# Trapped modes in linear quantum stochastic networks with delays

- Gil Tabak
^{1}Email authorView ORCID ID profile and - Hideo Mabuchi
^{}

**3**:3

https://doi.org/10.1140/epjqt/s40507-016-0041-9

© Tabak and Mabuchi 2016

**Received: **26 October 2015

**Accepted: **22 February 2016

**Published: **3 March 2016

## Abstract

Networks of open quantum systems with feedback have become an active area of research for applications such as quantum control, quantum communication and coherent information processing. A canonical formalism for the interconnection of open quantum systems using quantum stochastic differential equations (QSDEs) has been developed by Gough, James and co-workers and has been used to develop practical modeling approaches for complex quantum optical, microwave and optomechanical circuits/networks. In this paper we fill a significant gap in existing methodology by showing how trapped modes resulting from feedback via coupled channels with finite propagation delays can be identified systematically in a given passive linear network. Our method is based on the Blaschke-Potapov multiplicative factorization theorem for inner matrix-valued functions, which has been applied in the past to analog electronic networks. Our results provide a basis for extending the Quantum Hardware Description Language (QHDL) framework for automated quantum network model construction (Tezak *et al.* in Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 370(1979):5270-5290, 2012) to efficiently treat scenarios in which each interconnection of components has an associated signal propagation time delay.

## Keywords

## 1 Introduction

Just as in classical electrical and light-wave circuit design, there are many quantum network modeling scenarios in which it is necessary to capture the impact of time delays in the propagation of signals between components. For example in large-area communication networks there is an obvious need to analyze synchronization issues; in integrated photonic circuits the high natural bandwidth of nanoscale components may create problems of delay-induced feedback instability, and may support the design of devices (such as oscillators) that exploit finite optical propagation delays.

In considering how best to represent and simulate time delays in quantum networks we would like to strike an expedient balance between the need to minimize additional computational overhead and the desire to derive intuitive approximate models. Our main interest in this paper is to develop a systematic approach to modeling the leading-order effects of signal propagation time delays in a linear passive quantum optical network that can be specified naturally using the so-called SLH formalism of Gough, James and co-workers [2–7]. Whereas series and feedback interconnections of open quantum systems in the SLH formalism are generally treated as having vanishing signal propagation time delay, we seek to expand the formalism in a natural way that allows each interconnection to have an associated finite delay. Our approach utilizes additional degrees of freedom to capture the behavior of trapped resonant modes created by the system’s internal network of feedback pathways and time delays, targeting a specific frequency range that corresponds to the intrinsic bandwidths of components in the network.

The study of time-delay systems has a long history (see [8] for a thorough overview). In the context of quantum systems, a quantum control scenario incorporating time-delayed coherent feedback has been analyzed recently by Grimsmo [9]. The construction developed by Grimsmo appears most suitable for use in scenarios with very large feedback time delay, requiring a much higher computational overhead than should be necessary when propagation delays are relatively small. Very recently Pichler and Zoller [10] have described an approach to modeling the dynamics of a finite-delay quantum channel that exploits the Matrix Product State formalism for computational efficiency, and demonstrate its use in analyzing quantum feed-forward and feedback dynamics. Our work here is distinguished by showing how networks incorporating feedback via many coupled signal channels can be treated efficiently and by focusing on SLH-compatible modeling at the level of quantum stochastic differential equations (QSDEs). The method we present can straightforwardly be incorporated into the Quantum Hardware Description Language (QHDL) framework [1] for automated model construction for complex quantum networks.

## 2 Preliminaries

In the spirit of SLH/QHDL we assume that we are given a network of open quantum systems whose input ports, output ports, and static passive linear components are connected over some channels representing the signal propagating in the network. We additionally assume that an end-to-end propagation time delay is specified for each channel. If feedback loops are created by the network topology, trapped modes will be created that may need to be modeled dynamically in order to accurately simulate the overall behavior of the network.

The basic problem lies in choosing a procedure for embedding new stateful dynamics into the ‘space between components’ in an SLH network. Prior work such as [10] has addressed the question of how to model an individual channel with finite time delay efficiently, however, our philosophy here will be to work at the level of more complex sub-networks that mediate interconnections among multiple-input/multiple-output components. Our method is restricted to sub-networks that are linear and passive, and thus may include components such as beam-splitters and phase shifters but not, *e.g.*, gain elements or nonlinear traveling-wave interactions. We nevertheless gain a significant advantage by considering linear passive sub-networks in that we are able to recognize the creation of trapped modes by feedback with finite time delays, and can provide a systematic procedure for adding the stateful dynamics required to simulate the behavior of such modes within a frequency band of interest.

We treat channels as passive linear quantum stochastic systems [11–14] whose input-output behavior can be characterized by the relationship of the input and output annihilation fields only. The input-output relationship for a linear system must satisfy certain physical realizability conditions. Our systematic method preserves the physical realizability condition while allowing us to simulate the system with only a small number of degrees of freedom. Our resulting approximate system describing the dynamics of a passive linear sub-network can also be combined with other possibly nonlinear components using the standard SLH composition rules. Incorporating nonlinear components embedded within a network whose topology results in trapped modes may need a more delicate treatment. To do this, the interaction Hamiltonian between the trapped modes and the nonlinear components must be found. This is an issue we will write about in more detail in a future publication.

*N*input and output ports and

*M*oscillator modes \(a_{1},\ldots,a_{M}\) satisfying the canonical commutation relations

*A*,

*B*,

*C*,

*D*are complex-valued matrices of appropriate size. The \(B_{i}\) and \(B_{\mathrm{out},i}\) are the forward differentials of adapted quantum stochastic processes satisfying the commutation relations with their respective operator adjoints, in the sense of [15, 16]:

Our approach seeks to approximate the transfer function within a given frequency range by selecting only a finite subset of the original modes and generating a state-space representation using the information close to the zeros or poles of the modes. The resulting approximation is a passive linear system satisfying the physical realizability condition. We discuss a sufficient condition for our approximation to converge to the true transfer function for a large class of possible transfer functions.

There are other approaches that could be used to obtain a different set of modes that may approximate the system of interest. For example, one approach may involve approximating each delay term in the system using a symmetric Padé approximation, which would result in a physically realizable component (see Appendix D). Although this approach can be simple to use, the Padé approximation will not always introduce the zeros and poles of the transfer function at the correct locations, and may introduce spurious zeros and poles to the approximated transfer function. On the other hand, the zero-pole interpolation by construction adds zeros and poles to the approximated transfer function only when they are present in the original transfer function. This feature of our approach may be important in many physical applications because the locations of the zeros and poles have physically meaningful consequences, including the resonant frequencies and linewidths of the effective trapped cavity modes resulting in the network due to feedback. Nevertheless, as discussed in Section 8, there are passive linear systems for which the zero-pole interpolation is insufficient - in this case, a finite-dimensional state-space representation will necessarily have spurious zeros and poles.

## 3 Problem characterization

### 3.1 Frequency domain

*s*-domain unless otherwise noted). A function of time \(f(t)\) can be mapped to the frequency domain by the Laplace transform, resulting in a function \(F(z)\). The Laplace transform is given by

*transfer function*\(T(z)\). The transfer function is defined by the relation between the inputs \(\mathbf{I(z)}\) and outputs \(\mathbf{O(z)}\) by \(\mathbf{O}(z) = T(z) \mathbf{I}(z)\). For example, we can find the transfer function of the system described by Eq. (2) by taking the Laplace transform of the equation. After some algebra, the transfer function is found to be

### 3.2 Problem characterization in the frequency domain

We consider a linear system with *N* input and *N* output ports. We will primarily be interested in a system which is linear, passive, and has a transfer function \(T(z)\) that is unitary for all \(z\in i \mathbb{R}\). The last condition guarantees that the system conserves energy. We remark that more generally the loss of energy can be considered for example by adding additional ports. We assume throughout the transfer function is a meromorphic matrix-valued function. For simplicity, we assume that each pole has multiplicity one.

*T*has transfer function of the form \(e^{-zT}\). In general, any system of delays and beamsplitters can be written as

*N*-dimensional input and output signals, and

*x*are the internal signals of the system. The values of

*x*are taken along edges corresponding to delays before each signal is delayed. The \(M_{i}\) are constant matrices of the appropriate size determined by the specific details of the system. \(E(z)\) is a diagonal matrix whose diagonal elements are the transfer functions of the various delays in the system, \(e^{-zT_{1}},e^{-zT_{2}},\ldots,e^{-zT_{N}}\). The system is illustrated abstractly in Figure 1.

*z*satisfying \(\det (T(z)) = 0\).

Throughout, we will also assume that the system is asymptotically stable. In terms of the network transfer function, all the eigenvalues of \(M_{1}\) have norm less than 1, with the consequence that \(T(z)\) is bounded for \(\Re(z) \ge0\).

This transfer function has the important feature of being unitary whenever *z* is purely imaginary. That is, \(T(i\omega)T^{\dagger}(i\omega) = T^{\dagger}(i\omega)T(i\omega)= I\) whenever \(\omega\in\mathbb{R}\). This is exactly the physical realizability condition for a passive linear system. We will refer to this constraint throughout the paper as the *unitarity constraint*. One consequence of this condition that can be obtained by analytic continuation is that \(T(z) T^{\dagger}(-\overline{z}) = I\) except when a pole is encountered.

*Some observations.* We see that the poles and zeros occur in pairs *z*, −*z̅*. In this paper we will refer to such a pair as a *zero-pole pair*. In general, we observe that there may be infinitely many solutions to Eq. (11). If we take \(M_{1}\) to be a real matrix, *z* is a pole whenever *z̅* is also a pole. Furthermore, applying the maximum modulus principle shows the system is stable (see Appendix B). This implies that the poles appear in the left half-plane.

The solution to Eq. (11) can be found numerically within a bounded subset of \(\mathbb{C}\). There are dedicated algorithms that use contour integration to guarantee finding all the roots within a contour, which we briefly discuss in Section 6.1. In the special case when the time delays are commensurate, we can re-write the equation for the poles as a polynomial equation of the variable \(w = e^{-zT_{0}}\) for some \(T_{0}\). Doing this will make the root finding procedure much more simple.

If the system has passive linear components other than delays and beamsplitters, the transfer function shown above may be modified. It will still have the same important features that the poles appear on the left half-plane (except when the system is marginally stable) and the function restricted to the imaginary axis is unitary.

## 4 Specific example and approximation procedure: one time delay and one beamsplitter

*τ*has the transfer function

*N*, \(\tilde{T}_{N}(z)\) is a product over \(n=-N,-N+1,\ldots,N-1,N\), we can drop the \(c_{n}\) factors by symmetry considerations.

One interpretation of each term in the product is the frequency-domain representation of a cavity mode. We can obtain an approximation for \(T(z)\) by truncating the product with a finite number of poles. The resulting rational function can then be interpreted as the transfer function of a finite-dimensional system.

Notice that the procedure used to obtain \(\tilde{T}(z)\) does not guarantee that \(\tilde{T}(z) = T(z)\), although this is indeed the case for the example above. In general the procedure above may not capture some of the properties of the system. For example, if we added another delay with no feedback in sequence, we would obtain an additional phase factor dependent on *z*, while still satisfying the desired properties. For a SISO system satisfying the unitarity constraint with the same root-pole pairs, a phase dependence \(T(z) = \tilde{T}(z) e^{-\alpha z}\) (\(\alpha\ge0\)) is actually the most general modification we might need. For a system satisfying the same conditions but having *N* input and output ports, the \(e^{-\alpha z}\) term will be replaced by a similar singular function which we will refer to throughout as the *singular* term (see Section 5.1 for a discussion). An illustration of \(T(z)\) and the transfer function resulting when an additional delay is augmented to the system is illustrated in Figure 3.

Notice how the augmented delay substantially changes some of the properties of the transfer function in the complex plane. For instance, when \(\Re(z) \to-\infty\), the transfer function in Figure 3(a) approaches a constant, while that in Figure 3(b) diverges. In Section 8 we will be able to utilize the difference in behavior of transfer functions to determine whether they incorporate a nontrivial everywhere analytic term, like the exponential resulting from a time delay.

We can check to see if such a factor is needed in the factorization above. Assuming that \(\lim_{\Re(z) \to-\infty} \tilde{T}(z) = C \ne0\) (see Appendix E), we can take \(\lim_{\Re(z) \to -\infty} T(z)\) to check if the additional phase factor is present in \(T(z)\). We see \(\lim_{\Re(z) \to-\infty} T(z) = -1/r\), which shows in our example that no such additional term is needed, and therefore \(\tilde{T}(z) = T(z)\). It can be confirmed numerically that the values of \(T(z)\) and \(\tilde{T}(z)\) agree.

In the rest of the paper, we will show how a similar procedure can be applied more generally to multiple-input and multiple-output (MIMO) systems.

## 5 Factorization theorem and implications for passive linear systems

Certain kinds of matrix-valued functions can be factorized using the Potapov factorization theorem. A more general factorization theorem is discussed in Appendix C.2. Here we discuss a special case useful for our application. In Section 5.1 below we show how this special form can be obtained using the theorems in Appendix C.2.

### Theorem

*Let*\(T(z)\)

*be a meromorphic matrix*-

*valued function satisfying*\(T(z) T^{\dagger}(z) = I\)

*for*\(z\in i\mathbb{R}\)

*and bounded on the right half*-

*plane*\(\mathbb{C}^{+}\).

*Then we can represent*

*T*

*with the factorization*

*where*

*In the product*,

*each of the terms*\(B_{k}\)

*has the form*

*The*\(B_{k}\)

*terms are the Blaschke*-

*Potapov factors written in the*

*s*-

*plane formalism*(

*see Appendix*A),

*and*

*U*

*and the*\(V_{k}\)

*’s are some unitary matrices*.

*We use the symbol*\(\overset{\curvearrowright}{\int} \)

*to refer to the*multiplicative integral,

*or product integral*.

*The integral above is given by*

*where we take*\(0 = t_{0} \le t_{1}\le\cdots\le t_{n} = \ell\).

*In the integral*, \(K(t)\)

*is a summable non*-

*negative family of Hermitian matrices with*\(\operatorname {tr}[K(t)]=1\)

*on some interval*\([0,\ell]\)

*for some*

*ℓ*.

*The*\(P_{k}\)

*is an orthogonal projection*, \(A_{k}(z)= \frac{z-\lambda_{k} }{z + \overline{\lambda_{k}} } e^{i \phi_{k}}\),

*and*\(e^{i \phi_{k}}= \frac{\vert 1-\lambda_{k}^{2}\vert }{1-\lambda_{k}^{2}} \)

*is a phase factor*.

Notice that each Blaschke-Potapov factor has a zero at \(\lambda_{k}\) and pole at \(-\overline{\lambda_{k}}\). We will refer to the terms \(B(z)\) and \(S(z)\) above as the Potapov product and the singular term, respectively. Both \(B(z)\) and \(S(z)\) are inner functions. A function \(F(z)\) is said to be *inner* if it satisfies the unitarity condition \(F(z) F(z)^{\dagger}= I\) for \(z \in i \mathbb{R}\) and is contractive on the right half-plane (*i.e.*
\(F(z) F^{\dagger}(z) \le I\) for \(z \in\mathbb{C}^{+}\)). When \(B(z)\) is a finite product, the \(V_{k}\) terms can be redefined so that the phase factors \(\phi_{k}\) can be absorbed into the unitary matrix *U*.

### 5.1 Obtaining the special case of the factorization theorem for passive systems

In this section, we will show how the theorems in Appendix C.2 are used to obtain the special form we will use.

First, notice we assumed above that \(T(z)\) is a meromorphic matrix-valued function unitary for \(z\in i\mathbb{R}\) and bounded on \(\mathbb{C}^{+}\). These assumptions hold for the system described in Section 3. By Appendix B, it follows that \(T(z)\) is an inner function on \(\mathbb {C}^{+}\).

Because \(T(z)\) has no poles on \(\mathbb{C}^{+}\), the term \(\bar{B}_{\infty}(z)\) is trivial. Notice that \(S(z)\) resulting in our discussion may only have poles only for \(z \in i \mathbb{R}\), because of the form of the integrand of the multiplicative integral of the Potapov factorization.

### 5.2 Interpretation as cascaded passive linear network

We remark that the Blaschke-Potapov factorization of an inner function can be interpreted as a limiting case of a system of beamsplitters, feedforward delays, and cavity modes.

## 6 Approximation procedure - zero-pole interpolation

In order to reconstruct an approximation for the transfer function \(T(z)\) using only a finite number of modes, we will use a two-step procedure. The first step consists of finding the zero-pole pairs in a region of interest. The second step consists of examining the numerical values of the transfer function near the zeros or poles to obtain the correct form of each of the Blaschke-Potapov terms, which are determined up to a constant unitary factor. The product of the resulting terms will equal a truncated version of the Blaschke-Potapov product discussed in Section 5.1, and will approximate the transfer function in the region of interest.

We will take a transfer function *T* and obtain an *M*-dimensional approximation by identifying appropriate factors for a Blaschke-Potapov product. It is possible that the transfer function may have a nontrivial singular component (*i.e.* a nontrivial everywhere analytic term) as discussed in Section 5.1, in which case the zero-pole interpolation may not reproduce a converging sequence of approximations to the given transfer function \(T(z)\). In this section we assume that the singular term is trivial or otherwise unimportant. In any case we determine *U* in Eq. (15) of Section 5.1 using \(T(0)\).

A trivial example when the above approach might fail is a delay with no feedback at all. In this case, there are no poles to evaluate and the method fails. When a transfer function is entirely singular, or when its singular component cannot be neglected, a different approach will be needed, such as using the Padé approximation. This is discussed in Appendix D.

### 6.1 Identifying mode location

We remind the reader that we take our coordinate system in the *s*-domain. We assume for simplicity that we are interested in the behavior of the system near the origin. However, our procedure can be used to obtain approximations of the given transfer function for arbitrary regions in the *s*-plane. In order to identify the appropriate modes, we find roots of the transfer function of the full system, \(\lambda_{1},\ldots,\lambda_{M}\) (with corresponding poles \(p_{1},\ldots,p_{M}\)). Each root will represent a ‘‘trapped’’ resonant mode. In general, there will be infinitely many such roots in the full system, so it is important to have a criterion for selecting a finite number of roots. Each root will have an imaginary part, which will correspond to the frequency of the mode, and a real part, which is linked to the linewidth of the mode. One criterion might be to select root whose imaginary part falls in some range \([-\omega_{\max},\omega_{\max}]\), so that the approximation is valid for a particular bandwidth. This approximating system may be improved by increasing the maximum frequency, \(\omega_{\max}\). As the number of zero-pole pairs increases, the quality of the approximation increases, but in addition the approximated system will incur a greater number of degrees of freedom.

Luckily there is a well-known technique that can be used based on contour integration developed in [19]. This algorithm runs in a reasonable time and can essentially guarantee that it does indeed find all of the desired points. The latter point is an important feature that most typical root-finding algorithms do not have because they do not utilize the properties of analytic functions. For details about a more polished algorithm see [20]. Methods of this kind require a contour in the complex plane as the input in which the roots of the function will be found. This contour may be, for example, a rectangle in the complex plane. In practice we may make use of symmetries in the system and the known regions where poles and zeros are located.

*i.e.*effective cavity modes have a long lifetime) since the real part of each pole in the system corresponds to the exponent of decay of each mode. The system illustrated Figure 6 originates from Example 2 in Section 7.2. If the maximum possible real part of each root is determined for the system of interest, a computational advantage can be gained since the contour does not need to be extended beyond that value.

### 6.2 Finding the Potapov projectors

The procedure we use assumes that the given transfer function \(T(z)\) has a specific form guaranteed by the factorization theorem (see Eq. (15) in Section 5.1). For the purposes of this section, we neglect the contribution due to the singular term (the \(S(z)\) in Eq. (15)). This procedure is similar to the zero-pole interpolation discussed in [21]. We handle the singular term separately, as we will discuss in Section 8.

*p*. Based on the form of the Blaschke-Potapov factors, we can separate the transfer function into the product

*P*is in general the orthogonal projection matrix onto the subspace where the multiplication by the Blaschke factor takes place. We wish to extract the

*P*given the known location of the pole

*p*, which we assume to be a first-order pole for simplicity. We also assume for simplicity that

*P*is a rank one projection, and so it can be written as the product of a normalized vector

*p*, so \(\tilde{T}(z)\) will be analytic at

*p*. Therefore, the first term on the right hand side goes to zero. Taking \(L \equiv\lim_{z \to p} T(z) (z-p) \), we get

*P*is a rank one projector, have obtained an expression where

*L*must also be rank one. In order to find

*v*we can simply find the normalized eigenvector corresponding to the nonzero eigenvalue of

*L*. This task may be done numerically. Finally, we can find the \(\tilde{T}(z)\) from Eq. (22) above.

*M*desired roots of \(T(z)\) to obtain a factorization

*T*has only the

*M*roots picked. We can approximate \(T_{M}\) with a unitary factor that can be determined from

*T*and the product in Eq. (25) evaluated at some point \(z_{0}\) in the region of interest.

The computer code for this procedure can be found on [22].

## 7 Examples of zero-pole decomposition

*iω*for \(\omega\ge0\) in Figures 8 and 10, respectively. Along both examples, we also plot several approximate transfer functions determined by the zero-pole interpolation of Section 6. The approximate transfer functions correspond to a Blaschke-Potapov product that has been truncated to a certain order. In both examples we see that as we increase the number of terms, the approximation improves. The first example illustrates the case when the zero-pole interpolation converges to the correct transfer function. In the second example, while the zero-pole interpolation appears to converge, the function to which it converges deviates from the original transfer function. This suggests that the singular term \(S(z)\) in Section 5.1 makes a contribution for which the zero-pole interpolation does not account. In Section 8, we discuss a condition for convergence and show how the effects of the singular term may be separated from the rest of the system. Figure 10 also includes the transfer function once the singular term has been removed, demonstrating that the zero-pole approximations converge to that function.

### 7.1 Example 1. Zero-pole interpolation converges to given transfer function

The first example we discuss involves two inputs and two outputs. Figure 7 shows this network explicitly. In Figure 8 we see that the zero-pole interpolation appears to converge to the correct transfer function. We can check this by confirming that the \(M_{1}\) is nonsingular, as we will show in Section 8.

Here \(\tau_{1} = 0.1\), \(\tau_{2} = 0.23\), \(\tau_{3} = 0.1\), \(\tau_{4} = 0.17\), \(r_{1} = 0.9\), \(r_{2} = 0.4\), \(r_{3} = 0.8\).

### 7.2 Example 2. Zero-pole interpolation fails to converge to given transfer function

Here \(\tau_{1} = 0.1\), \(\tau_{2} = 0.039\), \(\tau_{3} = 0.11\), \(\tau_{4} = 0.08\), \(r = 0.9\).

In Figure 10, we see that the zero-pole interpolated transfer functions deviate from the true transfer function in the \((0,1)\) and \((1,1)\) phase components. This demonstrates how in general it is important to consider the singular function. On the other hand, for the systems in consideration it is possible to separate the Blaschke-Potapov product from the singular term, which corresponds to feedforward-only components, as discussed in Section 8.3. In black we graph the transfer function components resulting once the feedforward-only components have been removed. Up to a unitary factor, this function is equal to the infinite Potapov product. We see that the approximated transfer functions from the zero-pole interpolation converge to this function.

## 8 The singular term

In this section, we examine the factorization of the transfer function given in Eq. (15) in Section 5.1. In the form of the fundamental theorem by Potapov that we obtained, we had an infinite product of Blaschke-Potapov factors and a singular term. Although the zero-pole decomposition allowed us to extract the Blaschke-Potapov factors, it gave us no information regarding the singular term. In some systems, it may be crucial to include the singular term to obtain a good approximation of the system. To learn about this term, we will need a different method.

In this section, we give a condition for the singular term to be trivial. This condition can then be specialized to the network from Section 3. Based on this condition, we can develop a method to explicitly separate the network described by Eq. (8) in Section 3 into the Potapov product and the singular term.

### 8.1 Condition for the multiplicative integral term to be trivial

We examine the form of the singular term in the factorization theorem and notice that its determinant becomes large when \(\Re(z) \to-\infty\). To avoid mathematical details, we will assume here that the Blaschke-Potapov product \(\prod B_{k}(z)\) is well-behaved in the limit \(\Re(z) \to-\infty\) in the sense that the limit of the product converges (to a nonzero constant). Justification for this assumption is discussed further in Appendix E. We have the following observations.

### Observation

If \(\lim_{\Re(z) \to-\infty} T(z)\) is a constant, then the multiplicative integral in Eq. (15) of Section 5.1 is a constant.

This follows from the properties of the multiplicative integral defined in Eq. (45) of Appendix C.1.

### Observation

In particular, for the transfer function \(T(z)\) in Eq. (9) of Section 3, \(\lim_{\Re(z) \to-\infty} T(z)\) is a constant if and only if \(M_{1}\) in Eq. (9) is full-rank. This gives a sufficient condition for when the zero-pole expansion converges exactly.

To obtain this result, it is enough to consider the term \((E(-z) - M_{1})^{-1}\) in the limit \(\Re(z) \to-\infty\).

The above observations can be seen in the two examples discussed in Section 7. In Example 1, the \(M_{1}\) matrix is full-rank, while in Example 2 it is not.

### 8.2 Maximum contribution of singular term

For many applications we anticipate that we may be able to drop the contribution of the singular term altogether. One example is an optical cavity in certain regimes. If the lifetime of the modes in the cavity is long in comparison to the delays in the system, we would expect the delays to be less significant. We would like to be able to provide a justification for when it is acceptable to neglect the singular term.

First, we will obtain the maximum value for *ℓ* necessary in the multiplicative integral appearing in Eq. (16) of Section 5.1. This is an important result because it tells us that the lengths of the delays themselves determines the greatest contribution of the singular function.

### Remark

To apply the factorization in Eq. (15) of Section 5.1 to the transfer function in Eq. (9) of Section 3, it suffices to take \(\ell\le\sum_{k} T_{k}\).

This can be seen by noting the scaling of \(\det[(E(-z) - M_{1})^{-1}]\) in the limit \(\Re(z) \to- \infty\).

The above bound occurs in the case of several delays feeding forward in sequence.

We can give one condition under which the singular term can be dropped: \(\vert z \vert \ll 1 / \ell\). Furthermore, crude estimates for the error can now be found using the Taylor expansion of the exponential.

Intuitively, Potapov factors correspond to resonant modes while the singular function corresponds to feedforward-only components. With this interpretation, we see that the zero-pole interpolation yields a transfer function close to the true transfer function when the feedforward-only term can be neglected. We can interpret *ℓ* as an upper bound on the duration of time the signal can spend being fed-forward only. When \(1/\ell\) becomes large with respect to the size of the region of interest in the frequency domain, the feedforward-only terms become unimportant.

### 8.3 Separation of the Potapov product and the singular term in an example

In this section, we discuss how for a network of beamsplitters and delays the Blaschke-Potapov product and the singular term of Section 5.1 can be separated explicitly. We will give a systematic procedure at least with the simplifying assumption that the delays are commensurate (are rational multiples of one another). In practice, one can always approximate the delays to arbitrary precision with commensurate delays, resulting in a large but sparse network.

*k*parallel delays can be commuted with a given a unitary component

*U*of

*k*ports. This is illustrated in Figure 11(a).

Next, it becomes apparent in the new network shown in Figure 11(a) that one of the internal system nodes is unnecessary, since it is followed by a delay of duration zero (call it \(x_{0}\)). For this reason, \(x_{0}\) can be eliminated from the network. In the process, we can combine two of the unitary components preceding and following \(x_{0}\) to form a separate unitary component, illustrated in Figure 11(b). The network depicted in Figure 11(b) can be decomposed into a feedforward-only component followed by a network for which the \(M_{1}\) matrix is invertible. The feedforward-only component consists of the identity applied to In_{0} combined in parallel with the addition of the delay \(\tau_{2}\) to In_{1} - that is, the feedforward component only delays the input from port In_{1} by \(\tau_{2}\). The network following the feedback-only component results from the exclusion of the delay \(\tau_{2}\) in Figure 11(b). Since its \(M_{1}\) matrix is invertible, this network has trivial singular part.

### 8.4 Systematic separation of Potapov product and analytic term for passive delay networks

Suppose we have a system given in the form of Eq. (8) in Section 3. Our observation that in order for the multiplicative integral term to be trivial we need \(M_{1}\) to be invertible suggests that there may be a way to isolate the Potapov product term from the remaining analytic function. We now present a systematic way of doing this. For simplicity we will assume that all the delays are comensurate. Without a loss of generality, we can write the system in such a way that all the delays have equal duration, and therefore \(E(z)\) is a multiple of the identity.

*S*such that

*J*is the Jordan decomposition of \(M_{1}\) such that the zero eigenvalue block is at the right bottom block of

*J*. We introduce \(\bar{x} = S^{-1} x\) and rewrite the equation for

*x*in Eq. (9) of Section 3 as

*x̄*. Denoting the last column of

*S*by \(S_{1}\), the matrix of columns excluding the last as \(S_{\setminus1}\), and the matrix of rows of

*J*excluding the last as \(J_{/1}\), we can write

*i.e.*no signal feeds back to a node from which it originated). The second network takes the \(\tilde{x}_{\mathrm{in}}\) as inputs and yields the outputs

*J̃*is matrix resulting from dropping the last row and column in

*J*. We see that Eq. (35) has the same form to the original equation for

*x*in Eq. (9) in Section 3. The difference is that now the

*J̃*replaces \(M_{1}\), and has one fewer zero eigenvalue. Conveniently,

*J̃*is also in its Jordan normal form, so the procedure can be repeated until the matrix ultimately replacing \(M_{1}\) (call it \(\tilde{M}_{1}\)) has no zero eigenvalues left. In this case \(\tilde{M}_{1}\) is an invertible matrix, which is exactly the condition we needed for the transfer function of the network to consist of only the Potapov product and not the multiplicative integral.

## 9 Relationship to the \(ABCD\) and SLH formalisms

In this section we demonstrate how the approximating system our procedure designs is physically realizable. In particular we show how to extract the *ABCD* and SLH forms for a single term resulting in the truncated Blaschke-Potapov product designed to approximate the transfer function of the system. Since the transfer function is equal to a product of such terms, we can interpret the approximating system as a sequential cascade of single-term elements of this form.

*ABCD*model for a single Potapov factor, begin with the following factor

*B*and

*C*matrices may be chosen. In particular, one choice is also consistent with the form used for passive components in the SLH formalism. The

*ABCD*formalism is related to the SLH formalism in the following way for a passive linear system.

*A*. The only remaining component is the imaginary part of

*A*, which is multiplied by

*i*. Notice the Ω satisfies the condition of being Hermitian.

## 10 Simulations in time domain

We translate our model into the *ABCD* state-space formalism, as discussed in Section 9. Doing this allows us to run a simulation in the time domain. Notice that for linear systems this approach suffices for finding the dynamics in the time domain. We can apply an input field at some frequency and record the output. The relationship between the inputs and outputs at the steady-state will correspond to the value of the transfer function at the appropriate frequency.

*i.e.*\(\omega= 0\)) at one of the ports (port 0) and zero input in the other port (port 1). In the steady-state, the signal will be transmitted from the input port 0 to the output port 1. However, if the initial state of the system is different than the steady-state, we will observe some transient behavior in the system. This transient behavior is captured by our simulation and is demonstrated in Figure 12. Here, we show the outputs of the two ports based on different numbers of modes selected to approximate the cavity formed due to the delay. As the number of modes is increased, we see the signal from the output ports as a function of time approaches a step function, and we better reproduce the time-domain dynamics of the network with feedback loops. The jumps we see in the time-domain correspond physically to times when a propagating signal arrives at one of the ports. This physical interpretation is further explained in Figure 13.

## 11 Conclusion

In this paper we have utilized the Blaschke-Potapov factorization for contractive matrix-valued functions to devise a procedure for obtaining an approximation for the transfer function of physically realizable passive linear systems consisting of a network of passive components and time delays. The factorization in our case of interest consists of two inner functions - the Blaschke-Potapov product, a function of a particular form having the same zeros and poles as the original transfer function, and an inner function having no roots or poles (a singular function). The factors in the Potapov product correspond physically to resonant modes formed in the system due to feedback, while the singular term corresponds physically to a feedforward-only component. We also demonstrate how these two components may be separated for the type of system considered.

The transfer function resulting from our approximation can be used to obtain a finite-dimensional state-space representation approximating the original system for a particular range of frequencies. The approximated transfer function can also be used to obtain a physically realizable component in the SLH framework used in quantum optics. Our approach has the advantage that the zero-pole pairs corresponding to resonant modes are identified explicitly. In contrast, obtaining a similar approximation for a feedforward-only component requires introducing spurious zeros and poles. Our approach has the advantages that we may retain the numerical values of the zero-pole pairs of the original transfer function in our approximated transfer function and that we can conceptually separate these zero-pole pairs from spurious zeros and poles. These advantages may be important in applications and extensions of this work.

We hope that in the future our factorization procedure may be extended to a more general class of linear systems. We also hope to introduce nonlinear degrees of freedom in a similar way to atoms in the Jaynes-Cummings model and quantum optomechanical devices in the presence of modes formed due to optical cavities.

## Declarations

### Acknowledgements

This work has been supported by ARO under grant W911-NF-13-1-0064. Gil Tabak was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program and by a Stanford University Graduate Fellowship (SGF). We would also like to thank Ryan Hamerly, Nikolas Tezak, and David Ding for useful discussions while this work was being prepared.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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