Multiphonon Fock state heralding with single-photon detection

II. FORWARD BRILLOUIN SCATTERING

Brillouin scattering involves inelastic light-matter interactions, where incident light will shift frequency upon interaction with acoustic phonons. The more commonly known backward Brillouin scattering involves two counterpropagating light waves that can be coupled with a traveling sound wave via electrostriction [25]. Confined systems, such as resonators and waveguides, permit a form of forward Brillouin scattering (FBS) where a cutoff mechanical mode can transfer energy between copropagating optical resonances [21].

In FBS, this phonon mode with frequency mediates scattering between a ladder of optical modes with frequencies labeled ω m , where m is a general integer placeholder labeling each optical mode. This process is most efficient when the participating optical and mechanical waves satisfy phase-matching conditions, akin to energy and momentum conservation, given by

where k m and q are the wave vectors for the mth optical mode and phonon, respectively. Using the optical dispersion relation ω m = ck m /n m , where n m is the effective index evaluated at ω m , c is the speed of light, and noting that k m-1 ≈ k m -(ω mω m-1 )∂k m /∂ω, Eqs. ( 1) and (2) require that the phonon phase velocity is equal to the optical group velocity. While the phase velocity of sound is much less than light in bulk materials, Eqs. ( 1) and ( 2) can be satisfied in confined systems that support cutoff mechanical modes that can oscillate without propagating, i.e., = 0 when q ≈ 0 (see Figs. 1 and 2). Related systems comprising optical resonators that contain highly confined mechanical waveguides are suitable to realize FBS optomechanics when the mechanical frequency /(2π ) is an integer multiple of the optical free spectral range (FSR) (e.g., see Fig. 1). The Brillouin active mechanical mode for said systems is denoted in Fig. 2(a) as the white circle on the cutoff phonon dispersion curve. The optical dispersion and phase matching for these modes is shown in Fig. 2(b). While scattering to distant optical modes is physically limited by the optical dispersion, these effects are negligible for a broad variety of candidate systems over the relatively narrow frequency ranges of FBS [26]. For example, recent experiments demonstrate cascading to over 25 Stokes orders [27]. It is important to note that the dispersion curves in Fig. 2 would exhibit a continuum of allowable wave vectors for a linear waveguide. In addition, the following analysis assumes an optomechanical cavity system, whereas linear waveguide systems analytically require envelope operators [28]. In contrast with more familiar forms of Brillouin light scattering, FBS exhibits a phonon frequency and optomechanical coupling, which are highly tailorable with changes in the waveguide geometry, leading to selective resonant phonon frequencies and coupling strengths [23,29]. For example, tailored Brillouin shifts spanning 1-16 GHz have been experimentally demonstrated [30]. These properties have been explored in a broad variety of systems [31], including telecommunications [21], photonic crystal [32][33][34][35], and liquid-core [36] fibers; silicon waveguides [30,37,38]; microring [39] and microfluidic resonators [40]; and optical microspheres [41,42]. Forward scattering in these systems can be ensured as the dominant scattering mechanism when the Brillouin shift for backward scattering is not an integer multiple of the FSR of the optical cavity.

III. QUANTUM DYNAMICS

To model the quantum dynamics of an FBS optomechanical system, we use the Hamiltonian

where H 0 and H int describe the free evolution and interactions of the system, respectively, and H.c. denotes the Hermitian conjugate. Here, , b, and b † are the angular frequency, annihilation operator, and creation operator for the phonon mode, respectively. With ω p as a central resonant frequency, the first term in H 0 describes the collection of resonantly spaced optical modes, where a m is the annihilation operator for the mth optical mode that has the angular frequency ω m = ω p + m . As a consequence of the definition of ω m , ideal phase matching is assumed throughout, i.e., ω m = ω m-1 + . To reflect the independence of these distinct optical modes in the absence of FBS, the annihilation and creation operators satisfy [a m a † m ] = δ mm where δ mm is the Kronecker delta. The interaction Hamiltonian H int and associated constant coupling rate g quantify the strength of forward Brillouin interactions that can be produced by electrostriction and radiation pressure [23,29]. Important conserved quantities that arise from the Hamiltonian include the noninteracting energy (H 0 ) and the total photon number ( N = m a † m a m = m n m ), given that [H 0 , H] = 0 and [ N, H] = 0, where n m is the number operator for the mth optical mode.

By considering dynamics occurring at times much shorter than reported phonon decoherence times, we neglect loss on the mechanical mode in our analytical model. This regime can be accessed when the Brillouin coupling rate exceeds the mechanical decoherence rate. As one example, forward Brillouin coupling rates as high as g ∼ (2π )29 kHz and mechanical decoherence rates as low as T -1 2 ∼ (2π )1.2 kHz have been realized in silicon systems, where the significant contribution is from dephasing [8,30,37]. We confirm this point by simulating the dynamics with physically realized phonon dephasing and loss values, seeing exceptional agreement with our analytical model. To first build intuition for the dynamics of this system, optical losses are neglected, but are later analyzed by solving the Lindblad master equation. Other than decreasing the probability of a heralding event, optical dissipation does not alter the essential conclusions of the lossless model.

An analytical solution of the system's lossless dynamics can be found through application of the time evolution operator on an initial state: |ψ (t ) = U (t, 0)|ψ (0) , with U = exp{-iHt/ h}. Under the condition of phase-matched optical and mechanical modes, the FBS interaction described by H int conserves energy (i.e., [H 0 , H int ] = 0), permitting the time evolution operator to be factorized in the following form:

with U 0 = exp{-iH 0 t/ h} and U int = exp{-iH int t/ h}. Taking advantage of the infinite discrete symmetry of the system [i.e., the involvement of an infinite ladder of equally spaced optical modes and constant coupling g in Eqs.

(3) and ( 4)], U int can be factored further using the Glauber formula, a special case of the Zassenhaus expansion,

where X and Y are placeholders for any operators satisfying

For U int , the operators X and Y are given by

where

Noting that A and A † commute, we find [X, Y ] = g 2 t 2 AA † and [X, [X, Y ]] = [Y, [X, Y ]] = 0, permitting application of the Glauber formula and leading to the factorized form of U int given by

We assume that the phonon is initially found in the ground state. For standard dilution refrigerator systems, this assumption places constraints on the frequencies of the participating mechanical mode. For example, mechanical frequencies, /2π , exceeding 1 GHz provide >99.1% ground-state occupancy at 10 mK, frequencies and temperatures that are readily achieved [9,30,37]. Additionally, we consider a single photon injected into the m = 0 mode, providing an initial state of |ψ (0) = | . . . , 0 -1 , 1 0 , 0 1 , . . . ⊗ |0 ph . Here, the first ket describes the ladder of optical modes with each mode separated by a comma and where the suffix labels the optical mode. The second ket describes the state of the phonon, i.e., |0 ph . Explicit calculation of U (t, 0)|ψ (0) gives

where we have introduced the compact notation

This entangled superposition state quantifies how the input photon is Stokes shifted in frequency down the resonant optical ladder while simultaneously increasing the occupation number of the phonon mode. Equation (11) shows how measuring a photon of a specified frequency ω j can collapse the wave function and isolate a highly nonclassical phonon quantum state. To achieve this, the quantum state is spatially entangled by demultiplexing the emitted optical state as a function of frequency, where subsequent single-photon detection in thejth optical mode signals the collapse of the wave function into a phonon Fock state of known occupation number. An example of this heralding process is shown in Fig. 3, where |ψ in is the injected photon and |ψ out is the entangled state to be filtered by frequency into different transmission lines, leading to single-photon detectors. Owing to the large optical spacing determined by , the required demultiplexing is readily achieved in state-of-the-art, chip-scale filters [44,45]. In this example, the detected photon has been Stokes shifted to mode m = -3, corresponding to the generation of an n = 3 phonon Fock state.

Although permitting analytical simplicity, single-photon inputs present greater experimental challenges, requiring a heralding process to achieve the desired initial state. A more physically accessible initial state would be the input of a weak coherent state (faint laser), where the initial state of the m = 0 optical mode is given by |α ≈ |0 + α|1 (|α| 1), taking the place of |ψ in in Fig. 3. Applying the time evolution operator on |ψ (0) ≈ |vac ⊗ |0 ph + α|ϕ 0 ⊗ |0 ph gives the wave function at time t,

with |vac denoting vacuum for all optical modes. For this wave function, the vacuum state dominates in probability; however, the phonon manipulation relies on conditional measurement. While the detection probability is low, the entire state collapses upon detection, heralding a phonon Fock state in the same manner as before. Lower bounds on state synthesis fidelity, produced by corrections to the coherent state approximation above, are estimated in Sec. I of the Supplemental Material [46]. Having described the dynamics of the lossless system, we now analyze the effects of optical losses using the Lindblad master equation. When weakly coupled to a zero-temperature Markovian bath, the density matrix for the system satisfies

producing exponential decay for each optical mode at the decay rate γ [47]. Restricting our attention to single-photon input and accounting for losses modeled by Eq. ( 14), the density matrix can be represented by

where Eq. ( 14) provides the following equations for the timedependent coefficients α n,n and β n,n :

βn,n =γ δ nn α n,n . (

The solution to these two equations, given by

FIG. 4. Detection probabilities for the first few accessible states when FBS system is stimulated with a single photon, plotted over gt (g as the coupling rate, t as time). Optical loss is taken as γ = g. Solid lines correspond to Eq. ( 20) and circles correspond to simulations that include mechanical dephasing and loss. Inset: Heralding rates for Fock states n = 1-5 |n ph in Hz with pulsed source amplitude |α| = 0.2 and duty cycle T = 2/g.

can be shown to preserve the trace of the density matrix [i.e., tr(ρ) = 1], and gives the probability of single-photon detection in thejth mode (i.e., P j = α j, j ) and collapse to a phonon Fock state of number j,

This result shows that although the detection probability is suppressed by exp{-γ t}, the entanglement between singlephoton occupation and phonon Fock state is preserved. Thus, optical losses do not fundamentally alter the ability to herald complex phononic states (see Fig. 4 with γ = g as solid lines). Figure 4 also shows the impact of phonon losses and decoherence on the heralding probabilities. The circles represent simulated dynamics that include experimentally determined values for mechanical loss (∼ Hz) and dephasing (∼ kHz) [8], showing agreement with the analytical model. In this example, mechanical decoherence is dominated by dephasing, T -1 2 γ φ , which is modeled in the Lindblad master equation with the jump operator √ γ φ b † b. While large γ φ can impact the quantum evolution, pure dephasing does not change Hilbert space; i.e., phonon number and photon frequency remain correlated. Additionally, we find through simulation that the system's evolution is insensitive to increasing mechanical loss and dephasing by a few orders of magnitude (see Sec. II of the Supplemental Material [46]). We estimate the average heralding rates R n for each phonon Fock state to range from 10 to 90 Hz when stimulated with a weak coherent state, where |α| = 0.2. Here, R n |α| 2 P n /T , where T is the duty cycle of weak laser pulses and P n is the heralding probability averaged over the T . The heralding rate is limited by T , which must be long enough to allow for significant quantum evolution and suppress the chance of double photon occupancy.

After the detection event, readout and manipulation of the phonon state can be done through optomechanical π and π/2 pulses [20], where a beam splitter interaction between phonons and photons can be realized by intense optical pulses. In an FBS system, enabling this requires engineered mode suppression on a resonant mode of higher frequency than the input photon, where the occupation is forbidden. Otherwise, an intense optical pulse will simultaneously lead to beam splitting and two-mode squeezing interaction. Leveraging that the injected photon cannot scatter to frequencies higher than m = 0, Eq. ( 11) satisfies the Schrodinger equation with a single-photon input for suppression of any mode m = j > 0. Suppression of mode ω j removes the terms containing a j or a † j from the sum in Eqs. ( 3) and ( 4), leaving only an anti-Stokes sideband for the ω j+1 mode. Thus, strong pulses with frequency ω j+1 can transfer or superpose the heralded mechanical states into or with the optical domain at ω j+2 , where they can be isolated using bandpass filters and analyzed with balanced homodyne detection [3,24], given that the swap operation occurs at times before thermal decoherence times ( n -1 T 1 for Fock state |n at 20 mK [48]) after the heralding event. Physically, this mode suppression can be engineered in a multitude of ways, including structural changes to the resonator system [49][50][51][52] or Bragg gratings in linear waveguides [53,54].

In addition to verification of the nonclassical mechanical states, our findings indicate an approach to on-demand generation of multiphoton Fock states. Multiphoton states are a prerequisite for the synthesis of a variety of complex entangled states [55][56][57][58], which contribute to applications across quantum information processing [59][60][61]. Benefiting from long mechanical coherence times, an optomechanical π pulse can map a stored heralded phonon state to a photonic state in a well-defined spatial and spectral mode [5]. Although the heralding event is probabilistic, applying feedback techniques can approach near-deterministic generation [62,63].

In summary, we show how the quantum dynamics of intramodal spontaneous forward Brillouin scattering can be paired with single-photon detection to herald phonon Fock states of an arbitrary and selective number. Exploiting the entanglement generated by the system when stimulated with a single photon or a faint laser, single-photon detection collapses the mechanical state of definite phonon number, determined by the relative frequency shift of the detected photon with respect to the optical input. We discuss a method for quantum tomography via balanced homodyne detection of the mechanical-to-optical swapped quantum state, which by the same process suggests the use of these systems as an on-demand multiphoton source.

We have shown that mechanical decoherence can be neglected for times much shorter than the phonon lifetime, and although optical losses can be significant, a lost photon will only lower the probability of detection, leaving the heralded multiphonon Fock state unaffected. This result, in conjunction with long phonon quantum coherence times, the compatibility of these systems with telecommunication wavelengths of light, the large effective mass of the phonon modes, and access to a ladder of resonant frequencies of light, has the potential to enable new quantum devices, new approaches to state synthesis, and open new windows on fundamental physics questions.