# Generating nonclassical quantum input field states with modulating filters

- John E Gough
^{1}Email author and - Guofeng Zhang
^{2}

**Received: **14 January 2015

**Accepted: **1 June 2015

**Published: **19 June 2015

## Abstract

We give explicit constructions of quantum dynamical filters which generate nonclassical states (coherent states, cat states, shaped single and multi-photon states) of quantum optical fields as inputs to general quantum Markov systems. The filters will be quantum harmonic oscillators damped by the input fields, and we exploit the fact that the cascaded filter and system will have a Lindbladian that is naturally Wick-ordered in the filter modes. In particular the initialization of the modulating filter will determine the signal state generated.

## Keywords

## 1 Introduction

Historical, the concept of coloring white noises has enjoyed much application in control engineering, and in particular signal processing. A white noise input, say corresponding to the derivative of a Wiener process \(B(t)\), may be converted into a colored process \(Y(t) = \int h(t-s) \, dB(s)\) where *h* is a causal kernel function. In practice this convolution may be physically implementable by passing the input through a dynamical system, such as an electronic circuit, an obtaining *Y* as output. The resulting output will have a nonflat spectrum \(S_{Y} (\omega) \equiv| H(\omega) |^{2}\), where \(H( \omega)\) is the Fourier transform of the kernel *h*, see for instance [13]. However, the concept is still useful as a theoretical construct in modelling systems driven by colored noise, as it allows an extended model with a white noise input.

The idea has been extended to quantum systems, and at its simplest corresponds to cascading an ancillary system (the filter) in front of the system, a concept going back to Carmichael, [14]. A systematic study of quantum coloring filters was initiated in [15] by one of the authors. More recently, finite-level ancillas were proposed to generate multi-photon states for quantum input processes [16]. In this setting, the dynamical and filtering equations took on a matrix form determined by the ancilla space. However, due to the choice of couplings (raising and lowering of the ancilla levels) the class of multi-photon states obtained had a chronological ordering property of the photon one-particle states which was not the intended form of the multi-photon state. Or alternative here is to use linear quantum dynamical models as ancilla. We should also mention recent work of Xue *et al.* [17] who have treated the Belavkin filtering problem for Ornstein-Uhlenbeck noise input: this input may be readily modelled as output of a linear system such as a cavity mode driven by white noise quantum input processes.

### Notations

Denote by \(\mathscr{F}_{n}\) the span of all symmetrized vectors of the form \(f_{1}\,\hat{\otimes}\,\cdots\,\hat{\otimes }\,f_{n}=\frac{1}{n!}\sum_{\sigma}f_{\sigma(1)}\,\otimes\,\cdots\otimes f_{\sigma (n)} \) where \(f_{1},\ldots,f_{n}\) lie in a one-particle Hilbert space \(\mathscr{V}\), and the sum is over all permutations *σ* of the *n* indices. The Boson Fock space over \(\mathscr{V}\) is then the direct sum \(\mathscr{F}=\bigoplus_{n=0}^{\infty}\mathscr{F}_{n}\), with \(\mathscr{F}_{0}\) spanned by the *vacuum vector*
\(| \text{vac} \rangle\).

*k*.

*t*, and zero otherwise. The Itō differential \(dB_{t}\) has the action \(dB_{t}|\mathbf{n}\rangle=\sum_{k=1}^{\infty}\sqrt{n_{k}}|n_{1},\ldots,n_{k}-1,\ldots\rangle e_{k}(t)\,dt\), where, for convenience, we may assume an orthonormal basis of continuous test functions \(e_{k}\). For nonorthonormal states we have

For convenience we consider a single quantum input process.

*initial space*, then an open system is described by the triple of operators \(G\sim ( S,L,H ) \) on \(\mathfrak{h}_{0}\) - with

*S*the unitary

*scattering matrix*,

*L*the

*collapse, or coupling, operator*and

*H*the

*Hamiltonian*- which fixes the open dynamical unitary evolution \(U(t)\) on \(\mathfrak{h}_{0} \otimes\mathfrak{F}\) as the solution to the quantum stochastic differential equation [19]

*output processes*are given by the formula

Finally we recall that there is the natural factorization \(\mathfrak{F} = \mathfrak{F}_{t}^{-} \otimes\mathfrak{F}^{+}_{t}\) of the Fock space into past and future Fock spaces for each \(t>0\) [19].

### Definition 1

(from [19])

An operator on a tensor product \(\mathfrak{h}_{1}\otimes\mathfrak{h}_{2}\) is the ampliation of an operator \(X_{1}\) on \(\mathfrak{h}_{1}\) if it takes the form \(X_{1}\otimes I_{2}\). A quantum stochastic process \((X(t))_{t\geq0}\) is adapted if, for each \(t>0\), it is the ampliation of an operator on the past space \(\mathfrak{h}_{0}\otimes\mathfrak{F}_{t}^{-}\) to the full space \(\mathfrak{h}_{0}\otimes\mathfrak{F}\).

The unitary evolution process \((U(t))_{t\geq0}\) is adapted, as will be the Heisenberg dynamical process \((j_{t} (X) )_{t \geq0}\), for each initial choice of operator *X*. The following formula will be used extensively.

### Lemma

*Let*\((X(t))_{t\geq0}\)

*be a quantum stochastic integral process of the form*

*where the*\((x_{\alpha\beta}(t))_{t\geq0}\)

*are adapted processes*.

*Then*

The proof is a routine application of the quantum stochastic calculus [19]. We note that if we set the \((x_{\alpha\beta} (t) )_{t \geq0}\) equal to zero then we recover the standard Lindblad-Heisenberg equations of motion for \(U^{\ast}(t) (X_{0} \otimes I_{ \mathfrak{F}} ) U(t)\): that is the noisy dynamics of the system observable with initial value \(X_{0}\). Conversely setting \(X_{0}=0\) and taking the \((x_{\alpha \beta} (t) )_{t \geq0}\) to be constants leads to the input-output relation. Equation (5) therefore contains general information about evolution of both system observables and field observables.

## 2 Modulating filter

Our strategy is to employ a modulating filter *M* to process vacuum input and to feed this forward to the system. In principle, the modulator and system are run in series as a single Markov component driven by vacuum input, as in Figure 1. Tracing out the modulator degrees of freedom leads to an effective model which leads to the same statistical model as a nonvacuum input to the system. We shall show below how to realize different nonclassical driving fields in this way. In our proposal we consider a linear passive system as modulator: physically corresponding to modes in a cavity. The choice of (time-dependent) coupling operators describing the modulator will be important in shaping the output, however, in this set-up the crucial element determining nonvacuum statistics will be the initial state \(\phi_{0}\in\mathfrak{h}_{M}\) of the modulator.

Let us denote by \(\widetilde{U}_{t}\) the joint unitary generated by \(\widetilde{G}\). This is a unitary adapted process with initial space \(\mathfrak{h}_{0} = \mathfrak{h}_{M}\otimes\mathfrak{h}_{G}\).

### Definition 2

*G*determine an open quantum system and let \(\Xi\in\mathfrak{F}\) be a state of the input field. A modulator

*M*with initial state \(\phi_{0}\) and vacuum input is said to replicate the open system if we have (with \(|\Psi_{0} \rangle= | \phi_{0}\otimes\psi _{0}\otimes \text{vac}\rangle\))

### 2.1 The cascaded Lindbladian

### 2.2 Oscillator mode modulators

*a*(say a cavity mode) as modulator, and set

*λ*to be a complex-valued time-dependent damping parameter.

A key feature of equation (8) when \(L_{M}=\lambda (t)a \) is that the *a* and \(a^{\ast}\) appear in *Wick ordered form* about *A*. We now exploit this property.

### 2.3 Generating shaped 1-photon fields

*ξ*, and this means choosing the correct

*λ*. It will be required that

*ξ*be normalized, that is, \(\int_{0}^{\infty} \vert \xi (t)\vert ^{2}\,dt=1\). Now let us set \(\mathsf{w}(t)=\exp \{ -\int_{0}^{t}|\lambda(s)|^{2}\,ds \} \), then

*ξ*normalized, we see that the appropriate choice for

*λ*is

### 2.4 Replicating nonvacuum input

This follows from (5) where we replace the *S* and *L* with the cascaded operators \(I_{M}\otimes S\) and \(I_{M}\otimes L+\lambda a\otimes S\).

The modulator therefore replicates the nonvacuum input model.

### 2.5 Replicating coherent states

### Theorem 1

*The quantum open system*
\(G\sim ( S,L,H ) \)
*driven by input in the continuous*-*variable coherent state*
\(\Xi =\vert\beta\rangle\)
*is replicated by the single mode modulator of the linear form*
\(M\sim ( I_{M},\lambda a,\omega a^{\ast}a ) \)
*with the initial state*
\(\phi_{0}=|\alpha\rangle\)
*for the modulator and with*
\(\lambda ( t ) \)
*and*
\(\omega ( t ) \)
*chosen so that* (16) *holds*.

## 3 Replicating multi-photon input

### 3.1 Fock state input fields

*n*quanta with the same (normalized) one-particle test function \(\xi\in L^{2}[0,\infty)\) is

*n*photon state and an \(n-1\) photon state. This feature will be typical, and so it is convenient to introduce general matrix elements

*X*to have the form (4), we see that (15) leads to

### Theorem 2

*The quantum open system*
\(G\sim ( S,L,H ) \)
*driven by input in the nonclassical state*
\(\Xi =\xi^{\otimes n}\)
*is replicated by the single mode modulator of the linear form*
\(M\sim ( I_{M},\lambda(t) a,\omega(t) a^{\ast}a ) \)
*with the initial state*
\(\phi_{0}=|n\rangle\)
*for the modulator and with*
\(\lambda ( t ) \)
*and*
\(\omega ( t ) \)
*chosen so that* (11) *holds*.

### 3.2 General multi-photon input fields

*k*th cavity mode.

**n**if \(\mu=0\), and \((n_{1},\ldots, n_{k}-1,\ldots,n_{N})\) if \(\mu=1\).

### Theorem 3

*The quantum open system*
\(G\sim ( S,L,H ) \)
*driven by input in the nonclassical state*
\(\Xi (\mathbf{n})=\widehat{\bigotimes}_{k=1}^{N}\xi_{k}^{\otimes n_{k}} \)
*is replicated by the*
*N*
*mode modulator of the linear form*
\(M\sim ( I_{M},\sum_{k} \lambda_{k} (t) a,\sum_{k} \omega_{k} (t) a_{k}^{\ast}a_{k} ) \)
*with the initial state*
\(\phi_{0}=|\mathbf{n} \rangle\)
*for the modulator and with*
\(\lambda_{k} ( t ) \)
*and*
\(\omega_{k} ( t ) \)
*chosen so that* (21) *holds*.

## 4 Superposition principles

We now make a basic observation.

### Principle of Superimposed models

*For a fixed modulator*

*M*-

*that is*,

*a quantum open system with definite*\((I_{M},L_{M},H_{M})\) -

*suppose that initial states*\(| \phi_{0}^{A} \rangle, | \phi_{0}^{B} \rangle, \ldots \)

*replicate*\(| \Xi^{A} \rangle, |\Xi^{B} \rangle, \ldots \)

*respectively and are compatible in so far as*

*for each pair*

*A*,

*B*.

*Then if the modulator is prepared in a normalized superposition*\(|\phi _{0} \rangle= c_{A} | \phi_{0}^{A} \rangle+ c_{B} | \phi_{0}^{B} \rangle+ \cdots\)

*replicates the nonclassical state*\(| \Xi\rangle= c_{A} | \Xi^{A} \rangle + c_{B} | \Xi^{B} \rangle+ \cdots\) .

This follows automatically from the bra-ket structure of the matrix elements.

### 4.1 Replicating cat states

The principle of superposition therefore implies that the initial state \(\sum_{k} \gamma_{k} \vert\beta_{k} \rangle\) for the modulator will then replicate the cat state \(\Psi= \sum_{k} \gamma_{k} \vert\beta_{k} \rangle\). Note that the \(\beta_{k}\) that may be generated this way must take the form (25). This is somewhat restrictive since we cannot obtain independent pulse, only pulse which differ by the scale factors \(\alpha_{k}\). However this already gives a wide class of cat states for practical purposes.

## 5 Conclusion

As a conceptual tool, this opens up the prospect of extending known results on quantum trajectories for vacuum inputs to models with nonclassical inputs. One such feature which we will address in a future publication is the issue of filter convergence, that is, when does the estimated conditional density operator converge to the true conditional density operator when one starts with the wrong initial state \(\psi_{0}\) for the system - this has been treated for vacuum inputs [29], but is largely unknown in the case of nonclassical inputs.

## Declarations

### Acknowledgements

The authors acknowledge support through the Royal Academy of Engineering UK and China scheme, and National Natural Science Foundation of China (NSFC) grant (Nos. 61374057) and a Hong Kong RGC grant (No. 531213).

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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