Exact calculation of stimulated emission driven by pulsed light

Stimulated emission can be defined as the process when an incoming photon stimulates an additional quantum of energy from an atom into the same electromagnetic mode as the impinging photon. Hence, the two outgoing photons are identical. In a waveguide or free space, this intuition is typically found through Fermi's Golden rule, however, this does not properly account for the wave-like nature of the photons. Here, I present an exact solution to stimulated emission from a quantum two-level atom that properly accounts for the incoming and outgoing electromagnetic modes. This result is valid over a huge range of incident photon numbers. For a single incident photon, it shows how the photon must properly mode match the two-level system to cause stimulated emission. For a Fock state drive with large photon number, the exact solution shows how a two-level system Rabi oscillates with the traveling Fock mode as it passes by. I additionally use the same formalism to show that stimulated emission by a coherent pulse cannot be understood as an additional photon being emitted into the incident coherent state because the two-level system's state factorizes with the electromagnetic field's coherent state. Recent advances in superconducting circuits make them an ideal platform to test these predictions.

I present an exact solution to stimulated emission from a quantum two-level system based on quantum stochastic calculus, including for very large photon number. I use this expression to explore the intuitive limits of high driving photon number and short driving pulses. For a Fock state drive, this limit reduces to imagining the two-level system coherently exchanging energy with a cavity mode for a length of time equivalent to the pulse width. Notably, the limiting case for drive by a coherent state does not reduce to a two-level system briefly interacting with a cavity mode because the two-level system's state factorizes with the waveguide's coherent state.
Stimulated emission is one of the most fundamental effects in quantum optics, and is behind one of the most important inventions of the 20th century, the laser. In a laser, discrete electromagnetic modes representing a cavity interact with a gain medium consisting of few-level atoms. The atoms are incoherently pumped to their excited states and give their energy to the coherent field in the cavity modes via stimulated emission. This process has been rigorously studied and understood mathematically using non-perturbative density matrix theories [1].
In free space, structured materials, or waveguides, however, the dynamics of stimulated emission are less well understood. The most common viewpoint is that an incoming photon will stimulate emission from a two-level system prepared in its excited state, and cause the photon to be emitted into the same electromagnetic mode as the stimulating photon. To show this, most methods rely on perturbative approaches which have limited explanatory power [2]. In contrast, non-perturbative calculations have recently shown that a very specific incident photon shape is required to efficiently stimulate emission [3]. Many approaches have now shown this type of result theoretically, however, they are computationally intractable when more than a couple of photons stimulate the emission [4][5][6][7]. Hence, they are incapable of recovering all experimental intuitions about stimulated emission. Further, as nanophotonics [8] brings realizations of these thought experiments closer to reality [9][10][11], a better understanding of stimulated emission generally may be desired.
Here, I will develop a non-perturbative approach based on quantum stochastic calculus to build a fully accurate account of stimulated emission. The most basic model is built around a two-level atom coupled to a bath of electromagnetic modes (schematic as Fig. 1), which could represent at its simplest a nanophotonic waveguide with single transverse spatial profile [12]. I briefly note this causes no loss of generality in the model, which could easily be extended to more complicated electromagnetic baths so long as they are Markovian [13].
The two-level atom has a ground |g and excited state * kevinf@stanford.edu FIG. 1. Illustration of stimulated emission from a two-level system, initially prepared in its excited state |e and coupled to a unidirectional (chiral) waveguide at rate γ. A Fock state |n ξ has the potential to stimulate emission of a photon from a two-level system into the mode ξ. A coherent pulse driving the two-level atom cannot be thought of as stimulating emission into the same mode as the pulse.

|e , and Hamiltonian
with resonant frequency ω 0 and dipole operator σ = |g e|. Meanwhile in the white-noise limit, the electromagnetic bath is described by where b(t) is the continuous temporal-mode annihilation operator of the bath, with commutation [b(t), b † (s)] = δ(t − s) and b(t) |vac = 0. In the interaction picture, the total Hamiltonian is given by the dipolar energy exchange between the field and the atom at position x = 0 where γ is the interaction rate andb(t) = e iω0t b(t). With the white-noise limit, the Schrödinger equation for the unitary evolution operator U (t) |ψ(0) ≡ U (t, 0) |ψ(0) = |ψ(t) must be interpreted as a Stratonovich quantum stochastic differential equation (QSDE) [12,14] where dB t =b(t) dt is the quantum noise increment. Importantly, the Itō correction factor − γ 2 σ † σ dt results in the noise increments commuting with past evolution op- The Itō QSDE can be formally integrated to find the evolution operator and T indicates chronological time ordering. Below, we use this QSDE to consider two scenarios: stimulated emission under Fock state drive and under coherent state drive.
Fock state drive-the two-level atom is initially prepared in its the excited state |e , while the waveguide is prepared in a continuous-mode photon Fock state [15] The temporal mode of the Fock state is ξ, which is normalized to ds |ξ(s)| 2 = 1. Notably, the continuousmode Fock states are not unique, so for a different mode profile χ then n χ |n ξ = 1 necessarily. We denote the joint initial state of the system and waveguide |ψ 0 = |n ξ ⊗ |e ≡ |n ξ , e . Importantly, the Itō table for Fock state drive remains unchanged from the vacuum one, i.e. dB t dB † t = dt and all other products involving two increments in the same time bin [t, t + dt) are zero [16]. Hence, the Itō evolution Eq. 6 is still valid for the bath in a Fock state.
In order to compute the probability of stimulated emission, the fraction of the energy from the two-level system emitted into the same mode as the incident Fock state, we need to compute the overlap with the final state |ψ stim = |n + 1 ξ , g during and after emission. Specifically, we want to compute Because the initial and final states have definite photon number, the unitary propagator in Eq. 8 can be expanded and grouped by number of system-waveguide scattering interactions involving a potential transfer of energy from σ dB † ti −σ † dB ti to yield and t 0 = 0. Moving from Eqs. 8 to 9a required only a re-expression of the unitary propagator and the orthonormality of the Fock states n ξ |m ξ = δ nm , with the Kronecker-delta function. One of the key steps from Eqs. 9a to 9b is to understand how the nonlinearity of the two-level system requires an odd number of system-waveguide scattering interactions. Specifically, for the inner product between g|A|e = 0, the operator A ∝ σ or equivalently A ∝ σ(σ † σ) p , which imposes the order of alternating emission dB † t2i and absorption dB t2i−1 scattering events. Here, also I mention that e −γσ † σt/2 |g = |g and e −γσ † σt/2 |e = e −γt/2 |e . Lastly, obtaining Eq. 9c requires the relation dB t |n ξ = dt √ n ξ ξ(t) |n − 1 ξ , proved elsewhere [16]. The expressions derived using these manipulations, Eqs. work and provide a complete solution to the stimulated emission problem. My next step is to show how these expressions for stimulated emission can be evaluated in practice. After the stimulating wavepacket has finished interacting with the system, the light stimulated into the mode ξ keeps traveling along the waveguide uninterrupted. Hence, it will be convenient to work in the limit S = lim t→∞ U (t) where since the final stimulated state is an eigenstate of S. Now, I evaluate this expression for an exponentially decaying pulse shape with length τ : which was previously shown to be the optimal stimulating mode for a single photon when τ = 0.36/γ [3]. Then, there is a nice series expression for the coefficients The resulting curves for P stim versus pulse length are plotted in Fig. 2 for several different photon numbers. Previously studies have looked at the lifetime of the twolevel system to determine the optimal pulse length for maximum stimulation [3], but here we can find this length directly from the probability to stimulate emission into the driving mode. As the number of photons increases, the probability to stimulate emission begins to Rabi oscillate due to a strong coupling between the driving field mode and the two-level system. Further, for very large photon number and short pulses for the dominant terms in the sequence, yielding an intuitive expression for the stimulated emission probability P stim ≈ sin 2 4γτ (n ξ + 1) .
This expression is plotted for 144 driving photons (orange dotted curve) in Fig. 2 and matches very well with the exact curve, though for longer pulses spontaneous emission spoils the ability to emit into the driving mode. For very short pulses with a large number of photons, it is similarly possible to show that of the population is not stimulated and the field is left in its initial state immediately after the system-pulse interaction (but before spontaneous emission has time to occur, i.e. τ T 1/γ). This is the first definitive proof to my understanding, that when a two-level system is driven by a short pulse it is possible to stimulate oscillations between the bath mode ξ identically and the two-level system. I find the result quite remarkable, because the two-level system only ever exchanges one quantum of energy at a time with the waveguide during each bin [t, t + dt), but it does so in a completely coherent manner through different interference pathways that leads to emission of an additional photon into the mode ξ. Hence, stimulated emission can be viewed as a coherent interaction at rate g eff = 4γ/τ between the two-level system and a single bosonic mode at Rabi frequency ω R = g eff n ξ + 1 for a brief period of time, which was suggested in Ref. [17]. In between the energy exchange, the system and field become maximally entangled. These results have been recently suggested by an interesting technique of Fock-state master equations [16], however, the calculations trace over the past field states and hence could not definitely conclude stimulation into the exact same field mode ξ.
Coherent state drive-the two-level atom is initially prepared in its the excited state |e , while the waveguide is prepared in a continuous-mode coherent state For coherent drive, the authors in Refs. [18] have shown that stimulated emission takes a very different form for coherent drive. In a similar manner as our Fock state example, we will compute the overlap with a final coherent state |β during and after emission. Specifically, we want to compute Using Eq. 19 we can simplify this expression (21) There are three immediately obvious features from this expression: 1. As the pulse becomes longer, the Itō correction factor again causes the photons to be emitted into a bunch of modes other than in the initial state.
2. The probability of oscillation between ground and excited states is maximized by the choice of β = α.
3. The state completely factorizes into an atom part and a coherent field part.
Therefore, the coherent field cannot stimulate emission from the two-level system into a coherent state in a bath. Instead, the field remains mostly un-entangled with the system while the system undergoes Rabi oscillations.
The excited two-level system of course must emit its energy, but it does so in a somewhat unusual fashion involving multiple photon emissions [18]. In the short pulse limit for a square mode profile P e (t) = | α, e|ψ(t) | 2 = cos 2 ( √ γ|α|t) and P g (t) = | α, g|ψ(t) | 2 = sin 2 ( √ γ|α|t) during 0 < t < T . In summary, I have shown how to use quantum stochastic calculus to solve the stimulated emission problem exactly. In particular, under drive by a continuousmode Fock state the interaction with a two-level system has a particularly remarkable behavior in the short pulse and large photon number regime. Despite only exchanging one quantum of energy with the bath sequentially, the collective behavior could be modeled as an oscillating Jaynes-Cummings system given a cavity mode prepared with n photons. On the other hand, stimulated emission by a coherent pulse in free-space or a waveguide channel cannot be understood as an additional photon being emitted into the incident coherent state. This results from the factorizability of the waveguide coherent states with the system state at all times. In contrast, a twolevel system coupled to a single cavity mode prepared in a coherent state quickly produces maximal atom-cavity entanglement [19].
I thank Jelena Vučković for helpful feedback and discussions, and I also acknowledge helpful discussions with David AB Miller, Vinay Ramasesh, Joshua Combes, Daniil Lukin, and Rahul Trivedi, as well financial support from the Air Force Office of Scientific Research (AFOSR) MURI Center for Quantum Metaphotonics and Metamaterials.