Optical quantum frequency filter based on generalized eigenstates

Chia-Yi Ju; Ming-Hsun Chou; Guang-Yin Chen; Guang-Yin Chen; Yueh-Nan Chen; Yueh-Nan Chen; Yueh-Nan Chen; Yueh-Nan Chen

doi:10.1364/OE.395140

1. Introduction

It is well-known that quantum mechanics started with the assumption that the spectrum of energy is quantized/discretized which later lead to the birth of Heisenberg’s matrix mechanics. Ever since the invention of the Dirac’s formalism, it became clearer that Schrödinger’s wave mechanics is closely related to the matrix mechanics. However, Hilbert space alone is not sufficient for quantum mechanics since physical quantities are not always quantized, for example, position and momentum. To include continuous spectrum, the underlying mathematical quantum space has to be extended from Hilbert space to rigged Hilbert space [1–4] to accomodate both discrete and continuous spectrum.

Normally, the eigenvalues of the normalizable eigenstates form the discrete part of the spectrum and the ones of the unnormalizable eigenstates (generalized eigenstates in the rigged Hilbert space) form the continuous part. Nevertheless, von Neumann and Wigner discovered that some eigenvalues of normalizable states can be embedded in the continuous spectrum [5]. Loosely speaking, there can be some “discrete eigenvalues” sit inside the continuous part. This phenomenon is now called bound states in the continuum (BIC) [6,7]. Although initially it originated from an artificial mathematical construction, an optical waveguide array structure [8], hybrid plasmonic-photonic structure [9], qubit-waveguide composite systems [10], and some photonic systems [11] have shown that these phenomena indeed exist in the real physical systems.

Besides Hermitian quantum system, non-Hermitian quantum systems have gained a lot of attention since Bender and Boettcher’s work on parity-time ($\mathcal {PT}$)-symmetric quantum mechanics [12]. In some continuous $\mathcal {PT}$-symmetric quantum systems, the space has to be analytically continued to complex space inevitably. In fact, the notion of complex dimensions has been used in many physics fields, including quantum field theory [13,14], statistical mechanics [15–17], etc. One of the main reasons of complexifying dimensions is to regularize some seemly divergent quantities to a finite one. The most well-known regularized method among these regularization scheme is perhaps the Wick rotation [18], which rotates time dimension from real to imaginary axis. Besides regularizing some infinities, a bonus to the Wick rotation is that it often makes the physics more transparent and easier to deal with. For example, it transforms Minkowski space into Euclidean space, instantons into solitons, time into temperature, etc.

In this work, we find that the square integral of a few unnormalizable eigenstates can be regularized to a finite value after the Wick rotation if time dimension is extended to a complex plane. The eigenenergies of these square-integral-regularizable states correspond to the poles in the scattering matrix which is a feature of Gamow states [19–22]. However, this work will be focusing on the eigenstates with real energies, i.e., non-decaying states.

These regularizable states are part of the scattering states, their eigenenergies locate within the continuous spectrum just like BIC. However, the physical properties of the BIC and the Gamow states are significantly different since BIC are localized but the generalized states are not confined in any region. Therefore, an optical quantum frequency filter of scattering state is proposed by using these properties of the Gamow states.

2. Regularizing generalized (scattering) eigenstates

A quantum state can be represented as a vector in a Hilbert space; however, not every vector in the Hilbert space is compatible with all the observables in the system. When dealing with a quantum system that involves position and momentum operators, since they are both unbounded operators, only a non-trivial subset, $\Phi$, of the Hilbert space is compatible with these two operators. For example, assuming $|{\psi }\rangle$ is a vector in the Hilbert space, it is still possible that ${|\langle {\phi } X^n |{\phi }\rangle | \rightarrow \infty }$ or ${|\langle {\phi } P^n |{\phi }\rangle | \rightarrow \infty }$ for some $n$, then the expectation value of $X^n$ or $P^n$ is not well-defined for $|{\phi }$. Hence, for $|{\phi }\rangle$ to have well-defined expectation values of the observables, the corresponding wave function has to be smooth and approach zero faster than any polynomial at the infinity. In a less rigorous way of saying this is that the wave function has to be at least exponentially decaying to zero at the infinity.

The eigenstates of Hamiltonian $H$ can be found by solving the eigenvalue equation, $H |{n}\rangle = E_n |{n}\rangle$, where ${|{n}\rangle \in \Phi }$ is the eigenstate and $E_n$ is the corresponding eigenenergy. The eigenenergies of this kind form a discrete spectrum and the eigenstates are bound states. Nevertheless, there are (scattering) states which are physical, but non-localized and cannot be described by these bound states.

In order to include the scattering states, the eigenvalue equation is relaxed to

(1)$$\langle{\phi} H |{\cal{E}}\rangle = \cal{E} \langle{\phi}|{\cal{E}}\rangle,$$

for all $|{\phi }\rangle \in \Phi$. Besides the original eigenenergy and eigenstate pairs, ${(E_n,|{n}\rangle )}$, there are some generalized eigenstate pairs, ${(E,|{\psi (E)\rangle })}$, while the states do not belong to $\Phi$ but still are solutions to Eq. (1). These states are not normalizable but $|\langle {\phi }|{\psi (E)\rangle }| < \infty$ for all $|{\phi }\rangle \in \Phi$. The eigenenergies of these states, unlike bound states, form a continuous spectrum.

When the potential or the interaction in the Hamiltonian is localized in some region, the scattering states far from the region behave like (superposition of) free particle states in general. Therefore, the state far from the region can be split into incoming and outgoing parts. The relation between the incoming and outgoing states far away from the interacting region is encoded in the scattering matrix which can be derived from the Hamiltonian. The scattering matrix is usually well-behaved when dealing with the scattering state eigenenergies, and the poles are known to be the eigenenergies of bound states. This is understandable since they cannot propagate far from the system. Nevertheless, there are some scattering states called the Gamow states which also contribute to the poles.

On the other hand, if time dimension is treated as complex rather than pure real, the time axis can be Wick rotated to imaginary time axis by ${t \rightarrow - i t}$ which renders ${E \rightarrow i E}$ [23]. Interestingly, although the norm of the most scattering states are ill-defined, some of them becomes finite after the Wick rotation. In other words, ${\langle {\psi (E)}|{\psi (E)}\rangle \rightarrow \infty }$ for some scattering state $|{\psi (E)}\rangle$ with ${\langle {\psi (iE)}|{\psi (iE)}\rangle < \infty }$. Roughly speaking, these states are localized in complex time dimension. Interestingly, we will later see that the eigenenergies of these states are poles of the scattering matrix and, therefore, are the Gamow states (Fig. 1).

Fig. 1. Besides the typical bound states and scattering states, the norm of some scattering states can be regularized to a finite value if time dimension is treated as complex. These states that are not localized but also contribute poles to the scattering matrix and, therefore, are Gamow states. Some of these poles lie in the continuous real spectrum which is very similar to bound states in the continuum (BIC) from scattering matrix point of view. Nevertheless, unlike BIC, the wave function of these non-decaying Gamow states are non-localized.

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In this study, rather than the typical Gamow states with complex eigenenergies [24,25], we focus on the Gamow states with real eigenenergies. It is worth mentioning that the states of this kind can be found relatively easier by using the Wick rotation method since the states become square-integrable and, hence, some standard mathematical technique, e.g., Fourier transform, can be applied directly since they become functions in $\textrm {L}^2$ space. Moreover, these regularizable states are interesting because, unlike the decaying Gamow states, they are non-decaying which lead to its filtering nature. The explicit calculations are shown in Sec. 3 and some possible applications are provided in Sec. 4.

3. Field-TLS interaction

In this section, we work with a practical model — a field-TLS (two-level-system) interacting model. The Hamiltonian in the position space of the $N$ TLS interacting with the right/left going fields can be written as [26–29]

(2)$$\begin{aligned}H = & -i v \hbar \int dx \left[ c_{\textrm{ R}}^\dagger (x) \partial_x c_{\textrm{ R}}(x) - c_{\textrm{L}}^\dagger (x) \partial_x c_{\textrm{L}}(x)\right] \\ & + \sum_{j=1}^N \hbar \Big[ \Omega_j a_j^\dagger a_j + \kappa_j c_{\textrm{ R}}^\dagger (x_j) a_j + \kappa_j^* c_{\textrm{ R}} (x_j) a_j^\dagger \\ & + \kappa_j c_{\textrm{L}}^\dagger (x_j) a_j + \kappa_j^* c_{\textrm{L}} (x_j) a_j^\dagger \Big] , \end{aligned}$$

where $c_{\textrm { R/L}} (c_{\textrm { R/L}}^\dagger )$ represents the annihilation (creation) operator of the right/left going field at position $x$ and $v$ is the corresponding velocity, $a_j(a_j^\dagger )$ is the lowering (raising) operator of the $j^{\textrm {th}}$ TLS, and $\kappa _j$ is the coupling strength between the $j^{\textrm {th}}$ TLS and the fields. The $\hbar \Omega _j$ in the Hamiltonian is an effective energy of the $j^{\textrm {th}}$ TLS, a combination of energy difference between the two levels $\hbar \omega _j$ and the amount of dissipation/pumping rate [30] to the $j^{\textrm {th}}$ TLS is $g_j$, i.e. ${\Omega _j \equiv \omega _j + i g_j}$. The numbering of TLSs are ordered from left to right in the following discussions, i.e. ${x_1 \leq x_2 \leq \cdots \leq x_N}$.

Letting $|{\omega }\rangle$ be a single-particle state, namely

(3)$$\begin{aligned}|{\omega}\rangle & = \Bigg[ \int dx' \psi_{\textrm{ R}}(x') c_{\textrm{ R}}^\dagger (x') + \int dx' \psi_{\textrm{L}}(x') c_{\textrm{L}}^\dagger (x') \\ & \quad + \sum_j e_j a_j ^\dagger \Bigg] |{\textrm{vac}}\rangle, \end{aligned}$$

where $|{\textrm {vac}}\rangle$ is the vacuum state [31]. Using ${H |{\omega }\rangle = \hbar \omega |{\omega }}\rangle$, we find that the wave equations and the population amplitudes have to satisfy the following equations:

(4)$$\begin{aligned}& \left(- i v \partial_x - \omega \right) \psi_{\textrm{ R}} (x) = - \sum_{j=1}^N \kappa_j e_j \delta(x-x_j), \\ & \left(i v \partial_x - \omega \right) \psi_{\textrm{L}} (x) = - \sum_{j=1}^N \kappa_j e_j \delta(x-x_j), \\ & \left( \omega - \Omega_i \right) e_i = \kappa_i^* \left[ \psi_{\textrm{ R}} (x_i) + \psi_{\textrm{L}} (x_i) \right], \end{aligned}$$

where $\psi _{\textrm { R(L)}}$ is the wave function of the right (left) going fields and $e_j$ is the population amplitude of the $j^{\textrm {th}}$ TLS. By solving the above differential equations, we arrive at the following set of equations:

(5)$$\begin{aligned}& \psi_{\textrm{ R}} (x) = \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{i \frac{\omega (x - x_j)}{v}}\textrm{sgn} (x - x_j), \\ & \hspace{4em} + R e^{i \frac{\omega x}{v}} \end{aligned}$$

(6)$$\begin{aligned}& \psi_{\textrm{L}} (x) = \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{- i \frac{\omega (x - x_j)}{v}}\textrm{sgn} (x_j - x), \\ & \hspace{4em} + L e^{-i \frac{\omega x}{v}} \end{aligned}$$

(7)$$\left( \omega - \Omega_i \right) e_i = \kappa_i^* \left[ \psi_{\textrm{ R}} (x_i) + \psi_{\textrm{L}} (x_i) \right],$$

where $R$ and $L$ come from the homogeneous part of the differential equations. It is worth noting that since the Hamiltonian in Eq. (2) is first order in derivative, there should be no eigenstates with real eigenenergies since $\psi _{\textrm { R}}$ or $\psi _{\textrm {L}}$ does not decay fast enough at the infinity. Nevertheless, following the idea in the previous section and choose

(8)$$\left\lbrace\begin{array}{l} \displaystyle R = \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{i \frac{\omega (x - x_j)}{v}}\\ \displaystyle L = -\sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{- i \frac{\omega (x - x_j)}{v}} \end{array}\right.,$$

the state becomes:

(9)$$\psi_{\textrm{ R}} (x) = \frac{-i}{v} \sum_{j=1}^N \kappa_j e_j e^{i\frac{\omega (x-x_j)}{v}} \theta (x-x_j),$$

(10)$$\psi_{\textrm{L}} (x) = \frac{-i}{v} \sum_{j=1}^N \kappa_j e_j e^{-i\frac{\omega (x-x_j)}{v}} \theta (x_j-x),$$

(11)$$\left( \omega - \Omega_i \right) e_i = \kappa_i^* \left[ \psi_{\textrm{ R}} (x_i) + \psi_{\textrm{L}} (x_i) \right].$$

After the Wick rotation, $\omega \rightarrow i \omega$, the wave functions of Eqs. (9–10) becomes square-integrable and, therefore, the state is regularizable. In fact, these solutions can be obtained much easier by Wick rotating Eqs. (4) first. However, not every $\omega$ are compatible with Eqs. (9–11) without making the wave function trivially zero. To have a non-trivial wave function, the value of $\omega$ should satisfy

(12)$$0 = \begin{pmatrix} \Delta_1 & - e^{-i \theta_{12}} & \cdots & - e^{-i \theta_{1N}}\\ - e^{i \theta_{21}} & \Delta_2 & \cdots & - e^{-i \theta_{2N}}\\ \vdots & \vdots & \ddots & \vdots\\ - e^{i \theta_{N1}} & - e^{i \theta_{N2}} & \cdots & \Delta_N \end{pmatrix} \begin{pmatrix} \displaystyle e_1\\ \displaystyle e_2\\ \displaystyle \vdots\\ \displaystyle e_N \end{pmatrix} ,$$

where $\theta _{ij} \equiv \dfrac {\omega (x_i - x_j)}{v} ~ \textrm {and} ~ \Delta _k \equiv \dfrac {i v \left ( \omega - \Omega _k \right )}{|\kappa _k|^2} - 1$. We will denote the $\omega$ that satisfies the above equation as $\omega _{\textrm {R}}$ to distinguish it from the arbitrary scattering frequency $\omega$. The above equation can be found by insetting Eq. (11) with Eqs. (9–10).

The only solution to the matrix equation is when all the $e_j$’s are identically zero unless the rank of the square matrix is less than $N$, i.e.

(13)$$ 0 = \det \begin{pmatrix} \Delta_1 & - e^{-i \theta_{12}} & \cdots & - e^{-i \theta_{1N}}\\ - e^{i \theta_{21}} & \Delta_2 & \cdots & - e^{-i \theta_{2N}}\\ \vdots & \vdots & \ddots & \vdots\\ - e^{i \theta_{N1}} & - e^{i \theta_{N2}} & \cdots & \Delta_N \end{pmatrix}.$$

In a nutshell, the regularizable eigenenergy is a solution to Eq. (13). Note that Eq. (13) is a transcendental equation for $N > 1$. We work out the regularized eigenenergy for $N=1,2$ in the rest of the section and show that the scattering matrices in these cases indeed has a pole at $\omega = \omega _{\textrm { R}}$, which means that they are indeed Gamow states.

3.1 $N=1$ case

For a single TLS system, Eq. (13) degenerates into a simple algebraic equation:

(14)$$\omega_{\textrm{ R}} - \omega_1 - i \left(g_1 - \frac{|\kappa_1|^2}{v} \right) = 0.$$

It is obvious that to have a real $\omega _{\textrm { R}}$, the system needs ${v g_1 = |\kappa _1|^2}$. The eigenenergy of the state is the same as the energy difference of the TLS, i.e. ${\omega _{\textrm { R}} = \omega _1}$.

Using the general form of scattering matrix in Eq. (33), the scattering matrix for states with energy $E = \hbar \omega$ of $N=1$ case is written as

(15)$$S(E) = \frac{1}{d_1(\omega)} \begin{pmatrix} \omega - \omega_1 - i g_1 & - i \dfrac{|\kappa_1|^2}{v}\\ -i \dfrac{|\kappa_1|^2}{v} & \omega - \omega_1 - i g_1 \end{pmatrix},$$

where $d_1(\omega ) = \omega - \omega _1 - i \left (g_1 - |\kappa _1|^2 / v \right )$ is the overall denominator. It can be easily seen that when ${v g_1 = |\kappa _1|^2}$, $d_1(\omega ) = 0$ when ${\omega = \omega _{\textrm { R}} = \omega _1}$. In fact, ${E = \hbar \omega _1}$ is a simple pole for the transmission and reflection amplitudes, which means that the transmission and reflection coefficients blows up at the pole.

3.2 $N=2$ case

The previous example shows some important features of a regularizable eigenstate. However, the drawback is that the pole lies at the energy difference of the TLS. Using two TLSs, we can move the eigenenergy of the regularizable eigenstate away from the energy difference of the TLS. The evidence becomes stronger if the pole of the scattering matrix is at the eigenenergy of the regularizable eigenstate but not the energy difference of the TLSs.

In two-TLS case, Eq. (13) becomes a transcendental equation. After replacing the concise notation with the physical quantities, the equation becomes

(16)$$\begin{aligned} & \left(\omega_{\textrm{ R}} - \omega_1 - i g_1 + i \frac{|\kappa_1|^2}{v} \right) \left(\omega_{\textrm{ R}} - \omega_2 - i g_2 + i \frac{|\kappa_2|^2}{v} \right)\\ & + \frac{|\kappa_1|^2|\kappa_2|^2}{v^2} \exp[\frac{2 i \omega_{\textrm{ R}} (x_2-x_1)}{v}] = 0.\end{aligned}$$

It is not hard to see that we have more freedom to move $\omega _{\textrm { R}}$ away from both $\omega _1$ and $\omega _2$. With the proper choice of $\omega _{1/2}$, $\kappa _{1/2}$, $g_{1/2}$, $v$, and $x_2 - x_1$, the $\omega _{\textrm { R}}$ can become real, which is the regularizable eigenstate eigenenergy.

From the aspect of scattering matrix, we look for the parameters that render the transmission and reflection amplitudes diverge. Rather than showing the full messy scattering matrix, we focus on the overall denominator, which is proportional to

(17)$$\begin{aligned}d_2(\omega) = & \left(\omega- \omega_1 - i g_1 + i \frac{|\kappa_1|^2}{v} \right) \left(\omega - \omega_2 - i g_2 + i \frac{|\kappa_2|^2}{v} \right)\\ & + \frac{|\kappa_1|^2|\kappa_2|^2}{v^2} \exp[\frac{2 i \omega (x_2-x_1)}{v}]. \end{aligned}$$

Compared with Eq. (16), $\displaystyle \lim _{\omega \rightarrow \omega _{\textrm { R}}} d_2(\omega ) = 0$ if a regularizable eigenstate exists, which means the transmission and reflection coefficients tends toward infinity at this energy. To demonstrate this filtering effect, we plot in Fig. 2 the transmission spectrum with the parameters adopted from Rydberg-atom systems [32,33].

Fig. 2. The logarithm of the transmission coefficient in $N=2$ case with $\omega _1=9.95 \times 10^{-1}$ GHz, $\omega _2=1.005$ GHz, $|\kappa _1|^2 / v = 10$ MHz, $|\kappa _2|^2 / v = 8.1$ MHz, $g_1 = 10$ MHz, and $\theta _{21} \sim 0.6285$. As derived from Eq. (16), the pole appears when $g_2 = g_{\textrm { R}} \sim 4.702 \times 10$ MHz with eigenenergy $\omega _R = 1$ GHz. It can be seen from the figure that it tends to infinity at the eigenfrequency of the regularizable eigenstate in the parameter space. In plotting this figure, the parameters are adopted from Rydberg-atom systems.

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Although we only showed one- and two-TLS cases, this conclusion still holds in any arbitrary N-TLS systems. We provide the detailed proof in the Appendix B.

4. The physical meaning of the poles

The absolute value of the transmission or the reflection amplitudes tends to infinity might not seems physical at first. However, the unphysical output actually comes from the unphysical input. To really understand what these infinities mean, we take one step back and discuss the use of the scattering matrix. Roughly speaking, a scattering matrix is a machine that links the outgoing semi-free particle states to the incoming ones. That is to say it maps the incoming state to outgoing state as $|{O}\rangle _{\textrm {s}} = S |{I}\rangle _{\textrm {s}}$, where $S$ is the scattering matrix operator and the “s” in the subscript stands for scattering state. For example, given an incoming semi-free particle state, the scattering matrix maps the incoming state ${|{I(E)\rangle }_{\textrm {s}}}$ to the outgoing state ${[t(E) |{T(E)\rangle }_{\textrm {s}} + r(E) |{R(E)\rangle \rangle }_{\textrm {s}}]}$ where $t(E)$ and $r(E)$ are the transmission and reflection amplitudes and the $|{T(E)\rangle }_{\textrm {s}}$ and $|{R(E)\rangle }_{\textrm {s}}$ are the corresponding states. Since the scattering matrix has poles when $E$ approaches the regularizable eigenenergies at which both $|t(E)|^2$ and $|r(E)|^2$ tend to infinity.

However, the input in the above setting composes of a single generalized eigenstate, which is not a real physical state since it is not normalizable. Therefore, it should not be surprising that the output state is unphysical. A real physical incoming state needs to involve a range of energies, which means that it can be expanded with the generalized eigenstates, $|{I}\rangle = \int _0^\infty dE ~\! c(E) |{I(E)\rangle }_{\textrm {s}}$, where $c(E)$ is the energy distribution of the incoming state. We can now use the scattering matrix operator to find its outgoing state as follows:

(18)$$\begin{aligned}|{O}\rangle & = S |{I}\rangle = \int_0^\infty dE ~\! c(E) S |{I(E)\rangle}_{\textrm{s}} \\ & = \int_0^\infty dE ~\! c(E) \big[ t(E) |{T(E)\rangle}_{\textrm{s}} + r(E) |{R(E)\rangle}_{\textrm{s}} \big]. \end{aligned}$$

In the above expression, $c(E)t(E)$ can now be considered as the distribution of the transmitted state and $c(E)r(E)$ as the distribution of the reflected states. This means that the infinities in the scattering matrix are merely poles in the distribution of the outgoing state. There is a systematic treatment of these in mathematics, namely Cauchy contour integral. That is to say, Gamow states produces no unphysical infinities but are some of the main ingredients of the outgoing states.

Furthermore, if the distribution of the incoming state ${c(E)}$ decays fast enough at ${E \rightarrow 0}$, the incoming state can be approximated by extending the lower bound of integration from $0$ to $-\infty$, i.e.

(19)$$\begin{aligned}|{O}\rangle & \approx \int_{-\infty}^\infty dE ~\! c(E) \big[ t(E) |{T(E)\rangle}_{\textrm{s}} + r(E) |{R(E)\rangle}_{\textrm{s}} \big] \\ & = \sum_i \left[ C_{\textrm{T}i} |{T(\hbar \omega_{\textrm{ R}i})\rangle} + C_{\textrm{R}i} |{R(\hbar \omega_{\textrm{ R}i})\rangle}\right] \\ & \hphantom{=} + ~ \textrm{bound state contributions}, \end{aligned}$$

where

(20)$$\begin{aligned}C_{\textrm{T}i} &= \pi i \underset{E \rightarrow \hbar \omega_{\textrm{R}i}}{\textrm {Res}} c(E)t(E), \\ C_{\textrm{R}i} &= \pi i \underset{E \rightarrow \hbar \omega_{\textrm{R}i}}{\textrm {Res}} c(E)r(E), \end{aligned}$$

and $\omega _{\textrm {R}i}$ is the eigenenergy of the $i^{\textrm {th}}$ Gamow state providing ${|c(E)| \approx 0}$ for ${E < 0}$. In other words, if $c(E)$ decays fast enough at $E \rightarrow 0$, the outgoing state is almost made of the Gamow states when measured far from the interacting region since bound states are localized. Most of these states are decaying states, however, the real-eigenenergy Gamow states, like the ones discussed above, are stationary. Therefore, the poles on the real axis lead to a frequency filtering effect on the outgoing states, which suppresses most of the contributions from other generalized eigenstates other than the real-eigenenergy Gamow states. Since the eigenenergies of the Gamow states are determined by the parameters of the system, such as the TLS energy spacings and the coupling constants in the worked-out examples above, the filtering frequency can be tuned by changing the system set-up.

Actually, this filtering effect is quite similar to stimulated emission in a standard laser system. When the frequency of the incoming bosonic field and the dissipation/pumping rates match the condition forming the Gamow states, the incident field stimulates the “emission" of the TLS. The characteristic frequency in the bosonic field is therefore amplified.

A few remarks on experimental realizations should be addressed here. The real-eigenenergy Gamow states can be achieved experimentally through the scattering of the incident bosonic fields on the one-dimensional array of two-level atoms [34–36]. The non-Hermitian pumping leading to the amplification of the scattering coefficients can be implemented by degenerate four-wave mixing [37], index guiding [38], electrically pumping devices [39], and laser pumping [40–42]. To avoid the remnant contributions from the BIC and other bound states, the measurement device should be placed distant from where the scattering take place since these states are localized in some finite regions and do not radiate at all [10,43]. Moreover, due to the frequency-filtering nature of the real-eigenenergy Gamow states, the lineshape of the scattering spectrum [27–29,44] and its resonant frequency can help to distinguish the Gamow states from the ordinary scattering states. It should be noted that although it is not easy to precisely achieve the filtering condition introduced in Sec. 3, the frequency filtering effect emerges when the experimental parameters are fairly close to the theoretical ones. The two panels in Fig. 3 show the energy probability densities of the transmitted waves (the blue-solid curves) under the same incoming Gaussian-distributed wave (the red-dashed curves) with different pumping rates on the second TLS. Although none of them is at the exact ideal value, the peak in the lower panel, which has the rate closer to the ideal value compared with that of the upper panel, is already significant. This result provides an evidence that this filtering effect can be still observed even if the experimental condition is slightly deviated from the ideal value.

Fig. 3. Comparing the probability densities between the incoming and the outgoing states in an $N=2$ TLS. The incoming probability densities (the red-dashed curves) are $\left | c(\omega )/(\int d\omega ' |c(\omega ')|^2) \right |^2$, where $c(\omega )$ is assumed to be a Gaussian curve centered around $\omega _{\textrm {in}} = 1.5 \omega _{\textrm {R}}$ with standard deviation $\sigma = 0.5 \omega _{\textrm {R}}$. The outgoing ones (the blue-solid curves) are $\left | d(\omega ) /(\int d\omega ' |d(\omega ')|^2) \right |^2$, where $d(\omega ) = c(\omega ) t(\omega )$ and the transmission amplitude $t(\omega )$ is found in the scattering matrix. The parameters in both upper and lower panels are taken directly from Fig. 2 with $g_2$ differing from the ideal value $g_{\textrm {R}}$. The $g_2$ in the upper panel is about $1.411 \times 10$ MHz, while the $g_2$ in the lower panel is $\sim 2.821 \times 10$ MHz. In the upper panel, the majority of the outgoing frequencies are the same as the incoming ones with a small peak around $\omega _{\textrm {R}}$. On the other hand, the peak around $\omega _{\textrm {R}}$ in the lower panel becomes quite significant even though $g_2$ is not exactly at $g_{\textrm {R}}$.

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5. Conclusions

The distinction between bound states and scattering states is quite obvious: The norms of the former are finite and the latter are not. However, if the time dimension is complexified, the boundary of the two becomes fuzzy. Although rare, we discover the norm of some scattering states can be regularized using the Wick rotation. The eigenenergies of these states, like the ones of bound states, are poles of the scattering matrix and, therefore, are Gamow states. Moreover, with the right setup, these eigenenergies can be real and positive. That is to say these poles appear within the continuous spectrum of the scattering states, which is very similar to the case of BIC. However, BIC is an actual bound state and the regularizable states are still scattering states which are non-localized.

Stemming from the properties of these regularizable states, the frequency of the outgoing states will be made mostly of the eigenenergies of the regularizable states. A feasibly tunable quantum frequency filter is proposed utilizing the nature of the regularizable states.

Appendix A Scattering matrix for $N$-TLS

The scattering matrix for the $N$-TLS is the relation between the inputs ($I_{\textrm {f}}$ and $I_{\textrm {f}}$) and the outputs ($O_{\textrm {f}}$ and $O_{\textrm {f}}$) away from the system (see Fig. 4). From Eqs. (5–7), we find

(21)$$I_{\textrm{f}} = R - \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{-i\omega x_j},$$

(22)$$I_{\textrm{b}} = L - \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{i\omega x_j}, $$

(23)$$O_{\textrm{f}} = R + \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{-i\omega x_j}, $$

(24)$$ O_{\textrm{b}} = L + \sum_{j=1}^N \frac{\kappa_j e_j}{2iv} e^{i\omega x_j}. $$

Fig. 4. A schematic diagram for the general $N$-TLS.

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To find the scattering matrix $S$, let $\displaystyle {\mathcal {O} = \begin {pmatrix}O_{\textrm {f}} \\O_{\textrm {b}} \end {pmatrix}}$ and $\displaystyle {\mathcal {I} = \begin {pmatrix}I_{\textrm {f}}\\I_{\textrm {b}}\end {pmatrix}}$, then $S$, by definition, is

(25)$$\mathcal{O} = S \mathcal{I}.$$

Equation (7) together with Eq. (5) and Eq. (6) can be written as

(26)$$\begin{aligned}& \overbrace{\begin{pmatrix} \delta_1 & - \sin \theta_{12} & \cdots & - \sin \theta_{1N}\\ \sin \theta_{21} & \delta_2 & \cdots & - \sin \theta_{N1}\\ \vdots & \vdots & \ddots & \vdots\\ \sin \theta_{N1} & \sin \theta_{N1} & \cdots & \delta_N \end{pmatrix}}^{\displaystyle \mathcal{K}}~~~~ \overbrace{\begin{pmatrix} \displaystyle \kappa_1 e_1\\ \displaystyle \kappa_2 e_2\\ \displaystyle \vdots\\ \displaystyle \kappa_N e_N \end{pmatrix}}^{\displaystyle \mathcal{E}} \\ & = v \underbrace{\begin{pmatrix} \displaystyle e^{i \omega x_1} & \displaystyle e^{- i \omega x_1}\\ \displaystyle e^{i \omega x_2} & \displaystyle e^{- i \omega x_2}\\ \displaystyle e^{i \omega x_3} & \displaystyle e^{- i \omega x_3}\\ & \hspace{-1cm}\vdots & \end{pmatrix}}_{\displaystyle \mathcal{Q}} \underbrace{\begin{pmatrix} \displaystyle \vphantom{e^{ipx_1}} R\\ \displaystyle \vphantom{e^{ipx_1}} L \end{pmatrix}}_{\displaystyle \mathcal{U}}, \end{aligned}$$

where $\theta _{ij} \equiv \dfrac {\omega (x_i - x_j)}{v} ~ \textrm {and} ~ \delta _k \equiv \dfrac {v \left ( \omega - \Omega _k \right )}{|\kappa _k|^2}$.

From Eqs. (21–24) and Eq. (26), we find

(27)$$\mathcal{I} = \mathcal{U} - \frac{1}{2iv} \mathcal{Q}^\dagger \mathcal{E},$$

(28)$$\mathcal{O} = \mathcal{U} + \frac{1}{2iv} \mathcal{Q}^\dagger \mathcal{E},$$

and

(29)$$\mathcal{E} = v \mathcal{K}^{-1} \mathcal{Q} \mathcal{U}.$$

In the rare case that $\det \mathcal {K} = 0$ for some $\omega = \omega _0 \in \mathbb {R}$, one can still proceed with Eq. (29) formally and take $\omega \rightarrow \omega _0$ at the end of calculation. Rearranging the above equations, we find

(30)$$\mathcal{I} = \left( \mathbb{1}_{2 \times 2} - \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) \mathcal{U},$$

(31)$$\mathcal{O} = \left( \mathbb{1}_{2 \times 2} + \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) \mathcal{U},$$

where $\mathbb {1}_{2 \times 2}$ is a $2 \times 2$ identity matrix. It is obvious that

(32)$$\mathcal{O} = \left( \mathbb{1}_{2 \times 2} + \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) \left( \mathbb{1}_{2 \times 2} - \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right)^{-1} \mathcal{I}.$$

The scattering matrix $S$ for arbitrary $N$-TLS can be read out by comparing Eq. (25) and Eq. (32)

(33)$$S = \left( \mathbb{1}_{2 \times 2} + \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) \left( \mathbb{1}_{2 \times 2} - \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right)^{-1}.$$

Appendix B The pole condition of the scattering matrix for $N$-TLS

From Eq. (33), the poles of the scattering matrix comes from

(34)$$\det \left( \mathbb{1}_{2 \times 2} - \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) = 0,$$

where $\mathcal {K}$ and $\mathcal {Q}$ are defined in Eq. (26). This condition is only true when there is a zero eigenvalue. In other words, there is at least an eigenvector corresponds to the zero eigenvalue, i.e.,

(35)$$\left( \mathbb{1}_{2 \times 2} - \frac{1}{2i} \mathcal{Q}^\dagger \mathcal{K}^{-1} \mathcal{Q} \right) \Psi = 0,$$

where $\Psi$ is an $2 \times 1$ matrix. Applying a $\mathcal {Q}$ from the left renders

(36)$$\left( \mathbb{1}_{N \times N} - \frac{1}{2i} \mathcal{Q} \mathcal{Q}^\dagger \mathcal{K}^{-1} \right) \mathcal{Q} \Psi = 0$$

(37)$$\Rightarrow \left( \mathcal{K} - \frac{1}{2i} \mathcal{Q} \mathcal{Q}^\dagger \right) \Phi = 0,$$

where $\Phi = \mathcal {K}^{-1} \mathcal {Q} \Psi$ and the $\mathbb {1}_{N \times N}$ is an $N \times N$ identity matrix since $\mathcal {Q}$ is an $N \times 2$ matrix. The above equation shows that $\Phi$ is an eigenvector of $\mathcal {K} - \frac {1}{2i} \mathcal {Q} \mathcal {Q}^\dagger$ with eigenvalue zero. That is to say,

(38)$$\det \left( \mathcal{K} - \frac{1}{2i} \mathcal{Q} \mathcal{Q}^\dagger \right) = 0$$

(39)$$\Rightarrow \det \left( i \mathcal{K} - \frac{1}{2} \mathcal{Q} \mathcal{Q}^\dagger \right) = 0,$$

which is exactly Eq. (13). Therefore the regularizable condition corresponds to a pole in the scattering matrix for $N$-TLS systems.

Funding

Ministry of Science and Technology, Taiwan (107-2628-M-006-002-MY3, 108-2112-M-005-005, 108-2627-E-006-001); Army Research Office (W911NF-19-1-0081).

Acknowledgments

C.Y.J. and G.Y.C. acknowledge the support of the Ministry of Science and Technology, Taiwan, grant number MOST 108-2112-M-005-005. Y.N.C. acknowledges the support of the Ministry of Science and Technology, Taiwan [Grants No. MOST 107-2628-M-006-002-MY3, MOST 108-2627-E-006-001, and Army Research Office (Grant No. W911NF-19-1-0081)].

Disclosures

The authors declare no conflicts of interests.

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Optical quantum frequency filter based on generalized eigenstates

Abstract

1. Introduction

2. Regularizing generalized (scattering) eigenstates

3. Field-TLS interaction

3.1 $N=1$ case

3.2 $N=2$ case

4. The physical meaning of the poles

5. Conclusions

Appendix A Scattering matrix for $N$-TLS

Appendix B The pole condition of the scattering matrix for $N$-TLS

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (4)

Equations (39)

Optics Express