Feedback network models for quantum transport

Quantum feedback networks have been introduced in quantum optics as a framework for constructing arbitrary networks of quantum mechanical systems connected by unidirectional quantum optical fields, and has allowed for a system theoretic approach to open quantum optics systems. Our aim here is to establish a network theory for quantum transport systems where typically the mediating fields between systems are bidirectional. Mathematically, this leads us to study quantum feedback networks where fields arrive at ports in input-output pairs, making it a special case of the unidirectional theory where inputs and outputs are paired. However, it is conceptually important to develop this theory in the context of quantum transport theory—the resulting theory extends traditional approaches which tend to view the components in quantum transport as scatterers for the various fields, in the process allowing us to consider emission and absorption of field quanta by these components. The quantum feedback network theory is applicable to both Bose and Fermi fields, moreover, it applies to nonlinear dynamics for the component systems. We advance the general theory, but study the case of linear passive quantum components in some detail.

Figure 1

A component representing a quantum mechanical system driven by several input fields. There will be the same number of output fields. It is often convenient to think of grouped inputs with multiplicity greater than one, as in the lower figure.

Figure 2

Several SLH models run in parallel: They may be collected into one single SLH model.

Figure 3

We feed selected outputs back in as inputs to get a reduced model.

Figure 4

An arbitrary quantum feedback network.

Figure 5

Systems in series.

Figure 6

Single component with multiple lead contacts.

Figure 7

A bidirectional contact may be considered as an equivalent unidirectional input/output pair.

Figure 8

A two-lead quantum transport device is naturally modeled as a two-input, two-output SLH component.

Figure 9

The usual input/output description is modified to have inputs and outputs corresponding to a given lead all appearing grouped on one side.

Figure 10

A pair of cascaded quantum transport systems is reinterpreted as a quantum feedback network.

Figure 11

The algebraic loop appearing in the cascaded quantum transport setup in Fig. 10.

Figure 12

Several quantum transport components in a chain series.

Figure 13

A standard procedure in circuit theory is to terminate a cascade of devices with a terminal load $\mathcal{T}$. In the input-output representation, this amounts to a feedback arrangement as shown.

Figure 14

The chain-scattering representation of the terminal load.

Figure 15

Cascaded system with delay.

Figure 16

A cavity QED component is connected to a quantum dot (qubit system) so that the qubit is a nonlinear terminal load element. The chain-scattering representation is sketched underneath.

Figure 17

The equivalent quantum feedback network for the cavity-qubit system. Below it is the open-loop setup before the feedback connections are made—note that the outputs and input are labeled so that output 1 goes to input 3, output 3 goes to input 2, while output 2 and input 1 are the external fields that remain after the feedback reduction.