## 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).

## 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 (*1*–*3*) 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 (*6*–*9*) and inhibits thermalization (*10*–*12*). 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 matrix
(1)(*3*, *11*, *13*, *14*), where denotes a set of conserved quantities and is the partition function. The Lagrange multipliers λ* _{m}* 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 ^{87}Rb 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,

(2)where *z*_{1}, *z*_{2}, …, *z _{N}* 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 . Previously, this correlation function was studied in regions where the system is approximately translationally invariant (*21*, *26*), that is,* C*(*z*_{1}, *z*_{2}) ≡ *C*(*z*_{1} – *z*_{2}). Here, more comprehensive information about generic many-body states is obtained by mapping the full correlation function *C*(*z*_{1}, *z*_{2}) for any combination of the coordinates *z*_{1} and *z*_{2} (Fig. 1).

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 *z*_{1} = *z*_{2} 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*, *27*–*31*), which in the present case can be described as the dephasing of phononic excitations (*29*, *31*–*33*). The occupation numbers *n _{m}* of these excitations are the conserved quantities of the corresponding integrable model (

*25*).

With a knowledge of the conserved quantities, we can directly calculate the Lagrange multipliers λ* _{m}* for the GGE. In terms of the excitation energies ε

*they can be written as λ*

_{m}*= β*

_{m}*ε*

_{m}*, which defines an effective temperature 1/β*

_{m}*for every excitation mode.*

_{m}For the steady state illustrated in Fig. 2A, the proportionality factor β* _{m}* can be well described by β

*≈ β*

_{m}_{eff}= 1

*/k*

_{B}

*T*

_{eff}(where

*k*

_{B}is the Boltzmann constant). A fit yields

*k*

_{B}

*T*

_{eff}= 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

*T*

_{eff}(

*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 *z*_{1} = –*z*_{2}, 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 β_{2}* _{m}* = 1/[

*k*

_{B}(

*T*

_{eff}+ Δ

*T*)] for the even modes and β

_{2}

_{m}_{–1}= 1/[

*k*

_{B}(

*T*

_{eff}– Δ

*T*)] for the odd modes, respectively. Fitting the experimental data of the steady state with this model, we find

*k*

_{B}

*T*

_{eff}= 0.64 (±0.01) × μ, Δ

*T*= 0.48 (±0.01) ×

*T*

_{eff}, and a reduced χ

^{2}≈ 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 χ^{2} 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: *k*_{B}*T*_{eff} = 0.38 (±0.01) × μ, reduced χ^{2} ≈ 25], as visualized in Fig. 2C.

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 χ^{2} 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 χ^{2} (*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.

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

## References and Notes

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**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.