Exploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity
EExploring unresolved sideband, optomechanicalstrong coupling using a single atom coupled to acavity
Lukas Neumeier , Darrick E. Chang , ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science andTechnology, 08860 Castelldefels, Barcelona, Spain, ICREA-Instituci´o Catalana deRecerca i Estudis Avan¸cats, 08015 Barcelona, SpainE-mail: [email protected]
Abstract.
A major trend within the field of cavity QED is to boost the interactionstrength between the cavity field and the atomic internal degrees of freedom of thetrapped atom by decreasing the mode volume of the cavity. In such systems, itis natural to achieve strong atom-cavity coupling, where the coherent interactionstrength exceeds the cavity linewidth, while the linewidth exceeds the atomic trapfrequency. While most work focuses on coupling of photons to the internal degrees offreedom, additional rich dynamics can occur by considering the atomic motional degreeof freedom as well. In particular, we show that such a system is a natural candidate toexplore an interesting regime of quantum optomechanics, where the zero-point atomicmotion yields a cavity frequency shift larger than its linewidth (so-called single-photonoptomechanical strong coupling), but simultaneously where the motional frequencycannot be resolved by the cavity. We show that this regime can result in a number ofremarkable phenomena, such as strong entanglement between the atomic wave-functionand the scattering properties of single incident photons, or an anomalous mechanismwhere the atomic motion can significantly heat up due to single-photon scattering,even if the atom is trapped tightly within the Lamb-Dicke limit.PACS numbers: 42.50.-p, 42.50.Pq, 42.50.Wk a r X i v : . [ qu a n t - ph ] A p r xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity ω m exceeds the cavity linewidth κ , is required. For example,this enables cooling to the quantum ground state [5, 6], which represents a fiducialpure state preparation. In one remarkable theoretical work [7], it has been predictedthat the combination of sideband resolution and single-photon optomechanical strongcoupling – where the zero-point motional uncertainty induces a shift in the opticalresonance frequency larger than the cavity linewidth – would enable the generation ofnon-classical, anti-bunched light.Here, we study the complementary regime of single-photon optomechanical strongcoupling, but with unresolved sidebands [8, 9]. We show that interesting quantum effectsboth in the light and motion can be observed, at least when the mechanical system iswell-isolated and can be separately prepared in the ground state. A natural candidatesystem consists of a single atom [10, 11, 12, 13, 14, 15] or ion [16, 17, 18, 19, 20, 21] incavity QED, whose electronic transition is strongly coupled to a near-resonant opticalmode. To provide an intuitive picture, strong coupling within cavity QED [22, 23]implies that a point-like atom produces a shift in the cavity resonance frequency thatis larger than the cavity linewidth, when the atom is situated at a cavity anti-node.If the atom is displaced by a quarter wavelength to a node, this shift vanishes. Giventhe light mass, it is straightforward for a trapped atom to have a zero-point motion onthat scale, thus realizing single-photon optomechanical strong coupling. Furthermore,realistic trap frequencies for atoms are quite low ( (cid:46) MHz), and are naturally exceeded bythe cavity linewidth for small cavities [14, 15, 24, 25]. In this regime of optomechanicalstrong coupling and unresolved sidebands, the interesting physics arises because theresonance frequency of the cavity correlates strongly with the atomic position, and asthe reflection or transmission of a single photon depends on the resonance frequency,a strong entanglement between photon and motion ensues, which is visible in both ofthese degrees of freedom.In this work we begin by considering a single atom externally trapped inside a cavitymode that is driven with a coherent state. When the cavity frequency is detuned fromthe atomic resonance, we derive from the full Jaynes-Cummings model of cavity QEDan effective optomechanical Hamiltonian, which only depends on the atomic motionand cavity degrees of freedom. We proceed by tracing out the cavity degree of freedomand analytically derive an effective quantum master equation describing the motionaldynamics of the atom only. This master equation would allow for the calculationof motional energy eigenvalues and their lifetimes, and yields interesting insights inthe heating processes associated with entanglement between light and motion. Thisentanglement is also directly revealed by applying scattering theory to exactly solve xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 1.
An atom is trapped externally by a potential (blue) with equilibriumposition x inside a cavity with intensity mode profile u ( x ). Ψ ( x ) is the initial wavefunction of the atomic motion. Incident photons with frequency ω L arrive from theleft. The left mirror has a decay rate of κ r and the right mirror has a decay rate of κ t . for the joint atom-photon wave function following the scattering of a single incidentphoton. Using this formalism, we show that the properties of the scattered photon canbecome entangled with the atomic motion on length scales much smaller than eitherthe resonant wavelength or the atomic zero-point motion. As one consequence, once thephoton is traced out, the atomic motion is seen to heat up significantly, even if the atomis tightly trapped within the Lamb-Dicke limit. We also show that this entanglementcan manifest itself in the second-order correlation functions of the outgoing field givena weak coherent state input, or be used to produce a heralded single-phonon Fock stateof the atomic motion.
1. Cavity QED with motion
In this section, we introduce the Jaynes-Cummings (J-C) model [26] to describe theinteraction of a (moving) two-level atom with photons in a cavity mode with amplitude u ( x ) = cos( k c x ), where k c is the wavevector of the cavity mode as shown in figure 1. Inthe case where the atomic frequency ω is far detuned from the bare cavity resonance ω c , we eliminate the atomic internal degrees of freedom, to arrive at an effectiveoptomechanical interaction between the atomic motion and cavity. We further proceedto derive an effective master equation describing the atomic motion when the cavity isexternally driven by a coherent state with photon number flux E and frequency ω L .We note that such a procedure would give rise to, e.g., the usual optical cooling andheating rates in a conventional optomechanical system [27, 5, 6]. In our case, however,we neither linearize the cavity field around a steady-state solution nor the motion, owingto the potentially large coupling between motion and the cavity field, which leads tomuch richer effects.The full quantum master equation associated with the J-C model, in an interactionpicture rotating with the laser frequency ω L , is given by˙ ρ = − i [ H JC , ρ ] + ( L γ + L κ ) ρ ≡ Lρ. (1) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity H JC = ω m b † b − δ σ ee − δ c a † a + √ κ r E ( a + a † ) + g u ( x )( a † σ ge + h.c. ) . (2)It is written in terms of the detuning between laser and atom/cavity δ /c = ω L − ω /c ,respectively, and the mechanical frequency ω m of the external trap. Furthermore, a and b denote the photon and phonon annihilation operators, respectively, while σ αβ = | α (cid:105)(cid:104) β | ,where α, β = g, e correspond to combinations of the atomic ground and excited states. κ r denotes the decay rate of the left cavity mirror (reflection), which also serves as thesource of injection of photons. The right mirror has a decay rate of κ t (transmission).In addition to the external coupling, the cavity has an intrinsic loss rate κ in , such asthrough material absorption or scattering losses. The total cavity linewidth is thus κ = κ r + κ t + κ in . The last term of H JC describes the coupling between cavity and atomwith the coupling strength g u ( x ) depending on the atomic position x = x zp ( b + b † ),which can be written in terms of the zero-point motion x zp = (cid:112) (cid:126) / (2 mω m ) ( m beingthe atomic mass), and where g is the magnitude of the vacuum Rabi splitting at theanti-node of the cavity. The Lindblad L c operator describing cavity dissipation is givenby: L κ ρ = − κ (cid:0) a † aρ + ρa † a − aρa † (cid:1) (3)and the general Lindblad operator L γ for spontaneous emission into three dimensionsof the atom at a rate γ reads [28]: L γ ρ = − γ (cid:18) σ ee ρ + ρσ ee − (cid:90) d Ω (cid:126)u N f ( (cid:126)u ) σ ge e − i k c (cid:126)u · (cid:126)r ρe i k c (cid:126)u · (cid:126)r σ eg (cid:19) . (4)This process, additionally to the emission of a photon, causes a recoil of k = ω c ≈ k c opposite to the direction (cid:126)u of the emitted photon, which is integrated over solid angle( d Ω (cid:126)u ) and weighted by the distribution function N f ( (cid:126)u ) corresponding to the dipoleemission pattern. However, to provide a simpler model that qualitatively captures thecorrect behavior, we will just consider one single direction of spontaneous emission alongthe positive cavity axis ( x ). With a single spontaneous emission direction we can write L γ ρ = − γ (cid:0) σ ee ρ + ρσ ee − σ ge e − i k c x ρe i k c x σ eg (cid:1) . (5)Now we consider the dispersive regime ∆ = ω − ω c (cid:29) g , κ, γ , where the atom-cavitydetuning is large. Thus the single-excitation eigenstates of the J-C Hamiltonian areeither mostly atomic ( | ψ + (cid:105) ≈ | e, (cid:105) ) or photonic ( | ψ − (cid:105) ≈ | g, (cid:105) ), where 0,1 denotethe intra-cavity photon Fock state number. These eigenstates have correspondingeigenenergies E +1 ≈ ω + g ∆ u ( x ) and E − ≈ ω c − g ∆ u ( x ), respectively. Here, we focuson the case when the system is driven near resonantly with the photonic eigenstate. Inthat limit, the atom can approximately be viewed as a classical dielectric that providesa position-dependent cavity shift with an effective optomechanical coupling strength ∝ g ∆ . We will derive this effective optomechanical model now in more detail. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity For large laser-atom detunings δ (cid:29) g , the atomic ground state population isapproximately one, which allows for an effective elimination of the atomic excited state[28, 29] using the Nakajima-Zwanzig projection operator formalism [30, 31, 27] (detailsin Appendix A.1). The resulting effective master equation is given by˙ ρ = − i [ H om , ρ ] + L om ρ, (6)with an effective optomechanical Hamiltonian H om = ω m b † b − ∆ c ( x ) a † a + √ κ r E ( a + a † ) . (7)The position dependent cavity-laser detuning is given by∆ c ( x ) = δ c − g δ δ + γ u ( x ) , (8)which now accounts for the cavity shift arising from off-resonant coupling to the atomictransition. The system losses are given by the effective Liouvillian L om ρ = − κ (cid:0) a † aρ + ρa † a − aρa † (cid:1) − γ g δ + γ (cid:0) u ( x ) a † aρ + ρa † au ( x ) − au ( x ) e − i k c x ρe i k c x u ( x ) a † (cid:1) , (9)which describes the broadening of the cavity linewidth due to atomic spontaneousemission, κ ( x ) = κ + γ g δ + γ u ( x ) . (10)Aside from Appendix C, where we discuss in greater detail the corrections to andlimitations of the effective model, we will work in regimes where the atomic contributionis negligible compared to the (large) bare cavity linewidth.In typical treatments of optomechanical systems, the position-dependent shiftin equation (8) would only be treated to linear order in the displacement, withthe justification that the maximum possible displacement is very small. However,for atoms, the zero-point motion can be comparable to the optical wavelength (thescale over which u ( x ) varies), a ratio that can be characterized by the Lamb-Dickeparameter η LD ≡ k c x zp . For example, taking a recoil frequency ω rec = 2 π × . Ca + -ions and a trap frequency of ω m = 2 π × . η LD = (cid:112) ω rec /ω m ≈ .
26. For η LD ∼
1, the atomic wavepacket would have significantweight both in a cavity anti-node and node, with an associated cavity frequency shift of g om = − g δ δ + γ (11) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity δ (cid:38) g (seeAppendix C), one sees that strong optomechanical coupling g om (cid:38) κ can be achievedif the strong coupling regime of conventional cavity QED ( g > κ ) is realized.The standard optomechanical Hamiltonian (linearized in displacement) describinginteractions between single-photons and single-phonons is given by H oms = g m ( b † + b ) a † a ,where g m = ∆ (cid:48) c ( x ) x zp ∼ g om η LD . Thus, in order to achieve strong optomechanicalcoupling on the single-photon, single-phonon level ( g m (cid:38) κ ), additionally a sufficientlylarge Lamb-Dicke parameter η LD is required. Given the above considerations, wenext derive an effective master equation for the atomic motion alone that is valid forstrong and nonlinear optomechanical coupling, which can be viewed as a generalizationof the typical optically-induced cooling and heating rates obtained for linearizedoptomechanical coupling [27, 5, 6]. Our master equation also complements previouswork investigating intra-cavity optical forces on atoms in the semi-classical limit[32, 33, 28, 34, 35]. Starting with equation (6) we can use the Nakajima-Zwanzig technique to effectivelyeliminate the cavity degrees of freedom (Appendix A.2). Here, for simplicity we assumethat spontaneous emission can be ignored. The resulting master equation for atomicmotion in conventional Lindblad-form is then given by:˙ ρ = − i[ H m , ρ ] − (cid:0) J † J ρ + ρJ † J (cid:1) + J ρJ † . (12)The Hermitian Hamiltonian and jump operators are given respectively by H m = ω m b † b + κ r E ∆ c ( x )∆ c ( x ) + κ (13)and J = i √ κκ r E ∆ c ( x ) + i κ . (14)We will provide an intuitive picture of this master equation in section 2. Now we focuson the effective mechanical potential which arises in the Hamiltonian. We can alwaysrewrite a master equation in terms of an effective non-Hermitian Hamiltonian H c whichthen contains a complex potential:˙ ρ = − i( H c ρ − ρH † c ) + J ρJ † (15) H c = ω m b † b + V ( x ) (16)with V ( x ) = κ r E ∆ c ( x )∆ c ( x ) + κ − i2 J † J. (17)The real and imaginary parts of the complex potential V ( x ) are illustrated in figure 2. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 2. Quantum and classical mechanical potential arising from acoherently driven cavity mode
Real part Re[ V ( x )] (blue) and imaginary part Im[ V ( x )] (red) of the quantum potentialequation (17) as a function of position. Also plotted is the classical potential U ( x )(dahsed, green) derived by integrating the expectation value of the force acting on theatom. One can observe that the real part of the quantum potential is significantlydifferent from the classical expectation value. Here, we choose a laser frequency ω L such that the resonant position k c x r = π/
4, and Jaynes-Cummings parameters of g /κ ∼
20 and δ = − g (yielding an effective optomechanical coupling strength of g om ∼ κ ). The potentials are plotted in units of (cid:126) ( κ r /κ ) E . As the resonance frequency of the cavity depends on the position of the atom, therecan be atomic positions for which the cavity is resonant with the coherent drive. Thesepositions x r are called resonant positions and are defined by ∆ c ( x r ) = 0. Around thesepositions, the real part of the potential changes sign and the imaginary part has sinksindicating increased heating around those positions.It is also interesting to compare the “coherent” potential, Re[ V ( x )], with theclassical potential U ( x ) as derived from the average force F ( x ) = d (cid:104) p (cid:105) /dt = Tr( pρ )on the atom, and defined via dU/dx = − F ( x ). The result is given by U ( x ) = − κ r κ E arctan (cid:18) c ( x ) κ (cid:19) , (18)which agrees with a previous, completely classical analysis of a dielectric object trappedin a cavity [36]. The potential is illustrated in figure 2. For large g om /κ , U ( x ) is seen toapproach a square well, with the walls of the well aligning with the resonant positions ∼ x r where the large intracavity field results in a large classical restoring force. Bycomparing V ( x ) and U ( x ), it is clear that a significant contribution of the average forcemust arise from the stochastic process associated with the quantum jumps J . As oneconsequence, although it would be highly interesting to realize a square well for atoms(leading, e.g., to a highly anharmonic phonon spectrum), the direct quantization of U ( x )in this case is not meaningful. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity
2. Single-photon scattering theory: Optomechanical strong coupling withunresolved sidebands
A complementary physical picture of the optomechanical coupling between an atom andcavity can be gained by considering not a coherent external drive, but single incidentphotons. From equation (6), the effective non-Hermitian Hamiltonian associated withan undriven system is H eff = ω m b † b − (∆ c ( x ) + i κ a † a (19)where ∆ c ( x ) = ω L − ω c ( x ) is the position-dependent detuning between photon frequency ω L and cavity frequency ω c ( x ) = ω c − g om u ( x ). To be specific, we will consider singlephotons incident through the left mirror (see figure 1), which has a decay rate back intothe reflection channel of κ r . The right mirror is coupled to the controlled transmissionchannel with κ t . The total cavity linewidth is thus κ = κ r + κ t . For simplicity we ignorehere an intrinsic loss rate, although it is straightforward to include later on.A connection can be made between the eigenstates of H eff and the properties ofsingle-photon scattering via the S-matrix formalism. Formally, the S-matrix describesa coherent evolution mapping an input state ( t = −∞ ) to an output state ( t = + ∞ ): | Ψ out ( ω L ) (cid:105) = S | Ψ in ( ω L ) (cid:105) . (20)Here, we assume a single monochromatic photon with frequency ω L incident on the leftcavity mirror | Ψ in ( ω L ) (cid:105) = | ( ω L ) left , (cid:105) , (21)whereas the optomechanical system initially is in its ground state represented by thesecond entry in the ket state. Generically the output state will consist of a superpositionof n phonons in the mechanical state, which were excited by the incoming photon, andan outgoing photon of energy ω L − nω m in either the reflection port (r) or transmissionport (t): | Ψ out ( ω L ) (cid:105) = (cid:88) n S r , n ( ω L ) | ( ω L − nω m ) r , n (cid:105) + (cid:88) n S t ,n ( ω L ) | ( ω L − nω m ) t , n (cid:105) . (22)Due to a connection between the scattering matrix and the Heisenberg input-outputoperators [37] one can express the S-matrix elements in terms of the eigenvalues λ β and eigenstates | β (cid:105) of the effective Hamiltonian H eff [38]. We provide a detailedderivation of the S-matrix elements in Appendix B. In reflection, the output consists of asuperposition between a non-interacting propagating photon ( δ n, ) and photon emissionfrom the excited optomechanical system: S r , n ( ω L ) = δ n, + i κ r (cid:88) β (cid:104) c , n | β (cid:105) λ β (cid:104) β | c , (cid:105) . (23)Here, (cid:104) c , n | β (cid:105) is the projection of the eigenstates | β (cid:105) onto the basis states (cid:104) c , n | with1 c referring to a single photon inside the cavity mode. Similarly, the matrix elements xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity S t , n ( ω L ) = i √ κ t κ r (cid:88) β (cid:104) c , n | β (cid:105) λ β (cid:104) β | c , (cid:105) . (24)The matrix element for photon transmission lacks the contribution from the non-interacting propagating photon as the input channel on the transmitting side of thecavity is in the vacuum state. To proceed further, we assume in the following that adetector cannot effectively resolve the frequency of the outgoing photon. Then, we caneffectively write the outgoing state as | Ψ out ( ω L ) (cid:105) = S r ( ω L , x )Ψ ( x ) | r (cid:105) + S t ( ω L , x )Ψ ( x ) | t (cid:105) , (25)where | r/t (cid:105) indicates an outgoing reflected/transmitted photon, respectively, andΨ ( x ) is the initial motional wave function of the atom. The entanglement betweenthe photon frequency and the motional state has been suppressed, as we haveassumed that any projective measurement of a photon in either port is not frequency-resolving. Furthermore, we now assume that we operate in the sideband-unresolvedlimit κ (cid:29) ω m . The Hamiltonian H eff is approximately diagonal in the position basis,as the optomechanical interaction dominates over the free Hamilontian ω m b † b in H eff (equation 19). Thus, the eigenvalues of H eff are approximately λ ≈ − ∆ c ( x ) − i κ andthe scattering matrix elements can be simply written as S r ( ω L , x ) = 1 − i κ r ∆ c ( x ) + i κ (26)and S t ( ω L , x ) = − i √ κ t κ r ∆ c ( x ) + i κ . (27)As the shape of the mechanical wave function after the decay of a single photon intoone specific channel is the product between the corresponding S-matrix element and theinitial wave function Ψ ( x ), we observe that the shape of the mechanical wave functionafter one such scattering event is strongly entangled with whether the decaying photonis reflected or transmitted.Motivated by the observation that the scattering matrices S t and S r of Eqs. (26)and (27) are very similar to the jump operators J (equation 14), we express the masterequation (12) in a way that its jump operators correspond to the single photon scatteringmatrices: ˙ ρ = − i( H s ρ − ρH † s ) + E ( S r ρS † r + S t ρS † t ) (28)with the Hamiltonian H s = ω m b † b − i2 E . (29)Written in this form the connection between scattering theory and jump formalismbecomes clear. The non-Hermitian term in H s describes the rate that quantumjumps are applied to the motional wave function, which corresponds to the rate E xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 3. Reflection spectrum p r as a function of laser frequency ω L . Here,we take critical coupling ( κ r = κ/
2) and a trap equilibrium position of k c x = π/ a) If the zero-point motion is unresolved, the reflection spectrum (blue) just behaveslike the reflection spectrum of an empty cavity (green, dashed) but is shifted to a newresonance ω c ( x ). Here we choose g om = κ and η LD = 0 .
01, implying r zp = 0 . b) If the zero-point motion is resolved, the reflection spectrum is broadened by roughly g m and becomes shallower. Here we choose g om = 5 κ and η LD = 0 .
2, implying r zp = 2. of incident photons on the cavity. The jump operators themselves, J r / t = E S r / t , with( J † r J r + J † t J t = E ), are proportional to the single-photon scattering matrix elementsin reflection and transmission, encoding the two processes by which the original wavefunction can change by becoming entangled with a scattered photon. Interestingly, thecoherent part of the potential, Re[ V ( x )] in equation (17), is seen to arise from the term S r ρS † r in equation (28), and specifically from the interference between the incident andscattered components (first and second terms on the right of equation (26), respectively).
3. Quantum Effects due to Zero-Point motion
We have already seen that the scattering of a single photon on a cavity containingan atom leads to an entangled output state (25). This output state describes thecoexistence of the possibilities of photon reflection and photon transmission and howthe wave function of the atom gets modified for each of those events. We now proceedto describe some of the relevant observational consequences.We can expand the position-dependent cavity detuning around a resonant position x r (defined by ∆ c ( x r ) = 0) until linear order:∆ c ( x ) ≈ δ c + g om u ( x r ) − g om sin(2 k c x r ) k c ( x − x r ) . (30)This is a good approximation in the Lamb-Dicke regime η LD (cid:28)
1. In order to predictobservables, linearizing displacement is also a good approximation for g om (cid:29) κ , evenif η LD ∼
1, since then the cavity frequency shifts out of resonance for displacements k c δx (cid:28)
1. The term sin(2 k c x r ) indicates that the cavity frequency is most sensitive todisplacements if k c x r = ± π/
4, halfway between a cavity node and anti-node. Then itcan be seen that if the atomic wave function is centered around k c x = k c x r = π/
4, the xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 4. Resolution beyond zero-point uncertaintya)
For r zp = 2, the spatial width ∼ x zp of the atomic probability density | Ψ ( x ) | (blue) exceeds the spatial resolution R , which corresponds to the width of the absolutevalue of the scattering matrix | S r ( x ) | (red dashed). As the cavity is only resonantwith an incoming photon if the atom is located within R , there is a large probabilitythat the cavity is off-resonant, even though ω L = ω c ( x ). The probability of reflectionis calculated by the overlap of both plotted functions. b) Same as a), but the absolute value of the S-matrix for transmission (red, dashed). c) Probability of photon reflection p r (red) and transmission p t (green) as a functionof zero-point resolution r zp , for an incident photon that is resonant with the cavity inthe limit that atomic motion fluctuations are ignored (i.e., δ c = − g om u ( x )). Onesees that for large r zp , the probability of transmission becomes negligible, because theprobability of finding the atom within R (which would imply a resonant system andconsequent transmission) approaches zero for r zp (cid:29) cavity frequency shifts by a linewidth κ , if the atom moves a distance of k c R = κ/g om . Asthe transmission/reflection of a single, near-resonant photon changes significantly as itsfrequency varies over a cavity linewidth, R can be viewed as the spatial resolution overwhich the single photon ”learns” about the atomic position via its scattering direction.We will now define the zero-point resolution r zp ≡ (2 x zp ) /R = (2 g m ) /κ, (31)with g m = g om η LD being the single-photon, single phonon coupling strength as definedin section 1.1. The zero-point resolution tells us how much finer the resolution of anincident photon is compared to the width of the atomic wave function. It distinguishestwo regimes: unresolved zero-point motion r zp (cid:28)
1, which corresponds to the usualregime of weak optomechanical interactions, and the resolved zero-point motion regime r zp (cid:29)
1, where the resolution of the system becomes smaller then the zero-point motion,which is until now unexplored and which gives rise to novel effects as we will demonstratein the following.
Here, we assume the atom to be initially in its motional ground state Ψ ( x ) ∝ e − ( x − x ) /x with a trap equilibrium k c x = π/ κ r = κ/ ω L , is then xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity p r ( ω L ) = (cid:90) dx | S r ( ω L , x ) | | Ψ ( x ) | . (32)Figure 3(a) shows p r as a function of cavity detuning δ c = ω L − ω c for r zp (cid:28) κ . The blue solid line is calculated with equation (32) for r zp = 0 . p r ≈ | S r ( ω L , x ) | . One can see that it exhibits the same Lorentzian response asan empty cavity, but with a resonance frequency shifted by − g om u ( x ). Figure 3(b)shows the reflection spectrum p r for resolved zero-point motion r zp = 2. We observethat the probability of reflection is strongly increased for δ c = − g om u ( x ), comparedto the case of small r zp . This behavior can be understood from equation (30). Inparticular, the resonance frequency of the coupled atom-cavity system depends on theposition of the atom, and δ c = − g om u ( x ) corresponds to the resonance of the mostlikely atomic position. However, the large spread of the atomic wave function results ina large uncertainty of the resonance frequency, which increases the reflection probability.Conversely, an incident photon with frequency far from δ c = − g om u ( x ) sees a decreasedreflection probability (thus the broadening of the spectrum), as there is some chancethat the spread in atomic position allows the coupled system to be on resonance with thephoton. This is illustrated in figure 4(a), where we plot the atomic probability density | Ψ ( x ) | (blue) and the absolute value of the reflection S-matrix | S r ( x ) | (red dashed)(equation 26) as a function of position x and for r zp = 2. One can see, that the widthof the atomic wave function ∼ x zp exceeds the spatial resolution R , within which thecavity is resonant. For completeness, we also provide a plot of the absolute value ofthe transmission S-matrix | S t ( x ) | (red dashed) in figure 4(b). Figure 4(c) shows theprobability of reflection and transmission for δ c = − g om u ( x ) as a function of r zp . For r zp (cid:28) r zp it becomes less likelyto find the atom within the spatial resolution R within which the cavity is resonant,leading to an increase of p r . Finally, the reflection probability p r approaches unity for r zp (cid:29) r zp ∼
10 (Appendix D.1) whereas a current fiber cavity experiment reaches r zp ∼ p r , the zero-point resolution r zp can thenbe gradually decreased by increasing the atom-cavity detuning ω − ω c , increasing trapfrequency ω m or by moving the trap equilibrium x away from the position of maximaloptomechanical coupling k c x = ± π/
4. This procedure would experimentally reproduceparts of figure 4(c).
Having previously investigated the unconditional reflection spectrum of an incidentphoton, we now study more carefully the correlations that build up between the atomic xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 5. Illustration of a single photon scattering event for resolved zero-point motiona) Input state:
An incident photon (green) with a frequency ensuring x = x r isflying towards a cavity containing a trapped atom with probabilitiy density | Ψ ( x ) | (black). Due to its zero-point uncertainty, the system is in an effective superpositionof resonance frequencies. This input state is given by equation (21). b) Output state: Illustration of the entangled output state given by equation (25),which is a superposition of the photon being reflected, which implies an off-resonantsystem and a photon being transmitted, which implies a resonant system. Theplotted probability densities | Ψ r / t ( x ) | are the normalized product of | Ψ ( x ) | andthe respective scattering matrix | S r / t ( x ) | of figure 3(a) and 3(b), where r zp = 2. Forthis value, the probability of reflection is p r ≈ . motion and photon reflection or transmission for the case when the trap equilibriumfalls at the resonant position ( x = x r ). As the atom is in a coherent superposition ofbeing within the spatial resolution R and not, and an incoming photon gets transmittedif the atom is within that spatial resolution and reflected if otherwise, the resulting state(equation 25) is entangled. Given that the photon has been transmitted, the normalizedconditional wave function is given byΨ t ( x ) = p − / t S t ( x )Ψ ( x ) . (33)Its probability density is propotional to the product of | Ψ ( x ) | and | S t ( x ) | asindividually drawn in figure 4(b). Thus, for r zp (cid:29)
1, the transmission of a photonprojects the atom into a narrow spatial region ∆ x ∼ /R around the resonant position,which is consistent with the photon having seen a resonant cavity response.In contrast, the reflection of a photon projects the atom away from that same spatialregion, which results in a hole around x r with width ∆ x ∼ /R . This is consistent withthe photon having seen an off-resonant cavity. The normalized conditional wave functionafter a photon reflection is then given byΨ r ( x ) = p − / r S r ( x )Ψ ( x ) . (34) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity | Ψ ( x ) | and | S r ( x ) | . Figure 5(a) shows an illustration of the unentangled input state.The atom (black) is in its motional ground state, centered around x = x r , while a singlephoton (green) is incident and resonant with the atom-cavity system for this position.In figure 5(b) we illustrate the entangled output state for r zp = 2. We illustrate how thetransmission or reflection of a photon are entangled with atomic wave functions Ψ t ( x )or Ψ r ( x ) consistent with the respective scattering process, for the same parameters asin figures 3(a) and (b).Interestingly, in the unresolved zero-point motion regime r zp (cid:28) x : S r ( x ) ≈ − x/R . This leads to a finalconditional wave function Ψ r ( x ) ∝ x Ψ ( x ) which corresponds to a single-phonon Fockstate. This represents the high-fidelity generation of a single-phonon Fock state, whichis heralded on detection of a reflected photon (the probability of a single photon beingreflected itself is quite low, p r ≈ r ). This approach is distinct from previous proposalsfor heralded generation, involving the detection of a Stokes-scattered photon in thesideband resolved regime [39].The wave function after a transmission/reflection event adjusts in a way that itincreases the probability of a subsequent transmission/reflection. To demonstrate this,we calculate the conditional probability of photon transmission given that a photon hasjust been transmitted: p ( t | t ) = 1 p t (cid:90) dx | S t ( x, ω L ) | | Ψ ( x ) | . (35)Figure 6(a) shows p ( t | t ) (green) as a function of cavity detuning δ c for a fixedtrapping position k c x = π/
4. We plot the corresponding probability of transmission p t (blue) as well, which is seen to be lower than the conditional probability. Weuse parameters of an existing fiber cavity QED experiment with trapped Ca + -ions(Appendix D.2(ii)). We chose ω m = 2 π ×
50 kHz and ω − ω c = 4 g . The asymmetryof p ( t | t )) is due to the nonnegligible dependence of g om (equation (11)) and κ ( x )(equation (10)) on the laser frequency ω L (and thus δ c ) for those parameters. For2 δ c /κ = − (2 g om /κ ) u ( x ) ≈ − . x r = x ) a zero-point resolutionof r zp ≈ .
89 is obtained, which needs to be calculated with equation (C.8) ashere spontaneous emission cannot be neglected. As one consequence of the higherlikelihood of conditional transmission, the second-order correlation function g (2)tt (0) = p t (cid:82) dx | S t ( x, ω L ) | | Ψ ( x ) | of the transmitted field, given a weak coherent input state,would exhibit bunching, as shown in figure 6(b). Likewise, as reflection of a first photonsuppresses the probability of transmitting a second photon (and vice versa), second-order cross-correlations g (2)rt (0) = p t p r (cid:82) dx | S t ( x, ω L ) | | S r ( x, ω L ) | | Ψ ( x ) | between thereflected and transmitted field would exhibit anti-bunching (figure 6(c)). xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity - - - - - - - - Figure 6. Photon statistics due to wave function projection for a fixed trappingposition k c x = π/ (a) Probability of photon transmission p t (blue) as a function of cavity detuning δ c and the conditional probability of transmission, given that a photon just hasbeen transmitted p ( t | t ) (green). We observe that a transmitted photon increases theprobability of transmitting again. (b) The second-order correlation function g (2)tt (0) of the transmitted field as a functionof δ c shows bunching due to the increased likelihood of detecting a transmitted photonafter the transmission of a first photon. (c) The second-order cross-correlations g (2)tr (0) between the transmitted and thereflected field as a function of δ c shows anti-bunching.Here we use parameters of an existing fiber cavity QED experiment with trapped Ca + -ions with recoil frequency ω rec = 2 π × . g = 2 π ×
41 MHz, γ = 2 π × . κ = 2 π × ω m = 2 π ×
50 kHz and ω − ω c = 4 g . These values correspond to a zero-point resolutionof r zp ≈ .
89 for 2 δ c /κ ≈ − . Each projection of the atomic wave function is associated with an increase in energy.We will now show that this energy can vastly exceed the energy added in free space orin a trap. In free space a recoil momentum (cid:126) k L results in a kinetic energy change of ω rec (typically a few kHz). In a stiff trap ( ω rec (cid:28) ω m ) it is unlikely that a phonon can beexcited due to the insufficient energy associated with the recoil. In that case, it is well-known [40, 41] that the probability of exciting a phonon due to single-photon scatteringis suppressed as ω rec /ω m = η . However, here we show that for atoms trapped insidecavities, and in the regime of strong optomechanical coupling, it is possible for a singlescattered photon to produce a much larger heating effect, even when the atom is trappedtightly within the Lamb-Dicke limit ( η LD (cid:28) n r / t = (cid:104) Ψ r / t | b † b | Ψ r / t (cid:105) ofcreated phonons as a function of r zp after measuring a reflected/transmitted photon,respectively. For these plots we assume the atom to be initially in its ground state andthat the resonance position matches with the trap equilibrium ( x r = x ). We find that¯ n r ≈ r zp (cid:28)
1, which reflects the fact that the resulting conditional wave function xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure 7. Added phonons per photon
We assume critical coupling, the atom to be initially trapped in its ground stateand that the resonance position matches with the trap equilibrium ( x r = x ). a) Conditional expectation values of created phonons after scattering a single photon¯ n r / t . We find the scalings ¯ n r ≈ r zp (cid:28) n r ∝ r zp for r zp (cid:29) n t ∝ r for all values of r zp , leading to a very large number of added phonons for resolvedzero-point motion in the case of a measured transmitted photon. b) Total expectationvalue ¯ n (unconditional) of added phonons per photon as a function of r zp . The scalings¯ n ∝ r for r zp (cid:28) n ∝ r zp for r zp (cid:29) n = p t ¯ n t + p r ¯ n r . in this regime is a single-phonon Fock state, as explained in section 3.2. For r zp (cid:29) n r ∝ r zp , whereas ¯ n t ∝ r for all values of r zp . We now wantto give the intuition behind these scalings. Generally, the number of created phononsis the energy increase normalized with trap frequency: ¯ n = ∆ Eω m . The main contributionof added energy comes from the increase in momentum uncertainty, due to the narrowspatial features associated with the conditional wave functions after photon scattering(see figure 5(b)). Thus, the added energy after one scattering event is approximately∆ E ≈ (cid:104) Ψ | p | Ψ (cid:105) m . Transmitting a photon localizes the atomic wave function around theresonant position x r up to an uncertainty of ∆ x ∼ (cid:126) / ∆ p ∼ /r zp , which yields a kineticenergy increase corresponding to ¯ n t ∝ r . The scaling ¯ n r ∝ r zp for r zp (cid:29) κ t = 0 (but the argument holds generally). There, the photonexperiences a phase shift Φ( x ) = arg[ S r ( x )] ≈ arctan[(2( x − x r ) R ) / ( R − ( x − x r ) )]which depends on the atomic position. Φ( x ) only varies significantly for displacementssmaller than δx (cid:46) R ∝ /r zp and its slope reaches a maximum value of Φ (cid:48) ( x r ) ∝ r zp .The phase shift dominates the contribution to the added kinetic energy, ¯ n r ∝ (cid:104) Ψ | p | Ψ (cid:105) ∝ (cid:82) dx | Ψ ( x ) | (Φ (cid:48) ( x )) ∝ (cid:82) dx (Φ (cid:48) ( x )) ∝ r /r zp = r zp as for r zp (cid:29)
1, (Φ (cid:48) ( x )) peaks overa region much smaller than the width of the wavefunction, and has a width ∝ /r zp anda maximum value of ∝ r .In figure 7(b) we plot the unconditional number of added phonons per photon ¯ n (the photon is not measured after the interaction). As it is given by ¯ n = p t ¯ n t + p r ¯ n r ,it can be understood as a combination of figure 7(a) and 4(c). Thus, the scaling of ¯ n t dominates for r zp (cid:28)
1, whereas the scaling of ¯ n r dominates for r zp (cid:29) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity We have presented the theory of strong optomechanical coupling in nano/micro-cavities,where naturally the mechanical sidebands are unresolved. Possible candidate platformsare trapped atoms or ions in photonic crystal cavities or fiber cavities. We show thatthese platforms already reach a regime where the atomic zero-point motion is resolvedby incident photons, leading to strong entanglement between the photon and the atomicmotion. Signatures of this entanglement can be measured in the reflection spectrum,the second-order photon correlation functions, or in the number of added phonons perphoton. Furthermore, we showed that one can create non-Gaussian motional states fromGaussian states by reflecting a single photon, even for unresolved zero-point motion.Generally we want to emphasize that the presented theory of this work is relevant toany experiment where atoms are strongly coupled to cavities with small mode volumes.
Acknowledgments
Lukas Neumeier acknowledges financial support from the Spanish Ministry of Economyand Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellencein R&D (SEV-2015-0522). DEC acknowledges support from the Severo OchoaProgramme, Fundacio Privada Cellex, CERCA Programme / Generalitat de Catalunya,ERC Starting Grant FOQAL, MINECO Plan Nacional Grant CANS, MINECO ExploraGrant NANOTRAP, and US ONR MURI Grant QOMAND. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Appendix A. From the Jaynes-Cummings model including motion to aneffective model of motion only
Equation 1 of the main text describes the full master equation of a moving two-levelatom interacting with a cavity, in the presence of cavity losses and atomic spontaneousemission. In the limit where the cavity is driven near resonantly and the atom is far-detuned, the atomic excited state can be eliminated to yield an effective optomechanicalsystem involving just the atomic motion and the cavity mode. One can go a step furtherand eliminate the cavity mode, to yield the reduced dynamics of just the atomic motion.The procedure by which a certain degree of freedom can be eliminated from an opensystem is known as the Nakajima-Zwanzig projection operator formalism [30, 31, 27],which we now describe here.
Appendix A.1. Projecting out the atomic excited state
We first want to eliminate the atomic excited state from the full dynamics of equation 1.It is convenient to define a set of operators
P, Q , which project the entire system densitymatrix ρ = | g (cid:105)(cid:104) g | ρ gg + | g (cid:105)(cid:104) e | ρ ge + | e (cid:105)(cid:104) g | ρ eg + | e (cid:105)(cid:104) e | ρ ee , (A.1)into the subspace spanned by | g (cid:105)(cid:104) g | (which we want to project the dynamics into), andits orthogonal − | g (cid:105)(cid:104) g | . Here ρ ij = (cid:104) i | ρ | j (cid:105) are the reduced density matrices for thereduced Hilbert space, which still contain all other existing degrees of freedom. Thus,we define a projection operator P : P ρ = | g (cid:105)(cid:104) g | ρ gg (A.2)and its complementary Qρ = | g (cid:105)(cid:104) e | ρ ge + | e (cid:105)(cid:104) g | ρ eg + | e (cid:105)(cid:104) e | ρ ee . (A.3)It is straightforward to show P = P, Q = Q, QP = 0 , P + Q = . In figure A1 we drawa simple picture of the full Hilbert space of the internal degrees of freedom of the atomin order to visualize the part of the Hilbert space we are interested in (described by P ρ )and the part we are not (described by Qρ ). We will now divide the super-operator L up in parts according to the way they act on the Hilbert space describing the internaldegrees of freedom of the atom: L = L o + L a + L I + J. (A.4)Here, L o = L m + L c is composed of terms that do not act on the internal degrees offreedom, with L m and L c describing respectively the trapped atomic motion and thebare dynamics of the driven cavity mode: L m ρ = − i[ ω m b † b, ρ ] (A.5) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure A1.
The complete Hilbert space of the internal degrees of freedom of theatom.
P ρ is the part we are interested in and the remainder is characterized by theprojection operator Q . L c ρ = i δ c [ a † a, ρ ] − i √ κ r E [( a + a † ) , ρ ] − κ (cid:0) a † aρ + ρa † a − aρa † (cid:1) . (A.6)The super-operator L a ρ = i δ [ σ ee , ρ ] − γ { σ ee , ρ } (A.7)acts on | e (cid:105)(cid:104) g | , | g (cid:105)(cid:104) e | , | e (cid:105)(cid:104) e | (the subspace spanned by Q ) and just multiplies those termsby a c-number. It describes evolution and damping of the excited internal state of theatom. L I ρ = − i[ g ( x )( σ eg a + σ ge a † ) , ρ ] (A.8)acts on all the states and all Hilbert spaces, describing the interaction of the atom withthe cavity field and J ρ = γσ ge e − i k c x ρe i k c x σ eg (A.9)describes the spontaneous jump of the excited state of the atom into its ground stateaccompanied by a momentum recoil. In figure A2 we draw arrows showing how thesesuper-operators act on different parts of the Hilbert space of atomic internal degreesof freedom. We are interested in the dynamics of the subspace P ρ , while accountingfor fluctuations into Qρ . Thus, only closed loops which start and end in P ρ contributeto the evolution of the reduced density matrix
P ρ . To see how this works, we define v = P ρ and w = Qρ and insert P + Q = into equation (1):˙ v = P ˙ ρ = P Lρ = P LP ρ + P LQρ. (A.10)Let us first look at
P LP : P LP ρ = P ( L o + L a + L I + J ) P ρ. (A.11)To quickly identify vanishing terms we take advantage of figure A2 by following the paththe super-operators take us through the Hilbert space applying them from the right tothe left. Here are some examples: xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure A2.
The Hilbert space of the internal degrees of freedom of the atom. Thenotation is as follows: The label of an arrow corresponds to a Liouvillian, while thedirection of the arrow indicates the possible beginning and ending subspaces of theLiouvillian. For example, the red arrow indicates that the Liouvillian J acting on thesubspace | e (cid:105)(cid:104) e | takes this subspace to | g (cid:105)(cid:104) g | . Since we assume δ or γ to be much largerthan κ and ω m , we can neglect the action of L o = L m + L c during a fluctuation outof P ρ , which we indicate by crossing them out in the right-top corner and neglectingthem in equation (A.13). (i) The term
P L I P : P projects into the subspace | g (cid:105)(cid:104) g | , while L I maps a state from P to Q . Thus, acting again with P causes this term to vanish.(ii) P L a P : P projects into | g (cid:105)(cid:104) g | and we immediately see that L a does not act on it,so this term vanishes.(iii) P J P = 0 because J does not act on | g (cid:105)(cid:104) g | .After identifying all vanishing terms, we obtain:˙ v = L o v + P ( J + L I ) w (A.12)and ˙ w = QL I v + Q ( L o + L a + L I ) w. (A.13)Note that w describes the evolution of the fluctuations out of the subspace of interest.As the timescale of these fluctuations is set by δ and γ and we assume that either δ or γ is much larger than both ω m and κ , we can neglect the free evolution of the cavity ormotion during one of these fluctuations and approximate L o w ≈ w ( t ) = (cid:90) t dτ e Q ( L o + L a )( t − τ ) QL I w ( τ ) + (cid:90) t dτ e Q ( L o + L a )( t − τ ) QL I v ( τ ) (A.14) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity w (0) = 0 as the initial condition. Now we plug this equation twice intoequation (A.12) (iteratively) in order to catch a term of the order J L I :˙ v ( t ) = L o v + P ( J + L I ) (cid:90) t dτ e Q ( L o + L a )( t − τ ) QL I v ( τ )+ P ( J + L I ) (cid:90) t dτ e Q ( L o + L a )( t − τ ) QL I (cid:90) τ dτ (cid:48) e Q ( L o + L a )( t − τ (cid:48) ) QL I v ( τ (cid:48) ) . (A.15)Here we neglected the term proportional to w ( τ (cid:48) ) since it produces only terms ∝ L I orhigher. Again by following the path of how these super-operators act with figure A2, wecan quickly identify which terms vanish since all contributing terms need to have closedloops starting and ending in | g (cid:105)(cid:104) g | . So we are left with:˙ v ( t ) = L o v + P L I (cid:90) t dτ e ( L o + L a )( t − τ ) L I v ( τ )+ P J (cid:90) t dτ e ( L o + L a )( t − τ ) L I (cid:90) τ dτ (cid:48) e ( L o + L a )( t − τ (cid:48) ) L I v ( τ (cid:48) ) . (A.16)After extending the lower integral borders to −∞ (Markov approximation), we obtainequation (6) of the main text. Appendix A.2. Projecting out the cavity field
The next step is to find a master equation only containing motional degrees of freedom( p and x ) of the atom as operators. In order to find this equation we need to use theNakajima-Zwanzig technique to project out the cavity mode from equation (6). For thesake of simplicity we assume δ (cid:29) γ (and thus g δ + γ ≈ g δ ) and κ (cid:29) γ in the following, sowe can ignore the atomic decay channel for this derivation by approximating L om ≈ L κ .For weak driving, we can restrict ourselves to the photon subspace defined by | (cid:105) , | (cid:105) .Subsequently, we can adopt our projection operator formalism from above and writethe density operator as follows: ρ = | (cid:105)(cid:104) | ρ + | (cid:105)(cid:104) | ρ + | (cid:105)(cid:104) | ρ + | (cid:105)(cid:104) | ρ (A.17)with ρ ij = (cid:104) i | ρ | j (cid:105) being the reduced density matrix describing atomic motion. As weare interested in the subspace spanned by | (cid:105)(cid:104) | we define an projection operator P : P ρ = | (cid:105)(cid:104) | ρ (A.18)and Qρ = | (cid:105)(cid:104) | ρ + | (cid:105)(cid:104) | ρ + | (cid:105)(cid:104) | ρ . (A.19)We again decompose the total Liouvillian in parts according to the way they act: L = L m + L ca + L D + J (A.20) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure A3.
The Hilbert space of the single excitation subspace of the cavity. Thelabel of an arrow corresponds to a Liouvillian, while the direction of the arrow indicatesthe possible beginning and ending subspaces of the Liouvillian. For example, the redarrow indicates that the Liouvillian J acting on the subspace | (cid:105)(cid:104) | takes this subspaceto | (cid:105)(cid:104) | . As we assume κ (cid:29) ω m , we can neglect the time evolution due to the super-operator L m during a fluctuation out of P ρ . with L m defined in equation (A.5), L ca ≈ − i[ − ∆( x ) a † a, ρ ] − κ { a † a, ρ } (A.21)and L D ρ = − i √ κ r E [ a + a † , ρ ], which describes the interaction of the cavity mode withan external coherent laser drive. J ρ = κaρa † describes the spontaneous decay of thecavity mode. Now we draw in figure A3 a picture of the Hilbert space of the degreesof freedom of the cavity, including the arrows which illustrate how these defined super-operators act. A similar prodecure as in Appendix A.1 leads to the quantum masterequation (12) of the main text describing atomic motion. Appendix B. Single Photon Scattering Theory
Here we provide details of the derivation of equations (23) and (24) in the main text.Inserting equations (21) and (22) into equation (20) and multiplying with (cid:104) ( ω (cid:48) ) r / t , m | from the left gives us an equation for the S-matrix elements: S r / t , n ( ω L ) δ ( ω L − ω (cid:48) − nω m ) = (cid:104) ( ω (cid:48) ) r / t , n | S | ( ω L ) left , (cid:105) (B.1)where ω (cid:48) refers to the frequency of the reflected or transmitted photon. In the following,we will establish a connection between the S-matrix elements, and the standard input-output formalism of cavity QED [38]. Conveniently, this connection enables one tocalculate S-matrix elements based upon knowledge of the eigenvalues and eigenstates ofthe system Hamiltonian H eff . The input-output equation states that the output field ineach decay channel (reflection/transmission) is the sum of the input field and the fieldemitted by the scattering center. For example the input-output equation for photonreflection is given by a out ( t ) = a in ( t ) − i √ κ r a ( t ) (B.2) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity a in / out ( ω ) are connected to the input-output Heisenberg-Langevin operators a in / out ( t ) by a simple Fourier transform [37] a in / out ( ω ) = 1 √ π (cid:90) dte i ωt a in / out ( t ) . (B.3)Now we focus on the S-matrix for the process of photon reflection S r , n ( ω L ) δ ( ω L − ω (cid:48) − nω m ) = (cid:104) c , n | a out ( ω (cid:48) ) a † in ( ω L ) | c , (cid:105) (B.4)where we expressed the S-matrix in terms of scattering operators a † in ( ω L ) and a out ( ω (cid:48) )which create in- and out-going monochromatic scattering states [42]. Using the input-output equation, one can re-write a out in terms of the cavity field and input field, yielding S r , n ( ω L ) δ ( ω L − ω (cid:48) − nω m ) = δ ( ω L − ω (cid:48) ) δ n, − i √ κ r (cid:104) c , n | a ( ω (cid:48) ) a † in ( ω L ) | c , (cid:105) . (B.5)Now we replace the scattering operators with the Fourier transform of the correspondinginput-output operators. The matrix element (cid:104) c , n | a ( t (cid:48) ) a † in ( t L ) | c , (cid:105) vanishes for t L > t (cid:48) since [ a ( t (cid:48) ) , a † in ( t L )] = 0 for t L > t (cid:48) and (cid:104) c | a † in ( t L ) = 0. Thus, we introduce the timeordering operator T making sure that t (cid:48) > t L . Then we have (cid:104) c , n | T [ a ( t (cid:48) ) a † in ( t L )] | c , (cid:105) = − i √ κ r (cid:104) c , n | T [ a ( t (cid:48) ) a † ( t L )] | c , (cid:105) , (B.6)where we replaced a in ( t L ) with a ( t L ) using the input-output equation. The termcontaining the output operator vanishes as [ a ( t (cid:48) ) , a † out ( t L )] = 0 for t (cid:48) > t L (which isalready ensured by T) and (cid:104) c | a † out ( t L ) = 0. Finally, we arrive at S r , n ( ω L ) δ ( ω L − ω (cid:48) − nω m ) = δ ( ω L − ω (cid:48) ) δ n, − κ r τ n ( ω L ) (B.7)with τ n ( ω L ) = 12 π (cid:90) dt L dt (cid:48) e i( ω (cid:48) t (cid:48) − ω L t L ) (cid:104) c , n | T a ( t (cid:48) ) a † ( t L ) | c , (cid:105) . (B.8)For the S-matrix describing the process of photon transmission we obtain S t , n ( ω L ) δ ( ω L − ω (cid:48) − nω m ) = −√ κ r κ t τ n ( ω L ) (B.9)Note that the S-matrix of reflection S r includes the term δ ( ω L − ω (cid:48) ) δ n, describinginteraction-free reflection of photons. In contrast, in the S-matrix of transmission S t there is no such term, since the input field on the transmitting side of the cavity is inthe vacuum state and thus the transmitted field is built exclusively from the emissionof photons by the scattering center. We can write (cid:104) a ( t (cid:48) ) a † ( t L ) (cid:105) = Tr (cid:104) ae L s ( t (cid:48) − t L ) a † ρ (0) (cid:105) , (B.10)where ρ (0) = | c , (cid:105)(cid:104) c , | and L s ρ = − i[ H eff , ρ ] + κaρa † with H eff described byequation (19) from the main text. Since the term κaρa † reduces the number of photons, xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity a ( t ) is governed by H eff alone and for evaluating the S-matrix we caneffectively use a ( t ) = e i H eff t ae − i H eff t . (B.11)We further express (cid:104) c , n | T a ( t (cid:48) ) a † ( t L ) | c , (cid:105) = Θ( t L − t (cid:48) ) e i ω n nt L (cid:104) c , n | e − i H eff ( t L − t (cid:48) ) | c , (cid:105) (B.12)where e i ω n nt L counts the energy of the created phonons during the scattering processand the step function Θ( t L − t (cid:48) ) which vanishes for t L < t (cid:48) ensures time ordering. Inorder to express the S-matrix fully in terms of eigenvalues λ β and eigenstates | β (cid:105) of H eff with H eff | β (cid:105) = λ β | β (cid:105) we insert a unity operator 1 = (cid:80) β | β (cid:105)(cid:104) β | right before | c , (cid:105) .Therefore we write (cid:104) c , n | e − i H eff ( t L − t (cid:48) ) | c , (cid:105) = (cid:88) β (cid:104) c , n | β (cid:105) e − i λ β ( t L − t (cid:48) ) (cid:104) β | c , (cid:105) (B.13)where (cid:104) c , n | β (cid:105) is the projection of the eigenstates | β (cid:105) into the basis states (cid:104) c , n | . Afterevaluating the Fourier transform in equation (B.8) we are left with τ n ( ω L ) = − i δ ( ω L − ω (cid:48) − nω m ) (cid:88) β (cid:104) c , n | β (cid:105) λ β (cid:104) β | c , (cid:105) . (B.14)which together with equation (B.7) and (B.9) reproduces equation (23) and (24) in themain text. Appendix C. The full effective theory and its validity
Here we begin by generalizing our effective theory presented in the main text (Section 1and 2) by including spontaneous emission into the master equation (12) and the singlephoton scattering output state (25). Then we define the parameter space for which ourtheory is valid. We do this by comparing results of our effective theory with a numericalsimulation of the full Jaynes-Cummings model including motion (1) where the onlyassumption is the Lamb-Dicke regime η LD (cid:28) u ( x ). This approximation is only done for numerical purposes and we notethat our effective theory does not depend on the Lamb-Dicke parameter.For systems where κ (cid:29) γ is not true, we need to include the atomic decay channel.Doing so, the single photon scattering output state now generalizes to: | Ψ out (cid:105) = S r ( ω L , x )Ψ ( x ) | ( ω L ) r (cid:105) + S t ( ω L , x )Ψ ( x ) | ( ω L ) t (cid:105) + S at ( ω L , x )Ψ ( x ) | ( ω L ) at (cid:105) (C.1)where the scattering matrices for reflection, transmission and the scattering matrix forspontaneous emission are respectively given by: S r ( ω L , x ) = 1 − i κ r ∆ c ( x ) + i κ ( x )2 (C.2) xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity S t ( ω L , x ) = − i √ κ t κ r ∆ c ( x ) + i κ ( x )2 , (C.3) S at ( ω L , x ) = (cid:115) g δ + γ i √ γκ r ∆ c ( x ) + i κ ( x )2 u ( x ) e i k c x . (C.4)The scattering matrices conserve probability and obey | S r ( ω L , x ) | + | S t ( ω L , x ) | + | S at ( ω L , x ) | = 1 for all values of ω L and x . Note that we treat here for simplicityonly one direction of spontaneous emission which has a one dimensional decay channeldescribed by | ( ω L ) at (cid:105) . The resulting momentum kick qualitatively reproduces the maineffect that would occur in a full three-dimensional treatment of spontaneous emission.We also did not exclusively account for intrinsic cavity losses at a possible rate κ in ,however including this process would simply result in an additional term in the outputstate equation (C.1) with a corresponding S-matrix that looks like S t , but with κ t replaced by κ in . The total effective linewidth of the cavity is increased by the effectiverate of spontaneous emission κ ( x ) = κ r + κ t + γ g δ + γ u ( x ) , (C.5)which depends on the position of the atom. As explained in the main text, we canexpress the jump operators in terms of the scattering matrices such that they describeintuitive physical decay processes. The corresponding master equation describing acoherent drive is then given by:˙ ρ = − i( H e ρ − ρH † e ) + E ( S r ρS † r + S t ρS † t + S at ρS † at ) (C.6)with the Hamiltonian H e = ω m b † b − i2 E . (C.7)Note that by including spontaneous emission into the model the zero-point resolutionreads in good approximation r zp ≈ η LD g | δ | κ ( x )( δ + γ ) . (C.8)We have averaged the position dependent effective decay rate κ ( x ) ≈ (cid:104) Ψ | κ ( x ) | Ψ (cid:105) with the atomic wave function Ψ ( x ).In order to derive the single photon output state (C.1) and the master equation(C.6) we made two assumptions:(i) Large atom/laser detuning δ (cid:29) g , which allowed us to effectively eliminate theexcited state of the atom leading to an effective optomechanical master equation (6).Note that a large spontaneous emission rate γ (cid:29) g would allow this elimination aswell. However, here we are interested in strongly coupled systems, where g (cid:38) γ . xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity κ (cid:29) ω m which allowed us to derive the outputstate (C.1) and effectively eliminate the cavity mode in order to derive the masterequation (C.6).Now we will check the limits of these assumptions by numerically simulating a singlephoton scattering event with the full model (equation (1)). The numerical simulationis done by diagonalizing the Hamiltonian H D = ω m b † b − ( δ + i γ σ ee − ( δ c + i κ a † a + g ( u ( x ) + g η LD ( b † + b ))( a † σ ge + h.c. ) , (C.9)in the single-photon subspace and using the eigenvalues and eigenstates in the exactscattering matrices for reflection, transmittion and atomic decay constructed accordingto equation (23) and equation (24) which is described in Appendix B. One has to takecare that the unity operator as inserted in equation (B.13), is here = (cid:80) β | β (cid:105)(cid:104) β ∗ | ,with the eigenvectors normalized as (cid:104) β ∗ | β (cid:105) = 1, since the Hamiltonian H D is complexsymmetric due to losses rather than Hermitian. Appendix C.1. Limits of the assumption | δ | (cid:29) g We begin with the question of how large g | δ | can be, such that all approximationspreviously made are still valid. This is important to know, as the previously studiedregime of resolved zero point motion r zp (cid:29) r zp ∝ g om ∝∼ g | δ | . Thus, to reach this regime, it is beneficial to choose g | δ | as largeas possible. However, increasing this fraction, we will eventually leave the parameterspace in which our effective theory correctly predicts results. To understand when thishappens we will now compare our effective theory with a numerical simulation of thefull master equation (C.9) as a function of g / | δ | (and later as a function of ω m /κ forsimilar reasons). We will assume in the following that the atom is trapped in its motionalground state at a location with maximum intra cavity intensity slope k c x = π/ c ( x ) = 0, which implies x r = x . Figure C1(a) shows the probability ofphoton reflection p r (red), photon transmission (orange) and spontaneous emission p at (green) as a function of g / | δ | calculated with the effective theory: p r / t / at ( ω L ) = (cid:90) dx | S r / t / at ( ω L , x ) | | Ψ ( x ) | . (C.10)We use for | S r / t / at ( ω L , x ) | , equation (C.2), equation (C.3) and equation (C.4),respectively. We also use parameters from a recent fiber cavity experiment (AppendixD.2), where γ > κ and thus, one needs to account for spontaneous emission. Theblue dots correspond to the full numerical simulation of the Jaynes-Cummings modelincluding motion (equation (C.9)). We observe a great match for g / | δ | < / n r = (cid:104) Ψ r | b † b | Ψ r (cid:105) given areflected photon as a function of g / | δ | for the same parameters as a). Ψ r ( x ) is givenby equation (34) in the main text. We observe a great match for g / | δ | < xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure C1. Effective theory vs numerical simulation.
We assume the atom tobe initially in its motional ground state and that the incident photon is on resonancewith the atom-cavity system. a) Probability of photon reflection p r (red), photon transmission p t (orange) andspontaneous emission p at (green) as a function of g | δ | and calculated with the effectivetheory. Blue smaller dots: exact numerical simulation. Here, we have used parametersfrom a recente fiber cavity experiment with trapped Ca + -ions (see Appendix D.2,parameter set II). Here we choose κ t = 2 π × . κ r = 2 π × . η LD = (cid:112) ω rec /ω m = 0 . ω m = 2 π × . b) Conditional phonon expectation value ¯ n r given that a photon is reflected from thecavity for the same parameters as a). The effective theory (red) matches very wellwith the numerical simulation (blue). Appendix C.2. Limits of the assumption κ (cid:29) ω m Here we want to check the validity of the effective theory once sideband resolution isapproached. We plot the created phonon expectation value ¯ n r after reflecting a singlephoton in figure C2 as a function of ω m κ . Here, we take the vacuum Rabi splitting g = 2 π ×
10 GHz corresponding to a possible photonic crystal cavity (Appendix D.1),an atom-cavity detuning of ω − ω c = 100 g , and again we consider a resonant photonfor an atom trapped at k c x = π/
4. For illustrative purposes, we take an artificially lowvalue of κ = 2 π ×
20 MHz, which is distributed only between reflection and transmissionports (with κ r = 4 κ t ), and allow ω m to vary. We observe a reasonable match betweenthe exact numerical simulation and our effective model for ω/κ < / Appendix D. Experimental canditate systems for resolving zero-pointmotion
Appendix D.1. Photonic Crystal Cavities
The coupling of atoms to the mode of a photonic crystal cavity can be as large as g ∼ π ×
10 GHz [44] for Rubidium atoms. Rubidium atoms have a natural linewidthof γ ∼ π × ω rec ≈ π × . λ c ≈ Q ∼ are feasible inside photonic crystal nano-cavities [45], associated with xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure C2. Effective theory (blue) vs numerical simulation (red dots)approaching sideband resolution
We assume the atom to be initially in its motional ground state and that the incidentphoton is on resonance with the atom-cavity system. We plot the phonon expectationvalue ¯ n r after reflecting a photon as a function of ω m /κ . Parameters are chosen for anatom trapped inside a photonic crystal cavity as presented in Appendix D.1. We choosean atom-cavity detuning of ω − ω c = 100 g and an artificial value of κ = 2 π ×
20 MHz(with κ r = 4 κ t ) as we only want to check the validity of the effective theory oncesideband resolution is approached. a decay rate of rougly κ ∼ π × .
25 GHz. Since γ (cid:28) κ , spontaneous emission canbe ignored and experiments are very well described by the effective master equation(equation (12)) and the effective output state (equation (25)). The achievable zero-point resolution in photonic crystal cavities is r zp ∼
10 by taking η LD = 0 .
25 (calculatedwith equation (C.8)).
Appendix D.2. Fiber Cavities
Here we discuss a fiber cavity QED experiment with trapped Ca + -ions ( ω rec ≈ π × . γ = 2 π × . g and κ by changing the cavity length. Here we give twoexamples:(i) Parameter set I is given by: g = 2 π ×
41 MHz , κ = 2 π × g = 2 π ×
21 MHz and κ = 2 π × . r zp as a function of cavity-atom detuning ω − ω c for parameter set I (red) and set II (blue, dashed) calculated with equation (C.8).We choose ω m = 2 π × . k c x = π/ δ c is chosen in a way that the condition∆ c ( x r ) = 0 is satisfied, which implies x r = x . We observe that by choosing ω − ω c = 2 g one achieves r zp ≈ .
05 with parameters set I and r zp ≈ .
03 with parameter set II. Wealso demonstrate how to choose δ c in order to obtain k c x r = k c x = π/ δ c as a function of ω − ω c for parameter set I (red) and set II (blue, dashed). xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Figure D1. a)
Zero-point resolution r zp as a function of cavity-atom detuning ω − ω c for parameter set I (red) and set II (blue, dashed) of a tunable fiber cavityexperiment with trapped ions. For parameters see Appendix D.2. Here, we choose ω m = 2 π × . k c x = π/ δ c such that x r = x (see b)). b) Here we show how to choose δ c in order to ensure k c x r = k c x = π/
4. Plotted isthe cavity-laser detuning δ c as a function of ω − ω c for parameter set I (red) and setII (blue, dashed) satisfying the condition ∆ c ( x r ) = 0. Note that because the spontaneous emission rate γ is comparable to κ , the processof spontaneous emission cannot be neglected and the master equation (C.6) and singlephoton scattering output state (C.1) need to be applied in order to predict outcomes ofthis experiment. References [1] Aspelmeyer M, Kippenberg T J, and Marquardt F, 2014 Cavity Optomechanics.[2] Safavi-Naeini A H, Mayer Alegre T P, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T,Chang D E, and Painter O 2011 Electromagnetically induced transparency and slow light withoptomechanics
Nature (7341) 69.[3] Palomaki T A, Teufel J D, Simmonds R W, and Lehnert K W 2013 Entangling mechanical motionwith microwave fields
Science
Phys.Rev. A (3) 033804.[6] Marquardt F, Chen J P, Clerk A A, and Girvin S M 2007 Quantum theory of cavity-assistedsideband cooling of mechanical motion Phys. Rev. Lett. (9) 093902.[7] Rabl P. 2011 Photon Blockade Effect in Optomechanical Systems Phys. Rev. Lett
New J. Phys (5)053030.[9] Ojanen T, and Børkje K 2014 Ground-state cooling of mechanical motion in the unresolvedsideband regime by use of optomechanically induced transparency Phys. Rev. A (1) 013824.[10] Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E, and Kimble H J 2005 Photonblockade in an optical cavity with one trapped atom Nature (7047) 87. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity [11] Reiserer A, Kalb N, Rempe G, and Ritter S 2014 A quantum gate between a flying optical photonand a single trapped atom Nature (7495) 237.[12] Hacker B, Welte S, Rempe G, and Ritter S 2016 A photonphoton quantum gate based on a singleatom in an optical resonator
Nature (7615) 193.[13] Reiserer A, and Rempe G 2015 Cavity-based quantum networks with single atoms and opticalphotons
Rev. Mod. Phys. (4) 1379.[14] Volz J, Scheucher M, Junge C, and Rauschenbeutel 2014 Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom Nature Phot. (12) 965.[15] Shomroni I, Rosenblum S, Lovsky Y, Bechler O, Guendelman G and Dayan B 2014 All-opticalrouting of single photons by a one-atom switch controlled by a single photon Science (6199)903-6.[16] Guthhrlein G R, Keller M, Hayasaka K, Lange W, and Walther H A 2001 Single ion as a nanoscopicprobe of an optical field
Nature (6859) 49.[17] Mundt A B, Kreuter A, Becher C, Leibfried D, Eschner J, Schmidt-Kaler F, and Blatt R 2002Coupling a single atomic quantum bit to a high finesse optical cavity.
Phys. Rev. Lett. (10)103001.[18] Russo C, Barros H G, Stute A, Dubin F, Phillips E S, Monz T, Northup T E, Becher C, SalzburgerT, Ritsch H and Schmidt P O 2009 Raman spectroscopy of a single ion coupled to a high-finessecavity Appl. Phys. B (2) 205-12.[19] Leibrandt D R, Labaziewicz J, Vuleti´c V and Chuang I L 2009 Cavity sideband cooling of a singletrapped ion Phys. Rev. Lett. (10) 103001.[20] Sterk J D, Luo L, Manning T A, Maunz P and Monroe C 2012 Photon collection from a trappedion-cavity system
Phys. Rev. A (6) 062308.[21] Steiner M, Meyer H M, Deutsch C, Reichel J, and K¨ohl M 2013 Single ion coupled to an opticalfiber cavity Phys. Rev. Lett. (4) 043003.[22] Kimble H J 1998 Strong interactions of single atoms and photons in cavity QED
Phys. Scr. (T76) 127.[23] Maunz P, Puppe T, Schuster I, Syassen N, Pinkse P W, and Rempe G 2004 Cavity cooling of asingle atom
Nature (6978) 50.[24] Thompson J D, Tiecke T G, de Leon N P, Feist J, Akimov A V, Gullans M, Zibrov A S, Vuleti´c V,Lukin M D 2013 Coupling a single trapped atom to a nanoscale optical cavity
Science (6137)1202-5.[25] Aoki T, Dayan B, Wilcut E, Bowen W P, Parkins A S, Kippenberg T J, Vahala K J, and KimbleH J 2006 Observation of strong coupling between one atom and a monolithic microresonator
Nature (7112) 671.[26] Jaynes E T, and Cummings F W 1963 Comparison of quantum and semiclassical radiation theorieswith application to the beam maser
Proc. IEEE (1) 89-109.[27] Wilson-Rae I, Nooshi N, Dobrindt J, Kippenberg T J, and Zwerger W 2008 Cavity-assistedbackaction cooling of mechanical resonators New J. Phys. (9) 095007.[28] Domokos P, Horak P, and Ritsch H 2001 Semiclassical theory of cavity-assisted atom cooling J.Phys. B (2) 187.[29] Sch¨utz S, Habibian H, and Morigi G 2013 Cooling of atomic ensembles in optical cavities:Semiclassical limit Phys. Rev. A (3) 033427.[30] Nakajima S 1958 On quantum theory of transport phenomena, steady diffusion Prog. Theo. Phys. (6) 948-59.[31] Zwanzig R 1960 Ensemble method in the theory of irreversibility J. Chem. Phys. (5) 1338-41.[32] Horak P, Hechenblaikner G, Gheri K M, Stecher H, and Ritsch H 1997 Cavity-induced atom coolingin the strong coupling regime Phys. Rev. Lett. (25) 4974.[33] Domokos P, and Ritsch H 2003 Mechanical effects of light in optical resonators JOSA B (5)1098-1130.[34] Domokos P, and Ritsch H 2002 Collective cooling and self-organization of atoms in a cavity Phys. xploring unresolved sideband, optomechanical strong coupling using a single atom coupled to a cavity Rev. Lett. , (25) 253003.[35] Fern´andez-Vidal S, De Chiara G, Larson J, and Morigi G 2010 Quantum ground state of self-organized atomic crystals in optical resonators Phys. Rev. A (4) 043407.[36] Neumeier L, Quidant R, and Chang D E 2015 Self-induced back-action optical trapping innanophotonic systems New J. Phys. (12) 123008.[37] Fan S, Kocabas S E, Shen J T 2010 Input-output formalism for few-photon transport in one-dimensional nanophotonic waveguides coupled to a qubit Phys. Rev. A (6) 063821.[38] Caneva T, Manzoni M T, Shi T, Douglas J S, Cirac J I, and Chang D E 2015 Quantum dynamicsof propagating photons with strong interactions: a generalized input-output formalism New J.Phys. (11) 113001.[39] Galland C, Sangouard N, Piro N, Gisin N, and Kippenberg T J 2014 Heralded single-phononpreparation, storage, and readout in cavity optomechanics Phys. Rev. Lett. (14) 143602.[40] Wineland D J, and Itano W M 1979 Laser cooling of atoms
Phys. Rev. A (4) 1521.[41] Wolf S, Oliver S J, and Weiss D S 2000 Suppression of recoil heating by an optical lattice Phys.Rev. Lett. (20) 4249.[42] Taylor J R 2006 Scattering theory: the quantum theory of nonrelativistic collisions, CourierCorporation[43] Brandst¨atter B, McClung A, Sch¨uppert K, Casabone B, Friebe K, Stute A, Schmidt PO, DeutschC, Reichel J, Blatt R and Northup TE 2013 Integrated fiber-mirror ion trap for strong ion-cavitycoupling Rev. Sci. Inst. (12) 123104.[44] Thompson J D, Tiecke T G, de Leon N P, Feist J, Akimov A V, Gullans M, Zibrov A S, Vuleti´cV, and Lukin M D 2013 Coupling a single trapped atom to a nanoscale optical cavity Science (6137) 1202-5.[45] Asano T, Song B S, and Noda S 2006 Analysis of the experimental Q factors ( ∼ Opt. exp.14