Theory
This section summarizes the theoretical ideas behind QuantumInputOutput.jl, i.e. input-output theory for quantum pulses, its formulation via SLH networks, and the use of virtual cavities to handle traveling wave packets. We also explain how temporal output-mode bases are determined using two-time correlation functions and describe the interaction-picture, as well as pulse delay for traveling pulses.
Input-Output Theory with Quantum Pulses
The central goal of the input-output theory with quantum pulses is to describe the dynamics of a photon pulse, which interacts with a system. For such a system one is usually interested in the pulse shape (temporal mode) and the quantum state of the pulse, as well as the quantum state of the system during and after the interaction. We consider a local quantum system with Hamiltonian $H_s$ coupled to a traveling bosonic field $\hat b_{in}(t)$ through a system operator $\hat c$. Under standard Markov and dispersionless-propagation assumptions, the interaction can be written as
\[\hat V_{SB}(t) = i \sqrt{\gamma}\,\big(\hat c\,\hat b_{in}^\dagger(t) - \hat c^\dagger \hat b_{in}(t)\big).\]
The outgoing field satisfies the input-output relation
\[\hat b_{out}(t) = \hat b_{in}(t) + \sqrt{\gamma}\,\hat c(t).\]
Quantum pulses are not single-frequency modes but wave packets occupying a continuum of modes. A normalized temporal mode $u(t)$ defines a creation operator
\[\hat b_u^\dagger = \int dt\, u(t)\,\hat b^\dagger(t).\]
When the system interacts with such a pulse, the outgoing radiation can become multimode. The goal is to describe the full system–field dynamics in a tractable way, without explicitly discretizing the entire continuum.
Virtual Cavities and Pulse Modes
The key idea is to replace the traveling input and output pulses by virtual cavities, each with a time-dependent, complex coupling to the system determined by the corresponding temporal mode.
For a normalized input mode $u(t)$, the coupling
\[g_u(t) = \frac{u(t)}{\sqrt{1 - \int_0^t dt' \,|u(t')|^2}}\]
emits the initial intracavity state into the traveling mode $u(t)$. Conversely, for a chosen output mode $v(t)$, the coupling
\[g_v(t) = -\frac{v^*(t)}{\sqrt{\int_0^t dt' \,|v(t')|^2}}\]
absorbs that mode into the virtual output cavity. These expressions make it possible to treat input and output pulses as single oscillator modes in a cascaded network. Note, that this formalism can be extended to multiple input and output modes with modified couplings for each additional mode, see e.g. A. Kiilerich, et al., Phys. Rev. A 102, 023717 (2020).
SLH Networks for Cascaded Pulses
In the SLH formalism, each component is specified by a triple (S, L, H):
Scattering matrixLindblad operatorsHamiltonian
Networks are built using composition rules:
- Cascade: $G_1 \triangleright G_2$ connects the output of $G_1$ into the input of $G_2$
\[G_1 \triangleright G_2 = \left( S_2 S_1,\; L_2 + S_2 L_1,\; H_1 + H_2 + \frac{1}{2i}\left(L_2^\dagger S_2 L_1 - L_1^\dagger S_2^\dagger L_2\right) \right).\]
- Concatenation: $G_1 \boxplus G_2$ stacks channels side-by-side
\[G_1 \boxplus G_2 = \left( \begin{bmatrix} S_1 & 0 \\ 0 & S_2 \end{bmatrix},\; \begin{bmatrix} L_1 \\ L_2 \end{bmatrix},\; H_1 + H_2 \right).\]
- Feedback reduction: connecting output port $x$ back into input port $y$ of an $n$-port component
\[G = (S, L, H)\]
yields a reduced $(n-1)$-port model $[(G)]_{x \to y} = (\tilde S, \tilde L, \tilde H)$ with
\[\tilde S = S_{\bar x,\bar y} + S_{\bar x,y}(1 - S_{x,y})^{-1}S_{x,\bar y},\]
\[\tilde L = L_{\bar x} + S_{\bar x,y}(1 - S_{x,y})^{-1}L_x,\]
\[\tilde H = H + \frac{1}{2i}\left[ \left(\sum_{j=1}^n L_j^\dagger S_{j,y}\right)(1 - S_{x,y})^{-1}L_x - \mathrm{h.c.} \right].\]
Here $\bar x$ and $\bar y$ denote the remaining output and input ports after removing $x$ and $y$. This rule closes internal loops in an SLH network and is used whenever a network contains a direct coherent feedback path.
By modeling the input and output pulses as virtual cavities and cascading them with the physical system, we obtain an effective SLH triple for the full problem. This describes a master equation involving the system and auxiliary modes, which can be solved with standard Lindblad solvers.
Common SLH Elements
Appendix 1 of Combes et al. (2017) lists SLH triples for commonly used components. The most frequently used elements in this package are summarized below using the standard $(S, L, H)$ ordering.
Phase shifter (single input/output):
\[G_{\mathrm{PS}} = \left(e^{i\phi},\, 0,\, 0\right).\]
Beam splitter (two inputs/outputs, with unitary $S$):
\[G_{\mathrm{BS}} = \left( \begin{bmatrix} r_{11} & t_{12} \\ t_{21} & r_{22} \end{bmatrix},\, 0,\, 0 \right), \quad S^\dagger S = I.\]
Coherent drive (displaces the input field by $\alpha(t)$):
\[G_{\mathrm{coh}} = \left(1,\, \alpha(t),\, 0\right).\]
One-sided cavity (decay rate $\kappa$, detuning $\Delta_c$, mode operator $a$):
\[G_{\mathrm{1s}} = \left(I,\, \sqrt{\kappa}\,a,\, \Delta_c\,a^\dagger a\right).\]
Two-sided cavity / Fabry–Perot (decay rates $\kappa_1,\kappa_2$):
\[G_{\mathrm{2s}} = \left( I_2,\, \begin{bmatrix} \sqrt{\kappa_1}\,a \\ \sqrt{\kappa_2}\,a \end{bmatrix},\, \Delta_c\,a^\dagger a \right).\]
Two-level system side-coupled to a waveguide (guided decay $\kappa_g$ and unguided decay $\kappa_\perp$):
\[G_{\mathrm{TLS}} = \left( I_2,\, \begin{bmatrix} \sqrt{\kappa_g}\,\sigma_- \\ \sqrt{\kappa_\perp}\,\sigma_- \end{bmatrix},\, \frac{\omega}{2}\sigma_z \right).\]
Output Modes and the Correlation Function
The output field is generally multimode. To determine a proper basis of temporal output modes and their occupations, we compute the first-order correlation function
\[g^{(1)}(t_1, t_2) = \langle \hat L_s^\dagger(t_1)\, \hat L_s(t_2) \rangle,\]
where $\hat L_s$ is the output operator (e.g., $\hat L_s = g_u(t) \hat{a}_u + \sqrt{\kappa} \hat c$ for a single sided cavity interacting with a pulse$u(t)$). The eigen-decomposition
\[g^{(1)}(t_1, t_2) = \sum_i n_i\, v_i^*(t_1)\, v_i(t_2)\]
defines an orthonormal set of temporal modes $v_i(t)$ with mean occupations $n_i$. The most populated output modes correspond to the largest eigenvalue and are used as the physically relevant pulse shapes for further analysis. This procedure yields a principled way to extract a small number of relevant modes from a continuum and is central to the workflows implemented in QuantumInputOutput.jl.
Interaction Picture for Traveling Pulses
For pulse scattering problems (with large photon numbers) it is often advantageous to transform to an interaction picture that removes the pure mode-transfer dynamics between the virtual input and output cavities. Following Christiansen et al. (2023), one splits the total Hamiltonian into the system part and a coupling part that only exchanges excitations between the upstream and downstream virtual modes. For the two-mode case this is
\[\hat H_{uv}(t) = \frac{i}{2}\left[g_u(t) g_v^*(t)\,\hat a_u^\dagger \hat a_v - g_v(t) g_u^*(t)\,\hat a_v^\dagger \hat a_u\right].\]
The corresponding interaction-picture transformation affects only the field-mode operators. Writing $(\hat a_u(t), \hat a_v(t))^T = M(t)\,(\hat a_u(0), \hat a_v(0))^T$, the coefficient matrix solves
\[\frac{d}{dt} M(t) = A(t)\,M(t), \qquad A(t) = \frac{1}{2}\begin{bmatrix} 0 & g_u(t) g_v^*(t) \\ -g_u^*(t) g_v(t) & 0 \end{bmatrix},\]
with $M(0)=I$. In this picture the remaining Hamiltonian and dissipators are obtained by substituting the interaction-picture mode operators into the full cascaded model and subtracting $\hat H_{uv}(t)$ from the original Hamiltonian. The key practical benefit is that the virtual modes no longer undergo full emptying and refilling during the pulse; instead, only a narrow band of Fock states is populated at any time, which allows much smaller Hilbert-space truncations for numerical simulations.
The interaction picture can be generalized to multiple cascaded modes with
\[A(t) = \frac{1}{2}\begin{bmatrix} 0 & g_1 g_2^* & \cdots & g_1 g_N^* \\ -g_1^* g_2 & 0 & \ddots & \vdots \\ \vdots & \ddots & 0 & g_3 g_4^* \\ -g_1^* g_N & \cdots & -g_{N-1}^* g_N & 0 \end{bmatrix}\]
which corresponds to the interaction of the Hamiltonian obtained from $G_1 \triangleright ... \triangleright G_N$, where $G_i = (0, g_i \hat a_i, 0)$.
Pulse Delay
Propagation delays become essential when pulses traverse different path lengths as for example in a Mach-Zehnder interferometer. Such delays can be modeled without discretizing the entire field continuum by introducing a virtual delay cavity that first absorbs a chosen pulse and later re-emits it with a controlled delay. For long delays, the pulse can first be captured and then re-emitted. For short delays, however, capture and emission must occur simultaneously. In that case, the couplings are modified to account for both incoming and outgoing photons, yielding
\[\tilde g_{\mathrm{out},u,v}(t) = \frac{u^*(t)}{\sqrt{\int_0^t dt' \,|v(t')|^2 - \int_0^t dt' \,|u(t')|^2}} \qquad \tilde g_{\mathrm{in},v,u}(t) = -\frac{v^*(t)}{\sqrt{\int_0^t dt' \,|v(t')|^2 - \int_0^t dt' \,|u(t')|^2}}\]
The same formalism extends to interferometric settings by introducing one delay cavity per path and adjusting only the relative delays, which usually is the physically relevant quantity.
References
- Input-output theory with quantum pulses
- The SLH framework for modeling quantum inputoutput networks
- Interaction picture and pulse delay for quantum pulses