Coherent energy manipulation in single-neutron interferometry
S. Sponar, J. Klepp, R. Loidl, S. Filipp, G. Badurek, Y. Hasegawa, H. Rauch
aa r X i v : . [ qu a n t - ph ] D ec Coherent energy manipulation in single-neutron interferometry
S. Sponar , J. Klepp , , R. Loidl , S. Filipp , G. Badurek , Y. Hasegawa , , and H. Rauch Atominstitut der ¨Osterreichischen Universit¨aten, 1020 Vienna, Austria Institut Laue-Langevin, B.P. 156, F-38042 Grenoble Cedex 9, France PRESTO, Japan Science and Technology Agency (JST), Kawaguchi, Saitama, Japan (Dated: November 1, 2018)We have observed the stationary interference oscillations of a triple-entangled neutron state inan interferometric experiment. Time-dependent interaction with two radio-frequency (rf) fieldsenables coherent manipulation of an energy degree of freedom in a single neutron. The system ischaracterized by a multiply entangled state governed by a Jaynes-Cummings Hamiltonian. Theexperimental results confirm coherence of the manipulation as well as the validity of the description.
PACS numbers: 03.75.Dg, 03.65.Ud, 07.60.Ly, 42.50.Dv
Since the pioneering work of Einstein, Podolsky, andRosen [1] numerous experiments have exploited the con-cept of Nonlocality which tests local hidden variable the-ories (LHVTs). The LHVTs are a subset of a largerclass of hidden-variable theories namely the noncontex-tual hidden-variable theories (NCHVTs). Noncontextu-ality implies that the value of a measurement is inde-pendent of the experimental context, i.e. of previous orsimultaneous measurements [2, 3]. Noncontextuality is amore stringent demand than locality because it requiresmutual independence of the results for commuting ob-servables even if there is no spacelike separation [4].In the case of neutron experiments, entanglement isnot achieved between particles, but between differentdegrees of freedom. Since the observables in differentHilbert spaces commute with each other, the single neu-tron system is suitable for studying NCHVTs. Single-particle entanglement, between the spinor and the spa-tial part of the neutron wave function [5], as well as fulltomographic state analyses [6], have already been accom-plished. In addition, the contextual nature of quantumtheory [7] has been demonstrated using neutron interfer-ometry [8]. Aiming at the preparation of a single-particlemultiply entangled state, implementation of another de-gree of freedom to be entangled with the neutron’s spinand path degrees of freedom was a challenge.The neutron’s energy seems to be an almost ideal can-didate for this third degree of freedom, due to its exper-imental accessibility within a magnetic resonance field[9]. For this purpose the time evolution of the systemis described by a photon-neutron state vector, whichis an eigenvector of the corresponding modified Jaynes-Cummings (J-C) Hamiltonian [10, 11]. The J-C Hamilto-nian can be adopted for a system consisting of a neutroncoupled to a quantized rf-field [12].This letter reports on observation of stationary inter-ference patterns, confirming coherent energy manipula-tion of the neutron wavefunction. This technique pro-vides realization of triple-entanglement between the neu-tron’s path, spin and energy degrees of freedom.Since two rf-fields, operating at frequencies ω and ω/ H J-C = − ~ m ∇ − µB ( r ) σ z + ~ ( ωa † ω a ω + ω a † ω/ a ω/ )+ µ B ( ω )1 ( r ) √ N ω ( a † ω e σ + h. c. )+ B ( ω/ ( r ) p N ω/ ( a † ω/ e σ + h. c. ) ! . (1)with e σ = ( σ x + iσ y ). The first term accounts for thekinetic energy of the neutron. The second term leads tothe usual Zeeman splitting of 2 | µ | B . The third termadds the photon energy of the oscillating fields of fre-quencies ω and ω/
2, by use of the creation and annihila-tion operators a † and a . Finally, the last term representsthe coupling between photons and the neutron, where N ω j = h a † ω j a ω j i represents the mean number of photonswith frequencies ω j in the rf-field. Note that the firsttwo and the last terms concern the spatial | ψ ( r ) i and the(time-dependent) energy | E ( t ) i subspaces of neutrons,respectively [13].The state vectors of the oscillating fields are rep-resented by coherent states | α i , which are eigenstatesof a † and a . The eigenvalues of coherent states arecomplex numbers, so one can write a | α i = α | α i = | α | e iφ | α i with | α | = √ N . Using Eq. (1) one can definea total state vector including not only the neutron sys-tem | Ψ N i , but also the two quantized oscillating magneticfields: | Ψ i i = | α ω i⊗ | α ω/ i⊗ | Ψ N i . In a perfect Si-crystalneutron interferometer the wavefunction behind the firstplate, acting as a beam splitter, is a linear superposi-tion of the sub-beams belonging to the right ( | I i ) andthe left path ( | II i ), which are laterally separated by sev-eral centimeters. The sub-beams are superposed at thethird crystal plate and the wave function in the forwarddirection then reads as | Ψ N i ∝ | Ψ (I)N i + | Ψ (II)N i , where | Ψ (I)N i and | Ψ (II)N i only differ by an adjustable phase fac-tor e iχ ( χ = N ps b c λD , with the thickness of the phaseshifter plate D , the neutron wavelength λ , the coherentscattering length b c and the particle density N ps in thephase shifter plate). By rotating the plate, χ can be var-ied systematically. This yields the well known intensityoscillations of the two beams emerging behind the in-terferometer, usually denoted as O- and H-beam [8]. A FIG. 1: ( a ) Schematic view of the experimental setup forstationary observation of interference between two rf-fields.Showing the arrangement of two radio-frequency flip coils (thefirst within one path of the skew-symmetric Mach-Zehnder-type neutron interferometer and the other driven by the halffrequency behind the interferometer), accelerator coil and π/ b ) Energy leveldiagram of the two interfering sub-beams | I i , | II i during theirpassage through the different static field regions (B ,B / |↑i , |↓i andtaking into account the spin flips at rf-frequencies ω and ω/ sketch of the setup, split up into regions numbered from1 on the left to 6 on the right side, is depicted in Fig. 1.In our experiment, only the beam in path II is ex-posed to the rf-field of frequency ω , resulting in a spinflip process in region 3. The spin flip configuration of thefirst rf-field ensures an entanglement of spin and spatialdegree of freedom of the neutron state [5]. Interactingwith a time-dependent magnetic field, the total energyof the neutron is no longer conserved after the spin-flip[14, 15, 16, 17, 18]. Photons of energy ~ ω are exchangedwith the rf-field. This particular behavior of the neutronis described by the dressed-particle formalism [12, 19].Consequently the two sub-beams | I i and | II i now differin total energy (see Fig. 1(b)). Therefore the neutronstate can be considered to consist of the three subsys-tems, namely the total energy, path and spin degree offreedom. In principle, a spin-independent energy ma-nipulation of neutrons is also possible: for instance, theup- and the down-spin wavepackets, separated by a so-called longitudinal Stern-Gerlach effect[20, 21], undergosuccessive fast-activated DC-RF and RF-DC flippers re-spectively, resulting in a positive energy-shift.A coherent superposition of | I i and | II i results in the multiply entangled dressed state vector, expressed as | Ψ( t ) i ∝ | α ω i ⊗ | α ω/ i ⊗ √ (cid:16) | I i ⊗ | E i ⊗ |↑i + e iχ | II i ⊗ e iωt | E − ~ ω i ⊗ e iφ ω |↓i (cid:17) , (2)where |↑i , |↓i denote the neutron’s up and down spinstates referred to the chosen quantization axis. Thestate vector of the neutron acquires a phase ± φ ω dur-ing the interaction with the oscillating field, given by B ( t ) = B cos( ωt + φ ω ), induced by the action of theoperators a ω and a † ω in the last term of Eq. (1). Theneutron part of the total state vector is represented bya path-energy-spin entanglement within a single neutronsystem. At the last plate of the interferometer (region4) the two sub-beams are recombined, which is describedby the projection operator ˆ O (P) = (cid:0) | I i + | II i (cid:1)(cid:0) h I | + h II | (cid:1) .Due to the orthogonality of the energy and spin eigen-states the polarization is zero and no intensity modula-tions are observed in the H-beam, which is plotted inFig. 2. A time-resolved measurement (see [9]) can revealthe dynamic behavior of the polarization expressed as e P O ( t ) = (cid:16) cos (cid:0) χ − ωt − φ ω (cid:1) , sin (cid:0) χ − ωt − φ ω (cid:1) , (cid:17) . (3)This phenomenon has been measured separately [9], andis related to the spinor precession known from zero-fieldspin-echo experiments [15, 16].The beam recombination is followed by an interactionwith the second rf-field, with half frequency ω/
2, in re-gion 5. Mathematically the energy transfer is representedby the operator ˆ O (E) = √ | E − ~ ω/ i (cid:0) h E | +( h E − ~ ω | (cid:1) ,respectively. The total state vector is given by | Ψ f i ∝ | α ω i ⊗ | α ω/ i ⊗ (cid:0) | I i + | II i (cid:1) ⊗ | E − ~ ω/ i⊗ √ (cid:16) e iφ ω/ |↓i + e iωT e iχ e i ( φ ω − φ ω/ ) |↑i (cid:17) , (4)where φ ω and φ ω/ are the phases induced by the tworf-fields and ωT is the zero-field phase, with T being theneutron’s propagation time between the two rf-flippers[22]. The energy difference between the orthogonal spinstates is compensated by choosing a frequency of ω/ P f = (cos ∆ tot , sin ∆ tot , , (5)where ∆ tot = ( χ − φ ω/ + φ ω + ωT ), consists of the phasesinduced by the path (phase shifter χ ), spin (phases of thetwo rf fields φ ω , φ ω/ ), and energy manipulation (zero-field phase ωT ). The principle of energy compensation isvisualized in Fig. 1(b). As seen from ∆ tot in Eq.(5) each =0 ° =45 ° =90 ° =135° Phase Shift, χ (deg) I n t e n s i t y ( n e u t r o n s / s e c ) H-BeamO-Beam φ ω φ ω φ ω φ ω FIG. 2: Typical interference patterns of the H- and the O-beam. In the H-beam no interference fringes are observeddue to orthogonal spin states in the interfering sub-beams,whereas the O-beam exhibits time-independent sinusoidal in-tensity oscillations, when the phase shifter plate ( χ ) is ro-tated. A phase shift occurs on varying φ ω . of the three degrees of freedom can be manipulated inde-pendently and the associated observables are separatelymeasurable.The arrangement of two rf-flippers of frequencies ω and ω/ P (S) = |↑ih↑ | to the spin (region 6), the stationary interference oscilla-tions are given by I ∝ ν cos( χ + Φ+ ωT ), introducingthe fringe visibility ν and the relative phase Φ. The rel-ative phase can be calculated as Φ = φ ω − φ ω/ . In thefollowing experiment we demonstrate the coherence prop-erty of the modified J-C manipulation defined in Eq. (1)as well as the phase dependence expressed above.The experiment was carried out at the neutron in-terferometer instrument S18 at the high-flux reactorof the Institute Laue-Langevin in Grenoble, France.A monochromatic beam, with mean wavelength λ =1 .
91 ˚A(∆ λ/λ ∼ .
02) and 5x5 mm beam cross-section,is polarized by a bi-refringent magnetic field prism in ˆ z -direction [23], see Fig. 1(a) region 1. In a non-dispersivearrangement of the monochromator and the interferome-ter crystal the angular separation can be used such thatonly the spin-up (or spin-down) component fulfils theBragg-condition at the first interferometer plate (beamsplitter) in region 2. Behind the beam splitter the neu-tron’s wave function is found in a coherent superposi-tion of | Ψ (I)N i and | Ψ (II)N i , and only | Ψ (II)N i passes the first rf-flipper mounted in one path of the interferome-ter. Acting like a typical NMR arrangement, rf-flippersrequire two magnetic fields: A static field B · ˆ z with B = ~ ω rf / (2 | µ | ) and a perpendicular oscillating field B ( ω )1 cos( ωt + φ ω ) · ˆ y with amplitude B ( ω )1 = π ~ / (2 τ | µ | ),where µ is the magnetic moment of the neutron and τ isthe time the neutron requires to traverse the rf-field re-gion. The oscillating field is produced by a water-cooledrf-coil with a length of 2 cm, operating at a frequency of ω/ π = 58 kHz. The static field is provided by the uni-form magnetic guide field B ∼ B ( ω/ cos (cid:0) ( ω/ t + φ ω/ (cid:1) · ˆ y , and the strengthof the guide field was tuned to about 1 mT in order tosatisfy the frequency resonance condition.This flipper compensates the energy difference betweenthe two spin components, by absorbtion and emission ofphotons of energy E = ~ ω/
2. The phases of the two guidefields and the zero-field phase ωT were compensated byan additional Larmor precession within a tunable accel-erator coil with a static field, pointing in the ˆ z -direction.Finally, the spin is rotated back to the ˆ z -direction by useof a π/ z -direction due to the spin dependent reflection within aCo-Ti multi-layer supermirror. Typical interference pat-terns are depicted in Fig. 2. In the O-beam a fringe con-trast of 52.4(2) % is achieved, whereas no oscillation wasobserved in the H-detector, where no further manipula- -600-400-2000200400600 -600-400-2000200400600 ∆Φ + ∆Φ + ∆Φ − ∆Φ − (a) (b) R e l a t i v e P h a s e , ∆ Φ ( d e g ) rf Phase, φ (deg) ω rf Phase, φ (deg) ω/2 FIG. 3: Relative phase ∆Φ ± vs. ( a ) φ ω and ( b ) φ ω/ . Thesign of the phase depends on the chosen initial polarization. tions were applied.It is possible to invert the initial polarization simplyby rotating the interferometer by a few seconds of arc,thereby selecting the spin-down component to enter theinterferometer, which is expected to lead to an inver-sion of the relative phase. In order to observe a relativephase shift, in practice it is necessary to perform a ref-erence measurement. This is achieved by turning off therf-flipper inside the interferometer, thus yielding the rel-ative phase difference ∆Φ ± = ± φ ω ∓ φ ω/ , where ± de-notes the respective initial spin orientation. Figure 3(a)shows a plot of the relative phase ∆Φ ± versus φ ω , with φ ω/ = 0, and a phase shift ∆Φ ± caused by a variationof φ ω . As expected, the slope is positive for initial spinup orientation(1.007(8)), and negative for the spin downcase(-0.997(5)). In Fig. 3(b) φ ω/ is varied, while φ ω iskept constant, yielding slopes of -1.995(8) and 1.985(7),depending again on the initial beam polarization.At this point the geometric nature of ∆Φ ± should beemphasized. Within the rf-flipper that is placed insidethe interferometer, the neutron spin traces a semi-greatcircle from |↑i to |↓i on the Bloch sphere and returnsto its initial state |↑i when passing the second rf-flipper.This procedure is repeated along different semi-great cir-cles when varying φ ω or φ ω/ respectively. The two semi-great circles enclose an angle φ ω − φ ω/ and hence a solidangle Ω = 2( φ ω − φ ω/ ). The solid angle Ω yields a puregeometric phase Φ ± G = Ω / [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. ,777 (1935).[2] J. S. Bell, Rev. Mod. Phys. , 447 (1966).[3] N. D. Mermin, Rev. Mod. Phys. , 803 (1993).[4] C. Simon, M. Zukowski, H. Weinfurter, and A. Zeilinger,Phys. Rev. Lett. , 1783 (2000).[5] Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, andH. Rauch, Nature (London) , 45 (2003).[6] Y. Hasegawa, R. Loidl, G. Badurek, S. Filipp, J. Klepp,and H. Rauch, Phys. Rev. A , 052108 (2007).[7] Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, andH. Rauch, Phys. Rev. Lett. , 230401 (2006).[8] H. Rauch and S. A. Werner, Neutron Interferometry (Clarendon Press, Oxford, 2000).[9] G. Badurek, H. Rauch, and J. Summhammer, Phys. Rev.Lett. , 1015 (1983).[10] E. T. Jaynes and F. W. Cummings, Proc. IEEE , 89(1963).[11] B. W. Shore and P. L. Knight, J. Mod. Optic , 1195(1993).[12] E. Muskat, D. Dubbers, and O. Sch¨arpf, Phys. Rev. Lett. , 2047 (1987).[13] Factorization of wavefunctions is a common technique,e.g., to solve time-independent Schr¨odinger equations.[14] R. Golub, R. G¨ahler, and T. Keller, Am. J. Phys. ,779 (1994).[15] R. G¨ahler and R. Golub, Phys. Lett. A , 43 (1987).[16] S. V. Grigoriev, W. H. Kraan, and M. T. Rekveldt, Phys.Rev. A , 043615 (2004). [17] B. Alefeld, G. Badurek, and H. Rauch, Z. Phys. B ,231 (1981).[18] J. Summhammer, K. A. Hamacher, H. Kaiser, H. We-infurter, D. L. Jacobson, and S. A. Werner, Phys. Rev.Lett. , 3206 (1995).[19] J. Summhammer, Phys. Rev. A , 556 (1993).[20] N. Arend, R. G¨ahler, T. Keller, G. Georgii, T. Hills, andP. B¨oni, Phys. Lett. A , 21 (2004).[21] B. Alefeld, G. Badurek, and H. Rauch, Phys. Lett. , 32 (1981).[22] S. Sponar, J. Klepp, G. Badurek, and Y. Hasegawa, Phys.Lett. A , 3153 (2008).[23] G. Badurek, R. J. Buchelt, G. Kroupa, M. Baron, andM. Villa, Physica B , 389 (2000).[24] A. G. Wagh, G. Badurek, V. C. Rakhecha, R. J. Buchelt,and A. Schricker, Phys. Lett. A , 209 (2000).[25] A. Shapere and F. Wilcek, Geometric phases in physics ,vol. 5 of
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