Input-output theory for superconducting and photonic circuits that contain weak retro-reflections and other weak pseudo-cavities

For a system coupled to a single IO channel, e.g. a superconducting qubit terminating a 1D transmission line, the detection of an outgoing photon could have one of two possible origins. Either the qubit system made a transition creating an outgoing photon, or an incoming photon reflected off the terminating boundary condition. In a lossless system, a perfectly reflecting boundary can only result in a scattering phase shift between the incident and exiting fields. The Heisenberg equation of motion for the outgoing field operator will be the coherent sum of these two processes, i.e.

Here $\theta$ is the scattering phase shift and $L(t)$ is a system operator describing resonant emission.

As an example consider Fig. 1$(a)$. In order for an interaction Hamiltonian like Eq.(2) to implement a single channel model, the macroscopic boundary conditions that describe perfect reflection at the position $z_{0}$, require that $F(z)=0$ for all $z\leq z_{0}$. Setting $F(z_{0})$ in Eq.(9) to zero leads to the boundary condition

or in other words $\theta=-2k_{0}z_{0}+\pi$. This boundary has a physical effect on system’s emission, as the reflected portion of the left going part interferes with the right going part. In fact direct substitution into Eq.(10), with $N=1$, results in

where

Thus a change in $\theta$ has a real effect on the system field coupling, possibly transforming a system located at an anti-node to a node where $L_{1D}=0$. The importance of the above example is that we can utilize the lesson of the reflecting boundary condition at $z_{0}$ to apply multiple constraints across a network of guided modes.

Utilizing the lessons of microwave engineering, the linear passive response of a lossless multiport device is summarized by a single unitary scattering matrix $\mathbf{S}$, which maps the incoming traveling waves to the outgoing ones. Here we adopt the convention that the outgoing traveling wave exiting a port will have the same mode index as the field entering that port. Thus when $\mathbf{S}_{ij}=\delta_{ij}$ all incoming fields are perfectly reflected with no change of phase ($\theta_{i}=0$). This is in contrast to a convention where $\delta_{ij}$ corresponds to perfect transmission in some preferred direction. However, our results are ultimately independent from any particular choice of mode labels.

A pertinent example for a multiple port scatter is an imperfect circulator. Fig.1(c) shows two localized qubit systems connected via two circulators. Consider the circulator on the left, for which the spatial coordinates of the three ports are denoted by $z_{1}$, $z_{2}$, and $z_{3}$, with $z_{1}$ and $z_{3}$ lying on a vertical axis increasing from bottom to top and $z_{2}$ on a horizontal axis increase from left to right. If we collect the input and output fields of the three ports into vectors as

then the scattering matrix tells us how the input fields get split up and directed among the output fields:

Given how we have chosen to order the input and output fields within the vectors $\mathbf{b}_{\mbox{\scriptsize in}}(t)$ and $\mathbf{b}_{\mbox{\scriptsize out}}(t)$, if we write the scattering matrix $\mathbf{S}_{\mbox{\scriptsize circ}}$ as

then the elements $r_{ii}$ are the input retroreflections, the $t_{ij}$ are the circulating transmission coefficients, and $c_{jk}$ are the “cross-talk” coefficients.

A crucial point for the formalism we develop is that we only consider scattering matrices that describe classical boundary conditions and are independent from any quantum degrees of freedom. In other words, we assume that for any quantum system operator $A$, $[\mathbf{S},A]=0$. The original theory of Gough and James is more general in that they used the quantum probability theory of Hudson and Parthasarathy Hudson and Parthasarathy (1984) to include theoretical models in which $[\mathbf{S},A]\not=0$.

The many-input analogy of the single-field IO relation, Eq.(12), is made by first defining, for an $N$ port system, the vector of inputs (outputs),

where the propagation direction $\lambda_{i}$ is positive (negative) when the coordinate $z_{i}$ is increasing (decreasing) as the input field approaches the port. The corresponding output field then propagates in the opposite direction, thus $\bar{\lambda}_{i}\equiv-\lambda_{i}$. The general network IO relation is then

Here $\mathbf{L}$ is a vector composed of the operators via which the systems interact with the fields at each of their ports:

Each of the operators $L_{i}$ describes the resonant emission from a quantum system inside the scattering region. Note that the elements of $\mathbf{L}$ are labeled, not by the individual subsystems, but by the output ports. If our model contains multiple subsystems within the scattering region, then the $L_{i}(t)$ will in general be collective operators (e.g. $L_{\pm}$ in Eq.(11)).

Fig. 2 shows an example multimode network containing two crossed waveguides that form a general 4-input/4-output junction. The local scattering between inputs $b^{\mbox{\scriptsize in}}_{i}$ and $b^{\mbox{\scriptsize out}}_{j}$, $i,j=1\dots 4$, is made by a general 4-by-4 unitary matrix $\mathbf{S}_{\mbox{\scriptsize J}}$. The network also contains two superconducting qubits that are each coupled to one of the ports of the 4-port junction. To describe this network we first consider the two qubits and the 4-port junction as three separate components, each with their own input/output ports. Since each of the qubits has a single input/output port, together the three components have 6 inputs and 6 outputs.

We now define a scattering matrix $\mathbf{S}$ that describes the relationship between the 6 inputs and outputs before the three components are connected together to form the network. Since each of the components acts on its own inputs separately from the others, this scattering matrix is block-diagonal where each block describes the action of one of the components. Since each qubit has only one port, $\mathbf{S}$ has two 1-by-1 blocks and a single 4-by-4 block given by $\mathbf{S}_{\mbox{\scriptsize J}}$:

Physically, the fact that the three components are connected with two ports of the junction each terminated by a qubit makes the 6-by-6 model redundant. The right going field entering the junction at $z_{1}$, $F_{+}(z_{1})$, is simply the spatial translation of the right-going field that exited qubit a, $F_{+}(z_{a})$. The monochromatic approximation of Eq. 9 implies that so long as $|z_{1}-z_{a}|\ll\ell_{0}$, this translation is simply a change in phase, $F_{+}(z_{1})=e^{ik_{0}(z_{1}-z_{a})}F_{+}(z_{a})$, and there is only one relevant field operator, $b_{+}(t)$. Thus the connection from $a$ to the junction input at $z_{1}$ imposes the constraint

If $\mathbf{S}_{\mbox{\scriptsize J}}$ contains retroreflecting amplitudes at positions $z_{1}$ and $z_{3}$, then there is the distinct possibility of developing circulating power in the (hopefully weak) cavities formed in the intervals $[z_{a},z_{1}]$ and $[z_{3},z_{b}]$. The legitimacy of the monochromatic approximation ultimately depends upon these intermediate cavities being of poor quality with a rapid decay rate, see Section IV.5.

Fig. 2 contains a total of 4 constraints, as both right and left waveguides have bidirectional connections. For a general network consisting of $N$ inputs and $N$ outputs, if $M\leq N$ of the outputs are injectively routed to $M$ distinct inputs then the $M$ constraint equations can be easily written as a single matrix equation relating $M$ elements of $\mathbf{b}_{\mbox{\scriptsize out}}(t)$ to $M$ elements of $\mathbf{b}_{\mbox{\scriptsize in}}(t)$.

We will find it extremely convenient to utilize matrix projectors that isolate the $M$ connected “internal” modes from the remaining $N-M$ free “external” modes. In addition to distinguishing internal from external, we must also maintain the distinction between input and output modes, as they refer to traveling-wave fields in physically distinct regions. Thus we define the projectors onto the internal/external inputs as $\mathbf{I}_{\mbox{\scriptsize i}}$ and $\mathbf{X}_{\mbox{\scriptsize i}}$ and the projectors onto the internal/external outputs as $\mathbf{I}_{\mbox{\scriptsize o}}$ and $\mathbf{X}_{\mbox{\scriptsize o}}$. As they are orthogonal projectors they satisfy the relations

where $\boldsymbol{\mathbbm{1}}$ is the $N\times N$ identity matrix. We denote the internal/external partitioning of the vector of field operators as superscripts, i.e.

Given this notation, the constraints imposed by a given network (that is, which of the outputs are routed to which inputs) can be captured by a single connection matrix, $\mathbf{W}$ ¹¹1We have chosen to denote the connection matrix by $\mathbf{W}$ because in mathematical terminology it is a “weighted adjacency matrix”, via the relation

For the crossed-waveguide network depicted in Fig. 2, the connection matrix is explicitly

where the propagation phases are $\varphi_{ij}=k_{0}|z_{i}-z_{j}|$. Note that $\mathbf{W}$ is symmetric here because we are considering bidirectional (i.e. reciprocal) connections, which may not hold for more general networks. However, the rank of $\mathbf{W}$ is always equal to the number of internal connections, $M$. In fact we will repeatedly use the relations,

It then follows that $\mathbf{W}=\mathbf{I}_{\mbox{\scriptsize i}}\mathbf{W}\mathbf{I}_{\mbox{\scriptsize o}}$.

It is worth emphasizing that, at its most general, $\mathbf{S}$ is a unitary map that takes all inputs to all outputs. As a matrix, its columns index input modes while its rows index the outputs. In contrast, $\mathbf{W}$ is a one-for-one mapping between a subset of output modes to an equal number of inputs. So as a matrix, $\mathbf{W}$ has columns labeled by outputs, rows labeled by inputs, and is null on every unconstrained external mode. In a sense, $\mathbf{W}$ describes $M$ direct feedback connections, as it returns outputs to inputs.

A great amount of physical intuition can be gained by noting that every possible path through the network can be created by taking alternating products of $\mathbf{S}$ and $\mathbf{W}$. Consider the infinite sum of matrices

We have already discussed that matrix elements $[\mathbf{S}]_{jk}$ are the scattering amplitudes that directly map $b^{\mbox{\scriptsize in}}_{k}$ to $b^{\mbox{\scriptsize out}}_{j}$. The elements of the second term of the above series, which we can equivalently write as $\left[\mathbf{SI}_{\mbox{\scriptsize i}}\mathbf{WI}_{\mbox{\scriptsize o}}\mathbf{S}\right]_{jk}$, is the sum over all paths that map input $k$ to output $j$, while traversing the network exactly once. If $j$ is an internal output then the path will continue and traverse the network a second time. In general $\left[(\mathbf{SW})^{n}\mathbf{S}\right]_{jk}$ is the sum over all $k\mapsto j$ paths that traverse the network precisely $n$ times. Assuming this series converges (that is, longer paths have successively smaller amplitudes) we then concluded that the elements of the matrix

are the coherent sum over all scattering paths through the network. The subset of these paths that take external inputs to external outputs are thus the nonzero elements of

which is the effective scattering matrix of the reduced input/output model.

The vector of effective sources $\mathbf{L}_{\mbox{\scriptsize eff}}$ is interpretable along similar lines. Consider the sum

This is the coherent sum over the raw system emissions $\mathbf{L}$, the emissions having progressed through one round trip of the network, $\mathbf{SWL}$, the emissions after two round trips, and so on. Summing this series and projecting it onto the external outputs gives,

Thus we expect that

We now turn to identifying how the connection constraints effect the evolution of the local systems embedded in the network. We will begin by considering the Heisenberg-Langevin equation for the evolution of any system operator $A$, interacting with $N$ independent fields;

Here we have included an unspecified system Hamiltonian $H_{\mbox{\scriptsize sys}}$, to allow for the presence of external controls. We assume that $H_{\mbox{\scriptsize sys}}$, when transformed to the interaction picture, causes slow evolution of the system(s), ensuring that the system-field coupling remains quasi-resonant. The compact but possibly ambiguous vector notation introduced above should be interpreted as implicit sums:

The presence of the scattering matrix $\mathbf{S}$ in the equations of motion for the system comes from the fact that $L_{i}$ is coupled to the $i^{\mbox{\scriptsize th}}$ output and $\sum_{j}s_{ij}b^{\mbox{\scriptsize in}}_{j}$ is the free output field for that mode. (Note that this equation fails to hold when $[A,\mathbf{S}]\neq 0$.)

In the appendix we show that by imposing the connection constraints, $\mathbf{b}^{\mbox{\scriptsize int}}_{\mbox{\scriptsize in}}=\mathbf{Wb}^{\mbox{\scriptsize int}}_{\mbox{\scriptsize out}}$, the equation of motion for $A$, Eq.(41), is transformed not only by the replacement $(\mathbf{S},\mathbf{L})\mapsto(\mathbf{S}_{\mbox{\scriptsize eff}},\mathbf{L}_{\mbox{\scriptsize eff}})$, but also by the replacement of the total Hamiltonian, $H_{\mbox{\scriptsize sys}}$, with the effective Hamiltonian

The proof of this result is essentially an exercise in matrix algebra. Physically the second term in $H_{\mbox{\scriptsize eff}}$ is equal to the “imaginary part” of the coherent sum over all paths where a quanta is emitted via $\mathbf{L}_{j}$ and subsequently absorbed by $\mathbf{L}^{\dagger}_{i}$. Such terms account for the spin exchange rates for atoms coupled to a 1D wave guide, as well as the Lamb shift in either free space Asenjo-Garcia et al. (2017) or cavity like Yao et al. (2009) conditions.

We can now write down the complete effective input-output description for a network that contains “weak loops” for the fields. For ease of reference we now collect all the equations that define this effective description. Given that $\mathbf{W}$ is the matrix that defines the connections between the systems (the network topology), and that $\mathbf{b}_{\mbox{\scriptsize in}}^{\mbox{\scriptsize ext}}$ and $\mathbf{b}_{\mbox{\scriptsize out}}^{\mbox{\scriptsize ext}}$ are, respectively, the external inputs and outputs to this network, we have

where $\boldsymbol{\mathbbm{1}}$ is the $N$-dimensional identity matrix, and

with

The effective equation of motion for any system operator $A$ is

in which the effective Hamiltonian is given above in Eq.(43).

While the operators $H_{\mbox{\scriptsize eff}}$ and $\mathbf{L}_{\mbox{\scriptsize eff}}$ are all that are needed to specify the master equation in Lindblad form (for vacuum input fields), some physical insight can be gained by considering the non-hermitian dissipative “Hamiltonian”, $\widetilde{H}_{\mbox{\scriptsize loss}}\equiv H_{\mbox{\scriptsize eff}}-i\frac{1}{2}\mathbf{L}_{\mbox{\scriptsize eff}}^{\dagger}\mathbf{L}_{\mbox{\scriptsize eff}}$. An application of the tricky relation in Eq.(126) shows that,

Each term of $\widetilde{H}_{\mbox{\scriptsize loss}}$ has a clear and distinct meaning. The first and second show the “unconstrained” open system dynamics, consisting of the externally applied $H_{\mbox{\scriptsize sys}}$ and the dissipation induced by every output port. The third term shows the cumulative effect of the network. The coherent effects of $H_{\mbox{\scriptsize eff}}$ and the dissipation induced by $\mathbf{L}_{\mbox{\scriptsize eff}}^{\dagger}\mathbf{L}_{\mbox{\scriptsize eff}}$ combine as real and imaginary parts so that $\widetilde{H}_{\mbox{\scriptsize loss}}$ only involves the sum $\mathbf{SW}+\mathbf{SWSW}+\dots$ and not its “reversed” adjoint. If $\mathbf{W}$ is asymmetric and only connects system $a$ to system $b$ but not vice-versa, then this unidirectional flow is preserved in $\widetilde{H}_{\mbox{\scriptsize loss}}$.

Here we identify the conditions under which the effective input-output description, derived above, is a good approximation to the real network in which the fields pass through the loops multiple times. This involves identifying when the network geometry allows repeated interactions, while ensuring that the criteria for the monochromatic traveling-wave approximation remains valid. Consider the case depicted in Fig. 3 where a simple empty cavity is formed when a section of waveguide of length $l$ is capped by two partially reflecting boundary conditions. The scattering elements at these boundaries are defined by the frequency independent unitary matrices $\mathbf{S}_{\ell}$ and $\mathbf{S}_{r}$. In the monochromatic approximation, the in-to-out scattering boundary condition, and the constraint of free propagation of internal modes are imposed by the block diagonal matrices,

and

Outside of this approximation, we still have the two parameter field operators $b_{\pm}(z_{i},t)$, which can still be arranged into input/output vectors $\mathbf{b}_{\mbox{\scriptsize in}}(t)$ and $\mathbf{b}_{\mbox{\scriptsize out}}(t)$.

The free-propagation of the internal fields still give the constraints,

This implies that

Combining this with the general condition

shows that,

Note that this is simply the time-delayed version of Eq.(113).

Previously we have argued that so long as the spatial separation remains small when compared to the characteristic length $\ell_{0}=v_{p}/\kappa_{0}$ then the field operators are still well approximated by a single field operator that delta commutes in time. But here we have an infinite number of distances so that there always exists a number reflections $n_{\mbox{\scriptsize cut}}$ such that $n_{\mbox{\scriptsize cut}}l\sim\ell_{0}$. Nevertheless, the contribution of these significantly delayed paths are all attenuated by a factor of $(\mathbf{SW})^{n_{\mbox{\scriptsize cut}}}$. If the largest singular value of this matrix is small compared to 1, then we can be confident in the approximation.

In a general network the time delay for the different internal connections may be different. Rather than being merely $n$ multiples of a single traversal time, the cumulative delay will depend upon the specific path taken. For a given path consisting of $n$ internal connections followed by a final exit at an external port, the total delay $\tau_{n}$ is simply the sum over the individual delays, so that

and the weighting factor $w_{n}$ is the product of matrix elements

Ultimately, we require that any path that has a significant delay must also have an insignificant weight. We can therefore state a simple sufficient condition for the validity of the effective description as follows. If $\tau_{\mbox{\scriptsize min}}$ is the minimum relevant system dynamical timescale (e.g. $1/\kappa_{0}$) then

This condition captures the requirement that any effective cavities that are formed by loops in the network decay sufficiently fast, as discussed in Section III.3.

For example, consider the two qubit system of Fig 1(c), where two qubits are connected by a transmission line, via isolating circulators. Taking this as a model for a long distance communication channel, let the interconnecting distance, $L=|z_{4}-z_{2}|$, be generally larger than the distances between the bare qubits and their circulators, $\Delta z_{a}=|z_{1}-z_{a}|$ and $\Delta z_{b}=|z_{5}-z_{b}|$. The paths with the longest delay times are clearly those that traverse the interconnecting line. The shortest path traveling from qubit $a$ to qubit $b$ follows the sequence of outputs with locations $z_{a}\rightarrow z_{2}\rightarrow z_{5}\rightarrow z_{b}$. This path has a delay time $\tau_{1}=(L+\Delta z_{a}+\Delta z_{b})/v_{p}$ and carries a weight with magnitude $|w_{1}|=|t_{54}t_{21}|$. The paths following outputs $z_{a}\rightarrow z_{2}\rightarrow z_{4}\rightarrow z_{1}\rightarrow z_{a}$, and $z_{a}\rightarrow z_{2}\rightarrow z_{4}\rightarrow z_{2}\rightarrow z_{5}\rightarrow z_{b}$ have delays of $\tau_{2}=2(L+\Delta z_{a})/v_{p}$ and $\tau_{3}=(3L+\Delta z_{a}+\Delta z_{b})/v_{p}$. The corresponding weights have magnitudes $|w_{2}|=|c_{12}r_{44}t_{21}|$ and $|w_{3}|=|t_{54}r_{22}r_{44}t_{21}|$. An off-the-shelf circulator operating in the 4-8 GHz range (e.g. Low Noise Factory - LNF-ISC4_8A), specifies the parameters $|t_{ij}|\sim 0.98$, and $|r_{ij}|,|c_{ij}|\lesssim 0.08$. Thus for the delays $\tau_{1},\tau_{2},\tau_{3}$ the associated weights have magnitudes $|w_{1}|\sim 0.96$, and $|w_{2}|,|w_{3}|\sim 6\times 10^{-3}$.

Were we to follow the manufacturer’s lead and consider the weights $|w_{2}|$ and $|w_{3}|$ negligible, we then recover the standard implementation of cascading two systems in a unidirectional way. In this case, the time-of-flight delay, $\tau_{1}$ can be easily absorbed by evaluating qubit $b$ at the retarded time $t_{r}=t-\tau_{1}$ Gardiner (1993).

We identify the emission properties of the two-qubit system by first analyzing how the populations of different states are transferred via the so called “feeding terms” of the master equation. Specifically, consider the expression

The single-qubit terms (i.e. those for which $i=j$) show that the network has the effect of increasing the action of these terms by the factor