Quantum Speed Limit for Non-Markovian Dynamics

With RWA, non-Markovianity can indeed decrease QSL time τQSL.

“if someone draws the conclusions that non-Markovianity can decrease QSL time τQSL18, our results will provide important additions and amendments that with using RWA, non-Markovianity can indeed decrease QSL time τQSL.”

Deffner and Lutz derived geometric generalizations of both the Mandelstam-Tamm and Margolus-Levitin bounds.

“Deffner and Lutz [33] derived geometric generalizations of both, the Mandelstamm-Tamm bound as well as the Margolus-Levitin bound.”

Three independent approaches to quantifying the maximum speed of open quantum systems were proposed in 2013.

“In 2013 three letters proposed, in quick succession, three independent approaches of how to quantify the maximal quantum speed of systems interacting with their environments.”

For unitary dynamics induced by time-independent positive semi-definite Hamiltonians, the time-averaged trace norm of the generator equals the average energy.

“One easily convinces oneself that for unitary dynamics induced by time-independent, positive semi-definite Hamiltonians the time-averaged trace norm of the generator is identical to the average energy, L(ρ t ) tr = H [33].”

In figure 3(a), the quantum speed limit time is monotonically decreasing as a function of the coupling strength.

“As noted in [33] we observe that τ QSL is indeed monotonically decreasing as a function of the coupling strength, γ 0 .”

In the approach of reference [33], the bounds are not independent.

“However, in the approach chosen in [33] the bounds are not independent.”

The Hilbert-Schmidt bound is of Mandelstam-Tamm type because for Hamiltonian dynamics the Hilbert-Schmidt norm reduces to the energy variance.

“That equation ( 88) is a bound of the Mandelstam-Tamm type is obvious, since the Hilbert-Schmidt norm reduces for Hamiltonian dynamics to the variance of the energy [33].”

The operator, trace, and Hilbert-Schmidt bounds can be written together as a unified bound.

“Similarly to the case of isolated dynamics, see section 2, we obtain seemingly independent expressions for the quantum speed limit, which we can write together as [33] τ QSL = [sin (L(ρ 0 , ρ τ ))] 2 max 1 Λ op , 1 Λ tr , 1 Λ hs ,(89)”

For the damped Jaynes-Cummings model with a Lorentzian spectral density, the dynamics is analytically solvable and the decay rate is given by a specific expression.

“the dynamics is analytically solvable and the decay rate can be written as [33]”

Deffner and Lutz found that the quantum speed limit time is a monotonically decreasing function of the coupling ratio, suggesting that non-Markovian backflow can accelerate dynamics.

“Deffner and Lutz [33] then found that τ QSL is a monotonically decreasing function of γ 0 /ω 0 , which suggests that the non-Markovian backflow of information from the environment into the system can accelerate its dynamics.”

For the two-qubit state |Ψ1> = α|01> + sqrt(1-α^2)|10>, the QSL time ratio τ/τ_QSL is the same as in single-qubit cases, and reduction in QSL time (τ/τ_QSL > 1) occurs only when the environment has memory; otherwise τ/τ_QSL ≡ 1.

“Obviously, the QSL time ratio τ QSL of state | Ψ 1 〉 is the same as the exact form in single qubit cases discovered in Refs [30, 32], where the reduction in QSL time ( τ/τ QSL > 1) only occurs when the environment is of memory; otherwise the evolution will always be along the QSL time ( τ/τ QSL ≡ 1).”

In the damped Jaynes-Cummings model of a single qubit, the transition from no potential capacity of speedup to potential speedup (a reduction of QSL) is the critical point when the memoryless environment becomes of memory.

“It is found that in the damped JaynesCummings model of a single qubit, the transition from no potential capacity of speedup ( τ/τ QSL = 1) to potential speedup ( τ/τ QSL > 1), or say a reduction of QSL in quantum evolution is just the critical point when the memoryless environment becomes of memory [30, 32].”

A QSL time based on the Schatten p norm for an arbitrarily driven open system was presented by Deffner and Lutz.

“In particular, a QSL time based on the Schatten p norm for an arbitrarily driven open system was presented by Deffner and Lutz [30]”

Non-Markovianity in the dynamics process and the associated information backflow from the reservoir can lead to faster quantum evolution and hence to smaller QSLT.

“As noted in Refs. [25-27], the non-Markovianity in the dynamics process ( ρ 0 to ρ τ D ), and the associated information backflow from the reservoir, can lead to faster quantum evolution, and hence to smaller QSLT.”

A unified lower bound including both MT and ML types has been derived by Deffner and Lutz.

“A unified lower bound, including both MT and ML types, has been derived by Deffner and Lutz [25].”

The ML-type bound based on the operator norm (p = ∞) of the nonunitary generator provides the sharpest bound on the QSLT.

“And the ML-type bound based on the operator norm ( p = ∞ ) of the nonunitary generator provides the sharpest bound on the QSLT [25].”

In weak system-environment couplings, the QSLT equals the actual driving time, while strong system-environment couplings can reduce the QSLT below the actual driving time.

“the QSLT would equal to the actual driving time in the weak system-environment couplings, while the strong system-environment couplings can reduce the QSLT below the actual driving time [25].”

Our mechanism cannot be captured by the previous treatment with Lorentzian spectrum, where the lower limit of the frequency integration is artificially extended from 0 to -∞, and this approximation loses the availability of the bound state.

“Note that our mechanism cannot be captured by the previous treatment with Lorentzian spectrum [25, 26, 28, 33, 34], where the lower limit of the frequency integration in f ( t -τ ) is artificially extended from 0 to -∞ . This approximation is mathematically convenient but loses the availability of the bound state [40].”

For our TLS, the optimal pair of initial states to maximize N are ρ1,2(0)=|±><±|.

“For our TLS, it has been proven that the optimal pair of initial states to maximize N are ρ 1 , 2 (0) = |±〉〈±| [25]”

Non-Markovian effect characterized by non-Markovianity can speed up quantum evolution.

“Deffner and Lutz [25] showed that non-Markovian effect characterized by non-Markovianity [29-32] can speed up quantum evolution, which has been verified in different settings [26, 28, 33, 34].”

A unified bound of quantum speed limit time including both Mandelstam-Tamm and Margolus-Levitin types for non-Markovian dynamics can be formulated.

“It is shown that a unified bound of QSLT including both MT and ML types for non-Markovian dynamics can be formulated18.”

The unified bound for non-Markovian dynamics is applicable for a given driving time for pure initial states but is not feasible for mixed initial states.

“However, while this unified bound is applicable for a given driving time for the pure initial states, it is not feasible for mixed initial states.”

For a pure initial state, expression (7) recovers the unified bound of the quantum speed limit time obtained in Ref. 18.

“For a pure initial state ρτ = 0 = |ϕ0〉〈ϕ0|, the singular value , then . We can find that the expression (7) recovers the unified bound of the QSLT obtained in Ref. 18, which confirms the validity of our results.”

The speed-up reported here is physically equivalent to the quantum speed limit behavior predicted in Ref. [14], although this work studies field dynamics while Ref. [14] studied atoms.

“Before we continue the discussion with the experimental results, note that the speed-up proposed here is physically equivalent to the behavior of the QSL predicted in Ref. [14].However, here we are interested in the dynamics of the field, whereas Ref. [14] focused on the atoms.”

Reference [14] analyzes the damped Jaynes-Cummings model, which describes many cavity QED systems.

“Reference [14] analyzes the dynamics of the damped Jaynes-Cummings model, which describes many cavity QED systems.”

A recent theoretical study hinted that speed-ups of quantum evolution could be observed when an open quantum system undergoes environmental changes.

“On the theoretical side, a recent study [14] hinted at the possibility of observing speed-ups of the quantum evolution if an open quantum system is subject to environmental changes.”

The observed speed-up agrees with theoretical predictions that stronger coupling speeds up evolution in cavity QED, and it differs from adiabatically eliminated cases where the atom-number dependence does not appear in the nonclassical correlation function.

“This is in agreement with theoretical predictions [14] that observe a speed-up as the coupling increases in a cavity QED system and in contrast to the dependence of N in the adiabatically eliminated cases stated above, that do not show in the nonclassical correlation function.”

Numerous experiments have demonstrated strong coupling between N-V centers and the modes of gallium phosphide PCC, silicon nitride PCC, and PCC in monocrystalline diamond.

“On the other hand, numerous experiments have successively demonstrated the strong coupling between N-V centers and the modes of gallium phosphide PCC [35], silicon nitride PCC [36], and PCC in monocrystalline diamond [37], respectively.”

The composite planar PCC-N-V system is a suitable platform to test the results.

“These advances imply that this composite system is a suitable platform to test the results.”