1. Introduction

Ultracold atoms in periodically driven optical lattices, including shaken and amplitude modulated systems, have attracted much attention recent years, for they can bring out new interesting phenomena including realization of artificial gauge fields in different lattice geometries [1–3] and the coherent control of atomic wavefunctions [4–6]. In a shaken one dimensional optical lattice, the ferromagnetic transition in trapped Bose gas has also been observed [7] by coupling ground band s of the lattice system to the first excited band p. Specific to amplitude modulated lattice systems, although only bands with the same parity can be coupled, there are still researches on a wide range of problems, such as transfer of atoms from the ground band s to the second excited band d [8], detection of superfluid-Mott insulator transition [9] and study of the dynamical tunnelling of ultracold atoms with quantum chaos [10, 11]. The technique can also be applied in areas including realization of a velocity filter [12] and detection of gravity [13,14].

Normally, studies of amplitude modulated lattices are based on single frequency modulations that only involve processes with one photon emission or absorption. During a polychromatic modulation, not only the amplitudes, but also the phases of different frequency components in the modulation can be controlled independently, providing a more flexible way to coherently manipulate quantum states. In this paper, we coherently transfer atoms from the ground state s to the high excited g band via a double frequency modulation. The peaks of transfer rate with different lattice depths are measured experimentally, while influence of the modulation phase is demonstrated by performing an interference between two independent paths of excitations. These experiments completely investigated how frequencies and phases of different modulation frequency components would influence the excitation between Bloch states. Furthermore, we show an application of the double frequency modulation to build a large-momentum-transfer (LMT) beam splitter. Comparing with LMT beam splitters based on Bloch oscillation [8,15] or high order Bragg scattering [16], our method is easy in practise and would not introduce phase shift between two separated atom clouds.

The paper is organized as follows: In Sections 2 and 3 the polychromatic modulation theory based on Floquet method and our experimental system are introduced respectively. In Section 4, we study the effects of the modulation phase for the single frequency modulation. In Section 5 collaborative effect of different frequency components is studied, demonstrate the effect of both modulation frequencies and phases. Section 6 presents a way to realize a LMT beam splitter. Discussion and conclusion are in Section 7.

2. Theory for optical lattice with polychromatic modulation

For an atom in an amplitude modulated lattice system along $\widehat{x}$ axis, as schematically shown in Fig. 1, the time dependent Hamiltonian can be written as

 

Fig. 1 A sketch of lattice depth modulation in our system. The depth of lattice potential V₀ cos²(k_Lx) is driven by a polychromatic modulation ∑_iV_i cos(ω_it +ϕ_i).

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The first term in the right hand side is kinetic energy with M the atom’s mass and p_x its momentum along the x direction. The second term represents optical lattice without the modulation. V₀ is the constant part of the lattice depth, and the wave vector is k_L = 2π/λ with λ the laser wavelength. The last term expresses the amplitude modulation with the modulation amplitude V_i, the frequency ω_i and the phase ϕ_i of each frequency components.

Typically, time-periodic systems are described by Floquet’s theorem [17]. In our system, each of the modulation frequencies ω_i gives a period T_i = 2π/ω_i and the Hamiltonian has a period of H(t + T) = H(t), with T the lowest common multiple of T_i. By defining a time-evolution operator for one period as $\widehat{U}\left(T\right)$, solutions to this problem must satisfy

For a special case of single frequency modulation, considering the time and coordinate periodicity of the Floquet states $|u_{q,\alpha }〉$, by a Fourier transformation we can study the problem with a set of basis $|v_{lm}〉=e^i{}^{(2l{k}_{L}x-m{\omega }_{1}t)}$ in an extended Hilbert space $\mathrm{S}=\mathrm{H}\otimes \mathrm{T}$. Where $\mathrm{H}$ is the Hilbert space and $\mathrm{T}$ is the space of all functions with periodic T. By integration in one period, we can turn to a time-independent system.

To get rid of the phase ϕ₁, we perform an unitary transformation $U_{lm,lm}=e^{im{\varphi }_1}$ to the Hamiltonian by A′ = U⁻¹AU. There are three kinds of terms in the matrix of A′. The diagonal terms ${A^{\prime }}_{lm;lm}={\left(2l\overline{h}{k}_L\right)}^2/2M+m\overline{h}{\omega }_1$ are the energy of momentum states shifted by absorbing or emitting Floquet photons with energy $\overline{h}{\omega }_1$. The terms ${A^{\prime }}_{lm;l\pm 1,m}=V_0/4$ show stationary component of the lattice would coupling atoms with momentum difference $2\overline{h}{k}_L$. In addition, ${A^{\prime }}_{lm;l\pm 1,m\pm 1}=V_1/8$ terms show the time modulation of the lattice, which would coupling two momentum states separated by $2\overline{h}{k}_L$ while absorbing or emitting one Floquet photon.

When the modulation frequency is near-resonant with the energy difference between two specific bands and far detuned from the others, we can get an effective Hamiltonian by a rotating wave approximation.

The extension to polychromatic driven is straightforward. For example, a double frequency modulation induced two-photon process between s-g band is described by an effective Hamiltonian H_sg as:

The effective Hamiltonian H_sg is constructed by means of nearly degenerate perturbation technique [18], in which we include six nearly degenerate states considering four main processes in the excitation. The six states are |E_s〉 the s band, $|E_d-\overline{h}{\omega }_1〉$, $|E_d\overline{h}{\omega }_2〉$ the d band dressed by Floquet photon ω₁ or ω₂ and $|E_s-\overline{h}({\omega }_1+{\omega }_2)〉$, $|E_g-2\overline{h}{\omega }_1〉$ and $|E_g-2\overline{h}{\omega }_2〉$ the g band dressed by two Floquet photons. Using this basis a general state (v₁, v₂, v₃, v₄, v₅, v₆)^T gives complex coefficient of the six dressed states. Population of Bloch states s is |v₁|², while population on g band is $|v_4{e}^{i({\omega }_1+{\omega }_2)t}+v_5{e}^{2i{\omega }_{2}t}{|}^2$, given by coherent superposition of all g band states dressed with different Floquet photons. Solution of the model consists with the time dependent Schrödinger equation and the effective model provides us a better understanding of the mutiphoton process. However, in the calculation more states associated with higher order processes could be included to get a more accurate result, especially when the modulation amplitude is large.

3. Experimental system

Our experiment begins with a quasi-pure condensate of typically 1.5×10^{5 87}Rb atoms in the |F = 2,m_F = 2〉 hyperfine ground state, produced in a combined potential of a single-beam optical dipole trap and a quadruople magnetic trap. The trapping frequencies are ω_x = 2π × 28Hz,ω_y = 2π × 60Hz,ω_z = 2π × 70Hz. The optical lattice is formed by a retro-reflected red detuned laser beam, with lattice constant a = λ/2 = 426nm focused to a waist of 110μm. Density of atoms in our system is less than 5×10¹³cm⁻³, and the mean-field interaction can be omitted to capture the main physical mechanism in the excitation [19,20].

The lattice depth is calibrated by Kapitza-Dirac scattering and the modulation of lattice depth is controlled by an acousto-optic modulator(AOM). The modulation amplitudes V_i and phases ϕ_i are generated from a signal generator and the intensity of lattice laser is monitored by a photodetector. Experimental results are absorption images taken after 28ms time-of-flight (TOF). Occupation number at different momentum states $|2l\overline{h}{k}_L〉$ (l is integral momentum index) can be given from TOF images by n_l = N_l/N, with N_l the atom number at momentum state $|2l\overline{h}{k}_L〉$ and N the total atom number. The initial state for experiment is prepared non-adiabatically with numerically designed sequence of lattice pulses [21, 22]. For experimental convenience, the lattice pulses are carried out with the same depth as the constant part of the modulated lattice potential V₀. Typically, each of the pulses and the subsequent intervals are lasting for no more than 25μ_s, and the whole process can be finished within 60μ_s, thus the loading time is greatly reduced comparing with traditional adiabatic loading method.

4. The initial phase effect in single frequency modulation

Single frequency modulation can be seen as the basis of polychromatic driven. In this part, we present the preparation of a Floquet state in the single frequency driven system, which is shown to be highly related to the modulation phase.

Following the discussion in Sec. 2, Fig. 2 depicts a typical quasi-energy spectrum of the single frequency driven system, which is obtained by direct diagonalization of A′ at various quasi-momentum q. The same calculation also gives eigenvectors in the extended Hilbert space. The spectrum exhibits a complex structure as a result of the periodically repetition of high excited bands.

 

Fig. 2 Left side is the calculated Floquet spectra of a single frequency driven system, with parameters V₀ = 5E_r, V₁ = 0.5E_r, $\overline{h}{\omega }_1=5E_r$. In the figure the first seven bands are presented. The heavy lines depict states maximally overlapping with the s(blue), p (green) and d (red) Bloch bands respectively. Right side shows the details of two Floquet bands most overlapping with s and d bands. The two bands are separated by a band gap E_F at q = 0.

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To connect our calculation with the experiments, we should project the Floquet states |u_q,α〉 into momentum space. The occupation number at momentum states $2l\overline{h}{k}_L$ and its phase can be given by a summation of all the Fourier components with the same l as $c_l(t)|2l\overline{h}{k}_L〉={\sum }_m{e}^{-im{\omega }_1\tau }{\nu }_{lm}|2l\overline{h}{k}_L〉$, where ${\nu }_{lm}=〈{\nu }_{lm}|u_{q,\alpha }〉$ is coefficient of the Floquet state. τ is related to the modulation phase ϕ₁ and holding time t as $\tau =\frac{{\omega }_{1}t+{\varphi }_1}{2\pi }T$. It is also useful to define the overlapping between the Floquet state |u_q,α 〉 and a Bloch state |n_q〉 of the undriven potential as $P={\displaystyle {\int }_0^T|{\displaystyle {\sum }_l〈c_l}(t)|n_q〉{|}^{2}dt}$. Property of the Floquet band is typically characterized by its most overlapping Bloch band [23]. Without loss of generality, our experiments are restricted to quasimomentum q = 0, and energy gap between Bloch bands α and β are written as $\overline{h}{\omega }_{\alpha \beta }$. The technique can also be applied in systems with acceleration [12], which brings out phenomena different from our study.

When modulation phase ϕ₁ is given, a Floquet state can be projected into the momentum space, and such a state can be prepared by carrying out two lattice pulses with numerically designed pulse sequence [21, 22]. In the fast loading process, lattice depth of the two pulses is kept constant. For a target state, the fidelity of a prepared state can be given numerically for different pulse durations and time intervals, and the optimized pulse sequence is obtained by finding the maximum loading fidelity. Throughout this method we can get a loading fidelity of more than 95% experimentally.

With the initial modulation phase ϕ₁ = −π/2, a pure Floquet state is loaded by lattice pulses calculated with $\tau =\frac{{\varphi }_1}{2\pi }T=-T/4$. For the same state, we also present a modulation with ϕ₁ = π/2 to show the influence of modulation phase. The population on momentum components $|0\overline{h}{k}_L〉$ and $|\pm 2\overline{h}{k}_L〉$ for different initial phases ϕ₁ = −π/2 and ϕ₁ = −π/2 are shown in Fig. 3(a1) and 3(a2) respectively. The parameters are V₀ = −5.0E_r,V₁ = 0.5E_r and driven frequency is $\overline{h}{\omega }_1=5.0E_r$, red detuned from the band gap $\overline{h}{\omega }_{sd}=5.2E_r$. Figure 3(a1) is a Floquet state, which shows a time period of T = 2π/ω₁ = 62.75μs. The time evolution in momentum space would be quite different when the modulation phase is changing by π, as shown in Fig. 3(a2). Besides the oscillation with a frequency near ω₁, we observe a slow growth of n₁. Meanwhile, the amplitude of oscillations are also increasing following increase of the population on d band.

 

Fig. 3 Time evolution of n_l measured from the experiments with initial modulation phase (a1) ϕ = −π/2 and (a2) ϕ = π/2. Time averaged fraction 〈n_l〉 are also shown for (b1) ϕ = −π/2 and (b2) ϕ = π/2 respectively. n₀ is shown with black dots comparing to the numerical simulation in solid lines. n₁ and n₋₁ are shown in average with red circles the corresponding numerical result is shown in dashed lines. Each point is averaged by three experiments and the error bars indicate the standard deviation.

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The influence of modulation phase is shown more clearly when taking time average in one modulation period T. In Fig. 3(b1) with ϕ₁ = −π/2 the Floquet state shows a constant population in different momentum states by taking time average. While for ϕ₁ = π/2, Fig. 3(b2) shows a Rabi oscillation between s-d bands with Rabi frequency $E_F/\overline{h}$, where E_F is the gap between two Floquet bands. The observation can also be explained by Eq. (5). The phase of coupling terms would change with ϕ₁, thus the eigenstate is superposition of |s〉 and |d〉 with a relative phase determined by ϕ₁. When the modulation phase is changed, the loaded state is no longer an eignstate, and we can observe the Rabi oscillation.

5. Excitation of ground state via a double frequency modulation

The study of lattice modulation can be extended from single frequency to a more general form as described in Eq. (6). Specific to two-photon excitation between s-g band, the time dependent lattice is described as V_L(t) = V₀ +V₁ cos(ω₁t +ϕ₁) +V₂ cos(ω₂t +ϕ₂), which includes seven parameters V₀,V₁,V₂,ω₁,ω₂,ϕ₁,ϕ₂. During the experiments, we mainly focus on the influence of modulation frequencies and phases, while keeping other parameters constant.

5.1. Spectrum of two-photon excitation

The single frequency modulation shows that ϕ₁ is related to relative phase between s and d band components of the Floquet state. However, with a nearly pure s band, the phase of single modulation can be neglected, only the phase difference between two modulations is important. Therefore, we chose ϕ₁ = π while leaving ϕ₂ variable to control the relative phase between two modulations.

For s-g band coupling through a two-photon absorption process, the sum of the two modulation frequencies is chosen as ω₁ +ω₂ = ω_sg. This process is well described by Eq. (6), in which we have considered different pumping paths, as schematically shown in Fig. 4(a). There are two cases which would benefit the excitation process.

 

Fig. 4 (a) Two special cases in detecting the transfer population spectrum. In case 1, absorption of photons with ω₁(orange) or ω₂(red) is resonant with d band. In case 2, two frequencies are equal. (b) For s-g coupling ω₁ provides a two-photon process while ω₂ = 2ω₁ provides a one-photon process. Phases of two paths are controlled independently by modulation phases of ω₁ and ω₂.

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Case 1: Resonant two-photon process. When ω₁ = ω_sd or ω₂ = ω_sd, atoms are transferred from |s〉 to |g〉 with the assistance of d band as an intermediate band.

Case 2: Equal frequency two-photon process. When ω₁ =ω₂ =ω_sg/2, two modulations with the same frequency can be added together, and the coupling strength of the process is doubled.

With resonance condition ω₁ = ω_sd in case 1, when two modulation amplitudes are chosen as V₁Ω_sd = V₂Ω_dg the transfer rate would show a maximum resonant peak. And we keep this modulation amplitudes while sweeping ω₁. Modulation phase is chosen from numerical simulation to get a maximum transfer.

In the experiments, we sweep the frequency ω₁ for different lattice depths V₀ =5E_r, 10E_r and 14E_r. Population on momentum states $\pm 4\overline{h}{k}_L$ measured from the experiment are shown with rectangles in Fig. 5, comparing with the theoretical calculation shown in solid curves. Within the lattice depth we considered, g band is greatly concentrated on $|\pm 4\overline{h}{k}_L〉$ momentum states, thus n_±2 can reflect transfer rate to g band.

 

Fig. 5 Spectrum for the population on $\pm 4\overline{h}{k}_L$ states with increasing of modulation frequency ω₁. Population detected on $\pm 4\overline{h}{k}_L$ after a double frequency modulation for (a) V₀ = 5E_r(black) with V₁ = 1.4E_r, ±V₂ = 1.6E_r t = 300μs, (b) V₀ = 10E_r(blue) with V₁ = 2.8E_r, V₂ = 2.2E_r and t = 200μs, (c) V₀ = 14E_r(red) with V₁ = V₂ = 2.5E_r and t = 150μs are shown in rectangles with error bars. Solid lines are corresponding numerical simulation.

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Figure 5(a) shows the case of V₀ = 5E_r. For the lattice depth we have Ω_sd/Ω_dg = 1.11, correspondingly the modulation amplitudes are chosen as V₁ = 1.4E_r, V₂ = 1.6E_r, and t = 300μs. The holding time t may be chosen shorter if the maximum of numerical simulation is reached at an earlier time. In the figure, two peaks appear at ω₁ = ω_sd and ω₂ = ω_sd which follows case 1 we have discussed. And there the central peak at frequency ω₁ = ω₂ following case 2 is much lower than two peaks for case 1.

Figure 5(b) shows the spectrum with V₀ = 10E_r. With the increasing of V₀, the energy difference ω_sd is getting closer to ω_dg, and three peaks are overlapping. Comparing with V₀ = 5E_r the central peak is much higher, because the process of case 2 is also near resonance with d band.

For V₀ = 14E_r only one peak would be measured in the spectrum as shown in Fig. 5(c), which means case1 and case2 are fulfilled simultaneously. Under this condition, the coupling between s and g band is also greatly enhanced. Modulation amplitudes are V₁ = V₂ = 2.5E_r, for the two modulation frequencies are the same at case1 and can’t be distinguished.

The experimentally detected peaks are governed by two cases, which are within the description of Eq. (6). Thus in the numerical simulation we can neglect higher order processes of emission and absorption of Floquet photons. The discrepancy between experimental result and the theoretical simulation is probably due to the influence of interaction and initial momentum distribution of the condensate. These effects would destroy coherency during the modulation, and the measured population of excited state would be lower than the maximum value in the theoretical simulation.

5.2. The role of modulation phases

When ω₁ and ω₂ are incommensurate, the influence of relative phase is not prominent because there is only one path for the pumping process and no states could interfere with each other. However, the effect of modulation phases would be more pronounced when two modulation frequencies are commensurate.

As shown in Fig. 4(b), when 2ω₁ = ω₂, relative phase of modulations can be shown by performing a one-photon process simultaneously with a two-photon process. We choose the frequency ω₁ = ω_sg/2 at central peak of V₀ = 10E_r lattice and the second modulation is performed with ω₂ = ω_sg. Modulation amplitudes are V₁ = V₂ = 2.5E_r and t = 500μs. Similar to the process described by Eq. (6), in this problem we consider the interference between two states $|E_g-2\overline{h}{\omega }_1〉$ and $|E_g-\overline{h}{\omega }_2〉$. During the two-photon process with ϕ₁ = π, phase of state $|E_g-2\overline{h}{\omega }_1〉$ remains zero, while ϕ₂ determine phase of state $|E_g\overline{h}{\omega }_2〉$ as ϕ₂ −π. Relative phase of two modulations is defined as $({\varphi }_2-\pi )-\frac{{\omega }_2}{{\omega }_1}({\varphi }_1-\pi )={\varphi }_2-\pi$, and the interference can be changed from constructive to destructive with different ϕ₂.

Figure 6(a) depicts the population transferred to g band with the phase of one-photon process changing by 2π. Figure 6(b) shows how the depth of lattice is varying with time for four different modulation phases. With ϕ₂ = π, the population transferred to |g〉 through one-photon process and through two-photon process are in phase and the transfer rate is enhanced. When ϕ₂ is increasing, relative phase of two processes are deviating from zero, and the population on |g〉 would decrease. In the case of ϕ₂ = 0 the two process are out of phase, and only few atoms can be transferred to g band. Further increasing of ϕ₂ would increase the measured population, and when ϕ₂ is changed by 2π the population on g band reaches the maximum value again.

The single atom Hamiltonian in Eq. (1) can well explain the excitation in our system. However, for the 500μs modulation, decoherence from mean-field interaction and the momentum distribution becomes significant, thus it is necessary to carry out a simulation based on the time-dependent Gross-Pitaevskii equation,

 

Fig. 6 The excited population on g band shows the interference between two paths. (a) Population transferred to n_±2 is shown in black dots with error bars. The dashed line shows theoretical simulation for comparison. (b1)-(b4) V_L for different phases.

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The collaborative effect of different frequency components is studied in two aspects, demonstrate the effect of both modulation frequencies and phases of different frequency components. The usefulness of this study is not limited to double frequency modulation, within these two aspects, collaborate effect of more frequency components can also be well understood.

6. Application in LMT beam splitter

The double frequency modulation can also be applied to realize a LMT beam splitter by pumping atoms to higher momentum states, which is useful in experiments of atomic interferometry [24]. According to the resonant condition we have discussed, a preferred choice is V₀ = 14E_r with the modulation frequency ω₁ = ω_sd = ω_dg. The other modulation frequency is chosen as ω₂ = ω_gi (with i the 6^th excited Bloch band) to get a distribution at $\pm 6\overline{h}{k}_L$.

We begin with a condensate, the lattice is suddenly turn on, modulated with V₁ = V₂ = 2.5E_r and the preferable phases are found numerically. A TOF image of experimental result is shown in Fig. 7(a). Figure 7(b) shows the population of atoms on momentum states $|\pm 6\overline{h}{k}_L〉$ where we have subtracted the thermal gas. Near 80% of the atoms are coherently transferred into $\pm 6\overline{h}{k}_L$ momentum states within 160μs, and the maximum lattice depth needed is below V₀ +V₁ +V₂ = 24E_r. Comparing with a LMT beam splitter based on high order Bragg scattering [16], our method needs a much lower lattice depth and is easy in practice. Furthermore, the process is symmetric for both sides, and would not introduce phase shift between two separated atom clouds.

 

Fig. 7 A LMT beam splitter with a separation of $12\overline{h}{k}_L$. (a) TOF image of the LMT beam splitter. (b) Experimentally measured population of atom on momentum states $|\pm 6\overline{h}{k}_L〉$ are shown in black dots with error bars.

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A momentum splitting of $12\overline{h}{k}_L$ is not the limit of the amplitude modulation. More frequency components or another subsequent modulation can be introduced to reach a larger momentum splitting. For example, after the double frequency modulation, by preforming another single frequency modulation resonance with the energy difference between $|\pm 6\overline{h}{k}_L〉$ and $|\pm 8\overline{h}{k}_L〉$ momentum states, the atoms can be transferred to $|\pm 8\overline{h}{k}_L〉$ coherently.

7. Discussion and conclusion

Following Floquet-Bloch theory we have presented an experimental preparation of a Floquet state and study its property. The non-adiabatic loading method would take much less time and greatly reduce the heating problem. The loading method could also be extended to systems with a shaken lattice [7] or a combined modulation [6, 23], which provides a several millisecond longer lifetime for condensate in the experimental study on areas including the detection of Floquet topological states [17, 25, 26].

In conclusion, based on the study of single frequency modulation, we investigate double frequency modulation in detail. Experimental observations show that different modulation frequency components would influence each other in two ways. When two frequencies are resonant with subsequent excitation processes, the modulation can induce a resonant two photon process which can effectively transfer atoms to higher excited bands. With specific modulation frequencies, interference between a one-photon path and a two-photon path is observed, revealing influence of modulation phases when two frequencies are commensurate. The quantum interference can be used to enhance the excitation rate or destruct unwanted excitations. Using the technique of double frequency modulation, we also demonstrate an efficient way to realize a LMT beam splitter with low lattice depth. Our study provides a more flexible way to coherently control quantum states through an optical lattice.