Suppressing decoherence and heating with quantum bang-bang controls David Vitali and Paolo Tombesi Dip. di Matematica e Fisica and Unità INFM, Università di Camerino, Italy

Decoherence is the main limiting factor for quantum information processing Many different routes to decoherence control: - Coding methods: quantum error correction codes error avoiding codes (i.e., decoherence free subspaces) - Dynamical methods: closed-loop (feedback) techniques open-loop techniques, i.e., the application of suitably shaped time-varying control fields acting on the system alone Open-loop techniques are able to decouple an open quantum system from any environment. They can be seen as the quantum generalization of refocusing techniques used in NMR to eliminate undesired dephasing due to local magnetic field fluctuations

Decoupling via quantum bang-bang control System S interacting with an environment B: H 0 = H S + H B + H SB Cyclic control Hamiltonian H 1 (t), acting on system only, with period T c H TOT (t) = H 0 +H 1 (t) The stroboscopic evolution at t N = NT c, in the infinitely fast control limit T c  0, N  ∞, t N = NT c = const., is driven by an average Hamiltonian where

quantum bang-bang controls: application of Hamiltonians with very large strength and for very small times (full strength/fast switching scheme)  Application of a piecewise constant decoupling operator U 1 (t) = g j, j∆t ≤ t ≤ (j+1)∆t G = {g j } = group of unitary operators with |G| elements g1g1 g2g2 g |G|..... ∆t=T c /|G| Applying the quantum bang-bang control is equivalent to symmetrize the evolution with respect to the group G (Viola, Knill, Lloyd 99, Zanardi 99)

The system is perfectly decoupled from the environment when the interaction is symmetrized to zero,  c (H SB ) = 0 The effective system evolution is then governed by H S =  c (H S ) = projection of H S into the commutant of G If sufficient symmetry conditions are satisfied (for example, G or its commutant are sufficiently large) this result can be used for implementing universal fault tolerant quantum computation, using appropriate tricks (Viola, Lloyd, Knill, 99) or encoding into noiseless subsystems (Viola, Knill, Lloyd 2000, Zanardi 2000)

Example 1: K qubits with linear dissipation Decoupling is obtained through symmetrization with respect to the tensor power of the Pauli group, G = {1,  i  (i) x,  i  (i) y,  i  (i) z }, |G| = 4 (Duan, Guo, 99, Viola, Knill, Lloyd, 99) Quantum bang-bang cycle = four  pulses:  x,  -z,  -x,  -z Example 2: oscillator (a) with linear dissipation H SB = a †  + a  †  Decoupling is obtained through symmetrization with respect to the group G = Z 2 = {1, P (parity)}, |G| = 2 (Vitali, Tombesi, 99) Bang-bang control = “parity kicks” by pulsing the oscillation frequency 00  0 +   TCTC   =   P = exp{i  a † a}

The most stringent condition for decoupling is on timing: The infinitely fast control limit T c  0 cannot be met exactly in practice How small has T c to be to achieve significant decoupling ? First estimates (Viola-Lloyd 98, Vitali-Tombesi 99) showed that the significant timescale is determined by environment typical timescale,  c ≈  c -1, inverse of the environment frequency cutoff  c,  c T c ≤ 1 The decoupling technique can be used to eliminate “slow” environments, i.e., typical non-Markovian situations Open questions: What is the effect of bath temperature ? May the scheme be used to decouple from a high temperature bath and for example suppress heating of the vibrational motion in a linear chain of trapped ions ?

Vibrational mode coupled to a thermal bath Environment described as a set of independent bosons (Caldeira-Leggett, 81) It can be decoupled using “parity kicks” (Example 2) No Markovian appoximation, no bath elimination  numerical solution with a bath of 201 oscillators, with frequencies equally spaced by ∆, in the interval   [0,  c ],  c = 2  0 ; ohmic bath  i = (  ∆/2π) 1/2,  i,  = damping rate We use the fact that coherent states are preserved either with or without the decoupling parity kicks with the appropriate unitary matrix L ij (t)

Effect of parity kicks on heating The linear chain is first cooled to the ground state and it is then heated by the high temperature bath at temperature T The vibrational mode reduced density matrix at time t is a thermal state with mean vibrational number = (t)

T = 1K, N(  0 ) = Vibrational heating is suppressed for  c T c /2  < 1, for any temperature !  0 = 10 Mhz,  = 0.1 Mhz,  c = 20 Mhz T = 100 mK, N(  0 ) = T = 10 mK, N(  0 ) =

Effect of parity kicks on decoherence Vibrational mode initially in a Schrödinger cat state, which is then heated by the high temperature bath at temperature T The vibrational mode reduced Wigner function at time t is (t) describes again heating L 00 (t) describes amplitude decay  (t): fringe visibility function, describing decoherence

T = 1K, N(  0 ) = At variance with heating, decoherence is well suppressed only at low T !  0 = 10 Mhz,  = 0.1 Mhz,  c = 20 Mhz T = 100 mK, N(  0 ) = T = 10 mK, N(  0 ) =

This is due to the thermal acceleration of decoherence. Decoherence is affected not only by the environment timescale 2  /  c, but also by the temperature-dependent timescale t dec ≈ [  N(  0 )] -1 Decoherence is suppressed by parity kicks only when T c < min{2  /  c, [  N(  0 )] -1 } On the contrary, heating is affected only by the cutoff frequency and it can be suppressed by parity kicks when T c < 2  /  c In fact, in the Markovian limit This quantum bang-bang control can be used to avoid heating of the center-of-mass motion in linear ion traps

Effect of parity kicks on the Wigner function of a Schrödinger cat state with  0 = √5, in a thermal bath with T = 10mK t = 0 t = 1/  No parity kicks With parity kicks, with  c T c /2  = 1/16