 16 (2014)

2 OUTLINE Berry s transitionless quantum driving! the rotating frame for unitary evolutions! adiabatic theorem for open quantum systems! transitionless quantum driving in open quantum systems! examples

3 Berry s transitionless quantum driving Ĥ 0 (t) ' n (t)i = E n (t) ' n (t)i if the initial state of the system is an instantaneous eigenstate of a time dependent Hamiltonian Ho it will remain the the corresponding eigenstate at time t as long as Ho varies slowly enough and there are no level crossing Ĥ(t) =Ĥ0(t)+Ĥ1(t) Ĥ(t) ' n (t)i = i@ t ' n (t)i transitionless quantum driving: add a new time dependent hamiltonian term H1 so that the state becomes an exact solution regardless of the speed of change of the hamiltonian

4 a classical analog a spin following a magnetic field! whose direction changes in time adiabatic fast

5 the rotating frame Ĥ(t) = X i,j iihi Ĥ(t) jihj Û(t) = X i ' i (t)ihi U(t) diagonalizes the hamiltonian Û 1 (t)ĥ(t)û(t) Ĥd(t) = X i E i (t) iihi

6 the time dependent Schoedinger eq. i d = Û 1 i. time dependent unitary transformation Ĥ d (t)+i@ t Û 1 (t)û(t)] i d = i@ t i d [Ĥd(t)+Ĥ0 d(t)+ĥ0 nd(t)] i d = i@ t i d. Ĥ 0 d(t) =i X t Û 1 (t)û(t) iihi = i X h ' i ' i i iihi, Ĥ 0 nd(t) =i X i6=j t Û 1 (t)û(t) jihj = i X i6=j h ' i ' j i iihj,

7 transitionless quantum driving Berry connection Ĥ 0 d(t) =i X t Û 1 (t)û(t) iihi = i X h ' i ' i i iihi, non adiabatic term Ĥ 0 nd(t) =i X i6=j t Û 1 (t)û(t) jihj = i X i6=j h ' i ' j i iihj, transitional quantum driving Ĥ tqd (t) = Û(t)Ĥ0 nd(t)û 1 (t).

8 adiabatic approximation in open quantum systems L[%] = i[ĥ(t),%]+1 2 NX j=1 (2ˆj(t)%ˆ ˆ (t) {%, j j (t)ˆj(t)}) due to the coupling of the system with the environment, the energy-difference between neighbouring eigenvalues of the Hamiltonian no longer provides the natural time-scale! with respect to which a time-dependent Hamiltonian could be considered to be slowlyvarying.!! adiabaticity of open systems is reached when the evolution of the state of a system occurs without mixing the various Jordan blocks into which L can be decomposed.! Sarandy Lidar PRA 71, (2005)

9 a matrix reppresentation of L define a time independent basis in the D2-dimensional space of the density matrices.this could consist, for example, the three Pauli matrices and the identity matrix in the case of a single spin-1/2. B {ˆi} i = {1,...,D 2 }. the density operator becomes a vector %ii =( 1, 2,..., D 2), the Lindblad operator becomes a super matrix L(t) %ii = %ii. j =Tr[ˆ j %] L jk (t) =Tr[ˆ j (L t[ˆk])].

10 Jordan decomposition Although the supermatrix L(t) might be non-hermitian, in which case it cannot be diagonalized in general, it is always possible to find a similarity transformation C(t) such that L(t) is written in the canonical Jordan form L J (t) =C 1 (t)l(t)c(t) = diag[j 1 (t),...,j N (t)], C(t) = NX M X D,µ (t)iihh,µ, =1 µ =1 L(t) D,µ (t)ii = D,µ 1(t)ii + (t) D,µ (t)ii, D,0 (t)ii represents the eigenvector of L(t)! corresponding to the the eigenvalue (t)

11 transitionless open dynamics formal analogy with the unitary case (L J + L 0 J + L 0 nd) %ii J = %ii J, L 0 J = X,µ iihh,µ Ċ 1 C,µ iihh,µ L 0 nd = X 6= 0,µ iihh,µ Ċ 1 C 0,µ 0 0 iihh 0,µ 0 0 transitionless quantum driving L tqd = CL 0 ndc 1. the driving term can be unitary (hamiltonian)! or non unitary (a quantum channel) for one dimensional Jordan blocks! the off diagonal! () () i t L t j t i t j t = matrix term of the correction term are λ λ () () () j i.

12 rotating jump operators and unitary driving γ ρ = Γˆ ργˆ Γˆ Γˆ [ ] () t () t () t () t ρ 2 2,, { } k k k k k k ˆ = ˆ k Γˆ ˆ Γˆ t = Uˆ t Γˆ Uˆ t k () () () Γˆ k in the rotating frame 0 () () () ρ = Uˆ t ρ t Uˆ t ρ = γ ρ = ΓρΓ ˆ ˆ { ΓΓ ˆ ˆ ρ } ˆ ˆ () () ϱ 2 2 k k k k, i iu t U t, k ˆ the quantum unitary driving Hˆ t = iuˆ t Uˆ t. tqd () () ()

13 example 1: single spin amplitude damping L ad [%] = 2 [2ˆn %ˆ+n {ˆn ˆ+n, %}] ˆn =(ˆ+n) = #i n h" g the direction and { #i amplitude damping along a! time dependent direction n = ( sin θ cos ϕ, sin θ sin ϕ, cos θ) ctively. In order to write explicitly b precession around the z axis B ˆ ˆ, σˆ, σˆ, σˆ θ ϕ ( ) x y z. precesses around th Hˆ () = ( ) tqd t n n σˆ, a magnetic eld which ϕ = ωt ϱ = i H ˆ ϱ tqd t, [ ] () tqd city. Indeed, the correction t () ˆ ˆ H t = iuu ˆ, tqd ubits. We start by

14 a two qubit example Γ ( ) ( ) = Uˆ 0 1 ˆ Uˆ ; Γ = Uˆ ˆ 0 1 Uˆ where U is a Hadamardt gate followed by a C-NOT ( )( ) the liuvillian has the following fixed point: ψ = + state 00, where ˆ represent let s generalise by assuming U is an arbitrary single qubit rotation followed by a C-NOT. In this case the ψ fixed point is! () t = ( cos θ () t 00 + sin θ () t 11 ) ow, the system will remain in the instantaneo by rotating q one can drag the fixed point

15 a two qubit example by varying slowly one can drag the fixed point such dragging can be achieved exactly with no constrains on speed by adding the following coherent driving: H tqd = iuu ˆ ˆ = iθ θ 0 0 i 0 0 iθ 0 0 iθ e written as ( ) H = iθ h.c. tqd