aa r X i v : . [ qu a n t - ph ] A p r October 24, 2018
Wires with Quantum Memory
C. Chryssomalakos, H. Hernandez, D. Gelbwaser-Klimovsky and E. Okon
Instituto de Ciencias NuclearesUniversidad Nacional Aut´onoma de M´exicoApdo. Postal 70-543, 04510 M´exico, D.F., M ´EXICO chryss, hcoronado, david.gelbwaser, [email protected]
Abstract:
We show that quantum particles constrained to move along curves undergoing cyclicdeformations acquire, in general, geometric phases. We treat explicitly an example, involving particulardeformations of a circle, and ponder on potential applications. ires with Quantum Memory You leave a wire on a table in a locked room, and rumours soon have it that somebody entered the room,played with the wire, and left it at the exact same initial position — is it possible to detect the intrusion?Geometrical phases may appear in quantum systems the hamiltonian H of which depends on a setof external parameters ξ A , H = H ( ξ ). After a cyclic change of the parameters, along a loop C inparameter space, slow enough to guarantee that the system remains in an instantaneous eigenstate of H ,the wavefunction acquires a phase that, apart from the standard dynamical part of the “energy timestime” type, contains a contribution that only depends on C — hence the term “geometric” [1]. Observableeffects are obtained if the system starts out in a suitable superposition of energy eigenstates, each of whichacquires a different geometric phase — numerous experiments have confirmed these theoretical predictions(see, e.g. , [18, 3, 15, 19, 21, 16]).On a different vein, the dynamics of quantum particles constrained to move along a curve in threedimensional euclidean space, by a steep confining potential in the plane normal to the curve, is governedby an effective hamiltonian that depends on the curvature κ ( s ) and the torsion τ ( s ) of the curve, where s denotes arclength [7, 4, 17, 10] (see also [8, 9, 5, 12, 11]). Thus, the Fourier coefficients, for example, of κ and τ , can be considered as external parameters of the effective hamiltonian. Slow, cyclic changes inthe shape of the curve may then lead to the appearance of geometric phases, and it is the confirmationof this possibility that we report on here. Thus, the opening question is answered to the affirmative, atleast for a playful enough intruder (see below). Apart from limiting what intruders can get away with,the effect pointed out here opens up a wide arena for experimentation, both gedanken and real — weponder on some possibilities in the concluding section. Consider a hamiltonian H ξ , as above, where the ξ ’s are varying with time, tracing out a loop C in ξ -space.Suppose the physical system governed by H ξ starts out, at time t = 0, in the nondegenerate eigenstate | n, ξ i , which satisfies H ξ | n, ξ i = E ξ n | n, ξ i . Then, in the adiabatic approximation, in which the changein H is slow (in the time scale set by the energy difference of neighboring eigenstates), the system’s stateat time t is | n, t i = e − iα n ( t )+ iγ n ( t ) | n, ξ t i , (1)where | n, ξ t i is an instantaneous eigenket of the hamiltonian, H ξ t | n, ξ t i = E ξ t n | n, ξ t i , the phase α n ( t ) = R t dτ E ξ τ n is the expected dynamical one, and γ n ( t ) = i Z t dτ h n, ξ τ | ddτ | n, ξ τ i = i Z ξ t ξ dξ · h n, ξ |∇ ξ | n, ξ i , (2)is the geometrical phase — the latter form shows that it is time reparametrization invariant ( ∇ ξ denotesthe gradient in ξ -space). For a loop C in ξ -space, Stokes’ theorem may be used to cast (2) in the form γ n ( C ) = Z S dξ A dξ B K ( n ) AB , (3)where K ( n ) AB = i ( ∂ A h n, ξ | )( ∂ B | n, ξ i ) (4)is the Berry curvature, and S is any two-dimensional patch with C as its boundary [1, 14, 2] (seealso [6, 13]).When the initial state | n, t = 0 i is d -fold degenerate, the geometric phase factor e iγ n ( t ) generalizes toa unitary matrix [20] (the Wilczek-Zee effect) U n ( t ) = P exp i Z ξ t ξ dξ A ξn ! , (5) ires with Quantum Memory P exp is a path-ordered exponential, and A ξn is a hermitean d by d matrix with entries( A ξn ) ab = i h n, b ; ξ |∇ ξ | n, a ; ξ i (6)( a , b = 1 , . . . , d , range over the degenerate subspace). Consider a quantum particle constrained to move along a smooth curve in three dimensional euclideanspace — what confines the particle to the vicinity of the curve is a two-dimensional harmonic oscillatorpotential in the plane normal to the curve, of width η , with its minimum at the position of the curve. It isconvenient to use an adapted coordinate frame, with coordinates ( s, α, β ), where s is arclength along thecurve, and ηα , ηβ are distances along the normal n and the binormal b of the curve, respectively. Thenthe position vector r ( s, α, β ) of an arbitrary point in the vicinity of the curve is related to the positionvector R ( s ) of the curve itself by r ( s, α, β ) = R ( s ) + ηαn ( s ) + ηβb ( s ) . (7)Taking η ≪ κ − , guarantees that the frame is well defined in the region of physical interest — we doassume that κ ( s ) = 0. The hamiltonian for the particle, expressed in terms of the adapted coordinates,is given by H = − p | G | ∂ A G AB p | G | ∂ B + V ( α, β ) , (8)where A , B range over the adapted coordinates, G AB is the metric induced from the ambient euclideanone, G AB = ∂ A r · ∂ B r , G AB its inverse, and G = (1 − ηακ ) its determinant . For the confining potentialwe take V ( α, β ) = ( α + β ) / η . Notice that V only depends on the normal coordinates, so that,classically, the tangential motion of the particle is free. The normalization condition for the particle’swavefunction Φ is Z dsdαdβ p | G || Φ | = 1 , (9)which motivates working with a rescaled wavefunction Ψ = | G | / Φ, obeying Z dsdαdβ | Ψ | = 1 , (10)so that R dαdβ | Ψ | can be interpreted as the probability density for finding the particle at the position s along the curve. Accordingly, the hamiltonian undergoes a similarity transformation, H → ˜ H = | G | / H | G | − / . (11)Plugging in (8), and expanding in powers of η results in˜ H = 1 η H − + H + O ( η ) , (12)where H − = −
12 ( ∂ α + ∂ β ) + 12 ( α + β ) H = −
12 ( ∂ s − iτ L ) − κ , (13) We follow here the exposition in [12] — notice that the sign of η in the expression for G that follows from Eq. (3) inthat reference is opposite to the one given here — we believe our expression is the correct one. ires with Quantum Memory L = i ( β∂ α − α∂ β ) is the generator of rotations in the normal plane. Looking for ˜ H eigenstates,˜ H Ψ = ˜ E Ψ, in the factorized form Ψ( s, α, β ) = χ ( α, β ) ψ ( s ) , (14)one is led to consider simultaneous H − and L eigenkets χ ( n ) σ , H − χ ( n ) σ = ( n + 1) χ ( n ) σ , Lχ ( n ) σ = σχ ( n ) σ , (15)leading to − ψ ′′ σ + iστ ψ ′ σ + 12 (cid:18) iστ ′ + σ τ − κ (cid:19) ψ σ = E σ ψ σ (16)for the tangential wavefunction (primes denote derivatives with respect to s ). Then Ψ σ = χ ( n ) σ ψ σ and˜ E = ( n + 1) /η + E σ . Consider a particle constrained to move on a unit circle, then κ ( s ) = 1 and τ ( s ) = 0. For the normalket, take either of the degenerate doublet |±i = ( | i ± i | i ) / √
2, with H − eigenvalue 2, and satisfying L |±i = ±|±i (we use the standard notation | i ≡ a † α | i , | i ≡ a † β | i , where a † α , a † β are creationoperators of excitations along the axes α , β ). Eq. (16) then becomes − ψ ′′ σ − ψ σ = E σ ψ σ , (17)with σ = ± ψ ( k ) σ = 1 √ π e iks , E ( k ) σ = 4 k − , k ∈ Z . (18)We choose k = 0 — the corresponding 3D states Ψ (0) ± = χ ± ψ (0) ± form a degenerate doublet, with χ ± = ρe − ρ / e ± iφ / √ π , ˜ E (0) ± = 2 /η − / ρ , φ denote the standard polar coordinates in the normal plane).Consider now a two-parameter deformation of the circle, given by R ( s ; ξ, ζ ) = (cos s, sin s, ξ ( − cos s, sin s,
0) + ζ (0 , , cos 2 s ) (20)in cartesian coordinates ( x, y, z ), or R ( s ; ξ, ζ ) = (0 , − , ξ ( 12 sin 2 s, cos 2 s,
0) + ζ (0 , , cos 2 s ) (21)in the adapted frame ( t, n, b ), with ξ , ζ ≪ ξ ≡ ξ , ξ ≡ ζ ). The corresponding velocity field v = ∂R/∂ξ | ξ =0 satisfies the locally arclength preserving condition ∂ s v t − κv n = 0 (similarly for u = ∂R/∂ζ | ζ =0 ). This enables the physical identification of points of thecurve with the same s coordinate, for different values of the deformation parameters, so that derivativesof the wavefunction with respect to the latter can be meaningfully taken. The above deformation bringsalong changes in κ ( s ) and τ ( s ), which in turn result in the perturbation hamiltonian H = 34 ξ cos 2 s + 6 iζσ (sin 2 s ∂ s + cos 2 s ) , (22) ires with Quantum Memory ψ ± = 1 √ π (cid:18) − (cid:0) ξ ± i ζ (cid:1) cos 2 s (cid:19) . (23)We are still not in the position to use (4) though. The reason is that, on the one hand, the true, physicalwavefunction is the Φ that appears in (9), and not the rescaled Ψ we have been working with, while, on theother hand, our expressions involve the adapted coordinates, which depend implicitly on the parameters ξ and ζ , since the curve being deformed in (20) drags with it the adapted frame. Our task then, inprinciple, would be to express the adapted coordinates in terms of the cartesian ones, differentiate themwith respect to the parameters, treating the cartesian cordinates as constants, and reexpress the resultin terms of the adapted coordinates. Then the derivatives with respect to the parameters in, e.g. , (2),would contain contributions both from the explicit dependence of Φ( s, α, β ; ξ, ζ ) on the parameters, aswell as the implicit one through the adapted coordinates. Taking the above into consideration, we findthat the matrix A in (6) is diagonal, so that the nondegenerate expression for K , Eq. (4), may be used,giving K (0 ,σ ) ξζ = − σ . (24)The fact that the two states in the doublet, corresponding to σ = ±
1, pick up opposite geometricalphases, allows the detection of the latter by starting out the system in a suitable superposition, e.g. , inthe state | Ψ i t =0 = | , i = ( | , + i + | , −i ) / √
2, with wavefunction Ψ ∼ αe − ρ / — a plot of a constantprobability density surface appears in Fig. 1. Suppose now that the the systen is driven cyclically in the ξ - ζ plane, tracing the circle ξ = ǫ (1 − cos λt ), ζ = ǫ sin λt , with λ ≪
1, so that the adiabatic approximationis valid. For t = 0 the wire described by (20) is a unit circle in the x - y plane, centered at the origin, with n radially inwards, and b along z . As t increases, the wire sweeps out the self-intersecting surface shownin Fig. 2 (assuming ǫ = . | , ±i is ± ∆ φ , where ∆ φ = 2 πǫ K (0 , +) ξζ = − πǫ , (25)so that after m revolutions, the state vector is | Ψ i t = ( e im ∆ φ | , + i + e − im ∆ φ | , −i ) / √ . (26)As a result, the corresponding probability density profile rotates in the plane normal to the curve by anangle ∆ φ — Fig. 3 assumes an integer m such that ∆ φ ≈ π/ We pointed out a wide class of quantum systems exhibiting geometrical phases: particles constrained tomove along curves in three dimensional euclidean space, with the shape of the latter in the role of aninfinite dimensional parameter space. We presented a particular example, involving deformations of acircle, and leading to a nonzero Berry curvature, Eq. (24). The effect is observable, through changes in theprobability density distribution, when superpositions of states are considered, each of which accumulatesa different, in general, geometrical phase.There are obvious directions along which our result might be generalizable: higher dimension and/orcodimension of the constraint manifold, e.g. , the case of surfaces embedded in R . Along with such en-deavors, a detailed analytical study of the problem we considered here should provide general expressionsfor the curvature in terms of the initial shape of the wire and the velocity fields of the perturbations —we defer such an analysis to a longer article, currently in progress. ires with Quantum Memory Figure Plot of a constant probability density surface for the state | , i , corresponding to excitation ofthe 2D harmonic oscillator along n . The darker colored ring represents the (undeformed) wire. A question that we find particularly interesting is the extend to which the details of the deformationcan be encoded in the probability density distribution. It is clear that, no matter what the initial quantumstate of the particle is, there are deformations of the wire that pass unnoticed, e.g. , those that evolvealong an arbitrary curve in parameter space and then retrace their evolution along the same curve, backto the original shape. This means that one cannot, in general, reconstruct the deformation entirely froma comparison of the initial and final probability density distributions, but it is, nevertheless, plausiblethat, in general, more can be inferred than just the fact that there was some deformation.Experimental uses of the above effect in quantum wires and nanotubes might be possible, as tinyperiodic mechanical deformations are encoded cumulatively in a quantum system — this could providesensitive (periodic) motion detectors. In this context, it would be interesting to consider the potentialamplification of the effect, either in the present, or its higher dimensional variants alluded to above, byreplacing the single particle employed here by mesoscopic condensates. Another direction worth pursuingwould be applications to holonomic quantum computing,C. C. would like to thank his colleagues at NTUA and, in particular, George Zoupanos and Kon-stantinos Anagnostopoulos, for hospitality, and multifaceted support. The authors acknowledge partialfinancial support from DGAPA-UNAM projects IN 121306-3 and IN 108103-3, as well as the EPEAEKprogramme “Pythagoras II”, co-funded by the European Union (75%) and the Hellenic state (25%).
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