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Cited by 156Quantum optics and atomic interactionsQuantum Mechanics and ApplicationsAdvanced Thermodynamics and Statistical Mechanics

Quantum speedup in a memory environment

Open access
Xu, Zhen-Yu, Luo, Shunlong, Yang, W. L., Liu, Chen, Zhu, Shiqun·2014·Physical Review A
DOI 10.1103/physreva.89.012307
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Abstract

Memory (non-Markovian) effect is found to be able to accelerate quantum evolution [S. Deffner and E. Lutz, Phys. Rev. Lett. 111, 010402 (2013).]. In this work, for an atom in a structured reservoir, we show that the mechanism for the speedup is not only related to non-Markovianity but also to the population of excited states under a given driving time. In other words, it is the competition between non-Markovianity and population of excited states that ultimately determines the acceleration of quantum evolution in memory environments. A potential experimental realization for verifying the above phenomena is discussed by using a nitrogen-vacancy center embedded in a planar photonic crystal cavity under current technologies.

AI snapshot

Written from the paper’s full text.

TL;DR

For a two-level atom in a photonic crystal cavity, the paper shows that quantum speedup in a memory environment is governed not only by non-Markovianity but by its competition with the excited-state population at the driving time. It also proposes a nitrogen-vacancy center in a planar photonic crystal cavity as an experimental platform.

Key findings

  • The quantum speed limit time \(\tau_{\rm QSL}\) decreases as the atomic transition frequency approaches and enters the photonic band edge, signaling speedup; far outside the band gap, e.g. \(\delta/\beta=10\), \(\tau_{\rm QSL}\) equals the actual driving time \(\tau\).
  • The speedup is determined by the competition between non-Markovianity \(N(\Lambda)\) and the excited-state population \(P_\tau\); the transition from no-speedup to speedup occurs when the environment changes from Markovian to non-Markovian, since \(N(\Lambda)=0\) gives \(\tau_{\rm QSL}=\tau\).
  • For short driving time \(\tau=1\), speedup occurs only inside the band edge (\(\delta/\beta<0\)); for \(\tau=3,5,10\), the speedup region extends outside the band edge.
  • A possible N-V center in a planar photonic crystal cavity realization is discussed with \(d\sim 4\times10^{-30}\,\mathrm{C\cdot m}\), \(\omega_0\sim 2.9\,\mathrm{PHz}\), \(\beta\sim 1\,\mathrm{GHz}\), and Stark shift \(\Delta\sim 5\,\mathrm{GHz}\), giving \(\Delta/\beta\simeq 5\), enough to observe the no-speedup to speedup transition.

Methods

  • The model is a two-level atom embedded in an ideal isotropic photonic crystal cavity with a photonic band gap; the atom is coupled to the radiation field, and the reduced dynamics is obtained from a master equation with a decoherence function \(b_t\) calculated by inverse Laplace transform.
  • The quantum speed limit time is derived using the Deffner-Lutz unified bound for open systems, with initial state taken as the excited state \(\rho=|1\rangle\langle1|\).
  • Non-Markovianity is quantified by the Breuer-Laine-Piilo trace-distance backflow measure; because no general analytical method exists for the optimal initial state pair, numerical calculations over 2000 random pairs are used, with \((|0\rangle,|1\rangle)\) found optimal in the example \(\delta/\beta=-10\).

Limitations

  • The paper states that there is no general analytical method to find the optimal initial state pair for non-Markovianity, so numerical calculations are used.
  • The paper states that the work mainly focuses on a specific open-system model and that further study of more complicated non-Markovian systems, such as post-Markovian master equations, would be of interest.
  • The experimental realization is proposed and analyzed with current-technology parameters, not performed; the model also assumes an ideal isotropic photonic crystal and specific approximations, as evident from the text.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physreva.89.012307

Cites

Statistics

Nothing to recompute.

Claims the paper makes

0 of 19 cited sources checked · 0 supported · 0 flagged

  • Findingcitation
    § V. POSSIBLE EXPERIMENTAL REALIZATION · p. 4

    For the PCC-N-V system, the optical transition dipole moment d∼4×10^-30 C·m and typical transition frequency ω0∼2.9 PHz lead to β∼1 GHz.

    “For the relevant experimental parameters, the optical transition dipole moment and the typical transition frequency of the N-V center are observed to be d ∼ 4 × 10 -30 C · m [31, 32] and ω 0 ∼ 2 . 9PHz [33], which lead to β ∼ 1GHz .”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [31] A. M. Zaitsev, Optical Properties of Diamond (SpringerVerlag, Berlin, 2001). source

      Not checked in the free run.

    • cites [32] F. C. Waldermann et al. , Diamond and Related Materials 16 , 1887 (2007). source

      Not checked in the free run.

    • cites [33] N. B. Manson et al. , Proc. SPIE 7611 , 761105 (2010). source

      Not checked in the free run.

  • Findingown data
    § III. QUANTUM SPEED LIMIT TIME · p. 2

    The deeper the atomic transition frequency lies in the band edge, the smaller the QSL time will be in a general trend.

    “Obviously, the deeper the atomic transition frequency lies in the band edge, the smaller the QSL time will be in a general trend.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.
  • Findingreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 3

    For this optimal state pair, the trace distance of the evolved states can be written as D_t = |b_t|^2 = P_t.

    “It is easy to check that for this optimal state pairs, the trace distance of the evolved states can be written as D t = | b t | 2 = P t ( D t denotes the optimal trace distance in the following).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.
  • Findingown data
    § I. INTRODUCTION · p. 1

    In a two-level atom embedded in a photonic crystal cavity, the transition from no-speedup to speedup of quantum evolution is observed when the atomic frequency is approaching and going inside the band edge.

    “In this setting, the transition from no-speedup to speedup of quantum evolution is observed when the atomic frequency is approaching and going inside the band edge.”
    • Also stated in: § III. QUANTUM SPEED LIMIT TIME · p. 2 · boosted
      “In Fig. 1, we depict the QSL time τ QSL in unit of 1 /β (blue solid curves) together with different actual driving time τ (red dashed lines) as functions of δ/β, where the transition from no-speedup to speedup is clearly shown.”
    • Also stated in: § VI. DISCUSSION AND CONCLUSION · p. 5 · plain
      “In summary, for a model of two-level atom embedded in a PBG structured reservoir, the transition from nospeedup to speedup of quantum evolution has been observed.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.
  • Findingreasoning
    § I. INTRODUCTION · p. 1

    The QSL time is related to two quantities: non-Markovianity during the driving time and excited population at τ.

    “QSL time is found to be related to two quantities: non-Markovianity during the driving time and excited population at τ .”
    • Also stated in: § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 3 · boosted
      “Clearly, the QSL time is related to two quantities: the non-Markovianity N (Λ) within the driving time and the atomic excited population P τ .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.
  • May combine two claims. 7
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 4

    For driving time τ=10, when the atomic transition frequency is deeply inside the bandgap edge (δ/β=-10), the non-Markovianity becomes small, and τ_QSL is mainly dependent on Pτ.

    “Finally, if the atomic transition frequency is deeply inside the bandgap edge (e.g., δ/β = -10), the non-Markovianity becomes small, and τ QSL is mainly dependent on P τ .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.

  • May combine two claims.

    Findingreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 4

    For driving time τ=10, when the atom transition frequency is far outside the bandgap edge (δ/β=10), the population will not be trapped (Pτ=10≃0) and non-Markovianity approaches zero, therefore τ_QSL=τ.

    “When the atom transition frequency is far outside the bandgap edge frequency (e.g., δ/β = 10), the population will not be trapped, i.e., P τ =10 ≃ 0 , and the non-Markovianity also approaches zero, therefore, τ QSL = τ .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.

  • May combine two claims.

    Findingreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 4

    For driving time τ=10, when the atomic transition frequency is around the bandgap edge (δ/β ∈ [-1,0]), τ_QSL is dependent on both the population Pτ and non-Markovianity N(Λ), because the atomic excited population is trapped and the non-Markovianity is strong.

    “When the atomic transition frequency is around the bandgap edge (e.g., δ/β ∈ [ -1 , 0]), τ QSL is dependent on both the population P τ and non-Markovianity N (Λ) , for the atomic excited population is trapped, i.e., P τ =10 = 0 , and the non-Markovianity is strong.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § V. POSSIBLE EXPERIMENTAL REALIZATION · p. 4

    The initial state of NV center can be prepared by 532nm laser pulse with high fidelity.

    “In experiment, the initial state of NV center can be prepared by 532nm laser pulse with high fidelity [29, 30]”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [29] F. Jelezko et al. , Phys. Rev. Lett. 93 , 130501 (2004). source

      Not checked in the free run.

    • cites [30] M. W. Doherty et al. , Phys. Rep. 528 , 1 (2013). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § V. POSSIBLE EXPERIMENTAL REALIZATION · p. 4

    The Stark shift can tune energy levels with Δ∼5 GHz, so Δ/β≃5, which is large enough to observe the no-speedup to speedup transition.

    “We can tune the energy levels by the Stark shift with ∆ ∼ 5GHz [34] . Therefore, ∆ /β ≃ 5 , which is large enough to obverse the no-speedup to speedup transition (e.g., in Fig. 1(b), δ/β : 5 → 0).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [34] Ph. Tamarat et al. , Phys. Rev. Lett. 97 , 083002 (2006). source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § V. POSSIBLE EXPERIMENTAL REALIZATION · p. 4

    The final state can be tested with standard tomography technique.

    “and the finial state can be tested with standard tomography technique [2].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [2]

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 4

    For the leaky cavity model, when ̃N(Λ)=0 (Markovian), τ_QSL=τ; consequently, the non-Markovian effect becomes the unique reason for speeding up quantum evolution in this model.

    “In this case, when ˜ N (Λ) = 0 (Markovian), τ QSL = τ. Consequently, the nonMarkovian effect becomes the unique reason for speeding up quantum evolution in this model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 4

    When the driving time is short (τ=1), the speedup only takes place within the bandgap edge because the transition point from Markovian to non-Markovian environment just occurs inside the bandgap edge.

    “It is therefore easy to account for the phenomenon that the speedup only takes place within the bandgap edge when τ = 1 [Fig. 1(a)], for the transition point from Markovian to non-Markovian environment just occurs inside the bandgap edge when the driving time is short [Fig. 3(a)].”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.

  • Classified as probably not a substantial claim.

    Premisecitation
    § IV. MECHANISM FOR THE SPEEDUP OF QUANTUM EVOLUTION · p. 3

    There exists no general analytical method to find the optimal initial state pair.

    “There exists no general analytical method to find the optimal initial state pair ( ρ 1 , ρ 2 ) [24].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [24] S. Wißmann et al. , Phys. Rev. A 86 , 062108 (2012). source

      Not checked in the free run.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    For a damped Jaynes-Cummings model with an atom resonantly coupled to a leaky single mode cavity, non-Markovian effect can speed up quantum evolution, and QSL decreases when the non-Markovian effect becomes stronger.

    “Interestingly, it is discovered that non-Markovian effect can speed up the quantum evolution with a damped Jaynes-Cummings model for an atom resonantly coupled to a leaky single mode cavity: QSL decreases when the non-Markovian effect becomes stronger [16].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [16] S. Deffner and E. Lutz, Phys. Rev. Lett. 111 , 010402 (2013). source

      Not checked in the free run.

10 of 10

  • The authors state this as an assumption.

    The photon dispersion relation near the band edge is approximated by ω_k = ω_c + A(k-k0)^2 for an ideal isotropic photonic crystal.

    “In this work, we consider an ideal photonic crystal with isotropic photon dispersion relation approximated by ω k = ω c + A ( k -k 0 ) 2 near the band edge [20], where ω c is the upper band-edge frequency, and A = ω c /k 2 0 with k 0 ≃ ω c /c being a specific wave vector with respect to the point-group symmetry of the PCC.”
1 of 1

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The quantum speed limit time can be written as the total time times one minus the excited-state population divided by twice the non-Markovianity measure plus one minus the excited-state population.

    “It is then easy to see [146] that the quantum speed limit time, equation ( 97) reads, τ QSL = τ (1 -P τ ) 2 N + (1 -P τ ) .(100)”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [146]

  • 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 &gt; 1) only occurs when the environment is of memory; otherwise the evolution will always be along the QSL time ( τ/τ QSL ≡ 1).”

    From Quantum-speed-limit time for multiqubit open systems (2015) · cites this paper as [32]

  • 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 &gt; 1), or say a reduction of QSL in quantum evolution is just the critical point when the memoryless environment becomes of memory [30, 32].”

    From Quantum-speed-limit time for multiqubit open systems (2015) · cites this paper as [32]

  • 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.”

    From Classical-driving-assisted quantum speed-up (2015) · cites this paper as [27]

  • 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].”

    From Mechanism for quantum speedup in open quantum systems (2016) · cites this paper as [33]

The citation lists haven’t been fetched yet

References 75 · Cited by 156

Research works printed but not linked: 3

  1. M. A. Nielsen, 2000, Quantum Computation and Quantum Information
  2. L. Mandelstam, 1945, J. Phys. (USSR), 9, 249
  3. J. D. Joannopoulos, 2011, Photonic Crystals: Molding the Flow of Light, 10.1016/S0038-1098(96)00716-8 · DOI · Paper page

Authors
Xu, Z.-Y., Luo, S., Yang, W. L., Liu, C., & Zhu, S.
Affiliations
  • Chinese Academy of Sciences (CN)
  • Soochow University (CN)
ORCID
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2014-01-08
Volume
89
Issue
1
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.89.012307
OpenAlex
W1992837188
Cited by
156
References
75
Citations by year
2026: 2 · 2025: 10 · 2024: 12 · 2023: 15 · 2022: 14 · 2021: 12 · 2020: 18 · 2019: 17 · 2018: 9 · 2017: 16 · 2016: 14 · 2015: 10 · 2014: 7
FWCI
7.80
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • arXiv (Cornell University)
  • Bulletin of the American Physical Society
Funders
National Natural Science Foundation of China (11204196, 11274351, 11074184, 11375259, 61134008)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum optics and atomic interactions
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
Keywords
quantum advantage, non-Markovianity, memory effects, excited state population, structured reservoir, nitrogen-vacancy centers, photonic crystal cavities, open quantum systems

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