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by The Oscillation PublicationMarch 5th, 2024

This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira;

(2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering;

(3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University.

- Introduction
- Model and Protocol
- Different Regimes in d˜− Γ˜ Plain and Multiple QMPE
- QMPE in Region (d): Second-order Ep
- QMPE in Region (a1): Oscillations
- Summary
- Appendix A: Eigenvectors of L in region (d)
- Appendix B: Eigenvectors of L in region (a1)
- Appendix C: QMPE in region (b): purely exponential relaxation
- Appendix D: QMPE in region (e): 3rd order EP
- References

Using Eqs. (10) and (11) in Eq. (1), we have obtained the following expressions of the density matrix elements ρj (t) (j = 1, 2, 3, 4):

where the initial density matrix elements ρn(0) before quench are given by Eq. (5) and the elements ℓk,n are provided in Appendix A. Interestingly, in the presence of the second-order EP in region (d), the density matrix elements in Eq. (13) have picked up an additional algebraic term proportional to time t. This is remarkably different from the time evolution in regions (b) which has only purely exponential relaxations [see Appendix B] and region (a1) that contains damped oscillations [see Sec. V]. The linear t dependence originates from the off-diagonal defect in the Jordan normal form in Eq. (11).

To investigate QMPE, we focus on the ground state density matrix element ρgg(t) (≡ ρ4(t)). To detect the QMPE, we would like to find the intersections of the two copies I and II during their relaxation, i.e. solution(s) to the equation ∆ρgg(t) = 0. We obtain the following form of ∆ρgg(t)

where the co-efficients α1, α2, α3 are given by

The solutions to such transcendental equation in Eq. (14) are given by the Lambert W function, where the equation yey = x has only one solution y = W0(x) if x > 0 and two solutions y = W0(x) and y = W−1(x) if −1/e 6 x < 0. Indeed, such properties of Lambert W function imply that there can be more than one solutions to ∆ρgg(t) = 0, meaning multiple QMPE. More precisely, at most two intersection times i.e. double QMPE can be observed in region (d). Below we provide the criteria for double QMPE and single QMPE in this region:

The asymptotic expansions of the functions W0(x) and W−1(x) can be found in Ref. [105]. Note that Lambert W functions have found applications in several fields such as viscous flows [106], crystal growth [107], epitaxial film growth [108], phase separation of polymer mixtures [109], criticality in temporal networks [110] etc.

In Fig. 1, we plot the solutions t˜0 and t˜−1 [Eq. (17)] as functions of the driving strengths before quench. Indeed, we observe that there are regions with single QMPE where only t˜0 exists and there are other regions where both solutions prevail leading to double QMPE. In particular, we observe double QMPE for large values of ˜dI and ˜dII. Note that QMPE appears to be unavoidable in the ground state probability ρgg(t) in Fig. 1 i.e. there is no such region in the (d˜ I, d˜ II) where there is no QMPE.

Here we analyze QMPE in average energy E(t) = Tr[ρb(t)H] where H is the system Hamiltonian presented as a 2 × 2 matrix with elements H11 = 1, H22 = 0, H12 = H21 = ˜d/2 [95]. Consequently, E(t) can be expressed as

Note that the average energy directly captures the effect of coherence due to the presence of the off-diagonal density matrix elements in Eq. (18). To explore the QMPE in E(t), we consider the time evolution of the energies starting from the two different initial conditions I and II and look for their intersections during the relaxations. More precisely, we would like to find the solutions to ∆E(t) = 0 where ∆E(t) = EI(t) − EII(t). We obtain the following expression

where the elements α1, α2, α3 are given by Eq. (15). Note that Eq. (20) is a similar kind of transcendental equation as Eq. (14), only with different coefficients. Therefore, we can analyze the possibility of QMPE in energy using Lambert W functions as we did for the ground state probability. The two possible solutions to ∆E(t) = 0 are given by

where the superscript E denotes energy and is used to differentiate the solutions from Eq. (17) obtained for ground state probability.

Figure 2 exhibits the solutions in Eq. (21) as functions of the control parameters ( ˜dI, ˜dII). We observe region with single QMPE (only t˜E 0 exists) as well as region for double QMPE (both t˜E 0 and t˜E −1 exist). In addition, we observe regions where both the solutions are absent i.e. no QMPE. This crucial difference in Fig. 2 compared to Fig. 1 indicates that QMPE in ground state probability does not necessarily implies QMPE in average energy. However, Figs. 1 and 2 also show that there are common parameter values that lead to QMPE both in ρgg(t) and E(t). We present an explicit example in Fig. 3 where both of these quantities exhibit double QMPE, for the same set of parameters. However, the two intersection times for ρgg(t) and E(t) are different from each other. The intersection times for ρgg(t) are comparatively far from each other with the second QMPE being much weaker in magnitude [Fig. 3(a)], whereas the intersection times for E(t) are comparatively much closer to each other, with both the first and second QMPE being comparable in magnitude [Fig. 3(b)].

Next we consider the von Neumann entropy SvN(t), which is a function of ρb(t), defined as

The nonlinearity of SvN(t) as a function of the density matrix elements poses difficulties to perform analytical treatment of the QMPE in von Neumann entropy, in contrast to what we could achieve previously for ground state probability and energy. To observe QMPE in SvN(t), we consider the intersection time(s) for the quantity ∆SvN(t) = S I vN −S II vN. In Fig. 4, we present the behavior of ∆SvN(t) for the same set of parameter values used in Fig. 3. Indeed, we observe that ∆SvN(t) intersects zero twice, implying the existence of double QMPE in the von Neumann entropy, similar to ρgg(t) and E(t).

We use the definition of temperature T (t) following Ref. [65] to examine the possibility of thermal QMPE in Hatano’s model, as given below

where E(t) and SvN(t) are defined in Eqs. (18) and (22), respectively. The QMPE in system temperature has been

observed recently in Ref. [65] using a semi-classical approach. Naturally, it would be interesting to explore thermal QMPE in Hatano’s model that involves the effect of off-diagonal density matrix elements. However, we find that there is no thermal QMPE for the same set of parameters that have produced QMPE for ground state probability [Fig. 3(a)], energy [Fig. 3(b)] and entropy [Fig. 4]. To search for thermal QMPE, we consider a variable dissipation protocol that allows us to use Γ˜ I 6= Γ˜ II 6= Γ˜ (wider parameter space compared to the fixed dissipation protocol Γ˜ I = Γ˜ II = Γ). Indeed, using such protocol, we ˜ have obtained the thermal QMPE shown in Fig. 5 where we present the temperature difference ∆T (t) between two copies I and II. We observe that ∆T (t) becomes zero at some intermediate relaxation time that denotes the intersection of two copies, after which the initially hotter copy becomes colder, leading to thermal QMPE.

The Kullback-Leibler (KL) divergence is a measure of difference from steady state which is a monotonically decreasing function of time, and it is given by

To obtain QMPE in KL divergence, we use the variable dissipation protocol. In Fig. 6(a) we present an example where the copy starting at a higher distance from steady state relaxes faster compared to the other copy with lower initial distance. The copies I and II intersect each other leading to QMPE in DKL(t) and the copy starting at higher initial distance reaches the steady state faster. We have not observed multiple QMPE in KL divergence. It would be an interesting future question to understand if the absence of multiple QMPE in DKL(t) is its generic feature and if the reason is its monotonicity.

An important variable of interest in any relaxation process, be it classical or quantum, is the speed of the relaxation. Since the KL divergence gives the distance measure from steady state, a relevant relaxation speed can be defined as

where the negative sign is used to complement the fact that the KL divergence decreases with time. It would be interesting to see if the QMPE in DKL(t) is also manifested in the speed vKL(t). For the same set of parameters that produced QMPE in KL divergence [Fig. 6(a)], we investigate QMPE in the speed in Fig. 6(b). Indeed, we observe that the speeds of the two copies I and II intersect each other, leading to QMPE. The higher distant copy goes towards steady state with a higher speed and after the intersection its speed becomes lower. Notably, the intersection times for DKL(t) and vKL(t) are different. In fact, there is a time window where the initially higher distant copy becomes lower distant copy but still maintains its higher speed for some time. It would be intriguing to investigate whether there is some general upper bound on the relaxation speed vKL(t).

We end this section by mentioning that the analysis of the QMPE in region (c) would follow similar manner as in region (d) discussed elaborately in the present section. This is because region (c) is also a line of second-order EP just like region (d), the only difference is that the arrangement of eigenvalues for region (c) obeys 0 < λ2 < λ3 = λ4.

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