Vibrational enhancement of quadrature squeezing and phase sensitivity in resonance fluorescence

Vibrational environments are commonly considered to be detrimental to the optical emission properties of solid-state and molecular systems, limiting their performance within quantum information protocols. Given that such environments arise naturally it is important to ask whether they can instead be turned to our advantage. Here we show that vibrational interactions can be harnessed within resonance fluorescence to generate optical states with a higher degree of quadrature squeezing than in isolated atomic systems. Considering the example of a driven quantum dot coupled to phonons, we demonstrate that it is feasible to surpass the maximum level of squeezing theoretically obtainable in an isolated atomic system and indeed come close to saturating the fundamental upper bound on squeezing from a two-level emitter. We analyse the performance of these vibrationally-enhanced squeezed states in a phase estimation protocol, finding that for the same photon flux, they can outperform the single mode squeezed vacuum state.

Quadrature squeezed light has been identified as an important resource for continuous variable quantum information applications [1][2][3][4][5][6]. By minimising the variance in one expectation value of a quantum field at the expense of another, squeezed states can be used to reduce uncertainties in interferometric measurements for metrology applications [7], for secret encodings in quantum key distribution [5,8,9], and to provide the necessary nonclassicality for quantum computing schemes [4,6]. There now exist several mature methods that exploit optical non-linearities and light confinement in dielectric materials to produce squeezed states, which typically consist of many photons [10].
In 1981 Walls and Zoller [11] showed that quadrature squeezed light may also be generated through the resonance fluorescence of a single driven two-level emitter (TLE), where the emitted photons are simultaneously antibunched [12,13]. This was recently demonstrated experimentally using a semiconductor quantum dot (QD) platform [14], which offers the necessary high photon collection efficiency unobtainable in most atomic approaches [15][16][17]. The TLE scheme relies on the buildup of steady-state coherence between the ground and excited state, which then gives rise to a coherent superposition of the vacuum and first Fock-state of the emitted field, i.e. a state of the form |0 + α|1 with appreciable α [11]. This is achievable in the weak-driving Rayleigh (or equivalently Heitler) limit in which coherent (elastic) scattering dominates [15,16,[18][19][20][21][22][23]. However, as this occurs well below saturation of the transition, it restricts the photon flux that may be produced. Moreover, as we shall detail below, the obtainable level of squeezing is limited to values considerably smaller than the fundamental upper bound for a two-level system [11,24,25].
In this work we establish how to generate quadrature squeezed states instead at stronger driving above saturation, resulting in a bright source of single photons with a level of squeezing that can surpass the (atomic) Walls and Zoller maximum. Our approach harnesses the thermal environment commonly present in solid-state and molec-ular emitters to access states otherwise unreachable in conventional resonance fluorescence. In particular, we exploit thermalisation within the driving-induced dressed state basis to obtain steady-state coherence above saturation [19], where it would usually be strongly suppressed. As a concrete example, we illustrate this behaviour through a microscopic model of a driven QD coupled to a phonon environment [26][27][28][29]. We show that off-resonant driving can be used to access levels of squeezing close to the fundamental upper bound for a two-level system, beyond those possible in the Rayleigh scattering limit below saturation (and hence any parameters for which the thermal environment is absent). We thus identify a scenario in which thermal processes in solidstate emitters can be used to access regimes and generate photonic states unavailable to their atomic counterparts.
Let us begin by analysing squeezing from a TLE in the standard setting without any additional thermal environment [11]. The uncertainty in any two observables described by operators A and B is determined by the Heisenberg uncertainty principle ∆A∆B We say that the observable A is squeezed if its variance satisfies ∆A 2 < 1 2 | [A, B] |. Within the dipole approximation and the far field limit, the positive frequency component of the electric field describing emission from a TLE may be written in the Heisenberg picture (with = 1), [30,31]. The first term in this expression, E 0 (t) = l a l (0) exp(−iω l t), describes the free evolution of the field, with a l the annihilation operator for mode l. The second term describes the TLE emission, where σ = |g X| (ground state |g , excited state |X ) and Γ is the spontaneous emission rate. We then define the electric field relative to a phase reference ϕ as E(t, ϕ) = (e iϕ E (+) (t) + e −iϕ E (−) (t))/ √ 2. For this field, the steady-state squeezing condition is ∆E(t, ϕ) 2 < 1 2 | [E(t, ϕ), E(t, ϕ + π/2)] | and amounts to :∆X(ϕ) 2 : < 0, where we define the dimensionless normal ordered quantity :∆X(ϕ) 2 : = ∆X(ϕ) 2 − | σ z |. Here, X(ϕ) = e iϕ σ + e −iϕ σ † is the field quadrature and ex-arXiv:1806.08743v1 [quant-ph] 22 Jun 2018 pectation values are taken with respect to the steadystate. By writing σ = | σ |e −iφ we find :∆X(ϕ) 2 : = 1−| σ z |−4| σ | 2 cos 2 (φ−ϕ), and we see that the quadrature with the lowest variance is that with φ = ϕ.
Before we determine the conditions under which squeezing occurs, it is instructive to establish a relationship between the quadrature variance and quantities more commonly used in studies of resonance fluorescence: namely the power, the coherent scattering, and antibunching. To do so we recall that the steadystate spectrum of resonance fluorescence from a TLE is proportional to S(ω) = 1 π Re ∞ 0 dτ g (1) (τ )e −iωτ , with g (1) (τ ) = σ † (τ )σ the first order field correlation function. The coherent contribution is separated by writing S(ω) = S coh (ω) + S inc (ω) where S coh (ω) = g (1) coh δ(ω) and coh ]e −iωτ , with g [18][19][20][21]. The total radiated power can be similarly separated into coherent and incoherent contributions, giving P = ∞ −∞ dωS(ω) = σ † σ = P coh + P inc with P coh = | σ | 2 ≤ P . In terms of power contributions the quadrature variance is then written showing that squeezing only occurs when P coh is appreciable, and the total power P takes on values closest to its extremal values of 0 and 1. Finally, we note that the antibunching behaviour is captured by the probability to simultaneously detect two photons g (2) For a TLE σ 2 = 0, leading to g (2) (0) = 0 regardless of the parameter regime, showing that any squeezed emission from a TLE is simultaneously antibunched.
For a TLE without additional thermal interactions, the effectively flat frequency spectrum of the electromagnetic environment restricts the magnitude of the total and coherently scattered power, which in turn limits the level of squeezing. To see this, we consider a TLE driven by a coherent source with Rabi frequency Ω and detuning δ, for which a master equation can be written within a rotating frame and the rotating wave approximation and ρ is the TLE density operator. Solving the master equation in the steady state, i.e. whenρ = 0, we find P = S/(2(S + 1)) and P coh = P/(S + 1), leading to where we have defined the saturation parameter S = s/(1+d) in terms of a dimensionless driving s = 2(Ω/Γ) 2 and detuning d = 4(δ/Γ) 2 . The squeezing is greatest when Eq. (2) is minimised, which occurs for S = 1/3, at which point P = 1/8, P coh = 3/32, and :∆X(φ) 2 : = −0.125 [11]. This represents the theoretical maximum squeezing obtainable in resonance fluorescence for a TLE undergoing spontaneous emission into an unstructured environment [11]. However, as we shall see, this maxi-mum is a consequence of the limited set of states available in this simple model, and is not a fundamental bound.
To verify this we consider a TLE described by a completely generic density operator. We can then express the total power as P = σ † σ = (1/2)(1 + l cos θ) and the coherently scattered power as P coh = | σ | 2 = (1/4)l 2 sin 2 θ, where 0 ≤ l ≤ 1 is the TLE Bloch vector length and 0 ≤ θ ≤ π the polar angle in the Bloch sphere. These expressions represent a less restricted set of values for the power contributions than before. Minimising Eq. (1) now gives l = 1, and θ = π/3 (P coh = 3/16, P = 3/4) or θ = 2π/3 (P coh = 3/16, P = 1/4), both of which result in :∆X(φ) 2 : = −0.25. This is the true maximum level of squeezing obtainable from a TLE, and is limited only by the two level nature of the system.
How, then, can we obtain such a state within resonance fluorescence? We shall now show that naturally occurring thermal interactions in, for example, solid-state and molecular systems can help to drive a TLE into such a state, resulting in a level of squeezing close to the maximum value of :∆X(φ) 2 : = −0.25, and certainly greater than the value of :∆X(φ) 2 : = −0.125 obtainable in their absence. This can be understood qualitatively by considering equilibration of our system with respect to the additional thermal reservoir. At inverse temperature β = 1/k B T the resulting TLE thermal state is writ- Making the identifications l = tanh(βη/2) and θ = arctan(Ω/δ), we see that for suitable choices of the adjustable Hamiltonian parameters Ω and δ, we can satisfy the conditions above which lead to maximum squeezing. Namely, if we choose δ = ±Ω/ √ 3, the maximum squeezing of :∆X(φ) 2 : = 0.25 is achieved in the low temperature limit (βη → ∞) .
Let us now explore this process in greater detail and analyse the limitations of the simple argument given above. To do so we consider the example of a QD as a solid-state TLE with ground state |g and excited state |X describing a single exciton of energy ω 0 . The QD is driven by a semiclassical monochromatic laser. Within a frame rotating with respect to the laser frequency and the rotating wave approximation the system Hamiltonian may be written H S = δσ † σ + Ω 2 σ x , as before. The QD couples to both vibrational and electromagnetic environments, which results in the total Hamiltonian H = H S + H P H + H EM + H E . At low temperatures the electron-phonon interaction is dominated by a linear displacement coupling with Hamiltonian is the annihilation (creation) operator of phonon mode k with frequency ν k and coupling strength g k [21,[32][33][34]. The electronphonon coupling is characterised by the spectral density with coupling strength α and cut-off frequency ν c , which sets the timescale of lattice relaxation. Coupling to the electromagnetic field in the dipole and rotating wave approximations takes the form H EM = l h l σ † a l + h.c., where a l (a † l ) is the annihilation (creation) operator for mode l of the field, with frequency ω l and coupling strength h l . We assume the spectral density of the optical field varies slowly over the relevant energy scales of the system, allowing us to use the flat function J EM (ω) = l |h l | 2 δ(ω − ω l ) ≈ 2Γ/π, where Γ is again the spontaneous emission rate. Free evolution of the environments is described by To account for the electron-phonon coupling we make use of the variational polaron transformation [19,21,35] defined by the unitary . This leads to a QD state dependent displacement of the phonon environment, where the f k are chosen to minimise the Feynman-Bogoliubov bound on the free energy, defining an optimised basis in which perturbation theory can then be applied. In the variational polaron frame we derive a second order Born-Markov master equation, which is valid for both strong and weak exciton-phonon coupling, as well as from weak to strong driving strengths. This may be written compactly asρ where ρ V is the reduced density operator of the QD in the variational frame; coupling to phonons is accounted for in the dissipator K ph and the renormalised system parameters δ r and Ω r . Full details are given in the supplementary information.
To calculate the squeezing of the scattered field, we note that in the variational frame the dipole operator carries a displacement operator, such that σ → B − σ. This displacement operator leads to a phonon sideband, a consequence of non-Markovian lattice relaxation during the emission process [20,[36][37][38]. Including this effect, the field emitted by the QD becomes E(t) = E 0 (t) − 2Γ/πB − (t)σ(t), and we obtain a modification to the quadrature variance, which becomes [cf. Eq. (1)] Here the displacement operator thermal average is B = Having outlined our model, in Fig. 1 we show the total QD emitted power P and the coherently scattered power P coh as functions of the dimensionless driving s and the detuning δ, both including [a) and c)] and excluding [b) and d)] phonons as indicated [39]. As noted for resonant driving in Ref. [19], plot c) shows that in the presence of phonons we obtain a significant coherent scattering power when driving above saturation, which we here find extends across a broad range of detunings as well. As expected, this occurs as a result of phonon-mediated thermalisation of the QD exciton in the dressed state basis, which at low temperatures and strong fields leads to sustained steady-state coherence. At very high driving strengths the coherent scattering falls off, as here the Rabi frequency exceeds the extent of the phonon spectral density set by ν c , leading to a regime in which the exciton and phonons are effectively decoupled [35,40].
Looking at the cases without phonons in Figs. 1 b) and d), it is evident that for all detunings there exists a driving strength for which the power contributions are such that squeezing may occur, and indeed this was seen in  Fig. 1.  FIG. 3. Wigner functions for the vacuum state a), and three squeezed states generated in resonance fluorescence; on resonance and with weak driving b), and off resonant with strong driving c) and d). In cases c) and d) the squeezing occurs due to thermalisation, and can attain levels greater than in case b). Parameters as in Fig. 1. Eq.
(2), which shows that the power contributions and quadrature variance depend only on the generalised saturation parameter S. When phonons are included, however, the situation appears more complex. On or near resonance for weak to moderate driving, the excitonic system is dominated by the spontaneous emission processes as it samples the phonon spectral density at the small generalised Rabi frequency Ω 2 r + δ 2 r . The power contributions, and hence the quadrature variance, are then similar to that of an atomic system with no phonon coupling. This can be seen in Fig. 2 a), where the variance is shown as a function of the saturation parameter on resonance, δ = 0, calculated with (solid curve) and without (dotted curve) phonons included. We see that :∆X(φ) 2 : has a minimum for S(0) = s = 1/3 (log 10 s ≈ −0.5), in accordance with Eq. (2). This is the regime explored experimentally in Ref. [14]. Although the phonons are not playing a qualitatively significant role here, we do see that the minimum of :∆X(φ) 2 : is slightly higher with them included. This can be attributed to the phonon sideband present in the QD emission spectrum, responsible for the factor of B 2 ≤ 1 in Eq. (4), whose contribution is independent of the driving conditions [20,36,37].
Above saturation, we see that the power contributions with and without phonons markedly differ, as in this regime the phonon environment dominates over spontaneous emission due to its spectral density being sampled at larger generalised Rabi frequencies. The exciton then tends towards a thermal state as previously described, in which P coh can be significant, while positive and negative detunings lead to a low and high total power P , respectively. As anticipated, this then gives rise to two regimes with quadrature squeezing, as shown explicitly in Figs. 2 b) and c), where we plot the quadrature variance as a function of the generalised saturation parameter S for fixed detuning. The black dotted curves correspond to the approximate expression in Eq. (3), showing good agreement with the full phonon model until the driving strength becomes large enough that the decoupling regime is reached. For positive detuning we see that a level of squeezing close to the up-per bound :∆X(φ) 2 : = −0.25 can be obtained, and that this is only possible when phonons are included. For the negative detuning case, as the driving increases phonons lead to thermalisation towards a state with total power P > 0.5 due to steady-state population inversion [see Fig. 1 a)] [41][42][43]. This means that the quadrature variance first increases at small driving, then begins to decrease with a discontinuous derivative (not shown) as P passes through 0.5. The strong driving exciton-phonon decoupling regime also sets in sooner than for positive detuning. Nevertheless, the obtained levels of squeezing still surpass those possible in the absence of the thermal environment.
In order to elucidate the nature of the squeezed states of light produced, we now consider the emitted field Wigner function defined as W(x, p) = π −1 ∞ −∞ x + y|ρ EM |x − y exp[−2ipy]dy, where ρ EM is the state of the field. Following Ref. [14] we use the correspondence between the field operators and QD operators to associate the QD excited state |X with the first field Fock state |1 and the QD ground state |g with the field vacuum |0 , such that ρ EM = n,m=0,1 (ρ V ) nm |n m|. In doing so, we expect W(x, p) to provide a qualitative representation of the electromagnetic field Wigner function for a class of measurements [44].
In Fig. 3 we show Wigner functions for the vacuum a), the squeezed state generated in the weak resonant excitation regime below saturation b), and the squeezed states generated only in the presence of phonons above saturation for off-resonant driving c) and d). Interestingly, in the resonant case b), although the state generated is strictly speaking non-Gaussian, its Wigner function is nevertheless positive everywhere, and is not significantly dissimilar from a truly Gaussian displaced squeezed vacuum state [31]. However, in cases c) and d) the non-Gaussian nature of the field is quite apparent, with the Wigner functions taking on substantial negative values associated with non-classicality. Thus, in this regime a highly non-classical, quadrature squeezed and antibunched state is produced.
In summary, we have shown that interactions between a TLE and a thermal environment can be harnessed to produce a bright source of single photons that are squeezed to a level that would otherwise be impossible within resonance fluorescence. In fact, the obtainable squeezing can reach values very close to the fundamental bound for a two-level system. We have illustrated our findings with an explicit example of a QD coupled to phonons, which provides a feasible experimental platform to engineer such squeezed states of light.
Acknowledgements: The authors wish to thank Alistair Brash and Pieter Kok for useful discussions. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No.

SUPPLEMENTARY INFORMATION
In this supplement we outline the theoretical methods used in the main manuscript. We first define the variational polaron transformation for a driven two level system, before deriving the corresponding master equation. We then demonstrate how field expectation values must be modified in the variation polaron frame.

VARIATIONAL POLARON TRANSFORMATION
The starting point is the Hamiltonian describing a two level system (TLS) with ground and excited states |g and |X respectively, driven by a classical laser field with frequency ω l and Rabi frequency Ω. As introduced in the main text we have, where we have defined the Pauli operators in the usual way. The Hamiltonian is written in a frame rotating with respect to the laser frequency ω l and in the rotating wave approximation, with the laser and QD transition detuned by δ = ω X − ω l .
To model the dynamics of the TLS, we apply the variational polaron transformation to the above Hamiltonian, allowing us to derive a master equation valid outside of the weak electron-phonon coupling regime [1][2][3]. The variational polaron transformation is given by the state dependent displacement operator: where is a multi-mode displacement operator, and f k is a parameter which we shall use to variationally optimise the transformation in the next section.
Applying this transformation to the initial Hamiltonian yields Notice that we have introduced the renormalised detuning δ r = δ + R, with R = k ν −1 k f k (f k − 2g k ), and renormalised driving strength Ω r = ΩB, where B = tr B (B ± ρ P H B ) is the expectation value of the displacement operator with respect to the thermal state The interaction Hamiltonians also transform, such that: where Notice that the electron-phonon interaction now contains a mixture of displacement operators and linear coupling. In the next section we outline how the variational principle can be used to specify the contribution of each of these terms.

Minimising the free energy
As mentioned above, we now choose the displacement such that the transformed Hamiltonian minimises the free energy of the system. We do so by minimising the Feynman-Bogliubov upper bound on the free energy [3], that is: where H 0 = H S + H B . This procedure allows us to derive a master equation which is valid over a broad range of parameters, and does not suffer from the same pathologies as standard Polaron theory (see Refs. [2] and [3] for a detailed discussion). Thus, minimising A B with respect to f k , we obtain the expression Solving this equation, and substituting the minimised displacements into the expressions for the renormalised system parameters, we find that in the continuum limit we have: where we have introduced the variational function: Here η r = δ 2 r + Ω 2 r is the renormalised generalised Rabi frequency. These equations can be solved self-consistently to find the renormalised parameters that minimise the Feynman-Bogliubov free energy.

VARIATIONAL MASTER EQUATION
To describe the dynamics of the reduced state of the TLS, ρ V (t), we shall treat the interaction Hamiltonian H I = H EM I + H PH I to 2 nd -order using a Born-Markov master equation, which in the interaction picture takes the form [4]: where in the Born approximation we factorise the environmental density operators in the variational polaron frame such that they remain static throughout the evolution of the system. Note that correlations may be generated between the system and the phonon environment in the original representation. We shall assume that in the variational polaron frame the phonon environment remains in the Gibbs state defined above, while the electromagnetic environment remains in its vacuum state ρ EM B = m |0 m 0 m |. Since the trace over the chosen states of the environments removes terms linear in creation and annihilation operators, we may split the master equation into two separate contributions corresponding to the phonon and photon baths respectively [5], In the subsequent sections we shall analyse each of these contributions in turn.

Phonon contribution
To derive the contribution from the phonon environment, we follow Ref. [1]. We start by transforming into the interaction picture with respect to the Hamiltonian H 0 = δr 2 σ z + Ωr Using this transformation, the interaction Hamiltonian takes the form: Here The system operators can be formally written in the interaction picture as: σ α = jk σ jk α e iλ jk t |ψ j ψ k |, where |ψ i are the eigenstates of the system Hamiltonian satisfying H S |ψ i = ψ i |ψ i , λ ij = ψ i − ψ j , and σ ij α = ψ i |σ α |ψ j with α ∈ {x, y, z}. We can then write the phonon dissipator in the Schrödinger picture using the compact notation where we have defined the rate operators as: and we have introduced the bath correlation functions given by: with the two remaining cross terms Λ xy (t) = Λ xz (t) = 0. Here we have defined the terms: where we have defined ϕ(t) = ∞ 0 J P H (ν)F (ν) 2 ν 2 coth βν 2 cos ωt + i sin ωt dν.

Photon contribution
We now focus on the interaction between the electromagnetic field and the TLS. The interaction picture Hamiltonian for the field may be written as H EM I = σ † (t)e iω l t B + (t)A(t) + h.c., where A(t) = m h m a m e −iωmt and B + (t) is as given in the previous section. If we consider the interaction picture transformation for the system operators we have σ(t)e −iω l t = e i( δr 2 σz+ Ωr 2 σx)t σe −i( δr 2 σz+ Ωr 2 σx)t e −iω l t ≈ σe −iω X t , where we have used the fact that ω l Ω r , δ r for typical solid-state emitters to simplify the interaction picture transformation [1,6]. By substituting this expression into the photon contribution of the master equation, and assuming all modes of the field are in their vacuum state, ρ EM B = m |0 m 0 m |, we have where the spontaneous emission rate is given by [1,7,8] γ(ω X ) = Re Here Λ(τ ) = ∞ 0 dωJ EM (ω)e iωτ , with J EM (ω) = m |f m | 2 δ(ω − ω m ) being the spectral density of the electromagnetic environment. As discussed in the manuscript, the local density of states of the electromagnetic field does not vary appreciably over energy scales relevant to QD systems in bulk, which allows us to make the assumption that the spectral density is flat [1,6], J EM (ω) ≈ 2Γ/π. The electromagnetic correlation function may then be evaluated as Λ(τ ) ≈ Γδ(ω) + iP[1/τ ], where P denotes the principal value integral. Combining these expressions and resolving the remaining integral, we find that the spontaneous emission rate takes on the form γ(ω X ) ≈ Γ, where we have used the