Observation of Quantum Criticality with Ultracold Atoms in Optical Lattices

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Science  02 Mar 2012:
Vol. 335, Issue 6072, pp. 1070-1072
DOI: 10.1126/science.1217990


Quantum criticality emerges when a many-body system is in the proximity of a continuous phase transition that is driven by quantum fluctuations. In the quantum critical regime, exotic, yet universal properties are anticipated; ultracold atoms provide a clean system to test these predictions. We report the observation of quantum criticality with two-dimensional Bose gases in optical lattices. On the basis of in situ density measurements, we observe scaling behavior of the equation of state at low temperatures, locate the quantum critical point, and constrain the critical exponents. We observe a finite critical entropy per particle that carries a weak dependence on the atomic interaction strength. Our experiment provides a prototypical method to study quantum criticality with ultracold atoms.

In the vicinity of a continuous quantum phase transition, a many-body system enters the quantum critical regime, where quantum fluctuations lead to nonclassical universal behavior (1, 2). Quantum criticality not only provides novel routes to new material design and discovery (1, 36), but also provides a common framework for problems in condensed matter, nuclear physics (7, 8), and cosmology (1, 9). Quantum criticality plays a central role in strongly correlated systems such as heavy-fermion materials (5), spin dimer systems (10), Ising ferromagnets (11), and chromium at high pressure (12).

Ultracold atoms offer a clean setting for quantitative and precise investigation of quantum phase transitions (1316) and critical phenomena (17). For example, the superfluid-to-Mott insulator quantum phase transition can be realized by loading atomic Bose-Einstein condensates into optical lattices (13). In recent experiments, scaling behavior of physical observables was reported in interacting Bose gases in three (17) and two dimensions (18), and in Rydberg gases (19), where collective behavior is insensitive to microscopic details. In addition, suppression of the superfluid critical temperature near the Mott transition was observed in three-dimensional (3D) optical lattices (20). Studying quantum criticality in cold atoms on the basis of finite-temperature thermodynamic measurements, however, remains challenging and has attracted increasing theoretical interest in recent years (2124).

We report the observation of quantum critical behavior of ultracold cesium atoms in a two-dimensional (2D) optical lattice across the vacuum-to-superfluid transition. This phase transition can be viewed as a transition between a Mott insulator with zero occupation number and a superfluid, and can be described by the Bose-Hubbard model (25). Our measurements are performed on atomic samples near the normal-to-superfluid transition, connecting to the vacuum-to-superfluid quantum phase transition in the zero-temperature limit.

The quantum phase transition and quantum critical regime in this study are illustrated in Fig. 1. The zero-temperature vacuum-to-superfluid transition occurs when the chemical potential μ approaches its critical value μ0. Sufficiently close to the quantum critical point, the critical temperature Tc for the normal-to-superfluid transition is expected to decrease according to the following scaling (25)

Fig. 1

The vacuum-to-superfluid quantum phase transition in 2D optical lattices. At zero temperature, a quantum phase transition from vacuum (horizontal thick blue line) to superfluid occurs when the chemical potential μ reaches the critical value μ0. Sufficiently close to the transition point μ0, quantum criticality prevails (red shaded area), and the normal-to-superfluid transition temperature Tc [measurements (30) shown as empty circles] is expected to vanish as Tc ~ (μ − μ0)zν; the blue line is a guide to the eye. From the prediction zν = 1 (22, 23, 25), the linearly extrapolated critical chemical potential is μ0 = −3.6(6)t, consistent with the theoretical value −4t (28). Both the thermal energy scale kBT and the chemical potential μ are normalized by the tunneling t.

kBTct=c(μμ0t)zν (1)

where kB is the Boltzmann constant, t is the tunneling energy, z is the dynamical critical exponent, ν is the correlation length exponent, and c is a constant. In the quantum critical regime (shaded area in Fig. 1), the temperature T provides the sole energy scale, and all thermodynamic observables are expected to scale with T (25). Thus, the equation of state is predicted to obey the following scaling (21)

N˜=F(μ˜) (2)

in which F(x) is a generic function, and

N˜=NNr(kBTt)Dz+11zν and  μ˜=μμ0t(kBTt)1zν(3)

are the scaled occupation number and scaled chemical potential, respectively. Here, N is the occupation number, D is the dimensionality, and Nr is the nonuniversal part of the occupation number. For the vacuum-to-superfluid transition in the 2D Bose-Hubbard model, we have Nr = 0 and D = 2, and the predicted critical exponents are z = 2 and ν = 1/2, characteristics of the dilute Bose gas universality class (2, 22, 25). We note that in a 2D system, there can be logarithmic corrections to scaling functions, including those in this study, near the quantum critical point (2). Within the temperature range of our experiment, however, the measurement is consistent with the above scaling laws in the absence of logarithmic corrections. Scaling behavior of Tc in the quantum critical regime was also observed in 2D condensates of spin triplets (10).

Our experiment is based on 2D atomic gases of cesium-133 in 2D square optical lattices (26, 27). The 2D trap geometry is provided by the weak horizontal (r-) confinement and strong vertical (z-) confinement (27), with envelope trap frequencies fr = 9.6 Hz and fz = 1940 Hz, respectively. Typically, 4000 to 20,000 atoms are loaded into the lattice. The lattice constant is d = λ/2 = 0.532 μm and the depth is VL = 6.8 ER, where ER = kB × 63.6 nK is the recoil energy, λ = 1064 nm is the lattice laser wavelength, and h is the Planck constant. In the lattice, the tunneling energy is t = kB × 2.7 nK, the on-site interaction is U = kB × 17 nK, and the scattering length is a = 15.9 nm. The sample temperature is controlled in the range of 5.8 to 31 nK.

We determine the equation of state n(μ,T) of the sample from the measured in situ density distribution n(x,y) (18, 26). The chemical potential μ(x,y) and the temperature T are obtained by fitting the low-density tail of the sample where the atoms are normal. The fit is based on a mean-field model that accounts for interaction (2830). Equation of state measured near the quantum critical point can reveal essential information on quantum criticality, as proposed in (28).

We locate the quantum critical point by noting that at the critical chemical potential μ = μ0, the scaled occupation number N˜=Nt/kBT=nd2t/kBT is temperature-independent, as indicated by Eq. 2. Here, we have applied a predicted exponent ν = 1/2. We plot N˜ as a function of μ in the low-temperature range of 5.8 to 15 nK, and indeed observe a crossing point at μ0 = −4.5(6)t (Fig. 2A, inset). We identify this point as the critical point for the vacuum-to-superfluid transition, and our result agrees with the prediction −4t (28).

Fig. 2

Evidence of a quantum critical regime. (A) Scaled occupation number Embedded Image as a function of the scaled chemical potential Embedded Image; the blue solid line shows the average curve for the lowest four temperatures. Inset shows the low-temperature data in the range of T = 5.8 to 15 nK, and the critical chemical potential μ0 is identified as the crossing point; see text. The result, μ0 = −4.5(6) t, agrees with the prediction μ0 = −4t (28). (B) Determination of the dynamical critical exponent z and the correlation length exponent ν based on μ0 = −4.5t. The color represents the reduced chi-squared (χ2), and indicates how well the scaled equation of state can collapse into one single curve. The exponents are determined as Embedded Image and Embedded Image, and the uncertainties correspond to a 95% confidence level. The predicted values are z = 2 and ν = 1/2.

To test the critical scaling law, we compare the equation of state at different temperatures. On the basis of the expected exponents z = 2 and ν = 1/2, we plot the scaled occupation number N˜ as a function of the scaled chemical potential μ˜=(μμ0)/kBT (Fig. 2A). Below 15 nK, all the measurements collapse into a single curve, confirming the emergence of the quantum critical scaling law (Eq. 2) at low temperatures. Note that we observe scaling behavior at temperatures from 5.8 to 15 nK, which are high compared to the tunneling scale t/kB = 2.7 nK. Deviations become obvious at higher temperatures.

We examine the range of critical exponents z and ν that allow the scaled equation of state at low temperatures to overlap within experimental uncertainties. Taking μ0 = −4.5t and various values of z and ν in the range of 0 < z < 4 and 0 < ν < 1, we compute the corresponding scaled occupation numbers N˜ and scaled chemical potentials μ˜ based on Eq. 3. We then evaluate how well the scaled equation of state in the range of T = 5.8 to 15 nK can collapse to a single curve by computing the reduced chi-squared (30). The best-fit exponents (Fig. 2B) are determined as z=2.20.5+1.0 and ν = 0.520.10+0.09, and the uncertainties correspond to a 95% confidence level. On the basis of the theoretical value of μ0 = −4t, we find the exponents to be z= 2.60.6+1.2 and ν = 0.44(8). In the following analyses, we adopt z = 2, ν = 1/2, and μ0 = −4.5t.

Our measurements at different temperatures allow us to investigate the breakdown of quantum criticality at high temperatures. To quantify the deviations, we focus on the temperature dependence of the scaled occupation number N˜ at the critical chemical potential μ = μ0 (Fig. 3). Deviations from the low-temperature value are clear when the temperature exceeds T* = 22 nK ≈ 8t/kB. From this, we conclude that at μ = μ0, the upper bound of thermal energy for the quantum critical behavior in our system is kBT* ≈ 8t. Our result is in fair agreement with the prediction of 6t based on quantum Monte Carlo calculations (23).

Fig. 3

Finite-temperature effect on quantum critical scaling. Scaled occupation number Embedded Image at the critical chemical potential μ = μ0 as a function of the normalized temperature kBT/t. The black dashed line is an empirical fit, giving a temperature scale T* ≈ 8t/kB. For T < T*, Embedded Image is independent of the temperature; for T > T*, Embedded Image deviates from the low-temperature value.

From the equation of state, one can derive other thermodynamic quantities in the critical regime. We derive the entropy per particle S/N based on measurements in the temperature range of 5.8 to 15 nK, using a procedure similar to (31). The measured entropy per particle depends only on the scaled chemical potential μ˜ and monotonically decreases (Fig. 4A), indicating a positive specific heat. Near the critical point μ˜=0, the entropy per particle has an approximate linear dependence on the scaled chemical potential: S/NkB=abμ˜, with a = 1.8(1), b = 1.1(1). From this linear dependence, we derive an empirical equation of state analogous to the ideal gas law (30)

P=Cnx(kBT)y (4)

where P is the pressure of the 2D gas, x = 2/(1 + b) = 0.95(5), y = 2b/(1 + b) = 1.05(5), C = 0.8(2)(td2)w is a constant, and w = (1 − b)/(1 + b) = −0.05(5).

Fig. 4

Entropy per particle in the critical regime. (A) Entropy per particle S/N as a function of the scaled chemical potential Embedded Image, measured in the temperature range of 5.8 to 15 nK (same symbol and color scheme as in Fig. 2A). (B) Critical entropy per particle Sc/N as a function of the effective interaction strength g: measurements for Bose gases with 2D optical lattices (black circle) and without lattice [black triangles, extracted from data in (18)], mean-field calculations (blue line), and a power-law fit to the measurements, Sc/NkB = 1.6(1)g0.18(2) (red line).

Finally, we observe a weak dependence of the critical entropy per particle on the atomic interaction. Noting that a weakly interacting 2D Bose gas follows similar scaling laws near μ = 0 (18) because it belongs to the same underlying dilute Bose gas universality class (2, 32), we apply similar analysis and extract the critical entropy per particle Sc/N at four interaction strengths g ≈ 0.05, 0.13, 0.19, 0.26, shown together with the lattice data (g ≈ 2.4) in Fig. 4B. We observe a slow growing of Sc/N with g, and compare the measurements with mean-field calculations. The measured Sc/N is systematically lower than the mean-field predictions, potentially as a consequence of quantum critical physics. The weak dependence on the interaction strength can be captured by a power-law fit to the data as Sc/NkB = 1.6(1)g0.18(2).

In summary, on the basis of in situ density measurements of Bose gases in 2D optical lattices, we confirm the quantum criticality near the vacuum-to-superfluid quantum phase transition. Our experimental methods hold promise for identifying general quantum phase transitions, and prepare the tools for investigating quantum critical dynamics.

Supporting Online Material

Materials and Methods

Fig. S1

References (33, 34)

References and Notes

  1. Materials and methods are available as supporting online material on Science Online.
  2. Acknowledgments: We thank N. Prokof'ev and D.-W. Wang for discussions and numerical data; Q. Zhou, K. Hazzard, and N. Trivedi for discussions; and N. Gemelke and C. Parker for discussions and reading of the manuscript. The work was supported by NSF (grants PHY-0747907 and NSF-MRSEC DMR-0213745), the Packard foundation, and a grant from the Army Research Office with funding from the Defense Advanced Research Projects Agency Optical Lattice Emulator program. The data presented in this paper are available upon request sent to
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