# Tunable Non-Markovianity for Bosonic Quantum Memristors

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

## 3. Results

#### 3.1. Dynamical Non-Markovianity (DnM)

#### 3.2. Time-Dependent Decay Rate

#### 3.3. Bosonic Quantum Memristor

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DnM | Dynamical non-Markovianity |

## Appendix A

## References

- Li, C.-F.; Guo, G.-C.; Pillo, J. Non-Markovian quantum dynamics: What does it mean? EPL
**2019**, 127, 50001. [Google Scholar] [CrossRef] - Li, C.-F.; Guo, G.-C.; Pillo, J. Non-Markovian quantum dynamics: What is it good for? EPL
**2020**, 128, 30001. [Google Scholar] [CrossRef] - Lee, H.; Cheng, Y.-C.; Fleming, G.R. Coherence Dynamics in Photosynthesis: Protein Protection of Excitonic Coherence. Science
**2007**, 316, 1462. [Google Scholar] [CrossRef] - Chin, A.W.; Datta, A.; Caruso, F.; Huelga, S.F.; Plenio, M.B. Noise-assisted energy transfer in quantum networks and light-harvesting complexes? New J. Phys.
**2010**, 12, 065002. [Google Scholar] [CrossRef] - Fleming, G.R.; Huelga, S.F.; Plenio, M.B. Focus on quantum effects and noise in biomolecules. New J. Phys.
**2011**, 13, 115002. [Google Scholar] [CrossRef] - Alex, W.C.; Susana, F.H.; Martin, B.P. Quantum Metrology in Non-Markovian Environments. Phys. Rev. Lett.
**2012**, 109, 233601. [Google Scholar] - Mirkin, N.; Larocca, M.; Wisniacki, D. Quantum metrology in a non-Markovian quantum evolution. Phys. Rev. A
**2020**, 102, 022618. [Google Scholar] [CrossRef] - Sweke, R.; Sanz, M.; Sinayskiy, I.; Petruccione, F.; Solano, E. Digital quantum simulation of many-body non-Markovian dynamics. Phys. Rev. A
**2016**, 94, 022317. [Google Scholar] [CrossRef] - Barreiro, J.T.; Müller, M.; Schindler, P.; Nigg, D.; Monz, T.; Chwalla, M.; Hennrich, M.; Roos, C.F.; Zoller, P.; Blatt, R. An open-system quantum simulator with trapped ions. Nature
**2011**, 470, 486–491. [Google Scholar] [CrossRef] - Pfeiffer, P.; Egusquiza, I.L.; Di Ventra, M.; Sanz, M.; Solano, E. Quantum memristors. Sci. Rep.
**2016**, 6, 29507. [Google Scholar] [CrossRef] [PubMed] - Spagnolo, M.; Morris, J.; Piacentini, S.; Antesberger, M.; Massa, F.; Crespi, A.; Ceccarelli, F.; Osellame, R.; Walther, P. Experimental photonic quantum memristor. Nat. Photon.
**2022**, 16, 318–323. [Google Scholar] [CrossRef] - Sanz, M.; Lamata, L.; Solano, E. Quantum memristors in quantum photonics. APL Photonics
**2018**, 3, 080801. [Google Scholar] [CrossRef] - Shevchenko, S.N.; Pershin, Y.V.; Nori, F. Qubit-Based Memcapacitors and Meminductors. Phys. Rev. Appl.
**2016**, 6, 014006. [Google Scholar] [CrossRef] - Norambuena, S.; Torres, F.; Di Ventra, M.; Coto, R. Polariton-Based Quantum Memristors. Phys. Rev. Appl.
**2022**, 17, 024056. [Google Scholar] [CrossRef] - Pershin, Y.V.; Di Ventra, M. Neuromorphic quantum computing. Proc. IEEE
**2012**, 100, 2071. [Google Scholar] [CrossRef] - Pehle, C.; Wetterich, C. Digital quantum simulation of many-body non-Markovian dynamics. Phys. Rev. E
**2022**, 106, 045311. [Google Scholar] [CrossRef] [PubMed] - Xu, H.; Krisn, A.T.; Verstraelen, W.; Liew, T.C.H.; Ghosh, S. Superpolynomial quantum enhancement in polaritonic neuromorphic computing. Phys. Rev. B
**2021**, 103, 195302. [Google Scholar] [CrossRef] - De Vega, I.; Aloso, D. Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys.
**2017**, 89, 015001. [Google Scholar] [CrossRef] - Rivas, Á.; FHuelga, S.; BPlenio, M. Quantum non-Markovianity: Characterization, quantification and detection. Rep. Prog. Phys.
**2014**, 77, 094001. [Google Scholar] [CrossRef] [PubMed] - Breuer, H.P. Foundations and measures of quantum non-Markovianity. J. Phys. B: At. Mol. Opt. Phys.
**2012**, 45, 154001. [Google Scholar] [CrossRef] - Breuer, H.P.; Laine, E.M.; Piilo, J. Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems. Phys. Rev. Lett.
**2009**, 103, 210401. [Google Scholar] [CrossRef] - Rivas, Á.; Huelga, S.F.; Plenio, M.B. Entanglement and Non-Markovianity of Quantum Evolutions. Phys. Rev. Lett.
**2010**, 105, 050403. [Google Scholar] [CrossRef] [PubMed] - Luchnikov, I.A.; Vintskevich, S.V.; Grigoriev, D.A.; Filippov, S.N. Machine Learning Non-Markovian Quantum Dynamics. Phys. Rev. Lett.
**2020**, 124, 140502. [Google Scholar] [CrossRef] [PubMed] - Bastidas, V.M.; Kyaw, T.H.; Tangpanitanon, J.; Romero, G.; Kwek, L.C.; Angelakis, D.G. Floquet stroboscopic divisibility in non-Markovian dynamics. New J. Phys.
**2018**, 20, 093004. [Google Scholar] [CrossRef] - Liu, B.-H.; Li, L.; Huang, Y.-F.; Li, C.-F.; Guo, G.-C.; Laine, E.-M.; Breuer, H.-P.; Piilo, J. Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems. Nat. Phys.
**2011**, 7, 931–934. [Google Scholar] [CrossRef] - Bernardes, N.K.; Cuevas, A.; Orieux, A.; Monken, C.H.; Mataloni, P.; Sciarrino, F.; Santos, M.F. Experimental observation of weak non-Markovianity. Sci. Rep.
**2015**, 5, 17520. [Google Scholar] [CrossRef] - Li, B.-W.; Mei, Q.-X.; Wu, Y.-K.; Cai, M.-L.; Wang, L.; Yao, L.; Zhou, Z.-C.; Duan, L.-M. Observation of Non-Markovian Spin Dynamics in a Jaynes-Cummings-Hubbard Model using a Trapped-Ion Quantum Simulator. Phys. Rev. Lett.
**2022**, 129, 140501. [Google Scholar] [CrossRef] [PubMed] - García-Pérez, G.; Rossi, M.A.; Maniscalco, S. IBM Q Experience as a versatile experimental testbed for simulating open quantum systems. NPJ Quantum Inf.
**2020**, 6, 1. [Google Scholar] [CrossRef] - Chen, X.-Y.; Zhang, N.-N.; He, W.-T.; Kong, X.-Y.; Tao, M.-J.; Deng, F.-G.; Ai, Q.; Long, G.-L. Global correlation and local information flows in controllable non-Markovian open quantum dynamics. NPJ Quantum Inf.
**2022**, 8, 22. [Google Scholar] [CrossRef] - Tavis, M.; Cummings, W. Exact Solution for an N-Molecule-radiation-Field Hamiltonian. Phys. Rev.
**1968**, 170, 379. [Google Scholar] [CrossRef] - Retzker, A.; Solano, E.; Reznik, E. Tavis-Cummings model and collective multiqubit entanglement in trapped ions. Phys. Rev. A
**2007**, 75, 022312. [Google Scholar] [CrossRef] - Jäger, S.B.; Schmit, T.; Morigi, G.; Holl, M.J.; Betzholz, R. Lindblad Master Equations for Quantum Systems Coupled to Dissipative Bosonic Modes. Phys. Rev. Lett.
**2022**, 129, 063601. [Google Scholar] [CrossRef] [PubMed] - Van Woerkom, D.J.; Scarlino, P.; Ungerer, J.H.; Müller, C.; Koski, J.V.; Landig, A.J.; Reichl, C.; Wegscheider, W.; Ihn, T.; Ensslin, K.; et al. Microwave Photon-Mediated Interactions between Semiconductor Qubits. Phys. Rev. X
**2018**, 8, 041018. [Google Scholar] [CrossRef] - Casabone, B.; Friebe, K.; Brandstätter, B.; Schüppert, K.; Blatt, R.; Northup, T.E. Enhanced Quantum Interface with Collective Ion-Cavity Coupling. Phys. Rev. Lett.
**2020**, 114, 023602. [Google Scholar] [CrossRef] - Wang, Z.; Li, H.-K.; Feng, W.; Song, X.-H.; Song, C.; Liu, W.-X.; Guo, Q.-J.; Zhang, X.; Dong, H.; Zheng, D.-N.; et al. Controllable Switching between Superradiant and Subradiant States in a 10-qubit Superconducting Circuit. Phys. Rev. Lett.
**2015**, 124, 013601. [Google Scholar] [CrossRef] - Johansson, J.; Nation, P.; Nori, F. QuTiP 2: A Python framework for the dynamics of open quantum systems. Comput. Phys. Commun.
**2013**, 184, 1234. [Google Scholar] [CrossRef] - Grossmann, F.; Dittrich, T.; Jung, P.; Hänggi, P. Coherent destruction of tunneling. Phys. Rev. Lett.
**1991**, 67, 516. [Google Scholar] [CrossRef] - Neu, P.; Silbey, R.J. Tunneling in a cavity. Phys. Rev. A
**1996**, 54, 5323. [Google Scholar] [CrossRef] - Luo, X.; Li, L.; You, L.; Wu, B. Coherent destruction of tunneling and dark Floquet state. New J. Phys.
**2014**, 16, 013007. [Google Scholar] [CrossRef] - Hu, M.-L.; Lian, H.-L. Geometric quantum discord and non-Markovianity of structured reservoirs. Ann. Phys.
**2015**, 362, 795. [Google Scholar] [CrossRef] - Salmiletho, J.; Deppe, F.; Di Ventra, M.; Sanz, M.; Solano, E. Quantum Memristors with Superconducting Circuits. Sci. Rep.
**2017**, 7, 42044. [Google Scholar] [CrossRef] [PubMed] - Guo, Y.-M.; Albarrán-Arriagada, F.; Alaeian, H.; Solano, E.; Barrios, G.A. Quantum Memristors with Quantum Computers. Phys. Rev. Appl.
**2022**, 18, 024082. [Google Scholar] [CrossRef]

**Figure 1.**Diagram of the model: a cavity (bosonic mode) coupled to a set of qubits embedded in a Markovian reservoir. Each auxiliary qubit can be dynamically tuned, and the cavity can be classically driven.

**Figure 2.**Dynamical non-Markovainity of the resonator. (

**a**) One-qubit case. (

**b**) Five-qubit case. In both cases, the decay rate of qubit and resonator is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005$. We consider resonator frequency ${\omega}_{R}=1$, qubit frequency ${\omega}_{Q}/{\omega}_{R}\in [0.5,1.5]$, the coupling strength $g/{\omega}_{R}\in [0,0.1]$ and the initial state $|{\psi}_{0}\rangle =|{1}_{R}{0}_{Q}\rangle $.

**Figure 3.**(

**a**) The DnM of the resonator in terms of the number of qubits n. (

**b**) The log–log plot of DnM and the number of qubits. For (

**a**,

**b**), we consider three cases, $g=0.01,0.05,0.1$, with the resonant condition ${\omega}_{Q}={\omega}_{R}=1$. (

**c**) The exponent k of the power–law dependence as a function of the coupling strength $g/{\omega}_{R}$, with qubit and resonator in resonance. (

**d**) The exponent k of the power–law dependence as a function of the frequency of the qubit ${\omega}_{Q}/{\omega}_{R}$ and a fixed coupling strength $g/{\omega}_{R}=0.05$. For all cases, we consider decay rates ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005{\omega}_{R}$, the initial state of the resonator $|{\psi}_{0}\rangle =|{1}_{R}\rangle $, and all the qubits initialized in the ground state.

**Figure 4.**The non-Markovainity of the resonator. (

**a**) One-qubit case. (

**b**) Five-qubit case. Parameters: In both cases, the decaying rate of qubit and resonator is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005{\omega}_{R}$. The driving frequency of qubit ${\mu}_{Q}/{\omega}_{R}\in [0,1]$, the driving amplitude of the qubit ${\mathsf{\Omega}}_{Q}/{\omega}_{R}\in [0,1]$, the qubit frequency ${\omega}_{Q}/{\omega}_{R}=1$, and the coupling strength $g/{\omega}_{R}=0.1$. The initial state is $|{\psi}_{0}\rangle =|{1}_{R}{0}_{Q}\rangle $.

**Figure 5.**The minimum (

**a**) and maximum (

**b**) of DnM for the resonator with a different number of auxiliary qubits. In both cases, the decaying rate is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005{\omega}_{R}$. The frequency of qubit ${\omega}_{Q}/{\omega}_{R}=1$, and the initial state is of resonator $|{\psi}_{0}\rangle =|{1}_{R}\rangle $. The qubits are all initialized in the ground state.

**Figure 6.**Transition from non-Markovian to Markovian dynamics by changing the driving frequency over the auxiliary qubits. Parameters: the driving amplitude of qubit is ${\mathsf{\Omega}}_{q}=0.5$, the number qubit $n=1$, decaying rate is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005$, frequency ${\omega}_{Q}={\omega}_{R}$, and coupling strength $g=0.05$.

**Figure 7.**Dynamical non-Markovainity of the resonator varies with decaying rate. In the blue line, the decay rate of resonator ${\mathrm{\Gamma}}_{R}=0.005$, and qubit decay rate ${\mathrm{\Gamma}}_{Q}=\mathrm{\Gamma}$. In the green line, the decay rate of qubit ${\mathrm{\Gamma}}_{Q}=0.005$, and resonator decay rate ${\mathrm{\Gamma}}_{R}=\mathrm{\Gamma}$. In the orange line, the decay rate of qubit and resonator are the same, ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=\mathrm{\Gamma}$. We consider that qubit and resonator are in resonance (${\omega}_{Q}={\omega}_{R}=1$), the coupling strength $g/{\omega}_{R}=0.05$, and the initial state $|{\psi}_{0}\rangle =|{1}_{R}{0}_{Q}\rangle $.

**Figure 8.**Trace distance of the resonator and the DnM under different driving frequencies. Top (

**a**), the blue line—the frequency of driving ${\mu}_{1}=0.419$, the green line—the resonator’s decaying rate is constant $\mathrm{\Gamma}\left(t\right)=0.005$. (

**b**) The blue line—the frequency of driving ${\mu}_{2}=0.20$, the green line—the resonator’s decaying rate is ${\mathrm{\Gamma}}_{1}\left(t\right)=0.05\left(sin\left(0.023t\right)+0.09\right)$. (

**c**) The blue line—the frequency of driving ${\mu}_{3}=1$. The green line—the resonator’s decaying rate is ${\mathrm{\Gamma}}_{1}\left(t\right)=0.25\left(sin\left(0.079t\right)+0.021\right)$. Bottom (

**d**), the DnM of resonator in different driving frequencies ${\mu}_{Q}\in (0,1)$. Parameters: the number qubit $n=1$, decaying rate is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005$, frequency ${\omega}_{Q}={\omega}_{R}$, coupling strength $g=0.05$, and driving amplitude ${\mathsf{\Omega}}_{Q}=0.5{\omega}_{R}$.

**Figure 9.**Memristive behavior; the green line shows the dynamics when the auxiliary qubits are not driven and off-resonant, and the blue curve is when we add a driving over the auxiliary qubits. (

**a**) Larger-DnM case—the number of qubits $n=1$ and driving frequency ${\mu}_{c}=1$. (

**b**) Medium-DnM case—the number of qubits $n=1$ and driving frequency ${\mu}_{c}=0.2$. (

**c**) Larger-DnM case—the number of qubits $n=5$ and driving frequency ${\mu}_{c}=1$. (

**d**) Medium-DnM case—the number of qubits $n=5$ and driving frequency ${\mu}_{c}=0.2$. Parameters: the driving amplitude of qubit is ${\mathsf{\Omega}}_{q}=0.5$, decaying rate is ${\mathrm{\Gamma}}_{Q}={\mathrm{\Gamma}}_{R}=0.005$, frequency ${\omega}_{Q}={\omega}_{R}$, coupling strength $g=0.05$, the driving amplitude of cavity ${\mathsf{\Omega}}_{c}=0.2$, and frequency ${\mu}_{c}=0.5$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tang, J.-L.; Alvarado Barrios, G.; Solano, E.; Albarrán-Arriagada, F.
Tunable Non-Markovianity for Bosonic Quantum Memristors. *Entropy* **2023**, *25*, 756.
https://doi.org/10.3390/e25050756

**AMA Style**

Tang J-L, Alvarado Barrios G, Solano E, Albarrán-Arriagada F.
Tunable Non-Markovianity for Bosonic Quantum Memristors. *Entropy*. 2023; 25(5):756.
https://doi.org/10.3390/e25050756

**Chicago/Turabian Style**

Tang, Jia-Liang, Gabriel Alvarado Barrios, Enrique Solano, and Francisco Albarrán-Arriagada.
2023. "Tunable Non-Markovianity for Bosonic Quantum Memristors" *Entropy* 25, no. 5: 756.
https://doi.org/10.3390/e25050756