Point-coupling Hamiltonian for frequency-independent linear optical devices

We present the point-coupling Hamiltonian as a model for frequency-independent linear optical devices acting on propagating optical modes described as a continua of harmonic oscillators. We formally integrate the Heisenberg equations of motion for this Hamiltonian, calculate its quantum scattering matrix, and show that an application of the quantum scattering matrix on an input state is equivalent to applying the inverse of classical scattering matrix on the annihilation operators describing the optical modes. We show how to construct the point-coupling Hamiltonian corresponding to a general linear optical device described by a classical scattering matrix, and provide examples of Hamiltonians for some commonly used linear optical devices. Finally, in order to demonstrate the practical utility of the point-coupling Hamiltonian, we use it to rigorously formulate a matrix-product-state based simulation for time-delayed feedback systems wherein the feedback is provided by a linear optical device described by a scattering matrix as opposed to a hard boundary condition (e.g., a mirror with less than unity reflectivity).

Figure 1

Schematic of a linear optical device acting on $N$ optical modes. The frequency-domain and position-domain annihilation operator of the $n\text{th}$ optical mode are denoted by $a_n\left(\omega \right)$ and $a_n\left(x_n\right)$ respectively where $x_n$ is the position of the point under consideration in the coordinate system attached to the $n\text{th}$ optical mode (note that we want each optical mode to, in general, have its own independent coordinate system with the linear optical device being at $x_n=x_n^0$ in the coordinate system of the $n\text{th}$ optical mode).

Figure 2

Schematic figures showing the action of (a) phase shifter with imparted phase $\phi$ acting on an optical mode denoted by $a$, (b) beam splitter with parameters $(\theta ,\phi )$ acting on optical modes denoted by $a_1$ and $a_2$, and (c) an optical circulator designed to act on optical modes $a_1$, $a_2$, and $a_3$. The relationship between the input and output signals of the linear optical device is also shown in the diagrams.

Figure 3

Schematic figure of the time-delayed feedback system analyzed in this paper. A two-level system with deexcitation operator $\sigma$ couples to a waveguide which supports both forward propagating and backward propagating waveguide modes with frequency-domain (position-domain) annihilation operators $a_{+}\left(\omega \right)$ and $a_{-}\left(\omega \right)$ [$a_{+}\left(x_{+}\right)$ and $a_{-}\left(x_{-}\right)]$ respectively. A mirror, modelled as a beam splitter on the forward and backward propagating waveguide mode with parameters $(\theta ,\phi )$, is located as $x_{\pm }=0$ and the two-level system couples to both the waveguide modes at a distance $t_d$ from the mirror. The decay rates of the two-level system into the forward and backward propagating waveguide modes are denoted by ${\gamma }_{+}$ and ${\gamma }_{-}$ respectively. Finally, we also consider the dynamics of this system when the two-level system is driven by an external laser pulse—$\mathrm{\Omega }\left(t\right)$ denotes the complex amplitude of the laser pulse that drives the two-level system.

Figure 4

Validation of our MPS update implementation against the implementation introduced in Ref. [22] for an ideal mirror (i.e., $\theta =\pi /2$) for two simulation settings [this paper (solid), Ref. [22] (dashed)]: (a) Simulation of an undriven two-level system $\left[\mathrm{\Omega }\right(t)=0]$ which is initially in its excited state for different mirror phases $\phi$ and (b) simulation of a two-level system initially in its ground state and driven by an exponentially decaying pulse [$\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0{e}^{-\alpha t}$ for $t>0$] for different mirror phases $\phi$. $\left|\varepsilon \right(t\left)\right|$ is the probability amplitude of the two-level system being in the excited state computed using ${\left|\varepsilon \left(t\right)\right|}^2=\langle {\sigma }^{†}\sigma \rangle$. It is assumed that ${\gamma }_{+}={\gamma }_{-}=\gamma /2$, ${\delta }_e={\omega }_e-{\omega }_0=0$, ${\omega }_0{t}_d=\pi$, $\gamma t_d=2$, and $\alpha =2\gamma$. For the discretization into an MPS, we use $\gamma \mathrm{\Delta }t=0.05$, and truncate the dimensionality of the Hilbert space of each waveguide bin to 2 for both forward and backward propagating modes. A threshold of 0.01 is used in all the Schmidt decompositions performed while applying the swap gates and the short-range gates. Refer to Appendix pp3 for convergence studies of the MPS simulations.

Figure 5

Impact of mirror reflectivity on the dynamics of the time-delayed feedback system for two simulation settings: (a) Simulation of an undriven two-level system $\left[\mathrm{\Omega }\right(t)=0]$ which is initially in its excited state for different mirror reflectivities $cos\theta$ and (b) simulation of a two-level system initially in its ground state and driven by an exponentially decaying pulse [$\mathrm{\Omega }\left(t\right)={\mathrm{\Omega }}_0{e}^{-\alpha t}$ for $t>0$] for different mirror reflectivities $cos\theta$. $\left|\varepsilon \right(t\left)\right|$ is the probability amplitude of the two-level system being in the excited state computed using ${\left|\varepsilon \left(t\right)\right|}^2=\langle {\sigma }^{†}\sigma \rangle$. It is assumed that ${\gamma }_{+}={\gamma }_{-}=\gamma /2$, ${\delta }_e={\omega }_e-{\omega }_0=0$, ${\omega }_0{t}_d=\pi$, $\gamma t_d=2$, $\phi =0$, and $\alpha =2\gamma$. For the discretization into an MPS, we use $\gamma \mathrm{\Delta }t=0.05$, and truncate the dimensionality of the Hilbert space of each waveguide bin to 2 for both forward and backward propagating modes. A threshold of 0.01 is used in all the Schmidt decompositions performed while applying the swap gates and the short-range gates. Refer to Appendix pp3 for convergence studies of the MPS simulations.

Figure 6

Convergence studies on the MPS simulations for the time-delayed feedback system introduced in Sec. 3. The undriven two-level system $\left[\mathrm{\Omega }\right(t)=0]$ is initially in its excited state and is allowed to decay into the forward and backward propagating waveguide modes with an ideal mirror ($\theta =\pi /2,\phi =0$) providing feedback. $\left|\varepsilon \right(t)$ simulated using MPS update for (a) different discretization time steps $\mathrm{\Delta }t$ and (b) different tolerances that govern the number of Schmidt vectors retained after every Schmidt decomposition on the MPS. The dotted black line shows $\left|\varepsilon \right(t\left)\right|$ obtained on solving the ODE [Eq. (C1)] with a very small time step ($\gamma \mathrm{\Delta }t=0.001$). $\left|\varepsilon \right(t\left)\right|$ is the probability amplitude of the two-level system being in the excited state computed using ${\left|\varepsilon \left(t\right)\right|}^2=\langle {\sigma }^{†}\sigma \rangle$. In the simulations shown in (a) the Schmidt tolerance is assumed to be 0.01 and in the simulations shown in (b) $\gamma \mathrm{\Delta }t=0.05$. It is also assumed that ${\gamma }_{+}={\gamma }_{-}=\gamma /2$, ${\delta }_e={\omega }_e-{\omega }_0=0$, ${\omega }_0{t}_d=\pi$, and $\gamma t_d=2$.