Optomechanical strong coupling between a single photon and a single atom

PAPER • OPEN ACCESS Optomechanical strong coupling between a single photon and a single atom To cite this article: Javier Argüello-Luengo and Darrick E Chang 2022 New J. Phys. 24 023006 View the article online for updates and enhancements. You may also like Cavity optomechanics: Manipulating photons and phonons towards the singlephoton strong coupling Yu-long Liu, , Chong Wang et al. - Optomechanical crystal nanobeam cavity with high optomechanical coupling rate Yongzhuo Li, Kaiyu Cui, Xue Feng et al. - Comparing nonlinear optomechanical coupling in membrane-in-the-middle and single-cavity systems Roel Burgwal, Javier del Pino and Ewold Verhagen - This content was downloaded from IP address 79.157.202.244 on 22/03/2023 at 10:50 New J. Phys. 24 (2022) 023006 https://doi.org/10.1088/1367-2630/ac4c69 OPEN ACCESS RECEIVED 13 August 2021 REVISED 13 December 2021 ACCEPTED FOR PUBLICATION 18 January 2022 PUBLISHED 8 February 2022 Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. PAPER Optomechanical strong coupling between a single photon and a single atom Javier Argüello-Luengo1,∗and Darrick E Chang1,2 1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860, Castelldefels, Barcelona, Spain 2ICREA-Instituci´ o Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain ∗Author to whom any correspondence should be addressed. E-mail: [email protected] Keywords: strong coupling, cavity QED, optomechanics, single atom Abstract Single atoms coupled to a cavity offer unique opportunities as quantum optomechanical devices because of their small mass and strong interaction with light. A particular regime of interest in optomechanics is that of ‘single-photon strong coupling’, where motional displacements on the order of the zero-point uncertainty are sufficient to shift the cavity resonance frequency by more than its linewidth. In many cavity QED platforms, however, this is unfeasible due to the large cavity linewidth. Here, we propose an alternative route in such systems, which instead relies on the coupling of atomic motion to the much narrower cavity-dressed atomic resonance frequency. We discuss and optimize the conditions in which the scattering properties of single photons from the atom-cavity system become highly entangled with the atomic motional wave function. We also analyze the prominent observable features of this optomechanical strong coupling, which include a per-photon motional heating that is significantly larger than the single-photon recoil energy, as well as mechanically-induced oscillations in time of the second-order correlation function of the emitted light. This physics should be realizable in current experimental setups, such as trapped atoms coupled to photonic crystal cavities, and more broadly opens the door to realizing qualitatively different phenomena beyond what has been observed in optomechanical systems thus far. Quantum optomechanics has emerged as a field with numerous exciting prospects for fundamental science and applications [1,2]. Generically, such systems are characterized by some mechanical degree of freedom, whose small displacements alter the resonance frequency of a cavity. This results in rich backaction effects once the cavity is driven, which allows for applications that include sensing [3–6], cooling of the mechanical mode [7,8], generation of squeezed light [9–11] or the creation of nonreciprocal devices [12]. A key figure of merit is the vacuum optomechanical coupling strength, g0=(∂ωc/∂x)xzp, given by the product of the sensitivity of the cavity frequency to position displacements, and the zero-point motion of the resonator. In particular, the single-photon, single-phonon, strong coupling regime ensues when g0exceeds the linewidth of the cavity, such that the optical response and dynamics change drastically at the level of individual quanta. For example, it has been proposed that this can give rise to quantum optical nonlinearities [13,14]. While a number of schemes have been proposed to reach this strong coupling regime [15–19], state-of-the-art optomechanical cavities remain at least two orders of magnitude away from reaching this regime [20,21]. To pursue this goal, atoms constitute an interesting candidate for an optomechanical element, due to their low mass and anomalously large optical response (i.e. a scattering cross section much larger than its physical size). In an atomic ensemble, the single-photon coupling strength gets enhanced by the number of atoms in the detuned regime, entering the strong coupling regime for sufficiently large ensembles [22–24]. Here we show that the single-photon strong coupling regime can be realistically achieved using just a single atom coupled to a high-finesse cavity [25–29]. While macroscopic cavity architectures allow for sufficiently small linewidths to reach the strong coupling regime [30–32], a number of emerging platforms [33–38] focus on achieving small mode volumes with a © 2022 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang prohibitively large linewidth. Here, we show that optomechanical strong coupling effects can nonetheless emerge in these devices by working in a detuned atom-cavity regime and probing motional interactions on the narrower dressed atomic resonance of a single atom. The enabling mechanism is based on the scattering properties of an incoming photon, which highly depend on its detuning to the dressed resonance frequencythatinturnissensitivetotheatomicposition within the cavity field. As a consequence, we show that a scattered photon highly entangles with the resulting atomic motional state, carrying information about its position. This leads to a per-photon atomic heating larger than expected from single-photon recoil events and, as a more direct signature, we observe that detection of a reflected photon triggers motion-induced oscillations in time of the second-order correlation function of reflected light. We show that these effects are observable in realistic systems, even for a non-zero initial motional temperature. 1. The system Here we focus on the interaction of a single two-level atom with an optical transition between ground and excited states |↓,|↑, and a given mode of the electromagnetic field inside the cavity. The coherent interactions are described by the Jaynes–Cummings (J–C) Hamiltonian [39]forasingleatom, ˆ HJC =−(Δ+Δ 0)ˆ a†ˆ a−Δ0ˆσ†ˆσ+g(ˆ x)ˆσˆ a†+h.c.−iεˆ a†−ˆ a,(1) where ˆσ≡|↓↑|is the atomic lowering operator, and Δ=ω0−ωcthe energy difference between the bare atomic and cavity resonance frequencies. Here, we also allow for an external laser drive of the cavity with Δ0=ωl−ω0representing the laser-atom detuning, and εthe driving amplitude, which we will generally consider weak enough to only produce a few excitations. g(ˆ x)=g0sin(kcˆ x) denotes the position-dependent vacuum Rabi coupling strength, with kcbeing the cavity mode wavevector. Importantly, we will treat the atomic position ˆ xas a quantum dynamical degree of freedom, and assume that the atom is harmonically trapped with frequency ωmand equilibrium position x0, as schematically illustrated in figure 1(a). One can quantize the atomic motion around this point as δˆ x=ˆ x−x0=xzp(ˆ b+ˆ b†), where xzp denotes zero-point motion fluctuations, and ˆ b(†)is the annihilation (creation) operator of phonons in the trap, ˆ Htrap =ωmˆ b†ˆ b. Here and in what follows we use the convention that ≡1. Further including photonic losses from the cavity with decay rate κ, and atomic excited state spontaneous emission at a rate γ, one can describe the total evolution of the density matrix as, dˆρ dt=−iˆ HJC +ˆ Htrap,ˆρ+κL[ˆ a]( ˆρ)+γLe−ikcˆxˆσ(ˆρ),(2) where we define the Lindbladian L[ˆ a]( ˆρ)=1 22ˆ aˆρˆ a†−ˆ a†ˆ aˆρ−ˆρˆ a†ˆ a.Thee −ikcˆxterm represents the recoil momentum that is imparted onto the atom upon spontaneous emission of a photon. Strictly speaking, the recoil along the xdirection is a (non-uniform) random variable between (−kc,kc), accounting for the possibility of a photon to be emitted in any direction [40]. As only one atom is present, the free-space direction of the emitted photon does not influence furtherinteractionsinthesystemanddisregardingits angular component is sufficient to capture the salient physics and heating caused by atomic recoil [41,42]. Here, we will consider the regime relevant to a number of cavity QED systems, where κγ[35–37]. In order to access a strong optomechanical coupling, this motivates working in a detuned atom-cavity regime |Δ|κ,g0and focusing on the dressed atom-like excitation branch with narrower linewidth ∼γ.Wewill start by presenting some heuristic arguments to estimate the optimal conditions to reach this single-photon optomechanical strong coupling, which we will later show are rigorously correct. To simplify the discussion, we will also start by considering the case where the atom is initialized in the motional ground state, treating thermal states in section 7. 2. Heuristic derivation of strong coupling condition in the regime, |Δ|g0,κ One can start by considering a static atom with fixed position x.Intheabsenceofadrive(ε=0), one can block diagonalize the J–C Hamiltonian (1) in the total number of excitations, nexc ≡ˆσ†ˆσ+ˆ a†ˆ a,as illustrated in figure 1(b) for up to nexc =2. In the limit of large atom-cavity detuning Δ,oneofthe single-excitation eigenstates |0↑is mostly an atomic excitation |0↑, but with a shifted resonance frequency, ˜ω0(x)≈ω0+g2(x) Δ, and broadened linewidth, ˜γ(x)≈γ+κg2(x) Δ2, due to the interaction with the cavity [43]. We can consider the sensitivity of this resonance frequency to small (static) displacements 2 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang Figure 1. (a) A single atom is trapped in a harmonic oscillator potential of frequency ωmcentered at position x0,andcoupledto a cavity mode. The cavity mode is externally driven with input field ˆ ain and can decay through this driving channel (κ1), or other undetected routes (κ2). In addition, the excited atom can spontaneously emit into free space at a rate γ.(b)Schematic representations of the lowest energy levels of the J–C Hamiltonian in the absence of a drive (analogous to considering ε=0and ωl=0in(1)), showing up to 2 total excitations. For atomic positions away from the cavity nodes (kcx=0, π), the frequencies of the dressed eigenstates  nph,↑/↓experience a shift from the uncoupled levels  nph,↑/↓that depends on atomic position, ˜ Δ0≈g2(x)/Δ. The linewidth of these dressed levels is represented by shaded regions in the situation γκexplored in this work. (c) In the studied configuration, reflectance is tuned to be null when the atom is placed at the center of the trap x0,withthe spatial width of this reflection minimum given by . (d) After an incoming photon is scattered, the initial motional wave function |ψ0is strongly modified over the length ,tothestate|ψrconditioned on the reflection of a photon, or the state |ψother conditioned on scattering into other channels. x=x0+δx, which to lowest order yields a new resonance frequency ˜ω0(x0+δx)≈ω0+ g2 0 Δsin2(kcx0)+sin(2kcx0)kcδx.The maximum sensitivity to a displacement δxthen occurs halfway between a cavity node and anti-node, when kcx0=π/4(seefigure1(b)). Although we take xto be static, one can nonetheless intuitively deduce a single-photon optomechanical strong coupling parameter, β≡g2 0η Δ˜γ(x0), which characterizes how much the dressed resonance frequency shifts if the atom is displaced by the zero-point motion, in units of the dressed linewidth. Optimizing over Δ, one observes that the maximum strong coupling parameter is dictated by the cooperativity, C≡g2 0/(κγ), as βmax =η√C/√2, where η≡kcxzp is the Lamb–Dicke parameter. We now derive the reflection coefficient of a weak monochromatic, coherent input field, as a function of atomic position. For this, we distinguish the decay rate of the cavity into the port used to drive the system (κ1), from the decay into transmission or absorption channels (κ2), so that the total cavity decay rate reads as κ=κ1+κ2(see figure 1(a)). In particular, the input-output formalism [44,45] allows us to express the field ˆ aout leaving the cavity through the channel associated to κ1in terms of the input field, ˆ ain,as ˆ aout(t)=ˆ ain(t)+√κ1ˆ a(t), (3) which satisfies, ˆ ain(t), ˆ a† in(t)=δ(t−t), and ˆ a† in(out)ˆ ain(out) has units of photon number per unit time. For an atom statically located in position x, one can define Sr(x) as the steady-state reflection coefficient defined by the ratio between output and input fields [46,47], Sr(x)=ˆ aout ˆ ain≈1−iκ1 Δ+Δ 0+iκ/2−g(x)2 Δ0+iγ/2 ,(4) 3 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang and a corresponding reflectance, R(x)=|Sr(x)|2, as a function of the atomic position. Intuitively, efficient optomechanical coupling requires a large contrast in R(x) when the atom is displaced from position x0by a small amount, so that the event of detecting a reflected photon reveals significant information about the atomic position. Experimentally, one can optimize this by adjusting the driving frequency, the atom-cavity detuning, and the coupling to the detection channel, κ1. First, we choose to drive the atom-like resonance, Δ∗ 0=g2(x0) Δ∗.(5) Expanding now the reflectance of the cavity around x0,R(x)=R0+[(x−x0)/]2, one can enforce that the reflectance at position x0is exactly zero, R0=0, so that detection of a reflected photon ensures that the atom is not placed at that point. Imposing this, one obtains the optimal detuning, Δ∗=g(x0)κ1−κ2 γ,(6) which corresponds to critical coupling, where the (dressed) atomic excited state decays equally into the cavity output, κ1g2(x0) Δ2, and other channels, γ+κ2g2(x0) Δ2. Maximizing the effective single-photon coupling parameter βas a function of κ1for the previous choice of parameters, one obtains κ∗ 1=2κ2.This in turn yields the minimum displacement, ∗/xzp ≡√2/ηCin,(7) over which the dressed atomic frequency shifts by ˜γ, thus bringing the system off resonance with respect to the fixed external laser frequency (see figure 1(c)). The fact that , representing the length scale over which a single photon can discriminate the atomic position, depends inversely with the square root of the intrinsic cooperativity Cin ≡g2 0/√κ2γwill play a prominent role in our following discussion. In particular, we observe that the maximum strong coupling parameter previously defined scales as β∗∼xzp/∗. 3. Role of the atomic motional wave function Previously, we have established that if the atom was a perfectly localized point particle, a photon would be reflected with an amplitude and phase given by Sr(x). Intuitively, once the atomic motional state is given by awavefunction|ψ0=dxψ0(x)|x, one might expect that the state upon scattering a single photon is given by Sr|1r|ψ0+Sother |1other|ψ0,where|1rdenotes a reflected photon, and |1otherdenotes the scattering into some orthogonal channel (transmission or cavity absorption, see appendix A). While Sr|1r|ψ0=|1rdxS r(x)ψ0(x)|xhas the natural meaning that the amplitude and phase of the reflected photon depends on the atomic position, one can also observe that the atomic wave function conditioned on the detection of the reflected photon becomes, |ψr=Sr|ψ0 |Sr|ψ0|,(8) where Sr|ψ0=dxS r(x)ψ0(x)|xand the denominator relates to the average reflectance of the cavity, R≡|Sr|ψ0|2=dxR(x)|ψ0(x)|2. These results, which were up to now argued intuitively, can in fact be derived rigorously through an adiabatic elimination of the cavity degrees of freedom in the unresolved sideband regime, ωmγ,κ[32], where the dynamics of the atom-cavity interface is much faster than the mechanical evolution of the atom inside the trap and Srdefines a scattering matrix [46,47]thatisdiagonal in the position basis (appendix A). Intuitively, if the reflectance of the cavity was similar for different atomic positions, R(x)≈R, reflection would reveal no information and the wave function would remain unaffected by detection, |ψr≈|ψ0. In contrast, if Sr(x) contains any narrow spatial features, those features are now imprinted onto the atomic wave function itself. To quantify this, we observe that sets the characteristic width for the spatial features imprinted in the wave function (see figure 1(d)). For an atom initially in the ground state of the trap, the effect of detection will then be large when this critical displacement is smaller than the zero-point motion of the atom, xzp, which corresponds to the strong coupling regime, η√Cin 1. 4 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang Figure 2. (a) Ratio of the per-photon increase in phonons (J) of an atom in the cavity compared to the free-space result η2,for different choices of Δand Δ0.Hereκ1=2κ2and kcx0=π/4 (see main text). When not indicated otherwise, parameters compatible with reference [35] are used in the figures: g0=2π×0.73 GHz, ωm=2π×160 kHz, γ=2π×6MHz, κ2=2π×3.9GHz,η=0.24. The red dotted line follows a resonant driving with the dressed atomic frequency (5), and the crossed marker indicates the optimal atom-cavity detuning (6). (b) Value of J/η2, maximized over free values of Δand Δ0 (orange dashed line), as compared to the result associated to Δ∗,Δ∗ 0within our effective model (blue line), for increasing intrinsic cooperativity tuned by varying g0. The red dotted line corresponds to a master equation simulation of the open system (see main text). Dashed and dotted black lines follow the scalings J∼η2Cin and J∼η√Cin expected in the regimes η2Cin 1 (coloured in green), and η2Cin 1 (coloured in blue), respectively. 4. Unconventional heating As the initial atomic state |ψ0is modified by events associated to reflection or emission in other channels, its mechanical energy departs from the trap ground state energy. Following the example of equation (8), one can calculate the average number of phonons induced by a single incident photon, J=Jr+Jt+Ja, associated to scattering in the detection channel, cavity transmission/absorption, or atomic spontaneous emission, respectively, where Jα=Sαψ0|ˆ b†ˆ b|Sαψ0(9) for α∈{r,t,a}. Note that for each of these emission mechanisms, Jαthen represents the number of phonons in the resulting atomic state Sαψ0|ˆ b†ˆ b|Sαψ0/Sαψ0|Sαψ0, weighted by the probability that this event occurs, Sαψ0|Sαψ0(see appendix A). For conventional scattering from a tightly trapped atom in free space, the characteristic number of phonons that an incoming photon can excite is characterized by the ratio between the single-photon recoil energy, ωr, and the mechanical frequency of the oscillator. When expressed in terms of the Lamb–Dicke parameter, this translates to a per-photon increase in phonons of η2=ωr/ωm[45]. In our coupled atom-cavity system, this heating effect can now be enhanced. Based on our previous analysis,weexpectthatthelargestvaluesofJwill appear for the choices of cavity-laser and atom-cavity detunings derived in equations (5)and(6). To validate this, in figure 2(a) we numerically calculate J/η2 for different detunings Δ0and Δin a cavity satisfying κ1=2κ2. The rest of the parameters are compatible with current experimental platforms [35] where one can reach large intrinsic cooperativities in the order of Cin ∼23, and a Lamb–Dicke parameter η∼0.24. In agreement with our derivation, we observe that a driving frequency in resonance with the dressed atomic frequency (red dotted line, equation (5)) corresponds to the region of larger heating (J∼10η2, associated to lighter colors), and that the atom-cavity detuning that produces maximal heating is compatible with the prediction of equation (6) (crossed marker). In figure 2(b) we calculate J/η2for the choices of Δ0and Δpredicted to maximize optomechanical coupling (equations (5)and(6)), but now as a function of the intrinsic cooperativity Cin by allowing the vacuum Rabi coupling g0to vary, while maintaining the rest of the experimental parameters (ωm,γ,κ2,η) as before. We observe that, for each value of intrinsic cooperativity, the heating that arises at the optimal parameters Δ∗ 0and Δ∗(blue line) does match a fully numerical maximization of the average number of induced phonons over free values of Δ0,Δ(orange dashed line), quantitatively confirming our analysis. While these calculations were based on the scattering matrix given in equation (4) and appendix A,we additionally validate these results by performing a master equation simulation of the open system (red dotted line) in a truncated space of up to 2 photons and 50 phonons, where convergence is observed. For this, we evolve under equation (2) a density matrix with initially no excitations in the system (cavity photons, atomic excitations or phonons) until the population of the cavity stabilizes (variations smaller than 1%). Using input-output relations analogous to equation (3) and normalizing by the number of 5 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang incoming photons, the three heating contributions can be calculated as Jr/t=κ1/2Tr ˆ b†ˆ bˆ aˆρssˆ a†/ ε2/κ1,andJa=γTr ˆ b†ˆ bˆσe−ikcˆxˆρsseikcˆxˆσ†/ε2/κ1. Adding them up (red dotted line), we observe in figure 2(b) good agreement with the scattering matrix calculation for the considered weak driving amplitude ε2/κ1=0.01 MHz. Interestingly, we observe that the average number of induced phonons Jscales differently with the intrinsic cooperativity in the weak (η2Cin 1) and strong coupling regimes (η2Cin 1), which can be understood from the phase ϕα(x)ofthescatteringmatricesSα∼eiϕα(x)that gets imprinted onto the atomic wave function. For the optimal parameters (5)and(6), we note that up to linear order in δx/xzp the imprinted phase associated to emission in undriven channels scales linearly as ϕt/a(x)∼η√Cinδx/xzp (see equation (A4)), which corresponds to an added momentum of η√Cin/xzp =√2/.AlthoughSr(x) cannot be expressed as a phase term, atomic heating can only depend on the total cavity decay rate κ, and not on the specific channel contributing to this rate. Thus, as κ1=2κ2, it follows that the heating rate due to reflection is twice that of transmission/absorption, Jr≈2Jt. Adding these three contributions in the weak-coupling limit, (xzp , green shaded region of figure 2(b)), the imprinted momentum affects the entire wave function and the corresponding kinetic energy increase leads to a heating rate of J∼(xzp/)2∼η2Cin. In contrast, in the strong optomechanical coupling limit (xzp ,blueshadedregion),thephaseimprinting only applies to a small region of the entire wave function, where the cavity is actually sensitive to the atomic position. This leads to a heating rate of J∼(xzp/)2·(/xzp)∼η√Cin, matching the scalings observed in figure 2(b) (black dashed and dotted lines, respectively). The fact that the per-photon heating rate could be one or two orders of magnitude larger than the expected free space result could be relevant to experiments that probe around the dressed atomic resonance frequency. Separately, we note that the enhanced heating of an atomic ensemble has been experimentally observed in a complementary regime, driving around the dressed cavity resonance of a detuned atom-cavity system [48]. 5. Second-order time correlations We now consider how the strong optomechanical coupling can manifest itself in the second-order time correlations of the reflected field, g(2) rr (t)≡ˆ a† out(0)ˆ a† out(t)ˆ aout(t)ˆ aout(0) ˆ a† out(t)ˆ aout(t)ˆ a† out(0)ˆ aout(0), (10) which quantifies the relative likelihood of detecting a reflected photon at time t,giventheprevious detection of a reflected photon at time t=0. We first present an approximate theory, based on the scattering matrix and the dynamics of the motional wave function following detection of a first reflected photon (|ψr,equation(8)). This approach neglects contributions to g(2) rr that arise from the anharmonicity ∼Δof the J–C ladder between 0 →1and 1→2 excitations (represented by the red arrows in figure 1(b)). We will later show, by comparing with full master equation simulations, that the scattering matrix captures well important features of g(2) rr (t)and,in particular, oscillations due to strong optomechanical coupling. For our previous choice of detunings ((5) and (6), ensuring R(x0)=0), a central hole is imprinted in the conditional atomic motional state |ψr (blue wave function in figure 3(a)), reducing atomic population at positions where reflection is more unlikely. Note that some of the other experimental parameters (κ1/κ2and η) have been changed relative to previous figures to make the relevant effects more visible. To approximate g(2) rr (t), we consider the limit of a weakly driven cavity, such that the forces associated with the cavity field are negligible compared to the external trap. The subsequent dynamics of the atomic state are then dominated by the evolution purely in the trapping potential, |ψr(t)=e−iˆ Htrapt|ψrbefore further scattering events occur, as the atomic motion is highly isolated from its environment. Because of the overall mirror symmetry in |ψrfound for the discussed configuration, a revival of the wave function appears with periodicity in time π/ωmas the atomic state evolves (red and green wave functions in figure 3(a), see also the illustration in figure 3(b)). This time-evolving spatial distribution, combined with the sensitivity of the cavity response to the position of the atom, should result in a conditional time-dependent reflectance that manifests in g(2) rr (t)as, g(2) rr (t)≈|Sr|ψr(t)|2 R, (11) 6 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang Figure 3. (a) Spatial probability distribution |ψr(t)|2following the detection of a reflected photon at t=0, and under the assumption that the subsequent motional wave function only evolves under the external trapping potential for the configuration η=0.05, κ1/κ2=1.6, marked with a red cross in figure 4(a). Later times are indicated by vertical shifts of the atomic density, and the wave function at times π/ωmand 2π/ωmare coloured in red and green, respectively. Following equations (5)and(6), Δ0 and Δare chosen to satisfy R(x0)=0. Rest of parameters as in figure 2.(b)Schematicrepresentationof|ψr|2at time t=0 (blue) and t=π/ωm(dotted red). (c) Calculation of g(2) rr (t) along this evolution, using the scattering matrix approach (blue line) and a master equation simulation of the cavity system (orange dashed line). The inset highlights the fast decay of g(2) rr (t)atinitial times revealed by the master equation (dashed line), compatible with a decay rate ˜γ(x)(continuous line). (d)–(f) Analogous plots to (a)–(c), now for the case R(x0+xzp)=0 (see text). which compares the reflectance of the cavity at time tafter detection of a reflected photon to the initial reflectance Rof the cavity, considering that intermediate scattering events are unlikely over the observation time. In figure 3(c)(bluecurve),weplotthepredictedg(2) rr (t)fromequation(11), for the spatial dynamics illustrated in figure 3(a). We observe a bunching effect immediately after detection of the first reflected photon, as detection projects the atomic state into a configuration compatible with that event. The same cavity response is expected whenever the state revives, which for the symmetric configuration presented above, occurs with periodicity π/ωm. To validate these results in the weakly driven regime, we have also performed a full master equation simulation of the driven system (2) for a weak field input as described in section 3(orangedashedlinein figure 3(c)). We observe good agreement with the results provided by equation (11)attimest>1/˜γ.At shorter times, we note an additional contribution to g(2) rr (t) that can be understood from the anharmonicity Δin the J–C ladder of a motionless atom (see red arrows in figure 1(b)). For large cooperativity, this detuning exceeds the linewidth of the cavity, Δ∗/κ ∼√Cin (see equation (6)), and favours the reflection of two-photon components. In order to separate nonlinearities arising from motion versus the two-level structure itself, it is important to note that any transient feature arising in g(2) rr (0) due to the (dressed) atomic state will decay at the cavity-enhanced atomic emission rate ˜γ, as we further illustrate in the inset, which is much larger than typical atomic trap frequencies. Previous work on ‘single polariton optomechanics’ involves adding a two-level atom as a third degree of freedom to an optomechanical system which explicitly allows for non-Gaussian states to be generated [49,50]. Here, although we also use an atom, the non-trivial time-dependent features in g(2) rr (t) we observe beyond t1/˜γcanthenbeattributable to the single-photon strong coupling originating from atomic motion, rather than the two-level nature of the atom. Regarding the significance of these time-dependent oscillations in g(2) rr (t), we point out that they differ from oscillations in reflection that could be observed, for example, by applying a classical momentum kick on the atom. In particular, in the latter case, given an atom originally in a stationary state (such as the motional ground state or a thermal state), an additional optical pulse (or a sudden variation in the trapping field) could induce motional oscillations in the atom. These would be already visible as temporal oscillations in the cavity output field ˆ a† out(t)ˆ aout(t), given a weak probe input. Note that these oscillations 7 New J. Phys. 24 (2022) 023006 J Argüello-Luengo and D E Chang Figure 4. (a) Scattering matrix calculation of g(2) rr (0) when driving in resonance the dressed atomic frequency for the configuration R0=0 as defined in equations (5)and(6), as one varies the Lamb–Dicke parameter ηand the ratio κ1/κ2.For κ1<κ 2, where it is not possible to obtain R0=0, we numerically maximize g(2) rr (0) as a function of Δ. Rest of experimental parameters as in figure 2. Red line follows the relation (1 +R0)=6xzp, and red marker indicates the configuration κ1/κ2=1.6, η=0.05 explored in figure 3. The inset zooms into the region of ratios κ1/κ2≈1. (b) For the same parameter choices as (a), we illustrate the overall variation along a full mechanical oscillation, Δg(2) rr ≡maxtg(2) rr (t)−mintg(2) rr (t). (c) Reflectance of this atom-cavity system as a function of the position of a motionless atom trapped in an harmonic potential centered at x0with Lamb–Dicke parameter η=0.07 and different values of κ1/κ2(see legend). Coloured area indicates the region |x−x0|<. would only be significant if the kicking pulse contained many photons, given the small recoil energy of a single photon compared to the trapping frequency. In the presented scheme, the ‘kick’ comes from the detection of just a single photon and the large conditional change that it imparts on the motional wave function, which is the essence of strong single-photon optomechanical coupling. Furthermore, the conditional nature of this effect causes these oscillations to appear in the higher-order correlation of g(2) rr (t), rather than the unconditional reflectance itself. Furthermore, the period of oscillations can be modified by tuning the driving frequency such that R(x0+xzp)=0(e.g.replacingx0→x0+xzp in equations (5)and(6)). The detection of a reflected photon results in a conditional wave function whose probability amplitude is increased on one side of the trap, as illustrated in figure 3(d). After half a period, this state now oscillates to the opposite side of the trap (see figure 3(e)) which, in this configuration, manifests as antibunching (g(2) rr (π/ωm)<1), restoring the natural periodicity 2π/ωmof the correlator g(2) rr (t), as we show in figure 3(f). We now discuss the approximate conditions desired to observe large contrast in the time-dependent oscillations in g(2) rr (t). We begin by noting that our previous strong coupling conditions, based on achieving an effective length /xzp as small as possible (see equation (7)), do not directly translate into large oscillations in g(2) rr (t). In particular, large oscillations require a large difference between the unconditional and conditional reflectances. Note that in the best case scenario, the detection of a reflected photon completely conditions the atomic wave function to reflect a second photon, resulting in a conditional reflectance of unity. Thus, one wants to avoid that the unconditional reflectance is already too close to unity, R→1. This large unconditional reflectance would occur, for example, if /xzp →0, such that the atom is effectively never in the narrow spatial width ∼where the reflection would differ from unity, resulting in g(2) rr (0) →1. To better interpret how intermediate situations maybeoptimal,onecanexplore a simplified uniform response in reflectance R(x)=1−(1 −R0)Θ−|x|forahomogeneousmechanicalstate |ψ(x)|2=(2xzp)−1Θxzp −|x|;whereΘ[x] denotes the step function that is 1 for x>0and0otherwise. In this toy model, one obtains that the maximum value of g(2) rr (0) occurs when (1 +R0)∼xzp,which defines an optimal (non-zero) length for each choice of R0. The optimal configuration is a balanced cavity (R0=0), where one would desire ∼xzp. To further illustrate this, in figure 4(a) we calculate the scattering matrix approximation to g(2) rr (0) (11), using the same parameters for g0,γ,andκ2as in the experiment of reference [35]andfigure2(a). However, we now allow the Lamb–Dicke parameter (experimentally tunable through the intensity of the trapping potential) and the output port decay rate κ1to vary. Choosing for each set of ηand κ1the atom-cavity detuning that minimizes R(x0), and driving in resonance with the dressed atomic frequency for an atom positioned at x0, we heuristically observe that the largest values of g(2) rr (0) ∼3 appear in a region compatible with (1 +R0)=6xzp (red continuous line), which aligns with the intuition built from our toy model. In figure 4(b) we further show the overall variation of g(2) rr (t) along a full mechanical oscillation, Δg(2) rr ≡maxtg(2) rr (t)−mintg(2) rr (t), observing that the largest values Δg(2) rr ∼2 appear in a region compatible to those with larger g(2) rr (0). Here, one can also see a sharp change in g(2) rr (0) around κ1=κ2, which is more evident as η<0.1(see inset in figure 4(a)). In this latter regime, the effective atomic displacement over which the reflectance of the 8