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Abstract
We propose a formulation to obtain the exact susceptibility for system arbitrary operators to the external fields by means of the whole-system Hamiltonian (system plus reservoir) diagonalization methods, where the dissipative effects directly reflect the nature of the structured non-Markovian reservoir. This treatment does not make the Born-Markovian approximation in structured non-Markovian reservoir. The relations between linear response function and bound-states for the system as well as structured reservoir are found, which shows the photon bound-states and continuous energy spectrum can be readout from the susceptibility, respectively. These results are then used to examine the validity of second-order Born-Markovian approximation, where we find interesting features (e.g., bound-states) are lost in the approximate treatments for open systems. We study the dependence of the response function on the type (spectrum density) of interaction between the system and structured reservoir. We also give the physical reasons behind the disappearance of the bound-states in the approximation method. Finally, these results are also extended to a more general quantum network involving an arbitrary number of coupled-bosonic system without rotating-wave approximation. The presented results might open a new door to understand the linear response and the energy spectrum for non-Markovian open systems with structured reservoirs.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is well known that linear response theory is based on the first-order perturbation theories for a quantum system in thermal equilibrium, which is one of the most useful methods to connect physical quantities with the underlying theoretical description of a system. The quantum response can be the thermal conductivity, electrical conductivity, magnetic susceptibility and Hall conductance, depending on the chosen observable and the perturbation. The pioneering work [1] by Ryogo Kubo analytically gave the linear response formula of quantum systems with time-dependent perturbations. It has many applications in quantum optics and condensed matter physics. For example, it can be used to calculate the transport coefficient in electronic systems [2,3], and study the linear response in random potential [4], superfluids, magnetic systems [5], and topological quantum materials [6–12], determining quantum Fisher information [13], and probing the topology of density matrices [14].
The linear response theory of closed systems has attracted more and more attention in the quantum optics, biophysics, nanophysics, and condensed matter physics. In recent years, it was also extended to open systems (i.e., its steady state might not be a thermal equilibrium state). These extensions were formulated for the Hall conductance of topological insulators subjected to decoherence based on the master equation [15], the adiabatic response of an open quantum system with dephasing [16,17], the projection operator method [18,19], the Lindblad equation [20–27], the non-linear response [28–30], steady-state Hall response [31], and adiabatic and non-adiabatic response [32,33]. The results were also formulated for systems in arbitrary initial states [34], through the hierarchical equation of motion [35], postselected system [36,37] and bipartite systems [38], extended quantum systems [39].
Despite the growing body of literature on this subject, the analysis has almost exclusively been focused on discussions of the influence of the reservoir on system susceptibility [18,19,28,34]. To our knowledge, no complete theoretical framework has been established for the relations between bound-states and susceptibility in structured reservoir to date.
In this manuscript, based on the whole system Hamiltonian (system plus reservoir) diagonalization methods, we derive an exact susceptibility for system arbitrary operators to the external fields, which includes all the influences of the structured reservoir on the system. We show that the susceptibility connects closely to the structure of energy spectrum of the whole system (system plus structured reservoir). Moreover, we can readout bound-states of the whole system from the susceptibility, which might be experimentally observed [40–45]. We then make comparisons of this result with second-order Born approximation [46–48] and Markovian approximation [49–51]. We find that the susceptibility obtained by these approximations cannot reveal the bound-states and continuum energy spectrum of the whole system. The physical reasons behind the disappearance of the bound states in the approximation methods are discussed.
The rest of this work is organized as follows. In Sec. 2, we first present a model to describe the system under study and calculate the exact susceptibility based on whole system Hamiltonian diagonalization methods. In Sec. 3, we establish connection between the energy spectrum of the system and its susceptibility. In Sec. 4, we compare the exact result with that given by the second-order Born approximation and the Markovian approximation, and observe that the susceptibility with approximations cannot reveal the photon bound-states. In Sec. 5, we expand the susceptibility from quantum systems under the rotating wave to the general non-rotating wave ones. Sec. 6 is devoted to discussions and conclusions.
2. Susceptibility for the cavity coupled with structured reservoir
In this section, we first present a model to describe the system under study and diagonalize the whole system Hamiltonian (system plus reservoir). Then we calculate the change of the expectation value of an arbitrary observable caused by the external perturbation, assuming the system and its reservoir are in thermal equilibrium without the perturbation. We define a linear response (called susceptibility in later discussions) for arbitrary system observables and present a method to evaluate its exact value based on whole system Hamiltonian diagonalization methods.
2.1 Whole system Hamiltonian diagonalization
We consider here a single-mode photonic system interacting with a general non-Markovian reservoir which is modeled as a collection of infinite modes, where the single-mode cavity system could be a nanocavity in nanostructures or photonic crystals and the non-Markovian reservoir may be a structured photonic reservoir [52]. The Hamiltonian of the system can be expressed as a Fano-type model of a localized state coupled with a continuum [53,54]
First we seek dressed annihilation operators ${{\hat A}_j}$ to diagonalize Hamiltonian (1), which is given by
while the dressed annihilation operators $\hat A_j$ can be expressed in terms of the system and reservoir annihilation operators as follows: in which the dressed operator commutator satisfies Bosonic orthogonal-normalization properties $[ {{{\hat A}_j},\hat A_m^{\dagger} } ] = {\delta _{jm}}$. The bare quasi-mode and continuum annihilation operators can then be expressed as2.2 Analytical expression of susceptibility
In linear response, we present a formulation of the susceptibility of a system in contact with an structured reservoir due to a time-dependent perturbation. The linear response of an arbitrary operator $\hat s$ of the system to the time-dependent perturbation ${\hat H_e}(t) = \phi (t){\hat a^\dagger } + {\phi ^*}(t)\hat a$ gives the system susceptibility as follows [1,73],
2.3 Susceptibility with bound-states
We first introduce a Fano-type tight-binding model, which is an experimentally realizable microcavity system. Then we investigate the energy spectrum of the whole system by non-Markovian linear response susceptibility. Figure 1 is a schematic plot for a single-mode microcavity coupled to a coupled-resonator optical waveguide (CROW) structure. The Hamiltonian of the whole system is given by
Fig. 1. (a) A schematic plot of a single-mode microcavity coupled to a CROW structure. Microcavity field to the ${P}$-coupled-resonator at the first site in the waveguide with the coupling strength $\gamma$ which is also controllable experimentally by adjusting the distance between defects, and the photon hopping between two consecutive resonators in the waveguide structure with the controllable hopping amplitude $\lambda _0$. The cavity is small perturbed. (b) Dispersion relation in $k$ space for the system. (c) Spectrum density for the optical waveguide structure with $\omega _0=\lambda _0=\omega _c$, $\eta =1$.
The cavity could be a point defect produced in photonic crystals and the waveguide consists of linear defects in which light propagates by couplings of the adjacent defects, which can also be realized in different types of microresonators, such as Fabry-Perot microcavities and microring resonators, as well as different coupling [87]. But the dispersion relationships of different types of microcavities and micro-waveguides are very similar and are characterized only by the free spectral range, the quality factor of the resonator, and the coupling between the resonators [88]. Therefore, Eq. (24) does describe a large class of microcavities coupled to the micro-waveguide. We first transform the bosonic operator at site $j$ into the $k$ space, ${{\hat b}_m} = \sqrt {2/\pi } \sum \nolimits _k {\sin (mk){{\hat b}_k}}$, where $\hat b_k^{\dagger}$ and $\hat b_k$ are the creation and annihilation operators of the corresponding Bloch modes of the waveguide, and $0 \le k \le \pi$. Considering the following relations
The first term in Eq. (30) corresponds to bound-states, which denotes to the energy spectrum of the whole system (see discussion of Sec. III). These bound-states do not decay, which give dissipationless non-Markovian dynamics. And the second term in Eq. (30) is the continuous part of the spectrum that leads to the usual dissipation. The bound state produces dissipationless dynamics of the system so that the system can permanently retain some of its initial state information. Thus, time-dependent dynamics corresponding to the bound states do not allow the cavity to approach thermal equilibrium state [89,90].
When we study the evolution of a physical quantity $\hat s_{z_1z_2}$ (not commute with physical quantity $\hat a^{\dagger}$) under a time-dependent external field, the susceptibility is given by the Dirac delta function $\delta (\omega - E_i + E_j)$, which denotes a resonance between energy levels $E_i$ and $E_j$ of the system. In open systems, the energy levels form a discrete and continuous spectrum, and the complex susceptibility also has discrete and continuous peaks, which are different from cases of closed quantum systems. The line shape of the spectrum (the imaginary part of the complex susceptibility) has an intrinsic line width, which is attributed to the energy structure of the system. In Fig. 2, we plot the susceptibility ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ [imaginary part of ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$] as a function of the frequency $\omega$ of the external field with spectral density (28) and different coupling strength $\eta$. We can see that the dependence of the line shape on the susceptibility ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ has a line shape with the frequency $\omega$ marked by $L_1-L_4$. It is interesting that we find four sharp peaks under $\omega = 0.35\omega _c$, $\omega = 1.65\omega _c$, $\omega = -0.815\omega _c$, and $\omega = 3.35\omega _c$ marked, respectively, by A, B, C, and D, which are not the positions of the microcavity eigenfrequency. In the next section, we will give the physical mechanism of such sharp peaks in the structured non-Markovian reservoir.
Fig. 2. Imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with different system-reservoir coupling strength $\eta$. Here and hereafter, $\omega _0$, $\lambda _0$, and external field frequency $\omega$ are rescaled in units of $\omega _c$, and the time $t$ is then in units of $1/\omega _c$. Hence all parameters are dimensionless. The width $L_1-L_4$ emphasizes the width related to the continuous spectrum in the reservoir, while the points $A-D$ mark the positions of system-reservoir bound-states, respectively. Other parameter is $\lambda _0 = 0.5\omega _c$.
3. Energy spectrum analysis for the whole system
In this section, we show that the non-Markovian dynamics of an open system connects strangely with the features of the energy-spectrum of the whole system (system plus structured reservoir). From a physical point of view, this can also be explained by examining the bound-states of the whole system [56–72]. The bound-states are actually stationary states without decay during the evolution. If such a bound state is formed, it will lead to a dissipationless dynamics. Thus the study of the energy spectrum may supply us with meaningful insights to understand its dynamics. Since $\hat N_1 = {{\hat a}^{\dagger } }{{\hat a} } + \sum \nolimits _k {\hat b_k^{\dagger} } {{\hat b}_k}$ is conserved, the Hilbert space splits into independent subspaces with definite $\hat N_1$. Considering the eigenstate in the single-exciton subspace
Fig. 3. Structure of solutions of eigenEq. (8). (a) Below the critical coupling, no solution exists outside the energy band. (b) Over the critical coupling, two solutions exist outside the energy band. (c) Critical regimes for different types of bound-states based on Eq. (33). In the regime $R_1$, no bound-states; in the regimes $R_2$ or $R_3$, one bound state; in regime $R_4$, two bound-states.
We will use eigenEq. (8) to understand the interesting points shown in Fig. 2. We plot the bound-state eigenenergies arranged in asscending order, where the $k$ space is discretize into $101$ Bloch modes from $0$ to $\pi$ (i.e., $k = 0,\pi /100, \cdot \cdot \cdot \pi$). Two cases are considered: (i) the eigenenergies lie inside the energy band in Fig. 4(a) (correspond to regime $R_1$ in Fig. 3(c)), (ii) several discrete eigenenergies lie outside the energy band in Figs. 4(b)–4(d) (correspond to regime $R_2-R_4$ in Fig. 3(c)): in Fig. 2(b) when $\omega _0=1.5\omega _c$ and $\eta =1.5$, we have $E_-=0.35\omega _c$ given by Eq. (8); this explains point $A$ in Fig. 2(b). For Fig. 2(c) $\omega _0=0.5\omega _c$ and $\eta =1.5$, we have $E_+=1.65\omega _c$; this explains what we find at point $B$. For $\omega _0=1.5\omega _c$ and $\eta =4$, we have $E_\pm =-0.815\omega _c,3.35\omega _c$, which explains points $C$ and $D$. In addition, the bulk of the spectrum can be decided by ${\omega _0} - 2{\lambda _0} \le \omega \le {\omega _0} + 2{\lambda _0}$ (see $L_1-L_4$ in Fig. 4), which is consistent in eigenEq. (8). Localized states are also referred to as dressed bound-states, related to Fano resonances [92], because Hamiltonian (1) is a Fano-Anderson Hamiltonian. Since the energy bands of the two leads overlap, there are at most two bound-states.
Fig. 4. The distribution of numerically calculated eigenenergy spectrum of Hamiltonian (1) at optical waveguide for $\lambda _0 = 0.5\omega _c$. Eigenenergies are arranged in asscending order, where the $k$ space is discretize into $101$ Bloch modes from $0$ to $\pi$, i.e., $k = 0,\pi /100, \cdot \cdot \cdot \pi$ in $k$ space. The parameters of Fig. 4 are the same as those in Fig. 2, respectively. The bulk of the spectrum remains fixed (see $L_1-L_4$), whereas the bound-state eigenenergies are outside the boundaries (see points $A-D$).
From the viewpoint of dynamics, we show the three regimes of Fig. 3(c) corresponding to complete decoherence, decoherence suppression, and periodic oscillation, respectively. In Fig. 5(a) (correspond to regime $R_1$ in Fig. 3(c)), where the bound state vanishes, the exciton dynamics undergoes a full dissipation process. In Figs. 5(b)–5(c), the non-bound state parts will rapidly approach zero within the energy band according to the Lebesgue-Riemann lemma [93]. In this case, there is only one bound state, therefore the microcavity field amplitude (30) can be obtained after a long time,
The result shows that the microcavity field amplitude $| {{{\cal M} }(t)} |$ holds a nonzero steady value after a long time. This is also understandable based on the fact that the bound-state, as a stationary state of the whole system, has a vanishing decay rate and the coherence contained in it would be preserved during the time evolution.Fig. 5. The absolute value of microcavity field amplitude $|{\cal M}(t)|$ of the microcavity in photonic waveguide. The parameters chosen in Fig. 5 are the same as those in Fig. 2, respectively.
When we consider regime $R_4$ as shown in Fig. 3(c), the quantum interference effects between the two bound-states after long time $t$ lead to periodic oscillation behaviors of the dynamics. The amplitudes of periodic oscillations do not decrease in time. From Eq. (30), we obtain the microcavity field amplitude in the long-time regime:
In order to continuously reveal the bound-states, we plot the imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with different system-reservoir coupling strength $\eta$ and external field frequency $\omega$ in Fig. 6. The positions of the bound-states can be found in Fig. 6 (see red-solid lines). Two cases are shown: (i) the microcavity energy $\omega _c$ is at the middle of the band ($\Delta =0$, Fig. 6(a)), and (ii) far from the middle of the band ($\Delta \ne 0$, Figs. 6(b) and 6(c)). The energies of the bound-states lie outside of the band, thus they are localized (not decay). As $\eta \to 0$, bound-state energy $E_{ - ( + )}$ approaches the bottom (top) of the photonic band. If the microcavity energy coincides with the band center, the energies of the bound-states are symmetrically located. Otherwise, if the microcavity energy is below the center, i.e., $\Delta < 0$, the energy of the lower bound-state ${E_ - }$ moves away from the microcavity energy faster than the energy of the upper bound-state ${E_+ }$ does, and vice-versa. Therefore, the position of the microcavity energy with respect to the band center originates an asymmetry between $E_+$ and $E_-$.
Fig. 6. Imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with different system-reservoir coupling strength $\eta$ and external field frequency $\omega$. The parameters chosen are $\lambda _0 = 0.5\omega _c$, $\omega _0 = \omega _c$ for (a), $\omega _0 = 1.5\omega _c$ for (b), $\omega _0 = 0.5\omega _c$ for (c) , where the $k$ space is discretize into $101$ Bloch modes from $0$ to $\pi$, i.e., $k = 0,\pi /100, \cdot \cdot \cdot \pi$) in $k$ space.
4. Comparisons with the other approximation methods
There are several methods in the literature to explore the linear response theory for open quantum systems. In the weak-coupling limit, the non-Markovian master equation for quantum system coupled to a bosonic reservoir approximates to second-order in coupling strength between system and structured reservoir. This derivation treats the system-reservoir coupling perturbatively and hence it is only available for weak system-reservoir couplings. It is worth recalling that the other systematically perturbative non-Markovian master equation that is local in time can be derived from the time-convolutionless projection operator formalism [46–48]. Now, we discuss the details. Our results (34), going beyond those in second-order Born approximation (SOBA) [49,94,95], and Markovian approximation (MA) [49–51], is governed by memory kernel containing all the influences of the reservoir on the system. It is available to examine the validity of those perturbative approaches applied to the SOBA and MA methods.
A local non-Markovian master equation in Born approximation valid to second order in the system-reservoir interaction strength can be written as
Fig. 7. Comparison of three methods for imaginary part ${{\chi ^I}_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ of the system susceptibility ${{\chi }_{\hat a{{\hat a}^{\dagger} }}}(\omega )/\hbar$ for open systems with the external field applied to the system. The red-solid line stands for the exact linear response (34), blue-dotted line for the linear response in the second-order Born approximation (39), and yellow-dashed line for the linear response obtained by applying Markovian limit (40). In this figure, we set $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.01$ for (a); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.05$ for (b); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.1$ for (c); $\lambda _0 = 2\omega _c$, $\omega _0 = \omega _c$, $\eta =0.5$ for (d); $\lambda _0 = 0.5\omega _c$, $\omega _0 = 1.5\omega _c$, $\eta =1.5$ for (e); $\lambda _0 = 0.5\omega _c$, $\omega _0 = 1.5\omega _c$, $\eta =4$ for (f).
5. Susceptibility for coupled-Boson systems without rotating-wave approximation
In this section, we generalize the linear response susceptibility for the above results to more general quantum network consisting of an arbitrary number of coupled-Boson systems (e.g., charged-Brownian particles) coupled to thermal reservoirs without rotating wave approximation (see Fig. 8), whose Hamiltonian is given by
Fig. 8. The figure shows that the response functions can be readout by probed the susceptibility of quantum many-body system consisting of $M$ mutually coupled charged-Brownian particles ($R_{mn}$ denotes the coupling matrix) interacting with the reservoir described by harmonic oscillators with frequency $\omega _k$. The large-circle denotes the charged-Brownian oscillator, which is interacted with a large number of oscillators (the reservoir shown by small circles) coupling by interacting strengths $v_{n,k}$.
6. Discussion and conclusion
In summary, we have calculated the exact susceptibility for structured non-Markovian quantum systems by means of the whole Hamiltonian diagonalization methods, which connects the response function and the system-reservoir energy spectrum. Furthermore, we show that the discrete spectrum and continuous spectrum of the whole system (system and reservoir) can be acquired from the system susceptibility, in which bound-states can be experimentally observed [40–45]. We then made comparisons with the results given by second-order Born and Markovian approximations and find that the latter two methods cannot reveal the bound-states and continuum spectrum of the system and reservoir. Note that our methods are not limited to linear systems. For non-Markovian nonlinear quantum systems, we can address those issues by iteration of nonlinear operator equations. The presented study opens a new door to calculate the response of an open system to an external field, which makes it possible to better understand the relations between the linear response function and energy spectrum for non-Markovian open systems.
Applications of our presented results to many other physically relevant systems as well as its extension to linear responses of non-Markovian open quantum system, e.g., nonlinear couplings with the structured reservoirs, Kerr nonlinearity system, etc., deserve future investigations.
Funding
Fundamental Research Funds for the Central Universities (2412019FZ044); Science Foundation of the Education Department of Jilin Province during the 13th Five Year Plan Period (JJKH20190262KJ); National Natural Science Foundation of China (11534002, 11705025, 11775048).
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