Open Access
Add to BibSonomy
Abstract
We study dynamical quantum phase transitions in a 2-qubit system interacting with a transverse field and a quantized bosonic environment in the context of open quantum systems. By applying the stochastic Schrödinger equation approach, the model with a spin-boson type of coupling can be solved numerically. It is observed that the dynamics of the rate function of the Loschmidt echo in a 2-qubit system within a finite size of Hilbert space exhibit nonanalyticity when the direction of the transverse field coupled to the system is under a sudden quench. Moreover, we demonstrate that the memory time of the environment and the coupling strength between the system and the transverse field can jointly impact the dynamics of the rate function. We also supply a semi-classical explanation to bridge the dynamical quantum phase transitions in many-body systems and the non-Markovian dynamics of open quantum systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Conventional quantum phase transitions (QPT) identify the nonanalyticity in the ground state energy of complex quantum systems [1]. It takes place in equilibrium many-body systems, when the control parameter of the Hamiltonian, such as external magnetic fields, is changed slowly across a critical point, and is accompanied by the observation of singular behavior of observables. In the context of many-body systems and equilibrium physics, the major difficulty in studying quantum phase transitions arises from the complicated structure of energy diagrams of many-body systems as the function of the control parameter. A large number of pioneering works in this field, both theoretical and experimental, have been dedicated in the past decades [2–9] to address this difficulty. However, recent progress in studying open quantum systems (OQS), e.g., coherent quantum control, quantum measurement, quantum coherence and correlations [10], observation of non-Markovian dynamics in experiments, and steady-state dissipative quantum phase transitions [11,12], motivates researchers to understand the nonequilibrium dynamics in open quantum systems [13–17]. The consequent difficulties in examining OQSs arise from the lack of methods to deal with the infinitely large Hilbert space of the environment and in obtaining the dynamics of the reduced density matrix of the central system. Many techniques have been developed in response: Feynman path integrals [18], Nakajima–Zwanzig equation [19,20], Lindblad master equation [21], Redfield master equation [22], stochastic Schrödinger equation approach (SSE) [23,24], general non-Markovian time-local master equation [25–29], etc.
It is known that the temporal evolution of a many-body system is usually not included in the context of many-body systems, until a singular behavior in the time domain was discovered in Refs.[30–42], namely, the dynamical quantum phase transition (DQPT). It characterizes a new type of critical phenomena in the infinitely large many-body system, when the control parameter undergoes a sudden quench and can perfectly serve as a witness of universal aspects in the context of QPTs and OQSs jointly. The experimental setup in the original work is a 1D transverse field Ising model. For example, effective spin-1/2 particle can be encoded in the states of a $^{171}\textrm {Yb}^{+}$ ion, and the nearest neighbor coupling strength is approximately proportional to $1/d^\alpha$ with a tunable $\alpha$ [43–45]. Moreover, recent progress reveals the existence of continuous quantum phase transitions in the quantum Rabi model (QRM), which indicates that the size of the system is no longer the only criteria for observing the sharp nonanalytical behavior. Consequently, in a finite-sized system such as the QRM, it is also possible to observe DQPTs [46]. These finite component systems supply a good platform to investigate the overlap of conventional QPTs and the temporal evolution of physical observables, as well as the applications in quantum techniques, e.g., quantum measurement and quantum feedback control schemes. However, the above-mentioned discussions take place in a framework of infinitely large Hilbert space. Naturally, a question arises: could DQPTs be observed in finite-sized Hilbert-space systems? To answer this question, we consider a strongly coupled 2-qubit system, placed in an external field in the $XY$ plane, simultaneously interacting with a quantum environment-a spin-boson type of coupling (SBM).
Despite the lack of exact solutions for the SBM [4–9], we apply the stochastic Schrödinger equation (SSE) to derive the temporal evolution of the reduced density matrix of the central system. Particularly, we will discuss the impacts of the sudden quench of parameters, the prepared initial states, the coupling strength within the 2 qubits, and the interaction between the system and the environment. By properly choosing the parameters of the spectral function of the environment, we will also investigate the existence of DQPTs in a non-Markovian regime and within the Markov limit. The motivation is 2-fold: (1) In the strong coupling regime, non-Markovian dynamics characterize the evolution of the system without the limit of the Born-Markov approximation; (2) To apply the language of OQSs to study the collective behavior of the environment. Although strong coupling systems can be studied in both Markovian and non-Markovian regimes, DQPTs provide an additional aspect to examine the differences between the 2 processes. A semi-classical analysis on the competition between 2 mechanisms is demonstrated at the end of the discussion.
2. Non-Markovian dynamics of the spin-boson model
The well-known spin-boson model (SBM) has attracted considerable interests during recent decades due to its simple experimental setup and the extreme difficulty in obtaining the exact mathematical solution, and its wide applications in a variety of schemes in quantum optics [5], e.g., quantum coherent control on strongly coupled systems [47], quantum tunnelling in a double-well system [48,49], quantum dissipative systems with external laser pumps and the evolution of open quantum systems coupled to a finite-temperature environment. In this paper, we consider a 2-qubit system coupled to a $XY$ plane external field [44], interacting with a bosonic environment whose Hamiltonian can generally be written as in the interaction picture (setting $\hbar =1$)
When considering the 0 temperature case, the above correlation function can be simplified as $\alpha (t,s) = \int \frac {d\omega }{\pi }J(\omega )e^{-i\omega t}$. Without loss of generality, we consider the Lorentzian spectral density
where $\Omega$ is the central frequency of the environment, $\gamma$ defines the spectral width of the coupling, $1/\gamma$ indicates the memory time of the environment and $\Gamma$ is the global coupling strength. Furthermore, we assume that the environment is decoupled from the qubit initially. Then the corresponding zero-temperature correlation function readsThe reason for choosing the above Ornstein-Uhlenbeck type correlation function is that we look to observe different nonanalytical behaviors when the environment shifts from a Markovian to a non-Markovian regime. SSE approach is employed here to describe the temporal evolution of the qubit system. After mapping all environmental modes to a Gaussian stochastic process, $b_k|z_k\rangle = z_k|z_k\rangle$, the closed deterministic Schrödinger equation of the system and environment turns out to a SSE about a stochastic state vector, $\psi _z = \langle z|\Psi _{\rm tot}\rangle$, with $|z\rangle = \otimes _k|z_k\rangle$, in the Hilbert space of system
whereThe stochastic Gaussian process $z_t^*$ satisfies the following correlation relationships: $\mathcal {M}(z_t^*z_s)=\alpha (t,s)$ and $\mathcal {M}(z_tz_s)=0$. Here, the symbol $\mathcal {M}(\cdot )$ means the ensemble average. Therefore, every noisy evolution of $\psi _z$ is called a trajectory and the reduced density matrix of the system can be revealed by averaging over all trajectories, $\rho (t)=\mathcal {M}(|\psi _z\rangle \langle \psi _z|)$. In addition, $\bar {O}=\int _0^t ds\ \alpha (t,s)O(t,s)$ can be numerically determined by the evolution equation and the initial condition of the $O$ operator:
Although the $O$ operator of the SBM is complicated in terms of noise, it can be decomposed using the ascending order of noises basis, $O=\sum _k \int ds_1{\cdots }ds_k (\prod _1^k z_{s_j}^*) O_k$, with the noise-free operator factor $O_k$ of the kth order noise term. Therefore, Eq. (8) can be broken down into a set of hierarchical differential equations of operators $O_k$ [25,50],
3. Results
Next, we will numerically show the evidence of DQPTs in the SBM. In most OQSs, the central system is in a mixed state once the coupling with the environment is on. In this article, we use the Loschmidt echo rate function in [38] to describe the dynamical critical behavior appeared in the system, which reads
where $\rho (0)$ is associated with the parameter $p=0$, and $\rho (t)$ is the state of the central system when $p$ is suddenly changed to $1$. Due to the coupling with the environment, the central system is in a mixed state, which can be revealed as $\rho (t) = \mathcal {M}(|\psi _z\rangle \langle \psi _z|)$. In our scheme, the parameter $p$ controlling the direction of the external field changes suddenly from $p=0$ to $p=1$, indicating a quench that the external field is aligned from the $X$ direction to the $Y$ direction quickly.3.1 Impacts of memory time
Memory time is the crucial feature to categorize the types of environment. In the Ornstein-Uhlenbeck correlation function the parameter $1/\gamma$ indicates the memory time. When $\gamma$ shifts from a finite number to $\infty$, the environment is transferred from the non-Markovian to the Markovian regime. In OQSs, the Born-Markov approximation indicates 2 assumptions: (1) the operational time is much longer than the system relaxation time; (2) the number of degrees of freedom of the environment is much larger compared to that of the system. However, if the system and its environment are studied as a closed system, the dynamics of the system must consider the feedback from the counterpart environment. From this prospect, the memory time indicates not only the correlation length, but also the size of the Hilbert space of the entire system, because it takes time to receive the back-flow information in a large strongly coupled system. Consequently, in our discussion, the increase of the size of the system often used when studying DQPTs will be replaced by the memory time. So we focus on the model of a transverse field 2-qubit system coupled to a quantized environment and investigate the impact of the memory time on the critical dynamics of the Loschmidt echo $r(t)$. In this section, the initial state is prepared as $|\psi _0\rangle = (|11\rangle - |10\rangle -|01\rangle + |00\rangle )/2$.
In Fig. 1, the dynamics of the Loschmidt echo rate function are compared between varying magnitudes of the parameter $\gamma$, from $0.01$ to $5$. It is observed that the oscillations persist on a timescale for all decay rates $\gamma$, while the decreasing memory time leads to a damping of cusps. Furthermore, DQPTs are observed in the strong non-Markovian regime when $\gamma =0.01$ (solid black) and $\gamma = 0.05$ (solid red). The numerical results imply that the memory time indeed is related to the correlation length of the entire system, and that such a nonanalyticity existing in the many-body system can be observed in a smaller sized central system, a portion of the entire system.
Fig. 1. Dynamics of the Loschmidt echo rate function with different memory time factor $\gamma$. The coupling strength with the transverse field $\Delta =3$. When $\gamma$ takes the values of $0.01$ (black-solid), $0.05$ (red solid), $0.5$ (blue solid), $1$ (black dashed), $2$ (red dashed), and $5$ (blue dashed), the environment has the length of memory time changed from long to short. Other parameters are chosen as: $\omega =1$, $\Gamma =1$, $\Omega =0$.
Then, we study the dynamics of the rate function under a continuous change of the memory time factor $1/\gamma$, plotted in Fig. 2. In Figs. 2(a) and 2(b), the coupling strength $\Delta$ is set as $2$ and $1$ respectively. Note that the change of $\Delta$ will affect the eigen energy of the system and consequently modulate the frequency of periodic oscillations. But for a particular $\Delta$, the dynamics of the rate function still persist on a time scale. From Figs. 2(a) and 2(b), it is observed that there is a critical value of $\gamma$, by which 2 sets of dynamics are distinguished. When $\gamma$ is small, sharp peaks of the rate function emerge. Once $\gamma$ goes through the critical point, the dynamics of the rate functions is likely independent from the change of $\gamma$ and the cusps disappear.
Fig. 2. Dynamics of the Loschmidt echo with continuous changing $\gamma$, from strong non-Markovian to Markovian regime. In (a), the coupling strength is set to $\Delta = 2$ and in (b), $\Delta =1$. Other parameters are chosen as: $\omega =1$, $\Gamma =1$, $\Omega =0$.
3.2 Impacts of coupling strength
Next, we investigate the impacts of the coupling strength between the 2-qubit system and the external transverse field. As shown in the Hamiltonian in Eq. (1), the coupling strength $\Delta$ controls the phase of the system. When the ratio $\Delta /\omega \xrightarrow {}\infty$, the ground state of the system is pointing in the $XY$ plane, and the direction is determined by a $p$ factor. While the ratio $\Delta /\omega \xrightarrow {}0$, the ground state will be aligned in the $Z$-direction. As mentioned, we prepare the initial state as $|\psi _0\rangle = (|11\rangle - |10\rangle -|01\rangle + |00\rangle )/2$, then turn the external field from the $X$ direction to the $Y$ direction completely.
In Fig. 3, the dynamics of the rate function under different $\Delta$ values are compared in Markovian and non-Markovian conditions. In Fig. 3(a), the memory time related parameter $\gamma =2$ represents a close-to-Markov regime. In Fig. 3(b), $\gamma$ is set to $1$, a regime closer to the non-Markovian case. We observe that the height of peaks of the rate function increases significantly when $\gamma$ shifts from a Markovian regime to a non-Markovian one. In addition, there are higher nonanalytical peaks when the coupling strength is strong, around $\Delta =5$. It indicates that the emergence of nonanalyticity is a result of 2 mechanisms simultaneously. As discussed above, the nonanalyticity, which can be observed in many-body systems when the size of the system increases toward infinity, can also emerge in the strongly coupled OQS when the memory time and coupling strength satisfy some critical conditions together. In addition, we notice that the coupling strength $\Delta$ can be applied to modulate the frequency of the cusps, which can be useful in designing quantum measurement schemes.
Fig. 3. Dynamics of the Loschmidt echo rate function with different values of coupling strength, $\Delta = 5$ (black solid), $2$ (red solid), $1$ (blue solid), $0.5$ (green solid). In order to investigate the relationships between the coupling strength and the memory time, we set $\gamma =2$ in (a) and $\gamma =1$ in (b). DQPTs can be observed when the coupling with the transverse field is strong. Other parameters are chosen as: $\omega =1$, $\Gamma =1$, $\Omega =0$.
To further investigate the impacts due to the coupling strength, the rate function dynamics under continuous changing $\Delta$ is plotted in Fig. 4. In Fig. 4(a), for $\gamma =0.5$, there is a strong non-Markovian regime. As the result of the 2 control parameters $\Delta$ and $\gamma$, the dynamics of the rate function are distinct when $\Delta$ is located on the 2 sides of the critical point $\Delta _c =1$. When $\Delta >1$, the ground state of the system is aligned with the Y direction and the nonanalyticity of rate function can be observed. While $\Delta <1$, the $Z$-direction field takes the dominant role and thus there are no cusps in the dynamics. Moreover, when the coupling is weak, this model can be equivalently broken down to the Jaynes-Cummings model with the rotating wave approximation. Therefore, we can only see the oscillations, but not cusps.
Fig. 4. Dynamics of the rate function with continuous changing $\Delta$. The memory time related parameter $\gamma =0.5$. (a) When the model slowly changes from a weak to a strong regime with $\Delta$ changing from $0$ to $5$, 2 modes of dynamics of the rate function are observed even though the environment is non-Markovian at $\gamma =0.5$. (b) The environment is close to the Markovian limit $\gamma =5$. Other parameters are chosen as: $\omega =1$, $\Gamma =1$, $\Omega =0$.
However, if $\gamma$ takes a large value and the environment is Markovian, as plotted in Fig. 4(b), there is no nonanalyticity even when $\Delta =5$. To quantitatively characterize the competitions between the 2 mechanisms, we introduce a phenomenological quantity $\Delta /\gamma$ to categorize the 2 distinct modes of dynamics. In order to display the increasing trend of the discontinuity of cusps of the dynamics of the rate function, we practically set the size of cusps as the threshold, when $\frac {\Delta }{\gamma } = 1$. As a result, when the ratio $\frac {\Delta }{\gamma } > 1$, the dynamics show periodic oscillations. Generally, stronger coupling strength and longer memory time mean more nonanalytical behavior. On the contrary, if the ratio $\frac {\Delta }{\gamma } < 1$, the dynamics will be smooth and continuous.
4. Conclusions
We have shown that dynamical quantum phase transitions can be observed in open quantum systems, where the central system has a small Hilbert space dimension, such as a few-qubit system. Particularly, the central system is placed in a transverse field and coupled to a quantized bosonic environment, where the coupling between system and environment is of the spin-boson type. The sudden quench in our model is realized by changing the transverse field direction from the $X$ direction to the $Y$ direction. Depending on the coupling strength with the classical field $\Delta$, dynamical quantum phase transitions are observed when nonanalyticity behavior in the rate function emerges in a strong coupling regime only. Otherwise, the dynamics of the rate function follow dissipative dynamics. However, we also demonstrate that dynamical quantum phase transitions emerging in open quantum systems are not solely determined by the coupling strength with the transverse field, but also by the memory time of the environment. When the memory time $1/\gamma$ is very short and the environment is close to the Markovian limit, there is no presence of cusps in the rate function. On the contrary, in the non-Markovian regime where $1/\gamma$ is approaching infinity, nonanalyticity in the rate function is clear. A phenomenological explanation reveals the connection between the conventional many-body systems and open quantum systems, that strong coupling strength and long memory time together indicate the long correlation length of the entire system plus the environment Hamiltonian. Therefore, the critical point of the ratio $\Delta /\gamma$ can effectively categorize the 2 distinct modes of the dynamics of the rate function. Owing to the universality and the relevance of the spin-boson model in a variety of experimental realizations, our results open paths for the exploration of dynamical quantum phase transitions in small-sized open quantum systems, and to the study of nonanalyticity in nonequilibrium conditions and its applications in developing innovative methods to prepare quantum states, manipulate quantum gates, conduct error correction in quantum computing, etc.
Funding
Institutional Support of Research and Creativity (ISRC) grant provided by New York Institute of Technology.; US Department of Education Title III grant (P031C160038).
Acknowledgment
D.D. is partially supported by the Joseph E. Schuh scholarship at Saint Peter’s University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. Sachdev, Quantum phase transitions (Cambridge University, 2011).
2. R. Puebla, M.-J. Hwang, J. Casanova, and M. B. Plenio, “Probing the dynamics of a superradiant quantum phase transition with a single trapped ion,” Phys. Rev. Lett. 118, 073001 (2017). [CrossRef]
3. H.-J. Zhu, K. Xu, G.-F. Zhang, and W.-M. Liu, “Finite-component multicriticality at the superradiant quantum phase transition,” Phys. Rev. Lett. 125, 050402 (2020). [CrossRef]
4. M.-J. Hwang, R. Puebla, and M. B. Plenio, “Quantum phase transition and universal dynamics in the Rabi model,” Phys. Rev. Lett. 115, 180404 (2015). [CrossRef]
5. N. Lambert, S. Ahmed, M. Cirio, and F. Nori, “Modelling the ultra-strongly coupled spin-boson model with unphysical modes,” Nat. Commun. 10, 3721 (2019). [CrossRef]
6. L. Magazzù, P. Forn-Díaz, R. Belyansky, J.-L. Orgiazzi, M. A. Yurtalan, M. R. Otto, A. Lupascu, C. M. Wilson, and M. Grifoni, “Probing the strongly driven spin-boson model in a superconducting quantum circuit,” Nat. Commun. 9, 1403 (2018). [CrossRef]
7. Q.-J. Tong, J.-H. An, H.-G. Luo, and C. H. Oh, “Quantum phase transition in the delocalized regime of the spin-boson model,” Phys. Rev. B 84, 174301 (2011). [CrossRef]
8. A. Winter and H. Rieger, “Quantum phase transition and correlations in the multi-spin-boson model,” Phys. Rev. B 90, 224401 (2014). [CrossRef]
9. Y.-A. Yan and J. Shao, “Stochastic description of quantum Brownian dynamics,” Front. Phys. 11, 110309 (2016). [CrossRef]
10. M.-L. Hu, X. Hu, J. Wang, Y. Peng, Y.-R. Zhang, and H. Fan, “Quantum coherence and geometric quantum discord,” Phys. Rep. 762-764, 1–100 (2018). [CrossRef]
11. M.-J. Hwang, P. Rabl, and M. B. Plenio, “Dissipative phase transition in the open quantum Rabi model,” Phys. Rev. A 97, 013825 (2018). [CrossRef]
12. P.-G. Zhong, C. Li, Y. Wang, J. Song, S.-T. Liu, Y.-Y. Jiang, and Y. Xia, “Quantum phase transitions triggered by a four-level atomic system in dissipative environments,” Phys. Rev. A 99, 043829 (2019). [CrossRef]
13. H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford University, 2010).
14. S. Gröblacher, A. Trubarov, N. Prigge, G. D. Cole, M. Aspelmeyer, and J. Eisert, “Observation of non-Markovian micromechanical brownian motion,” Nat. Commun. 6, 7606 (2015). [CrossRef]
15. Q. Ding, P. Zhao, Y. Ma, and Y. Chen, “Impact of the central frequency of environment on non-Markovian dynamics in piezoelectric optomechanical devices,” Sci. Rep. 11, 1814 (2021). [CrossRef]
16. R. Carballeira, D. Dolgitzer, P. Zhao, D. Zeng, and Y. Chen, “Stochastic Schrödinger equation derivation of non-Markovian two-time correlation functions,” Sci. Rep. 11, 11828 (2021). [CrossRef]
17. T. Ma, Y. Chen, T. Chen, S. R. Hedemann, and T. Yu, “Crossover between non-Markovian and Markovian dynamics induced by a hierarchical environment,” Phys. Rev. A 90, 042108 (2014). [CrossRef]
18. R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948). [CrossRef]
19. S. Nakajima, “On quantum theory of transport phenomena: Steady diffusion,” Prog. Theor. Phys. 20, 948–959 (1958). [CrossRef]
20. R. Zwanzig, “Ensemble method in the theory of irreversibility,” J. Chem. Phys. 33, 1338–1341 (1960). [CrossRef]
21. D. Manzano, “A short introduction to the Lindblad master equation,” AIP Adv. 10, 025106 (2020). [CrossRef]
22. A. Redfield, “The theory of relaxation processes,” Adv. Magn. Opt. Reson. 1, 1–32 (1965). [CrossRef]
23. L. Bouten, M. Guta, and H. Maassen, “Stochastic Schrödinger equations,” J. Phys. A: Math. Gen. 37, 3189–3209 (2004). [CrossRef]
24. L. Diósi, N. Gisin, and W. T. Strunz, “Non-Markovian quantum state diffusion,” Phys. Rev. A 58, 1699–1712 (1998). [CrossRef]
25. Y. Chen, J. Q. You, and T. Yu, “Exact non-Markovian master equations for multiple qubit systems: Quantum-trajectory approach,” Phys. Rev. A 90, 052104 (2014). [CrossRef]
26. Y. Chen and T. Yu, Non-Markovian Dynamics of Qubit Systems: Quantum-State Diffusion Equations Versus Master Equations (Springer International Publishing, 2016), chap. 25.
27. Y. Chen, J. Q. You, and T. Yu, “Non-Markovian quantum interference in multilevel quantum systems: Exact master equation approach,” Quantum Inf. Comput. 18, 1261–1271 (2018). [CrossRef]
28. Y. Chen, Q. Ding, W. Shi, J. Jun, and T. Yu, “Exact entanglement dynamics mediated by leaky optical cavities,” J. Phys. B: At., Mol. Opt. Phys. 53, 125501 (2020). [CrossRef]
29. Y. Qiao, J. Zhang, Y. Chen, J. Jing, and S. Zhu, “Quantumness protection for open systems in a double-layer environment,” Sci. China: Phys., Mech. Astron. 63, 250312 (2020). [CrossRef]
30. M. Heyl, “Dynamical quantum phase transitions in systems with broken-symmetry phases,” Phys. Rev. Lett. 113, 205701 (2014). [CrossRef]
31. M. Heyl and J. C. Budich, “Dynamical topological quantum phase transitions for mixed states,” Phys. Rev. B 96, 180304 (2017). [CrossRef]
32. U. Divakaran, S. Sharma, and A. Dutta, “Tuning the presence of dynamical phase transitions in a generalized xy spin chain,” Phys. Rev. E 93, 052133 (2016). [CrossRef]
33. S. Bandyopadhyay, S. Laha, U. Bhattacharya, and A. Dutta, “Exploring the possibilities of dynamical quantum phase transitions in the presence of a Markovian bath,” Sci. Rep. 8, 11921 (2018). [CrossRef]
34. A. Khatun and S. M. Bhattacharjee, “Boundaries and unphysical fixed points in dynamical quantum phase transitions,” Phys. Rev. Lett. 123, 160603 (2019). [CrossRef]
35. R. Jafari, “Dynamical quantum phase transition and quasi particle excitation,” Sci. Rep. 9, 2871 (2019). [CrossRef]
36. P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B. P. Lanyon, M. Heyl, R. Blatt, and C. F. Roos, “Direct observation of dynamical quantum phase transitions in an interacting many-body system,” Phys. Rev. Lett. 119, 080501 (2017). [CrossRef]
37. K. Xu, Z.-H. Sun, W. Liu, Y.-R. Zhang, H. Li, H. Dong, W. Ren, P. Zhang, F. Nori, D. Zheng, H. Fan, and H. Wang, “Probing dynamical phase transitions with a superconducting quantum simulator,” Sci. Adv. 6, eaba4935 (2020). [CrossRef]
38. V. Link and W. T. Strunz, “Dynamical phase transitions in dissipative quantum dynamics with quantum optical realization,” Phys. Rev. Lett. 125, 143602 (2020). [CrossRef]
39. T. H. Kyaw, V. M. Bastidas, J. Tangpanitanon, G. Romero, and L.-C. Kwek, “Dynamical quantum phase transitions and non-markovian dynamics,” Phys. Rev. A 101, 012111 (2020). [CrossRef]
40. H. Lang, Y. Chen, Q. Hong, and H. Fan, “Dynamical quantum phase transition for mixed states in open systems,” Phys. Rev. B 98, 134310 (2018). [CrossRef]
41. R. Puebla, A. Smirne, S. F. Huelga, and M. B. Plenio, “Universal anti-Kibble-Zurek scaling in fully connected systems,” Phys. Rev. Lett. 124, 230602 (2020). [CrossRef]
42. M. Syed, T. Enss, and N. Defenu, “Dynamical quantum phase transition in a bosonic system with long-range interactions,” Phys. Rev. B 103, 064306 (2021). [CrossRef]
43. J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, “Many-body localization in a quantum simulator with programmable random disorder,” Nat. Phys. 12, 907–911 (2016). [CrossRef]
44. B. P. Lanyon, C. Hempel, D. Nigg, M. Müller, R. Gerritsma, F. Zähringer, P. Schindler, J. T. Barreiro, M. Rambach, G. Kirchmair, M. Hennrich, P. Zoller, R. Blatt, and C. F. Roos, “Universal digital quantum simulation with trapped ions,” Science 334, 57–61 (2011). [CrossRef]
45. J. W. Britton, B. C. Sawyer, A. C. Keith, C.-C. J. Wang, J. K. Freericks, H. Uys, M. J. Biercuk, and J. J. Bollinger, “Engineered two-dimensional ising interactions in a trapped-ion quantum simulator with hundreds of spins,” Nature 484, 489–492 (2012). [CrossRef]
46. R. Puebla, “Finite-component dynamical quantum phase transitions,” Phys. Rev. B 102, 220302 (2020). [CrossRef]
47. J. P. Reithmaier, G. Seę, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, T. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot–semiconductor microcavity system,” Nature 432, 197–200 (2004). [CrossRef]
48. H. Grabert and U. Weiss, “Quantum tunneling rates for asymmetric double-well systems with ohmic dissipation,” Phys. Rev. Lett. 54, 1605–1608 (1985). [CrossRef]
49. M. P. Fisher and A. T. Dorsey, “Dissipative quantum tunneling in a biased double-well system at finite temperatures,” Phys. Rev. Lett. 54, 1609–1612 (1985). [CrossRef]
50. D.-W. Luo, C.-H. Lam, L.-A. Wu, T. Yu, H.-Q. Lin, and J. Q. You, “Higher-order solutions to non-markovian quantum dynamics via a hierarchical functional derivative,” Phys. Rev. A 92, 022119 (2015). [CrossRef]
