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corral with a Kondo resonant cobalt atom placed at one focus of the ellipse. The resonant scattering that arises from such an arrangement causes a Kondo peak to emerge at both the focal point containing the cobalt atom and at the second, empty, focal point (both of which were coloured purple in the resulting images, to show the presence of the Kondo signature).

The beautiful properties of the ellipse, and also the Kondo effect, are again exploited in the present experiments. In this case, the ellipticity of the 40-atom corral is adjusted so that there are degenerate pairs of states (with very different spatial character) in the corresponding hard-walled ellipse at certain energies. This degeneracy is then exploited to manipulate the mixing angle between the two degenerate states by moving an atom around inside the ellipse. Other states nearby in energy also get in on the act. The breeding is accomplished with a roving fence post (they call it a moveable gate) in the interior of the corral; this is a strange sort of quantum breeding wherein the horse (an electron) breeds with several versions of itself. This is effected with a roving post that makes coherent linear combinations of 5 or 7 total parents (quantum states), breeding the same number of offspring (the parents are transformed into the offspring). One offspring is selected as a show horse and is imaged. The trick required to do this is extremely clever: It is easy to show that given N degenerate independent states, N1 linear combinations (offspring) of them may be made to vanish at a given spot, that is, the interior post; therefore, the post has no effect on them, being much smaller than a

Fermi wavelength. They never visit it. The Nth state however is different. It has a large probability of visiting the post, but it roams the corral too. If the post is freshly painted purple, the selected show horse rubs off

some paint (the paint here is the signature of a Kondo resonance) and with a purple filter (selecting for the Kondo resonance signature imparted by the cobalt moveable post) the purple show horse is thus imaged (Fig. 1). The purple horse is quite a different beast, depending on where the post is, and this is much of the control the breeders have in determining the genetics (state superposition) of the show horse. There are interesting caveats and variations on the purple horse idea, having to do with nodal line crossings and energy widths of the states (resonances) in the corral, but we leave them to the paper itself.

Coherent control of matter is the key to new atomic-scale technologies. The structures reported by Moon, Lutz, and Manoharan3 are nanometres in size, built atom by atom, with the purpose of quantum-coherent manipulation. This is the ultimate atomic-scale engineering; there will be never be anything smaller than this that can be manipulated and stays put. Like our ancestors who thousands of years ago engineered the first corrals and trapped a few wild horses, we have crossed a threshold. The challenge now is to break the horses in to get them to do something useful.references1. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Nature

363, 524527 (1993).2. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Science

262, 218220 (1993).3. Moon, C. R., Lutz, C. P. & Manoharan, H. C. Nature Phys.

4, 454458 (2008).4. Heller, E. J., Crommie, M. F., Lutz, C. P. & Eigler, D. M. Nature

369, 464466 (1994).5. Manoharan, H. C., Lutz, C. P. & Eigler, D. M. Nature

403, 512515 (2000).6. Fiete, G. A. et al. Phys. Rev. Lett. 86, 2392 (2001).

Figure 1 the principle underlying the selective breeding of quantum states from a combination of degenerate states. a,b, An ideal square corral (which for illustrations sake is composed of sharp, continuous and impenetrable walls unlike those of a real quantum corral) possesses many pairs of degenerate levels, in this case (nx; ny) = (3; 2) and (2; 3) (a and b, respectively). c,d, Placing a kondo atom (cobalt) as shown splits the degeneracy slightly and paints one of the states (c) with kondo resonance character, as the amplitude of its eigenfunctions is large at the position of the atom. But because the other state (d) has a node (that is, it vanishes) at the position of this atom, it is unaffected by its presence. so although the resonance of the states is too broad to isolate them from each other, the interaction of the atom with the first of the states is still able to paint it, and not the second, with a kondo-like (purple) character.

tHErmodYnAmics

Limited adiabaticity

Wilhelm Zwergeris in the Physikdepartment, Technische Universitt Mnchen, James Franck Strae 85748 Garching, Germany.

e-mail: zwerger@ph.tum.de

T he notion of an adiabatic process is central to thermodynamics. Its applications range from determining the efficiency of an idealized

Carnot cycle and the dry adiabatic lapse rate of air when it expands at higher altitude to the decrease of temperature of the cosmic background radiation during expansion of the Universe1. Usually, adiabaticity that is, the absence of heat exchange Q = 0 is also understood to imply that the process is quasistatic. In this case the system under consideration has a well-defined temperature T at

any instant and Q = TdS. Adiabatic and quasistatic processes thus conserve entropy S. On page 477 of this issue, Polkovnikov and Gritsev2 examine the requirements for reaching the thermodynamic adiabatic limit in interacting quantum systems.

There is a general argument for why entropy should be conserved in a sufficiently slow process, provided that

The standard assumption in thermodynamics that a sufficiently slow change of external parameters will generate no entropy turns out to be wrong for low-dimensional, gapless systems. Its breakdown may be tested with ultracold gases.

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rethermalization with the environment can be neglected3. It relies on the assumption that the entropy difference Sif between an initial and final equilibrium state has a Taylor expansion in powers of the rate of change = /t of an external parameter that varies by a finite amount within a time interval t. As the entropy change has to be non-negative, by the second law of thermodynamics, it cannot depend on the sign of . Thus, to lowest order, one expects Sif = b2 with a positive constant b. This elementary argument indicates that the entropy change vanishes (at least) quadratically with the rate at which external parameters vary in time. The argument is compelling, however, it requires solving a concrete non-equilibrium problem to prove the validity of the expansion and to calculate the associated microscopic parameter b.

In simple textbook examples, such as the expansion of an ideal gas, it is straightforward to show that a quadratic dependence indeed holds1. The adiabatic limit is reached for expansion velocities much smaller than thermal velocities, usually not a very restrictive condition. When it comes to interacting systems, however, in particular those in quantum statistical physics and field theory, it is a highly non-trivial problem to derive quantitative criteria for when a process is adiabatic. Here, it is important to emphasize that adiabatic processes in thermodynamics do not require the change of the total extensive entropy of the system to be zero identically. Exact conservation of the microscopic statistical entropy is required, for example in the context of adiabatic quantum computation4, where a hamiltonian, whose ground state is simple, is adiabatically transformed into one that has non-trivial correlations. The possibility of performing such changes with a well-defined outcome relies on the adiabatic theorem in quantum mechanics. From the well-known LandauZener argument, the accuracy with which the adiabatic limit is attained here varies exponentially as e/ in the ramp speed, where is the minimum value of the gap parameter along the path in parameter space.

In a thermodynamic context, the adiabatic theorem of quantum mechanics is irrelevant. Indeed, for the large systems required to apply the concept of entropy in the first place, the macroscopic energy level density, which is proportional to ~eS, is of order eN. Thermodynamic systems thus have energy gaps in their many-body spectrum that vanish like eN, precluding the applicability of

the quantum adiabatic theorem for any realistic timescales. Requiring an adiabatic change of the microscopic density matrix in which Sif 0 is, however, a much too restrictive condition. In thermodynamics, instead, it is sufficient that the Boltzmann entropy per volume, which is a measure of the lack of information in the relevant degrees of freedom1, remains unchanged in the limit 0.

The most surprising result of Polkovnikov and Gritsev2 is that there are simple low-dimensional systems where the adiabatic limit may never be reached in practice, because it would require ramp speeds smaller than the inverse system size. To make their point, the authors consider systems whose low-energy excitations can be described by a set of harmonic oscillators. This choice sounds rather restrictive. But, as is well-known from the theory of lattice vibrations, any solid at low temperature can be described in terms of non-interacting phonons.

More generally, by the Goldstone theorem, a description by a set of non-interacting, gapless bosons is possible whenever there is a broken continuous symmetry. Beyond solids, this applies to magnetically ordered systems or superfluids, for instance. Dealing with a harmonic system has the advantage that even the non-equilibrium quantum dynamics can be solved analytically. There is a drawback, though, which is related to the fact that a gas of non-interacting quasiparticles is an integrable system. As a result, there is no equilibration after an external parameter has been changed in time. The entropy is therefore not defined in the final state even if its initial value was. As argued by Polkovnikov and Gritsev, the relevant quantity in this case is the change in energy if per volume induced by a slow variation of an external parameter. For small ramp speeds, it should again have an expansion of the form if = b2 if the system behaves in an analytic fashion expected from the adiabatic theorem. In particular, independent of the issue of equilibration, this implies that in the limit 0 no work is done on the system in a cyclic process.

In their calculations, Polkovnikov and Gritsev determine the density of excitations induced by slow changes in the parameter of the quadratic hamiltonian. It turns out that, generically, for three-dimensional systems, the expectation if ~ 2 from the simple argument given above turns out to be true independent of whether the initial

state is at zero or at finite temperature. A rather different behaviour, however, occurs when the ramp changes the nature of the low-energy spectrum from an initial quadratic dependence ~ q2 on wavevector q to a linear one with ~ q (this is the case of 0 = 0 discussed in the paper). A situation like that is realized, for instance, when interactions are slowly turned on from zero to a finite value in superfluids. In conventional many-body physics, doing this remains a theorists dream.

In the context of ultracold gases, however, it can be realized experimentally by a suitable change of, for example, magnetic field to tune the scattering length from zero to a finite positive value near a Feshbach resonance5. As shown by Polkovnikov and Gritsev, the energy change per volume if ~ || in a process of this type typically vanishes in a non-analytic fashion as the ramp speed approaches zero, with a power that is smaller than two. The adiabatic limit is therefore much harder to reach, but still possible. Surprisingly, the situation becomes very odd if one considers the same situation in one or two dimensions at a finite temperature. In this case, the change in energy under a slow variation of the interaction from zero to a finite value has the form if ~ ||L, with a non-trivial exponent > 0. As a result, reaching the adiabatic limit is now a matter of system size L and becomes impossible in the thermodynamic limit L .

Beyond the interest in the limits of thermodynamic adiabaticity from a conceptual point of view, a crucial question is of course whether these findings have implications that can be tested in an experiment. As pointed out by the authors, it is possible at least in principle with ultracold gases. In fact, there are two properties that are crucial in this context. First of all, with cold atoms it is possible to change the strength of interactions dynamically in a well-controlled manner. This may be done, for instance, by changing the depth of an optical lattice or, more directly, by changing the scattering length near a Feshbach resonance5. The second point is that cold gases are well-isolated systems, so rethermalization is not an issue over the timescales of interest.

A specific experiment that would test the predictions of Polkovnikov and Gritsev requires a slow increase of the repulsive on-site interaction for bosonic atoms in an optical lattice from zero to a finite value, still below the critical value for the transition to the Mott-insulating state. The energy of the resulting final

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state can be inferred, for instance, from the quality of the interference pattern obtained from a superfluid state after a time-of-flight experiment6. In a three-dimensional situation, the energy increase after the ramp should depend only on the ramp speed, but not on the system size. By contrast, in one or two dimensions, the energy added per atom at a given ramp speed will increase with system size. Numerical calculations by Polkovnikov and Gritsev on a slowly driven (non-integrable) BoseHubbard model support these predictions in a quantitative way.

It is remarkable that even for such basic problems as the adiabatic limit, sometimes new and surprising features are found. In a different context and for classical

systems, Jarzinsky7 showed that the work done in a non-equilibrium process is directly related to the equilibrium free-energy difference between the initial and final states. This idea has been applied successfully to determine free-energy landscapes of complex molecules from unfolding experiments with an atomic force microscope8. The present findings of Polkovnikov and Gritsev on the limits of adiabaticity in quantum systems may open a new avenue for investigation of non-equilibrium phenomena in many-body physics. This area has become accessible with cold gases but little work has been done so far. Perhaps, as indicated in their paper, the ideas on the breakdown of adiabaticity will eventually also be of

relevance in different areas, for instance the dynamics of the scalar field that drives inflation in the early-stage expansion of the universe9.

references1. Balian, R. From Microphysics to Macrophysics (Springer,

Berlin, 1991).2. Polkovnikov, A. & Gritsev, V. Nature Phys. 4, 477481 (2008).3. Landau, L. D. & Lifshitz, E. M. Statistical Physics (Pergamon,

Oxford, 1980).4. Farhi, E. et al. Science 292, 472476 (2001).5. Bloch, I., Dalibard, J. & Zwerger, W. Rev. Mod. Phys. (in the

press); preprint at (2007).6. Greiner, M., Mandel, O., Esslinger, T., Hnsch, T. W. & Bloch, I.

Nature 415, 3944 (2002).7. Jarzinsky, C. Phys. Rev. Lett. 78, 26902693 (1997).8. Bustamente, C., Bryant, Z. & Smith, S. B. Phys. Today

54, 4651 (2003).9. Mukhanov, V. Physical Foundations of Cosmology (Cambridge

Univ. Press, Cambridge, 2005).

Has lightning struck twice?

As the rush continues to push Tc ever higher, what of the physics of these superconductors? At first sight, Fe is an unusual ingredient, as ferromagnetism and superconductivity are generally viewed as being antagonistic. Placing a superconductor in a magnetic field will kill the superconductivity eventually.

And to complicate matters, antiferromagnetic correlations are present as well. Experimental probes58 detect a structural phase transition near 150 K and an antiferromagnetic transition with long-range spin-density wave order closer to 130 K in undoped LaOFeAs. With increasing F-doping, both of these features move to lower temperature, weakening and eventually disappearing. As superconductivity appears while the

John Bardeen, who would have celebrated his 100th birthday on 23 May, was the only person to have won two Nobel prizes in physics one for the transistor and the other for the microscopic explanation of superconductivity, known as the BardeenCooperSchrieffer theory (BCS). For about 30 years, all superconductors more or less behaved according to BCS, none of them violating the predicted 30 K maximum transition temperature Tc. However, in 1986, along came the copper-oxide-based superconductors, with Tc rapidly reaching 164 K (under pressure). According to David Pines, Bardeen was delighted, as he had proposed several (non-BCS) models for superconductivity of purely electronic origin with spin fluctuations replacing phonons as the pairing glue and he thought a novel mechanism had to be at work in the cuprates.

Recently, a new and unexpected family of superconductors, based on FeP or FeAs layers, has seemed to further challenge BCS. In the case of non-superconducting LaOFeAs, with alternating stacks of LaO and FeAs layers, fluorine doping into the LaO layer induces superconductivity at 26 K (ref. 1). Under pressure2, the superconductivity survives up to 43 K. Chemically exchanging a smaller ion for La mimics applied pressure, and indeed, with Sm substitution, the transition temperature increases to 43 K (ref. 3). With further tweaking4, SmO1xFxFeAs reaches a Tc of 55 K. Oxygen vacancies also seem to have an important role in raising Tc.

other two wane, the magnetism and superconductivity do seem to be competing states.

The similarities between these superconductors and the cuprates are suggestive, but its too early to jump to conclusions, or say whether the FeAs family will dethrone the cuprates in terms of Tc. From Bardeens approach to superconductivity, Pines ventures that Bardeen would have suspected that a magnetic mechanism could be at work, but he would have wanted some direct evidence for this. After considering various experimental probes, he would likely have encouraged his colleagues to produce sufficiently pure samples that NMR experiments could look for a build-up of antiferromagnetic spin fluctuations in the normal state, analogous to that seen in the cuprates, and perhaps, as a bonus, provide a tentative identification of an unconventional pairing state. Then, he would have waited for the experimental results before embarking on detailed model calculations.

references1. Kamihara, Y. J., Watanabe, T., Hirano, M. & Hosono, H.

J. Am. Chem. Soc. 130, 32963297 (2008).2. Takahashi, H. et al. Nature 453, 376378 (2008).3. Chen, X. H. et al. Nature doi:10.1038/nature07045 (2008).4. Ren, Z. A. et al. Preprint available at (2008).5. Dong, J. et al. Preprint available at (2008).6. de la Cruz, C. et al. Nature doi: 10.1038/nature07057 (2008).7. Kitao, S. et al. Preprint available at (2008).8. Klauss, H.-H. et al. Preprint available at (2008)May Chiao

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