RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing by ⌬ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding fixed, then increasing by (1). At low Reynolds number, the effects of in space can be accomplished through cyclic, ⌬ while holding fixed, decreasing , then inertia are negligible and, in the absence of engineered changes in the shape of a body. decreasing , which brings the system back external forces, bodies are at rest. Nevertheless, Motion in space can be accomplished without to the original relative configuration. In those as a body changes its shape, its location and thrust or external forces. phases of the cycle in which is held fixed, orientation generally change. A cyclic change in Swimming with spherical caps. The two- the system does not rotate. The rotation of the the shape of a body can lead to a net translation sphere example can be generalized to two rigid system during the two phases in which is or rotation. The net translation or rotation does bodies of arbitrary shape with a common fixed changed do not balance because is differ- not depend on the speed with which the shape point (3). Consider two circular spherical caps ent. For small ␥ (small caps), small (small changes are carried out; it is a consequence of of angular radius ␥, with uniform surface den- separation), and small ⌬ and ⌬ (small the geometry of the sequence of shapes, a clas- sity and mass m, on a sphere of radius R. The deformations), the net motion of the system sical example of geometric phase (2). For ex- relative configuration is specified by two angles per cycle of deformation is approximately ample, the cilia of a paramecium effectively (Fig. 1). Let xˆ, yˆ, and zˆ be a right-handed 1 define and allow changes in its shape, and loco- orthonormal basis with origin at the center of ⌬ ϭ ␥2 ⌬⌬ (2) motion is accomplished through cyclic changes the sphere. The caps are symmetrically dis- 2 in its shape through motion of its cilia. placed above and below the x-y plane by the Having made everything small compared Consider two concentric spheres in flat angle , and they are rotated oppositely about with the size of the sphere, it is apparent that space with equal moments of inertia connect- their centers by the angle . The angles and the essence of the matter is not that these are ed at their centers (3–5). If one sphere is are deformation coordinates (6). As the relative rigid bodies, but that the system lives on a rotated with respect to the other, then the configuration changes, the orientation of the curved manifold. orientation of the system adjusts appropriate- system adjusts dynamically. Because of the Swimming on curved manifolds. Rigid ly. The adjustment can be determined using symmetry of the system, it will move by rotat- bodies, defined by a large number of redundant the fact that angular momentum is conserved. ing around the zˆ axis. Let be the longitude of distance constraints between constituents, can- If the total angular momentum is zero, the the system, the angular displacement of the not, in general, move on a manifold of noncon- angular velocities of the spheres are equal but system around the zˆ axis. Assuming zero total stant curvature because the redundant con- opposite. Therefore, if one sphere is rotated angular momentum, the equation of motion of straints cannot remain consistent. Consider three about an axis by an angle with respect to the is masses on the vertices of a triangle, with geo- other sphere, then the first sphere rotates in desics of fixed length connecting them. Now D͑t͒Csin͑t͒ space about that axis by /2 and the other ͑ ͒ ϭ place a fourth mass in the middle of the triangle, D t ͑ ͑ ͒͒2 ϩ ͑ ͑ ͒͒2 rotates by Ϫ/2. If the two spheres undergo a A cos t C sin t equidistant from each of the three masses. The sequence of rotations with respect to each (1) distance to those masses will depend on the other and are then brought back to their orig- curvature of the manifold. This system of four inal relative orientation, then their orientation where D is the derivative operator, C is the masses cannot move to a region of different in space can undergo a net rotation. The net moment of inertia of each cap about the radial curvature without some of the constraint dis- rotation of the system does not depend on the line through its center, and A is the moment of tances changing. This is a consequence of the speed with which these relative rotations oc- inertia about any line through the center of the fact that the constraints are redundant—more cur; it depends only on the sequence of rela- sphere perpendicular to this line. The denomina- constraints are specified than are needed to tive orientations. This is another classical tor is half the moment of inertia of the system specify the relative location of the masses. If example of geometric phase. about the zˆ axis; the numerator is half the zˆ instead of specifying that the fourth mass is Generalizing the latter example, I show here component of the angular momentum due to the equidistant from the three masses it is specified that it is possible for quasi-rigid bodies to swim twisting (nonzero D) of the cap. Each cap makes that it is a certain distance from two of the on frictionless curved manifolds simply by way the same contribution to the zˆ moment of inertia vertices, then all of the constraints can be main- and the zˆ component of the angular momentum, tained even as the curvature changes. Such a Massachusetts Institute of Technology, Cambridge, so the additional factors of two in the numerator rigid body with irredundant constraints will be MA 02139, USA.E-mail: [email protected] and denominator cancel. Note that the counterro- called a quasi-rigid body. A quasi-rigid body has
www.sciencemag.org SCIENCE VOL 299 21 MARCH 2003 1865 R ESEARCH A RTICLE a well-defined configuration even though the metric, q is a tuple of manifold coordinates, and For the sphere, choose spherical coordi- distances between all the constituents are not v is a tuple of the associated generalized veloc- nates, with the equator bisecting the struts (Fig. fixed. One particular class of quasi-rigid bodies ities (7). A Lagrangian for a quasi-rigid body 2). The longitude of the vertex at time t is (t). has a tree-like topology. From each vertex ex- can be obtained by defining generalized coor- The length of the geodesic struts, which are tend an arbitrary number of branches or struts. dinates for the system that incorporate the time- spherical arcs, is specified by the angular extent At the end of any strut, more struts may be dependent constraints among the constituent e(t), measured from the center of the sphere. attached. Masses may be attached to the verti- particles, deriving the associated transformation The separation angle between the struts and the ces. The configuration of the system is deter- of the generalized velocities, and rewriting the line of symmetry is ␣(t). Conservation of mo- mined if the lengths of all the struts and the free Lagrangians for the constituent masses in mentum, with zero initial momentum, is angles they make with the connecting struts are terms of these system coordinates (8). enough to determine the motion of the system specified. The first example shows that more Consider the dynamics of such a quasi- as it deforms general quasi-rigid bodies are allowed; they rigid body on two illustrative manifolds: the D͑t͒ ϭ need not be tree-like and the struts need not be plane and the sphere (9). Because of the ͑ ␣͑ ͒ ͑ ͒ ͑ ͒ ␣͑ ͒ Ϫ ␣͑ ͒ ͑ ͒͒ geodesics. Struts of quasi-rigid bodies can be symmetry of the body and the manifold, these 2m1 D t sine t cose t sin t cos t De t ϩ ͑ ␣͑ ͒͒2 ϩ ͑ ␣͑ ͒͒2͑ ͑ ͒͒2 constructed from a local lattice or truss of nearly three-mass quasi-rigid bodies will move only m0 2m1 cos t 2m1 sin t cose t rigid rods. If the rods are small compared with along the direction bisecting the two struts. (4) the intrinsic curvature of the manifold, then they The calculation can be simplified by choos- can maintain the local constraints of the truss, ing one of the manifold coordinates to coin- The net translation can be written as a line without much strain, while conforming to the cide with this direction. integral of a real-valued 1-form along the defor- manifold on a larger scale. For the plane, choose rectangular coordi- mation parameter path, or by Stokes’s theorem For a quasi-rigid body to swim on a curved nates, with the x-axis bisecting the struts. The as an integral over the region enclosed by the manifold, it must undergo changes in its shape. x-coordinate of the vertex at time t is x(t), the path of the 2-form that is the exterior derivative For such a change to result in a net rotation or length of the strut is l(t), and the angle between of the 1-form. In this case, the real-valued translation of the body on the manifold, the the struts and the x-axis is ␣(t). Momentum 1-form is not closed and there is a net transla- shape changes must go through a nontrivial conservation, with zero momentum, leads to tion. For small deformations and for bodies with cycle of deformation. A simple quasi-rigid body Dx͑t͒ ϭ small extent relative to the size of the sphere, the that satisfies these requirements is a body con- translation per cycle is approximately 2m1 sisting of one mass point with mass m connect- ͑l͑t͒sin␣(t)D␣(t) Ϫ cos␣͑t͒ Dl͑t͒͒ 0 m ϩ 2m ⌬ ϭ ed to two other mass points, each with mass m , 0 1 1 4m m by geodesic struts of given length separated by a (3) Ϫ 0 1 ͑ ͒2 ␣⌬ ⌬␣ ͑ ϩ ͒2 sine sin e given angle. The body can be deformed by For a cycle of deformation in the parameter m0 2m1 ␣ ⌬ changing the length of the struts or the angle plane (l, ), the net translation, x, can be (5) between them; nontrivial contractible cycles en- written as an integral of a real-valued 1-form. close area on the deformation parameter plane. In this case, the 1-form is closed, so there is Swimming on manifolds without sym- A Lagrangian for the free geodesic motion no net translation for a cycle of deformation: metry. The examples so far have relied on of a particle on a manifold is mg(q)(v, v)/2, ⌬x ϭ 0. momentum conservation to deduce the iner- where m is the mass of the particle, g is the tial motion of the system from changes in the deformation parameters. If the momentum is not conserved, this does not work. However, if the parameters are varied quickly, then the response of the system is almost the same as if the momenta were conserved. Assume this is the case and consider henceforth the case for which the momentum is zero, whether or not it is exactly conserved or only approxi- mately conserved for fast deformations. The generalized momentum of the quasi- rigid body is obtained by taking the partial derivative of the system Lagrangian with re- spect to its generalized velocity argument. If the free Lagrangians are homogeneous quadratic forms in the velocities, then the momenta are sums of terms that are linear in the system generalized velocities and of terms that are linear in the rate of change of the deformation parameters. This is a set of linear equations that Fig. 2. A quasi-rigid body on the sphere.The two deformation parameters are the length l and can be solved to give the generalized velocities separation angle ␣.The geodesic struts subtend Dq(t) in terms of the momenta and q(t), c(t), Fig. 1. Two spherical caps on a sphere.The two the angle e ϭ l/R at the center of the sphere, and Dc(t), where q(t) are the generalized coor- deformation parameters are and .The caps where R is the radius of the sphere, and the dinates of the system, c(t) are the deformation twist oppositely by the angle , and they are separation angle ␣ is the angle between the struts parameters, and Dc(t) are their rates of change, symmetrically displaced by the angle above and the symmetry line, measured on the plane all at time t. For zero momentum, the result is and below the equator.The angle is the tangent to the sphere at the vertex.The angle longitude of the system.The deformation pa- is the longitude of the system.The deformation linear in Dc(t) rameters follow a specified schedule, and the parameters follow a specified schedule, and the longitude of the system adjusts dynamically. longitude of the system adjusts dynamically. Dq(t) ϭ A(c(t), q(t))Dc(t) (6)
1866 21 MARCH 2003 VOL 299 SCIENCE www.sciencemag.org R ESEARCH A RTICLE This may be viewed as defining a vector- i͑͒͑ ͒ ϭ i ͑ ͒a ϩ i ͑ ͒b A qua- B c,q Aa c,q Ab c,q Swimming in curved spacetime. valued 1-form that takes tangent vectors si-rigid body can swim on a manifold with (15) along the deformation parameter path, with intrinsic curvature through cyclic deforma- components Dc(t), to the generalized velocity The field strength can be used to determine tions of shape. General relativity portrays components. This vector-valued 1-form is a whether a contractible cyclic deformation re- spacetime as a curved four-dimensional man- vector gauge potential (3). sults in translation for more complicated ge- ifold. Is it possible to swim in spacetime Consider piece-wise linear parameter ometries than considered thus far, whether or through cyclic deformations? paths. For a segment with tangent Dc(t) ϭ, not symmetries are present. A nonzero field In relativity, forces of constraint move with the evolution is governed by the equations strength Fi implies there is a change in coor- finite velocity, so if one part of the system re- i Dc͑t͒ ϭ dinate q as the parameters traverse the loop ceives an impulse then there will be a delay in the specified by the two deformation parameter response of other parts of the system. Naturally ͑ ͒ ϭ ͑ ͑ ͒ ͑ ͒͒ Dq t A c t , q t (7) vectors and . In the cases considered thus occurring bodies are not described by pure posi- or, collectively, far, where the translation occurred along a tional constraints. However, there is no obstacle single coordinate, the corresponding compo- to engineering a quasi-rigid body that does main- Ds(t) ϭ G(s(t)) (8) nent of the field strength reduces to the real- tain positional constraints, as long as the schedule where s(t) ϭ (c(t), q(t)). Let valued 2-form specified earlier. of deformations of the body is known sufficiently in advance. In this case, the internal stresses that L F ϭ DF G (9) Swimming with intrinsic curvature. G Swimming on a frictionless plane cannot be are required to maintain the positional constraints
define the operator LG that gives the rate of accomplished by cyclic deformation of shape, can be precomputed and prespecified, and then change of state functions F along solution but it is possible to swim on a frictionless executed simultaneously. The engineered quasi- paths of G. Exponentiating this operator ad- sphere. The plane has no intrinsic or extrinsic rigid body is choreographed for a particular vances state functions (8). Advancing the curvature. The sphere has both. Therefore, it is frame, which defines simultaneity. Ballet is not coordinate selector gives the coordinates interesting to consider whether swimming on a Lorentz invariant. It is choreographed so that
⌬ cylinder is possible. The cylinder has extrinsic dancers make simultaneous movements in the ͑ ϩ ⌬ ͒ ϭ ͑ tLG ͒͑ ͑ ͒ ͑ ͒͒ q t0 t e Q c t0 , q t0 curvature but no intrinsic curvature. The exam- frame of the audience. In other frames, the danc- (10) ple illustrates the use of the field strength to ers would be out of sync, but those observers are determine the net translation. invited to slow down and enjoy the performance. where Q(c, q) ϭ q. Construct a body from three point masses This idea of engineered quasi-rigid bodies is in- Consider the evolution of the system result- as follows. The cylindrical coordinates of the troduced so that the analysis follows the previous ing from a small loop in deformation parameter vertex of the body, with mass m0, are ( , z), examples; it seems likely that it would not be space around a parallelogram specified by two where measures the angle around the cyl- strictly required to be able to swim in spacetime. vectors and . The loop is traversed by moving inder and z the distance along the axis. The Of most interest is Schwarzschild geom- first along , then , then Ϫ, then Ϫ, which radius of the cylinder is R. Two other point etry, the curved spacetime around a nonrotat-
brings the system back to the initial point. The particles, with mass m1, are connected to the ing mass (12). The Schwarzschild metric is coordinate tuple after evolution around the par- vertex by geodesic struts of length l(t). The 2GM allelogram is (10) angle between each strut and the bisector is g͑q͒͑ , ͒ ϭϪͩ1 Ϫ ͪ c2 t t ␣ 0 1 c2r 0 1 q͑t ϩ ⌬t͒ ϭ (t). The orientation of the body is specified 0 Ϫ Ϫ by , the angle from the horizontal (which is ͑eLGeLGe LG e LGQ)(c(t ), q(t )) (11) 1 0 0 perpendicular to the axis of the cylinder) to ϩ r r 0 1 For small loops, the lowest order change in 2GM the bisector of the struts. ͩ1 Ϫ ͪ the coordinates is given by a commutator The components of the gauge potential are c2r ͑ ␣ ͒ ϭ 2 2 ͑ ϩ ⌬ ͒ ϭ Al l, ; , z, ϩ ͑ ϩ ͑ ͒ ͒ q t0 t r 0 1 sin 0 1 (18) Ϫ ␣ Ϫ ␣ ͑ ͒ ϩ ͓͑ 2m1 cos cos 2m1 sin cos ϭ q t0 LG ,LG]Q)(c(t0), q(t0)) (12) ͩ ͪ where q (t, r, , ) are Schwarzschild , ,0 R͑m0 ϩ 2m1͒ R͑m0 ϩ 2m1͒ ϭ t r coordinates, i ( i, i, i , i ) are compo- The commutator defines a vector-valued A␣͑l, ␣; , z, ͒ ϭ nents of tangent vectors with respect to the 2-form of the vectors and . This 2-form Schwarzschild coordinate basis vectors, M is 2lm cos sin␣ 2lm sin sin␣ is the field strength associated with the 1 1 the Schwarzschild mass, G is the gravitation- ͩ ͑ ϩ ͒ , ͑ ϩ ͒ ,0ͪ vector gauge potential. Let dca and dcb be R m0 2m1 m0 2m1 al constant, and c is the speed of light. dual basis 1-forms on the parameter plane. (16) According to general relativity, test particles The components of the vector potential are The fact that the component of the gauge follow geodesics on this curved spacetime man- i ϭ i a ϩ i b A (c, q) Aa(c, q) dc Ab(c, q) dc with potentials is zero means the tilt of the body ifold. Geodesics are solutions of the Lagrange i running through the component selectors does not change as the body deforms. The equations for the Lagrangian g(q)(u, u)/2, where of q˙. The components of the field strength other components are nonzero, so there is g is the Schwarzschild metric, q the tuple of are (11) some motion on the cylinder. However, the Schwarzschild coordinates, and u the tuple of the F i͑,͒͑c,q͒ field strength is identically zero associated generalized velocities. The indepen- ͑ ͒͑ ␣ ͒ ϭ ͑ ͒ dent variable is proper time. The magnitude of ϭ ͓͑L , L ͔Q͒͑c, q͒ (13) F , l, ; , z, 0, 0, 0 G G the generalized velocity is conserved and equal ϭ ͑͑Ѩ i ͑ ͒͒ ͑ ͒ (17) 1Ab c, q Aa c, q to c. This conserved quantity can be used to Ϫ ͑Ѩ i ͑ ͒͒ ͑ ͒ ͑ ab Ϫ ba͒ for arbitrary deformation vectors and . eliminate the proper time in favor of Schwarzs- 1A c, q Ab c, q ) a Therefore, small cyclic deformations of this child time t as the independent variable. This ϩ ͑Ѩ ͑ i͑͒͒͑ ͒͒ 0 B c, q body on a cylinder do not give a net transla- allows the “dance” of the swimmer to be cho- Ϫ ͑Ѩ ͑ i͑͒͒͑ ͒͒ 0 B c, q (14) tion. Apparently, intrinsic curvature is re- reographed in a frame of constant t. quired for swimming on curved manifolds, The reduced Lagrangian for a particle of where and extrinsic curvature does not help. mass m is (13)
www.sciencemag.org SCIENCE VOL 299 21 MARCH 2003 1867 R ESEARCH A RTICLE ͑ ˙ ͒˙ what stresses to apply so that the system main- ⌬ ϭ Lr t; r, , ; r˙, , r tains the required positional constraints. Consider the following spacetime swim- 3m m GM 2GM Ϫ 0 1 2 ␣⌬ ⌬␣ ϭ mc c2ͩ1 Ϫ ͪ mer (Fig. 3). Place one point mass, with ͑ ϩ ͒2 l 2 3 sin l c2r m0 3m1 c r mass m0, at the vertex. Then extend three equal length struts, with proper length l(t) (20) r˙2 Ϫ in the Schwarzschild frame. In a local sta- Therefore, it is indeed possible to swim in 2GM tionary Lorentz frame, let ␣(t) be the angle spacetime (16). Translation in space can be ͩ1 Ϫ ͪ c2r each of these struts makes with an axis accomplished merely by cylic changes in 1/ 2 defining the orientation of the body, and shape, without thrust or external forces. distribute the struts equally in angle about The curvature of spacetime is very slight, ϩ 2 ˙2 ϩ ͑ ͒ 2˙ 2 r ( sin ) (19) the axis. For three struts, the angle between so the ability to swim in spacetime is unlikely them is 2/3 radians (15). At the end of to lead to new propulsion devices. For a each of these three struts, place a point meter-sized object performing meter-sized
The independent variable is Schwarzschild mass of mass m1. The deformation param- deformations at the surface of the Earth, the time t. The coordinates are the Schwarzschild eters are l(t) and ␣(t). The system Lagrang- displacement is of order 10Ϫ23 m(17). Nev- spatial coordinates, and the generalized ve- ian is obtained from the individual free ertheless, the effect is interesting as a matter locities are the rates of change of the Schwarz- Lagrangians, as before. of principle. You cannot lift yourself by pull- schild coordinates with respect to t. The calculation is simplified by introduc- ing on your bootstraps, but you can lift your- The active struts are constructed at fixed ing one additional assumption. The free La- self by kicking your heels.
Schwarzschild time. They are taken to be geo- grangian Lr is not quadratic in the velocities, desics of specified proper length on the sub- so the momenta are not linear in the general- manifold of constant t. The struts must be con- ized velocities. In this case, the solution for stantly monitored to make sure that the con- the generalized velocities in terms of the rates References and Notes 1.A.Shapere, F.Wilczek, J. Fluid. Mech. 198, 557 straints are maintained. This monitoring may be of change of the deformation parameters (1989). done locally along the strut by surveying neigh- would involve the solution of nonlinear equa- 2.Geometric phase is also called anholonomy.Some boring points on the strut (14). The monitoring tions. But in the limit, where the velocities restrict the term geometric phase to quantum phases and use the term anholonomy for classical may be done sufficiently quickly by giving each are small compared with the velocity of light, processes. surveyor responsibility for an arbitrarily small the mass matrix is constant. For simplicity, 3.A.Shapere, F.Wilczek, Eds.,in Geometric Phases in segment. Each surveyor must know in advance this assumption is made here. Physics (World Scientific, London, 1989) pp.449– The goal is to show that it is possible to 460. 4.A.Guichardet, Ann. Inst. Henri Poincare´ 40, 329 swim in spacetime. So it is enough to con- (1984). sider a special orientation of the body. If the 5.R.Montgomery, in The Geometry of Hamiltonian axis of the body is oriented radially away Systems, T.Ratiu, Ed.(Springer, New York, 1991). from the central mass, then the symmetry of 6.Contrast the fact that here the deformation is specified by coordinates (the associated basis vector fields com- the Schwarzschild geometry and the three- mute) with the fact that the successive angles of rota- fold symmetry of the swimmer guarantee that tion used in the two-sphere example are not coordi- any translation due to cyclic deformation will nates (the associated vector fields do not commute). occur only in the radial direction. The prob- 7.A tuple is an ordered list. 8. G.J.Sussman, J.Wisdom, with M.E.Meyer, Structure lem is, then, to compute the radial component and Interpretation of Classical Mechanics (MIT Press, of the deformation field strength, which is a Cambridge, MA, 2001). real-valued 1-form, as in the example of the 9.It is informative to consider these manifolds even though they have constant curvature, so the assump- plane and sphere. tion of quasi-rigidity is not required. The calculation follows the same outline 10.Note that functions compose in the reverse order to as the simpler examples presented above. the operators that generate them: (eLG I) ⅙ (eLG I) ϭ eLG eLG I, where I is the identity function. ץ The general form of the displacement can 11.In this expression, the structure produced by the 1 be anticipated, given Eq. 5. The displace- contracts with the vector components of the fol- ment will be proportional to the square of lowing A.Similarly, the structure generated by the ץ the ratio of the size of the object to the 0 contracts with the components of the following or (8).Equation 14 is fully expanded for clarity. radius of curvature of the manifold. For The first term of Eq.14 is the product of two Schwarzschild geometry, the components factors.The first factor can be written in terms of of the Riemann curvature tensor are pro- a commutator of vector fields on the configuration 2 3 manifold with coefficients Aa(c, q) and Ab(c, q); the portional to GM/(c r ), which may be second factor is the area of the parallelogram thought of as the inverse of the square of a formed by and on the deformation parameter characteristic radius of curvature. There- plane. fore, the displacement should be propor- 12. C.W.Misner, K.S.Thorne, J.A.Wheeler, Gravitation 2 2 3 (W.H.Freeman, San Francisco, 1970). tional to l GM/(c r ). In addition, the 13.E.Bertschinger, notes for MIT class 8.962,“General Fig. 3. A quasi-rigid spacetime swimmer.From displacement should be proportional to the Relativity” (1999). 14.A passive fixed-length strut moving uniformly with the vertex, with mass m0, extend three geodesic change in length, to the change in separa- struts of proper length l.In a local Lorentz frame tion angle, and to a factor that is homoge- respect to the Schwarzschild frame would appear at the vertex, each strut is tilted by the angle ␣ Lorentz contracted in the Schwarzschild frame.A neous of degree zero in the masses. De- surveyor moving with the strut would have to ac- from the axis of the swimmer.The three struts tailed calculation confirms these expecta- count for this and add to the schedule of deforma- are equally spaced around this axis.The axis is tions. For large r, where the body is small tions a change in the length of the strut so that the pointing radially away from the central mass, length in the Schwarzschild frame, the frame of cho- which is to the left.The spatial Schwarzschild compared to the radius of curvature of reography, is as expected. coordinates of the vertex are (r, , ). spacetime, the displacement is found to be 15.If the axis of the body were aligned with the North
1868 21 MARCH 2003 VOL 299 SCIENCE www.sciencemag.org R ESEARCH A RTICLE
pole, then ␣ would be the co-latitude of the struts swimmer.However, the calculation that is present- tion.I thank H.Abelson, E.Bertschinger, D.Finkel- and the longitudinal angle between the struts would ed is enough to show the existence of the effect stein, R.Hermann, P.Kumar, G.J.Sussman, J. be 120°. that is surely also present in more complicated Touma, and F.Wilczek for helpful and pleasant 16.The presence of the factor of c 2 confirms that this is situations.Perhaps the most interesting case to conversations. a relativistic effect.There is no swimming effect in consider would be a swimmer in a circular orbit, the analogous Newtonian problem. where the swimming effect could be used to grad- 17.The fact that the swimming displacement per ually increase the radius of the orbit. 11 December 2002; accepted 12 February 2003 stroke is so small means that, strictly speaking, one 18.I thank J. Touma for infecting me with his Published online 27 February 2003; should consider the swimming effect relative to interest in geometric phase and for bringing the 10.1126/science.1081406 the ordinary nonswimming geodesic motion of the articles of A.Shapere and F.Wilczek to my atten- Include this information when citing this paper. REPORTS evaporation of Al, shows the three in-line Coherent Quantum Dynamics of Josephson junctions together with the small loop defining the qubit in which the persistent current can flow in two directions, as shown a Superconducting Flux Qubit by arrows (Fig. 1A). The area of the middle I. Chiorescu,1* Y. Nakamura,1,2 C. J. P. M. Harmans,1 J. E. Mooij1 junction of the qubit is ␣ϭ0.8 times the area of the two outer ones. This ratio, together ϭ 2 We have observed coherent time evolution between two quantum states of a with the charging energy EC e /2C and the ϭ superconducting flux qubit comprising three Josephson junctions in a loop. The Josephson energy EJ hIC/4 e of the outer superposition of the two states carrying opposite macroscopic persistent cur- junctions (where IC and C are their critical rents is manipulated by resonant microwave pulses. Readout by means of current and capacitance, respectively), deter- switching-event measurement with an attached superconducting quantum in- mines the qubit energy levels (Fig. 2A) as a ␥ terference device revealed quantum-state oscillations with high fidelity. Under function of the superconductor phase q ␥ ϭ strong microwave driving, it was possible to induce hundreds of coherent across the junctions (Fig. 1B). Close to q oscillations. Pulsed operations on this first sample yielded a relaxation time of 900 nanoseconds and a free-induction dephasing time of 20 nanoseconds. These results are promising for future solid-state quantum computing.
It is becoming clear that artificially fabricated freedom, thereby reducing the decoherence. solid-state devices of macroscopic size may, un- Quantum oscillations of a superconducting two- der certain conditions, behave as single quantum level system have been observed in the Cooper particles. We report on the controlled time-depen- pair box qubit using the charge degree of freedom dent quantum dynamics between two states of a (4). An improved version of the Cooper pair box micron-size superconducting ring containing bil- qubit showed that quantum oscillations with a lions of Cooper pairs (1). From a ground state in high quality factor could be achieved (5). In ad- which all the Cooper pairs circulate in one direc- dition, a qubit based on the phase degree of tion, application of resonant microwave pulses freedom in a Josephson junction was presented, can excite the system to a state where all pairs consisting of a single, relatively large Joseph- move oppositely, and make it oscillate coherently son junction current-biased close to its critical Fig. 1. (A) Scanning electron micrograph of a flux between these two states. Moreover, multiple current (6, 7). qubit (small loop with three Josephson junctions of critical current ϳ0.5 A) and the attached SQUID pulses can be used to create quantum operation Our flux qubit consists of three Josephson (large loop with two big Josephson junctions of sequences. This is of strong fundamental interest junctions arranged in a superconducting loop critical current ϳ2.2 A).Evaporating Al from two because it allows experimental studies on deco- threaded by an externally applied magnetic flux different angles with an oxidation process between ⌽ ϭ herence mechanisms of the quantum behavior of a near half a superconducting flux quantum 0 them gives the small overlapping regions (the Jo- macroscopic-sized object. In addition, it is of h/2e [(8); a one-junction flux qubit is described in sephson junctions).The middle junction of the great importance in the context of quantum com- (9)]. Varying the flux bias controls the energy qubit is 0.8 times the area of the other two, and the ratio of qubit/SQUID areas is about 1:3.Ar- puting (2) because these fabricated structures are level separation of this effectively two-level rows indicate the two directions of the persistent attractive for a design that can be scaled up to system. At half a flux quantum, the two lowest current in the qubit.The mutual qubit/SQUID in- large numbers of quantum bits or qubits (3). states are symmetric and antisymmetric super- ductance is M Ϸ 9 pH.( B) Schematic of the Superconducting circuits with mesoscopic Jo- positions of two classical states with clockwise on-chip circuit; crosses represent the Josephson junctions.The SQUID is shunted by two capacitors sephson junctions are expected to behave accord- and anticlockwise circulating currents. As ϳ ing to the laws of quantum mechanics if they are shown by previous microwave spectroscopy ( 5 pF each) to reduce the SQUID plasma fre- quency and biased through a resistor (ϳ150 ohms) separated sufficiently from external degrees of studies, the qubit can be engineered such that to avoid parasitic resonances in the leads.Symme- the two lowest eigenstates are energetically try of the circuit is introduced to suppress excita- 1Quantum Transport Group, Department of Nano- well separated from the higher ones (10). tion of the SQUID from the qubit-control pulses. Science, Delft University of Technology and Delft Because the qubit is primarily biased by The MW line provides microwave current bursts Institute for Micro Electronics and Submicron Tech- magnetic flux, it is relatively insensitive to inducing oscillating magnetic fields in the qubit nology (DIMES), Lorentzweg 1, 2628 CJ Delft, Neth- loop.The current line provides the measuring pulse 2 the charge noise that is abundantly present in erlands. NEC Fundamental Research Laboratories, 34 I and the voltage line allows the readout of the Miyukigaoka, Tsukuba, Ibaraki 305-8501, Japan. circuits of this kind. b switching pulse Vout.The Vout signal is amplified, *To whom correspondence should be addressed.E- The central part of the circuit, fabricated and a threshold discriminator (dashed line) detects mail: [email protected] by electron beam lithography and shadow the switching event at room temperature.
www.sciencemag.org SCIENCE VOL 299 21 MARCH 2003 1869