Introduction
In recent decades, small open quantum systems (coupled to large quantum environments) have been studied extensively in various fields of physics [1–4]. Basically, such a system is described by its reduced density matrix (RDM), which may approach a steady state in many cases. As an example of interesting properties of such systems, it is now well known that the phenomenon of decoherence, due to interaction with huge quantum environments, may happen in such a way that the RDM becomes approximately diagonal in a so-called preferred (pointer) basis [5–7].
Roughly speaking, the decoherence mechanism has been studied well in two extreme situations, i.e. for pure-dephasing interactions, with a preferred basis given by the eigenbasis of self-Hamiltonian [7, 8], and for very strong interactions under complex environments, with a preferred basis given by the eigenbasis of the interaction Hamiltonian [8, 9]. But, under a generic dissipative system-environment interaction,4(4Here, for a total system undergoing a unitary Schrödinger evolution, dissipativeness of system-environment interaction means that the interaction Hamiltonian is not commutable with the Hamiltonian of the central system.) there may exist interplay between relaxation and decoherence and, as a result, the situation with decoherence is much more complicated. When the interaction is sufficiently weak such that the interplay can be studied by a first-order perturbation theory, it was found that the system's eigenbasis is approximately a preferred basis under a quantum chaotic environment [10]. Sometimes, the eigenbasis of a renormalized Hamiltonian may be considered [11].
More recently, the long-time averaged RDM has been studied for a generic small central system S, which is locally coupled to a large many-body chaotic environment ${ \mathcal E },$ with the total system undergoing a Schrödinger evolution [12]. Under a local interaction Hamiltonian, whose environmental part satisfies the so-called eigenstate thermalization hypothesis (ETH) ansatz [13–18] with diagonal elements in the ansatz treated as a constant within the energy region of relevance, it was found that a preferred basis is given by the eigenbasis of a renormalized self-Hamiltonian that includes the certain impact of the system-environment interaction when steady states exist.
In [12], due to the requirement of constant diagonal elements, its result is usually valid when the interaction is relatively weak. In fact, since the slope of the diagonal elements within the relevant energy region is approximately proportional to the interaction strength, the elements may not be treated as a constant under a sufficiently strong interaction. The purpose of this paper is to go further to study numerically a case of stronger interaction, by extrapolating the result of [12].
Setup
We consider a total system, which is composed of a two-level system (TLS) S as the central system and a many-body quantum chaotic system ${ \mathcal E }$ as the environment. The environment consists of N particles (N ≫ 1), whose Hilbert space has a dimension ${d}_{{ \mathcal E }}$. The total Hamiltonian H is written as
$\begin{eqnarray}H={H}_{S}+{H}_{{ \mathcal E }}+{H}_{S{ \mathcal E }},\end{eqnarray}$
where HS is for the TLS with a level-splitting ΔS, $\begin{eqnarray}{H}_{S}={{\rm{\Delta }}}_{S}{S}_{z},\end{eqnarray}$
${H}_{{ \mathcal E }}$ for the environment, and ${H}_{S{ \mathcal E }}$ is the interaction Hamiltonian, $\begin{eqnarray}{H}_{S{ \mathcal E }}=\lambda {S}_{x}\otimes {H}_{I{ \mathcal E }},\end{eqnarray}$
where ${H}_{I{ \mathcal E }}$ is a local observable of the environment. Here, Sx,z indicate Pauli matrices divided by 2 for the central system. Normalized eigenstates of HS and of ${H}_{{ \mathcal E }}$ are denoted by ∣α⟩ and ∣i⟩, respectively, with labels α and i as positive integers starting from 1. The corresponding eigenenergies are denoted by ${E}_{\alpha }^{S}$ and Ei, respectively, both in the increasing-energy order $\begin{eqnarray}{H}_{S}| \alpha \rangle ={E}_{\alpha }^{S}| \alpha \rangle ,\end{eqnarray}$
$\begin{eqnarray}{H}_{{ \mathcal E }}| i\rangle ={E}_{i}| i\rangle ,\end{eqnarray}$
where for brevity we have omitted a superscript ${ \mathcal E }$ for Ei.We use ${{\rm{\Delta }}}_{{ \mathcal E }}$ to indicate the main energy domain of the environment, excluding levels at the two edges of the spectrum. The mean level spacing is approximately given by ${{\rm{\Delta }}}_{{ \mathcal E }}/{{d}}_{{ \mathcal E }}$. The environmental operator ${H}_{I{ \mathcal E }}$ satisfies the ETH ansatz, i.e.
$\begin{eqnarray}\langle i| {H}_{I{ \mathcal E }}| j\rangle =h(E){\delta }_{{ij}}+{\rho }_{\mathrm{dos}}^{-1/2}(E)g(E,\omega ){R}_{{ij}},\end{eqnarray}$
where h(E) is a slowly-varying function of E, ρdos(E) is the density of states, g(E, ω) is some smooth function of its variables, E = (Ei + Ej)/2 and ω = Ej − Ei, and the quantity Rij has certain random feature with a normal distribution (zero mean and unit variance) [19, 20].5(5The randomness of Rij comes from the so-called Berry's conjecture, which was proposed based on random motions of classical trajectories in the phase space.)At time t = 0, we prepare the total system in a separate initial state $| {\rm{\Psi }}(0)\rangle =| {\phi }_{S}\rangle \otimes | { \mathcal E }\rangle $ for a general system state ∣φS⟩ = ∑αcα∣α⟩. The environment state $| { \mathcal E }\rangle $ is set as a typical state within a narrow energy shell Γ0, which is centered at E0 with a width δE, namely Γ0 = [E0 − δE0/2, E0 + δE0/2]. The total system evolves as ∣$\Psi$(t)⟩ = e−iHt∣$\Psi$(0)⟩. By definition, the RDM of the system S, denoted by ρS(t), is given by ${\rho }^{S}(t)={\mathrm{Tr}}_{{ \mathcal E }}\rho (t)$, where ρ(t) = ∣$\Psi$(t)⟩⟨$\Psi$(t)∣. We study the long-time averaged RDM of the TLS
$\begin{eqnarray}{\overline{\rho }}^{S}:= \mathop{\mathrm{lim}}\limits_{T\to \infty }(1/T){\int }_{0}^{T}{\rho }^{S}(t){\rm{d}}t.\end{eqnarray}$
Extrapolation of a result of [12]
The main result of our previous work [12] is as follows: If the condition of8 ) depends on neither the central system's initial condition, nor that of the environment (under the condition of equation (7 )); hence, it is robust. It turns out that this relation implies the existence of a preferred basis, which is given by the eigenbasis of the following renormalized self-Hamiltonian
$\begin{eqnarray}| {\rm{\Delta }}h/{h}_{0}| \ll 1,\end{eqnarray}$
is fulfilled, where h0 = h(E0), Δh is the maximum difference between h0 and h(Ei) of Ei ∈ [E0 − δE/2, E0 + δE/2], then, elements of the long-time averaged RDM satisfy the following relation, $\begin{eqnarray}{\overline{\rho }}_{\alpha \beta }^{S}\simeq \displaystyle \frac{\lambda {h}_{0}/2}{{E}_{\beta }^{S}-{E}_{\alpha }^{S}}({\overline{\rho }}_{\beta \beta }^{S}-{\overline{\rho }}_{\alpha \alpha }^{S}).\end{eqnarray}$
Here, $\delta E=\delta {E}_{0}+2{{\rm{\Delta }}}_{S}+{w}_{\max }$, where ${w}_{\max }$ is the maximum width of the eigenfunctions of the eigenstates of H in the energy basis of the uncoupled system. The relation in equation ( $\begin{eqnarray}{\widetilde{H}}_{S}={H}_{S}+\lambda {h}_{0}{S}_{x},\end{eqnarray}$
which includes certain averaged impact of the system-environment interaction.Furthermore, the condition of equation (7 ) has a complex dependence on the reduced coupling strength Λ, which is a dimensionless parameter from the first-order perturbation[21–23], given by the ratio of a relevant typical off-diagonal matrix element of the interaction Hamiltonian to the mean level spacing (or, equivalently, to ${\rho }_{\mathrm{dos}}^{-1}$). In the models considered here, making use of equation (5 ), one finds that such a typical element is given by $\lambda g(E,{{\rm{\Delta }}}_{S})/2\sqrt{{\rho }_{\mathrm{dos}}(E)}$. Thus
$\begin{eqnarray}{\rm{\Lambda }}=\lambda g(E,{{\rm{\Delta }}}_{S})\sqrt{{\rho }_{\mathrm{dos}}(E)}/2\approx \displaystyle \frac{\lambda g\sqrt{{d}_{{ \mathcal E }}}}{2\sqrt{{{\rm{\Delta }}}_{{ \mathcal E }}}}.\end{eqnarray}$
Making use of this reduced strength, one may divide the whole coupling region into three (sub)regimes, regarding some parameter of the order of magnitude of 1, which we denote by a. More exactly, The regimes include a weak coupling regime with Λ ≪ a, a strong coupling regime with Λ ≳ a, and an intermediate coupling regime between the above two. The exact value of a may be model-dependent and, as mentioned above, one may expect that 0.1 ≲ a ≲ 10.Loosely speaking, for small ΔS, the condition of equation (7 ) may be valid in the weak and part of the intermediate coupling regime as is shown in [12]. In this paper, we study an opposite case, in which Λ lies in the strong coupling regime, with ΔS ∼ ξ, where ξ indicates the averaged per-particle energy. In this case, equation (7 ) was found invalid and some of the techniques, which were used in the derivation of equation (8 ), become invalid.
We have performed lots of numerical simulations. Phenomenologically, to generalize equation (8 ) to the case considered here, the simplest method is to change h0 to quantities that depend on the labels of α and β. More exactly, we have numerically studied the following possible extrapolation of equation (8 ), i.e.11 ) reduces to equation (8 ). Note that eα is initial-state dependent.
$\begin{eqnarray}{\overline{\rho }}_{\alpha \beta }^{S}=\displaystyle \frac{\lambda /2}{{E}_{\beta }^{S}-{E}_{\alpha }^{S}}({h}_{\beta }{\overline{\rho }}_{\beta \beta }^{S}-{h}_{\alpha }{\overline{\rho }}_{\alpha \alpha }^{S}),\end{eqnarray}$
where hα ≡ h(eα), with eα determined by $\begin{eqnarray}{e}_{\alpha }={E}_{0}+| {c}_{\beta }{| }^{2}({E}_{\beta }^{S}-{E}_{\alpha }^{S}).\end{eqnarray}$
Clearly, if hα ≃ hβ ≃ h0, equation (Numerical tests
To check equation (11 ) numerically, we have employed two models for the environment. The first model is a defect Ising chain, which consists of N $\tfrac{1}{2}$-spins lying in an inhomogeneous transverse field of periodic boundary condition (PBC), with a Hamiltonian written as
$\begin{eqnarray}{H}_{{ \mathcal E }}={h}_{x}\displaystyle \sum _{l=1}^{N}{S}_{x}^{l}+{d}_{1}{S}_{z}^{1}+{d}_{5}{S}_{z}^{5}+{J}_{z}\displaystyle \sum _{l=1}^{N}{S}_{z}^{l}{S}_{z}^{l+1},\end{eqnarray}$
where Sxl and Szl indicate Pauli matrices divided by 2 at the lth site. Here, hx, Jz, d1 and d5 are parameters, which are adjusted such that the defect Ising chain is a quantum chaotic system. That is, for levels not close to the edges of the energy spectrum, the nearest-level-spacing distribution P(s) is close to the Wigner–Dyson distribution ${P}_{W}(s)=\tfrac{\pi }{2}s\exp (-\tfrac{\pi }{4}{s}^{2})$, the latter of which is almost identical to the prediction of RMT [24–26]. Exact values of the parameters used are hx = 0.9, Jz = 1.0, d1 = 1.11, and d5 = 0.6.The second model is a one-dimensional mixed-field Ising chain with PBC
$\begin{eqnarray}\begin{array}{rcl}{H}_{{ \mathcal E }} & = & {h}_{x}\displaystyle \sum _{l=1}^{N}{S}_{x}^{l}+{h}_{z}\displaystyle \sum _{l=2(\ne 5)}^{N}{S}_{z}^{l}+{h}_{1}{S}_{z}^{1}+{h}_{5}{S}_{z}^{5}\\ & & +{J}_{z}\displaystyle \sum _{l=1}^{N}{S}_{z}^{l}{S}_{z}^{l+1},\end{array}\end{eqnarray}$
where Jz = 1.0, hx = 0.9, hz = 0.5, h1 = 1.11, and h5 = 0.6. This model is also quantum chaotic. Both two models fulfill the ETH ansatz [27, 28]. The TLS is coupled to the kth spin of the two chains with $\begin{eqnarray}{H}_{S{ \mathcal E }}=\lambda {S}_{x}\otimes {S}_{x}^{k}.\end{eqnarray}$
In numerical simulations, we took ΔS = 0.3.We have computed the off-diagonal element of the long-time-averaged RDM, namely ${\overline{\rho }}_{12}^{S}$, as a function of the coupling strength λ for ${ \mathcal E }$ as the defect Ising chain. In figures 1(a) and (b), for two initial states of the central system, one sees that equation (11 ) works quantitatively well almost in the whole coupling regime from weak to strong, except in regions around the peculiar valleys, where the agreement is qualitative. In particular, the predictions of equation (11 ) are very close to the exact values in the strong coupling regime with λ ≳ 0.3. And, a similar phenomenon has been observed, when the mixed-field Ising chain is taken as the environment (figure 2).
Figure 1. Values of $| {\overline{\rho }}_{12}^{S}| $ (black dots) versus the coupling strength λ in the logarithm scale, for the central system's initial states as (a) (c1, c2) = (0, 1) and (b) $({c}_{1},{c}_{2})=(1/2,\sqrt{3}/2)$. The solid lines (red) represent predications of equation ( |
Figure 2. Similar to figure 1, but for mixed-field Ising model and N = 13, other parameters are the same. |
In the defect Ising chain, as is known, $({g}^{2}/{\rho }_{\mathrm{dos}}){d}_{{ \mathcal E }}\,={g}^{2}/{{\rm{\Delta }}}_{E}\approx 1$ [12]. This gives that ${\rm{\Lambda }}\approx \tfrac{1}{2}\lambda \sqrt{{d}_{{ \mathcal E }}}=64\lambda $. Thus, Λ ≈ 19 for λ = 0.3, belonging to the strong coupling regime discussed previously.
Moreover, we found that, at larger value of the number N of spins in the environment, equation (11 ) may work at smaller λ. This is understoodable in diffusive one-dimensional systems, in which $g(E,{{\rm{\Delta }}}_{S})\sim {\left(N/{{\rm{\Delta }}}_{S}\right)}^{1/2}$ for ΔS ∼ ξ, and $\rho (E)\sim {d}_{{ \mathcal E }}/N$ [29–34]. In fact, the dimensionless coupling strength ${\rm{\Lambda }}\sim \lambda \sqrt{{d}_{{ \mathcal E }}/{{\rm{\Delta }}}_{S}}$ with ${d}_{{ \mathcal E }}={2}^{N}$, implying that smaller λ should correspond to larger N at a fixed Λ.
Conclusions and discussions
To summarize, we have studied the long-time averaged RDM ${\overline{\rho }}^{S}$ of a TLS, which is locally coupled to quantum chaotic Ising chains, with the total system undergoing a Schrödinger evolution. A phenomenological relation among elements of ${\overline{\rho }}^{S}$ in the eigenbasis of the TLS is proposed for a dissipative interaction in the strong coupling regime and has been tested numerically for two Ising chains.
The relation may be useful in the study of decoherence. At least it suggests that, under dissipative interactions, the following naive picture may need some modification. That is, decoherence may always reduce off-diagonal elements of the RDM in the energy basis to small values, as long as the environment is sufficiently large and undergoes a sufficiently irregular motion.
Some further comments for steady states: For systems of the type studied in this paper, it is already known that the RDM ρS(t) is close to ${\overline{\rho }}^{S}$ for most of the times in the long-time limit [35–40]. Moreover, although analytical demonstration of the emergence of steady states is a subtle topic, numerical simulations show that such states may emerge in many situations.