Control and amplification of Bloch oscillations via photon-mediated interactions

We propose a scheme to control and enhance atomic Bloch oscillations via photon-mediated interactions in an optical lattice supported by a standing-wave cavity with incommensurate lattice and cavity wavelengths. Our scheme uses position-dependent atom-light couplings in an optical cavity to spatially prepare an array of atoms at targeted lattice sites starting from a thermal gas. On this initial state we take advantage of dispersive position-dependent atom-cavity couplings to perform non-destructive measurements of single-particle Bloch oscillations, and to generate long-range interactions self-tuned by atomic motion. The latter leads to the generation of dynamical phase transitions in the deep lattice regime and the amplification of Bloch oscillations in the shallow lattice regime. Our work introduces new possibilities accessible in state-of-the-art cavity QED experiments for the exploration of many-body dynamics in self-tunable potentials.


I. INTRODUCTION
Bloch oscillations (BO) [1] are center-of-mass oscillations or coherent breathing experienced by independent particles in a periodic lattice potential in the presence of a constant force (e.g.gravity).Although it has been hard to directly control BO in conventional electron systems, they have been observed in tailored semiconductor systems [2] as well as ultracold atom systems trapped in optical lattices [3,4].Nevertheless, for the latter, the lattice potential is by implementation rigid and therefore not a good test bed example of the underlying physics in real materials where the phonons of the crystal dynamically interact with the electron motion.Furthermore, inter-atomic interactions have always been a competing mechanism which damp the oscillations.
Here we propose a scheme to control and amplify atomic BO via photon-mediated interactions in a gravitytilted optical lattice supported by a standing-wave optical cavity with incommensurate lattice and cavity wavelengths.In our case, photons can actively modify the periodic potential experienced by the atoms and therefore resemble the role of phonons in a real solid state environment.Even though experiments that track BO in optical cavities have been implemented before using a Bose Einstein Condensate (BEC) [5][6][7][8][9], here we propose to use inhomogeneous atom-light couplings to prepare an array of atoms on specific lattice sites and initialize the dynamics [10].This can be achieved via position-dependent dispersive atom-light couplings to map the motion of the atoms under BO into the frequency shift of the cavity resonance.Our protocol not only avoids the ultracold degenerate initial states required in non-destructive measurements of BO, but also provides flexible self-tunability of the cavitymediated long-range interactions by the atomic motion.Moreover, in contrast to prior experiments where the periodic potential was generated by the probe laser field itself [5][6][7][8][9] or separate probe field for site-independent atom-light coupling [11], we use an additional lattice potential that traps the atoms and controls the degree of delocalization of the underlying Wannier-Stark (WS) states [12] in our system.In this setting, different to the well-studied case of contact interactions [13][14][15][16][17][18][19][20], the photon-mediated interactions can modify BO depending on the position of other atoms in the array.Taking advantage of this feature we show versatile many-body phenomena can be realized in different parameter regimes of this system: In the deep lattice region, we find dynamical phase transitions (DPT) related to the Lipkin-Meshkov-Glick (LMG) model [21,22], which potentially enables rapid generation of spin-squeezed states [23][24][25] with WS states directly, bypassing the need for Raman transitions [12]; In the shallow lattice, we find the amplification of Bloch oscillation amplification originating from the pair production [26][27][28][29] process from the central to adjacent WS states.We also discuss feasible implementations in state-of-the-art cavity QED experiments [30,31].

II. MODEL
We consider an ensemble of N ultracold atoms with mass M trapped in a standing-wave optical cavity along the vertical direction ẑ as shown in Fig. 1(a).The atoms are confined in the lowest band of the one-dimensional (1D) optical lattice supported by the cavity, with local gravitational acceleration ⃗ g generating an additional M gz potential between sites separated by a vertical distance z.Here we consider the pure 1D model for simplicity and discuss the modification by the radial modes in [32].A single internal level |g⟩ in the atomic ground manifold, is coupled to an atomic excited state |e⟩ with a transition energy ℏω 0 via a single cavity mode â with frequency ω c and wavelength λ c .The atom-cavity coupling has spatial dependence G(z) = G 0 sin(k c z), where k c = 2π/λ c and G 0 is proportional to single atom vacuum Rabi splitting.The cavity mode is coherently driven by an additional laser with frequency ω p thus detuning ∆ c = ω p −ω c from the bare cavity mode, which generates a net injected field in the cavity with amplitude η p .The cavity has a finite linewidth κ.Due to atomic motion, the cavity resonance will be shifted by G 2 0 N eff (t)/∆0, with N eff (t) defined in Eq. ( 3).(d) N eff (t) displays oscillatory behavior reflecting single-particle atomic BO, generated by a sudden quench on lattice depth from 15ER to 8ER.
We work in the dispersive regime of the atom-light interaction, where both the cavity mode and the external drive are far detuned from the atomic resonance, i.e., ∆ 0 ≫ G 0 ⟨â † â⟩ with ∆ 0 = ω p − ω 0 .In such limit, we can adiabatically eliminate the excited state and only consider the atomic motion in the ground state, which results in the following second-quantized Hamiltonian, where Ĥ0 = dz ψ † g (p 2 /2M + V 0 sin 2 (k l z) + M gz) ψg includes the kinetic energy, lattice potential, and gravitational potential experienced by the atoms.Here, V 0 is the lattice depth, k l = 2π/λ l is the wavenumber of the lattice beam that sets the atomic recoil energy E R = ℏ 2 k 2 l /2M , where λ l is the lattice wavelength.The field operator ψg (z) annihilates a ground state atom at position z.The second term in Eq. ( 1) describes the dispersive atom-light coupling after the adiabatic elimination of the excited state.The cavity Hamiltonian is given by Ĥcav /ℏ = −∆ c â † â + η p â † + η * p â.The eigenstates of Ĥ0 are the so-called Wannier-Stark (WS) states.In the tight-binding limit, the wave function for a WS state centered at lattice site n takes the form of ϕ n (z) = m J m−n (2J 0 /M ga l ) w (z − ma l ) [12,33], which is a superposition of localized ground-band Wannier functions w(z) [See Fig. 1(b)].Here J n is the Bessel function of the first kind, J 0 /ℏ is the nearest-neighbour tunneling rate, and a l = λ l /2 is the lattice spacing.The eigenenergy of |ϕ n ⟩ is nℏω B , where ω B = M ga l /ℏ is the Bloch frequency and T B = 2π/ω B the corresponding Bloch period.We expand the field operator in the WS basis, ψg (z) = n ĉn ϕ n (z), where the operator ĉn annihilates an atom in the WS state ϕ n .In this basis, Eq. ( 1) can be rewritten as where Here, J m,n = dzϕ m (z)ϕ n (z) sin 2 (k c z) describes the overlap between the WS states ϕ m , ϕ n weighted by the cavity field mode function.Neff can be understood as the effective number of atoms coupled to the cavity, which are responsible for generating a frequency shift G 2 0 N eff /∆ 0 on the cavity resonance, where N eff = ⟨ Neff ⟩.This dispersive term allows us to either perform non-destructive probing or many-body control of the atomic motion, depending on the operating parameter regime.
Assuming the cavity field adiabatically follows the atomic motion, which is valid since the cavity field dynamics (∆ c ∼ MHz) is much faster than the time evolution of the atomic field (ω B ∼ kHz), one can replace the cavity field operator by â ≈ η p /(∆ c − G 2 0 Neff /∆ 0 ).This leads to the following effective atom-only Hamiltonian [32], where Vcav ( Neff ) = −(V N/β)/(1 + β Neff /N ) is the cavity-induced potential depending on the atomic motion.Here, is the maximum AC Stark shift on the atoms introduced by the bare cavity mode, β = −N G 2 0 /(∆ 0 ∆ c ) is the ratio between the maximum cavity shift and the bare cavity detuning.We assume β > 0 (∆ 0 and ∆ c have opposite signs) to avoid hitting a cavity resonance.

III. SINGLE-PARTICLE DYNAMICS
First we consider the simplest case where the cavity is used as a probe and does not affect the single-particle dynamics set by Ĥ0 , valid in the regimes V ≪ ω B .We consider the case where atoms are initially loaded in an almost localized WS state in a deep lattice at sites n minimally coupled to the cavity (k c na l /π = r with n, r ∈ Z).
Then we suddenly quench the lattice depth to a shallow depth, and the atoms start hopping to the nearestneighbour sites [see Fig. 1(a)].Since the initially localized state corresponds to a superposition of WS states of the shallow lattice [see Fig. 1(b)], after the quench, each WS state acquires a phase that evolves at a rate set by ω B .The interference of different WS states induces tunneling away from the initially populated site, resulting in coherent breathing behavior at the BO frequency ω B .
To probe the BO, we use the fact that atoms at different sites coupled differently to the cavity.Therefore tunneling out and back into the initial site leads to a periodic oscillation in N eff (t) at frequency ω B as shown in Fig. 1(d), which can be measured by tracking the cavity frequency shift G 2 0 N eff (t)/∆ 0 .Note that a technique to initially prepare atoms at lattice sites with low initial coupling to the cavity mode has been demonstrated in [10].Instead of an initially localized state, we can also use amplitude modulation of the lattice depth [19] to prepare a superposition of WS states.In this case a similar behavior can be observed as detailed in [32].
For the numerical simulations throughout this letter, we consider the case of 87 Rb atoms with cavity wavelength λ c = 780 nm and lattice wavelength λ l = 532 nm.However, the discussion can be easily adapted to other type of atoms discussed in [32].

IV. DEEP LATTICE REGIME
The interplay between single-particle atomic motion and cavity-mediated interactions occurs if V ∼ ω B .Here we focus on the deep lattice regime (V 0 = 20E R ) where WS states are almost localized at individual lattice sites.If atoms are prepared at site n = 0, and V > 0, the differential cavity induced AC Stark shift (first order in β in the limit β ≪ 1) between the n = 0 and n = −1 sites ∝ V (J 0,0 − J −1,−1 ) can compensate for their energy difference ℏω B as shown in Fig. 2(a), restoring tunnelling between these two sites.Since the atomic motion is restricted to take place between these two states, we map them to an effective spin 1/2 degree of freedom: ĉ−1 as ĉ⇑ , ĉ0 as ĉ⇓ as well as the spin oper- In the limit of β ≪ 1, one can expand Ĥeff [Eq.( 4)] in a power series of β, and keep only the leading order terms.The Hamiltonian simplifies to, This approximated Hamiltonian [Eq.( 5)] is equivalent to the LMG model [12, 21-25, 32, 34],  sphere, Ŝα = R † Ŝα R, where R = exp(iθ Ŝy ), and tan θ = ∆ −1 /Ω −1 [32], which enables fast entanglement state generation under particular choice of χ, Ω, δ [23][24][25].The LMG model features a DPT from a dynamical ferromagnetic (FM) to a dynamical paramagnetic phase (PM), signaled by a sharp change in the behavior of the long-time average of the excitation fraction [21,22].In our model [Eq.( 4)], the long-time average of the signal N eff /N = lim T →∞ T 0 dtN eff (t)/(T N ) plays the role of the dynamical order parameter.We also show that the DPT exists in our model [Eq.( 4)] even beyond the β ≪ 1 limit as we discuss below.
To find the DPT, we solve the mean-field equations of motion for s x,y,z = 2⟨ Ŝx,y,z ⟩/N .Such non-linear dynamics can be further reduced to ( Ṅeff /N ) 2 + f (N eff /N ) = 0 with f (J 0,0 ) = 0, and we can associate the DPT with an abrupt change in the number of real roots of the effective potential f (N eff /N ) [32].This leads to the distinct dynamical behaviors of N eff /N tuned by varying V and β as shown in Fig. 2(b,c,d).When the dynamics are dominated by interaction effects, the system is in the FM phase where the Bloch vector features small oscillations around the south pole, also shown as small amplitude oscillations in N eff (t)/N .This phase is separated by a DPT to a PM phase where the Bloch vector exhibits large excursions around the Bloch sphere, also shown as large amplitude oscillations in N eff (t)/N .For β < 0.32 [32], the DPT transforms into a smooth crossover and the dynamics becomes dominated by single-particle tunneling processes.The dynamical phase boundary is plotted in Fig. 2(b) with the full model (solid line) and the LMG model (dashed line).The LMG model is unable to capture the phase boundary beyond the β ≪ 1 limit.

V. SHALLOW LATTICE REGIME
In a shallow lattice, the WS states extend over a few adjacent lattice sites.In this case, one can obtain a significant suppression of differential AC Stark shifts generated by the cavity by operating near the so-called magic lattice depth (V 0 = 6E R for the Rb parameters we use) [12], where J n,n is nearly a constant and the energy difference between nearest-neighbour WS states is roughly ℏω B [see Fig. 3(a)].Thus the dynamics features BO even in the presence of strong cavity-mediated interactions.In fact, after preparing the atoms in the WS state |ϕ 0 ⟩ and thus in an eigenstate of the single particle Hamiltonian, one can observe the generation and amplification of BO due to cavity-mediated interactions in a window around the magic depth as shown in Fig. 3(b,c,d).Since the short-time dynamics occurs mainly between the WS states centered at n = 0, ±1, we can concentrate only on these states and simplify the dynamics via the undepleted pump approximation (UPA): To the leading order, one can replace the operators for the initially occupied states as c-numbers, ĉ0 , ĉ † 0 ∼ √ N , and keep the operators for unoccupied states (ĉ ±1 , ĉ † ±1 ) to the second order while absorbing the linear term generated by singleparticle tunneling via a displacement of a coherent state, ĉ±1 = α ±1 + ĉ′ ±1 .In this way, Ĥeff [Eq.( 4)] simplifies into a quadratic form [32], We analyze the exact dynamics of Eq. ( 6) via the Bogoliubov-de Gennes method, in which the Heisenberg equation of motion for operators Ĉ = (ĉ ′ 1 , ĉ′ −1 , ĉ′ † 1 , ĉ′ † −1 ) T takes the form i∂ t Ĉ = H BdG Ĉ.The matrix H BdG can have either real or complex eigenvalues, which leads to distinct dynamical behaviors as shown in Fig. 3(c).When all the eigenvalues are real (normal regime), the populations ρ 0 and ρ ±1 , with ρ n = ⟨ĉ † n ĉn ⟩, feature stable small amplitude oscillations; on the other hand when all the eigenvalues are complex, then ρ ±1 feature an exponential growth associated with the correlated pair production of atoms at WS centered at n = ±1, which leads to the amplification of the BO signal until UPA breaks down.The transition between the real and complex eigenvalues of H BdG is marked by dashed lines in Fig. 3

(b).
To quantify the population transfer, we define A dip = 1 − min{ρ 0 } with min{ρ 0 } as the minimum of ρ 0 during t ∈ [0, 40T B ].A large A dip signals efficient population transfer.In Fig. 3(b), we show A dip as a function of the lattice depth V 0 and the cavity parameter β.The region of amplified BO lies within the two dashed boundaries.The left boundary is fixed at the magic lattice depth (V 0 = 6E R ) and the right boundary pushes to larger β as V 0 increases.Inside the amplification region A dip ̸ = 0, while outside A dip ≈ 0. The evolution of N eff (t)/N is shown in Fig. 3(d), where the enhanced population transfer induced by the cavity-mediated interactions lead to the growth of the BO amplitude in the amplification regime.

VI. EXPERIMENTAL CONSIDERATION
The predicted behavior should be achievable in stateof-the-art cavity QED systems with N ∼ 10 4 87 Rb atoms.We focus on the unitary dynamics in this letter while the main decoherence sources come from cavity loss and spontaneous emission from the excited states.The cavity loss generates collective dephasing processes at a rate V βκ/∆ c and spontaneous emission generates offresonant photon scattering processes at a rate V γ/∆ 0 , where γ is the spontaneous emission rate.For an optical cavity with cooperativity C = 4G 2 0 /γκ ∼ 0.5, κ/∆ c ∼ 0.05, γ/∆ 0 ∼ 0.01, one obtains negligible dissipation within experimentally relevant time scales (∼ 50 BO periods).Our scheme does not require BEC while utilizes site-selection to prepare the initial state, which is robust to the radial thermal noise up to T ∼ 1µK [32].Contact interactions between atoms can also be ignored for the dilute quantum gas used here (∼ 50 atoms per site).Moreover, our model can be realized with other species of alkali atoms (D 2 transition) and alkaline earth atoms ( 1 S 0 → 3 P 1 transition) with appropriate choices of lattice wavelength [32].In particular, contact interactions can be further suppressed using 88 Sr atoms featuring negligible scattering lengths or any type of fermionic atoms interacting only via the p-wave channel.

VII. CONCLUSION AND OUTLOOK
In summary, we proposed a scheme to perform manybody control of atomic BO in an optical cavity.Our work opens new possibilities for Hamiltonian engineering in many-body systems by taking advantage of the interplay between atomic motion, gravity and cavity-mediated interactions.For example, although so far we only focused on a single internal level, by including more levels and more cavity modes, it should be possible to engineer dynamical self-generated couplings between WS states via cavity-mediated interactions, which could be used to study dynamical gauge field [35][36][37] in a synthetic ladder without the overhead of Raman beams.Furthermore, although most of the calculations so far have been limited to regimes where the mean-field dynamics are a good description of the system, by loading the atoms in 2D or 3D lattice, one should be able to increase the role of beyond mean-field effects and enter the regimes where quantum correlations dominate the dynamics.In the main text, we considered N ultracold atoms trapped in a standing-wave optical cavity along the vertical direction ẑ.The atoms are assumed to be confined in the ground band of the one-dimensional lattice with lattice depth V 0 and wave vector k l = 2π/λ l .A single internal level |g⟩ in the atomic ground manifold is coupled to an atomic excited state |e⟩ with a transition energy ℏω 0 = ℏ(ω e − ω g ), via a single cavity mode â with angular frequency ω c and wavelength λ c .The atom-cavity coupling has spatial dependence G(z) = G 0 sin(k c z), with k c = 2π/λ c .The cavity mode is coherently driven by an external light field with detuning ∆ c = ω p − ω c from the bare cavity mode, which generates a net injected field in the cavity with amplitude η p .The full atom-cavity Hamiltonian is given as H = Ĥatom + Ĥlight + Ĥint .each of the terms can be written as : Here V (z) = M gz + V 0 sin 2 k l z describes the external potentials experienced by the atoms.ψ † e(g) (z) is the field operator that creates an atom in the state e(g) at position z, ω e(g) .Under the rotating frame of the pump field (set by the Hamiltonian H 0 = ℏω p â † â + ℏω p dz ψ † e (z) ψe (z)), the system's Hamiltonian takes the following form: where we defined the detuning of the pump from the atomic transition as ∆ 0 = ω p − ω 0 .Furthermore, under the assumption ∆ 0 ≫ G 0 ⟨â † â⟩ and ∆ 0 ≫ γ with γ the excited state spontaneous emission rate, the excited state population remains negligible during the relevant time scales.In this limit we can adiabatic eliminate the excited state |e⟩ ( ψe (z) ≈ G 0 â ψg (z) sin k c z/∆ 0 ), which leads to the following effective Hamiltonian acting on the ground state |g⟩ manifold only, In the tight-binding limit, the resulting single-particle eigenstates of the Hamiltonian p 2 /2m + V (z) become the so-called Wannier-Stark (WS) states |ϕ n ⟩ (n ∈ Z): Here J n denotes the Bessel function of the first kind, J 0 denotes the nearest-neighbor couplings in the ground band, a l = λ l /2 is the lattice spacing and w(z) is the ground band Wannier function.We will also use E R = (ℏk l ) 2 /2M for the atomic recoil energy.The field operator, when written in the WS basis takes the form, ψg (z) = n ĉn ϕ n (z), where ĉn annihilates an atom in the state |ϕ n ⟩.In this basis we can rewrite the Hamiltonian [Eq.(S6)] as: where J m,n = dzϕ m (z)ϕ n (z) sin 2 k c z describes the overlap between the WS states |ϕ m ⟩, |ϕ n ⟩ weighted by the cavity field mode function.In Fig. S1 we show the value of these couplings for the typical lattice depths we work in this paper.We define the effective particle number: as the effective number of atoms coupled to the cavity, which shifts the cavity resonance frequency by G 2 0 Neff /∆ 0 .

B. Adiabatic elimination of cavity field
Here we study the dynamics via Heisenberg equations of motion using a Markovian approximation.We adiabatic eliminate the cavity field using the fact that ∆ c sets the largest frequency scale and derive the effective atom-only Hamiltonian.To do that, we formally integrate the Heisenberg equation of motion of the cavity mode operator â and photon number operators â † â, then plug them back into the Hamiltonian [Eq.(S8)].We remove the fast rotating terms which relax much faster than the time it takes an atom to perform a BO.
The Heisenberg-Langevin equation of the motion for the cavity mode â is given by: with κ for the cavity decay rate.The above equation captures the dissipative dynamics generated by κ along with the quantum Langevin noise operator f , which gives the formal solution for the cavity field operator: Here f ′ is another quantum Langevin noise operator.Below we consider the regime ∆ c , κ ≫ ω B where the cavity-field dynamics evolve much faster than the atomic dynamics, thus it follows the latter adiabatically.As a result, we can obtain the formal solution for the cavity photon number operator: One more time ĝ represents a different quantum Langevin noise operator.For the last approximation above, we focus of the regime ∆ c ≫ κ where the unitary dynamics dominates and we can ignore the dissipation process to leading order.
If we insert the above solution of the cavity field into Eq.(S8), the effective atom-only Hamiltonian in the Schrodinger picture can be written as: where Vcav ( Neff ) = −(V N/β)/(1 + β Neff /N ) is the dynamical potential induced by the cavity which depends on the atomic motion.Vcav is parameterized by the maximum AC Stark shift introduced by the bare cavity mode, , as well as by the ratio between the maximum cavity shift and the bare cavity detuning, β = −N G 2 0 /(∆ 0 ∆ c ).We assume β > 0 (∆ 0 and ∆ c have opposite signs) to avoid hitting a resonance.As a benchmark for the effective atom-only Hamiltonian derived in Eq. (S14), we compare the exact dynamics for 6 particles in 3 WS states under Eq.(S8) and Eq.(S14) in Fig. S2(a).In the simulation, we choose ∆ c = 400ω B , κ = 20∆ c , η p = ∆ c /10 as well as G 2 0 /∆ 0 = −100ω B in the atom-cavity simulation (dashed lines), which corresponds to 5 in the atom-only simulation (solid lines).The simulation results match well with each other for the lattice depth in the normal regime (red curves) and amplification regime (blue curves), which verify the effectiveness of the atom-only Hamitlonian.

C. Equations of motion for atoms
To simulate the dynamics under Eq.(S14), we can calculate the equations of motion for the field operators ĉm as: Then we can simplify the equations above with ĉm , p,q J p,q ĉ † p ĉq = n J m,n ĉn : Finally, we apply the mean-field approximation to the operators n J mn ĉn Neff ≈ ⟨ n J mn ĉn ⟩ Neff , and obtain the following equations of motion: All the results in the main text were obtained by solving the mean-field equations of motion written above.Meanwhile, the mean-field equations for the atom-cavity Hamiltonian [Eq.(S8)] is given by, with α = ⟨â⟩ .We compare the mean-field dynamics Eq. (S17) and Eq.(S18) in Fig. S2(b).In the simulation, we choose reasonable experimental parameters N = 2 × 10 4 atoms, ∆ c = 2π × 2 MHz, κ = ∆ c /20, η p = 3∆ c as well as G 2 0 /∆ 0 = −2π×100 Hz in the atom-cavity simulation [Eq.(S18)], which corresponds to Still, the simulations for both normal regime and amplification regime match with each other pretty well with the difference can be ignored, which again verifies the validation of the effective atom-only Hamiltonian.In Fig. S2(c), we plot the evolution of cavity photon number 2 which follows the atomic motion adiabatically.

S2. DYNAMICAL PHASE TRANSITION WITH WANNIER-STARK STATES
In this part, we consider the deep lattice regime and discuss how to map the atom-only Hamiltonian to a spin model.As discussed in the main text, for a deep lattice V 0 = 20E R , the WS states approach the Wannier orbitals which are localized.The overlap integral J m,n for V 0 = 20E R is shown in the left panel of Fig. S1 with J 0,0 ≈ 0 ≪ J 1,1 ≈ J −1,−1 and J 1,0 ≈ −J 0,−1 ≈ 0. As a result non-trivial dynamics happens only for V (J n,n − J 0,0 ) + nω B ≈ 0 when starting from |ϕ 0 ⟩.Here we consider V > 0 and deal with two bosonic modes ĉ0 , ĉ−1 .For simplicity, we define Ω n = J n,n+1 as well as ∆ n = (J n,n − J 0,0 )/2.The spin operators are defined as follows, FIG. S4.Single particle Bloch oscillation with amplitude modulation scheme (pink curve, V0 = 8ER and V1 = 0.4ER) and Quench scheme (grey curve, quench from V0 = 15ER to V0 = 8ER as main text).

S4. EXPERIMENTAL CONSIDERATIONS A. Single-particle Bloch oscillations
In this section, we discuss the protocols to observe single-particle Bloch oscillations in the experiment.The main idea is to prepare a superposition of different WS states which accumulate different phases under Ĥ0 .In the main text, we discussed the quench scheme where the initial localized WS state ϕ 0 becomes a superposition of delocalized WS states.An alternative way to probe Bloch oscillations is to amplitude modulate the lattice depth as: which has been demonstrated in [S3].Here we define the tunnelling rate between ϕ m+n and ϕ n as: . Moreover, we can choose ω ≈ mω B , m ∈ Z to drive the mth sideband (between ϕ m+n and ϕ n ) and ignore the fast rotating terms: As a result, starting from ϕ 0 and performing the amplitude modulation for time τ , we obtain the initial state to be a superposition of WS states {ϕ n×m }.In Fig. S4 (pink curve), we simulate the case with lattice depth V 0 = 8E R and modulation strength V 1 = 0.4E R , also the first sideband transition (ω = ω B ). Different from the quench scheme (grey curve), after the modulation the single particle wavefunction can have a non-zero coupling to the cavity field (N eff /N ̸ = 0).
In the experiment, there may be higher bands populated in the quench protocol we discussed in the main text.In other words, the WS basis describing the ground band for the shallow lattice (after quench) is not necessarily complete to describe the initial localized state.Higher bands population will inevitably introduce other frequency components to N eff (t) disrupting the BO signal.However, in the simulations we performed for the main text (quench from V 0 = 15E R to V 0 = 8E R ), 98% atoms remained in the ground band and the higher band population can be ignored.Similarly, in the amplitude modulation schemes, we also choose V 1 to be much smaller than the band gap to avoid higher bands population.

B. Experimental parameters
Here we discuss the parameters for the specific case of 87 Rb with incommensurate lattice wavelength (λ l = 532 nm, ω B = 2π × 557 Hz) and cavity wavelength (D 2 transition with λ c = 780 nm).We are interested in the parameter regime with V ∼ ω B and β ∼ O(1), where the dynamics is mostly unitary and the dissipative processes can be ignored as we explain below.Another requirement is that the gap (27ω B for λ l = 532 nm and V 0 = 6E R ) should be much larger than V if we want to only work with the ground band WS states.Moreover, the cavity decay rate κ ∼ 2π × 0.1 MHz, the atom-light coupling strength G 0 ∼ 2π × 0.3 MHz, and atomic transition decay rate γ ∼ 2π × 10 MHz give the cavity cooperativity C = 4G 2 0 /γκ ∼ 0.36, which can be tuned even larger for larger G 0 and smaller κ, γ.The cavity loss generates collective dephasing processes at a rate V βκ/∆ c , while spontaneous emission generates off-resonant photon scattering processes at a rate V γ/∆ 0 as mentioned in the main text.Under κ/∆ c ∼ 0.05 and γ/∆ 0 ∼ 0.01, one obtains negligible dissipation within the experimentally relevant time scales and β ∼ O(1).For the maximum AC Stark shift, we first find that G 2 0 /∆ 0 ∼ 2π × 100 Hz with the parameters listed above, then |η p | 2 /∆ 2 c can be tuned between 1 to 10 for V ∼ ω B .
Our proposal works with a single internal level in the ground state manifold for atoms hopping between motional states (WS states here).Since interactions are mediated by photons, quantum statistics are not important in our scheme.As a result, even though above we considered the case of Rb, our model can be realized with other species of alkali atoms (D 2 transition) and alkaline earth atoms ( 1 S 0 → 3 P 1 transition) i.e. 87 Sr (boson), 88 Sr (fermion), 171 Yb (fermion) with appropriate choices of lattice wavelength and magic lattice depth summarized in table S1.Note that both 88 Sr, 171 Yb have very small scattering lengths in the ground states.
The single particle Bloch oscillations and dynamical phase transition in the deep lattice doesn't set too much limit on the choice of λ l and λ c .We only want the near-neighbour coupling coefficient J m,m+1 to be larger, while the overlaps between ϕ m (z), ϕ m+1 (z) and sin 2 (k c z) become tiny when λ l ≈ λ c , so we want to choose different λ l and λ c .While for the amplification of BOs in the shallow lattice region, we need to perform the experiment around magic lattice depth thus too shallow magic depth (such as 171 Yb) isn't favorable.

C. Radial mode thermal distribution
In this section, we discuss the effect of the Radial thermal motion following Ref.[S4].The Gaussian geometry of the laser beams in experiments inevitably couples the vertical and radial wave functions.The Gaussian profile of the lattice and cavity beams causes atoms in different radial modes to have different tunneling rate, resulting in a slightly different overlap integral J m,n for atoms in different radial modes.This effect can also be understood as fluctuations of the lattice potential V 0 due to radial thermal excitation.
First, we focus on the Gaussian beam profile of a 1D lattice, which leads to the following trapping potential: where w l is the beam width.In the presence of additional radial trapping potential V r (r, z) = M ω 2 r r 2 /2 [S5] we can expand the total trapping potential V (r, z) = V 0 (r, z) + V r (r, z) to second order of r and obtain: here ω r0 = 4V 0 /M w 2 l .The first term describes the lattice potential along the axial direction with the characteristic Bloch functions as eigenstates.The second term describes the radial harmonic trapping with eigenstate ϕ nx,ny (r) = ϕ nx (x)ϕ ny (y) and eigenenergies E nx,ny = ℏω r (n x + n y + 1/2).The third term describes the coupling between axial and radial degrees of freedom.The correction of J 0 is given by [S4]: Here the function f is the characteristic Mathieu value of type A for q ∈ (−ℏk l , ℏk l ), and the characteristic Mathieu value of type B for q = ±k l .We define q = q/ℏk l and v 0 = V 0 /E R .One can take such J0 into Eq.(S7) to calculate ϕ n (z), which causes inhomogeneity for the coupling matrix J m,n .We can estimate the contribution from different radial eigenmode with Boltzmann distribution p nx,ny (S51) In Fig. S5(a), we plot the standard deviation of the tunneling rate J 0 as a function of ω r and different temperature T .In Fig. S5(b), we plot the standard deviation of the coupling coefficient J 0,0 due to the correction of the tunneling rate.Similar behavior for other coupling coefficients J m,n .As a result, one can suppress the effect of the radial modes occupation by increasing the total radial trapping frequency ω r or lowering the temperature.The standard deviation ∆J 0,0 ≈ 0.01J 0,0 up to temperature T ∼ 1 µK as well as ω r = 2π × 1 kHz, thus the radial thermal noise only has a tiny effect on many-body dynamics we predict.

D. Atoms loading
In this section, we discuss the real process of atoms loading in the experiment.In the main text, we mention first loading atoms at position k c z/π = r, r ∈ Z which atoms-cavity coupling becomes perfect zero.However one can only set a threshold for the atom-cavity coupling during the loading process i.e. load all the atoms with sin 2 k c z < ϵ in the real experiment.Such loading error makes J m,n deviate from expected values, which brings additional inhomogeneity.In Fig. S5(c), we plot the standard deviation of the coupling coefficient J 0,0 as a function of error ϵ.We consider the total lattice length to be 1 mm and assume atoms load into all the sites n which satisfy sin 2 (k c na l ) < ϵ uniformly.These imperfect sites cause tiny inhomogeneity in the coupling coefficient J m,n up to ϵ ∼ 5%.

FIG. 1 .
FIG. 1. Model system.(a) An ensemble of N atoms are trapped in the lowest band of an optical lattice supported by an optical cavity aligned with gravitational acceleration ⃗ g.Considering the atoms are initially localized in a Wannier orbital (grey dashed line), hopping to the nearby sites (grey solid line) can lead to a change of atom-cavity coupling due to incommensurate lattice (λ l ) and cavity (λc) wavelengths.The cavity has a finite linewidth κ.(b) The initially localized Wannier orbitals can also be written as a superposition of partially delocalized Wannier-Stark states which accumulate different phases due to gravity.(c) Frequencies of atomic transition (ω0), external pump (ωp) and cavity resonance (ωc).Due to atomic motion, the cavity resonance will be shifted by G 2 0 N eff (t)/∆0, with N eff (t) defined in Eq. (3).(d) N eff (t) displays oscillatory behavior reflecting single-particle atomic BO, generated by a sudden quench on lattice depth from 15ER to 8ER.
by a rotation along the y-axis of the Bloch

FIG. 2 .
FIG. 2. Dynamical Phase Transition (DPT) in the deep lattice regime (V0 = 20ER).(a) For the case of V > 0 we define an effective spin-1/2 degrees of freedom: |⇑⟩ (|ϕ−1⟩) and |⇓⟩ (|ϕ0⟩).The cavity-mediated interactions generate energy shifts to balance the potential energy of these two sites (red curve), as well as dynamical couplings between them (orange arrow).(b) Phase diagram of the DPT determined by the long-time average Neff /N .The phase boundary separating the paramagnetic (PM) and ferromagnetic (FM) phase is predicted by the full model (solid line) and LMG model (black dashed line).The smooth crossover regime is below the gray dashed line.(c) Mean-field dynamics with V = 1.9ωB, β = 0.5 (green) and V = 2.2ωB, β = 0.5 (red).The upper panel shows the mean-field trajectories on the Bloch sphere, and the lower panel displays the normalized signal N eff (t)/N .The solid (dashed) line show predictions of the full (LMG) model respectively.(d) Horizontal cut of the phase diagram in (b) for β = 0 (solid), β = 0.5 (dashed), β = 2 (dot dashed).

FIG. 3 .
FIG. 3.Cavity-mediated amplification of Bloch oscillations in the shallow lattice regime.(a) The red lines show the gravity plus optical lattice potential.Around V0 ≈ 6ER, WS states can extend to the nearest-neighbour lattice sites.The orange vertical lines represent the cavity-induced onsite shift of the energy levels and the orange arrows illustrate the cavity-mediated tunneling process shown in Eq. (6).(b) Transition between amplification regime and normal regime indicated by A dip = 1 − min{ρ0}.V is fixed to be 2ωB.The black dashed line shows the predicted boundary from UPA. (c) Mean-field dynamics of ρn with initial state |ϕ0⟩ and V = 2ωB, β = 3.Nearly no dynamics happen in the left panel (purple square, V0 = 5.8ER) while large population transfer to |ϕ1⟩ and |ϕ−1⟩ (pink circle, V0 = 6.2ER) is observed in the right panel.(d) Mean-field simulations for the normalized signal N eff (t)/N for the same parameters described in (c).The purple line stays almost constant while the pink line signals the cavity enhancement of the BO.
FIG. S1.The coupling coefficient Jm,n for 87 Rb atoms (λ l = 532 nm, λc = 780 nm).Left: V0 = 20ER and right: V0 = 6ER.Start from |ϕ0⟩, the many-body dynamics mainly happens within the dashed square for either two-level model (left, deep lattice region for dynamical phase transitions) and three-level model (right, shallow lattice region for amplification of Bloch oscillations).

FIG. S2 .
FIG. S2.Benchmarks of the atom-cavity Hamiltonian [Eq.(S8)] with the effective atom-only Hamiltonian [Eq.(S14)].The red curves for V0 = 5.8ER (normal regime) and the blue curves for V0 = 6.2ER (amplification regime) are used for all the simulations in the figure.(a) Exact Diagonalization (ED) simulation of the dynamics for 6 particles in 3 WS states with the initial state (ĉ † 0 ) 6 |vac⟩.Populations ĉ † 0 ĉ0 (start from 6) as well as ĉ † 1 ĉ1 (start from 0) for these two lattice depths are plotted.The solid lines are the exact simulations under the Hamiltonian Eq. (S8) (with the photon space ncut = 10) and the dashed lines under the Hamiltonian Eq. (S14).(b) Mean-field dynamics of ρn with initial state |ϕ0⟩.The solid lines are simulated with the atom-cavity mean-field equations of motion [Eq.(S18)] and the dashed lines are simulated with atom-only equations of motion [Eq.(S17)].Populations ρ0 (start from 1) as well as ρ1 (start from 0) for these two parameters are plotted.The differences between the atom-cavity and atom-only simulations can be ignored for both (a) and (b).(c) Mean-field evolution for the cavity photon number with the same parameters for the red and blue curves as in (b).

FIG. S5 .
FIG.S5.The standard deviations of (a) the ground band tunneling rates, (b) the coupling coefficient J0,0 as a function of radial trapping ωr with fixed T = 0.1 µK (blue curve), T = 1 µK (orange curve) as well as T = 10 µK (green curve).We use the beam width w l = 50µm and lattice potential V0 = 6ER in the calculation.(c) The standard deviations of J0,0 as a function of loading error rate ϵ.

1
JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado 80309, USA 2 Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA (Dated: February 14, 2024)

TABLE S1 .
Atomic species λ l (nm) λc (nm) Magic lattice depth (ER) Summarized lattice, cavity wavelength and magic lattice depth for different atomic species.