Enhanced continuous generation of non-Gaussianity through optomechanical modulation
Sofia Qvarfort, Alessio Serafini, André Xuereb, Dennis Rätzel, David Edward Bruschi
EEnhanced continuous generation of non-Gaussianitythrough optomechanical modulation
Sofia Qvarfort , Alessio Serafini , Andr´e Xuereb , DennisR¨atzel , David Edward Bruschi , Department of Physics and Astronomy, University College London, GowerStreet, WC1E 6BT London, United Kingdom Department of Physics, University of Malta, Msida MSD 2080, Malta Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, 12489 Berlin, Germany Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna,Austria Institute for Quantum Optics and Quantum Information - IQOQI Vienna,Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, AustriaE-mail: [email protected], [email protected]
Abstract.
We study the non-Gaussian character of quantum optomechanicalsystems evolving under the fully nonlinear optomechanical Hamiltonian. By usinga measure of non-Gaussianity based on the relative entropy of an initially Gaussianstate, we quantify the amount of non-Gaussianity induced by both a constant andtime-dependent cubic light–matter coupling and study its general and asymptoticbehaviour. We find analytical approximate expressions for the measure of non-Gaussianity and show that initial thermal phonon occupation of the mechanicalelement does not significantly impact the non-Gaussianity. More importantly,we also show that it is possible to continuously increase the amount of non-Gaussianity of the state by driving the light–matter coupling at the frequencyof mechanical resonance, suggesting a viable mechanism for increasing the non-Gaussianity of optomechanical systems even in the presence of noise. a r X i v : . [ qu a n t - ph ] M a r nhanced continuous generation of non-Gaussianity through optomechanical mod...
1. Introduction
Understanding nonlinear, interacting physical systems is paramount across many areasin physics. Specifically, “nonlinear” (or “anharmonic”) dynamical systems include allthose whose Hamiltonian cannot be expressed as a second-order polynomial in thequadrature operators. Crucially, these systems allow us to generate non-Gaussianstates, which cannot be done given only quadratic couplings. One family of systemwhere this is possible are optomechancial systems, where light interacts with amechanical element through a cubic interaction term.In recent years, the intrinsic value of nonlinear systems, as opposed to theaforementioned limitations that linear systems face, has been made clearer andmore rigorous. It has been shown that non-linearities in the form of non-Gaussianstates constitute an important resource for quantum teleportation protocols [1],universal quantum computation [2, 3], quantum error correction [4], and entanglementdistillation [5, 6, 7]. This view of non-Gaussianity as a resource for information-processing tasks has inspired recent work on developing a resource theory basedon non-Gaussianity [8, 9, 10]. In addition, it has been found that non-Gaussianityprovides a certain degree of robustness in the presence of noise [11, 12].In the context of quantum information and computation, there has been adrive towards the realisation of anharmonic Hamiltonians as well as more generalmethods and control schemes capable of generating and stabilising non-Gaussianstates [13, 14, 15, 16, 17, 18]. On the one hand, this is motivated by the fact that, inorder to obtain effective qubits from the truncation of infinite dimensional systems,one needs unevenly spaced energy levels, such that only the transition between the twoselected energy levels may be targeted and driven. In turn, this requires a sufficientlyanharmonic Hamiltonian. On the other hand, it has always been clear that protocolsentirely restricted to Gaussian preparations, manipulations and read-outs, throughquadratic Hamiltonians and general-dyne detection, are classically simulatable, astheir Wigner functions may be mimicked by classical probability distributions [19]. ‡ In optomechanical systems [20], where electromagnetic radiation is coherentlycoupled to the motion of a mechanical oscillator, the light–matter interaction inducedby radiation pressure is inherently non-linear [21, 22, 23]. The nonlinear features ofoptomechanical systems have been frequently explored in the context of metrology,such as force sensing [24] and gravimetry [25, 26]. In particular, the nonlinearcoupling enables the creation of optical cat states in the form of superpositions ofcoherent states [21, 22]. These cat states can also be transferred to the mechanics [27],which opens up the possibility of using massive superpositions for testing fundamentalphenomena such as collapse theories [28] and, potentially, signatures of gravitationaleffects on quantum systems at low energies [29, 30]. In addition to cat states, othernon-Gaussian states such as compass states [31, 32] and hypercube states [33], andalso all been found to have excellent sensing capabilities. This combination of sensingwith nonlinear state and fundamental applications makes it imperative to explore thenonlinear properties of the mechanical systems.A number of different optomechanical systems have been experimentallyimplemented, including Fabry–P´erot cavities with a moving-end mirror [34], as ‡ We should note here that, since uncertainties in quantum Gaussian systems are fundamentallybounded by the Heisenberg uncertainty relation, which in principle does not hold for classical systems,Gaussian operations are in fact sufficient to run some protocols requiring genuine quantum features,such as continuous variable quantum key distribution. nhanced continuous generation of non-Gaussianity through optomechanical mod... Figure 1.
An optomechanical system consisting of a moving end mirror. Theoperators ˆ a and ˆ a † denote the creation and annihilation operators for the lightfield, and ˆ b and ˆ b † denote the motional degree of freedom of the mirror. Otherexamples of optomechanical systems include levitated nanobeads and cold–atomensembles. shown in Figure 1, levitated nano-diamonds [35, 36], membrane-in-the-middleconfigurations [37] and optomechanical crystals [38, 39]. While several experimentshave demonstrated genuine nonlinear behaviour (see for example [40, 41, 42, 43]), mostexperimental settings can however be fully modelled with linear dynamics [44, 43], andit is generally difficult to access the fully nonlinear regime. As a result, significant efforthas been devoted to the question of how the non-linearity can be further enhanced.Most approaches focus on the few-excitation regime, where increasing the inherentlight–matter coupling allows for detection of the non-linearity. This enhancement canbe achieved, for example, by using a large-amplitude, strongly detuned mechanicalparametric drive [45], or by modulating the spring constant [46]. Similar work hasshown that the inclusion of a mechanical quartic anharmonic term can be nearlyoptimally detected with homodyne and heterodyne detection schemes, which arestandard measurements implemented in the laboratory [47].A natural question that arises considering the approaches above is: Are thereadditional methods by which the amount of non-Gaussianity in an optomechanicalsystem can be further increased ? One such proposal was put forward in [48] where itwas suggested that the nonlinearity in electromechanical systems could be enhancedby several orders of magnitude by modulating the light–matter coupling. This isachieved by driving the system close to mechanical resonance and takes a simpleform in the rotating-wave approximation. Here, we seek to fully quantify the non-Gaussianity of the exact, nonlinear optomechanical state for both ideal and opensystems. More precisely, given an initial Gaussian state evolving under the standardoptomechanical Hamiltonian, we quantify how non-Gaussian the state becomes asa function of time and the parameters of the Hamiltonian in question. To do so,we make use of recently developed analytical techniques to study the time-evolutionof time-dependent systems [49], and employ a specific measure of non-Gaussianitybased on the relative entropy of the state [50]. Our results include the fact thatthe non-Gaussianity of an optomechanical system initially in a coherent state scaleswith the inherent light–matter coupling, as expected. We also find that the non-Gaussianity scales logarithmically with the coherent state parameter of the opticalsystem, and we illustrate how this behaviour differs for small and large coherentstates. Most importantly, however, we find that the non-Gaussianity of the state can nhanced continuous generation of non-Gaussianity through optomechanical mod...
2. Dynamics
We start our work by presenting the necessary mathematical tools needed to solve thedynamics. We refer the reader to Appendix A for a more extensive introduction onthe techniques presented below.
We begin by considering two bosonic modes corresponding to an electromagneticmode and a mechanical oscillator. Without loss of generality, we shall henceforthrefer to the electromagnetic mode as the optical mode. The operators ˆ a and ˆ b of the cavity and mechanical modes respectively, obey the canonical commutationrelations [ˆ a, ˆ a † ] = [ˆ b, ˆ b † ] = 1, while all other commutators vanish. The radiationpressure induces a nonlinear interaction between the light and mechanics, and thewhole systems is modelled by the following Hamiltonian:ˆ H = (cid:126) ω c ˆ a † ˆ a + (cid:126) ω m ˆ b † ˆ b − (cid:126) g ( t ) ˆ a † ˆ a (cid:0) ˆ b + ˆ b † (cid:1) , (1)where ω c and ω m are the frequencies of the cavity mode and the mechanical moderespectively, and g ( t ) drives the, potentially time-dependent, nonlinear light–mattercoupling. The light–matter coupling strength g ( t ) takes on different functional formsfor different optomechanical systems, and we also note that this Hamiltonian governsthe evolution of many similar systems, including electro–optical systems [51].To simplify our notation and expressions, we rescale the laboratory time t by the frequency ω m , therefore introducing the dimensionless time τ := ω m t ,the dimensionless frequency Ω := ω c /ω m , and the dimensionless coupling ˜ g ( τ ) := g ( tω m ) /ω m . This choice will prove convenient throughout the rest of this work,and dimensions can be restored when necessary by multiplying by ω m . This nhanced continuous generation of non-Gaussianity through optomechanical mod... τ and the Hamiltonianˆ H/ ( (cid:126) ω m ) = Ω ˆ a † ˆ a + ˆ b † ˆ b − ˜ g ( τ )ˆ a † ˆ a (cid:0) ˆ b + ˆ b † (cid:1) . (2)To determine the action of this Hamiltonian on initial states, we now proceed to solvethe dynamics induced by (2). We now need an expression for the time evolution operator ˆ U ( τ ) for a system evolvingwith (2). The unitary time-evolution operator readsˆ U ( τ ) := ← T exp (cid:20) − i (cid:90) τ dτ (cid:48) ˆ H ( τ (cid:48) ) (cid:21) , (3)where ← T is the time-ordering operator [52]. This expression simplifies dramaticallywhen the Hamiltonian ˆ H is independent of time, in which case one simply hasˆ U ( τ ) = exp (cid:0) − i ˆ H τ (cid:1) . As we will here consider time-dependent light–matter couplings˜ g ( τ ), we instead seek to solve the full dynamics of the time-dependent Hamiltonian.To do so, we make the ansatz that the time-evolution operator can be written asa product of a number of operator ˆ U j . This is possible if there exists a finite set ofgenerators ˆ G j that form a closed Lie algebra under commutation [52]. We thus writethe evolution operator (3) asˆ U ( τ ) = (cid:89) j ˆ U j ( τ ) = (cid:89) j e − iF j ( τ ) ˆ G j , (4)where F j ( τ ) are generally time-dependent coefficients determining the influence ofthe generator ˆ G j on the quantum state. Our task is to find these coefficients, whichhas been done in [49] for an analogous setting. We note that compared with [49],the operators ˆ a and ˆ b have been swapped and that in our case the coupling ˜ g ( τ ) ispreceded by a minus-sign. For clarity, we therefore present the full derivation for thecase considered here in Appendix A.With these techniques, we find that the time-evolution operator ˆ U ( τ ) can be castinto the convenient formˆ U ( t ) = e − i ˆ N b τ e − i F ˆ N a ˆ N a e − i F ˆ Na ˆ B + ˆ N a ˆ B + e − i F ˆ Na ˆ B − ˆ N a ˆ B − , (5)where the operators, given by,ˆ N a := ˆ a † ˆ a ˆ N b := ˆ b † ˆ b ˆ N a := (ˆ a † ˆ a ) ˆ N a ˆ B + := ˆ N a (ˆ b † + ˆ b ) ˆ N a ˆ B − := ˆ N a i (ˆ b † − ˆ b ) , (6)form a closed Lie algebra under commutation, and where the coefficients thatdetermine the evolution in (5) are given by F ˆ N a = 2 (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) sin( τ (cid:48) ) (cid:90) τ (cid:48) d τ (cid:48)(cid:48) ˜ g ( τ (cid:48)(cid:48) ) cos( τ (cid:48)(cid:48) ) ,F ˆ N a ˆ B + = − (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) cos( τ (cid:48) ) , and F ˆ N a ˆ B − = − (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) sin( τ (cid:48) ) . (7) nhanced continuous generation of non-Gaussianity through optomechanical mod... c ˆ N a in order toneglect the phase-term e − i Ω τ . Given an explicit form of ˜ g ( τ ), it is then possible towrite down a full solution for ˆ U ( τ ). The decoupling techniques necessary to obtain thiscompact solution have a long tradition in quantum optics [53] and were generalisedand refined recently [52]. Finally, before we proceed, we also define the following twoparameters: θ a = 2 (cid:16) F ˆ N a + F ˆ N a ˆ B + F ˆ N a ˆ B − (cid:17) F = F ˆ N a ˆ B − + iF ˆ N a ˆ B + . (8)These quantities will be useful when discussing features of the non-Gaussianity. Let us show here that this method reproduces the standard evolution operator forthe optomechanical Hamiltonian. For this specific case, the light–matter interactionis held constant with ˜ g ( τ ) = ˜ g . The functions (7) simplify to F ˆ N a = − ˜ g (cid:2) − sinc(2 τ ) (cid:3) τ,F ˆ N a ˆ B + = − ˜ g sin ( τ ) ,F ˆ N a ˆ B − = ˜ g (cid:2) cos ( τ ) − (cid:3) . (9)These coefficients allow us to write the time evolution operator ˆ U ( τ ) asˆ U ( τ ) = e − i ˆ N b τ e i ˜ g [1 − sinc(2 τ )] τ ˆ N a e i ˜ g sin ( τ ) ˆ N a ˆ B + e i ˜ g (1 − cos ( τ )) ˆ N a ˆ B − . (10)This expression matches that found in the literature (see e.g. equation 3 in [22],which can be obtained with some rearrangement of the terms in (10)). In this work, we will examine the non-Gaussianity of the evolved state given two initialstates: a coherent state and a thermal coherent state.i)
Coherent states.
We start by considering the case when both the optical andthe mechanical modes are in a coherent state, which we denote | µ c (cid:105) and | µ m (cid:105) respectively. These states satisfy the relations ˆ a | µ c (cid:105) = µ c | µ c (cid:105) and ˆ b | µ m (cid:105) = µ m | µ m (cid:105) . For the optical field, this is a readily available resource, since coherentstates model laser light quite well. The mechanical element in optomechanicalsystems is most often found in a thermal state or, assuming perfect preliminarycooling, in its ground state, with µ m = 0. The initial state | Ψ(0) (cid:105) of the compoundsystem will therefore be | Ψ(0) (cid:105) = | µ c (cid:105) ⊗ | µ m (cid:105) . (11)i) Thermal coherent states.
The assumption that the mechanics is in theground state is not always justified, and therefore we shall also consider thenon-Gaussianity of cases where the mechanics is in a thermal coherent state [54].Such state is obtained simply by integrating over the coherent state parameterwith an appropriate kernel [55]. We define the thermal state ˆ ρ th asˆ ρ th = 1¯ nπ (cid:90) d β e −| β | / ¯ n | β (cid:105) (cid:104) β | , (12) nhanced continuous generation of non-Gaussianity through optomechanical mod... n is the average thermal phonon occupation of the state, | β (cid:105) is a coherentstate, and the integration occurs over the full complex space. Assuming that theoptical mode in the coherent state | µ c (cid:105) , the full initial state is therefore given byˆ ρ (0) = | µ c (cid:105) (cid:104) µ c | ⊗ ˆ ρ th (13)By starting in an initial Gaussian state, we ensure that any non-Gaussianityrevealed by our work is due to the nonlinear coupling in Eq. (2). Indeed, the only wayan initially Gaussian state may at any time be non-Gaussian is for the correspondingHamiltonian to induce some nonlinear evolution [56]. We do however note that whilethe measure of non-Gaussianity that we shall make use of has a clear and operationalnotion of the measure for pure states, it is harder to make statements about the non-Gaussianity of states that are mixed, such as the coherent thermal state ˆ ρ th . SeeSection 7 for a discussion of the properties of the relative entropy measure. All realistic systems experience decoherence. In optomecahnical systems, thismanifests as photons leaking from the cavity or as damping of the mechanical motion.Given a sufficiently weakly coupled environment, we can model the open dynamicsof the system with the help of the Lindblad equation [57]. We note, however, thatthere is increasing evidence that the standard master equation treatment breaks down,especially in the strong coupling regime [58]. Here, we shall only consider weakcoupling in the presence of noise, and so the Lindblad equation is given by˙ˆ ρ = − i (cid:104) ˆ H, ˆ ρ (cid:105) + ˆ L c ˆ ρ ˆ L † c + ˆ L m ˆ ρ ˆ L † m − { ˆ L † c ˆ L c , ˆ ρ } − { ˆ L † m ˆ L m , ˆ ρ } , (14)where ˆ L c and ˆ L m are the Lindblad operators for the optics and mechanics, respectively.To model photon and phonon decay, we assume that ˆ L c = √ κ c ˆ a and ˆ L m = √ κ m ˆ b ,where κ c is the optical decoherence rate and κ m is the phonon decoherence rate.While analytic solutions for this particular choice of ˆ L m were obtained in [22],photon decay can currently only be modelled numerically. We therefore make use ofthe Python library QuTiP to simulate the noisy state evolution and its effect on thenon-Gaussianity of the resulting state. We shall examine the non-Gaussianity for opensystems in Sections 5 and 6, but first, we define the measure of non-Gaussianity.
3. Measures of deviation from Gaussianity
Given a Hamiltonian ˆ H , and an initial Gaussian state ˆ ρ (0), we ask the followingquestion: can we quantify how much the state ˆ ρ ( τ ) deviates from a Gaussian state attime τ ? This question stems from the following observation. The dynamics of oursystem is non-linear. Therefore, we expect an initial Gaussian state, characterisedby a Gaussian Wigner function, to become a non-Gaussian state at later times. Infact, the only way for a Gaussian state to preserve its Gaussian character would beto evolve through a linear transformation, which is induced by a Hamiltonian with atmost quadratic terms in the quadrature operators [59].To answer our question we first need to find a suitable measure of deviation fromGaussianity. In this work we choose to employ a measure for pure states, which wedenote δ , that is based on the comparison between the entropy of the final state andthat of a suitably chosen reference Gaussian state [50]. A similar measure has beenused to compute features of mixed systems [60]. nhanced continuous generation of non-Gaussianity through optomechanical mod... Let us detail here the construction of the non-Gaussianity quantifier δ ( τ ) for ournonlinear dynamics. First, our initial state ˆ ρ (0) evolves into the state ˆ ρ ( τ ) at time τ .With our full solutions for the dynamics in Section 2.2, we can find analytic expressionsfor the first and second moments of ˆ ρ ( τ ). Then, we construct a state ˆ ρ G ( τ ), which isthe Gaussian state defined by the first and second moments that coincide with thoseof ˆ ρ ( τ ). Now, we recall that a Gaussian state is fully defined by its first and secondmoments. Therefore, if two Gaussian states ˆ ρ and ˆ ρ have equal first and secondmoments they are the same state [56, 59]. In general, our state ˆ ρ ( τ ) at time τ will notbe Gaussian and therefore cannot be specified fully by its first and second moments.This implies that we can introduce a measure δ ( τ ) that quantifies how ˆ ρ ( τ ) deviatesfrom ˆ ρ G ( τ ): δ ( τ ) := S (ˆ ρ G ( τ )) − S (ˆ ρ ( τ )) , (15)where S (ˆ ρ ) is the von Neumann entropy of a state ˆ ρ , defined by S (ˆ ρ ) := − Tr(ˆ ρ ln ˆ ρ ).This measure has been shown to capture the intrinsic non-Gaussianity of the system,and it vanishes if and only if ˆ ρ ( τ ) is a Gaussian state [50]. In other words, if at alltimes δ ( τ ) = 0 this implies that the state is Gaussian and the dynamics is fully linear.We now note that the time evolution is unitary. This means that S (ˆ ρ ( τ )) = S (ˆ ρ (0)). If we start from a pure state, then S (ˆ ρ (0)) = 0 and δ ( τ ) = S (ˆ ρ G ( τ )). Wediscuss the case where the initial state is mixed in Section 7. Since ˆ ρ G ( τ ) is a Gaussian state, its entropy can be exactly computed using thecovariance matrix formalism [56, 59]. The covariance matrix consists of the secondmoments of a quantum state, and can be used to fully characterise a Gaussian state(along with its first moments). This is convenient, as the construction of ˆ ρ G involvesfinding the first of second moments of ˆ ρ anyway. While we could compute the entropyfor ˆ ρ G by finding a diagonal basis in the Hilbert space, there exists a straight-forwardmethod within the covariance matrix formalism.To compute the entropy, we introduce the 4 × σ ( τ ) of thestate ˆ ρ G ( τ ). This matrix contains the second moments of the state ˆ ρ ( τ ) which in ourspecific choice of basis ˆ X = (ˆ a, ˆ b, ˆ a † , ˆ b † ) T is defined through its elements σ nm ( τ ) := (cid:104){ ˆ X n , ˆ X † m }(cid:105) − (cid:104) ˆ X n (cid:105)(cid:104) ˆ X † m (cid:105) , where {• , •} is the anti-commutator, we have defined theexpectation value of an operator (cid:104)•(cid:105) := Tr {• ρ G } , and where for the sake of simplernotation, we have chosen not to write out the time-dependence explicitly. To computethe entropy S (ˆ ρ G ( τ )) we require the symplectic eigenvalues {± ν + ( τ ) , ± ν − ( τ ) } of σ ( τ ),where the property ν ± ( τ ) ≥ i Ω σ ( τ ), where Ω is the symplecticform given by Ω = diag( − i, − i, i, i ) in this basis. The von Neumann entropy S ( σ ) isthen given in this formalism by S ( σ ) = s V ( ν + ) + s V ( ν − ), where the binary entropy s V ( x ) is defined by s V ( x ) := x +12 ln (cid:0) x +12 (cid:1) − x − ln (cid:0) x − (cid:1) . In summary, the state ˆ ρ ( τ )is non-Gaussian at time τ if and only if δ ( τ ) > We shall now infer some general characteristics of the measure of non-Gaussianity δ ( τ ).The following analysis only holds when the system is pure, which in our case means nhanced continuous generation of non-Gaussianity through optomechanical mod... ν ± satisfy ν ± ≥ ν ± = 1 + δν ± , where δν ± ≥ τ = 0 then it follows that ν ± (0) = 1 [59]. If the evolutionis linear, then it is also the case that ν ± ( τ ) = 1 for all τ . For closed dynamics, thesymplectic eigenvalues may only change if the evolution is non-linear. In this case, wewould define ν ± = ν ± + δν ± with ν ± >
1. Then, we would have that ν ± (0) = ν ± and, again, linear evolution would imply that ν ± ( τ ) = ν ± . The preceding statementsimply that δν ± are functions of the nonlinear contributions alone. Thus, when thenon-linearity tends to vanish, then δν ± →
0. Among the possible asymptotic regimeswe have that δν ± → + ∞ or that it becomes constant.These observations are important. To understand their implications we use theexpression ν ± = 1 + δν ± to compute the general deviation from Gaussianity as δ ( τ ) = s V (1 + δν + ) + s V (1 + δν − ). Using this form, we see that in the nearly linear(Gaussian) regime with only small contributions from the nonlinear dynamics, we willhave δν ± (cid:28) δ ( τ ) ≈ − δν + δν + − δν − δν − . (16)On the contrary, in the highly nonlinear (non-Gaussian) regime we have δν ± (cid:29) δ ( τ ) ≈ ln δν + + ln δν − . If the symplectic eigenvalues depend on alarge parameter x (cid:29)
1, then one will in general find that they have the asymptoticform ν ± ∼ x N ± (cid:80) N ± n =0 ν ( n ) ± x − n for some appropriate real coefficients ν ( n ) ± , where N ± constitutes the upper limits of the sum [61].A careful asymptotic expansion of the measure of nonlinearity in this regime gives δ ( τ ) ∼ ( N + + N − ) ln x. (17)These general results allow us to anticipate that these general behaviours will beconfirmed by the explicit analytical and numerical computations below. A detailedcomputation can be found in Appendix C.
4. General results
We now proceed to evolve the initial state ˆ ρ (0) with the evolution operator ˆ U ( τ ) in (5).To compute the amount of non-Gaussianity of this state δ ( τ ), we must first find theelements σ nm of the covariance matrix σ , which we construct form the first and secondmoments of ˆ ρ ( τ ). We do so in full generality, meaning that the light–matter coupling˜ g ( τ ) can take the form of any time-dependent function.We have computed the second moments and the covariance matrix σ in AppendixB. The second moments for the mechanical being in an initial coherent state and in aninitial coherent thermal state can be found in (B.5) and (B.9) respectively. These canthen be used to compute the covariance matrix elements σ nm where n, m take values0 , , ,
3. We have explicitly computed the elements of σ for a mechanical coherentstate in (B.6).Our challenge now is to compute the symplectic eigenvalues ν ± , given theexpressions (B.6). The process of computing the eigenvalues can be simplified byusing the expression 2 ν ± = ∆ ± (cid:112) ∆ − σ ), which is based on the existence ofsymplectic invariants [59]. The definition of ∆ is given in Appendix A.2. The full nhanced continuous generation of non-Gaussianity through optomechanical mod... ν ± of σ in (B.6) is too long andcumbersome to be printed here. It also does not yield any immediate insight intothe behaviour of δ ( τ ). Instead, we will proceed to derive two analytic expressions forthe two different regimes we identified in Section 3.3; small and large coherent stateparameters respectively. We begin by looking at the case where | µ c | (cid:28) ν + ∼ (cid:16) − | F | e −| F | (cid:17) | µ c | ν − ∼ (cid:16) − e −| F | (cid:17) | F | | µ c | . (18)This implies that the behaviour of δ ( τ ) for small | µ c | goes as δ ( τ ) ∼ − (cid:16) (cid:16) − e −| F | (cid:17) | F | (cid:17) | µ c | ln | µ c | , (19)in perfect agreement with (16). This approximation suggests that δ ( τ ) scales with ∼ | F | | µ c | ln | µ c | to leading order.These expressions do not hold if the mechanical element is initially mixed.However, we will find in the next section that initial phonon occupation onlymarginally affects the non-Gaussianity. We now investigate the case where | µ c | (cid:29)
1. Our goal is to derive an analyticexpression for the non-Gaussianity that can be used to analyse the overall featuresof δ ( τ ). Before making any quantitative evaluation, we recall that the measure willhave the form (17), where now x ≡ | µ c | . Let us proceed to demonstrate this resultanalytically for this specific case.For large µ c and for the mechanics in the ground-state µ m = 0, it is clear thatwhenever θ a (cid:54) = 2 πn for integer n , the matrix elements σ , σ and σ in (B.6) vanish,due to the exponentials containing the factor | µ c | . Therefore, far (enough) from thetimes where θ a = 2 πn we are left with the following covariance matrix elements σ ∼ σ = 1 + 2 | µ c | (cid:16) − e − | µ c | sin θ a / e −| F | (cid:17) σ ∼ σ = 2 | µ c | | F | + 1 σ ∼ σ ∗ = 2 | µ c | e − iτ F ∗ , (20)and all other elements are zero. We have kept the full expression for σ becauseit reproduces some key elements of δ which we shall discuss later. Therefore, we donot expect the thermal occupation of the mechanics to significantly affect the non-Gaussianity that can be accessed in this system. Note also that we need to keep thenext leading order in each element of (20), which is a constant in the case of σ and σ . Naively neglecting of this element would give an incorrect result when computingthe entropy, as the neglected factor becomes significant in the logarithm [61]. If the nhanced continuous generation of non-Gaussianity through optomechanical mod... (cid:104) a (cid:105) changes slightly and σ in (20) will look different. However, if we approximate σ as σ ≈ | µ c | , which follows from that (cid:104) a (cid:105) ∼ µ c , the non-zerocovariance matrix elements of the coherent thermal mechanical state are the same asin (20). We therefore conclude that an initially thermal coherent mechanical statewill also exhibit most of the non-Gaussianity we will examine for coherent mechanicalstates.With this simplified matrix, we are able to find a simple and analytic expressionfor the symplectic eigenvalues, which reads ν + ∼ | µ c | (cid:16) − e − | µ c | sin θ a / e −| F | (cid:17) ν − ∼ (cid:112) | µ c | | F | + 1 . (21)We note that both eigenvalues grow with | µ c | , as expected from our analysis inSection 3.3. The amount of non-Gaussianity for large µ c is now given by the followingexpression δ ( τ ) ∼ s V (cid:16) | µ c | (cid:16) − e − | µ c | sin θ a / e −| F | (cid:17)(cid:17) + s V ( (cid:112) | µ c | | F | + 1) , (22)which scales asymptotically as δ ( τ ) ∼ ˜ δ ( τ ) := 4 ln | µ c | , in perfect agreement with (17).Note that (22) is also valid for a time-dependent light–matter coupling ˜ g ( τ ). Inall cases, the nonlinearity grows as ln | µ c | to leading order. In Sections 5.2 and 6.3 wewill compare the asymptotic measure ˜ δ ( τ ) with the full measure δ for different cases.
5. Applications: Constant coupling
Let us now move on to a quantitative analysis of the evolving non-Gaussianity indifferent contexts. We begin by considering the case where the nonlinear light–matterinteraction is constant: ˜ g ( τ ) = ˜ g . To a large extent, this is the case for mostexperimental systems. The coefficients which determine the time-evolution are thosefound in (9), and we note that the function F , defined in (8), which appears in thecovariance matrix elements σ nm is now given by F = ˜ g (1 − e − iτ ).We now proceed to compute the exact measure of non-Gaussianity δ ( τ ) forconstant coupling ˜ g and with the system initially in two coherent states. The exactexpression is again too long and cumbersome to be reprinted here, but we plot theresults in Figure 2 and Figure 3. In Figure 2a we plot the measure of non-Gaussianity δ ( τ ) as a function of time τ for different values of the coherent state parameter µ c ,over the period 0 < τ < π . The other parameters are set to ˜ g = 1 and µ m = 0. Itis known that the full nonlinear dynamics is periodic (or recurrent) with period 2 π ,see [25], whenever ˜ g is an integer and this is clearly reflected in our plot.At τ = 2 π , the optics and mechanics are no longer entangled, and while themechanics returns to its initial coherent or coherent thermal state (see SupplementalNote 1 in [25] for an explicit proof), the final optical state will depends on the valueof ˜ g . For example, when ˜ g = 0 .
5, the cavity state becomes a superposition ofcoherent states at τ = 2 π , also known as a cat state [22]. However, if ˜ g is integer,we obtain a phase factor of e πi ˜ g = 1 in the optical state, and the optics returnsto an initial state as well. This is the case in Figure 2, where δ (2 π ) = 0. We willmake use of the asymptotic measure defined in Section 4.2 to analyse this behaviour, nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b)(c) Figure 2.
The measure of non-Gaussianity δ ( τ ) vs. time τ for systems withconstant nonlinear coupling ˜ g . (a) A plot of δ ( τ ) as a function of time τ fordifferent coherent state parameters µ c . The rescaled coupling is ˜ g = 1 and themechanics is in the ground state with µ m = 0. (b) A plot of δ vs. time τ near τ = π for varying µ c . The measure displays a local minimum centered around τ = π that becomes sharper with larger µ c . Here ˜ g = 1 and µ m = 0. (c) Aplot of log δ ( τ ) at very small times τ for different coherent state parameters µ c ,˜ g = 1 and µ m = 0. The measure increases exponentially at first before it plateaustowards a constant value, which is the overall behaviour we observe in (a). see Section 5.2. Furthermore, while it might seem that the non-Gaussianity peaks at τ = π , the measure δ exhibits a local minimum which grows increasingly narrow withlarger µ c . This is apparent from Figure 2b where we have shown a close-up of δ around τ = π for increasing values of µ c , and for ˜ g = 1 and µ m = 0. The dip occurs becauseat τ = π , we find that θ a = − π ˜ g and F = − g . Thus, for integer ˜ g , we havesin θ a / σ becomes σ = 1 + 2 | µ c | (cid:16) − e − g (cid:17) . The non-zero exponentcauses the non-Gaussianity to temporarily decrease, and the same behaviour occursin the other covariance matrix elements, resulting in the dip.As already noted, increasing µ c yields a logarithmic increase in δ ( τ ), which isevident from the approximation in Eq. (22). Figure 2a also implies that for closeddynamics, the nonlinear system will almost immediately become maximally non-Gaussian. It will then retain approximately the same amount of non-Gaussianity until τ = 2 π , meaning there will be a rapid decrease of non-Gaussianity before the systemrevives again. The appearance of these plateaus shows that the maximum amountof non-Gaussianity available during one cycle can be accessed almost immediately nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b) Figure 3.
The behaviour of the measure of non-Gaussianity δ ( τ ) at τ = π for systems with constant nonlinear coupling ˜ g starting in coherent states. (a) A log–log plot of δ ( τ ) vs. the rescaled coupling ˜ g . As ˜ g increases, the statebecomes more and more non-Gaussian, polynomially at first but then it quicklytends towards a constant value. (b) A log-plot of δ ( τ ) vs. the coherent stateparameter µ c for different values of ˜ g . δ ( τ ) first increases quickly, then plateaustowards a single value. without requiring the system to evolve for a long time. As a side remark, we notethat the functional form of δ ( τ ) in Figure 2a closely resembles the linear entropy ofthe traced-out subsystems as found in [22, 25].To better understand the behaviour of δ ( τ ) for small times τ , we plot thebehaviour of log δ ( τ ) for τ (cid:28) µ c in Figure 2c. We notethat δ ( τ ) increases quickly at first, but soon tends to a near-constant value. Thismeans that δ ( τ ) grows linearly for an interval of small times, which can be seen as theincreasing and decreasing parts in Figure 2a.Finally, we proceed to examine the scaling behaviour of δ ( τ ) at fixed time τ = π .Figure 3a shows a log –log plot of the measure δ ( τ ) as a function of the nonlinearcoupling ˜ g for different values of µ c . As ˜ g increases, the amount of non-Gaussianityfirst grows linearly in the logarithm, then plateaus as ˜ g increases further. The samebehaviour occurs for larger µ c , only more rapidly. This suggests that if we wish toincrease the non-Gaussianity substantially, it will become increasingly difficult to doso by increasing ˜ g . As such, focusing on increasing the coupling ˜ g will only givemarginal returns. Similarly, 3b shows log δ ( τ ) as a function of increasing µ c forvarious values of ˜ g . For a small amplitude coherent state of the optics, with | µ c | (cid:28)
1, and with themechanics in a coherent state, we found in (19) that δ ( τ ) scales with ∼ | F | | µ c | ln | µ c | .Given the explicit form of F , we see that it scales with F ∝ ˜ g . Since δ ( τ ) in thisregime is proportional to | F | , it follows that δ ( τ ) grows quadratically with the light–matter coupling in this regime. We derived an asymptotic form of δ ( τ ) in (22) for the case | µ c | (cid:29)
1, which wecalled ˜ δ ( τ ). As argued before, the behaviour of the measure δ ( τ ) in this regime nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b) Figure 4.
Comparing the measure of non-Gaussianity δ ( τ ) (solid lines) withthe asymptotic form computed in Eq. (22) (dashed lines) for coherent states. (a) Exact measure (solid line) vs. the approximation for different values of µ c . As µ c increases, the approximation grows increasingly accurate. In this plot, ˜ g = 1. (b) Exact measure (solid line) vs. the approximation for large µ c for increasingvalues of ˜ g and µ c = 10. The approximation becomes increasingly accurate as˜ g increases, even towards the beginning and end of one oscillation period. depends crucially on the distance of θ a from the value 2 π . In our present case wehave that θ a ∼ τ for τ (cid:28) θ a ∼ − ˜ g τ for τ (cid:29)
1. The functions thatwe decided to ignore (except for σ ) in the derivation of ˜ δ ( τ ) are of the form f | µ c | ( θ ) = (1 − exp[ − β | µ c | sin( θ a / f | µ c | ( θ ) = (1 − exp[ − β (cid:48) | µ c | sin ( θ a / β and β (cid:48) are irrelevant numerical constant of order 1. We focus on f | µ c | ( θ ) andnote that a similar argument applies for the other function as well. Finally, we ignorethe transient regime of τ (cid:28) θ a ∼ − ˜ g θ .To see how well the asymptotic form ˜ δ ( τ ) in (22) approximates the exact measure,we have plotted both the exact form of δ ( τ ) (solid lines) with the asymptotic form(dashed lines) in Figure 4. We note that, even for | µ c | ∼
1, the asymptotic measure˜ δ ( τ ) well approximates the exact value of δ ( τ ). In fact, it becomes even more accurateas the optical coherent state parameter µ c increases, which is to be expected giventhe nature of the approximation. The asymptotic form also becomes more accurateonce we also increase ˜ g , as evident in Figure 4b. For ˜ g = 10 , the approximation isalmost entirely accurate. This occurs because the function θ a increases with ˜ g , whichfurther suppresses the off-diagonal covariance matrix elements at the beginning andend of each cycle.Let us discuss the fact that the measure recurs with τ = 2 π for integer ˜ g which wecan now address analytically by examining the asymptotic covariance matrix elementsin (20). We find that F = 0 for all τ = 2 πn with integer n . This means that σ = 0and that σ = 1. We also find that θ n (2 π ) = − π ˜ g . Thus, if ˜ g is integer, we findthat sin θ a / σ = 1. This resultsin σ = diag(1 , , ,
1) which corresponds to a coherent state, which is fully Gaussian.As a result, the non-Gaussianity vanishes. When ˜ g is not an integer, some non-Gaussianity will be retained, but the fact that F = 0 will still result in a reduction at τ = 2 π . nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b) Figure 5.
Non-Gaussianity of open optomechanical systems with constant light–matter coupling starting in a coherent state. (a)
Non-Gaussianity δ vs. time τ for a system with increasing values of photon decoherence ¯ κ c for ˜ g = 1, µ c = 0 . µ m = 0. (b) Non-Gaussianity δ vs. time τ for a system with increasingvalues of phonon decoherence ¯ κ m for ˜ g = 1, µ c = 0 . µ m = 0. A populatedmechanical coherent state µ m (cid:54) = 0 does not affect the non-Gaussianity. Any realistic system will suffer from decoherence. In Figure 5 we have plotted thenon-Guassianity δ ( τ ) as a function of time for an optomechanical system with opendynamics. Here, the cavity state and the mechanics are both in initial coherentstates (11). Figure 5a shows the non-Gaussianity for increasing values of the photondecoherence rate ¯ κ c = κ c /ω m with Lindblad operator ˆ L c = √ ¯ κ c ˆ a and values µ c = 0 . g = 1 and µ m = 0. We have chosen a low value of µ c to ensure high numericalaccuracy of the simulation, as larger values quickly lead to numerical instabilities. Wenote that the non-Gaussianity δ ( τ ) tends towards a steady value, which is clear fromthe fact that the higher values of decoherence start to coincide around τ = 5 π . We alsonote that around τ = 2 πn , for integer n the inclusion of noise appears to temporarilyincrease the non-Gaussianity. This could, however, be due to the fact that the relativeentropy measure cannot distinguish between non-Gaussianity induced as a result ofgenuinely nonlinear dynamics or as a result of classical mixing of the states [62]. Wediscuss this further in Section 7. Similarly, in Figure 5b we have plotted the non-Gaussianity δ ( τ ) for increasing values of phonon decoherence rate ¯ κ m with Lindbladoperator ˆ L m = √ ¯ κ m ˆ b and the same values as before.
6. Applications: Time-dependent coupling
In all physical systems, such as optomechanical cavities, the confining trap is not ideal.This means that, in general, the coupling ˜ g ( τ ) is time-dependent as a consequence of,for example, trap instabilities. Time-dependent variations such as phase fluctuationsin the laser beam used to trap a levitated bead will modulate the coupling.In this work, we want to exploit the possibility of controlling the coupling ˜ g ( τ ) byconsidering its periodic modulation in time. In practice, such time-dependent controlwould be achievable for an optically trapped and levitated dielectric bead that interactswith a cavity field by controlling the optical phase of the trapping laser field. In fact, nhanced continuous generation of non-Gaussianity through optomechanical mod... We shall model a time-dependent light–matter coupling ˜ g ( τ ) by assuming that thecoupling has the simple form˜ g ( τ ) = ˜ g (1 + (cid:15) sin(Ω τ )) . (23)Here, ˜ g is the expected value of the coupling, (cid:15) is the amplitude of oscillationand Ω := ω /ω m is the dimensionless frequency that determines how the couplingoscillates in time. We can insert this ansatz in the general expressions (7) and obtainan explicit form for this case. The full expressions for the coefficients in (7) are againvery long and cumbersome, and we do not print them here. They are listed in (D.1).We can now compute δ ( τ ) for this time-dependent coupling for initial coherentstates, and we display the results in Figure 6. In 6a we plot δ ( τ ) vs. τ for differentvalues of the oscillation frequency Ω . Note that we here include a larger range of τ to capture potentially recurring behaviour. In the limit Ω → (cid:54) = 0 we see that we canachieve higher values for the nonlinear measure δ ( τ ). This is especially pronouncedas Ω →
1, where the trap oscillation frequency is equal to the mechanical frequency ω m , for which δ ( τ ) ceases to oscillate periodically, but instead steadily increases. Wediscuss this case in detail in the following section. The functions (D.1) contain denominators of the form Ω −
1. Therefore, amongall possible values of Ω , we can ask what happens on resonance , i.e., when Ω =1. Figure 6a already provides evidence that the system should behave markedlydifferently.At resonance, where Ω = 1, the functions (D.1) take the relatively simple form F ˆ N a = −
116 ˜ g (cid:2) τ − τ ) + (cid:15) (32 −
36 cos( τ ) + 4 cos(3 τ ))+ (cid:15) (cid:0) τ − τ ) + sin(2 τ ) cos(2 τ ) (cid:1)(cid:3) F ˆ N a ˆ B + = − ˜ g sin( τ ) (cid:16) (cid:15) τ ) (cid:17) F ˆ N a ˆ B − = ˜ g (cid:15) (sin(2 τ ) − τ ) − g sin (cid:16) τ (cid:17) (24)We have plotted in Figure 6 the exact measure of non-Gaussianity δ ( τ ) in the resonantcase for initially coherent states and for different values of µ c . As anticipated, herewe no longer have recurrent behaviour. Instead, the non-linearity increases as ln τ .Formally, this growth can continue for arbitrarily large times τ , however, the maximum nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b)(c) Figure 6.
The measure of non-Gaussianity δ ( τ ) for systems with time-dependentcoupling ˜ g ( τ ) = ˜ g (1 + (cid:15) sin (Ω τ )), where (cid:15) is the amplitude and Ω = ω /ω m isthe modulation frequency. (a) Plot of δ ( τ ) vs. rescaled time τ for different valuesof Ω . The case Ω = 0 (blue line) corresponds to the time-independent setting.At resonance, with Ω = 1 (green line), the system displays a drastically differentbehaviour. Other parameters are ˜ g = (cid:15) = 1 and µ m = 0. (b) Plot of δ ( τ ) vs.rescaled time τ at resonance Ω = 1 for various values of coherent state parameter µ c . The system no longer exhibits closed dynamics. Other parameters include˜ g = (cid:15) = 1 and µ m = 0. (c) A plot of δ ( τ ) vs. time τ for increasing oscillationfrequency (cid:15) at resonance Ω = 1 and with µ c = 1. δ ( τ ) increases slowly with (cid:15) .Again, we have set µ m = 0. time τ that can be achieved in practice is limited by the coherence time of theexperiment. Similarly, we plotted δ ( τ ) for various values of (cid:15) in Figure 6c. We notethat δ ( τ ) oscillates increasingly rapidly with larger (cid:15) but with decreasing amplitudefor increasing τ , as | F | ∼ ˜ g (cid:15) τ becomes the dominant term for τ (cid:29) (cid:104) ˆ a † ˆ a (cid:105) is conserved, the coupling acts as a photon number displacement. Ifthis coupling is time-dependent, this means that the photon pressure displaces witha time-dependence. When this occurs at resonance, this linear displacement growslinearly in time. See also [48] for further insight once the rotating wave approximationhas been applied. nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b) Figure 7.
A comparison between the full measure δ ( τ ) (solid line) and theapproximate measure (dashed lines) for time-dependent couplings ˜ g ( τ ). (a) Plotshowing the accuracy of the approximation for different values of µ c at Ω = 0 . µ c increases. (b) Plot comparingthe accuracy of ˜ δ for a different values of (cid:15) at µ c = 10. The approximationbecomes more accurate as (cid:15) increases. Using the explicit form of the coefficients (24), we note that | F | = F N a ˆ B − + F N a ˆ B + has the asymptotic behaviour | F | ∼ ˜ g (cid:15) τ for τ (cid:29)
1. This implies thatexp[ −| F | ] (cid:28) τ and therefore we expect, as it happened in Section 4.2, thatmost covariance matrix elements will vanish and will not contribute to the asymptoticform of δ ( τ ). This observation allows us to compute the symplectic eigenvalues, whichread ν + = 1 + 2 | µ c | (cid:16) − e − | µ c | sin θ a / e −| F | (cid:17) and ν − = (cid:112) | µ c | | F | , andthey match the expressions (21). We again stress that we have retained the exactexpression for σ to capture some crucial features of the non-Gaussianity, such as δ (0) = 0.In Figure 7, we compare the exact measure δ ( τ ) at resonance with the asymptoticform derived in (22). The solid lines represent the exact measure δ ( τ ) and the dashedlines represent the asymptotic expression. In Figure 7a we compare them for differentvalues of µ c . We note that, except for at very small τ , the asymptotic form is entirelyaccurate and gets even more precise for increasing values of µ c . This is a consequence,as we noted before, of the exponentials in (B.6) that suppress some elements for large µ c , unless θ a = n π . Similarly, in Figure 7b we have plotted δ ( τ ) and its asymptoticform for different values of the oscillation amplitude (cid:15) . Again, the suppression of theexponentials with increasing (cid:15) means that larger values of (cid:15) yield a more accurateexpression. If it is possible to continuously increase the non-Gaussianity, the system might have acertain tolerance to noise. That is, there is a level of noise at which the non-Gaussianityessentially reaches a steady-state. In Figure 8 we have plotted the non-Gaussianity δ asa function of time for different values of photon and phonon decoherence. Figure 8ashows the system at resonance with photons leaking from the cavity with a rate¯ κ c = κ c /ω m for parameters µ c = 0 .
1, ˜ g = 1, (cid:15) = 0 . µ m = 0. We note that nhanced continuous generation of non-Gaussianity through optomechanical mod... (a) (b) Figure 8.
Non-Gaussianity for open optomechanical systems at mechanicalresonance. (a)
Non-Gaussianity δ vs. time τ for a system with increasing valuesof photon decoherence ¯ κ c for ˜ g = 1, µ c = 0 . (cid:15) = 0 .
5, and µ m = 0. (b) Non-Gaussianity δ vs. time τ for a system with increasing values of phonondecoherence ¯ κ m for ˜ g = 1, µ c = 0 . (cid:15) = 0 .
5, and µ m = 0. Changing to µ m (cid:54) = 0does not affect the results. ¯ κ c = 0 . κ m = κ m /ω m .
7. Discussion and practical implementations
We have employed a measure of non-Gaussianity δ ( τ ) in order to quantify the deviationfrom linearity of an initial Gaussian state induced by the Hamiltonian (1). Our resultsshow that, for a constant light–matter coupling ˜ g , the non-Gaussianity δ ( τ ) scalesdifferently in two contrasting regimes: (i) For a weak optical input coherent state | µ c | ,the nonlinear character of the state grows as ˜ g | µ c | ln | µ c | if the mechanics is alsoin a coherent state, (ii) conversely, for large | µ c | , the nonlinear character of the stategrows logarithmically with the quantity ˜ g | µ c | , which also holds when the mechanicalelement is not fully cooled. The same general scaling with | µ c | occurs when ˜ g ( τ ) istime-dependent.Crucially, we also find that the amount of non-Gaussianity can be continuouslyincreased by driving the light–matter coupling at mechanical resonance. This becomesespecially useful in the presence of noise. We will now discuss these results in thecontext of concrete experimental setups, and specifically discuss how the modulatedlight–matter coupling can be engineered. First, however, we will discuss the measureof non-Gaussianity that we have used in this work. In this work, we chose to work with a relative entropy measure of non-Gaussianity(see Section 3) which was first defined in [50]. This measure has previously beenextensively used to compute the non-Gaussianity of various states [66], as well asin an experimental setting where single photons were gradually added to a coherentstate to increase its non-Gaussian character [62]. Several additional measures for thequantification of non-Gaussianity have been proposed in the literature, linking it to nhanced continuous generation of non-Gaussianity through optomechanical mod... µ c to infinity yields lim µ c →∞ δ = ∞ . Assuch, it is only possible to state that one state is more non-Gaussian than another.However, for pure states, there is the relation of the measure to the Hilbert-Schmidtmeasure. As such, the non-Gaussianity δ ( τ ) of pure states has strong operationalimplications [8].For mixed states, the operational meaning is not clear because the measure cannotdetect the difference between classical mixtures of Gaussian states, which can be easilyprepared by classical mixtures of Gaussian states, and inherent non-Gaussianity dueto some nonlinear evolution of pure states [62]. This means that the measure oftenneeds to be used together with a measure of non-classicality, such as the negativityof the Wigner function. We know from previous work [22, 25] that for a constantcoupling, the system is maximally entangled at τ = π , which satisfies the occurrenceof non-classicality in conjunction with the non-Gaussianity. The state is however fullydisentangled at τ = 2 π , and in the case of open system dynamics, this feature of themeasure becomes apparent. We note that the non-Gaussianity plotted in Figure 5spikes at times τ = 2 πn for integer n , which is when we usually have no entanglement.This implies that the addition of non-Gaussianity most likely comes from a classicalmixture of coherent states that have slightly decohered. There are two relevant experimental regimes for optomechanical systems. They aredetermined by the magnitude of the light–matter coupling g compared to the otherfrequencies in the system. In the weak single-photon optomechnical coupling regime,the light–matter coupling g is small compared to the resonant frequency ω m and theoptical decoherence rate κ c . Such experiments usually involve a strong laser drive,which tends to wash out the non-linearity. In the strong single-photon couplingregime, nonlinear effects are in practice small but more significant. Under theseconditions, a single photon displaces the mechanical oscillator by more than its zero-point uncertainty and weak optical fields tend to be used [71]. In summary, mostapproaches fall into one of two categories: (i) small g and linearised dynamics and (ii)large g and low number of photons.Our work suggests that we can further increase the amount of non-Gaussianity bymodulating the light–matter coupling. We emphasize that this scheme is applicablein both the weak and strong coupling regimes. This sets it apart from other schemes,which usually focus on enhancing the non–Gaussianity in one of the two categoriesmentioned above.Let us also briefly discuss our results with regard to linearised dynamics. Thislinearisation of dynamics is fundamentally different to the scenarios considered in thiswork. When linearising the dynamics, the system is opened and the field operators nhanced continuous generation of non-Gaussianity through optomechanical mod... a are treated as flucutations around a strong optical field as such: ˆ a → ˆ a = α + ˆ a (cid:48) ,where ˆ a (cid:48) are the fluctuations. In this work, we have retained the nonlinear dynamics,even when considering open system dynamics. Thus, while we observe that a largecoherent state parameter µ c increases the non-Gaussianity, we cannot generalise thisresult to the linearised dynamics. We saw in Section 6 that the amount of non-Gaussianity in the system increases whenthe light–matter coupling ˜ g ( τ ) is modulated. An explanation of this phenomena wasprovided in [48]. Consider the force (cid:126)F exerted by the photons on the mechanics. Fora number of n photons, this force is proportional to (cid:126)F ∝ ( n + 1 / / g ( τ ),the photon-pressure force (cid:126)F acts periodically on the mechanics, and is amplified whenpushing in tandem with the mechanical resonance.While engineering the modulation is challenging, we shall explore several methodsthat can achieve it. The question is whether the modulation can be performed atmechanical resonance. As a basis for this discussion, we present a derivation of atime-dependent light–matter coupling for levitated nanobeads in Appendix E, whichis based on the work in [72]. There are several practical ways in which one mayenvisage to increase the non-linearity by modulating the coupling, depending on thenature of the trap at hand:i) Optically-trapped levitating particles.
The effect that we are looking forcan be realised by modulating the phase of the trapping laser beam (which, inturn, can be achieved through an acousto–optical modulator). In our derivation inAppendix E, this phase is denoted by ϕ ( τ ) and it affects the light–matter couplingstrength by determining the particle’s location with respect to the standing waveof the cavity field. Thus if we let ϕ ( τ ) = π (1 + (cid:15) sin Ω τ ), with Ω = ω /ω m ,and where ω is the phase modulation frequency, we obtain the expression usedin Section 6. If, then, the phase frequency is resonant with Ω = 1, it should bepossible to increase the non-Gaussianity even further.ii) Paul traps.
The shuttling of ions has been demonstrated [65, 64] using Paultraps, which are customarily used for ions but which have also recently been usedfor trapping nanoparticles [63, 73, 74]. These works indicate that a modulationof the particle’s position, and hence, a modulation of the coupling as per pointi), can be obtained in a Paul trap as well.iii)
Micromotion in hybrid traps.
Paul traps display three different kinds ofparticle motion. Firstly, we have thermal motion , whereby the particle movesaround the trap. Secondly, and most importantly to our scheme, we have micromotion , which induces small movements around the potential minimum.Finally, there is mechanical motion , which is the harmonic motion in the trap,here denoted by ω m . Since the micromotion moves the bead around the potentialminimum with a frequency ω d , this already modulates the light–matter coupling,and is, in a way, an equivalent implementation to the “shaking” of the trap. Ifthe micromotion can be engineered to occur with a frequency ω d equal to ω m ,then one could, instead of averaging it out, adopt the micromotion’s variables toincrease the non–Gaussianity with the scheme we propose in Section 6.2. To date, nhanced continuous generation of non-Gaussianity through optomechanical mod... ω d ≤ ω m ,but current experimental efforts appear promising.There are potentially many more ways in which the light–matter coupling couldbe modulated, including with optomechanically induced transparecny [75, 76] and byusing the Kerr effect to change the refractive index of the oscillator.We conclude that the enhancement of the non-linearity predicted by our workcan be realised in experiments, given the capabilities mentioned above. There are, ofcourse, many challenges to be overcome. In fact, to take advantage of the rather slowlogarithmic scaling with time τ , one must keep the system coherent for longer, whichis difficult. However, although our analytical results are restricted to Hamiltoniansystems, we note that there is no reason to expect that this enhancement shoulddisappear in a noisy setting. In practise, how would one proceed to measure the amount of non-Gaussianity in thelaboratory? As shown in [62], the measure of non-Gaussianity used in this work hasbeen measured for the addition of single photons to a coherent state. This requires fullstate tomography and is thus an expensive process. There are however others waysto proceed. In [77] a witness of non-Gaussianity was proposed based on boundingthe average photon number in the system from above. While they apply to a singlesystem, they can probably be extended to bipartite systems as well.Finally, we here suggest a simple method by which non-Gaussianity can bedetected for pure states. We note that the von Neumann entropy S (ˆ ρ AB ) of abipartite state ˆ ρ AB is bounded by S (ˆ ρ AB ) ≥ | S (ˆ ρ A ) − S (ˆ ρ B ) | , through the Araki–Liebinequality [78] where ˆ ρ A and ˆ ρ B are the reduced states of the optical and mechanicalsubsystems, respectively. Therefore, the measure of non-Gaussianity δ ( τ ) that wedefined in (15) is lower-bounded by δ ( τ ) ≥ | S (ˆ ρ A ) − S (ˆ ρ B ) | − S (ˆ ρ (0)) . (25)In this sense, this reduced measure acts as a sufficient (but not necessary) conditionfor non-Gaussianity. That is, finding that the measure is non-zero does tell us thatthe state is non-Gaussian, however it does not tell us the full magnitude of thenon-Gaussianity. Furthermore, to compute this measure, one would still have tomeasure the second moments of the optical and mechanical subsystems. This does,however, require fewer measurements than full state tomography on the joint opticaland mechanical system.
8. Conclusions
We have quantified the non-Gaussianity of initially Gaussian coherent states evolvingunder the standard, time-dependent optomechanical Hamiltonian. We used a measureof non-Gaussianity based on the relative entropy of a state to characterise the deviationfrom Gaussianity of the full system. Our techniques allowed us to derive asymptoticexpressions for small and large optical coherent-state amplitudes, see Equation (19)and Equation (22) respectively. We found that for coherent states with amplitude | µ c | ≥
1, the amount of non-Gaussianity grows logarithmically with the input averagenumber of excitations | µ c | and with the light–matter coupling. At resonance, we find nhanced continuous generation of non-Gaussianity through optomechanical mod... Acknowledgments
We thank Fabienne Schneiter, Daniel Braun, Nathana¨el Bullier, Antonio Pontin, PeterF. Barker, Ryuji Takagi, Francesco Albarelli, Marco G. Genoni, James Bateman andthe reviewers for helpful comments and discussions.SQ acknowledges support from the EPSRC Centre for Doctoral Training inDelivering Quantum Technologies and thanks the University of Vienna for itshospitality. DR would like to thank the Humboldt Foundation for supporting hiswork with their Feodor Lynen Research Fellowship. This work was supported bythe European Union’s Horizon 2020 research and innovation programme under grantagreement No. 732894 (FET Proactive HOT).
Data availability statement
The figures in this work were generated using
Mathematica . The numerical results inSection 5.3 and 6.4 were obtained using the
Python library
QuTiP . The code used togenerate the data can be found at https://github.com/sqvarfort/QM-Nonlinearities.
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Physical Review A nhanced continuous generation of non-Gaussianity through optomechanical mod... Appendix A. Derivation of the dynamics and general tools
In this appendix, we will derive the coefficients in (7) that determine the time-evolutionof the system. This follows the derivation in [49]. We will also show the explicit time-dependence of the second moments and discuss some methods related to computingthe symplectic eigenvalues of the covariance matrix.
Appendix A.1. Properties of of the nonlinear Hamiltonian
Firstly, we remind the reader that the laboratory time t is rescaled by ω m . Findinga simple expression for ˆ U ( τ ) is straight-forward when the light–matter coupling˜ g = g/ω m is not time-dependent. If ˜ g ( τ ) = g ( ω m τ ) /ω m is time-dependent we requirea more rigorous framework. This is what we present here.We will here follow the derivation in Appendix A in [49]. We note that comparedwith [49], we have here swapped the definition of ˆ a and ˆ b , and we have a minus-signin front of ˜ g ( τ ).For the time-dependent Hamiltonian ˆ H in (1), the time-evolution operator isgiven by ˆ U ( τ ) := ← T exp (cid:20) − i (cid:90) τ d τ (cid:48) ˆ H ( τ (cid:48) ) (cid:21) , (A.1)where ← T is the time-ordering operator.The basis for decoupling the operator is finding a Lie algebra of generators ˆ G i that induce the time-evolution. This Lie algebra must be closed under commutation,that is, either [ ˆ G j , ˆ G k ] ∝ ˆ G l , or [ ˆ G j , ˆ G k ] = c where c is a scalar. This will allow forthe terms in ˆ U ( t ) to be moved with the Baker–Campbell–Hausdorff formula such thatˆ U ( t ) can be written in a simpler form.We start with the ansatz that the evolution operator (3) can be written asˆ U ( τ ) = (cid:89) j ˆ U j ( τ ) = (cid:89) j e − iF j ˆ G j , (A.2)where F j are coefficients corresponding to each of the generators ˆ G j . Our task is nowto find the coefficients F j .We begin by defining the operators ˆ G j in the algebra:ˆ N a := ˆ a † ˆ a ˆ N b := ˆ b † ˆ b ˆ N a := (ˆ a † ˆ a ) ˆ N a ˆ B + := ˆ N a (ˆ b † + ˆ b ) ˆ N a ˆ B − := ˆ N a i (ˆ b † − ˆ b ) , (A.3)It can be verified that the operators in (A.3) form a closed Lie algebra undercommutation. With these operators, our ansatz can be written asˆ U ( τ ) = ˆ U a ( τ ) ˆ U b ( τ ) ˆ U (2) a ( τ ) ˆ U + ( τ ) ˆ U − ( τ ) , (A.4)where we identifyˆ U a ( τ ) = e − i F ˆ Na ˆ N a ˆ U b ( τ ) = e − i F ˆ Nb ˆ N b ˆ U (2) a ( τ ) = e − i F ˆ N a ˆ N a ˆ U + ( τ ) = e − i F ˆ Na ˆ B + ˆ N a ˆ B + ˆ U − ( τ ) = e − i F ˆ Na ˆ B − ˆ N a ˆ B − . (A.5) nhanced continuous generation of non-Gaussianity through optomechanical mod... ← T exp (cid:20) − i (cid:90) τ d τ (cid:48) ˆ H ( τ (cid:48) ) (cid:21) = e − i F ˆ Na ˆ N a e − i F ˆ Nb ˆ N b e − i F ˆ N a ˆ N a e − i F ˆ Na ˆ B + ˆ N a ˆ B + e − i F ˆ Na ˆ B − ˆ N a ˆ B − . (A.6)Differentiating both sides brings down the Hamiltonian (1) on the left, which we herewrite in terms of the generators (6). We then multiply both sides by U † to obtain thefollowing differential equation:Ω ˆ N a + ˆ N b − ˜ g ( τ ) ˆ N a ˆ B + = ˙ F ˆ N a ˆ N a + ˙ F ˆ N b ˆ N b + F ˆ N a ˆ N a + ˙ F ˆ N a ˆ B + ˆ U b ( τ ) ˆ N a ˆ B + ˆ U † b ( τ )+ ˙ F ˆ N a ˆ B − ˆ U b ( τ ) ˆ U + ( τ ) ˆ N a ˆ B − ˆ U † + ( τ ) ˆ U † b ( τ ) (A.7)where ˙ F i = ddt F i . This is the equation that determines the coefficients. We can nowcommute all the operators through, where we findˆ U b ˆ N a ˆ B + ˆ U † b = cos ( F ˆ N b ) ˆ N a ˆ B + − sin ( F ˆ N b ) ˆ N a ˆ B − ˆ U b ˆ N a ˆ B − ˆ U † b = cos ( F ˆ N b ) ˆ N a ˆ B − + sin ( F ˆ N b ) ˆ N a ˆ B + ˆ U + ˆ N a ˆ B − ˆ U † + = ˆ N a ˆ B − + 2 F ˆ N a ˆ B + ˆ N a , (A.8)By inserting this into (A.7), we are able to determine the coefficients by linearindependence. Integrating, we obtain: F ˆ N a = Ω τ,F ˆ N b = τ,F ˆ N a = 2 (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) sin( τ (cid:48) ) (cid:90) τ (cid:48) d τ (cid:48)(cid:48) ˜ g ( τ (cid:48)(cid:48) ) cos( τ (cid:48)(cid:48) ) ,F ˆ N a ˆ B + = − (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) cos( τ (cid:48) ) , and F ˆ N a ˆ B − = − (cid:90) τ d τ (cid:48) ˜ g ( τ (cid:48) ) sin( τ (cid:48) ) , (A.9)where Ω = ω c /ω m and τ = ω m t . Depending on the form of ˜ g ( τ ), we can now usethese equations to find a simplified form of ˆ U ( t ). Appendix A.2. Computing determinants of symplectic matrices
When computing the amount of non-Gaussianity in (15), it is useful to consider thesymplectic eigenvalues of a Gaussian state [59]. In short, for an arbitrary covariancematrix σ , they are the eigenvalues of the matrix i Ω σ , where Ω = diag( − i, − i, i, i )is the symplectic form. There are other ways to define the symplectic eigenvaluesthough. In the following, we have to switch the basis of the operators to a moreconvenient one, but this does not affect the final result. The correct definition can befound in [56]. Let us write an arbitrary covariance matrix σ in the particular basisˆ X = (ˆ q a , ˆ p a , ˆ q b , ˆ p b ) T as σ = (cid:18) A CC T B (cid:19) , (A.10) nhanced continuous generation of non-Gaussianity through optomechanical mod... A = A T , B = B T and all matrices are 2 × a := det( A ), b := det( B ), c + c − := det( C ), and µ − := det( σ ) = ( ab − c )( ab − c − ).Finally, we introduce the parameter ∆ as ∆ := det( A ) + det( B ) + 2 det( C ). Thesymplectic eigenvalues ν ± are then given by2 ν ± :=∆ ± (cid:112) ∆ − σ )= a + b + 2 c + c − ± (cid:112) ( a − b ) + 4( a c + + b c − )( a c − + b c + ) . (A.11) Appendix B. Evolution of first and second moments
In the Heisenberg picture, the time evolution of the mode operators ˆ a and ˆ b induced bythe Hamiltonian considered here is simply ˆ a ( t ) := ˆ U † ( t ) ˆ a ˆ U ( t ) and ˆ b ( t ) := ˆ U † ( t ) ˆ b ˆ U ( t ).In terms of the generators of the Lie algebra defined in (A.3), we explicitly haveˆ a ( t ) := e − i F ˆ N a e − i ( F ˆ N a + F ˆ Na ˆ B + F ˆ Na ˆ B − ) ˆ N a e − i F ˆ Na ˆ B + ˆ B + e − i F ˆ Na ˆ B − ˆ B − ˆ a ˆ b ( t ) := e − i τ (cid:104) ˆ b + ( F ˆ N a ˆ B − − i F ˆ N a ˆ B + ) ˆ N a ) (cid:105) . (B.1)This expression can also be rewritten in more compact notation asˆ a ( t ) = e − iθ a ( ˆ N a +1 / ˆ D ˆ b ( F ∗ ) ˆ a ˆ b ( t ) = e − i τ (cid:104) ˆ b + F ∗ ˆ N a (cid:105) , (B.2)where ˆ D ˆ b ( F ∗ ) is a Weyl displacement operator and where we have introduces thequantities θ a = 2 (cid:16) F ˆ N a + F ˆ N a ˆ B + F ˆ N a ˆ B − (cid:17) F = F ˆ N a ˆ B − + iF ˆ N a ˆ B + . (B.3)These expectation values can then be used to compute the elements of the covariancematrix σ , which in our basis are given by σ = σ = 1 + 2 (cid:104) ˆ a † ˆ a (cid:105) − (cid:104) ˆ a † (cid:105) (cid:104) ˆ a (cid:105) σ = 2 (cid:104) ˆ a (cid:105) − (cid:104) ˆ a (cid:105) σ = σ = 1 + 2 (cid:104) ˆ b † ˆ b (cid:105) − (cid:104) ˆ b † (cid:105) (cid:104) ˆ b (cid:105) σ = 2 (cid:104) ˆ b (cid:105) − (cid:104) ˆ b (cid:105) σ = σ = 2 (cid:104) ˆ a ˆ b † (cid:105) − (cid:104) ˆ a (cid:105) (cid:104) ˆ b † (cid:105) σ = σ = 2 (cid:104) ˆ a ˆ b (cid:105) − (cid:104) ˆ a (cid:105) (cid:104) ˆ b (cid:105) , (B.4)where we have suppressed the time-dependence for notational convenience.We now compute the expectation values for initial optical coherent states andcoherent and thermal coherent states of the mechanics. nhanced continuous generation of non-Gaussianity through optomechanical mod... Appendix B.1. Mechanical coherent states
For the initial coherent state | Ψ( t = 0) (cid:105) = | µ c (cid:105) ⊗ | µ m (cid:105) in (11) and ignoring the globalphases e − i Ω τ , which can be done by transforming into a frame rotating with Ωˆ a † ˆ a ,we obtain (cid:104) ˆ a ( t ) (cid:105) := e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m µ c (cid:104) ˆ b ( t ) (cid:105) := e − iτ µ m + e − iτ F ∗ | µ c | (cid:104) ˆ a ( t ) (cid:105) := e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m ) µ (cid:104) ˆ b ( t ) (cid:105) := e − iτ (cid:0) µ m + F ∗ | µ c | (cid:1) + e − iτ F ∗ | µ c | (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) := | µ c | (cid:104) ˆ b † ( t )ˆ b ( t ) (cid:105) := (cid:12)(cid:12) µ m + F ∗ | µ c | (cid:12)(cid:12) + | F | | µ c | (cid:104) ˆ a ( t )ˆ b ( t ) (cid:105) := e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m µ c e − iτ (cid:2) µ m + (cid:0) | µ c | e − iθ a + 1 (cid:1) F ∗ (cid:3) (cid:104) ˆ a ( t ) ˆ b † ( t ) (cid:105) : = e − iθ a e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m µ c e iτ (cid:2) µ ∗ m + | µ c | e − iθ a F (cid:3) . (B.5)where we have introduced F := F ˆ N a ˆ B − + iF ˆ N a ˆ B + and θ a := 2( F ˆ N a + F ˆ N a ˆ B + F ˆ N a ˆ B − ).The covariance matrix elements are given by σ = σ = 1 + 2 | µ c | (cid:16) − e − | µ c | sin θ a / e −| F | e F ∗ µ ∗ m − F µ m (cid:17) σ = 2 µ e − iθ a e −| F | (cid:18) e − iθ a e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m ) − e | µ c | ( e − iθa − e F ∗ µ ∗ m − F µ m ) (cid:19) σ = σ = 1 + 2 | µ c | | F | σ = 2 e − i τ | µ c | F ∗ σ = σ = 2 F µ c | µ c | ( e − iθ a − e − i θ a e i τ e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m σ = σ = 2 F ∗ µ c (cid:0) | µ c | ( e − iθ a −
1) + 1 (cid:1) e − i θ a e − i τ e | µ c | ( e − iθa − e − | F | e F ∗ µ ∗ m − F µ m . (B.6) Appendix B.2. Mechanical thermal coherent states
In Section 2.4 we noted that the mechanical state is most often found in a thermalstate. We assume that the initial state is a coherent thermal state of the formˆ ρ th = 1¯ nπ (cid:90) d β e −| β | / ¯ n | β (cid:105) (cid:104) β | , (B.7)where ¯ n is the average thermal phonon occupation number. The cavity is still in thecoherent state | µ c (cid:105) . Several of the expectation values can then simplified by notingthat (cid:90) d β β = 0 , (cid:90) d β β = 0 . (B.8) nhanced continuous generation of non-Gaussianity through optomechanical mod... (cid:104) ˆ a ( t ) (cid:105) := 1¯ nπ (cid:90) d β e −| β | / ¯ n e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ β ∗ − F β µ c (cid:104) ˆ b ( t ) (cid:105) := e − iτ F ∗ | µ c | (cid:104) ˆ a ( t ) (cid:105) := 1¯ nπ (cid:90) d β e −| β | / ¯ n e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ β ∗ − F β ) µ (cid:104) ˆ b ( t ) (cid:105) := e − i τ F ∗ | µ c | (cid:0) | µ c | (cid:1) (cid:104) ˆ a † ( t )ˆ a ( t ) (cid:105) := | µ c | (cid:104) ˆ b † ( t )ˆ b ( t ) (cid:105) := | F | | µ c | (cid:0) | µ c | (cid:1) (cid:104) ˆ a ( t )ˆ b ( t ) (cid:105) := 1¯ nπ (cid:90) d β e −| β | / ¯ n e − i θ a e | µ c | ( e − iθa − e − | F | e F ∗ β ∗ − F β µ c × e − i τ (cid:2) β + (cid:0) | µ c | e − iθ a + 1 (cid:1) F ∗ (cid:3) (cid:104) ˆ a ( t ) ˆ b † ( t ) (cid:105) : = 1¯ nπ (cid:90) d β e −| β | / ¯ n e − iθ a e | µ c | ( e − iθa − e − | F | e F ∗ β ∗ − F β µ c × e i τ (cid:2) β ∗ + | µ c | e − iθ a F (cid:3) . (B.9)The resulting covariance matrix elements can be computed from here. Appendix C. Derivation of the asymptotic form of the symplecticeigenvalues
The symplectic eigenvalues ν ± have the expression ν ± = 1 + δν ± . We would like tosee what is the form of the function f ( x ) = x +12 ln x +12 − x − ln x − when we compute f ( ν ± ) and δν ± (cid:28) δν ± (cid:29) δν ± (cid:28)
1, and we have f ( ν + ) = f (1 + δν + )= 2 + δν + δν + − δν + δν + (cid:18) δν + (cid:19) ln (cid:18) δν + (cid:19) − δν + δν + (cid:18) δν + (cid:19) δν + − δν + δν + O (cid:32)(cid:18) δν + (cid:19) (cid:33) = − δν + δν + O (cid:18) δν + (cid:19) . (C.1)An analogous computation can be done for ν − . The last line of (C.1) is a consequenceof the fact that − x ln x (cid:29) x for x (cid:28) nhanced continuous generation of non-Gaussianity through optomechanical mod... δν ± (cid:29)
1, therefore f ( ν + ) = f (1 + δν + )= 2 + δν + δν + − δν + δν + (cid:18) δν + (cid:19) ln (cid:18) δν + (cid:19) − δν + δν + (cid:18) δν + (cid:19) ln δν + (cid:18) δν + (cid:19) − δν + δν + (cid:18) δν + (cid:19) ln δν + (cid:18) δν + (cid:19) ln (cid:18) δν + (cid:19) − δν + δν +
2= ln δν + (cid:18) δν + (cid:19) ln (cid:18) δν + (cid:19) + O (cid:32)(cid:18) δν + (cid:19) (cid:33) = ln δν + (cid:18) δν + (cid:19) δν + + O (cid:32)(cid:18) δν + (cid:19) (cid:33) = ln δν + O (cid:18) δν + (cid:19) , (C.2)which concludes the proof of the claim, since δν ± (cid:29) δν + (cid:29)
1. Ananalogous computation can be done for ν − . Appendix D. Coefficients for time-dependent light–matter coupling
In this appendix, we compute the coefficients used in Section 6. Starting from (A.9),we assume that the coupling has the functional form ˜ g ( τ ) = ˜ g (1 + (cid:15) sin τ Ω ), wherewe have set ˜ g ( τ ) = g ( ω m t ) /ω m . The algebra is straightforward, although cumbersome, nhanced continuous generation of non-Gaussianity through optomechanical mod... F ˆ N a = − ˜ g (cid:2) τ − sin( τ ) cos( τ ) (cid:3) + 2 (cid:15) ˜ g Ω (cid:20) sin ( τ ) cos( τ Ω) − (cid:16) τ (cid:17)(cid:21) − (cid:15) ˜ g (1 + Ω ) 4 sin(2 τ ) sin(Ω τ ) − (cid:15) ˜ g (1 − Ω ) 8 cos( τ ) sin (cid:18) (1 − Ω ) τ (cid:19) + (cid:15) ˜ g (1 + Ω ) (2 τ − τ ) cos(Ω τ )(cos( τ ) cos(Ω τ ) − (cid:15) ˜ g (1 − Ω ) (cid:18) τ ) cos(Ω τ )(cos( τ ) cos(Ω τ ) −
2) + 8 cos( τ ) sin(Ω τ )+ (1 − τ )) sin(2 Ω τ ) − τ (cid:19) + (cid:15) ˜ g (1 − Ω ) (cid:18) sin( τ ) cos(Ω τ ) − sin(2 τ ) cos(2 τ Ω ) − τ ) sin(Ω τ ) + 2 cos(2 τ ) sin(2 τ Ω ) (cid:19) F ˆ N a ˆ B + = − ˜ g (cid:15) sin( τ ) sin(Ω τ ) + 2 Ω ˜ g − Ω (cid:15) sin (cid:18) (1 − Ω ) τ (cid:19) − ˜ g sin( τ ) F ˆ N a ˆ B − = − ˜ g (cid:15) (sin( τ ) cos(Ω τ ) + sin((1 + Ω ) τ )) + ˜ g − Ω (cid:15) sin((1 − Ω ) τ ) − g sin (cid:16) τ (cid:17) (D.1)It can be seen from these expressions that there are some resonances expected, namelya drastic change in the behaviour of (some of) the functions in the limit Ω →
1, whichoccurs when ω = ω m .It is straight-forward to see how the terms F ˆ N a ˆ B + and F ˆ N a ˆ B − simplify as Ω → Ω → sin τ (1 − Ω / (1 − Ω ) = τ /
2. The long expression for F ˆ N a ismore challenging. We note that the terms independent of (cid:15) remain unchanged withΩ . Thus, at resonance, these coefficients read: F ˆ N a = −
132 ˜ g (cid:2) (cid:15) (cid:0) τ − τ ) + sin(4 τ ) (cid:1) + (cid:15) (64 −
72 cos( τ ) + 8 cos(3 τ )) + 32 τ −
16 sin(2 τ ) (cid:3) F ˆ N a ˆ B + = − ˜ g sin( τ ) (cid:16) (cid:15) τ ) (cid:17) F ˆ N a ˆ B − = ˜ g (cid:15) (sin(2 τ ) − τ ) − g sin (cid:16) τ (cid:17) . (D.2) Appendix E. Derivation of the modulated light–matter coupling
In this appendix, we will show how the time-dependent term used in Section 6 can bederived for levitated nanobead systems. In [72], a fully general theory of light–mattercoupling is presented. We will recount some of the derivation here and show how thecavity volume can be modulated in a manner such that it is useful to our scheme.Given a number of assumptions regarding the light–matter interaction (see [72]for a full description) the full Hamiltonian that describes the light–matter interaction nhanced continuous generation of non-Gaussianity through optomechanical mod... H tot = ˆ H fm + ˆ H fc + ˆ H fc + ˆ H f out + ˆ H f free + ˆ H i cav − out + ˆ H i diel . (E.1)The term ˆ H fm = ˆ p / M , where M is the total mass of the system, is the kinetic energyof the centre-of-mass position along the cavity axis. ˆ H Fc = (cid:126) ω c ˆ a † ˆ a is the energy ofthe cavity mode. ˆ H f out and ˆ H f free are terms describing an open system, which we shallignore in this work. We likewise ignore ˆ H i cav − out which describes a coupling betweenthe cavity input and the output mode.The last term ˆ H i diel describes the light–matter coupling and can be written in thegeneral form ˆ H i diel = − (cid:90) V ( r ) d x P ( x ) ˆ E ( x ) , (E.2)where P ( x ) is the polarization of the levitated objects (which we assume to be a scalarquantity) and ˆ E ( x ) is the total electric field, which can be obtained from solvingMaxwell’s equations given a set of well-defined boundary conditions. The quantisedmodes of the electric field can thus be written as [59]ˆ E ( x ) = i (cid:88) s,m E m (cid:0) a s,m − a † s,m (cid:1) χ s,m ( x ) , (E.3)where s is the spin-polarization index and m signifies the field-mode number, and E m = (cid:113) ω m (cid:126) (cid:15) V c is the field amplitude with V c being the cavity mode volume. Thefunctions χ s,m must obey the spatial solutions to the wave-equations, where the fullclassical solutions separate into E ( r , t ) = χ ( r ) T ( t ).If we assume that the polarization is given by P ( x ) = (cid:15) c (cid:15) E ( x ), we obtain thesimpler expression ˆ H i diel = − (cid:15) c (cid:15) (cid:90) V ( r ) d x [ ˆ E ( x )] , (E.4)where (cid:15) c = 3 (cid:15) r − (cid:15) r +2 , and where (cid:15) r is the relative dielectric constant of the nanodiamond.We now assume that the electric field operators are displaced by a classical part:ˆ a → (cid:104) ˆ a (cid:105) + ˆ a . The classical part (cid:104) ˆ a (cid:105) will form the optical trapping field, while thequantum part describes the light–matter interaction.Thus the classical contribution to the electrical field is given by E ( x ) = i (cid:114) ω c (cid:15) V c ( αf ( x ) − α ∗ f ∗ ( x )) , (E.5)where α is a complex prefactor and f ( x ) is a complex function which describe thestanding waves inside the cavity. We now write our full electric field as ˆ E tot ( x ) =ˆ E ( x ) + E ( x ), where ˆ E ( x ) is the quantum contribution containing ˆ a and ˆ a † , and E ( x )is the classical part. The full Hamiltonian is nowˆ H i diel = − (cid:15) c (cid:15) (cid:90) V ( r ) d x [ ˆ E ( x ) + E ( x )] = − (cid:15) c (cid:15) (cid:90) V ( r ) d x [ ˆ E ( x ) + E ( x ) + 2 ˆ E ( x ) E ( x )] . (E.6) nhanced continuous generation of non-Gaussianity through optomechanical mod... E ( x ) will yield a trapping frequency, while the operatorterms ˆ E ( x ) will yield the light–matter interaction term for the levitated sphere. Thecross-term, ˆ E ( x ) E ( x ) will generate elastic scattering processes inside the cavity whichconverts cavity photons and tweezer photons into free modes [72]. We shall ignore themhere and focus on the generation of the trapping frequency ω m and the coupling g ( t ).We begin with the trapping frequency. Appendix E.1. Mechanical trapping frequency
We now assume that the classical field has a Gaussian profile which extends in the y -direction for a cylindrical geometry. The cavity extends along the z -direction. Wehere follow the derivation presented in [79].If we denote the radius of the cylinder by r , we can write down the trapping fieldas E ( y, r ) = E W W ( y ) exp (cid:18) − r W ( y ) (cid:19) , (E.7)where E = (cid:113) P t (cid:15) cπW , P t is the trapping laser power and W is the beam waistwith the full beam as a funtion of y being W ( y ) = W (cid:113) y λ π W . It follows thatthe narrowest part of the beam W occurs at y = 0, which is the minimum in thepotential where the nanobead is trapped.We can now expand [ E ( y, r )] to second order in r and y around the origin y = r = 0. We start with the exponential, which we expand as[ E ( y, r )] ≈ E W W ( y ) (cid:18) − r W ( y ) (cid:19) . (E.8)Next, we expand the inverse beam width to second order in y :1 W ( y ) ≈ W (cid:18) − y λ π W (cid:19) . (E.9)Combining the two expressions give us[ E ( y, r )] ≈ E (cid:18) − y λ π W (cid:19) (cid:18) − r W (cid:18) − y λ π W (cid:19)(cid:19) ≈ E − E y λ π W + r E (cid:18) y λ π W − W (cid:19) . (E.10)If we now assume that y (cid:28) W , meaning that the beam waist is much larger than theregion we consider, we can approximate the above as[ E ( y, r )] ≈ E − r E W . (E.11)We then insert this now constant expression into the integral for the Hamiltonianand we drop all constant terms as they are just constant energy shifts. To performthis integral, we now assume that the radius R of the bead is much smaller than thewavelength of the light. This is often referred to as the ‘point–particle approximation’,or the Rayleigh approximation. Essentially, this means that the field inside the bead nhanced continuous generation of non-Gaussianity through optomechanical mod... x and y ). Thus we canassume that wherever the sphere is located in the field, the integral just simplifies tothe volume of the sphere times the field amplitude. For a derivation which includesarbitrary particle sizes, see [80].This gives H trap ≈ (cid:15) c (cid:90) V ( r ) d x r E W ≈ r (cid:15) c E W V, (E.12)where V is the integration volume. The result is a harmonic trapping of the form12 mω r = (cid:15) c E W V r , (E.13)where we identify the trapping frequency as ω = 2 m (cid:15) c E W V = 12 Imρc(cid:15) c W (cid:18) (cid:15) r − (cid:15) r + 2 (cid:19) , (E.14)where ρ = mV is the density of the levitated object and where we have used E = Ic(cid:15) ,where I is the intensity of the laser beam, and (cid:15) c = 3 (cid:15) r − (cid:15) r +2 . Appendix E.2. The light–matter interaction term
We now come to the most important term, which is the light–matter interaction termdenoted g in this work. We will continue to follow the derivation in [72] to showexactly where time–dependence could potentially be included.If the sphere is sufficiently small, we can choose a TEM 00 (transverseelectromagnetic mode) as the cavity mode, which is aligned in the z -direction. Inthis mode, the cross-section in x and y is perfectly Gaussian, and it is one of the mostcommonly used modes in experiments. If the sphere is smaller than the laser waistand if it is placed close to the centre of the cavity, we can approximate the field at thecentre of the beam by[ E ( x )] ≈ ω c (cid:15) V c (cid:18) − x + y ) W c (cid:19) cos ( k c z − ϕ ) ˆ a † ˆ a. (E.15)Here, the laser waist is given by W c = (cid:113) λL (2 π ) , L is the cavity length. λ is the laserwavelength. We assume that the wave-vector k c points in the z -direction, along theaxis of the cavity, and ϕ is a generic phase which determines the minimum of thepotential seen by the bead. For laser-trapped nanobeads, this phase can be madetime-dependent, whereas for a Paul trap, it is static. We will leave out the time-dependence for now for notational simplicity. Finally, ˆ a and ˆ a † are the annihilationand creation operators of the electromagnetic field.To obtain the Hamiltonian term, we now integrate over the full energy within thevolume of the nanobead. For a bead situated at r = ( x, y, z ) leads toˆ H diel = − (cid:15) c (cid:15) (cid:90) V ( r ) d x [ E ( x )] = − (cid:15) c (cid:15) (cid:90) V ( r ) d x ω c (cid:15) V c (cid:18) − x + y ) W c (cid:19) cos ( k c z − ϕ )ˆ a † ˆ a. (E.16) nhanced continuous generation of non-Gaussianity through optomechanical mod... R is much smaller than the wavelengthof the light, such that k c R (cid:28)
1. As mentioned above, this is the ‘point–particleapproximation’, or the Rayleigh approximation.Thus the integral simplifies toˆ H diel = − (cid:15) c (cid:15) (cid:90) V ( r ) d x ω c (cid:15) V c (cid:18) − x + y ) W c (cid:19) cos ( k c z − ϕ ) ˆ a † ˆ a = ω c f ( r ) ˆ a † ˆ a, (E.17)where we have defined the function f ( r ) as f ( r ) = − V (cid:15) c V c (cid:18) − x + y ) W c (cid:19) cos ( k c z − ϕ ) . (E.18)Now, we assume that the sphere is trapped at position r = ( x , y , z ) T , which wetake to be the origin of the cavity with x = 0, y = 0 and z = 0. For smallperturbations to z , which we will later quantize, we can expand (E.18) around z = 0to first order. For this to be valid, we must also expand ϕ to first order. We writecos ( k c z − ϕ ) = [cos( k c z ) cos( ϕ ) + sin( k c z ) sin( ϕ )] ≈ (cid:20)(cid:18) − k c z (cid:19) (cid:18) − ϕ (cid:19) + k c zϕ (cid:21) ≈ [1 + k c zϕ ] ≈ k c zϕ. (E.19)We note the linearised z -coordinate here, which will later become our quantumoperator. We can then write down the full expression ω c f ( r )ˆ a † ˆ a = − ω c V (cid:15) c V c (1 + 2 k c zϕ ) ˆ a † ˆ a. (E.20)From this term, we note that the light-interaction yields a constant reduction of thecavity resonant frequency ω c of the form ω c → ˜ ω c = ω c (cid:18) − (cid:15) c V V c (cid:19) . (E.21)The first-order correction in z can now be quantised by promoting z to an operator z → ˆ z = (cid:113) (cid:126) ω m m (ˆ b † + ˆ b ) so that we find the interaction termˆ H int = − ω c V (cid:15) c V c k c ϕ ˆ z. (E.22)We now use the fact that k c = ω c c to writeˆ H int = − (cid:114) (cid:126) ω m m ω c V (cid:15) c ϕ V c c ˆ a † ˆ a (cid:16) ˆ b † + ˆ b (cid:17) , (E.23)where we can define the final expression for the light–matter coupling: g = (cid:114) (cid:126) ω m m ω c V (cid:15) c ϕ V c c . (E.24) nhanced continuous generation of non-Gaussianity through optomechanical mod... ϕ = π .In optical traps, we can now modulate ϕ → ϕ ( t ), to change the light–mattercoupling. If we let ϕ ( t ) = π (1 + (cid:15) sin ω tt