week ending PRL 101, 220501 (2008) PHYSICAL REVIEW LETTERS 28 NOVEMBER 2008
Quantum Simulator for the Ising Model with Electrons Floating on a Helium Film
Sarah Mostame1 and Ralf Schu¨tzhold1,2,* 1Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, 01062 Dresden, Germany 2Fachbereich Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, Germany (Received 16 June 2008; published 26 November 2008) We propose a physical setup that can be used to simulate the quantum dynamics of the Ising model in a transverse field. Building on currently available technology, our scheme consists of electrons which float on a superfluid helium film covering a suitable substrate and interact via Coulomb forces. At low temperatures, the system will stay near its ground state where its Hamiltonian is equivalent to the Ising model and thus shows phenomena such as quantum criticality.
DOI: 10.1103/PhysRevLett.101.220501 PACS numbers: 03.67.Ac, 75.10.Hk
Introduction.—Richard Feynman’s observation [1] that the study of quantum phase transitions and percolation classical computers cannot effectively simulate quantum theory [5], spin glasses [5,6], as well as quantum annealing systems bred widespread interest in quantum computation. [7,8] etc. Although the Hamiltonian (1) is quite simple and He thought up the idea of a quantum processor which uses can be diagonalized analytically, the Ising model is con- the effects of quantum theory instead of classical physics. sidered a paradigmatic example [5] for second-order quan- As an example, Feynman proposed a universal quantum tum phase transitions and is rich enough to display most of simulator consisting of a lattice of spins with nearest- the basic phenomena near quantum critical points. For neighbor interactions that are freely specifiable and can J, the ground state is paramagnetic j !!! ...i efficiently reproduce the dynamics of any other many- with all spins polarized along the x axis. In the opposite particle quantum system with a finite-dimensional state limit J, the nature of the ground state(s) changes space [1]. Although such universal quantum computers qualitatively and there are two degenerate ferromagnetic of sufficient size (e.g., number of qubits, i.e., spins) are phases with all spins pointing either up or down along the z not available yet, it is possible to design a special quantum axis j""" ...i or j### ...i. The two regimes are separated by a J system in the laboratory which simulates the quantum quantum phase transition at the critical point cr ¼ , dynamics of a particular model of interest. Such a designed where the excitation gap vanishes (in the thermodynamic quantum system can then be regarded as a special quantum limit n "1) and the response time diverges. As a result, computer (instead of a universal one, which is more chal- driving the system through its quantum critical point at a lenging) which just performs the desired quantum simula- finite sweep rate entails interesting nonequilibrium phe- tion, see, e.g., [2–4]. nomena such as the creation of topological defects, i.e., In the following, we present a design for a quantum kinks [9]. Furthermore, the transverse Ising model can also simulator for the Ising spin chain in a transverse field and be used to study the order-disorder transitions at zero tem- demonstrate that it could be feasible with near-future tech- perature driven by quantum fluctuations [5,7]. Finally, two- nology, i.e., electrons floating on a thin superfluid helium dimensional generalizations of the Ising model can be film. A similar idea based on trapped ions has been pursued mapped onto certain adiabatic quantum algorithms (see, in [2]. Nevertheless, it is still worthwhile to study an e.g., [10]). However, due to the evanescent excitation en- alternative set-up, since different experimental realizations ergies, such a phase transition is rather vulnerable to possess distinct advantages and drawbacks. For example, decoherence, which must be taken into account [11]. the number of coherently controlled ions in a trap is rather The analogue.—In order to reproduce the quantum dy- limited at present [2], whereas our proposal is expected to namics of the (1 þ 1)-dimensional Ising model (1), we be more easily scalable to a large number of electrons— propose trapping a large number of electrons on a low- which is important for exploring the continuum limit and temperature helium film of thickness h (e.g., h ¼ 110 nm) scaling properties, etc. adsorbed on a silicon substrate [12]. Because of the polar- The model.—Our aim is to simulate the quantum dy- izability " 1:06 of the helium film, the electrons are namics of the one-dimensional Ising chain consisting of n bound to its surface (i.e., in z direction) via their image spins with nearest-neighbor interaction J plus a transverse charges and the large potential barrier (around 1 eV) for field along the x direction (@ ¼ 1) penetration into the helium film [13]. Since the binding Xn energy of around 8 K is much larger than the temperature T H x J z z ; ¼ f j þ j jþ1g (1) (below 1 K) and the width of the electron wave packet in z j ¼1 direction (of order 8 nm) is much smaller than all other x y z where j ¼ð j ; j ; jÞ are the spin-1=2 Pauli matrices relevant length scales, the electron motion is approxi- acting on the jth qubit. This model has been employed in mately two-dimensional (x, y plane).
0031-9007=08=101(22)=220501(4) 220501-1 Ó 2008 The American Physical Society week ending PRL 101, 220501 (2008) PHYSICAL REVIEW LETTERS 28 NOVEMBER 2008
In our scheme, each single electron on top of the helium WKB approximation [14] Z film is trapped by a pair of (grounded) gold spheres of ! y0 radius a (e.g., a ¼ 10 nm) and distance d (e.g., d ¼ 2 ¼ EA ES exp dyjpðyÞj : (4) y 60 nm) attached to the silicon substrate (i.e., on the bottom 0 of the helium film cf. Fig. 1). Depending on its position x, Here ! is the oscillation frequency (within one well) and y y, the electron will also induce image charges in the two 0 are the two inner (classical) turningpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi points cf. Fig. 1. p x; y m E U x; y gold spheres (which act as a pair of quantum dots) and The integrand is given by ð Þ¼ 2 e½ 0 ð Þ , hence experience a double-well potential where we can set x ¼ 0 since the tunneling probability away from the x ¼ 0 axis is strongly suppressed. Finally, ae2ðx2 þ y2 þ 2 þ 2Þ=4 " E m Uwðx; yÞ¼ ; (2) the energy 0 determines the turning points and e is the ðx2 þ y2 þ 2 þ 2Þ2 4 2y2 electron mass. For the parameters above, each valley can be well approximated by a harmonic oscillator Uwðx; y with ¼ d=2 þ a and 2 ¼ h2 a2. Since this potential y Þ ae2ðx2 þ½y y 2Þ=ð4 " 4Þ, and thus we is quite deep and symmetric Uðx; yÞ¼Uðx; yÞ cf. Fig. 1, min pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimin E ae2= "m 4 ! the ground-state wave function c ðx; yÞ is given by the obtain 0 2 e . þ x symmetric superposition of the two Wannier states So far, we derived the term j in Eq. (1) via Eqs. (2)– c x; y c x; y 0ð Þ while the first excited state ð Þ is the (4). In order to simulate the remaining part, we propose to antisymmetric combination line up the pairs of quantum dots at equal distances (e.g., ¼ 600 nm), where the parameters are supposed to obey c x; y c x; y 0ð Þ 0ð Þ j"i j#i the following hierarchy c ðx; yÞ¼ pffiffiffi ! pffiffiffi : (3) 2 2 h>d a: (5) For a sufficiently high potential barrier between the two Within this limit, the interaction between the electrons will wells, the Wannier state c 0ðx; yÞ is strongly concentrated be dominated by the direct CoulombP repulsion between the in the left well and models the spin state j"i and vice versa. n j;jþ1 U ðx; yÞ¼ Uc n The tunneling between the two states is then described by nearest neighbors c j¼1 with denoting the Pauli operator x with xj"i ¼ j#iand xj#i ¼ j"isuch the number of electrons floating on the helium film. For that the tunneling rate, given by the difference of the d, we may Taylor expand the Coulomb interaction into powers of y= due to y d=2. The zeroth-order eigenenergies EA ES of c þ and c , corresponds to the transverse field in Eq. (1). In the limit of strong term is constant and thus irrelevant while the first-order localization (i.e., weak tunneling), the energy splitting contributions vanish (up to boundary terms) after the sum j EA ES between the two levels can be estimated via the over sites . Thus, the leading term is bilinear in the electron positions e2 Xn Ucðx; yÞ yjyj ; (6) " d a 3 þ1 2 0ð þ þ 4 Þ j¼1
FIG. 1 (color online). Sketch of the proposed analogue quan- tum simulator and the induced double-well potential UðyÞ with FIG. 2 (color online). Sketch of various arrangements: (a) the four turning points for the energy E0. Electrons (e ) are floating line setup of Fig. 1 and (b) the alternative ladder setup discussed on a low-temperature helium film of height h adsorbed on a after Eq. (6), both for the one-dimensional Ising model (1), as silicon substrate. A double-well potential for each single electron well as their two-dimensional combination (c). Using the same is created by a pair of (grounded) gold spheres of radius a and distances ¼ 600 nm and 0 ¼ 560 nm as in (a) and (b), setup distance d on the bottom of the helium film. The double wells at (c) corresponds to the symmetric ferromagnetic interaction z z z each site provide the two lowest states of the electron and model J i j to lowest order—if we use the same sign for j within the spin states j"i and j#i at each site j. The tunneling rate one (horizontal) line and alternating signs in neighboring lines. between the two wells corresponds to the transverse field term After a checkerboard sign redefinition, the nearest-neighbor x 0 j . The electrons are lined up at distances and interact via interaction becomes antiferromagnetic. Varying the ratio = , z z x J Jx Jy Coulomb forces, which creates the term j jþ1. Alternative we may induce an asymmetry . The tunneling term j scenarios are sketched in Fig. 2. is the same in all cases. 220501-2 week ending PRL 101, 220501 (2008) PHYSICAL REVIEW LETTERS 28 NOVEMBER 2008 J z z z and precisely corresponds to the j jþ1 term in Eq. (1) perturbation Hamiltonian j corresponding to a longitu- J e2 d a 2= " d a 3 x with ¼ ð þ 2 Þ ð8 0½ þ þ 4 Þ. As an alter- dinal field (in addition to the transversal one j )of ¼ native setup, one may arrange the pairs of quantum dots in Oð10 3 KÞ, i.e., a weak perturbation . Deep in the parallel (i.e., as a ladder) cf. Fig. 2. Adjusting the distance paramagnetic phase J, the response of the system to 0 z accordingly (e.g., ¼ 560 nm instead of ¼ 600 nm), this weak perturbation is rather small h ji = . J we obtain the same coupling strength at lowest order, Approaching the phase transition, however, the static sus- z z provided that j is identified with the electron position in = ceptibility ¼ lim !0h ji grows and finally diverges alternating order. Combining the line and the ladder design at the critical point. In the broken symmetry phase, the per- then facilitates the realization of the two-dimensional Ising z z z turbation j lifts the degeneracy j ! j and hence model; cf. Fig. 2. the response is nonanalytic, i.e., independent of the small- Experimental parameters.— z For the example values ness of : e.g., for J , we have h i¼sgnð Þ¼ 1. given in the text, we obtain 0:1Kfor the tunneling j This signal h zi indicating the phase transition can be rate and the same value J 0:1Kfor the effective cou- j pling; i.e., we are precisely in the quantum critical regime. picked up by measuring the voltage difference induced by However, deviations from this critical point should be easy the position of the electron. If the spheres are not grounded to realize experimentally by varying the height h of the but isolated, the electron induces a voltage difference of up helium film, e.g., via changing the effective chemical to 13 mV between the two spheres. Clearly, this would potential difference between the thin helium film and the constitute a large disturbance and thus one should put a helium reservoir (which can be done by depleting the resistor or capacitor between them in order to reduce the reservoir, for example). This is possible because the tun- voltage difference to a few V. However, since the asso- neling rate depends strongly—in fact, exponentially—on h ciated charge transfer is small (a fraction of the elementary e (e.g., varying h by 10%, changes by 50%), whereas the charge ), a site-by-site readout is probably very hard to Coulomb force J remains approximately constant. In order realize. Nevertheless, collecting signals from many elec- to see quantum critical behavior, i.e., to avoid thermal trons should lead to measurable currents. Preferably, this fluctuations, the temperature should ideally be well below measurement should be done using other spheres than the this value 0.1 K (or at least not far above it). one on which the external voltage has been applied. E.g., Furthermore, the Coulomb repulsion energy between one could apply the external voltage of one V on half of two electrons (zeroth-order term) of about 11 K (for ¼ the electrons and measure (with suitably adapted internal z 600 nm) would tend to destabilize the electron chain. resistances) the internal response h ji on the other half— Fortunately, this effect is compensated by the binding or in large enough subsets. energy between the electron and its image on the sphere, Comparing the signals from the different partitions then z z which is around 13 K and thus stabilizes the electron chain. yields information about the correlator h i ji. In addition Note that the image charge on the sphere is much smaller to the static case, one could also study the time-resolved z 0 than 1 e due to h a. The probability for the electron to response h jðtÞi to a varying voltage ðt Þ, which is deter- z 0 z penetrate the helium film by tunneling to one of the gold mined by the dynamical correlator h i ðt Þ jðtÞi in lowest- 16 spheres is extremely small (of order 10 ) and can be order response theory. Even in the absence of an externally E : neglected. Finally, the ground-state energy 0 1 4K imposed voltage, the chain induces spontaneous voltage (within the harmonic oscillator approximation) is reason- fluctuations in the electrodes, which are strongest deep in U : ably well below the barrier height 0 3 1Ksuch that the ferromagnetic phase. The variance of these fluctuations the WKB approximation should provide a reasonable esti- z z again yields information about the correlator h i ji which mate. (The tunneling probability of 0.08 is also small is an order parameter for the phase transition and allows us enough.) On the other hand, E 1:4Kis a measure of 0 to detect topological defects (i.e., kinks) which might have the distance between the two lowest-lying states in Eq. (3) been produced during the sweep to the ferromagnetic and the remaining excited states in the double-well poten- phase. tial. As a result, these additional states do not play a role for Disorder and decoherence.—In a real experimental temperatures well below 1 K and thus the Hamiltonian (1) setup, the Hamiltonian will not be exactly equivalent to provides the correct low-temperature description. (1) due to imperfections such as electric stray fields, var- Read-out scheme.—Having successfully simulated the iations in the film thickness h and further geometric pa- Ising Hamiltonian (1), one is led to the question of how to rameters a, d, and etc. Therefore, the originalP expression actually measure its properties, e.g., how to detect signa- x (1) will typically be altered to H ¼ f j þ tures of quantum critical behavior. As one possibility, let us j j J z z z J imagine that the gold spheres are not grounded, but con- j j jþ1 þ j jg, where j ¼ þ j and j ¼ nected to small wires which allow us to address them J þ Jj. Assuming that the disorder parameters j, individually or in suitable partitions. For example, apply- Jj, and j are much smaller than the excitation gap ¼ ing a voltage of one V between the spheres associated 2jJ j of the undisturbed system (in the continuum with spin up j"iand spin down j#i, respectively, induces a limit), the impact of these imperfections will be sup- 220501-3 week ending PRL 101, 220501 (2008) PHYSICAL REVIEW LETTERS 28 NOVEMBER 2008 pressed. Near the critical point J ; however, this argu- transition by changing h—which might even be feasible in ment fails. Nevertheless, for a finite number n of electrons, a time-dependent manner, cf. [9]. The quantum critical one retains a minimum gap (within the symmetric or anti- behavior and the created topological defects (kinks) could symmetric subspace, respectively) of order J=n. Exploiting be detected via measuring the voltages induced on the this gap might be suitable for a reasonably small systems, spheres. Furthermore, a suitable generalization to two but for n 100 electrons, the required accuracy on the spatial dimensions might be relevant for adiabatic quantum sub-percent level is probably hard to achieve experimen- algorithms, see, e.g., [10]. Note that the realization of a tally. E.g., decreasing the diameter of the gold spheres by sequential quantum computer based on a set of electrons 10% with the other values remaining the same, the tunnel- floating on a helium film has been proposed in [15]. In ing rate increases by 50%. contrast, our proposal is not suited for universal computa- For a sufficiently large number of electrons, the rele- tions, but (as one would expect) should be easier to realize vance of the disorder induced by imperfections near the experimentally. Exploring a different limit, where many critical point depends on the dimensionality of the system. eigenstates of the double-well potential contribute, the In one spatial dimension, the critical exponent ¼ 1 of the proposed set-up could simulate the lattice version of inter- Ising model indicates that the renormalization flow is acting field theories such as the 4-model in (1 þ 1) directed away from the homogeneous situation, see, e.g., dimensions. [5], which means that disorder becomes important at large S. M. acknowledges fruitful discussions with R. scales n !1. In this case, one would expect effects such Farhadifar and R. S. is indebted to G. Volovik and P. as local paramagnetic regions inside the global ferromag- Leiderer for valuable conversations. This work was sup- netic phase and percolation transitions, etc. Therefore, ported by the Emmy-Noether Programme of the German turning this drawback into an advantage, one might gen- Research Foundation (DFG) under Grant SCHU 1557/1- erate these imperfections on purpose in order to study the 2,3 and by DFG Grant SCHU 1557/2-1. impact of disorder on the phase transition. In contrast to the original Hamiltonian (1), the above form in the presence of disorder is no longer analytically solvable and hence much less is known about its properties. In two spatial dimen- sions (cf. Fig. 2), the relevance of disorder is much weaker *[email protected] and hence it might be possible to observe quantum critical [1] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982); Found. behavior. Phys. 16, 507 (1986). Finally, in a real setup, the system will also experience [2] D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 decoherence due to the inevitable coupling to the environ- (2004); A. Friedenauer et al., Nature Phys. 4, 757 (2008). [3] R. Schu¨tzhold and S. Mostame, JETP Lett. 82, 248 (2005). ment [11]. These effects could be incorporated by J [4] T. Byrnes et al., Phys. Rev. Lett. 99, 016405 (2007); A. Yu. operator-valued variations j, j, and j associated to Smirnov, S. Savele´v, L. G. Mourokh, and F. Nori, the degrees of freedom of the environment. According to Europhys. Lett. 80, 67008 (2007); E. Manousakis, J. [15], the main decoherence channels in the setup under Low Temp. Phys. 126, 1501 (2002). consideration are due to the coupling to ripplons, i.e., [5] S. Sachdev, Quantum Phase Transitions (Cambridge surface waves on the helium film. At T ¼ 0:1K, their ther- University Press, Cambridge, England, 1999). mal fluctuations have a typical amplitude of h ¼ [6] K. H. Fischer and J. A. Hertz, Spin glasses (Cambridge pffiffiffiffiffiffiffiffiffiffiffiffiffiffi University Press, Cambridge, England, 1993). kBT= 0:06 nm, where is the surface tension. Since h h [7] A. Das and B. K. Chakrabarti, Quantum Annealing and is extremely small compared to ¼ 110 nm, the cou- Related Optimisation Methods, Lecture Notes in Physics pling terms and energy shifts induced by these height Vol. 679 (Springer-Verlag, Heidelberg, 2005). variations are negligible here. Furthermore, the energy ! [8] G. E. Santoro et al., Science 295, 2427 (2002); T. of the ripplons itself depends on their wave number k via Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 !2 k3 =%, where % is the density of the helium film. For (1998). typical wavelengths of the ripplons coupled to the electrons [9] J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005). of order 100 nm, the ripplon energy is below 0.01 K and [10] IEEE Spectrum online, Tech Talk, February 13th (2007). hence also negligible. Consequently, even though the rip- [11] S. Mostame, G. Schaller, and R. Schu¨tzhold, Phys. Rev. A plons might induce significant dephasing and decay of 76, 030304(R) (2007). excited states (T and T cf. [15]), they basically do not [12] J. Angrik et al., J. Low Temp. Phys. 137, 335 (2004). 2 1 [13] Two Dimensional Electron Systems on Helium and Other affect the quantum ground state. Cryogenic Substrates, edited by E. Y. Andrei (Academic Summary.—We have proposed a design for the simula- Press, New York, 1991). tion of the quantum Ising model with a system of electrons [14] M. Razavy, Quantum Theory of Tunneling (World floating on a liquid helium film adsorbed on a silicon Scientific, Singapore, 2003). substrate. Since the energy level splitting (tunneling rate [15] P. M. Platzman and M. I. Dykman, Science 284, 1967 ) depends exponentially on the thickness of the helium (1999); see also S. A. Jackson and P.M. Platzman, Phys. film h, we may tune the system through the quantum phase Rev. B 24, 499 (1981).
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