Interferometric modulation of quantum cascade interactions

We consider many-body quantum systems dissipatively coupled by a cascade network, i.e., a setup in which interactions are mediated by unidirectional environmental modes propagating through a linear optical interferometer. In particular we are interested in the possibility of inducing different effective interactions by properly engineering an external dissipative network of beam splitters and phase shifters. In this work we first derive the general structure of the master equation for a symmetric class of translation-invariant cascade networks. Then we show how, by tuning the parameters of the interferometer, one can exploit interference effects to tailor a large variety of many-body interactions.

Figure 1

Pictorial representation of the system studied: interactions between the quantum systems $\mathcal{S}:=\{S_1,S_2,...\}$ are mediated by a network of multimode bosonic chiral environmental channels $\mathcal{E}:=\{{\mathcal{E}}^{\left(1\right)},{\mathcal{E}}^{\left(2\right)},...\}$ which interfere through a collection of beam splitters ${\text{BS}}_{ij}$ (yellow squares in the figure) while progressing, from top to bottom, through the various levels of the network (thin horizontal lines). The X symbols in the figure identify the first entry of the various levels.

Figure 2

Input-output mapping in Heisenberg representation (2) induced by the beam splitter ${\text{BS}}_{m{m}^{\prime }}$ that couples the channels ${\mathcal{E}}^{\left(m\right)}$ and ${\mathcal{E}}^{\left(m^{\prime }\right)}$ ($m<m^{\prime }$).

Figure 3

Schematic representation of the paths (bold curves) that contribute to the coupling constants ${\zeta }_{m,m^{\prime }}$; see Eq. (25). (a) The single path which enters into the definition of the coupling ${\zeta }_{4,5}$ between $S_4$ and $S_5$; (b) those of $S_2$ and $S_4$; (c) those of $S_2$ and $S_5$; (d) those of $S_1$ and $S_5$.

Figure 4

Schematic representation of the model in the presence of loss: the signal from the subsystems along the network have probability $\nu$ of being lost, i.e., a probability $1-\nu$ of continuing their journey in the interferometric network.

Figure 5

Regular network case: as indicated by the labels the beam splitters (square elements of the figure) laying on the same diagonal of the network are assumed to be identical. For instance, the elements ${\text{BS}}_{12}, {\text{BS}}_{23}, {\text{BS}}_{34}, {\text{BS}}_{45}$,...possess the same transmissivity ${\tau }_{k=1}$ and the same relative phase ${\varphi }_{k=1}$; see Eq. (29). Similarly the elements ${\text{BS}}_{13}, {\text{BS}}_{24}, {\text{BS}}_{35}, {\text{BS}}_{46}$,...are characterized instead by transmissivity ${\tau }_{k=2}$ and by relative phase ${\varphi }_{k=2}$.

Figure 6

Setting the transmissivity ${\tau }_1=0$ the sites are coupled via a single chiral channel. This induces an effective coupling between the sites mediated by a fully connected Hamiltonian with coupling strengths whose intensity is independent from the distance between the various elements.

Figure 7

Plot of the modulus of the first four coupling constants ${\xi }_k$ for the case of a regular network with ${\tau }_2=0$ as a function of ${\tau }_1$ and ${\varphi }_2$. Notice that for ${\tau }_1=0$ all the coupling constants have modulus 1 in agreement with (37). For ${\tau }_1=1$ the odd terms nullify, while the even one maintains maximum value; see Eq. (44). In all the plots we set ${\varphi }_1=0$.

Figure 8

Pictorial representation of the components of the vectors ${\vec{V}}_{\ell }$ of Eq. (B1) for the cases $K=3$: for $\ell$ and $k$ integer, $V_{\ell }^{\left(k\right)}$ describes the amplitude probability associated with the path connecting the first entry of level 1 to the $k\mathrm{th}$ entry of the level $\ell$. In the lower panel we highlight the paths that contribute to the amplitude probability that yields the value of $V_3^{\left(2\right)}$.

Figure 9

Illustration of the amplitudes $W_{\ell }^{\left(k\right)}$ which label the horizontal lines of the network: for $\ell$ and $k$ integer they represent the probability amplitude associated with the propagation of a signal emerging from $S_1$ and reach the $\ell \mathrm{th}$ level of the $k\mathrm{th}$ horizontal step of the network. As an example in the right panel we highlight the paths which enter in the definition of $W_4^{\left(3\right)}$.

Figure 10

Left: Scheme of the possible paths when eliminating second-neighbor interactions, i.e., nullifying ${\xi }_2$. The signal coming from $S_1$ (green path) splits away in the first beam splitter, following two distinct paths. They then recombine in the beam splitter with ${\tau }_1$at level 2 of the network, just above $S_3$, both ending up in the channel propagating on the right because of interference effects. Right: The same as in the left panel, but for third-neighbor interactions. Here we notice that, once we have eliminated second-neighbor interactions, no signal coming from $S_1$ can make its way up to $S_3$, i.e., no signal arrives at level 3 of the network. This implies that all the signals coming from $S_1$ must end up entirely in channel ${\mathcal{E}}_2$ in order to eliminate also third-neighbor interactions.