Nonlinear quantum input-output analysis using Volterra series
Quantum input-output theory plays a very important role for analyzing the dynamics of quantum systems, especially large-scale quantum networks. As an extension of the input-output formalism of Gardiner and Collet, we develop a new approach based on the quantum version of the Volterra series which can be used to analyze nonlinear quantum input-output dynamics. By this approach, we can ignore the internal dynamics of the quantum input-output system and represent the system dynamics by a series of kernel functions. This approach has the great advantage of modelling weak-nonlinear quantum networks. In our approach, the number of parameters, represented by the kernel functions, used to describe the input-output response of a weak-nonlinear quantum network, increases linearly with the scale of the quantum network, not exponentially as usual. Additionally, our approach can be used to formulate the quantum network with both nonlinear and nonconservative components, e.g., quantum amplifiers, which cannot be modelled by the existing methods, such as the Hudson-Parthasarathy model and the quantum transfer function model. We apply our general method to several examples, including Kerr cavities, optomechanical transducers, and a particular coherent feedback system with a nonlinear component and a quantum amplifier in the feedback loop. This approach provides a powerful way to the modelling and control of nonlinear quantum networks.
onlinear quantum systems, Volterra series, quantum input-output networks, quantum coherent feedback control, quantum control.
There has been tremendous progress in the last few years in the fields of quantum communication networks and quantum internet [1, 2, 3], quantum biology , quantum chemistry, hybrid quantum circuits , quantum computing and quantum simulation [6, 7], and quantum control [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. These progresses pave the way to the development of large-scale quantum networks. Although scalable quantum networks exhibit advantages in information processing and transmission, many problems are still left to be solved to model such complex quantum systems. Different approaches have been proposed to analyze quantum networks, among which the input-output formalism of Gardiner and Collet [27, 28] is a useful tool to describe the input-output dynamics of such systems. In fact, using the input-output response to analyze and control the system dynamics is a standard method in engineering. The quantum input-output theory [27, 28] has been extended to cascaded-connected quantum systems [29, 30] and even more complex Markovian feedforward and feedback quantum networks, including both dynamical and static components [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. Two main different formulations are proposed in the literature to model such quantum input-output networks: (i) the time-domain Hudson-Parthasarathy formalism , which can be considered as the extension of the input-output theory developed by Gardiner and Collet; and (ii) the frequency-domain quantum transfer function formalism [47, 48].
In the existing literature, quantum input-output theory is mainly applied to optical systems, in which the “memory” effects of the environment are negligibly small and the nonlinear effects of the systems are weak and thus sometimes omitted. These lead to the development of the Markovian and linear quantum input-output network theory [31, 47, 49]. In more general cases, such as in mesoscopic solid-state systems, both the linear and Markovian assumptions may not be valid. Recently, the Markovian quantum input-output theory has been extended to the non-Markovian case for single quantum input-output components  or even quantum networks . However, how to model and analyze nonlinear quantum input-output systems is still an open problem.
Recent experimental progresses [36, 39, 52, 53, 54, 55, 56], especially those in solid-state quantum circuits [40, 57, 58, 59, 60], motivate us to find some ways to analyze nonlinear quantum input-output networks. It should be pointed out that the Hudson-Parthasarathy model  can in principle formulate particular nonlinear quantum input-output systems. However, it cannot be applied to more general cases, such as those with both nonlinear components and quantum amplifiers. An additional problem yet to be solved is the computational complexity for modelling large-scale quantum input-output networks. The Hudson-Parthasarathy model gives the input-output response in terms of the internal system dynamics, and this will lead to an exponential increase of the computational complexity when we apply it to large-scale quantum networks composed of many components. For most cases, this exponentially-increased large-scale model contains redundant information. Not all the internal degrees of freedom are necessary to be known. For example, let us consider a quantum feedback control system composed of the controlled system and the controller in the feedback loop. We may not be interested in the internal dynamics of the controller, but only concern how the controller modifies the signal fed into it. This can be obtained by an input-output response after averaging over the internal degrees of freedom of the controller, which may greatly reduce the computational complexity of the quantum network analysis. For linear quantum networks, such an input-output response can be obtained by the quantum transfer function model. However, it may not be applied to nonlinear quantum networks.
To solve all these problems, we establish a nonlinear quantum input-output formalism based on the so-called Volterra series [61, 62, 63, 64, 65]. This formalism gives a simpler form to model weak-nonlinear quantum networks. This paper is organized as follows: a brief review of quantum input-output theory is first presented in Sec. 2, and then the general form of the Volterra series for m-port quantum input-output systems is introduced in Sec. 3. The Volterra series approach is extended in sec. 4 to more general cases in the frequency domain to analyze more complex quantum input-output networks with multiple components connected in series products and concatennation products. Our general results are then applied to several examples in Sec. 5. Conclusions and discussion of future work are given in Sec. 6.
2 Brief review of quantum input-output theory
2.1 Gardiner-Collet input-output formalism
The original model of a general quantum input-output system is a plant interacting with a bath. Under the Markovian approximation (in which the coupling strengths between the system and different modes of the bath are assumed to be constants for all frequencies [27, 28]), an arbitrary system operator satisfies the following quantum stochastic differential equation (QSDE)
with , where ’s are the system operators representing the dissipation channels of the system coupled to the input fields. Let and output field be the time-varying input fields that are fed into the system and the output fields being about to propagate away, one has the relation
This is the standard Gardiner-Collet input-output relation.
2.2 Hudson-Parthasarathy model
The Gardiner-Collet input-output theory can be extended to more general case to include the static components such as quantum beam splitters by the Hudson-Parthasaraty model . A more general multi-input and multi-output open quantum systems can be characterized by the following tuple of parameters
where is the internal Hamiltonian of the system; is a unitary scattering matrix induced by the static components. The notations given in Eq. (3) can be used to describe a wide range of dynamical and static systems. For example, the traditional quantum input-output systems represented by Eqs. (1) and (2) can be written as , and the quantum beam splitter with scattering matrix can be represented by .
To obtain the dynamics of the input-output system given by Eq. (3), we first introduce the quantum Wiener process and the quantum Poisson process as
which are defined by
In the Heisenberg picture, the system operator satisfies the following quantum stochastic differential equation
where the Liouville superoperator is defined by
which is of the standard Lindblad form. Similar to Eq. (2), we can obtain the following input-output relation
where and are the output fields corresponding to the quantum Wiener process and Poisson process .
2.3 Quantum transfer function model
The Gardiner-Collet input-output theory, or the more general Hudson-Parthasarathy model introduced in subsections 2.1 and 2.2, can be used to represent a large class of quantum input-output systems. However, the Hudson-Parthasarathy model is complex if the interior degrees of freedom of the system are very high. The quantum transfer function model can be applied to some cases that the Hudson-Parthasarathy model is invalid or inefficient.
Different from the Hudson-Parthasarathy model, which is in the time domain, the quantum transfer function model is a frequency-domain approach [47, 48] and can only be applied to linear quantum input-output systems. The system we consider is composed of harmonic oscillators , which satisfy the following canonical commutation relations
We are interested in a general linear quantum system, which, in the notation given by Eq. (3), satisfies the following conditions: (i) the dissipation operators ’s are linear combinations of , i.e., can be written as ; and (ii) the system Hamiltonian is a quadratic function of , i.e., . Under these conditions, we can obtain the following equivalent expression of Eq. (3):
where . Such kind of linear equations can be solved in the frequency domain. To show this, let us introduce the Laplace transform which is defined for by
Then, we can obtain the input-output relation of the whole system
where is the transfer function of the linear quantum system, which can be calculated by
3 Nonlinear response by Volterra series
The Gardiner and Collet’s quantum input-output
model [27, 28], or more generally the
Hudson-Parthasarathy model, give a general form of the quantum
input-output response, but there are internal degrees of freedom
determined by Eq. (1). Sometimes, it is not easy to use these models to
describe an input-output system, especially for nonlinear systems
of which the interior degrees of freedom are extremely high or
even infinite-dimensional. However, the complexity of the quantum
input-output model can be greatly reduced if we average out the
interior dynamics. For linear quantum systems, such a reduction
process leads to the quantum transfer function model, in which the
quantum input-output response is represented in the frequency
domain as a linear input-output relation with a proportional gain
called the quantum transfer function. As an extension of
this method, we will show that the Volterra series can be used to
describe the input-output response for more general nonlinear
quantum input-output systems.
(Nonlinear quantum input-output response by Volterra series)
The quantum input-output relation of a general m-port quantum system with input field and output field can be expressed as the following Volterra series
where and ; and ; and are the conjugate operators of and ; and are the kernel functions of the -port input-output system. Here, we omit the sum of the indices and by the Einstein summation convention.
To prove the theorem, let us first assume that the dynamical Lie algebra of the dynamical system given by Eq. (1) is , where ’s are the basis elements of the Lie algebra which satisfy the following commutation relation
’s are the structure constants of the Lie algebra . Let us define an operator vector , of which the entries come from the basis elements of the Lie algebra . From Eq. (1), we can obtain the following dynamical equation for the operator vector
From Eq. (19), we have the following formal series solution for the above equation
where is the operator vector in the Schrödinger picture. By solving in the integral of Eq. (20), we can obtain the iterative solution
Solving the above equation in the same way, we can obtain the following series solution of Eq. (19)
Since is the basis of the dynamical Lie algebra of the quantum input-output system, the system operator in the output equation (1) can be written as the linear combination of , i.e.,
where . Let us then assume that the total input-output system composed of the internal degrees of freedom and the external input field is initially in a separable state , where and are, respectively, the initial states of the internal system and the external input field. If we average over the internal degrees of freedom of the quantum input-output system, the output equation (2) can be rewritten as
As shown in Eq. (17), the system input-output response is fully determined by the set of parameters called Volterra kernels. It is also shown in the proof of theorem 1 that these kernel functions are just determined by the high-order quantum correlations of the interior dynamics of the quantum input-output system. Notice that the quantum Volterra series is different from the classical Volterra series because the terms like do not commute with each other and the vacuum fluctuations in the input field should be considered when we analyze quantum input-output response.
Different from the Volterra series used for for classical systems, the kernel functions of the quantum Volterra series method we present here should satisfy additional physically-realizable conditions [31, 48] constrained by the theory of quantum mechanics. For example, for a Markovian input-output system , the commutation relation should be preserved from the input field to the output field, i.e., . This leads to additional equality constraints for the kernel functions .
Although the right side of Eq. (17) is an infinite series, i.e., a series with infinitely many terms, we can use its finite truncations to represent the input-output response under particular conditions. In fact, for linear systems, there are only linear terms in Eq. (17). Motivated by this consideration, we then study a weak-nonlinear quantum system with internal modes given by the annihilation (creation) operators (). For such a weak nonlinear quantum system, the system Hamiltonian and the dissipation operator in Eq. (1) can be expressed as
where and are quadratic and linear functions of
the annihilation and creation operators and ;
and are higher-order nonlinear terms of
and ; and is a parameter introduced to
determine the nonlinear degree of the system. For a weak-nonlinear
system, we have . The following theorem shows that we
can use the finite truncation up to low-order terms and omit
higher-order nonlinear terms in the Volterra series for
weak-nonlinear systems satisfying Eq. (24).
(Volterra series for weak-nonlinear systems)
For a weak-nonlinear quantum input-output system with Hamiltonian and dissipation operators given by Eq. (24), the Volterra series for this quantum input-output system can be written as
The proof of the theorem is given in the appendix. It can be easily seen that Eq. (25) is just the traditional convolution representation, or equivalently the transfer function representation, when , which corresponds to linear quantum systems.
4 Frequency analysis of nonlinear quantum input-output networks
The Volterra series can be expressed as a simpler form in the frequency domain, especially for quantum networks with several components. In the frequency domain, the quantum input-output relation can be rewritten as
and is the -th order Fourier transform of the kernel function defined by
The coefficients can be seen as the quantum version of the -th order nonlinear susceptibility coefficients.
The main merit of the Volterra series approach is that it can greatly reduce the computational complexity of quantum input-output network analysis in the frequency domain. In fact, from Eq. (26), we can find that the input-output response of a multi-input nonlinear component is fully determined by the quantum susceptibility coefficients . The following theorem shows that the quantum susceptibility coefficients of a large-scale quantum network with several components can be expressed as the polynomial functions of the lower-order quantum susceptibility coefficients of each component.
(Susceptibility coefficients for networks)
The -th order quantum susceptibility coefficients of a multi-component quantum network can be expressed as polynomials of lower-order quantum susceptibility coefficients of each component.
To prove our main results, we can see that an arbitrary nonlinear quantum network can be decomposed into two basic types of connections between different components, i.e., the concatenation product and the series product . Thus, we only need to verify the main results for these two types of basic quantum networks.
The concatenation product describes two components that are simply assembled together without any connection between them [see Fig. 2(a)]. Let us assume that and are the quantum susceptibility coefficients of the two components and are the quantum susceptibility coefficients of the total system, then it can be easily verified that
The series product can be used to describe two cascade-connected components [see Fig. 2(b)], i.e., the output of the first system is taken as the input of the second system. The -th quantum susceptibility coefficients of the quantum network in the series product can be calculated by the following equation
where and are the quantum susceptibility coefficients of the two components.
From Eq. (4) and Eq. (29), we can see that the -th quantum susceptibility coefficients of a quantum network in the concatenation product and series product can be expressed as the polynomials of lower-order quantum susceptibility coefficients of the components in the quantum network. That completes the proof of the theorem.
Theorem 3 shows that the computational complexity to describe the input-output response of a multi-component nonlinear quantum network increases linearly with the number of the components in the quantum network, in comparison to the traditional exponentially increasing complexity for describing a complex quantum input-output network.
As examples of multi-component nonlinear quantum networks, let us consider a quantum network in which a linear component is cascade-connected to a nonlinear component. This can be divided into two different cases (see Fig. 3):
(i) The input field is first fed into a linear component with the quantum transfer function  , and then transmits through a nonlinear component with quantum susceptibility coefficients . The quantum susceptibility coefficients of the total system can be calculated by
(ii) The input field is first fed into a nonlinear component with quantum susceptibility coefficients , and then guided into a linear component with quantum transfer function . The quantum susceptibility coefficients of the series-product system can be expressed as
Note that the above examples are quite useful for modelling a large class of important quantum input-output networks, such as the network with a nonlinear component cascaded connected to a quantum amplifier, that cannot be modelled appropriately by the existing approaches.
The Volterra series approach we introduce here can be applied to various linear and nonlinear quantum input-output systems, especially those with weak nonlinearity. To show this, we study the input-output relation of some conventional nonlinear components, which can be taken as the basic elements of more complex quantum networks.
As a first example, we consider a Kerr cavity with free Hamiltonian and dissipation operator coupled to the input field, where and are the annihilation and creation operators of the cavity. Here , , and are the frequency of the fundamental mode, the nonlinear Kerr coefficient of the cavity, and the coupling strength between the cavity and the input field (see Fig. 4).
Let us consider the weak-nonlinear assumption such that , then from theorem 2 we can expand the quantum Volterra series up to the third-order terms. If we further assume that the cavity is initially in the vacuum state, there is only one nonzero first-order Volterra kernel
and four nonzero third-order Volterra kernels
See the derivations in the appendix.
For this example, the Volterra series approach gives a more exact description of the quantum input-output response compared with other approximation approaches, such as the truncation approximation approach in the Fock space which is mainly used for low-excitation quantum systems and the semiclassical approximation, which is traditionally introduced to study highly-excited systems. To show this, let us see the simulation results given in Fig. 7. Given the system parameters , the output trajectory obtained by the quantum Voterra series approach matches more perfectly well with the ideal trajectory compared with those obtained by the few-photon truncation in the Fock space and the semiclassical approximation.
In the second example, let us concentrate on a single-mode cavity parametrically coupled to a mechanical oscillator (see Fig. 6), which received a high degree of attention recently [66, 67, 68, 69, 70, 71]. These systems can be used as sensitive detectors to detect spin and mass, or a sensitive mechanical transducer. The Hamiltonian of this system can be written as , where () and () are the annihilation (creation) operators of the cavity mode and the mechanical oscillator; and are the angular frequencies of these two modes; and is the optomechanical coupling strength. The cavity mode is coupled to the input field with damping rate , and is the damping rate of the mechanical mode.
Let us assume that the optomechanical coupling is weak enough such that . Note that determines the nonlinearity of the optomechanical systems, thus the above assumption means that the nonlinearity of the optomechanical system we consider is weak. From theorem 2, we can expand the quantum Volterra series to the third-order terms and omit higher-order terms. If we further assume that the cavity and the mechanical oscillator are both initially in the vacuum states and note that and , there are only one non-zero first-order Volterra kernel and two non-zero third-order Volterra kernels
(Nonlinear coherent feedback network with weak Kerr nonlinearity)
The Volterra series approach gives a simpler way to analyze nonlinear quantum coherent feedback control systems [44, 68, 72, 73, 74, 75, 76]. To show this, let us consider a simple coherent feedback system in Fig. 7(a). In this system, the controlled system is a linear cavity , and in the feedback loop there are a quantum amplifier and a Kerr nonlinear component . Notice that this coherent feedback system cannot be modelled by the existing approaches such as the Hudson-Parthasarathy model and the quantum transfer function model, but we can describe it by our approach. The total system can be seen as a cascade-connected system . The system dynamics can be obtained from Eqs. (30), (31), and example 1. This quantum coherent feedback loop induces an interesting phenomenon: the nonlinear component in the coherent feedback loop changes the dynamics of the linear cavity and make it a nonlinear cavity. This nonlinear effect is additionally amplified by the quantum amplifier in the feedback loop. Let , be the effective frequency and damping rate of the controlled linear cavity. , , are the effective frequency, nonlinear Kerr coefficient, and damping rate of the nonlinear Kerr cavity in the feedback loop. is the power gain of the quantum amplifier in the feedback loop. Under the condition that , the controlled cavity can be seen as a nonlinear Kerr cavity with effective Kerr coefficient . This amplified nonlinear Kerr effects leads to nonlinear quantum phenomena in the controlled cavity. For example, if the initial state of the controlled cavity is a coherent state, this state will evolve into a non-Gaussian state, which is highly nonclassical. In Fig. 7(b), we use the measure
to evaluate the non-Gaussian degree of the quantum states generated in the controlled cavity , where is a Gaussian state with the same first and second-order quadratures of the non-Gaussian state . Simulation results in Fig. 7(b) show that higher-quality non-Gaussian states can be obtained if we increase the power gain of the quantum amplifier in the feedback loop. We should point out that we have predicted a similar quantum feedback nonlinearization phenomenon in Ref. . But in that paper, the nonlinearity is induced by the nonlinear dissipation interaction between the controlled system and the mediated quantum field, and the feedback loop is linear. Here, we show that nonlinear coherent feedback loop can induce quantum nonlinearity , which can be further amplified by the quantum amplifier in the feedback loop.
We have introduced a new formalism of quantum input-output networks using the so-called Volterra series. It gives a simpler way to describe large-scale nonlinear quantum input-output networks especially in the frequency domain, and can be also used to analyze more general quantum networks with both nonlinear components and quantum amplifiers that cannot be modelled by the existing methods such as the Hudson-Parthasarathy model and the quantum transfer function model. An application to quantum coherent feedback systems shows that it can be used to show the quantum feedback nonlinearization effects, in which the nonlinear components in the coherent feedback loop can change the dynamics of the controlled linear system and these quantum nonlinear effects can be amplified by a linear quantum amplifier. Our work opens up new perspectives in nonlinear quantum networks, especially quantum coherent feedback control systems.
Proof of the theorem 2: Let us assume that is an operator vector of which the components are linear terms of the annihilation and creation operators and , , and is another operator vector of which the components are higher-order nonlinear terms of and . The system dynamics can be fully determined by the vector . From Eq. (1) and the special form of the system Hamiltonian and dissipation operator in Eq. (24), we can obtain the dynamical equations of and as follows
Derivations of Eqs. (32) and (1): Under the weak nonlinearity assumption , we can expand the system dynamics of the Kerr cavity up to the third-order terms of and and omit higher-order terms. Then, the system dynamics can be expressed as the following quantum stochastic differential equation
where . The coefficient matrices , , and can be given by Eq. (6), where .