Non-equilibrium scale invariance and shortcuts to adiabaticity in a one-dimensional Bose gas

We present experimental evidence for scale invariant behaviour of the excitation spectrum in phase-fluctuating quasi-1d Bose gases after a rapid change of the external trapping potential. Probing density correlations in free expansion, we find that the temperature of an initial thermal state scales with the spatial extension of the cloud as predicted by a model based on adiabatic rescaling of initial eigenmodes with conserved quasiparticle occupation numbers. Based on this result, we demonstrate that shortcuts to adiabaticity for the rapid expansion or compression of the gas do not induce additional heating.

throughout our experiments to remove hot atoms. The cloud is probed by standard absorption imaging techniques after a 4 ms to 10 ms long phase of time-of-flight expansion.
The geometry of the trap is governed by the current flow through a central Z-shaped wire and two U-shaped control structures on the atom chip, as shown in figure 1(a). Panels (b)-(e) show two different trapping potentials calculated for currents tuned to I Z~2 A and I U~0 A, as well as I Z~1 :5 A and I U~1 A. Varying I Z and I U results in traps with axial confinement ranging from v a~2 p|16 Hz to v a~2 p|7 Hz, and radial confinement from v r~2 p|600 Hz to v r~2 p|1100 Hz. A rapid change of the current ratio I U =I Z constitutes a quench of the trapping potential and induces excitations.
In our first set of experiments we probe the dynamical scaling of the phonon ensemble in the presence of an axial quadrupole-mode collective excitation 26 induced by such a quench. To this end, we employ a linear ramp from v a~2 p|12:1 Hz to 2p|8:2 Hz, and from v r~2 p|630 Hz to 2p|990 Hz, respectively, of duration t. The ramps of the trapping potential were designed to avoid transverse excitations. We chose to maintain a constant transverse position to avoid inducing a corresponding sloshing of the cloud. The ramp duration was chosen to be longer than t<5 ms so that adiabaticity with respect to the change of transverse trap frequency is fulfilled. Axial dipole oscillations are suppressed by the symmetric arrangement of the control wires.
We probe phononic excitations in the quasicondensate using a thermometry scheme based on the analysis of density correlations in free expansion 27,28 , as shown in the inset of figure 2(a). To extract the temperature we compare the measured density correlation functions with the results of a stochastic model 29 . Our analysis accounts for the effects of the collective excitation on the free expansion (see methods section below), and for the finite resolution of our imaging system. Figure 2 summarises our temperature measurements following a quench. We show data for ramp times of 10 and 30 ms and mean atom numbers of 11000 and 16000, compared to the behaviour expected from a scaling model building upon the results of Ref 14 .
The scale invariance of the underlying Hamiltonian allows to calculate time-dependent correlation functions: In the Thomas-Fermi regime, the density profile exhibits self-similar scaling described by n(z,t)~n with a time-dependent scale factor b(t)~R(t)=R 0 . Here, R 0 and n 0 denote the initial Thomas-Fermi radius and peak density, respectively,   H is the Heaviside function and z represents the axial coordinate. The scale factor obeys an Ermakov-like equation 31 € bzv 2 a (t)b~v Using the rescaled mean-field density (1), we can write the linearised hydrodynamic equations for density and velocity fluctuations dn and dv, disregarding the quantum pressure term, as and To solve these equations we introduce an ansatz of rescaled eigenmodes for density and phase fluctuations. This approach yields a set of uncoupled equations and hence no mixing of modes, finally predicting an adiabatic time evolution of the corresponding occupation numbers. For a thermal state, the initial phonon occupation numbers are given by a Bose distribution Adiabaticity results in a constant ratio v l (t)=T(t)~v l (0)=T(0). The spectrum at t~0 is given by 32 with mode index l and initial sound velocity c 0 . For tw0, it scales as v l (t)~v l (0)b {3=2 , due to the time-dependence of the sound velocity c(t)~c 0 = ffiffi ffi b p and radius R(t)~R 0 b(t). Hence, for an initial state in thermal equilibrium, we obtain the temperature scaling The density correlations in free expansion that our thermometry scheme relies on are governed by the coherence function. For a thermal state with homogeneous density, as realised in the vicinity of the cloud center, it has the form 20,32 : where n(z,0) denotes the density at time t~0 and k B the Boltzmann constant. Based on our model, the coherence function is expected to scale as  www.nature.com/scientificreports the initial temperature and plotted against the scale parameter b(t)~R(t)=R 0 , the datasets collapse onto a single line. This illustrates a scaling behaviour that is universal in sense that it is independent of absolute temperature, density or quench time. To validate our results we furthermore performed numerical simulations based on a stochastic Gross-Pitaevskii equation (SGPE) [33][34][35][36] , showing excellent agreement with the scaling model ( fig. 3).
So far, we considered the dynamics induced by a linear ramp of the trapping potential. In the following, we demonstrate the conservation of phonon occupation numbers during shortcuts to adiabaticity 31,18,17 for the rapid expansion and compression of a 1d quasi-BEC. To implement these shortcuts, we make use of an optimal control approach that is in spirit similar to the method proposed in ref. 37. We numerically solve the time-dependent 1d GPE with a suitable parametrisation of the trap which is subject to a global optimization procedure based on a genetic algorithm 38,39 . The ramp speed is limited by the requirement of adiabaticity in the transverse degree of freedom. This constraint also guarantees that the gas remains in the 1d hydrodynamic regime, and that the interaction strength varies slowly with time. The properties of the ultracold gas therefore remain consistent with the conditions necessary for the validity of the microscopic scaling laws 14 throughout the ramp.
The upper panel in figure 4 shows a comparison between simulation and experiment for a linear and a shortcut ramp performing a decompression within 30 ms from a trap with frequencies v 0 a~2 p|11:5 Hz and v 0 r~2 p|764 Hz to v f a~2 p|7 Hz and v f r~2 p|1262 Hz. The subsequent dynamics is observed www.nature.com/scientificreports SCIENTIFIC REPORTS | 5 : 9820 | DOI: 10.1038/srep09820 throughout a period of 170 ms, each picture taken after a short free expansion time of 5 ms, showing excellent agreement with simulations. It is interesting to note that our shortcut ramps are similar to theoretical results derived from a counter-diabatic driving method reported recently 19 .
For the STA, we expect an adiabatic state change, defined by T=T 0~v f a =v 0 a . The temperature measurements, corrected for the measured heating rate, are in good agreement with the adiabatic prediction of T=T 0 <0:609 for the implemented decompression shortcut, confirming that there is no additional heating during the applied procedure.

Conclusion
In summary, we have characterised the temperature of the phonon ensemble in a breathing quasi-1d Bose gas for different initial conditions, and used it to test the predicted dynamical scale invariance in the excitation spectrum of a quasi-1d Bose gas. Following these scaling laws, we have experimentally demonstrated rapid adiabatic expansion and compression of a 1d Bose gas in the hydrodynamic regime, allowing fast transformation of the trapped cloud without additional heating.
Our work is only the beginning for studies of many-body scaling solutions and shortcuts to adiabaticity. The existence of scaling solutions has been proposed for a large class of cold atom systems 14 . In principle, this opens up the interesting possibility to apply the techniques applied here to a variety of settings, such as fermionic systems or the 1d Bose gas with intermediate or strong interactions. We expect that studying the effect of quasiparticle interactions on the implementation of shortcuts to adiabaticity will shed new light on the complex many-body dynamics in these systems, in addition to providing novel tools for their controlled manipulation.
We expect that such extensions to studies in regimes of greater interaction strength, and to systems out of thermal equilibrium, will benefit from the tools presented in this work.

Methods
Condensate preparation and detection. We employ standard cooling and magnetic trapping techniques 40 to prepare ultracold quasi-one-dimensional samples of 87Rubidium atoms in the jF~2,m F~2 w state on an atom chip 20,41 . Atom chips feature microfabricated wire structures to create fields for atom trapping and manipulation 42 . The structures used in our experiments are produced by masked vapor depositon of a 2 mm gold layer on a silicon substrate, with a width of both trapping and control wires of 200 mm. For detection, we employ resonant absorption imaging 43 using a high quantum-efficiency CCD camera (Andor iKon-M 934 BR-DD) and a diffraction-limited optical imaging system characterised by an Airy radius of 4.5 mm. The RF shield at 12 kHz above the bottom of the trap is used to limit the number of atoms in the thermal background cloud populating transverse excited states of the trap, which would otherwise adversely affect our thermometry scheme by reduction of interference contrast in free expansion.
Characterization of the breathing mode. We characterise the breathing mode excited by a linear trap frequency ramp from v a~2 p|12:1 Hz to 2p|8:2 Hz, and v r~2 p|630 Hz to 2p|990 Hz, respectively, in figure 5. As an example, the upper panel shows the time evolution of the cloud radius after a ramp with duration t 5 12.5 ms. Fitting data as presented here allows us to extract frequencies, damping rates and amplitudes of the breathing mode. The frequency v b is influenced by the total atom number in the trap, and is expected to vary with the axial trap frequency between v b =v a~ffi ffi ffi 3 p in the 1d limit, and v b =v a~ffi ffiffiffiffiffi 2:5 p representing the elongated 3d regime 26 . The amplitude strongly depends on the duration and shape of the trap frequency ramp. The lower panel in figure 5 shows a comparison of measured breathing amplitudes for different ramp times between 2 ms and 100 ms with results calculated with a 1d Gross-Pitaevskii equation (GPE), taking into account corrections to the interaction term relevant in the 1d/3d crossover regime 44 , and shows good agreement in the chosen parameter range.
Thermometry. In this work we use the thermometry scheme proposed and demonstrated in ref. 28,27 based on the analysis of density correlations in freely expanding phase-fluctuating quasi-1d condensates and comparison with numerically calculated density profiles 29 .
Breathing contributes a velocity field characterized by the derivative of the scale parameter _ b, leading to an additional axial compression or expansion of the density profile during free expansion. This effect can be accounted for by an additional phase factor in the numerics, where b and _ b are determined by fits to the measured breathing oscillations. The error on the temperature measurements is estimated by a bootstrapping method as outlined in ref. 30.
Derivation of the temperature scaling. The general conditions for the existence of a scaling solution are stated in reference 14 . For the 1d Bose gas, they are fulfilled in the presence of contact interactions, as well as a harmonic, linear or vanishing axial trapping potential. Given that our system is a 1d quasicondensate, and the trapping potential is harmonic, we can derive the corresponding hydrodynamic scaling relations for correlation functions. Our starting point is the self-similar scaling of the density profile: H denotes the Heaviside function, R 0 the initial Thomas-Fermi radius and b the scale parameter. Similar to the discussion of the corresponding equilibrium problem 32 , a scaling solution in terms of eigenmodes for density and velocity fluctuations dn and dv can be formulated as with the Legendre polynomials P l (z), the interaction constant g, rescaled coordinates z~z=R~z=(R 0 b), and time-dependent amplitudes A l sin g l and A l cos g l . g l denotes the frequency of the oscillation between the quadratures of the mode l. Correspondingly, the initial equilibrium spectrum scales as v l (t)~v l (0)b {3=2 : Substituting dn and dv into the linearised Euler equations where we have disregarded the quantum pressure term, yields www.nature.com/scientificreports SCIENTIFIC REPORTS | 5 : 9820 | DOI: 10.1038/srep09820