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Alignement of magnetized_accretion_disks_and_relativistics_jets_with_spinning_black_holes
Alignment of Magnetized Accretion Disks and Relativistic Jets with Spinning Black Holes Jonathan C. McKinney et al. Science 339, 49 (2013); DOI: 10.1126/science.1230811 This copy is for your personal, non-commercial use only. If you wish to distribute this article to others, you can order high-quality copies for your colleagues, clients, or customers by clicking here. Downloaded from www.sciencemag.org on January 4, 2013 Permission to republish or repurpose articles or portions of articles can be obtained by following the guidelines here. The following resources related to this article are available online at www.sciencemag.org (this information is current as of January 4, 2013 ): Updated information and services, including high-resolution figures, can be found in the online version of this article at: http://www.sciencemag.org/content/339/6115/49.full.html Supporting Online Material can be found at: http://www.sciencemag.org/content/suppl/2012/11/14/science.1230811.DC1.html This article cites 88 articles, 20 of which can be accessed free: http://www.sciencemag.org/content/339/6115/49.full.html#ref-list-1Science (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. Copyright2013 by the American Association for the Advancement of Science; all rights reserved. The title Science is aregistered trademark of AAAS.
REPORTS viscosity aligns a very thin disk out to some dis-Alignment of Magnetized Accretion tance [estimated to be out to r ~ 10rg to 105 rg, where rg is a gravitational radius (19), dependingDisks and Relativistic Jets with on assumptions] from the BH. Whereas the vis- cosity has been thought to result from turbulenceSpinning Black Holes driven by the magneto-rotational instability (MRI) that amplifies weak small-scale (<~H, the disk height) magnetic fields (20), magnetohydrody-Jonathan C. McKinney,1,2* Alexander Tchekhovskoy,3 Roger D. Blandford1 namical (MHD) simulations of weakly magnet- ized disks have not yet produced any BP alignmentAccreting black holes (BHs) produce intense radiation and powerful relativistic jets, which are effect (21). The BP effect and LT precession re-affected by the BH’s spin magnitude and direction. Although thin disks might align with the main commonly invoked mechanisms to under-BH spin axis via the Bardeen-Petterson effect, this does not apply to jet systems with thick stand how tilt affects the evolution of BH massdisks. We used fully three-dimensional general relativistic magnetohydrodynamical simulations and spin (22, 23), how merging BHs are affectedto study accreting BHs with various spin vectors and disk thicknesses and with magnetic flux by any nearby plasma (24), and how disks andreaching saturation. Our simulations reveal a “magneto-spin alignment” mechanism that causes their jets are oriented (25, 26).magnetized disks and jets to align with the BH spin near BHs and to reorient with the outer Large-scale electromagnetic (EM) fields mightdisk farther away. This mechanism has implications for the evolution of BH mass and spin, BH also affect the jet’s and disk’s orientations via ex- ternal confinement forces (27, 28). Estimates based Downloaded from www.sciencemag.org on January 4, 2013feedback on host galaxies, and resolved BH images for the accreting BHs in SgrA* and M87. on the presumption that large-scale magnetic fields strophysical black holes (BHs) operate the angle between the BH spin axis and the disk’s are weaker than turbulent disk fields suggestedA as engines that convert the gravitational binding energy of accreting plasmas intointense radiation (1) and release BH spin energy angular momentum axis at large distances, influ- ences the intensity of the radiation via changes in the gravitational potential felt by the plasma; it that EM forces are insufficient to align the disk with the BH (27) or the BH with the disk (29, 30). Simulations without disks have given ambiguous(2–4) into powerful relativistic jets (5, 6). Rel- also influences what an observer at different results for the EM jet direction. For a uniform ver-ativistic jets from accreting BHs are commonly viewing angles sees as a result of disk warping tical magnetic field and no disk, the jet is directedobserved to emerge from active galactic nuclei and jet bending. along the magnetic field direction rather than along(AGN) or quasars, x-ray binaries that behave as One mechanism known to possibly affect the the BH’s tilted spin axis (31), whereas isolated mag-microquasars, and gamma-ray burst (GRB) orientation of a disk or jet is the Bardeen-Petterson netic threads tend to align with the BH spin axisevents. GRB jets allow one to probe the earliest (BP) effect (15–18), where Lense-Thirring (LT) when there is no disk to restrict their motion (32).epochs of star formation, whereas radiation and forces induced by the BH frame-dragging cause a We have used fully three-dimensional (3D)jets from AGN play a direct dynamical role via misaligned disk to precess and warp until a local general relativistic (GR) MHD simulations (33)feedback that suppresses star formation in theirhost galaxies (7). BHs are also intrinsically interesting because Table 1. Tilted black hole disk-jet systems. The simulation models are listed by model name, whichthey act as laboratories for probing Einstein’s identifies the approximate BH spin of j (values following “A”), something about the magnetic field choicesgeneral relativity theory and for testing theories (values following “B” and “N”) (28), and the initial relative tilt (qtilt,0 in radians) between the BH spin axisabout accreting BHs and jets. Astrophysical BHs and the disk rotation axis (values following “T”; if T is absent, qtilt,0 = 0). Successive columns give theare characterized primarily by their mass (M) and dimensionless BH spin ( j); the evolved quasi–steady-state value of the disk height/radius ratio H/Rdimensionless spin angular momentum ( j). BHs between r ~ 20rg and r ~ 30rg; the initial relative tilt (qtilt,0) between the disk + jet (having the same tilthave been measured to have masses of tens to initially) and the BH spin axis; the evolved relative tilt between the BH spin axis and the disk and jet, respectively, at r = 4rg; and the same measurements at r = 30rg. If the relative tilt is 0.0, the disk or jetbillions of solar masses M◉; the mass of the BH remained aligned with the BH spin axis; if the tilt is equal to the initial tilt, the disk or jet was unaffected.in M87 is ~6 × 109 M◉ (8). Spins have also beenmeasured and span over many of the possible Disk tilt Jet tilt Disk tilt Jet tiltvalues, including near the maximal value of j ~ 1 Model name BH j Disk H/R Initial tilt r = 4rg r = 4rg r = 30rg r = 30rgin the BH x-ray binary GRS1915+105 with A0.94BfN40 0.9375 0.6 0 0 0 0 0M ~ 14 M◉ and in the AGN MCG-6-30-15 with A0.94BfN40T0.35 0.9375 0.6 0.35 0.0 0.0 0.2 0.2M ~ 3 × 106 M◉ (9, 10). Structures on a few BH A0.94BfN40T0.7 0.9375 0.6 0.70 0.0 0.0 0.4 0.3event horizon length scales have recently been A0.94BfN40T1.5708 0.9375 0.6 1.5708 0.0 0.1 0.5 0.7resolved by Earth-sized radio telescope interfer- A-0.9N100 –0.9 0.3 0 0 0 0 0ometry for SgrA* (11, 12) and M87 (13, 14). A-0.9N100T0.15 –0.9 0.3 0.15 0.1 0.1 0.1 0.2 Although the BH’s present angular momen- A-0.9N100T0.3 –0.9 0.3 0.30 0.2 0.2 0.2 0.2tum axis is set by the history of plasma accretion A-0.9N100T0.6 –0.9 0.3 0.60 0.2 0.3 0.4 0.3and mergers with other BHs, the gas being cur- : A-0.9N100T1.5708 –0.9 0.3 1.5708 0.2 0.4 0.9 0.8rently supplied (at mass accretion rate M ) to the A0.9N100 0.9 0.3 0 0 0 0 0BH can have an arbitrarily different angular mo- A0.9N100T0.15 0.9 0.3 0.15 0.0 0.0 0.1 0.1mentum axis. This relative tilt, given by qtilt for A0.9N100T0.3 0.9 0.3 0.30 0.1 0.1 0.2 0.2 A0.9N100T0.6 0.9 0.3 0.60 0.1 0.1 0.3 0.31 Kavli Institute for Particle Astrophysics and Cosmology, Stanford A0.9N100T1.5708 0.9 0.3 1.5708 0.2 0.3 0.7 0.6University, Stanford, CA 94309, USA. 2Department of Physics A0.9N100 0.99 0.3 0 0 0 0 0and Joint Space-Science Institute, University of Maryland, Col- A0.99N100T0.15 0.99 0.3 0.15 0.0 0.1 0.1 0.1lege Park, MD 20742, USA. 3Center for Theoretical Science, A0.99N100T0.3 0.99 0.3 0.30 0.1 0.1 0.2 0.2Princeton University, Princeton, NJ 08544, USA. A0.99N100T0.6 0.99 0.3 0.60 0.1 0.1 0.3 0.4*To whom correspondence should be addressed. E-mail: A0.99N100T1.5708 0.99 0.3 1.5708 0.1 0.1 0.6 email@example.com www.sciencemag.org SCIENCE VOL 339 4 JANUARY 2013 49
REPORTS of accreting BHs to show that near the BH, both Over the horizon and in the jet, U e 10 for Our self-consistent fully 3D general rela- the disk angular momentum and jet direction our thinner disk models and U e 17 for our tivistic magnetohydrodynamic (GRMHD) simu- reorient and align with the BH’s spin axis. Our thick disk models (28). Also, for both thicknesses, lations started with a disk around an untilted BH simulations were designed so that the magnetic (r/rg)aeff ~ 15 and roughly constant with radius, where the BH spin axis, disk rotational axis, and field built up to a natural saturation strength with, and vr /vf ~ 1 near the horizon (28). So, at all dis- emergent jet’s direction all pointed in the vertical roughly, the disk’s thermal + ram + gravitational tances, the jet’s EM forces lead to tEM/tLT ~ 2 (z) direction. As the simulation proceeded, the forces balancing the disk’s and jet’s magnetic for our thinner disk models and tEM/tLT > ~5 for mass and magnetic flux readily advected from forces such that the trapped large-scale magnetic our thick disk models. Hence, we expect EM forces large distances onto the BH. The magnetic flux field threading the BH and disk became strong to dominate LT forces for both our thinner and versus radius saturated on the BH and within the relative to the disk’s turbulent field. The saturated thick disk models [including for small spins (33)]. disk near the BH after magnetic forces balanced field strength has been shown to be (i) indepen- EM alignment forces are effective when they the disk’s thermal + ram + gravitational forces. dent of the strength of the initial magnetic field are larger than forces associated with the newly ac- Magnetic braking causes such disks to become when the surrounding medium has a sufficient creted rotating plasma with torque per unit area of : even more sub-Keplerian than weakly magnetized supply of magnetic flux, (ii) weakly dependent on tacc e Mvf =ð2prÞ. Therefore, tEM =tacc e U2 WF = thick disks (28), which means that the classical BH spin, and (iii) proportionally dependent on disk ð4rvf Þ, and when these torques are equal one thin disk innermost stable circular orbit position thickness (4, 28, 34). We considered various BH obtains an implicit equation for a “magneto-spin is even less applicable than for weakly magne- spins (35), BH tilts, and disks with a quasi– alignment” radius of tized thick disks. The simulations were evolved steady-state height/radius ratio (H/R) of ~0.6 WF r2 U2 g for a long time period so that the disk reached a rmsa e ð5Þ Downloaded from www.sciencemag.org on January 4, 2013 for thick disks and ~0.3 for thinner disks. Nu- quasi-stationary magnetically saturated state out 4vf merical convergence of our results was determined to about r ~ 40rg (28, 33). on the basis of convergence quality measures for (with rg reintroduced), within which EM forces Then, the BH spin axis was instantly tilted by how well the MRI and turbulence were resolved can torque the accreting dense material. For suf- an angle of qtilt,0 (see Table 1 for tilts used for (table S2) as well as by explicit convergence test- ficiently small values of U or j, no alignment can different spins and disk thicknesses). The tilted ing (33). occur. We obtain rmsa > ~30rg for our thinner and disk-jet system underwent a violent rearrangement Let us motivate these MHD simulations by thick disk models that are sub-Keplerian by a for the larger tilts. The frame-dragging forces estimating whether EM forces are expected to factor of 0.5 to 0.1, respectively (28), although caused the nearly split-monopole BH magneto- dominate LT forces on the rotating heavy disk. accurate estimates require performing more sim- sphere to align with the BH spin axis, as expected Imagine a toy model with a flat heavy disk tilted ulations or accounting for more physics that could because the misaligned angular momentum was and pushed up against the magnetized jet gen- lead to much different rmsa (33). radiated away as part of the electromagnetic erated directly by the rotating BH. For a magnetic field B bending on scale r, the EM torque per unit area is tEM ~ rBrBf/4 (33) for a jet magnetic field that has both radial (Br) and toroidal (Bf ~ rBr ΩF) components and rotates with an angular frequen- cy of ΩF [where rgΩF/c ~ j/8 for j ~ 1 (28)]. The radial field is written in terms of a dimensionless magnetic flux given by 0:7F U ≈ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : ð1Þ 4pr2 Mcg for magnetic flux F ~ 4pr2Br for B in Gaussian units (rg and c reintroduced for dimensional clarity) that is consistent with measurements in our previous work (28). This gives r2 : g tEM e MWF U2 ð2Þ 8pr2 Meanwhile, the LT torque per unit area is tLT ~ ΩLTL with LT precession rate ΩLT ~ 2j/r3, disk angular momentum per unit area L ~ Srvf , and disk sur- : face density S e M=ð2prvr Þ. This gives : jcrg Mvf 2 tLT ð3Þ Fig. 1. 3D snapshot for an evolved model with j = 0.99, initial relative tilt qtilt,0 ≈ 90°, and disk thickness e pr3 v r H/R ~ 0.3. The rotating BH sits at the center of the box of size r = −40rg to r = +40rg in each dimension. The snapshot shows the disk near the BH (yellow isosurface, which is mostly flat in the figure plane), the The ratio of the EM to LT torques for j ~ 1 is then highly magnetized jet region (blue isosurface, with magnetic energy per unit rest-mass energy equal to tEM =tLT e U2 rvr =ð64rg vf Þ, with rg reintroduced about 70), the rotational axis of the disk both initially and at large distances (orange cylinder), outer disk for dimensional clarity. Far beyond the horizon, (green-yellow volume rendering, more aligned with disk rotational axis at large distances), magnetic field vectors (like iron filings on that surface) for a cross section of the jet (cyan vectors), and jet magnetic field tEM 1 2 raeff H 2 lines (white lines) that trace from the BH out to large distances. The disk and jet near the BH are aligned tLT e 64 U r R ð4Þ with the BH spin axis and point mostly in and out of the figure plane, whereas at larger distances the jet g points roughly halfway between the BH spin axis and the disk’s rotational axis (pointing along the orange for an effective viscosity aeff ≡ vr/[(H/R)2vf]. cylinder).50 4 JANUARY 2013 VOL 339 SCIENCE www.sciencemag.org
REPORTSoutflow on Alfvén time scales. The magnetic torque S1), where more tilt led to reduced efficiency due to accretion at either very low (37) or very highthen caused the heavy disk to lose its misaligned to more spatially and temporally irregular mass rates when H/R ~0.5] are expected. For ex-component of angular momentum and so reorient inflow. ample, for SgrA* and M87, if the BHs rotatewith the BH’s rotating magnetosphere. The time Our most extreme case of a tilted BH accre- sufficiently rapidly (33), then we expect the photonscale for alignment seems to be roughly the tion disk and jet system is the j = 0.99 model with spectra, temporal behaviors, and resolved imagesAlfvén crossing time near the heavy disk and a full tilt of qtilt,0 = 1.5708 ≈ 90° and disk thick- of their jets and disks to be affected by nonzeroBH. This “magneto-spin alignment” occurs be- ness H/R ~ 0.3. Even in this extremely tilted case, relative tilts as a result of disk warping and jetcause the magnetic field built up to a natural the evolved disk and jet near the BH aligned with bending near the BH.saturation strength on the horizon where U e 10 the BH spin axis (Fig. 1, fig. S1, and movie S1). Tidal disruption flare events such as Swift(depending on the thickness and tilt), which led The jet’s magnetic field wound around the per- J164449.3+573451 are thought to be producedto the BH magnetosphere’s forces dominating the sistent relativistic jet, and the magnetic field was by very high accretion rates onto BHs, whichdisk dynamics and LT forces. well ordered even for this highly tilted case. The launch fairly persistent jets that dissipate and All simulations (including with zero tilt) were jet was not symmetric around the jet axis, and give the observed emission (38). Our results sug-then further evolved in time until all the tilted instead there was a broad wing (with opening gest that the inner disk and inner jet are bothsimulations reached a new quasi-stationary state half-angle of about 25° by r = 40rg) and a narrow aligned with the BH spin axis, but the observedout to r ~ 40rg. This ensured that any differences wing (with opening half-angle of about 5° by r = jet dissipating at large distances need not pointat later times in the disk and jet between tilted and 40rg). At large distances from the BH, the jet along the BH spin axis. EM forces do not direct-untilted simulations were due to the BH tilt. This drilled its way through the disk material and grad- ly cause any precession (27), so the lack of LT Downloaded from www.sciencemag.org on January 4, 2013also ensured that each of the tilted and untilted ually got pushed away from the disk (Fig. 2 and precession-induced variability does not alone im-simulations reached their own quasi–steady state movie S2). The magnetic field wound around with ply that the jet is necessarily driven by the BHvalues for magnetic flux near the BH, magnetic a pitch angle of about 45° near the BH and a spin power (26). Further, quasi-periodic oscilla-flux in the disk, mass accretion rate, etc. We then smaller pitch angle at larger distances. By r ~ tions (39) and long-term dips seen in this sys-measured the evolved relative tilt between the 300rg, the jet had become parallel with (but offset tem’s light curve might be explained by oscillationsBH spin axis and the disk and jet at r = 4rg and from) the outer disk rotational axis, and so the jet in the disk-jet magnetospheric interface (28) orr = 30rg (Table 1). For all our models, the disk and counterjet were also offset. by periods of magnetic flux accumulation and re-and jet aligned with the BH spin axis near the Thus, our simulations have revealed a “magneto- jection by the BH (4) (both occurring for untiltedBH, whereas the disk axis and jet direction de- spin alignment” mechanism that aligns the disk systems) rather than by LT precession.viated at larger distances. Such deviations are ex- and jet axes with the BH spin axis near the BH Jet dissipation and emission (e.g., in blazars)pected because the jet interacted with circulation once the magnetic field has saturated on the BH might be due to the jet ramming into the diskwith stronger mass inflows at larger distances and within the disk (36). Unlike the BP effect, the until the jet aligns with the disk rotational axis at(33). Despite the tilts and jet deviations, the BH’s mechanism actually works best for thick disks, large distances. Measurements of BH spin inefficiency (defined as the ratio of energy out to and so the magneto-spin alignment mechanism AGN and x-ray binaries might be affected byenergy in) was roughly 100% for j ~0.9 (table should control jet systems where thick disks [due assumptions about the alignment between the disk, BH spin, and jet (9, 10). The cosmological evolution of BH mass and spin and AGN feed- back for accretion at high rates might be affected by the higher BH spin-down rates and jet effi- ciencies relative to standard thin disk spin-down rates and radiative efficiencies (28) and also by how the jet aligns the disk material before LT torques can be effective, thus possibly leading to less change in the BH spin direction relative to the BP effect. References and Notes 1. N. I. Shakura, R. A. Sunyaev, Astron. Astrophys. 24, 337 (1973). 2. R. D. Blandford, R. L. Znajek, Mon. Not. R. Astron. Soc. 179, 433 (1977). 3. S. Koide, K. Shibata, T. Kudoh, D. L. Meier, Science 295, 1688 (2002). 4. A. Tchekhovskoy, R. Narayan, J. C. McKinney, Mon. Not. R. Astron. Soc. 418, L79 (2011). 5. R. V. E. Lovelace, Nature 262, 649 (1976). 6. M. C. Begelman, M. J. Rees, R. D. Blandford, Nature 279, 770 (1979). 7. T. Di Matteo, V. Springel, L. Hernquist, Nature 433, 604 (2005). 8. K. Gebhardt et al., Astrophys. J. 729, 119 (2011). 9. C. S. Reynolds, A. C. Fabian, Astrophys. J. 675, 1048 (2008). 10. J. E. McClintock et al., http://arxiv.org/abs/0911.5408 (2009). 11. S. S. Doeleman et al., Nature 455, 78 (2008).Fig. 2. 3D snapshot similar to Fig. 1 but showing the outer disk (density isosurface in purple) at large 12. V. L. Fish et al., Astrophys. J. 727, L36 (2011).distances from the BH in a box of size r = −350rg to r = +350rg in each dimension. The jet (blue 13. K. Hada et al., Nature 477, 185 (2011).isosurface, here with magnetic energy per unit rest-mass energy equal to ~4, still corresponding to the jet 14. S. S. Doeleman et al., Science 338, 355 (2012). 15. J. M. Bardeen, J. A. Petterson, Astrophys. J. 195, L65spine) is aligned with the BH spin near the BH but gradually gets pushed by the disk material and becomes (1975).parallel to (but offset from) the disk rotational axis at large distances. The strong interaction between the 16. S. P. Hatchett, M. C. Begelman, C. L. 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REPORTS 17. S. Kumar, J. E. Pringle, Mon. Not. R. Astron. Soc. 213, 29. H. Kim, H. K. Lee, C. H. Lee, J. Cosmol. Astropart. Phys. Acknowledgments: J.C.M. thanks R. Narayan, J. Dexter, 435 (1985). 2003, 1 (2003). and P. C. Fragile for useful discussions, and R. Kaehler at 18. R. P. Nelson, J. C. B. Papaloizou, Mon. Not. R. Astron. 30. A. R. King, in Magnetic Fields in the Universe: From KIPAC (SLAC/Stanford) for the artistic rendering in fig. S1. Soc. 315, 570 (2000). Laboratory and Stars to Primordial Structures, Supported by NASA Fermi grant NNX11AO21G ( J.C.M.), a 19. We typically set GM = c = 1, where c is the speed of light, E. M. de Gouveia dal Pino, G. Lugones, A. Lazarian, Princeton Center for Theoretical Science fellowship (A.T.), G is the gravitational constant, and M is the mass of Eds. (American Institute of Physics, Melville, NY, 2005), and NSF Extreme Science and Engineering Discovery the BH, so that rg ≡ GM/c2 = 1. For dimensional clarity, pp. 175–182. 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Dotti, M. Colpi, M. Volonteri, Mon. Not. R. 35. Spins of j ~ 0.9 give mid-range BH rotation rates. See Astron. Soc. 399, 2249 (2009). Supplementary Materials the Physical Models section in the supplement. 24. T. Bogdanović, C. S. Reynolds, M. C. Miller, Astrophys. J. www.sciencemag.org/cgi/content/full/science.1230811/DC1 661, L147 (2007). 36. The magnetic field built up via direct magnetic flux advection, but the buildup might also occur via Materials and Methods 25. P. Natarajan, P. J. Armitage, Mon. Not. R. Astron. Soc. Fig. S1 309, 961 (1999). dynamo generation, as seen in our prior untilted simulations that showed emergent large-scale Tables S1 and S2 26. N. Stone, A. Loeb, Phys. Rev. Lett. 108, 061302 Movies S1 and S2 (2012). dipolar flux patches. References (40–98) 27. A. R. King, J. P. Lasota, Astron. Astrophys. 58, 175 37. R. Narayan, I. Yi, R. Mahadevan, Nature 374, 623 Downloaded from www.sciencemag.org on January 4, 2013 (1977). (1995). 27 September 2012; accepted 7 November 2012 28. J. C. McKinney, A. Tchekhovskoy, R. D. Blandford, 38. J. S. Bloom et al., Science 333, 203 (2011). Published online 15 November 2012; Mon. Not. R. Astron. Soc. 423, 3083 (2012). 39. R. C. Reis et al., Science 337, 949 (2012). 10.1126/science.1230811 Negative Absolute Temperature for In Fig. 1A, we schematically show the rela- tion between entropy S and energy E for a ther- mal system possessing both lower and upper Motional Degrees of Freedom energy bounds. Starting at minimum energy, where only the ground state is populated, an increase in energy leads to an occupation of a larger number S. Braun,1,2 J. P. Ronzheimer,1,2 M. Schreiber,1,2 S. S. Hodgman,1,2 T. Rom,1,2 of states and therefore an increase in entropy. As I. Bloch,1,2 U. Schneider1,2* the temperature approaches infinity, all states be- come equally populated and the entropy reaches Absolute temperature is usually bound to be positive. Under special conditions, however, its maximum possible value Smax. However, negative temperatures—in which high-energy states are more occupied than low-energy the energy can be increased even further if high- states—are also possible. Such states have been demonstrated in localized systems with finite, energy states are more populated than low-energy discrete spectra. Here, we prepared a negative temperature state for motional degrees of ones. In this regime, the entropy decreases with freedom. By tailoring the Bose-Hubbard Hamiltonian, we created an attractively interacting energy, which, according to the thermodynamic ensemble of ultracold bosons at negative temperature that is stable against collapse for definition of temperature (8) (1/T = ∂S/∂E), re- arbitrary atom numbers. The quasimomentum distribution develops sharp peaks at the upper sults in negative temperatures. The temperature is band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. discontinuous at maximum entropy, jumping from Negative temperatures imply negative pressures and open up new parameter regimes for positive to negative infinity. This is a consequence cold atoms, enabling fundamentally new many-body states. of the historic definition of temperature. A con- tinuous and monotonically increasing tempera- bsolute temperature T is one of the cen- with energy. If we were to extend this formula to ture scale would be given by −b = −1/kBT, also A tral concepts of statistical mechanics and is a measure of, for example, the amount of disordered motion in a classical ideal gas. There- negative absolute temperatures, exponentially in- creasing distributions would result. Because the distribution needs to be normalizable, at positive emphasizing that negative temperature states are hotter than positive temperature states, i.e., in thermal contact, heat would flow from a negative fore, nothing can be colder than T = 0, where temperatures a lower bound in energy is re- to a positive temperature system. classical particles would be at rest. In a thermal quired, as the probabilities Pi would diverge for Because negative temperature systems can ab- state of such an ideal gas, the probability Pi for a Ei → –∞. Negative temperatures, on the other sorb entropy while releasing energy, they give particle to occupy a state i with kinetic energy Ei hand, demand an upper bound in energy (1, 2). In rise to several counterintuitive effects, such as is proportional to the Boltzmann factor daily life, negative temperatures are absent, be- Carnot engines with an efficiency greater than cause kinetic energy in most systems, including unity (4). Through a stability analysis for thermo- Pi º e−Ei =kB T ð1Þ particles in free space, only provides a lower en- dynamic equilibrium, we showed that negative ergy bound. Even in lattice systems, where kinet- temperature states of motional degrees of free- where kB is Boltzmann’s constant. An ensemble ic energy is split into distinct bands, implementing dom necessarily possess negative pressure (9) and at positive temperature is described by an occu- an upper energy bound for motional degrees of are thus of fundamental interest to the description pation distribution that decreases exponentially freedom is challenging, because potential and in- of dark energy in cosmology, where negative pres- teraction energy need to be limited as well (3, 4). sure is required to account for the accelerating 1 So far, negative temperatures have been realized expansion of the universe (10). Fakultät für Physik, Ludwig-Maximilians-Universität München, in localized spin systems (5–7), where the finite, Cold atoms in optical lattices are an ideal Schellingstraße 4, 80799 Munich, Germany. 2Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, discrete spectrum naturally provides both lower system to create negative temperature states be- Germany. and upper energy bounds. Here, we were able to cause of the isolation from the environment and *To whom correspondence should be addressed. E-mail: realize a negative temperature state for motional independent control of all relevant parameters firstname.lastname@example.org. degrees of freedom. (11). Bosonic atoms in the lowest band of a52 4 JANUARY 2013 VOL 339 SCIENCE www.sciencemag.org