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Abstract
Manipulation of ultracold atoms in optical lattices is one of the optimal ways to observe phase transitions of the Hubbard model which is useful in a variety of condensed-matter systems. Bosonic atoms in this model experience a phase transition from superfluids to Mott insulators by tuning systematic parameters. However, in conventional setups, phase transitions take place over a large range of parameters instead of one critical point due to the background inhomogeneity caused by the Gaussian shape of optical-lattice lasers. To probe the phase transition point more precisely in our lattice system, we apply a blue-detuned laser to compensate for this local Gaussian geometry. By inspecting the change of visibility, we find a sudden jump point at one particular trap depth of optical lattices, corresponding to the first appearance of Mott insulators in inhomogeneous systems. This provides a simple method to detect the phase transition point in such inhomogeneous systems. We believe it will be a useful tool for most cold atom experiments.
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1. Introduction
Ultracold atoms in optical lattices provide a versatile platform for simulating the Hubbard model which is an important description of many condensed-matter systems [1–12]. Most optical lattices are formed by the interference of laser beams. The interference patterns serve as a periodic potential to neutral atoms through optical dipole force. Such systems can be described by the Bose-Hubbard model, in which the superfluid-Mott insulator quantum phase transition has been observed. Each atom can tunnel between neighboring sites and two atoms in one site provide an on-site interaction, and the competition between the tunneling (kinetic energy) and on-site interaction energy drives the system to undergo a continuous second-order phase transition from long-range coherence (superfluid) to localized phase (Mott insulator) [13]. This phenomenon was first observed in 2002 [14], and still being studied globally [15–19].
However, due to the Gaussian shape of laser beams, the intensity distribution of the optical lattices is not uniform, resulting in a Gaussian-shape background superimposed over the periodic potential. Owing to the ultracold temperatures of atoms, the Gaussian-shape trap approximately converts into a harmonic trap in the central region [5–7]. Then the inhomogeneity induces a non-constant chemical potential distribution, extending the phase transition point into a range of parameters [20,21]. In Fig. 1, we show a typical quantum phase diagram with a non-constant chemical potential. It shows that the inhomogeneity will induce the coexistence of different phases where the central part contains different Mott-insulator shells and the most outside part or transition parts contain superfluid. The coexistence of different phases induces the mixture of signals from different phases. Therefore, it is hard to probe a sudden jump of observables when the system experiences the phase transition, especially in a three-dimensional (3D) system.
Fig. 1. An illustration of the Bose-Hubbard phase diagram with an inhomogeneous background potential. (a) shows a typical phase diagram with a non-constant chemical potential. The red arrow shows a range of the chemical potential in the trap. When the Mott insulator first appears, the corresponding trap depth is at 13$E_r$ for rubidium 87. (b) shows a typical structure of phase coexistence during the phase transition. The central parts contain Mott-insulator shells corresponding to different $n$ where $n$ is the atom number per site. At the most outside part or in the regime between different Mott-insulator shells, the atoms are in superfluid where the atom number per site is not a well-defined quantum number. When we observe the phase transitions, the signals from the superfluid and Mott insulators both contribute during the coexistence.
Recently, there are new techniques to create homogeneous potentials [22,23] including spatial light modulators [24–26], digital micromirror devices [27] and specialized fixed optics [28–31]. These have been applied to create uniform degenerate quantum gases [28,32] achieving varieties of new phenomena [33–35]. However, these techniques require precisely-shaped laser beams and advanced-manufactured optics. Intuitively, one can cancel the harmonic confinement by imposing an anti-trap formed by lasers with opposite detunings from the atomic transition frequency. This approach has been used to increase cycles of quantum phase revivals [36] and observe the Bose-Fermi oscillations in a 1D system [37].
In this work, we develop a simpler but more sensitive approach to detect the QPT point in our 3D optical-lattice system [38–40]. Inspired by the compensation approach mentioned above, we add an external blue-detuned anti-trap to partially compensate for the harmonic background of the trap potential. We focus on the visibility of interference patterns of atoms released from lattice confinement. We find the visibility will drop when the anti-trap power is increased. When the power is above a certain threshold, a kick point appears, after which the visibility decreases much faster. If we look at the threshold versus the trap depth of optical lattices, there will be a sudden jump at one particular trap depth where the Mott insulators start to appear in the central region of the trap. This sudden jump shows a higher sensitivity to the QPT point compared to the previous methods [15,16,41,42]. Besides, we confirm this jump signal by observing the response of the fraction of coherent atoms. By tuning the anti-trap, we discover different response behaviors between superfluids and Mott insulators. We believe this approach can be used to probe quantum phase transitions in similar inhomogeneous systems.
2. Experimental setup
The bosonic atoms in optical lattices can be described by a Bose-Hubbard model, with a Hamiltonian written as
Here $a_i$ is the annihilation operator of one particle at lattice site $i$, and $n_i=a^\dagger _i a_i$ is the particle number operator. $J$ is the tunneling strength, and $U$ is the on-site interaction. $J$ and $U$ can be expressed by the trap depth $V_L$ of optical lattices over a range from 8$E_\text {r}$ to 30$E_\text {r}$ [41], as
Here $a_s$ is the s-wave scattering length, $\lambda _0$=1064 nm is the wavelength of the lattice lasers. $V_L$ is the trap depth of lattice laser beams in the unit of $E_r$ while $E_\text {r}=h^2/{2m\lambda _{0}^2}=h\times 2$ kHz is the recoil energy for rubidium 87 atoms. The Gaussian shape of lattice beams provide an isotropic harmonic trap $\frac {1}{2}m\omega ^2 r^2_i$ locally, and the vibrational frequency $\omega$ is $2\pi \times$ 44 Hz at the trap depth $V_L=5E_\text {r}$ in our systems, while $\omega$ has a scaling $\omega \propto \sqrt {V_L}$.
Our system is a three-dimensional simple cubic optical lattice filled with around $N=1.1(2)\times 10^5$ ultracold $^{87}$Rb atoms. We have three pairs of retro-reflected red-detuned laser beams at a wavelength of 1064 nm shooting along $x$, $y$ and $z$ axis and two blue-detuned laser beams at a wavelength of 532 nm propagating along the diagonal directions (45$^\circ$ and 135$^\circ$) in the $x$-$y$ plane as labeled in Fig. 2(a). Before loading atoms into lattices, we prepare a degenerate atomic sample with a radius of $\sim 10\mu$m, a temperature below 50 nK, and a condensate fraction higher than 88%.
Fig. 2. The experimental setup and time sequences. (a) shows the experimental setup of our optical-lattice system. The red-detuned optical lattices are held along the $x$, $y$, and $z$ axis, and the blue-detuned anti-trap is placed along the 45$^\circ$ ($\hat x$) and 135$^\circ$ ($\hat y$) directions. (b) shows the experimental time sequences. The red line denotes the ramping curve of red-detuned optical lattices and the green lines denote the curves of blue-detuned anti-trap with optical power ranging from 1 mW to 100 mW. (c) shows the potential combination along the z-axis. The red line denotes the envelope of the lattice potential, and the blue solid line represents the anti-trap potential. The black solid line denotes the effective potential after the combination. The left panel describes the combination when $P_a/P_L=0.125$, and the right part represents the case $P_a/P_L=0.0625$
When we load atoms into optical lattices, we both ramp up the lattice beams and the blue-detuned anti-trap lasers, the ramping curves are shown in Fig. 2(b). In the loading procedure, we start with an exponential ramping-up curve in 80 ms according to Ref. [14], which is close to an adiabatic process. Then we hold atoms 20 ms for relaxation at the target lattice depth $V_L$. Here the lattice trap depth can be calibrated within 1.3% relative errors by the multiple pulses Kapitza-Dirac diffraction along with the BPNN algorithm [43]. When the detuning is large enough compared to the excited state fine structure, $V_L$ can be evaluated through:
Here constant 4 denotes that the lattice depth is four times larger than the depth of the dipole trap without retro-reflection, c is the light speed, $\omega _0$ is the atomic resonant frequency, $\Gamma$ is the spontaneous decay rate, $\delta _L$ is the detuning relative to the center of the D1 and D2 line, $P_L$ is the optical power of the lattice lasers and $w_L$ is the beam waist radius of 140(1) $\mu m$.
During this process, we simultaneously ramp up the anti-trap lasers, and this provides an anti-trap depth $V_a$ with $V_a=3\pi \text {c}^2/(2 \omega _0^3) \cdot (\Gamma /\delta _a)\times 2 P_a/(\pi w_a^2)$, where $P_a$ is the optical power of the anti-trap lasers, $\delta _a$ is the detuning, and $w_a$=42(1) $\mu m$ is the beam waist of anti-trap lasers.
Taking all lasers into consideration, we obtain the effective potential $V_\text {eff}$ written as
Here $\hat {x}=(x+y)/\sqrt {2}$ denotes the 45$^\circ$ direction, and $\hat {y}=-(x-y)/\sqrt {2}$ marks the 135$^\circ$ direction. $V_\text {offset}=3V_L-2V_a$ is regarded as the offset of the potential at the trap center. In Fig. 2(c), we show the effective potential along $z$ direction with $x=y=0$. In the central region of the trap, the effective potential can be further approximated as
Here the vibrational frequency $\omega =\sqrt {8V_L/(m w_L^2)}$, and $\omega _a=\sqrt {8V_a/(m w_a^2)}$. When the anti-trap is increased from zero, the harmonic confinement can be cancelled firstly in $z$ direction. Note that when $P_a/P_L=0.0605$, $\omega =\omega _a$, there will be an effective “flat” potential over a range of about 40 $\mu m$ in the central region along $z$ axis as shown in Fig. 3. The trap depth of $V_\text {eff}$ at the position labeled by the dashed vertical line is 2% of the lattice depth, which is close to the interaction energy $U$. When $P_a/P_L>0.0605$, the potential in the center is not the minimal point anymore.
Fig. 3. The effective “flat” potential distribution along $z$ axis. The red curve denotes the lattice potential without anti-trap, and the black one represents the effective “flat” potential after the compensation. The dashed vertical lines define the region of “flat” potential. The space frequency shown in this figure is 6 times the real lattice constant $d=\lambda _0/2=532\,\text {nm}$.
3. Measurement of visibility and coherent fraction
Once the lattice sample is prepared, we suddenly turn off all trap lasers and the atoms start to expand freely for the time-of-flight measurement (TOF). After 30 ms, the absorption images are taken along $z$ axis showing the matter-wave interference patterns in $x$-$y$ plane as a momentum distribution $n(k_x,k_y)$. In Fig. 4, we show the momentum distributions at the trap depth from 5$E_\text {r}$ (a superfluid) to 20$E_\text {r}$ (a Mott insulator) without the blue-detuned anti trap. To further analyze the data quantitatively, we normalize the distribution $n(k_x,k_y)$ with the particle number $N$, integrate the distribution along the $y$ axis, and obtain a one-dimensional distribution $n(k_x)$ along the $x$ axis. Then we use a combined Gaussian curve to fit the distribution as
Here $A_0$, $\sigma _0$ denote the height and width of the central peak, $A_l$, $\sigma _l$ ($A_r$, $\sigma _r$) denote the height and width of the left (right) peak at the momentum $2\hbar k_{0}$ (-$2\hbar k_0$). $A_d$, $\sigma _d$ denote the height and width of the Gaussian background consists of incoherent atoms. Based on the fitting, we calculate the incoherent atom number $N_d$ denoted by the red shadow areas shown in Fig. 4(a) and (c). By substracting the incoherent atoms, we obtain the coherent part $N_0=N-N_d$ from which we can derive the coherent fraction.
Fig. 4. Superfluid-to-Mott insulators transition without anti-trap. (a) and (c) show a series of momentum distributions $n(k_x)$ along the $x$ direction, obtained from integration of 2D interference patterns $n(k_x,k_y)$ shown in (b) and (d). From left to right, the lattice depths are labeled as white texts, showing a gradient phase transition from superfluids to Mott insulators. Red solid lines denote the experimental data, and black dashed lines denote our combined Gaussian curve fitting. The red shadow areas denote the distributions of the incoherent atoms obtained from the fitting results mentioned in main text. Each data is averaged from 5 to 10 samples.
Besides the analysis of coherent and incoherent atoms, we analyze the visibility $\mathcal {V}$ of the-matter wave interference pattern [41], which is defined as
Here $N_\boxplus$ denotes the sum of the atom numbers in four boxes at the position of the first order diffraction peaks and $N_\boxtimes$ denotes four boxes at the same distance from the center along diagonal directions as shown in Fig. 5. The full width of the counting boxes is chosen to be 15% of $2\hbar k_0$.
Fig. 5. Extraction of visibility. The central peak corresponds to the zeroth-order momentum peak while the first-order diffraction peaks locate at $\pm 2\hbar k_0$ along $x$ or $y$ direction. The black boxes circle out the region for calculating $N_\boxplus$ and $N_\boxtimes$.
Based on these two observations, we can find a phase transition process when the system moves into Mott insulators from superfluids. While due to the inhomogeneity of the chemical potential distribution, the change becomes gradual as shown in Fig. 6. In such observations, one can not easily point out where the phase transition start to appear.
Fig. 6. Gradual changes during the SF-MI transition. (a) shows the fraction of coherent atoms versus the lattice depths. (b) shows the visibility versus the lattice depths.
4. Results and discussion
For further investigation, we start to ramp the anti trap at the same time of lattice loading. We change the optical power $\Delta$ of the anti trap, over a range from 1 mW to 100 mW, and we measure the response of visibility $\mathcal {V}$ and the fraction of coherent atoms $N_0/N$, with results shown in Fig. 7 and Fig. 8.
Fig. 7. Kick point measurement and different behaviors in response to $\Delta$ between SF and MI. (a) shows the visibility versus $\Delta$, the optical power of the anti-trap. We prepare steady states at 7, 9, 11, 13 and 15$E_\text {r}$ with adiabatic loading of adjustable anti-traps. After applied a piecewise linear fitting in the logarithmic scale, each curve is divided into two parts by a kick point labeled as a black solid diamond. In (b), we plot the kick points $\Delta _k$ versus the lattice depths $V_L$. The black solid line denotes the linear fitting result of kick points at lattice depths below 15$E_\text {r}$, and the red line represents the predicted values of $\Delta$ we need to compensate the harmonic trap into a “flat” one. (c) shows the decay rates $\alpha$ obtained from the fitting results in (a). The dashed vertical line denotes the lattice depth $V_L$ equals to 13$E_\text {r}$, which is the quantum phase transition point of the 3D Bose-Hubbard model. All the error bars denote one standard deviation, and the horizontal error bars are smaller than the marker size.
Fig. 8. Measurement of fraction $N_0/N$ versus the anti-trap depth $V_a$. (a) shows the fraction of coherent atoms $N_0/N$ versus $V_\text {a}(\Delta )$ (in the unit of interaction energy U). The black empty diamonds denote the same values of $\Delta _k$ as the solid ones shown in Fig. 7(a), and each point divides the corresponding curve into a linear changing part and an exponential decay. (b) shows the slopes $k$ of the linear parts in the panel (a). The grey shaded area is $95\%$ confidence region for linear fit, and the dashed vertical line denotes the quantum phase transition point. All the error bars denote one standard deviation, and the horizontal error bars are smaller than the marker size.
In Fig. 7(a), we list the data of visibility measurements at five lattice trap depths, 7, 9, 11, 13, and 15$E_\text {r}$. For each depth, we acquire the response curve of visibility by gradually increasing the anti-trap. From each curve, we find a faster exponential decay when the anti-trap is tuned above a certain threshold. We call this thresholds as kick points. To determine the kick points, we piecewise fit the curves as two linear parts with different slopes in the logarithmic scale,
Here $\mathcal {V}_0$ is the offset of the visibility and $\Delta _k$ is the optical power of kick points labeled as black solid diamonds in Fig. 7(a). Therefore, $\mathcal {V}_k=\mathcal {V}_0 -s\ln \Delta _k$ is the visibility at the kick point. From the fitting results, we find $\Delta _k$ linearly dependent on the lattice depths $V_L$ before the quantum many-body states across the phase transition point, 13$E_\text {r}$. We find that the values of $\Delta _k$ agree with the predicted values of $\Delta$ needed for obtaining the “flat” potential against different lattice depths $V_L$, corresponding results are shown in Fig. 7(b). However, there is an obvious deviation when the system reaches a Mott insulator at 15$E_\text {r}$. It means that the visibility of a insulator turns to drop faster before the effective potential becomes “flat”. If we focus on the decay rates of the latter parts after the kick points, there is a prominent difference between superfluids and Mott insulators as shown in Fig. 7(c). The deviation and the difference together provide a sudden jump signal revealing the QPT point, 13$E_\text {r}$.
According to previous researches [36,41], $\mathcal {V}$ is monotonically related to the theoretical quantity $\sum _i|\langle \hat {a}_i\rangle |^2$, which is strongly dependent on the tunneling strength $J$ of the lattice system. When the anti-trap is turned on, distribution of the order parameter $\langle \hat {a}_i\rangle$ is changed along each lattice site, determining the local or global depression on the tunneling.
Before the trap becomes “flat”, the repulsion force of the blue-detuned dipole anti-trap depress the tunneling in local domains, resulting in the slightly decay at first. In superfluid states, tunneling plays an important role in ground states, thus global existence of $\langle \hat {a}_i\rangle$ can barely break down before anti-trap reaches kick point. While in Mott insulator states, on-site interaction $U$ become dominant and the correlation length is confined within local sites, which depress the relation between tunneling and trap flatness. As a result, we can see a strong linear relation between $\Delta$ and $V_L$ over a range from 7 to 13$E_\text {r}$ in Fig. 7(b), while an obvious deviation appears at 15$E_\text {r}$ denoting the appearance of a Mott insulator.
When the anti-trap power crosses the kick point, the combined potential becomes convex upward forming a global depression on the tunneling, which is the reason of the obvious decay afterward. We then plot the decay rates $\alpha$ versus trap depths $V_L$ in Fig. 7(c), we find the decay rates in Mott insulator states much larger than that in superfluid states. It’s the consequence of the competition between the scale of the correlation length and the scope of global anti-trap depression. In superfluids, atoms can tunnel through several lattice sites, even wave-like spread over the whole lattice area in a shallow lattice trap. The depression from anti-trap in such case can be reduced, resulting in a small decay rate $\alpha$. Once the system reaches the critical state, the correlation length is suddenly suppressed no longer than the width of a single site, the tunneling depression from anti-trap becomes extremely significant, and this leads to an obvious rapid decay.
Then, we analyze the fraction of coherent atoms $N_0/N$ versus the anti-trap depth $V_a$ in the unit of interaction energy U which can be evaluated through Eq. (3), the results are shown in Fig. 8(a). The fraction is influenced by the presence of the anti-trap. At shallow lattice depths, the fraction $N_0/N$ drops instantly when the anti-trap is on. While at deeper depths, the fraction fluctuates around some constants or even rises up when the anti-trap is increased below certain thresholds, after which all the response curves begin to drop. Inspired by the analysis of visibility, we take kick points obtained above as fixed points, bringing in another piecewise fitting including a linear region before the kick point and an exponential decay afterwards. As a result, we find the linear slopes $k$ linearly dependent on the lattice depths $V_L$, especially when the system reaches the QPT point, the influence on the fraction $N_0/N$ from the anti-trap vanishes to zero as shown in Fig. 8(b). The intersection of the confidence region in Fig. 8(b) and the horizontal line $k = 0$ tells us the critical region with 95% confidence. This result confirms the phase transition point at $12.82(26)E_r$ where the error bar corresponds to one standard deviation.
Here the reduction of fraction $N_0/N$ is affected by different reasons for different phases. For superfluid, flattening the trap is reducing the external confinement of ultracold atoms while the atoms will expand with a larger effective volume. This decreases the phase space density, and the coherent fraction $N_0/N$ reduces as well. However, in the Mott insulator region, atoms are localized in each lattice site with an integer occupation number, so the condensed atoms only take the lattice sites among shell-like insulator domains caused by the original harmonic trap. When the effective volume becomes larger, the trap cannot hold these localized atoms because there are more available sites for atoms to tunnel. Therefore, the shell-like insulator starts to soften. The atoms tend to tunnel around and this increases the coherent fraction. Especially when the many-body states cross the critical point, the decrease of the phase space density and the reduction of insulator domains caused by trap flatness reach a balance, which leads to the result $k=0$.
5. Conclusions
We add an external blue-detuned anti trap to bosonic ultracold atoms trapped by red-detuned optical lattices. In superfluids, the visibility of interference pattern begins to fall when we increase the optical power of the anti-trap. We find an obvious kick point appear when the composite trap potential becomes “flat” in trap center. When we raise the lattice potential into Mott insulator region, the kick point suddenly becomes smaller than the predicted value. This deviation is related to the appearance of Mott insulators in Bose-Hubbard models. Besides, we obtain similar jumping signals through the response of both decay rates of visibility and fractions of coherent atoms versus the anti trap depths. Through these signals, we can determine a narrow range or a more precise value of the quantum phase transition point in an inhomogeneous system. This method can be widely used varieties of ultracold atom experiments. Meanwhile, we believe it can also be extended to probe local phase transitions or dynamics with irregular potential or phase coexistences by such a focused laser beam [44], including the systems such as two-dimensional materials [45,46] and nitrogen-vacancy centers [47,48].
Funding
National Natural Science Foundation of China (11920101004, 11934002, 11974202, 92165203); National Key Research and Development Program of China (2021YFA0718300, 2021YFA1400900).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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