Trapped modes in linear quantum stochastic networks with delays

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.

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.

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.

We will consider the example illustrated in Figure 2, where a single beamsplitter and time delay are combined to form a single-input and single-output (SISO) system. The example here is analogous to the one considered in Section VII B of [5].

In this particular example, we use the convention that a beamsplitter has the transfer function

$$\begin{aligned} T_{BS} = \begin{pmatrix} t & -r \\ r & t \end{pmatrix}, \end{aligned}$$

where \(r^{2} + t^{2} =1\), and a time delay of length τ has the transfer function

$$\begin{aligned} T_{\tau}(z) = e^{-\tau z}. \end{aligned}$$

The transfer function of the system drawn in Figure 2 is given by

$$\begin{aligned} T(z) = \frac{ e^{-\tau z} - r}{1-r e^{-\tau z}}. \end{aligned}$$

(12)

This transfer function is illustrated in Figure 3(a). The poles \(p_{n}\) for \(n \in\mathbb{Z}\) are found to be

$$\begin{aligned} p_{n} = \frac{1}{\tau} \bigl( \ln(r) + 2 \pi i n \bigr). \end{aligned}$$

(13)

Notice that the real part is negative, as it should be for a stable system. Each of the poles in the system here corresponds to a cavity mode. The imaginary part corresponds to the mode frequency and the real part determines the linewidth properties. With this interpretation, we can readily relate the free spectral range to the delay length. In an attempt to approximate the system, we can consider a product of the following form of terms having the same (simple) poles as \(T(z)\), satisfying the unitarity condition, and agreeing in value with \(T(z)\) at \(z = 0\)

$$\begin{aligned} \tilde{T}(z) =-\prod _{n \in\mathbb{Z}}c_{n} \biggl( \frac{z+\overline{p_{n}} }{z-p_{n} } \biggr). \end{aligned}$$

(14)

The factors \(c_{n}\) are phases to ensure the convergence of the product. If we evaluate the product as a limit of products \(\tilde{T}_{N}(z)\) as \(N\to\infty\) such that for a fixed 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.

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.

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.

5.2 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.

First, we will interpret the Blaschke-Potapov product in the optical setting. Each unitary matrix appearing in the factorization can be interpreted as a generalized beamsplitter. Each Blaschke factor with zero \(\lambda_{k}\) has the form \(B_{k}(z)\) of Eq. (17) in Section 5.1. We can interpret \(A_{k}(z)\) in Eq. (17) as the transfer function of a single cavity mode. The location of the zero \(\lambda_{k}\) in the complex plane determines the detuning and linewidth of the mode. The modes resulting from the Blaschke-Potapov product can be visualized as a sequence of components of the form portrayed in Figure 4.

Next, we shall interpret the singular term represented as the multiplicative integral in Eq. (18) of Section 5. Approximating the multiplicative integral over intervals \(\Delta t_{k} = t_{k+1}-t_{k}\), we obtain a product of terms that can each be represented by

$$\begin{aligned} e^{-z K(t_{k})(t_{k+1} - t_{k})} = e^{-z K(t_{k})\Delta t_{k}} = U_{k} e^{-zD_{k}} U_{k}^{\dagger}. \end{aligned}$$

(21)

Here \(K(t_{k}) \Delta t_{k} \ge0\) and \(U_{k}\) are some unitary matrices resulting in the diagonalization \(K(t_{k}) \Delta t_{k}= U_{k} D_{k} U_{k}^{\dagger}\). Each term of the form (21) can be interpreted as a component consisting of parallel feedforward delays inserted between two unitary components as illustrated in Figure 5. The sum of the delays across the parallel ports for each such term is given by \(\operatorname {tr}( K(t_{k}) \Delta(t_{k})) = \Delta t_{k}\). It is possible to further approximate the feedforward delays in the factorization with modes, but we will refrain from doing this here for conceptual clarity

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.

In Figure 6 we illustrate the step of our procedure for finding roots or poles. The various contours in dashed lines represent areas where roots and poles will be found. Notice in this plot that the roots and poles lie along a strip close to the imaginary axis. This is a typical feature of highly resonant systems (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.

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.

8.3 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.

To intuitively motivate the procedure we will use, we first give a demonstration for the case of the example in Section 7.2. In this network, extracting a single feedforward delay is sufficient for obtaining the separation of the two terms we desire. We will assume that \(\tau_{2} < \tau_{4}\). The important observation is that a collection of 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 In₀ combined in parallel with the addition of the delay \(\tau_{2}\) to In₁ - that is, the feedforward component only delays the input from port In₁ 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.

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.

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.

As a simple example, we consider the input-output relationship of a Fabry-Pérot cavity with a constant input (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.

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.