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Experimental observation of a generalized Gibbs ensemble

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Science  10 Apr 2015:
Vol. 348, Issue 6231, pp. 207-211
DOI: 10.1126/science.1257026

Detecting multiple temperatures

Most people have an intuitive understanding of temperature. In the context of statistical mechanics, the higher the temperature, the more a system is removed from its lowest energy state. Things become more complicated in a nonequilibrium system governed by quantum mechanics and constrained by several conserved quantities. Langen et al. showed that as many as 10 temperature-like parameters are necessary to describe the steady state of a one-dimensional gas of Rb atoms that was split into two in a particular way (see the Perspective by Spielman).

Science, this issue p. 207; see also p. 185

Abstract

The description of the non-equilibrium dynamics of isolated quantum many-body systems within the framework of statistical mechanics is a fundamental open question. Conventional thermodynamical ensembles fail to describe the large class of systems that exhibit nontrivial conserved quantities, and generalized ensembles have been predicted to maximize entropy in these systems. We show experimentally that a degenerate one-dimensional Bose gas relaxes to a state that can be described by such a generalized ensemble. This is verified through a detailed study of correlation functions up to 10th order. The applicability of the generalized ensemble description for isolated quantum many-body systems points to a natural emergence of classical statistical properties from the microscopic unitary quantum evolution.

Information theory provides powerful concepts for statistical mechanics and quantum many-body physics. In particular, the principle of entropy maximization (13) leads to the well-known thermodynamical ensembles, which are fundamentally constrained by conserved quantities such as energy or particle number (4). However, physical systems can contain many additional conserved quantities, which raises the question of whether there exists a more general statistical description for the steady states of quantum many-body systems (5).

Specifically, the presence of nontrivial conserved quantities puts constraints on the available phase space of a system, which strongly affects the dynamics (69) and inhibits thermalization (1012). Instead of relaxing to steady states described by the usual thermodynamical ensembles, a generalized Gibbs ensemble (GGE) was proposed to describe the corresponding steady states via the many-body density matrixEmbedded Image (1)(3, 11, 13, 14), where Embedded Image denotes a set of conserved quantities and Embedded Image is the partition function. The Lagrange multipliers λm associated with the conserved quantities are obtained by maximization of the entropy under the condition that the expectation values of the conserved quantities are fixed to their initial values. The emergence of such a maximum-entropy state does not contradict a unitary evolution according to quantum mechanics. Rather, it reflects that the true quantum state is indistinguishable from the maximum-entropy ensemble with respect to a set of sufficiently local observables (5).

The GGE is a direct generalization of the usual thermodynamical ensembles and is formally capable of describing a wide range of dynamically emerging steady states (15). For example, in the case where only the energy is conserved, the GGE reduces to the standard canonical or Gibbs ensemble, with temperature being the only Lagrange multiplier (4). Moreover, the GGE famously provides a description for the steady states of integrable systems, which have as many independent conserved quantities as they have degrees of freedom (9, 11, 14). Even if quantities are only approximately conserved, the GGE description was formally shown to be valid for extended time scales (16). The GGE has also been suggested as a description for many-body localized states (17). Numerical evidence for the emergence of a GGE has been provided for many systems, but a direct experimental observation has been lacking.

Here, we experimentally studied the relaxation of a trapped one-dimensional (1D) Bose gas. Our system is a close realization of the Lieb-Liniger model describing a homogeneous gas of 1D bosons with contact interactions, which is one of the prototypical examples of an integrable system (18, 19). In the thermodynamic limit, its exact Bethe Ansatz solutions imply an infinite number of conserved quantities, which provides a memory of an initial non-equilibrium state, forcing the gas to relax to a GGE. Recent results have also shown that trapped 1D Bose gases can be approximately integrable over very long time scales, enabling the detailed investigation of integrable dynamics (6, 7, 10, 20, 21). To demonstrate the emergence of a GGE, we prepared such a 1D Bose gas in different initial non-equilibrium states and observed how they each relaxed to steady states that maximize entropy according to the initial values of the conserved quantities.

The experiments started with a phase-fluctuating 1D Bose gas (22) of 87Rb atoms that was prepared and trapped using an atom chip (23). We initialized the non-equilibrium dynamics by transversally splitting this single 1D gas coherently into two nominally identical 1D systems, each containing half of the atoms on average. Information about the total system was extracted using matter-wave interferometry between the two halves (6, 20, 21, 24). This enabled the time-resolved measurement of individual two-point and higher-order N-point phase correlation functions,

Embedded Image (2)where z1, z2, …, zN are N coordinates along the length of the system, and φ(z) is the relative phase between the two halves (25). As described below, these correlation functions reveal detailed information about the dynamics and the steady states of the system.

We start with the two-point correlation function Embedded Image. Previously, this correlation function was studied in regions where the system is approximately translationally invariant (21, 26), that is, C(z1, z2) ≡ C(z1z2). Here, more comprehensive information about generic many-body states is obtained by mapping the full correlation function C(z1, z2) for any combination of the coordinates z1 and z2 (Fig. 1).

Fig. 1 Experimental concept.

A non-equilibrium system is prepared by splitting a 1D Bose gas into two halves. After an evolution time t, matter-wave interference is used to extract the local relative phase profile φ(z) between the two halves. This is accomplished by measuring the local position of the fluctuating interference fringes. Subsequently, the phase profile is used to calculate the two-point correlation function Embedded Image as a function of all possible coordinates z1 and z2 along the length of the measured phase profile. For z1 = z2 (e.g., green triangle on the diagonal), C(z1, z2) = 1. Coordinates with z1 = –z2 (e.g., red point) are located symmetrically around the center of the system and are found on the antidiagonal of the correlation function.

Our observations after a typical splitting, which is fast relative to the dynamics of the system and therefore realizes a quench (25), are summarized in Fig. 2A. As every point in the system is perfectly correlated with itself, the correlation functions exhibit a maximum on the diagonal z1 = z2 for all times. Away from the diagonal, the system shows a light cone–like decay of correlations (21) leading to a steady state. From a theoretical point of view, the emergence of this steady state is a consequence of prethermalization (6, 2731), which in the present case can be described as the dephasing of phononic excitations (29, 3133). The occupation numbers nm of these excitations are the conserved quantities of the corresponding integrable model (25).

Fig. 2 Two-point phase correlation functions C(z1, z2) for increasing evolution time.

(A and B) Different initial states were prepared using two different splitting protocols. Both states show a characteristic maximum on the diagonal and a decay of correlations away from the diagonal. We used a χ2 analysis to quantify the agreement of our theoretical model and the experiments. The steady state and the dynamics in (A) can be well described by a single temperature Teff. (C) The single-temperature model fails for the steady state and the dynamics in (B), which require more temperatures to explain additional correlations on the antidiagonal (see text). The observation of different temperatures in the same steady state constitutes our observation of a GGE. The center of the system is located at z1 = z2 = 0; color marks the amount of correlations between 0 and 1 and the local χ2 contribution between 0 and 150. The uncertainty of the correlation functions is estimated via bootstrapping over approximately 150 experimental realizations (25).

With a knowledge of the conserved quantities, we can directly calculate the Lagrange multipliers λm for the GGE. In terms of the excitation energies εm they can be written as λm = βmεm, which defines an effective temperature 1/βm for every excitation mode.

For the steady state illustrated in Fig. 2A, the proportionality factor βm can be well described by βm ≈ βeff = 1/kBTeff (where kB is the Boltzmann constant). A fit yields kBTeff = 0.50 (±0.01) × μ, which is independent of m and in very good agreement with theory (6, 33). Here, μ denotes the chemical potential in each half of the system. Although it is a GGE in principle, for our experiment (which observes the relative phase between the two halves of the system) it becomes formally equivalent to the usual Gibbs ensemble with a single temperature Teff (25).

To obtain direct experimental signatures of a genuine GGE, we modified the initial state so that it exhibited different temperatures for different excitation modes. This was accomplished by changing the ramp that splits the initial gas into two halves (25). The results are shown in Fig. 2B. In addition to the maximum of correlations on the diagonal, a pronounced second maximum on the antidiagonal can be seen. This corresponds to enhanced correlations of the points z1 = –z2, which are located symmetrically around the center of the system. These correlations are a direct consequence of an increased population of quasi-particle modes that are even, and a decreased population of quasi-particle modes that are odd, under a mirror reflection with respect to the longitudinal trap center. Consequently, the observations can be described to a first approximation by the above theoretical model but with different temperatures—that is, with β2m = 1/[kB(Teff + ΔT)] for the even modes and β2m–1 = 1/[kB(Teff – ΔT)] for the odd modes, respectively. Fitting the experimental data of the steady state with this model, we find kBTeff = 0.64 (±0.01) × μ, ΔT = 0.48 (±0.01) × Teff, and a reduced χ2 ≈ 6.

More detailed insights and a more accurate description of the experimental data can be gained by fitting the steady state with the individual mode occupations as free parameters. The results yield a reduced χ2 close to 1 and thus a very good description of the experimental data (see Fig. 2B). As expected from our intuitive two-temperature model, the fitting confirms that the occupation of even modes is strongly enhanced, whereas the occupation of odd modes is reduced (see Fig. 3A). Fixing these occupation numbers extracted from the steady state, our theoretical model also describes the complete dynamics very well. This clearly demonstrates that the different populations of the modes were imprinted onto the system by the splitting quench and are conserved during the dynamics. In contrast, a simple model based on a usual Gibbs ensemble with just one temperature for all modes clearly fails to describe the data [best fit: kBTeff = 0.38 (±0.01) × μ, reduced χ2 ≈ 25], as visualized in Fig. 2C.

Fig. 3 Statistical analysis of the fitting procedure.

(A) Occupation numbers nm (in units of εm/μ) of the quasi-particle modes with index m, for different fitting procedures applied to the data from Fig. 2B. The color of the points encodes results where modes up to m = Nmax are fitted freely, whereas all higher modes are fit with the same occupation number. The plot reveals that the occupation of the lowest even modes is increased, and that of the lowest odd modes is decreased, relative to the single-temperature state from Fig. 2A (dashed line). (B) Scaling of the reduced χ2 value, and of the corresponding P value with Nmax. The occupation numbers of the lowest nine modes and a single occupation number for all higher modes are sufficient to describe the experimental data to very good accuracy.

Notably, our fitting results for the GGE exhibit strong correlations between the different even modes and the different odd modes, respectively. This demonstrates the difficulty in fully and independently determining the parameters of such complex many-body states. In fact, any full tomography of all parameters would require exponentially many measurements. The results thus clearly show the presence of a GGE with at least two, but most likely many more temperatures.

Our work raises the interesting question of how many Lagrange multipliers are needed in general to describe the steady state of a realistic integrable quantum system. In analogy to classical mechanics, where N conserved quantities exist for a generic integrable system with N degrees of freedom, integrability in quantum many-body systems has been proposed to be characterized by the fact that the number of independent local conserved quantities scales with the number of particles (5). Here we conjecture that many experimentally obtainable initial states evolve in time into steady states, which can be described to a reasonable precision by far fewer than N Lagrange multipliers (8, 34). This would have the appeal of a strong similarity to thermodynamics, where also only a few parameters are needed to describe the properties of a system on macroscopic scales.

To illustrate this in our specific case, we investigated how many distinct Lagrange multipliers need to be considered in the GGE to describe our data with multiple temperatures (Fig. 3). Including more and more modes in the fitting of the experimental data, we found that the reduced χ2 values decrease and settle close to unity for nine included modes, with all higher modes being fitted by one additional Lagrange multiplier. Looking at the P value for the measured χ2 (35) shows that only a limited number of Lagrange multipliers needs to be specified to describe the observables under study to the precision of the measurement. A simple numerical estimate based on the decreasing contribution of higher modes to the measured correlation functions and the limited imaging resolution leads to approximately 10 Lagrange multipliers, which is in good agreement with our observations (25). Moreover, comparing this result with the single-temperature steady state discussed earlier illustrates that the complexity of the initial state plays an important role for the number of Lagrange multipliers that need to be included in a GGE.

In general, deviations of steady states from the GGE description are expected to manifest first in higher-order correlation functions. To provide further evidence for our theoretical description and the presence of a GGE, Fig. 4 shows examples of measured 4-point, 6-point, and 10-point correlation functions of the steady state. Like the 2-point correlation functions, they are in very good agreement with the theoretical model and clearly reveal the difference between the GGE and the usual Gibbs ensemble. This confirms that the description based on a GGE with the parameters extracted from the 2-point correlation functions also describes many-body observables, at least up to 10th order.

Fig. 4 Examples of experimental 4-, 6-, and 10-point correlation functions.

(A and B) Differences between the steady state described by a single temperature (A) and the steady state described by multiple temperatures (B) are significant and can be captured by the theoretical model. (C) The single-temperature model cannot describe the state with multiple temperatures, which is reflected in high values of the local χ2. From left to right, we plot C(z1, 10, z2, 10), C(z1, –12, z2, 14), C(z1, 10, 10, z2, –20, 10), C(z1, –8, 8, z2, –24, –20), C(z1, 4, 10, z2, –8, z2, –22, –18, 10, –4), and C(z1, –22, –8, z2, –22, –26, –22, z2, –26, –24). All coordinates are given in μm and were chosen as representative cases for our high-dimensional data.

Our results clearly visualize, both experimentally and theoretically, how the unitary evolution of our quantum many-body system connects to a steady state that can be described by a classical statistical ensemble. We expect our measurements of correlation functions to high order to play an important role in new tomography techniques for complex quantum many-body states (36). Moreover, the observed tunability of the non-equilibrium states suggests that our splitting process could in the future be used to prepare states tailored for precision metrology (37).

Supplementary Materials

www.sciencemag.org/content/348/6231/207/suppl/DC1

Materials and Methods

Supplementary Text

References (3844)

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We acknowledge discussions with E. Demler, E. Dalla Torre, K. Agarwal, J. Berges, M. Karl, V. Kasper, I. Bouchoule, M. Cheneau, and P. Grisins. Supported by the European Union (SIQS and ERC advanced grant QuantumRelax), the Austrian Science Fund (FWF) through the Doctoral Programme CoQuS (W1210) (B.R. and T.S.), the Lise Meitner Programme M1423 (R.G.) and project P22590-N16 (I.E.M.), Deutsche Forschungsgemeinschaft grant GA677/7,8 (T.G.), the University of Heidelberg Center for Quantum Dynamics, Helmholtz Association grant HA216/EMMI, and NSF grant PHY11-25915. T.L., T.G., and J.S. thank the Kavli Institute for Theoretical Physics, Santa Barbara, for its hospitality.
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