Abstract
Ramsey interferometers using internal electronic or nuclear states find wide applications in science and engineering. We develop a matter wave Ramsey interferometer for trapped motional quantum states exploiting the s and dbands of an optical lattice and study it both experimentally and theoretically, identifying the different dephasing and decoherence mechanisms. Implementing a band echo technique, employing repeated πpulses we suppress the dephasing evolution and significantly increase the coherence time of the trapped state interferometer by one order of magnitude. Thermal fluctuations are the main mechanism for the remaining decay of the contrast. Our demonstration of an echoRamsey interferometer with trapped quantum states in an optical lattice has potential application in the study of quantum manybody lattice dynamics, and motional qubits manipulation.
Introduction
Interferometers employing separated oscillating fields to create and probe superpositions of states, also known as Ramsey interferometry (RI), have originaly been developed for magnetic resonance to measure transition frequencies^{1,2} and were then extended to a general tool of spectrocopy and matter wave interferometry^{3,4,5}. Typical sequences consist of two π/2 pulses separated by an time for free evolution or a π/2 − π − π/2 sequence. Ramsey interferometers using internal electronic or nuclear states already have played an important role in accurate quantum state engineering and quantum metrology, such as in nuclear magnetic resonance^{6}, atomic clocks^{7}, quantum information^{8} and quantum simulation^{9}. In general, echo techniques are used in RIs to suppress dephasing for significantly increasing the coherent time^{10,11,12}. Even though motional states of particles are on the same footing as internal states in current quantum technologies, i.e., motional qubits^{13}, motional logic gates^{14,15,16}, motional quantum error correction^{17}, quantum entanglement and coherent control^{18,19}, quantum metrology with nonclassical motional states^{20}, and communication multiplexing^{21}, conventional echoRIs rarely exploit the quantum interference of trapped motional states.
Recently, an RI with trapped motional states of a Bose–Einstein condensate (BEC) trapped in an anharmonic potential has been demonstrated with 92% contrast for several cycles^{22}. This proofofprinciple experiment holds great promises for studying quantum manybody physics out of equilibrium, quantum metrology with nonclassical motional states and quantum information processing with motional qubits. Recently, this new technique has been used to investigate decoherence and relaxation dynamics^{23}, or to measure the phononic Lamb shift^{24}.
Ultracold atoms trapped in an optical lattice (OL) are an ideal test platform for studying quantum manybody dynamics^{25}, and have also been widely used as a highprecision metrology tool^{26}. Conventionally, these atoms are prepared in the lowest band of the OL. Over last few years, there is an increasing experimental and theoretical interest to prepare atoms in higher bands^{27,28}, opening a new way to simulate exotic orbital physics in strongly correlated matter with rich degrees of freedom, i.e., the formation of multiflavor systems^{29}, supersolid quantum phases in cubic lattices^{30}, quantum stripe ordering in triangular lattices^{31} and Wigner crystallisation in honeycomb lattices^{32}. A wellcharacterised RI with a clear understanding of the coherence mechanisms has potential applications in the study of coherence of trapped states in lattices^{33}, nonequilibrium many body physics^{34}, and its applications in metrology^{35}.
In this paper, we demonstrate a RI with trapped motional quantum states (TMQS) of atoms employing the S and Dbands in an OL. Due to the lack of selection rules for lattice band transition, a key challenge for constructing this RI is to realise π and π/2pulses analogous to those in conventional RIs. Using a shortcut loading method^{36,37}, we have designed sequences of optical pulses^{38,39}, analogous to a π or π/2pulse, to efficiently prepare a superposition of atoms in S and Dband states in the OL at zero quasimomentum with high fidelity within tens of microseconds, which is much shorter than the characteristic time scales of the decay process. Keeping the lattice on, we observe state interference and measure the decay of the coherent oscillations.
We have identified the mechanisms leading to the RI contrast reduction as following: the the homogeneity of the optical lattice depth, interactioninduced transverse expansion after loading the atoms from the harmonic trap into the optical lattice, laser intensity fluctuation and thermal fluctuations at finite temperature. We then implement a matterwave band echo technique to significantly suppress all the contrast decay effects except for quantum and thermal fluctuations, increasing the coherence time to 14.5 ms compared to 1.3 ms without echo at condensate temperature of 50 nK and 10E_{r} lattice depth.
Results
Experimental implementation
Our experiment start with a BEC of ^{87}Rb prepared in a hybrid trap formed by a singlebeam optical dipole trap with wavelength 1064 nm and a quadruple magnetic trap (Details are presented in our former works^{40,41}). A nearly pure condensate of about 1.0 × 10^{5} atoms at the temperature 50 nK is achieved with the harmonic trapping frequencies (ω_{ x }, ω_{ y }, ω_{ z }) =2π × (24, 48, 58) Hz, respectively. The system’s temperature can be controlled by the evaporative cooling process (see Methods). After preparation of the condensate, a one dimensional optical lattice is formed by two counterpropagating laser beams with wavelength 852 nm resulting in a lattice constant d = 426 nm along x axis, as shown in Fig. 1(a). Using a shortcut control method^{38}, the BEC is loaded into the lowest lattice band with the quasimomentum q = 0 within a few microseconds with nearly 100% fidelity. The interaction energy of the condensate in the ground state of OL with a lattice depth of 10E_{r} is 1.2 kHz (\(E_{\mathrm{r}} = \left( {\hbar k} \right)^2/2m\) is the recoil energy, k = 2π/426 nm^{−1} is the wave vector associated with the lattice and m is the atomic mass).
In the experiment, we construct the Ramsey interferometer with Bloch states ϕ_{i,q} with energy ε_{i,q}. We use the lowest band ϕ_{S,0} and the second excited band ϕ_{D,0} at quasimomentum q = 0, denoted as S〉, D〉 respectively in the following, to form a superposition state ψ = a_{ S }S〉 + a_{ D }D〉. The two bands of S and D are considered a twostate system (spin1/2 system), i.e., the two states can be expressed as \(\left( {\begin{array}{*{20}{c}} {a_{\rm{S}}} \\ {a_{\rm{D}}} \end{array}} \right)\). The scheme for the band energy and the superposition states are shown in Fig. 1b, c, respectively, with the lattice depth V_{0} = 10.0E_{r}. During the lattice pulse sequence, an accustooptic modulator controlled by an RF switch is used to switch on and off the lattice potential (light) quickly.
Manipulating the pseudospin system has its own challenges: unlike conventional RI where selection rules can be used to prepare population in two sates, the lattice band transition, similar to transition for vibration states in molecules^{42}, has no selection rules. Thus a π/2pulse analogous to those in conventional RIs had to be numerically designed, so that the atoms in Sband and Dband are to be transferred into the target states \(\left {\psi _1} \right\rangle = \left( {\left S \right\rangle + \left D \right\rangle } \right)/\sqrt 2\) and \(\left {\psi _2} \right\rangle = \left( {  \left S \right\rangle + \left D \right\rangle } \right)/\sqrt 2\), respectively. Extending our optimal control method^{38,39}, we construct a π/2 pulse from a sequence of lattice pulses with special selected timing and duration as shown in Fig. 1(d). We achieve a fidelity of 98.5% and 98.0%, for atoms initially on the S and Dband respectively. (see Methods).
The energy difference between S and Dband is changing with quasimomentum q, and the pulse sequences are desighed for q = 0. For a 10 E_{ r } lattice the applied shortcut pulses give more than 90% fidelity only in the region of q < 0.06ℏk.
The full time sequence for RI is shown in Fig. 1e. First the atoms in the harmonic trap are transferred into the Sband by a fast loading process, then the first pulse \(\hat R\left( {\pi /2}\right)\) is applied to prepare an initial superposition state \(\hat R\left( {\pi /2} \right)\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right) = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right)\). The final state, after evolution in the OL for time t_{ OL } and a second π/2 pulse can be expressed as:
with \(\hat R\left( \alpha \right) = \left( {{\mathrm{cos}}\frac{\alpha }{2}  i\,{\mathrm{sin}}\frac{\alpha }{2}} \right)\hat \sigma _y\), and the evolution operator \(\hat U\left( t \right) = \left( {{\mathrm{cos}}\,\omega t + i\,{\mathrm{sin}}\,\omega t} \right)\hat \sigma _z\), here ω = (ε_{D,0} − ε_{S,0})/ℏ is the frequency difference between two states, and \(\hat \sigma\) the Pauli matrix, α denotes rotation angle around an axis in the Bloch sphere.
We then apply band mapping^{27} to read out the state of the RI. For band mapping, the lattice depth is exponentiallyramped down in the form e^{−t/η} with a characteristic decay time η = 100 μs for a total length of 500 μs. Then absorption imaging is used to measure the population in the different bands after 31 ms time of flight (TOF) (see “Methods” section). The atoms originally occupying S〉 state populate a narrow Gaussian distribution around 0ℏk. The atoms originally occupying the D〉 are detected at ±2ℏk in the side zone. The total atom number we detected consists of condensed atoms and thermal atoms. From a bimodal fitting to each peak^{28}, we can quantitatively determine the population of condensate, and we note the condensed atom number as N_{S} (N_{D}) for Sband (Dband) (see “Methods” section).
Ramsey Interferometer in an optical lattice
Experimental results on the time evolution of the atom population in the Dband p_{D}(t_{OL}) = N_{D}/(N_{S} + N_{D}) with evolution time in the lattice t_{ OL } are shown in Fig. 2a for the time sequence \(\hat R\left( {\pi /2} \right)  \hat U\left( {t_{{\rm{OL}}}} \right)  \hat R\left( {\pi /2} \right)\), V_{0} = 10E_{r} and T = 50 nK. Each solid point with error bar is the mean of three measurements, shown as red circles. The red solid line is a fit of an damped oscillation with a period of 41.1 ± 1.0 μs, which is consistent with the reciprocal of the band gap energy of about 40.8 μs. The details of the early RI oscillation for t_{ OL } < 100 μs is shown in Fig. 2b, displaying a nearly perfect oscillation between the Sband and the Dband with amplitude close to 1. At P_{1} nearly all the atoms are transmitted from the Sband to the Dband by two π/2 pulses which can be seen as a π pulse. At P_{2}, the two π/2 pulses offset each other due to phase evolution of two states in the lattice, and the atoms are transferred to the Sband. However, when t_{OL} gets longer, the oscillation amplitude decays, as shown in Fig. 2a. The contrast C(t_{OL}) at t_{ OL } can be obtained by fitting the amplitude of oscillation p_{ D }(t_{OL}) in Fig. 2a with
Figure 2c shows the measured contrast decay versus time t_{OL} for the different initial temperatures of condensates, where each contrast value is fitted from about 20 experimental points with a time step 5 μs and each point is the mean value of three measurements. The errorbars are given by 95% confidence bounds. The horizontal dashed line in Fig. 2c indicates the contrast drops to the value of 1/e, which is used to define a coherence time τ. This time decreases from 1.3 ms to 0.8 ms when the temperature increases from 50 nK to 180 nK.
Contrast decay mechanisms
In order to improve the performance of the RI, we now investigate the mechanisms that lead to RI signal attenuation. We first study the effect of imperfect design of the π/2 pulse on the fidelity by solving Schrödinger equation with an uniform lattice potential, and the unbalanced population between the S and Dbands (see “Methods” section). The numerical result (brown dashed line in Fig. 3a) shows that the imperfectly designing of π/2 pulses and the unbalanced population have negligible effects on the contrast decay during the evolution in lattice potential.
In our further theoretical analysis, we replace the ideal lattice potential by a nonuniform potential distribution to account for the Gaussian beam in the radial direction, as shown in Fig. 1a, and include the quasimomentum distribution for the condensate distributed in the harmonic trap^{43}. These two effects result in inhomogeneous broadening of the transition frequency ω between the S and Dbands, and lead to to dephasing and contrast decay. By solving the zerotemperature Gross–Pitaevskii equation(GPE) with the real inhomogeneous potential^{44} (see “Methods” section), we obtained the contrast as shown by the blue dotted line in Fig. 3a.
The atomatom interaction leads to transverse expansion after the fast (nonadiabatic) loading of the atoms from the harmonic trap into the 1d optical lattice. This expansion leads to a significant reduction on contrast (blue dashed line in Fig. 3a).
Moreover, stability of lattice depth in experiment would also influence the contrast we measured. In the GPE simulation, we investigate the influence of both the variation of intensity in a single run and the variation between different experimental runs. We find that the contrast decay shows a stronger dependence on the variation of the mean lattice depth in between different experimental runs. The variation within a single run has little influence if the mean laser power is unchanged. This is because the variation of lattice depth is small and adiabatic, and the contrast at long holding time is reduced by a variation of the mean value. Quantum fluctuations at zero temperature is further added using a truncated Wigner method (see “Methods” section) and the related results shown in green dashed line nearly overlap with the dashdotted line. In our experiment, the atom number is high enough such that the influence by the quantum fluctuation is not significant.
Finally, we take into account the thermal fluctuations using a finite temperature truncated Wigner calculation (see “Methods” section). The result is shown in the orange solid line in Fig. 3a, agreeing well with the experimental results shown in black dots.
The relation of the coherence time to lattice depth is presented in Fig. 3b. When the lattice depth increases, the confinement in xaxis gets tighter, accordingly, the effects of the expansion in radial direction will increase due to the increased interaction energy. In addition, the nonuniformity of the lattice potential (the difference of the lattice potential from center to edge) also increases with the increasing of lattice depth. Due to these two effects, the coherence time τ decreases with increasing lattice depth. The theoretical curve in Fig. 3b, including all the decay mechanisms (orange solid line) fits well with an experimental measurement.
An EchoRamsey Interferometer with TMQS of atoms
To further improve the contrast of the RI with TMQS, we develop a matterwave band echo technique. A π pulse is designed which swaps the atom population in S and Dband. The pulse schema of our echoRI scheme is shown in Fig. 4. In between the initial and final π/2 pulses we insert n identical π pulses. A single π pulse is realised with the operation \(\hat U\left( {t_{{\rm{OL}}}/2n} \right)\hat R\left( \pi \right)\hat U\left( {t_{{\rm{OL}}}/2n} \right)\). The full Echo–Ramsey pulse sequence is then: \(\hat R\left( {\pi /2} \right)\left[ {\hat U\left( {t_{{\rm{OL}}}/2n} \right)\hat R\left( \pi \right)\hat U\left( {t_{{\rm{OL}}}/2n} \right)} \right]^n\hat R\left( {\pi /2} \right)\). The contrast decay C(t_{OL}) of a 4 × π pulse echoRI versus the holding time at various temperatures between 50 nK and 180 nK is shown in Fig. 5a. The solid lines are fits with an exponential function, which allow to extract the coherence time τ.
The relationship between the coherence time τ and the number of π pulses n is shown in Fig. 5b for three different temperatures (T = 50, 110 and 180 nK). The zerotemperature theoretical results of the coherence time can again be obtained by solving the GPE including the nonuniform lattice potential and the radial expansion, as shown by the circles in Fig. 5b. When further taking into account of the laser intensity fluctuation, the calculated coherence time is shown with the cross points. After implementing one π pulse, the two theoretical points merger together, which shows that the intensity fluctuation can be well suppressed with one π pulse; however, in order to eliminate the effects of nonuniform lattice potential and transverse expansion, more echo pulses are needed; after applying n = 6 echo pulses, the theoretical curves get nearly flat, showing that the two effects are well suppressed. There remains a significant discrepancy between the experimental and the zero temperature theoretical calculations, which we attribute to thermal fluctuations. Lowering the condensate temperature, the thermal fluctuations get weaker, and the coherence time can be greatly increased with multiple echopulses. The measured coherence time τ for the RI at T = 50 nK is 14.5 ms, which is one order of magnitude longer than the case without echopulses.
We will now look at the remaining fluctuations that cannot be suppressed by the echo technique: We fit the experimental contrast with C_{0}(t_{OL})exp[−βt_{OL}], where C_{0} is the calculated contrast for zero temperature and the decay rate β characterises the additional decay. For different temperatures, the fitted values of β are shown in Fig. 5c with red dots, showing a linear dependence on the temperature. This is a strong indication that the remaining decay of the interference contrast of the echo–Ramsey interferometer is caused by thermal fluctuations. To confirm this, we conducted finite temperature truncated Winger calculations^{45,46} (see “Methods” section)
Finally, we look at the performance of the echoRI for different lattice depth. As shown in Fig. 6a, the contrast decay rate strongly depends on the lattice depth for a condensate at 50 nK and using a sequence with six π pulses. The decay rate is small at medium optical lattice depth, and it becomes bigger for the shallow (<5E_{r}) and deep lattices (<10E_{r}). For the case of V_{0} = 4E_{r}, the separation between D and Pbands is small and it is hard to form a closed cycle between S and Dband without any transition to Pband in practice. We should note that for the two pulse RI, transition to other bands is not so evident due to the short lattice time. When the echo technique is applied, the coherence time is much increased, and transition to other bands induced by atomatom interaction become important. This can be seen in Fig. 6b, where the momentum distribution for 4E_{r} has components (marked with a red square) different from ±2ℏk.
When the lattice potential gets deeper, the band separation increases, and the coherence time increases accordingly. However, when the lattice depth is more than 10E_{r}, the coherence time decreases quickly. In fact the probability for atoms to be excited to higher momentum states by the pulse is proportional to the lattice depth. For a very deep lattice the atoms can easily obtain higher momenta^{47}. In Fig. 6b, the momentum distributions clearly show the components at 4ℏk for 20 E_{r}. Since the Rabi transition is designed for the transition between the S and Dband at zero quasimomentum, the higher states will lead to a reduced pulse fidelity and a faster decay of the contrast.
Discussion
In summary, we have demonstrated a Ramsey interferometer for atoms within an OL. We further employ a matter wave band echo technique to significantly enhance the coherence time by one order of magnitude. We have identified the mechanisms leading to the contrast decay for the RI signal. The contrast decay is closely related to the homogeneity of the optical lattice, laser intensity fluctuation and interactioninduced transverse expansion, and finite temperature dynamics of a BEC. All except for the effects of finite temperature can be suppressed by a matterwave band echo sequence. Thus, the damping from thermal fluctuations is well uncovered in this way.
So far, the π pulses and π/2 pulses for echoRI using TMQS are designed based on zero temperature single atom dynamics. In future developments, it would be interesting to unveil quantum many body dynamics in OL, like loop structures and swallowtails^{48,49,50}. These deliberate quantum control technologies could be applied in quantum information and precise measurements based on TMQS of atoms.
Methods
Design of Pi/2 and Pipulses
In a 1D optical lattice along xdirection, the eigenstates of the atom, i.e. the Bloch states, can be expressed as the superposition of momentum states \(\left {i,q} \right\rangle = \mathop {\sum}\nolimits_{l =  \infty }^{ + \infty } c_{i,l}\left( q \right)\left {2l\hbar k + q} \right\rangle ,\) where 2lℏk + q〉 is the basis in momentum space, i is the band index and q is the quasimomentum, c_{i,l}(q) is the superposition coefficient with l = 0, ±1, ±2,.... While the target state is considered to be the superposition of Bloch states, \(\psi = \mathop {\sum}\nolimits_i A_i\left {i,q} \right\rangle = \mathop {\sum}\nolimits_i A_i\mathop {\sum}\nolimits_l c_{i,l}\left( q \right)\left {2l\hbar k + q} \right\rangle\) with A_{ i } is the superposition coefficient.
Varying the pulse parameters, we numerically find a time sequence {t_{1} = 0, t_{2}, t_{3},...} of optical lattice pulses
that creates a \(\frac{\pi }{2}\) pulse in the Ramsey interferometer, i.e., after this operation, the atoms on Sband and Dband are transferred into target states \(\left( {\left S \right\rangle + \left D \right\rangle } \right)/\sqrt 2\) and \(\left( {  \left S \right\rangle + \left D \right\rangle } \right)/\sqrt 2\), respectively. In our designed pulse sequences, both the laser intensity and the duration can be changed. From the numerical simulation, we find that with a fixed laser intensity our designed pulse sequence can still reach high fidelity and is easier to realize in the real experiment. So we keep the lattice depth constant and only change the time sequence in the design for experimental simplicity without losing of fidelity.
The evolution operator for t_{2i−1} < t < t_{2i} is
and
for the free evolution operator at t_{2i} < t < t_{2i+1}, where p and m are,, respectively, the momentum and mass of a single atom. After applying the pulse sequence, the final states \(\left {\psi _{f_1}} \right\rangle\) and \(\left {\psi _{f_2}} \right\rangle\) can be written as
By changing the parameters of the pulse sequence, one can maximise the two fidelities F_{1} = \(F_1 = \left {\left\langle {\psi _{A_1}\psi _{f_1}} \right\rangle } \right^2\) and \(F_2 = \left {\left\langle {\psi _{A_2}\psi _{f_2}} \right\rangle } \right^2\). For V_{0} = 10E_{r}, the time sequence for π/2 pulse as {0, 56.2, 84.2, 106.8, 129.9}(μs) with F_{1} = 96.9%,F_{2} = 97.9%.
The method for designing π pulse is similar to the case of π/2 pulse, but with other target states. After the sequence of optical lattice pules, the atoms should be transferred from S〉 or D〉 into the target states D〉 or −S〉, respectively. For V_{0} = 10E_{r}, the time sequence for π pulse as {0, 49.2, 101.7, 123.8, 150.2}(μs), and the transfer of S〉 and D〉 to the target states are with the fidelity 98.5% and 98.0%, respectively.
During design of the pulse sequences, we have neglected atomatom interaction because the duration of pulses is short enough and atomatom interaction strength in our system is weak^{38}. Numerical results of the GPE with atomatom interaction show that the interaction leads to a change of fidelity <1% compared with our designed time sequence.
Data analysis
After the band mapping, we release the condensate from the trap and let it expand freely for 31 ms to perform time of flight (TOF) imaging. The TOF image shows the atoms from the Sband distributed in the center 0ℏk, while atoms from the Dband are distributed on the side zone ±2ℏk. We apply an algorithm to remove the background fringes^{51}, and integrate the atom distribution in ydirection of the TOF image. Then the atomic distribution is fitted with a bimodal function
with a Gaussian thermal distribution F_{ th } and a Thomas Fermi like the condensate. i = 1,2,3 corresponds to the three atom clouds for −2ℏk, 0ℏk, and 2ℏk, respectively. We also add a restriction to all six amplitude terms H_{ i } and G_{ i } as H_{1}/G_{1} = H_{3}/G_{3}. From the amplitude of each component H_{ i } and the width of condensate part χ_{ i },, we can determine the population of condensate in the Sband N_{ S } and in the Dband N_{ D }, and further calculate the population of the Dband as p_{D}(t_{OL}) = N_{D} / (N_{S} + N_{D}). The experimental contrast at a certain time is given by fitting the measured amplitude of oscillation p_{D} for 100 μs (about 2.5 periods) with a time step 5 μs using a cosine function, where the errorbar is given by the fitting error with 95% confidence bounds.
Contrast decay induced by atomatom interaction
When atoms collide in the excite band they can decay to a lower band^{52,53}. We calculate the collision decay of atom population in different bands using a simple rate equation model^{39},
where the factor K_{S(D)} is given by summation of crosssection of the twoatom inelastic collision from different channels. Solving Eq.(10), the population from the Dband N_{D}(t) decays as 1/(1 + t/τ_{D}), with the time constant of atom decay from the Dband given by 1/τ_{D} = K_{D}N_{D}. N_{S}(t) can similarly be given from the collision rate of atoms between the S and the Dbands.
We measure the collisional decay rate of atoms in different bands by band mapping without the final π/2 pulse and get τ_{D} = 1.9 ± 0.3 ms, τ_{S} = 5.1 ± 0.8 ms.
During the collision decay, the potential energy of the Dband is transferred to kinetic energy in radial direction. This energy is large enough so that the majority of the decayed atoms will not be counted in for the condensed peaks. After the population decay atom number N_{ S }(t) and N_{D}(t) is unbalanced, which leads to a reduction of the contrast by a factor \(\frac{{2\sqrt {N_SN_D} }}{{N_S + N_D}}\). From the measured N_{S} and N_{D}, we find that the influence of this population unbalance to the calculated contrast C_{0}(t_{ OL }) is less than 1% for an experiment with t_{ OL } < 2 ms as shown as the brown dashed line in Fig. 3a.
Applying the repeated πpulses, the coherence time is much longer than this decay time: (i) The πpulses reverse the population in the S and Dbands, and therefore prevent a strong imbalance. (ii) In the data analysis, extract the remaining condensed part by bimodal fitting. Both together dramatically reduce the effect of collisional decal on the contrast of the observed interference between the remaining coherent parts, and the collisional decay time does not limit the coherence time of our interferometer signal, as long as the remained population in condensed part is large enough for detection.
The contrast decay induced by the quasimomentum distribution and the nonuniform of optical lattice potential in radial direction
For these, we have to turn to numerical calculations taking the real potentials and their variation into account. Considering that the trapping frequencies in y,zdirections are very similar in our system, we can use use a mean field model (GPE) in cylindrical coordinates to describe the system at zero temperature. The evolution of the wave function Φ(r, t) is governed by
where r = (x, r, θ), V_{ext} is the external potential, and \(U_0 = \frac{{4\pi \hbar ^2a_s}}{m}\) is the interaction term with a_{ s } the scattering length.
In our case, the radial part of wavefunction is in the ground state and uniform in θ coordinate. The lattice potential itself depends on the radial position r. We first neglect the kinetic term in radial direction and separate the wavefunction into radial part and axial part as \({\mathrm{\Phi }}\left( {r,t} \right) = \frac{1}{{\sqrt {2\pi } }}\psi \left( {x,t} \right)\phi \left( r \right)\), then Eq. (11) can be simplified to the following 1d GPE at a certain value r = r_{ i } as
\(V_{{\mathrm{ext,r}}_{\mathrm{i}}}\) is the combination of both harmonic potential \(V_{{\mathrm{trap}}} = \frac{m}{2}\omega _x^2x^2\) and lattice potential \(V_{{\mathrm{latt}},r_i} = V_0Q\left( t \right)e^{  2r_i^2/w_{{\mathrm{latt}}}^2}{\rm{cos}}^2\left( {kx} \right),\) where Q(t) takes value 0 or 1 depending on the time sequence, and w_{latt} is the waist of lattice laser. \(\rho _{r_i} = \frac{{{\mathrm{exp}}\left[ {  m\omega _rr_i^2/\hbar } \right]}}{{\sqrt {\pi \hbar /m\omega _r} }}N\) is the linear density. We solve Eq. (12) to get the ψ(x, t) and the population in the Dband \(p_{D_i}\) for position r_{ i }. We then calculate 30 different radial positions and take their weighted average according to the atom number distribution \(\mathop {\sum}\limits_i {\left( {2\pi r_i\rho _{r_i} \cdot p_{D_i}} \right)} /\mathop {\sum}\limits_i {\left( {2\pi r_i\rho _{r_i}} \right)}\). Finally, the oscillation amplitude of the average p_{ D } is fitted to get a contrast as shown in dotted line in Fig. 3a.
In this simulation, we consider the wavefunction’s distribution instead of using a single atom model, thus the influence of quasimomentum distribution is also included automatically.
Contrast decay induced by radial motion of the condensate
In the above calculation, we consider the distribution of lattice potential in radial direction, however, the radial size of he wave function also changes with time. To consider this effect, we need to estimate the expansion speed of the atom cloud in this direction.
We can take each site of the lattice as a small independent BEC with about 1.5 × 10^{3} atoms (about 65 lattice sites), and calculate how it spreads after the trapping potential changed during switch on and loading into the lattice. When the lattice is turned on, the trapping frequency in xdirection increases to 20 kHz for 10E_{r}, which is much larger than the harmonic trap of 2π × 24 Hz, this sudden increase of trap frequency induces the spread in the radial direction, which can be calculated as^{54},
where \(\omega _x,\omega _r = \sqrt {\omega _y\omega _z}\) are the frequency of the effective trapping potential in x and rdirections for the small BECs in each lattice site, respectively, and λ_{ r } = r_{ i }(t)/r_{ i }(0) is the expansion with r_{ i }(0) the initially radial position. Using this timedependent radial expansion, we then follow a procedure like above to calculate the average p_{ D } and contrast as shown in dashed line in Fig. 3a of the main text.
Contrast decay induced by laser intensity fluctuations and thermal fluctuations
The laser intensity in our experiment fluctuates by about 0.1% during the holding time. The laser intensity changes are slow and do not cause excitation between the different bands. Simulations like above are then repeated with different laser intensities sampled from the measured fluctuations. The averaged result is shown in Fig. 3 of the main text with the purple dashdotted line.
To calculate that contrast decay induced by thermal fluctuations we apply the finitetemperature truncated Wigner method^{45}. We solve the 1d GPE with a stochastic initial wave function \(\psi \prime \left( x \right) = \psi + \mathop {\sum}\nolimits_j {\psi _j}\), where ψ is the zerotemperature condensate wavefunction within the onedimensional optical lattice, ψ_{ j } corresponding to the thermal fluctuations that is given by \(\psi _j = A\left( {r_0,\theta _0} \right)\left[ {u_j\left( x \right)\beta _j  v_j^ \ast \left( x \right)\beta _j^ \ast } \right]\), where \(A\left( {r_0,\theta _0} \right) = \sqrt {n_0\left( {0,r,\theta } \right)/{\int\!\!\!\int} n_0\left( {0,r,\theta } \right)r{\mathrm{d}}r{\mathrm{d}}\theta }\), with the condensation density n_{0}(x, r, θ). For simplicity and without loss of generality, we choose r_{0} = 0,θ_{0} = 0. Here β_{ j } is a complex number with random phase which satisfies \(\beta _j^ \ast \beta _j = \bar N_j + 1/2\), where the quantum fluctuations are included by the 1/2 term. in which \(\bar N_j = \mathop {\sum}\limits_i {1/\left( {e^{E_{ji}/k_BT}  1} \right)}\) and E_{ ji } = E_{ j } + Λ_{ i } is the summation of the energy of the j^{th} Bogoliubov mode E_{ j } plus the i^{th} eigenenergy Λ_{ i } of the radial harmonic potential. u_{ j } and v_{ j } are the j^{th} Bogoliubov modes solved from the onedimensional Bogoliubov de Gennes equation as^{45},
with
In the calculation, we use 300 excitation modes and repeat our simulation 15 times with different stochastic initial states. From the wave functions and the population distributions, we then obtain the contrast C(t_{OL}), calculated at the different holding times t_{OL}, which can then be compared to the experiment. In a separated calculation, we also calculate for the quantum fluctuations by taking \(\bar N_j = 0\), the result is shown in Fig. 3a of the main text with green dashed line, which is close to the result without quantum fluctuations, where the coherent time is only reduced by 0.2%.
Data availability
The authors declare that the main data supporting the findings of this study are available within the article. Extra data are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank Z. K. Chen for the calculation of pulse time sequences and discussion. Thank P. Zhang, Q. Zhou, and B. Wu for helpful discussion. This work is supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301501), and the NSFC (Grant Nos. 11334001, 61727819, 61475007, 91736208). G. J. Dong acknowledges the support by the National Science Foundation of China (Grant No. 11574085 and No. 91536218), and 111 Project (B12024), and the National Key Research and Development Program of China (Grant No. 2017YFA0304200). JS acknowledges support by the European Research Council, ERCAdG QuantumRelax.
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X.Z. designed the experiment, D.H., S.J. and L.N. performed the measurement, L.N. and D.H. analysed the data supervised by X.Z. and J.S., S.J. and L.N. did the theoritical calculations and modelling supervised by X.Z., J.S., G.D. and X.C.. All authors contributed to the interpretation of the result and writing of the manuscript. D.H. and L.N. contributed equally to this work.
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Hu, D., Niu, L., Jin, S. et al. Ramsey interferometry with trapped motional quantum states. Commun Phys 1, 29 (2018). https://doi.org/10.1038/s4200501800307
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