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Protocol for Hybrid Entanglement Between a Trapped Atom and a Semiconductor Quantum Dot

We propose a quantum optical interface between an atomic and solid state system. We show that quantum states in a single trapped atom can be entangled with the states of a semiconductor quantum dot through their common interaction with a classical laser field. The interference and detection of the resulting scattered photons can then herald the entanglement of the disparate atomic and solid-state quantum bits. We develop a protocol that can succeed despite a significant mismatch in the radiative characteristics of the two matter-based qubits. We study in detail a particular case of this interface applied to a single trapped \Yb ion and a cavity-coupled InGaAs semiconductor quantum dot. Entanglement fidelity and success rates are found to be robust to a broad range of experimental nonideal effects such as dispersion mismatch, atom recoil, and multi-photon scattering. We conclude that it should be possible to produce highly entangled states of these complementary qubit systems under realistic experimental conditions.


I. INTRODUCTION
Quantum entanglement, long considered to be the most puzzling aspect of quantum mechanics [1,2], is now realized to be a potential resource for enhanced processing and communication of information. The field of quantum information science exploits quantum entanglement for tasks that are otherwise impossible or inefficient using conventional information processing approaches [3]. Recent advances in the control of physical systems, ranging from isolated atoms and photons to individual degrees of freedom in condensed matter, have shown great promise in the development of quantum information hardware [4].
The majority of work to date has been centered around entanglement of identical quantum systems such as identical atoms and photons. There is great interest in extending entanglement over disparate quantum systems. Such hybrid entanglement can exploit advantages of each individual system to enhance capabilities of quantum technology. An important example is the hybrid entanglement between matter and photonic quantum systems. This type of entanglement, which has been demonstrated using both trapped ions [5] and atomic ensembles [6], enables one to combine the advantages of the long coherence times of atomic systems with the ability of photonic systems to transport quantum information. Another important example is the recent entanglement of two different species of trapped atomic ions, where one species (Al + ) has excellent coherence properties and the other (Be + ) allows efficient qubit measurement [7].
Here, we theoretically investigate the possibility of creating hybrid entanglement between semiconductor and atomic quantum systems. Specifically, we propose a protocol for entangling a quantum dot (QD) in a microcavity with a trapped atom through a common photonic interaction. Such hybrid entanglement is expected to stimulate new concepts for distributed quantum computation that exploit the long coherence times of trapped ions with the fast dynamics and strong atom-photon interactions of a QD. We show that a common photonic channel can link these disparate systems to achieve hybrid-matter quantum entanglement despite significant mismatches between atomic and semiconductor qubits.
Photon mediated entanglement has been proposed for entangling like systems such as atoms [6,8], NV centers in diamond [9], and cavity-coupled QDs [10,11]. It has been experimentally demonstrated between electrostatically trapped ions [12] and atomic ensembles [13][14][15]. To extend such ideas to entangle different matter qubits requires a protocol that is robust to significant mismatch in spectral and temporal properties. In this paper we propose such a protocol in which the atomic system is coupled to a free-space field by elastic scattering, while the QD is coupled via cavity interaction. We extensively investigate a particularly promising implementation of this protocol in which an electrostatically trapped 171 Yb + ion is entangled with a cavity coupled indium arsenide (InAs) QD. We address the central challenges in realizing this hybrid quantum link, including the mismatch in the optical spectra of these systems and decoherence processes that are particular to each node of the hybrid circuit. In particular, we analyze in detail "which-path" decoherence due to atom recoil and multi-photon scattering and derive an analytical expression for entanglement fidelity and entanglement rate.
In section II, we give a low-level description of the proposed entanglement protocol.
Section III provides detailed calculations of the atom scattering amplitudes, and analyzes the effect of dispersion. Section IV considers atom motion and recoil, and calculates the fidelity under realistic experimental conditions. In section V we consider the effects of leakage of which-path information due to multi-photon scattering. The final section investigates the validity of the weak excitation approximation which is extensively used to calculate both atomic and scattering and reflection from the cavity-QD system. Fig. 1a illustrates the proposed method for entangling a QD with an atom. The QD is coupled to a resonant microcavity while the atom is electromagnetically confined at a remote location. Each system consists of a qubit represented by the two states, |0 and |1 , where state |0 is coupled to a third excited state |e via an optical transition. We consider the case where this transition in the QD and atom occur at similar (not necessarily identical) optical wavelengths. The entanglement method we consider is general, and can apply to a broad range of atomic and semiconductor systems. However, to perform calculations we specialize to the case of an InGaAs QD that is entangled with an electrostatically trapped 171 Yb + ion.

II. BASIC PROTOCOL
The assumed level structure for these two qubit systems is shown in Fig. 1b. We focus on 171 Yb + because of its optical 2 D 3/2 to 3 [3/2] 1/2 transition, which occurs at 935 nm and is therefore compatible with the near infrared transition wavelength of InGaAs QDs. For the QD, the qubit states can reside in the Zeeman split spin states of a single charge carrier, with shown in the inset, and a common pump beam excites both systems. The reflected light from the cavity is mixed with the scattered light from the atom on a 50/50 beamsplitter. Path lengths are set for constructive interference at detector 1. A detection event at detector 2 places the two systems in an entangled state. (b) Specific level structure for the QD and atom. The spin states of a singly charged QD can serve to create the desired qubit states, while the positive or negative trion state (|t ± )serves to couple the qubit to the cavity. The two qubit states are split in energy by ∆ B due to an applied magnetic field. For the atom we consider the specific example of a 171 Yb + ion and use the hyperfine |1, 1 and |2, 2 states (with hyperfine splitting ∆ A = 0.86 GHz) in the 2 D 3/2 manifold to store quantum information, while optical transition to the |1, 1 state in the 3 [3/2] 1/2 manifold results in elastic scattering of the pump field. Selection rules ensure that scattering will occur only when the atom is in state |2, 1 .
an optical coupling to an excited trion state, as recently demonstrated [16,17]. The loading of a single spin into a microcavity coupled QD has also been recently achieved [18]. For the atom, the qubit states are represented by long-lived electronic/nuclear hyperfine states, with an optical coupling to a dipole-allowed excited electronic state [19]. We consider the hyperfine qubit states residing in the metastable 2 D 3/2 level, with a coupling to the excited 3 [3/2] 1/2 level at 935 nm [20].
Entanglement is established by first initializing the state of each of the two qubits to superposition state |ψ i = (|0 + |1 )/ √ 2. The QD system can be initialized using experimentally demonstrated single [21] and two-laser coherent control techniques [22,23], while the atom can be controlled using optical pumping and microwave or stimulated Raman transitions [19,20]. Following initialization, a laser pulse is coherently split into two components directed to the two quantum systems. One of the components is reflected off of the cavity containing a QD while the second component drives the atom off resonance, resulting in an elastically scattered field that is phase coherent with the input field. Phase coherence of the elastically scattered component has been experimentally verified through Young's interference [24], and has also been theoretically investigated [25]. We define |α as the coherent field reflected from the cavity, and |β as the field elastically scattered from the atom. The two fields are combined on a beamsplitter whose path lengths are set for constructive interference at detector 1 (shown in Fig. 1a), requiring optical interferometric stability over the system. A detection event at detector 2 will collapse the state of the atom and QD into an entangled state.
To understand the entanglement formation process, we first assume that both fields are quasi monochromatic, phase coherent, and sufficiently weak that they can be expanded to first order in photon number. In later sections we will derive the entanglement fidelity under more realistic experimental conditions. We define ω ↑ and ω ↓ as the trion resonant frequencies for the two different spin states of the charged QD, which are detuned by ∆ B due to an applied external magnetic field. The cavity resonant frequency is defined as ω c , and input field frequency as ω. The reflection coefficient r and transmission coefficient t of the cavity are [10,26,27] where δ qd = ω − ω ↑,↓ is the detuning between the cavity and the QD (which depends on the spin state of the QD), and ∆ = (ω − ω c )/κ is the laser-cavity detuning scaled to the cavity linewidth κ. The QD exciton decay rate is given by γ qd , C = 4g 2 /(γ qd κ) is the QD-cavity cooperativity, g is that QD-cavity coupling strength, and L(δ, γ) = γ/(γ −iδ) is a Lorentzian profile [10]. The amplitude of the reflected field is given by α = α in r(ω), where α in is the incident field amplitude.
The incident field is set to be resonant with the cavity (ω = ω c ) so that ∆ = 0. We also set ω ↑ = ω c so that the QD is resonantly coupled to the cavity mode when it is in state |0 = | ↑ . In this case we have δ qd = 0, and in the limit of large atomic cooperativity |C| 1, the cavity reflectivity approaches r(ω c ) = 1. If the QD is instead in state |1 = | ↓ there is little coupling between the QD and cavity either due to selection rules (g = 0) or large detuning (δ qd g 2 /κ). Thus, CL(δ qd , γ qd ) → 0 and t(ω c ) → 1, so all of the light is transmitted through the cavity. Therefore, the qubit state of the QD will switch the cavity from being highly reflecting to highly transmitting. This operation can be viewed as a controlled-NOT gate between the state of the QD and the propagation direction of the scattered light. Controlled reflectivity of a cavity via a single QD has been reported in several works [18,28,29].
For the atom, we assume that the driving field is detuned from the resonant transition frequency ω a by δ a = ω−ω a . If the atom is in state |0 , it will induce off-axis elastic scattering via the near resonant optical transition to state |e . The scattered field is collected by an objective lens and coupled into a single mode fiber. We define β as the coherent state amplitude of the field scattered into the fiber. In contrast, when the atom is in state |1 it will not scatter any light since it cannot make an optical transition due to selection rules. Under the assumption that the reflected field from the cavity and scattered field from the atom are sufficiently small so that they can be expanded to first order in photon number, a detection event at detector 2 will collapses the state of the QD and atom onto where D 2 = (|α − β| 2 + |α| 2 + |β| 2 ). By tuning the phase and amplitude of the input fields so that α = β, the above state becomes a maximally entangled Bell state |ψ f = It should be noted that, so long as the quasi-monochromatic limit is valid, a perfect entangled state can be generated even when the QD and atom have different resonant frequencies and decay rates.

III. FIDELITY UNDER PULSED EXCITATION
The ideal protocol described in section II considers the quasi-monochromatic limit, where the input pulses are sufficiently long in time to be considered as single frequency. In a real experiment, the entanglement operation must be completed before the QD and atom have had time to decohere, which requires excitation with short optical pulses. The coherence time of the atomic hyperfine states is long (53 ms), but the QD spin coherence time is much shorter (10 ns) due to hyperfine interactions with nuclear spin [30]. Recent progress in optical locking of nuclear spin polarization suggests that much longer lifetimes in the microsecond regime may be possible [31]. Nevertheless, it is important to consider the effect of short pulses (0.1 − 10 ns) on the fidelity of the generated entangled state.
Because we consider the weak excitation regime, the cavity-QD system and trapped atom are linear scatterers. Therefore, we may analyze the effect of time varying excitation by fourier decomposing the time dependent input fields and looking at the scattering behavior of each fourier component independently. Such approach would not be valid in the strong excitation limit where nonlinear scattering occurs due to absorption saturation. We defer that discussion to section VI.
In order to analyze the spectral properties of the field scattered by the atom, we need a more precise expression for its scattering amplitude. To derive this amplitude, we begin with the standard system Hamiltonian for the atom given by In the above equationb k is a bosonic operator for a free space mode obtained using box boundary conditions, ω a is the resonant frequency of the atom, σ z is the population difference operator for atom scattering transition, and σ − is the transition lowering operator. The scattered modes are coupled to the atomic dipole by the coupling strength g k = −d · k ω k / 0 V where V is the quantization volume and d = dẑ is the atomic dipole moment.
We assume that the input field is sufficiently bright so that it may be treated as a set of classical Rabi frequencies Ω(ω). The input field is assumed to be largely collimated and propagating along the x-axis so k ω =xω/c. The potential V trap is the harmonic trapping potential of the atom, while r is the location of the atom in the trap.
We first consider the stationary atom limit, where r is a real vector. Atomic recoil, which requires us to account for the quantized operator nature of r, will be addressed in the next section. When the atom is in qubit state |0 the Heisenberg equations of motion for the atomic and cavity field operators are given by where we have added the decay term γ a which accounts for the total dipole decay, including dephasing. In Eq. 5 we assume that the scattered field is very weak as compared to the strong classical field Ω, so its driving term is ignored in the equation. We also make the weak excitation approximation that the atom remains mostly in the ground state, so σ z → −1 [10].
Eq. 5 can be solved by direct integration to give where δ a = ω − ω a and, as before, L(ω, γ) = γ/(iω + γ). In deriving the above equation we consider only the forced response of the atom driven by the classical Rabi frequency, and ignore the transient response due to turn-on of the input field. Plugging into Eq. 5 and integrating, we subsequently obtain The classical pump field that drives the atom is derived from a laser source and is therefore a coherent state. The atom is a linear scatterer so the scattered field must also be a coherent state whose quantum state is fully characterized by the scattered amplitude In Appendix A we show that the total number of scattered photons N s = k |β k | 2 is given by where ω 0 is the center frequency of the incident field, n i is the input photon flux density given by n i = I/ ω 0 where I is the input field intensity, γ r is the radiative decay rate of the atom to state |0 , and σ 0 = 3λ 2 /2π is the atomic cross section.

Collection Region
where s k are a set of complex scattering coefficients representing a unitary transformation.
To proceed, we need a model for the collection optics. Here we consider a simple model, Our calculations focus on the paraxial limit, where the collected solid angle ∆ o is small, so that the collected light propagates nearly parallel to the input beam. We focus on this range of collection angles because it is known to minimize the effect of atomic recoil [24,25], as we will analyze in more detail in the next section. Using the simple model for the collection optics, we shown in Appendix B that in the paraxial limit with In the above equation the integral is taken over the collection area, while is the area that the collection region occupies on the unit sphere and L x is the length of the quantization volume in the x direction. For simplicity we assume that ω k can be replaced by its average value ω 0 and k 0 = ω 0 /c. We do not make this substitution in the Lorentzian function, however, because near resonance this function will vary rapidly even over a narrow bandwidth of interest.
In the monochromatic limit, we could always achieve the matching condition α = β by changing the amplitude and phase of the incident fields. But under pulsed excitation, the reflection coefficient of the QD-cavity system has a spectral profile described by Eq. 2, while the atom scattering amplitude follows primarily a Lorentzian profile. Thus, each spectral component will require a different matching condition. Since we cannot satisfy the matching condition perfectly for each frequency we expect that the generated state will no longer be perfectly entangled. It has previously been shown that the spectral width over which high reflection is achieved when a QD is coupled to a cavity is given by the inverse modified spontaneous emission lifetime of the cavity enhanced QD transition [10], typically 10-50 GHz. This bandwidth represents a response time which is much faster than any decay rates of the atom. Thus, we expect the fidelity of the generated entangled state to be dominated by the dispersive properties of the atom, which are much narrower in frequency the those of the QD.
To calculate the fidelity of the generated entangled state under pulsed excitation, we begin with the state of the system after the field from the QD-cavity system is mixed with the scattered field from the atom on the beamsplitter, as shown in Fig. 1. We once again assume that the scattered fields are sufficiently weak so that they may be expanded to first order in photon number. After the beamsplitter the state of the fields, atom and QD can be written as In the above state the operatorv ω is the bosonic operator for a photon in the mode detected by detector 2, and state |f is an un-normalized state representing the remaining components of the wavefunction that do not contain any photons in the detection mode. From the expressions for the reflection coefficients we have where α 0 and β 0 are complex amplitudes that can be adjusted by selecting the amplitude and phase of the input pulse. In the ideal case we would have α(ω) = β(ω) for all frequencies, which would reproduce the ideal fidelity of the monochromatic limit. Unfortunately, dispersion prohibits us from achieving this for all values of ω. The best we can do is adjust the amplitudes so that α(ω 0 ) = β(ω 0 ), where ω 0 is the center frequency of the pulse.
The fidelity of the entangled state can be determined by first calculating the reduced density matrix of the QD-atom system conditioned on a detection event at detector 2. This reduced density matrix is given by where P is a projector onto the subspace containing at least one photon in the detection mode for detector 2. The fidelity can be calculated using F = ψ − |ρ|ψ − , where |ψ − is the ideal spin singlet entangled state. The expression for fidelity is given by Larger δ a enables shorter pulses before fidelity drops.
In the long pulse limit, the fidelity approaches its ideal value all atomic detunings. However, for each value of atomic detuning the fidelity makes a rapid transition from F = 1 to F = 0.25 at some critical pulse duration. Thus, the range of pulses for which the monochromatic approximation is valid will depend on δ a . At detuning of δ a = 0.1 GHz, the transition occurs roughly at 10 ns pulse duration, which is on the order of the coherence time of the QD. By increasing the detuning to 1 GHz, the fidelity transitions at 1 ns pulse duration, well below the QD decoherence time. By further increasing the detuning to 10 GHz it is possible to use 100 ps pulses. Thus, by increasing the atomic detuning we can use shorter pulses to achieve high fidelity. This tradeoff occurs because the dispersion, dominated by a Lorentzian function, is maximum near resonance and tails off with larger detuning.
The disadvantage of going to larger detuning is that we need to use more pump power line plots pump intensity required to achieve 0.1 scattered photons in the pulse. As ∆ω a increases, we can use shorter pulses but require higher excitation energies to achieve the same scattering rate.
If the coherence time of the QD is 10 ns, the shaded region represents the parameter regime where entanglement can be achieved.
in order to achieve the same scattering rate. The number of scattered photons is given by Eq. 9. From this equation one can see that as δ a increases the Lorentzian decays in amplitude, forcing us to increase the incident photon density n i to attain the same number of scattered photons. The tradeoff between pulse duration and pump intensity is shown in One can see from the figure that pump intensities as low as 2 W/cm 2 would be sufficient to enable the use of 10 ns pulses with the specified scattering rate.

IV. DECOHERENCE DUE TO ATOM RECOIL
Entanglement between the atom and QD relies on the assumption one cannot distinguish whether a photon was reflected from the cavity or scattered from the atom, even in principle.
Atomic recoil can serve to betray this "which-path" information. When the atom scatters a photon there is a probability that it will recoil, leaving residual kinetic energy in the motional degrees of freedom of the center-of-mass wavefunction. To achieve high fidelity we require that this recoil probability is small.
Atomic recoil is already present in the expression for the scattered field amplitude given in Eq. 11 and Eq. 12. Previously we assumed that the position r of the atom was fixed, and therefore the integral term in the expression was simply a complex constant. To include the effect of recoil, we must keep r as the position operator and trace it out over the motional degrees of freedom of the atom. We define the initial state of the atom-QD system by the density matrix where |ψ ni = 1 2 (|00 + |11 + |10 + |01 ) |n |vac (19) In the above equations, |n is the harmonic oscillator eigenstate for the center of mass motion of the atom, |vac is the vacuum field for all optical modes, and p(n) is assumed to be a thermal distribution. After interaction with the two input fields, the state in Eq. 19 is transformed into with The state |v represents a single photon in the detection mode that is monitored by detector 2, while |f n is once again a wavefunction orthogonal to |v representing the state of the QD, atom, and all fields when there are no photons in modev. Conditioned on a detection event at detector 2, the final density matrix of the system is given by where Tr n represents a trace over all degrees of freedom except for the qubit states of the atom and QD.
The fidelity F of the final state can be defined as the overlap between the actual state of the system and the desired state |ψ − = (|01 − |10 )/ √ 2. Thus, Using the definition of ρ f in Eq. 22 we attain the following expression for the fidelity: where β = n p(n) n|β|n (25) The expressions in Eq. 26 and Eq. 27 can be evaluated under the assumption that the ion occupies a thermal distribution. In the case, we can then write [24,33,34] e i(k−k )·r = e −η 2 (1−cosθ cos φ)(n+1/2) (28) wheren is the average excitation number of the atom in the harmonic potential, and η = k /2m atom ω t with m atom being the mass of the atom and ω t the trap frequency. The dimensionless constant η is called the Lamb-Dicke parameter and determines the extent to which an atom will recoil. Since we are primarily interested in small anglesθ and φ, we expand the cosine terms in the exponent to second order in these angles. Using this approximation, we show in Appendix C that The above expression is derived in the limit that ∆ i → 0 and ∆ o = ∆. The matching condition α = β cannot be satisfied because β now fluctuates due to recoil. The fidelity is maximizes when α = − |β| 2 which indicates that α is set to the average amplitude of β and has the same phase. Under this optimal condition, the fidelity is given by This expression allows us to calculate the fidelity of the entangled state in the monochromatic limit as a function of the atom temperature (n) and the collection angle.

V. PHOTONIC LEAKAGE OF WHICH-PATH INFORMATION
The tradeoff between fidelity and collection angle implies that there will be a tradeoff between fidelity and entanglement success probability, because larger collection angle results in higher collection efficiency of scattered photons. The entanglement success probability is given by The factor of 4 in the exponent of the above equation accounts for the fact that each photon has a 25% probability of triggering detector 2.
To investigate the entanglement success probability we need to determine how large we can make N s . The expression for fidelity in Eq. 31 does not depend on N s because it was derived under the assumption that the scattered field is sufficiently weak so that it could be expanded to first order in photon number. If we make N s too large this assumption will no longer be valid. Thus, we need to derive a more precise expression for the fidelity that accounts for both recoil and the probability that more than one photon is scattered by the atom.
If the atom scatters more than one photon, this can lead to leakage of "which-path" information. This information leakage can be understood from the following simple argument. First, we note that in general the collection efficiency of scattered light is small. Even with a collected solid angle of 45 o , which corresponds to a numerical aperture of 1, only 20% of the photons are collected into the fiber. Reflection and absorption losses from the optics will serve to degrade this collection efficiency even more. Thus, if the atom scatters two photons it is much more likely that only one of the photons is collected than it is that both photons are collected. The photon which fails to be collected is never mixed on the beamsplitter, and therefore retains the information that the atom caused a scattering event causing the entanglement to decohere. Thus, in order to achieve high fidelity entanglement the probability of scattering two photons must remain low.
To analyze multi-photon scattering we retain the amplitude of the field scattered by the atom to all orders in photon number, but expand the field collected into the fiber to first order in photon number. This approximation is valid because, as noted previously, fiber collection efficiency is small in the limit we consider so even if the atom scatters many photons the probability that more than one of them is collected into the fiber is still small.
After the atom scatters the input field the state of the system becomes In the above equation the state |00 |χ denotes the atom and the QD are both in qubit state |0 , and χ = − N s (1 − 3∆ 2 /8) is the coherent state amplitude of the uncollected field.
Because we consider only small collection efficient χ ≈ − √ N s . Other states are defined analogously, and state |n is once again the harmonic oscillator state of the atom center of mass. Following the same procedure as in section IV, the fidelity can be obtained by tracing over both the field and atom center of mass motion to obtain The above equation gives the entanglement fidelity due to both recoil and multi-photon scattering in the monochromatic limit. One can see the interference term is degraded by e −Ns/2 , which means that to achieve high fidelity we need N s 1. The number of photons collected into the fiber will be subsequently much smaller, which highlights the need for efficient collection of photons.
For a fixed fidelity F , Eq. 34 gives a relationship between the scattered photon number N s and the collected solid angle ∆. Thus, the entanglement success probability given in Eq. 32 becomes a function only of ∆. Fig. 6 plot P success as a function of ∆ for several different trap temperatures with the same atom trap parameters used in Sec. IV, where the entanglement fidelity is held fixed at F = 0.9. For all trap temperatures other thann = 0, an optimal rate exists for a specific collection angle ∆. This optimal rate is determined by a balance between recoil and photon collection efficiency. Forn = 0 the atom is in the Lamb-Dicke regime where recoil is not a factor, so the entanglement rate is limited only by the fraction of light that can be collected by the optics. Additional losses to reflection from optics and imperfect fiber coupling would serve to reduce this number.

VI. VALIDITY OF WEAK EXCITATION LIMIT
In addition to multi-photon scattering, we must also consider the validity of the weak excitation approximation. All of our calculations so far assumed that the atom and cavity-QD system are driven with sufficiently weak excitation such that σ z → −1, where σ z is the population inversion operator. For the QD, it has been previously shown that the weak excitation limit is valid so long as N ref /τ p 1/τ mod , where N ref is the number of photons reflected from the cavity, τ p is the input pulse duration, and τ mod is the modified spontaneous emission lifetime of the QD [11]. For any N ref one can in principle satisfy weak excitation by make τ p sufficiently long. It should also be noted that from the matching condition α = β and the condition that N s 1 (due to multi-photon scattering) we know that N ref 1. In addition, for the input pulse spectrum to fit within the high-reflection window given in Eq. 2, τ p > τ mod [10]. When these two conditions are combined they automatically guarantee that weak excitation is satisfied for the QD. Thus, for the QD weak excitation does not impose any additional constraints on the entanglement success probability and can be generally satisfied by using sufficiently long input pulses.
where Ω(r, t) = dE(r, t)/ . The weak excitation approximation is valid so long as σ + σ − 1. This condition can be recast into a more recognizable form. The easiest way to do this is to assume that the input optical pulse is a square pulse of duration τ starting at t = 0, with electric field amplitude E.
where the last equality comes from Poynting's theorem. Eq. 35 becomes We attain a condition for the atom which states that the rate of scattered photons should be small compared to the atom decay rate. Although this condition was attained using the assumption of a square pulse, we expect this relation to hold for most pulse shapes.
As was the case for the QD, the limitations imposed by weak excitation restrict only the rate of emitted photons, not on total photon number. In this way, weak excitation provides a weaker restriction than multi-photon scattering, and can in general be well satisfied by picking sufficiently long pulses. In a practical experiment the clock cycle for generating entanglement will almost always be long compared to the atom decay rate. This ensures that all transients have decayed between consecutive cycles of the experiment. If we combine this requirement with the restriction placed by multi-photon scattering that N s 1, then Eq. 37 is automatically satisfied. Thus, at the point where nonlinearities become important the system will already have decohered due to multi-photon scattering.

VII. CONCLUSION
We conclude that it appears feasible to entangle a QD and an atom by weakly scattering light from each system and interfering these fields to produce an appropriate single-photon detection event that heralds the entanglement. Differential dispersion, atom recoil, and the multi-photon scattering can all be managed by properly selecting the input pulse duration, collected solid angle, and input pump power. It is noted that to implement the proposed protocol it is necessary to overcome difficult experimental challenges. For example, the protocol requires phase-locking of all optical pulses for qubit rotation of both atom and QD, which will require pulling all optical pulses from a common laser source or using multiple phase-locked lasers, adding to the experimental difficulty. In addition, decay of the atomic 171 Yb + to other transitions will require periodic re-pumping into the 2 D 3/2 manifold. Nevertheless, the work we present suggests that entanglement between an atomic and semiconductor system is within the reach of presently available technological capabilities.
The amount of energy in the pump beam can be determines using Poynting's theorem: where A is the cross sectional area of the pump beam. Using the definition we attain where N i is the total number of photons in the pump and ω 0 is the center frequency of the quasi-monochromatic pump beam. Defining the photon density n i = N i /A we then have Plugging this expression back into Eq. A3, and using leads directly to Eq. 9.

APPENDIX B: FIELD AMPLITUDE COLLECTED INTO OPTICAL FIBER
For each frequency ω we define a fiber mode b ω . The collection lens and fiber are linear optical components, which means that the fiber mode is related to the free space modes by the linear transformation The sum is carried out over all free space modes that have the same energy as the fiber mode due to linearity. Unitarity requires that k=ω/c |s k | 2 = 1 (B2) We define ∆θ as the angle between the k and the input field propagation direction which is assumed to be along the x axis. We adopt a simplified model that the collection optics collect all k-vectors satisfying ∆ i < ∆θ < ∆ o . Thus, for k-vectors satisfying this condition s k = C, where C is a constant, while s k = 0 for all other k-vectors.
The constant C must be determined from the condition Eq. B2, which results in k=ω/c |C| 2 = 1 (B3) To evaluate the above sum we make the additional assumption that the lens collects k vectors propagating very close to the x-axis. In this paraxial wave limit, the sum can be converted to an integral as k=ω/c |C| 2 = |C| 2 L y L z (2πc) 2 dθ dφω 2 sin θ (B4) where L y and L z are the length of the bounding box from the quantization of the free space modes in the y and z directions, and angles θ and φ are defined in Fig. 2. From the above equation we attain where A = 2π(cos ∆ i − cos ∆ o ) ≈ π(∆ 2 o − ∆ 2 i ). Using this expression we then have whereb k is given in Eq. 7. We again turn the sum into an integral and perform some algebraic manipulation to attain where angles θ and φ are illustrated in Fig. 2. We assume that the input pulse has a narrow bandwidth centered around ω 0 so that we may make the substitutions ω 3/2 ≈ ω 3/2 0 and exp(k ω − k) ≈ exp[k 0 (x −k)]. We do not, however, make this approximation for L(δ a , γ a ) which is a rapidly varying function of ω near resonance. With these approximations we attain the result stated in Eq. 11 and Eq. 12.

APPENDIX C: CALCULATION OF AVERAGE SCATTERING AMPLITUDE DUE TO ATOMIC RECOIL
We assume that the collection optics are frequency independent over the bandwidth of the collected signal. We assume quasi-monochromatic input so that L(δ a , γ a ) = γ a /[γ a − i(ω a − ω 0 )]. We can construct a fiber mode of the form where χ is a normalization constant. This is the only mode that the collected field will couple to. To understand why, we first note that we can construct a complete basis using the above mode along with a set of other orthogonal modes that can be calculated using Schmidt decomposition. We can then calculate the field amplitude using Since β ω is proportional to the complex conjugate of the expansion coefficient χE * (ω)e iωt , it will have a maximum overlap with the mode in Eq. C1, and will be orthogonal to all other modes, ensuring the mode in Eq. C1 is the only one that contains photons.