## Abstract

The phase transition of Bose-Einstein condensation was studied in the critical regime, where fluctuations extend far beyond the length scale of thermal de Broglie waves. We used matter-wave interference to measure the correlation length of these critical fluctuations as a function of temperature. Observations of the diverging behavior of the correlation length above the critical temperature enabled us to determine the critical exponent of the correlation length for a trapped, weakly interacting Bose gas to be ν = 0.67 ± 0.13. This measurement has direct implications for the understanding of second-order phase transitions.

Phase transitions are among the most striking phenomena in nature. At a phase transition, minute variations in the conditions controlling a system can trigger a fundamental change of its properties. For example, lowering the temperature below a critical value creates a finite magnetization of ferromagnetic materials or, similarly, allows for the generation of superfluid currents. Generally, a transition takes place between a disordered phase and a phase exhibiting off-diagonal long-range order, which is the magnetization or the superfluid density in the above cases. Near a second-order phase transition point, the fluctuations of the order parameter are so dominant that they completely govern the behavior of the system on all length scales (*1*). In fact, the large-scale fluctuations in the vicinity of a transition already indicate the onset of the phase on the other side of the transition.

Near a second-order phase transition, macroscopic quantities show a universal scaling behavior that is characterized by critical exponents (*1*) that depend only on general properties of the system, such as its dimensionality, symmetry of the order parameter, or range of interaction. Accordingly, phase transitions are classified in terms of universality classes. Bose-Einstein condensation in three dimensions, for example, is in the same universality class as a three-dimensional XY model for magnets. Moreover, the physics of quantum phase transitions occurring at zero temperature can often be mapped onto thermally driven phase transitions in higher spatial dimensions.

The phase transition scenario of Bose-Einstein condensation in a weakly interacting atomic gas is unique, as it is free of impurities and the two-body interactions are precisely known. As the gas condenses, trapped bosonic atoms of a macroscopic number accumulate in a single quantum state and can be described by the condensate wave function, the order parameter of the transition. However, it has proven to be experimentally difficult to access the physics of the phase transition itself. In particular, the critical regime has escaped observation because it requires an extremely close and controlled approach to the critical temperature. Meanwhile, advanced theoretical methods have increased our understanding of the critical regime in a gas of weakly interacting bosons (*2*–*5*). Yet a theoretical description of the experimental situation, a Bose gas in a harmonic trap, has remained elusive.

We report on a measurement of the correlation length of a trapped Bose gas within the critical regime just above the transition temperature. The visibility of a matter-wave interference pattern gave us direct access to the first-order correlation function. Exploiting our experimental temperature resolution of 0.3 nK (0.002 times the critical temperature), we observed the divergence of the correlation length and determined its critical exponent ν. This direct measurement of ν through the single-particle density matrix complements the measurements of other critical exponents in liquid He (*6*–*8*), which is believed to be in the same universality class as the weakly interacting Bose gas.

In a Bose gas, the physics of fluctuations of the order parameter is governed by different length scales. Far above the phase transition temperature, classical thermal fluctuations dominate. Their characteristic length scale is determined by the thermal de Broglie wavelength λ_{dB}, and the correlation function can be approximated by , where *r* is the separation of the two probed locations (*9*) (Fig. 1). Nontrivial fluctuations of the order parameter Ψ close to the critical temperature become visible when their length scale becomes larger than the thermal de Broglie wavelength. The density matrix of a homogeneous Bose gas for *r* > λ_{dB} can be expressed by the correlation function (1) (*10*, *11*), where ξ denotes the correlation length of the order parameter. The correlation length ξ is a function of absolute temperature *T* and diverges as the system approaches the phase transition (Fig. 1). This results in the algebraic decay of the correlation function with distance 〈Ψ^{†}(*r*)Ψ(0) 〉 ∝ 1/*r* at the phase transition. The theory of critical phenomena predicts a divergence of ξ according to a power law, (2) where ν is the critical exponent of the correlation length and *T*_{c} is the critical temperature. The value of the critical exponent depends only on the universality class of the system.

Although for noninteracting systems the critical exponents can be calculated exactly (*1*, *12*), the presence of interactions adds richness to the physics of the system. Determining the value of the critical exponent through Landau's theory of phase transitions results in a value of ν = ½ for the homogeneous system. This value is the result of both a classical theory and a mean-field approximation to quantum systems. However, calculations by Onsager (*13*) and the more recent techniques of the renormalization group method (*1*) showed that mean-field theory fails to describe the physics at a phase transition. Very close to the critical temperature—in the critical regime—the fluctuations become strongly correlated and a perturbative or mean-field treatment becomes impossible, making this regime very challenging.

Consider a weakly interacting Bose gas with density *n* and the interaction strength parameterized by the s-wave scattering length *a* = 5.3 nm in the dilute limit *n*^{⅓}*a* « 1. In the critical regime, mean-field theory fails because the fluctuations of Ψ become more dominant than its mean value. This can be determined by the Ginzburg criterion (*14*, *15*). Similarly, these enhanced fluctuations are responsible for a nontrivial shift of the critical temperature of Bose-Einstein condensation (*2*–*4*, *16*). The critical regime of a weakly interacting Bose gas offers an intriguing possibility to study physics beyond the usual mean-field approximation (*17*), which until now has been observed in cold atomic gases only in reduced dimensionality (*18*–*21*).

In our experiment, we let two atomic beams, which originate from two different locations spaced by a distance *r* inside the trapped atom cloud, interfere. From the visibility of the interference pattern, the first-order correlation function (*22*) of the Bose gas above the critical temperature and the correlation length ξ can be determined.

We prepared a sample of 4 × 10^{6 87}Rb atoms in the |*F* = 1, *m*_{F} = –1 〉 hyperfine ground state in a magnetic trap (*23*). The trapping frequencies were (ω_{x}, ω_{y}, ω_{z}) = 2π × (39, 7, 29) Hz, where *z* denotes the vertical axis. Evaporatively cooled to just below the critical temperature, the sample reached a density of *n* = 2.3 × 10^{13} cm^{–3}, giving an elastic collision rate of 90 s^{–1}. The temperature was controlled by holding the atoms in the trap for a defined period of time, during which energy was transferred to the atoms as a result of resonant stray light, fluctuations of the trap potential, or background gas collisions. From absorption images, we determined the heating rate to be 4.4 ± 0.8 nK s^{–1}. Using this technique, we covered a range of temperatures from 0.001 < (*T* – *T*_{c})/*T*_{c} < 0.07 over a time scale of seconds.

For output coupling of the atoms, we used microwave frequency fields to spin-flip the atoms into the magnetically untrapped state |*F* = 2, *m*_{F} = 0 〉. The resonance condition for this transition is given by the local magnetic field, and the released atoms propagate downward because of gravity. The regions of output coupling are chosen symmetrically with respect to the center of the trapped cloud and can be approximated by horizontal planes spaced by a distance *r* (*22*); the two released atomic beams interfere with each other. For the measurement, we typically extracted 4 × 10^{4} atoms over a time scale of 0.5 s, which is about 1% of the trapped sample. We detected the interference pattern in time at single-atom resolution with the use of a high-finesse optical cavity placed 36 mm below the center of the magnetic trap. An atom entering the cavity mode decreases the transmission of a probe beam resonant with the cavity. The geometry of our apparatus is such that only atoms with a transverse momentum (*p*_{x}, *p*_{y}) ≈ 0 are detected, resulting in an overall detection efficiency of 1% for every atom output coupled from the cloud. From the arrival times of the atoms, we determined the visibility *V*(*r*) of the interference pattern (*24*). From repeated measurements with different pairs of microwave frequencies, we measured *V*(*r*) with *r* ranging from 0 to 4 λ_{dB} (where λ_{dB} ≈ 0.5 μm).

With the given heat rate, a segmentation of the acquired visibility data into time bins of Δ*t* = 72 ms allowed for a temperature resolution of 0.3 nK, which corresponds to 0.002 *T*_{c}. The time bin length was chosen to optimize between shot noise–limited determination of the visibility from the finite number of atom arrivals and sufficiently good temperature resolution. For the analysis, we chose time bins overlapping by 50%.

Figure 2 shows the measured visibility as a function of slit separation *r* very close to the critical temperature *T*_{c}. The visibility decays on a much longer length scale than predicted by the thermal de Broglie wavelength λ_{dB}. We fit the long distance tail *r* > λ_{dB} with Eq. 1 (solid line) and determined the correlation length ξ. The strong temperature dependence of the correlation function is directly visible. As *T* approaches *T*_{c}, the visibility curves become more long-ranged, and similarly the correlation length ξ increases. The observation of long-range correlations shows how the size of the correlated regions strongly increases as the temperature is varied only minimally in the vicinity of the phase transition.

Figure 3 shows how the measured correlation length ξ diverges as the system approaches the critical temperature. Generally, an algebraic divergence of the correlation length is predicted. We fit our data with the power law according to Eq. 2, leaving the value of *T*_{c} as a free fit parameter, which has a typical relative error of 5 × 10^{–4}. Therefore, our analysis is independent of an exact calibration of both temperature and heating rate, provided that the heating rate is constant. The resulting value for the critical exponent is ν = 0.67 ± 0.13. The value of the critical exponent is averaged over 30 temporal offsets 0 < *t*_{0} < Δ*t* of the analyzing time bin window, and the error is the reduced ξ^{2} error. Systematic errors on the value of ν could be introduced by the detector response function. We found the visibility for a pure Bose-Einstein condensate to be 100% with a statistical error of 2% over the range of *r* investigated. This uncertainty of the visibility would amount to a systematic error of the critical exponent of 0.01 and is neglected as compared to the statistical error. The weak singularity of the heat capacity near the λ-transition (*1*) results in an error of ν of less than 0.01.

Finite size effects are expected when the correlation length is large (*25*, *26*), and they may lead to as light underestimation of ν for our conditions. Moreover, the harmonic confining potential introduces a spatially varying density. The phase transition takes place at the center of the trap, and nonperturbative fluctuations are thus expected within a finite radius *R* (*5*). Using the Ginzburg criterion as given in (*14*), we find *R* ≈ 10 μm, whereas the root-mean-square size of the thermal cloud is 58 μm. The longest distance we probed in our experiment is 2 μm, which is well below this radius *R*.

To date, in interacting systems the critical exponent ν has been determined for the homogeneous system. The λ-transition in liquid He is among the most accurately investigated systems at criticality. One expects to observe the same critical exponents even though the density differs by 10 orders of magnitude. In the measurements with liquid He, the critical exponent of the specific heat α has been measured in a spaceborne experiment (*8*). Through the scaling relation α = 2 – 3ν, the value of the critical exponent ν ≈ 0.67 is inferred, in agreement with theoretical predictions (*27*, *28*). Alternatively, the exponent ζ ≈ 0.67 (which is related to the superfluid density ρ_{s} =|Ψ|^{2} instead of the order parameter Ψ) can be measured directly in second-sound experiments in liquid He (*6*, *7*). Although it is believed that ν = ζ (*29*), a measurement of ν directly through the density matrix has so far been impossible with He. Further, this unique access to spatial correlations opens up new possibilities to study phase transitions using quantum gases of variable dimensionality or with tunable interactions.