Breaking rotational symmetry in a trapped-ion quantum tunneling rotor
BBreaking rotational symmetry in a trapped-ion quantum tunneling rotor
Ryutaro Ohira, ∗ Takashi Mukaiyama,
1, 2 and Kenji Toyoda Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, Japan Quantum Information and Quantum Biology Division,Institute for Open and Transdisciplinary Research Initiatives,Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka, Japan (Dated: January 28, 2020)A trapped-ion quantum tunneling rotor (QTR) is in a quantum superposition of two differentWigner crystal orientations. In a QTR system, quantum tunneling drives the coherent transitionbetween the two different Wigner crystal orientations. We theoretically study the quantum dynamicsof a QTR, particularly when the spin state of one of the ions is flipped. We show that the quantumdynamics of an N -ion QTR can be described by continuous-time cyclic quantum walks. We alsoinvestigate the quantum dynamics of the QTR in a magnetic field. Flipping the spin state breaks therotational symmetry of the QTR, making the quantum-tunneling-induced rotation distinguishable.This symmetry breaking creates coupling between the spin state of the ions and the rotationalmotion of the QTR, resulting in different quantum tunneling dynamics. INTRODUCTION
A trapped-ion quantum rigid rotor has recently beenrealized, and is referred to as a quantum tunneling ro-tor (QTR) system [1]. Due to the presence of micro-motion, it is not straightforward to cool the collectivemodes of two-dimensional (2D) ion crystals to the mo-tional ground state. However, ground state cooling ofthe rotational mode, which is a collective mode of a 2Dion crystal, has been demonstrated [1]. The unique prop-erty of the QTR is that the rotational motion of the QTRis driven by the quantum tunneling effect. In a previousstudy, a quantum-tunneling-induced transition betweentwo stable orientations of a Wigner crystal was realized[1]. In addition, quantum interference induced by theAharonov–Bohm (AB) effect [2] was observed [1].In this paper, we consider the quantum tunneling dy-namics of the QTR when the spin state of one of the ionsin the QTR is flipped. We first formalize the quantumstate of an N -ion QTR including spin degrees of freedom.The quantum dynamics of such a QTR can be consid-ered as a 2 N -site cycle graph. Therefore, we describe thequantum dynamics of the QTR as continuous-time cyclicquantum walks [3, 4]. Continuous-time cyclic quantumwalks require a quantum system where a quantum prop-agates into neighboring sites via quantum tunneling, andthe QTR system is an ideal quantum system for imple-menting continuous-time cyclic quantum walks.In addition, we investigate the quantum dynamics ofa QTR when a magnetic field passes through the QTRsystem. Flipping the spin state breaks the rotationalsymmetry in the QTR system, which then exhibits com-pletely different dynamics. In particular, when the ionsin the QTR couple to a vector potential, the difference inthe quantum tunneling dynamics becomes marked due toquantum interference induced by the AB effect [1]. Wefind that this spin-dependent quantum interference in-duces coupling between the spin state of the ions and the ∗ [email protected] rotational motion of the QTR, which may experimen-tally realize a quantum spin filter [5] or a cat state of thespin states of the ions and the Wigner crystal structures[6]. In terms of controlling the quantum tunneling dy-namics with the spin degrees of freedom of the ions, ourwork is analogous to studies on the quantum dynamicsof a double-well bosonic Josephson junction coupled to asingle atomic ion [7, 8]. TRAPPED-ION QTR
We first review the dynamics of the QTR. N ions withmass m and charge e are trapped in a harmonic potential.The trap frequencies are ω x and ω z (cid:28) ω y , so that themotion along the y direction is considered to be frozenout. The total Hamiltonian of this effective 2D systemis H Total = (cid:80) Ni =1 p i / m + V , where p i represents themomentum of the i -th ion. The potential V is derivedfrom the harmonic potential and the Coulomb repulsion; V = (cid:80) Ni =1 m ( ω x x i + ω z z i ) / (cid:80) Ni>j e / πε r ij , where r ij is the distance between the i -th and j -th ions. Weassume that the trap frequencies are ω z (cid:28) ω x so thatan ion chain along the z direction is formed. Then, bydecreasing the confinement along the x direction, the ionsgradually form a 2D crystal structure. When the trapfrequencies ω x and ω z are comparable, an almost regularpolygon crystal is created.When the ions have a high mean energy, rotation inthe x – z plane is observed [1, 9]. If the kinetic energy ofthe ions is lower than the energy barrier for the rotation,the ions are localized at local minima. However, undersome conditions, rotation can be driven by the quantumtunneling effect [1]. Specifically, when the motional meanenergy is low enough for the ions to be pinned and therotational barrier is quite low, quantum tunneling occurs.This can be achieved by cooling the rotational mode tothe motional ground state and making ω x slightly higherthan ω z .We take a QTR with three trapped ions as an exam-ple. We suppose that three Yb + ions are trapped and a r X i v : . [ qu a n t - ph ] J a n (a) (b)(c) (d) Quantum tunneling
FIG. 1. (a) Normalized six lowest collective frequencies of three ions as a function of the ratio of ω x to ω z . The inset showsthe eigenvector of each collective mode when ω x /ω z = 1.001. (b) Effective potential for ω x /ω z = 1.001 ( ω z = 2 π × | ψ up (cid:105) and | ψ down (cid:105) as a function of the angle θ . (d)Quantum-tunneling-induced transition of the Wigner crystal in the QTR system. their internal states are the same. As mentioned above,there are two steps to create a QTR [1]: . Motionalground state cooling of the rotational mode, and . adi-abatically ramping down ω x (adiabatic cooling). Figure1(a) shows a numerical calculation of the frequencies ofthe six lowest collective modes of the three ions. Thered curve (mode1) in Fig. 1(a) corresponds to the rota-tional mode. The other collective modes are also shownfor reference. In the following discussion, the frequencyof the rotational mode is denoted as ω Rot , and the trapfrequency along the z direction is fixed to ω z = 2 π × ω x /ω z = 1.001 are alsoshown in Fig. 1(a).To create the QTR, the rotational mode needs to becooled to the motional ground state; otherwise the ionsstart rotating due to the high kinetic energy. Therefore,we first set ω x to the point where ω Rot is high enoughto cool the rotational mode to the ground state. Aftermotional ground state cooling, adiabatic cooling [1, 10–12] is employed to increase the population of the groundstate of the rotational mode. As indicated in Fig. 1(a), ω Rot gets smaller as ω x is ramped down. By reducingthe confinement along the x direction under the condi-tion dω Rot dt / ω Rot (cid:28)
1, adiabatic cooling is realized. ω x isadiabatically ramped down to the point where the trap frequencies along the x and z directions are compara-ble. In this article, we discuss the dynamics of a QTRassuming perfect ground state cooling of the rotationalmode, so that the classical rotation caused by the otherquantum motional states, which have the higher kineticenergy than the effective rotational barrier, can be safelyignored. Figure 1(b) shows the effective potential createdby the harmonic potential and the Coulomb interactionbetween ions for ω x /ω z = 1.001. It is clear from Fig. 1(b)that there are two stable ion crystal orientations. We de-fine the wavefunctions of those stable ion structures as | ψ up (cid:105) and | ψ down (cid:105) .Figure 1(c) shows the effective potential and the prob-ability amplitudes for both wavefunctions, | ψ up (cid:105) and | ψ down (cid:105) , for ω x /ω z = 1.001. As can be seen from Fig. 1(c),the wavefunctions | ψ up (cid:105) and | ψ down (cid:105) overlap. Since | ψ up (cid:105) and | ψ down (cid:105) are not orthogonal, quantum tunneling canoccur.The QTR changes its crystal orientation due to thequantum tunneling effect, as shown in Fig. 1(d). TheCoulomb interaction prevents the exchange of ions whenquantum tunneling occurs. Therefore, the QTR can beconsidered to be a quantum rigid rotor system. It shouldbe noted that this protocol to create the QTR is notrealistic if the number of ions is even because the barrierfor rotation is significantly high. CYCLIC QUANTUM WALKSFormation of QTR, including the spin state
An ion has an internal state, which is considered an ef-fective spin state. For Yb + ions, the hyperfine states |↓(cid:105) ≡ | F = 0 , m F = 0 (cid:105) and |↑(cid:105) ≡ | F = 1 , m F = 0 (cid:105) are usu-ally used as the effective spin system. If the internalstates of the ions are the same, |↓(cid:105) , the internal statesare simply ignored and the quantum state of the N -ionQTR, | ψ (cid:105) , can be expressed as the superposition of | ψ up (cid:105) and | ψ down (cid:105) : | ψ (cid:105) = α | ψ up (cid:105) + β | ψ down (cid:105) , (1)where α and β are complex coefficients satisfying | α | + | β | = 1. Since the QTR is the superposition oftwo different polygon Wigner crystals, the N -ion QTRis viewed as a 2 N -site cycle graph, as shown in Fig. 2.We define the wavefunction of the k -th ion at the l -thsite as | ψ k,l (cid:105) . As is evident from quantum interference inthe QTR system induced by the AB effect [1], the parti-cles constituting the QTR are considered to be identical.Since Yb + is bosonic, the quantum states of | ψ up (cid:105) and | ψ down (cid:105) are given as follows: | ψ up (cid:105) = 1 √ N !perm | ψ , (cid:105) · · · | ψ , n − (cid:105) · · · | ψ , N − (cid:105) ... . . . ... . . . ... | ψ k, (cid:105) · · · | ψ k, n − (cid:105) · · · | ψ k, N − (cid:105) ... . . . ... . . . ... | ψ N, (cid:105) · · · | ψ N, n − (cid:105) · · · | ψ N, N − (cid:105) , (2) | ψ down (cid:105) = 1 √ N !perm | ψ , (cid:105) · · · | ψ , n (cid:105) · · · | ψ , N (cid:105) ... . . . ... . . . ... | ψ k, (cid:105) · · · | ψ k, n (cid:105) · · · | ψ k, N (cid:105) ... . . . ... . . . ... | ψ N, (cid:105) · · · | ψ N, n (cid:105) · · · | ψ N, N (cid:105) . (3)When the ions are fermionic, | ψ up (cid:105) and | ψ down (cid:105) can beexpressed using Slater determinants. There are two di-rections of the rotations induced by quantum tunnelingin the QTR system. Therefore, the Hamiltonian of theQTR is given as follows: H = (cid:126) j ( ˆ J CL + ˆ J CCL ) , (4)where (cid:126) and j are the Planck constant and the quantumtunneling rate. ˆ J CL and ˆ J CCL are defined asˆ J CL ≡ N (cid:79) k =1 2 N (cid:88) l =1 | ψ k,l +1 (cid:105) (cid:104) ψ k,l | , (5) N N-1 NN +1 N +2 N -1 N+ FIG. 2. Schematic representation of the N -ion QTR. Sincethe QTR is the superposition of two different polygon Wignercrystal orientations | ψ up (cid:105) and | ψ down (cid:105) , the QTR is consideredto be a 2 N -site cycle graph. ˆ J CCL ≡ N (cid:79) k =1 2 N (cid:88) l =1 | ψ k,l (cid:105) (cid:104) ψ k,l +1 | , (6)with | ψ k, N +1 (cid:105) = | ψ k, (cid:105) . Thus, ˆ J CL and ˆ J CCL satisfyˆ J CL | ψ up(down) (cid:105) = | ψ down(up) (cid:105) and ˆ J CCL | ψ down(up) (cid:105) = | ψ up(down) (cid:105) , representing clockwise and counter-clockwiserotation, respectively. It is possible to experimentallydistinguish these two states using a projective measure-ment [1], i.e., there are two distinguishable states.Next, we flip the internal state of the ion at the l -thsite ( l = 2 n or 2 n -1). By introducing the wavefunction | ψ up , − (cid:105) = 1 √ N !perm |↓(cid:105) | ψ , (cid:105) · · · |↑(cid:105) | ψ , n − (cid:105) · · · |↓(cid:105) | ψ , N − (cid:105) ... . . . ... . . . ... |↓(cid:105) | ψ k, (cid:105) · · · |↑(cid:105) | ψ k, n − (cid:105) · · · |↓(cid:105) | ψ k, N − (cid:105) ... . . . ... . . . ... |↓(cid:105) | ψ N, (cid:105) · · · |↑(cid:105) | ψ N, n − (cid:105) · · · |↓(cid:105) | ψ N, N − (cid:105) (7)and | ψ down , (cid:105) = 1 √ N !perm |↓(cid:105) | ψ , (cid:105) · · · |↑(cid:105) | ψ , n (cid:105) · · · |↓(cid:105) | ψ , N (cid:105) ... . . . ... . . . ... |↓(cid:105) | ψ k, (cid:105) · · · |↑(cid:105) | ψ k, n (cid:105) · · · |↓(cid:105) | ψ k, N (cid:105) ... . . . ... . . . ... |↓(cid:105) | ψ N, (cid:105) · · · |↑(cid:105) | ψ N, n (cid:105) · · · |↓(cid:105) | ψ N, N (cid:105) , (8)including the spin degrees of freedom, the quantum stateof the QTR can be given as follows: | ψ (cid:105) = α (cid:48) N (cid:88) n =1 c n − | ψ up , − (cid:105) + β (cid:48) N (cid:88) n =1 c n | ψ down , (cid:105) , (9)where α (cid:48) , β (cid:48) , c n − and c n are the complex coeffi-cients satisfying | α (cid:48) | + | β (cid:48) | = 1, (cid:80) Nn =1 | c n − | = 1 and (a) Spin state
Time
Quantum tunneling (b) (c) !" $ %& ’ ( ) * $$ + $ , $ - $$ . $$ / $$ * +0*.*0.0 *2002302/02-02+020 ) ( !7!" $ $ : ; < (d) !" $ %& ’ ( ) * $$ + $ , $ - $ . $ / $ $ $ $ *3 $ * +3*.*3.3 *5335135/35-35+353 ) ( !9!" : $ ; $ < = > FIG. 3. (a) Time evolution of the 3-ion QTR. The spinstates |↑(cid:105) and |↓(cid:105) are represented as red and blue circles, re-spectively. (b) Time evolution of the QTR shown in Fig. 3(a),considered as a 6-site cyclic graph. The ion in |↑(cid:105) propa-gates into the neighboring sites via quantum tunneling. (c,d)Continuous-time cyclic quantum walks. The time evolutionof the probability distribution of the ion in |↑(cid:105) is calculatedfor (c) 3-ion and (d) 5-ion QTR. (cid:80) Nn =1 | c n | = 1. This equation clearly shows that thereare distinguishable N states for each crystal orientation.Therefore, a QTR with an ion in the internal state |↑(cid:105) at the l -th site is considered to be a 2 N -site cyclic graph.Figure 3(a) shows the time evolution of a 3-ion QTR withone ion in |↑(cid:105) . As is clear from the above discussion, asshown in Fig. 3(b), this model is considered to be equiv-alent to the 2 N -site cycle graph where the ion with inter-nal state |↑(cid:105) at the n -th site evolves into the neighboringsites via quantum tunneling. By introducing the nota-tion | n − (cid:105) = | ψ up , − (cid:105) and | n (cid:105) = | ψ down , (cid:105) , the quantum state of the QTR can be rewritten as | ψ (cid:105) = N (cid:88) n =1 γ n | n (cid:105) , (10)where γ n is the complex coefficient satisfying (cid:80) Nn =1 | γ n | = 1. The Hamiltonian of this systemcan also be described as H = (cid:126) j N (cid:88) n =1 (ˆ a n ˆ a † n +1 + ˆ a † n ˆ a n +1 ) , (11)where ˆ a = ˆ a N +1 . ˆ a † n and ˆ a n are creation and annihila-tion operators of an ion in |↑(cid:105) at the n -th site.We show the quantum dynamics of the 3-ion QTRin Fig. 3(c). In the simulation, we prepared the initialstate in | (cid:105) , as shown in Fig. 3(b). Note that the evo-lution time is normalized to the unit of the quantumtunneling rate. Here, by considering the ion in |↑(cid:105) tobe a quantum walker, the time evolution of the proba-bility distribution of the ion in |↑(cid:105) can be described bycontinuous-time quantum walks. The quantum dynam-ics of the cyclic quantum walks using a 5-ion QTR is alsoshown in Fig. 3(d) (see appendix for more details). SPIN-DEPENDENTQUANTUM INTERFERENCE
We now consider a magnetic flux Φ = SB threadingthrough a QTR consisting of identical ions, as shown inFig. 4(a). Here, S and B are a closed area of the QTR anda magnetic field passing through the closed area, respec-tively. The ions in the QTR couple to the vector poten-tial. Therefore, the AB effect introduces a relative phasedifference 2 θ AB between wavefunctions rotating clockwiseand counterclockwise, as shown in Fig. 4(b) [1]. The ABeffect induced phase shift is θ AB = π Φ φ , where φ = (cid:126) e .Here, the Hamiltonian of the QTR is expressed as H = (cid:126) j ( ˆ J CL e iθ AB + ˆ J CCL e − iθ AB ) . (12)Fig. 4(c) shows the numerically calculated time evo-lution of the probability of finding | ψ up (cid:105) as a functionof time. In the numerical simulation, we prepared aninitial state of the 3-ion QTR in | ψ up (cid:105) . Then, we calcu-lated the probability of finding | ψ up (cid:105) based on Eq. (12).Since the spin states of the ions in the QTR are thesame, the rotational direction of the transition of theQTR is indistinguishable. Therefore, the quantum in-terference is induced by the AB effect [1]. When theAB effect phase shift is π/
2, the clockwise and counter-clockwise rotating wavefunctions counteract each other,resulting in suppression of quantum tunneling. These re-sults clearly show that it is possible to control the quan-tum tunneling probability by changing the amount ofmagnetic flux passing through the QTR. By combiningthe quantum interference effect and the time-dependentmagnetic field, it is possible to realize arbitrary superpo-sition of two different Wigner crystal orientations, such (a) (b)(c)
FIG. 4. (a) Magnetic field B passing through the 3-ion QTR.(b) Ions in the QTR system coupled to the vector potential,introducing a phase difference between wavefunctions rotatingin two opposite directions. (c) Probability of finding | ψ up (cid:105) foreach AB phase shift as a function of time. The evolution timeis normalized to the unit of the quantum tunneling rate. as | ψ (cid:105) = √ ( | ψ up (cid:105) + e iθ | ψ down (cid:105) ), where θ is the relativephase shift.We next consider the quantum dynamics of a QTR forwhich one of the spin states of the ions is flipped. Accord-ing to the above discussion and Eq. (11), the Hamiltonianof this QTR system is described as follows: H = (cid:126) j N (cid:88) n =1 (ˆ a n ˆ a † n +1 e iθ AB + ˆ a † n ˆ a n +1 e − iθ AB ) , (13)We show a simulation of the 3-ion QTR dynamicswhen the AB phase shifts are θ AB = 0 , π , π , and π in Fig. 5(a–d). Note that the time is normalized to theunit of the quantum tunneling rate and the initial state isprepared in | (cid:105) . The dynamics are slightly modulated bythe corresponding θ AB . In particular, when θ AB = π , de-structive interference always occurs at the opposite site,i.e., at 4-th site, resulting in the probability of finding | (cid:105) being always zero, as shown in Fig. 5(d).We find that the spin degrees of the trapped ions ina QTR can change the quantum tunneling dynamics ofthe QTR. This property can be used to realize spin-dependent motion of the QTR. For example, we considerthe dynamics of a 3-ion QTR having identical ions. Byapplying the magnetic field which induces θ AB = π , theQTR does not rotate due to the destructive interference,as shown in Fig. 4(c). However, if we flip the spin stateof one of the ions in the QTR, the QTR then starts torotate, as shown in Fig. 5(d). The dynamics of the QTRwith an ion in |↑(cid:105) when θ AB = π is exactly the same (a) !" $ %& ’ ( ) * $$ + $ , $ - $$ . $$ / $$ * +0*.*0.0 *2002302/02-02+020 ) ( !7!" $ $ : ; < (b) !" $ %& ’ ( ) * $$ + $ , $ - $$ . $$ / $$ * +0*.*0.0 *2002302/02-02+020 ) ( !7!" $ $ : ; < (c) !" $ %& ’ ( ) * $$ + $ , $ - $$ . $$ / $$ * +0*.*0.0 *2002302/02-02+020 ) ( !7!" $ $ : ; < (d) !" $ %& ’ ( ) * $$ + $ , $ - $$ . $$ / $$ * +0*.*0.0 *2002302/02-02+020 ) ( !7!" $ $ : ; < FIG. 5. (a–d) Time evolution of the probability distributionof an ion in |↑(cid:105) . The initial state is prepared in | (cid:105) . The evolu-tion time is normalized to the unit of the quantum tunnelingrate. (a) θ AB = 0. (b) θ AB = π . (c) θ AB = π . (d) θ AB = π . as Fig. 5(d). This is analogous to spin-motion coupling,such as the Stern–Gerlach experiment [13] or the quan-tum spin filter [5]. In addition, if one of the ions is pre-pared in √ ( |↑(cid:105) + |↓(cid:105) ), the superposition of two differentQTRs is realized, for which one QTR rotates and the an-other does not. Such a quantum system may realize theentangled state of the spin degrees of freedom and theWigner crystal structures [6]. DISCUSSION
To observe quantum tunneling, the quantum coherenceof the QTR is important. One of the biggest sourcesof decoherence is heating of the rotational mode. Thequantum tunneling rate of the QTR is relatively slow.For example, the numerically calculated quantum tunnel-ing rate using the wavefunctions and effective potentialshown in Fig. 1(c) is 4.95 Hz. Therefore, the heating rateof the rotational mode needs to be sufficiently suppressed.As previously discussed [1], the lack of adiabaticity of thetrap potential control and the fluctuation of the RF volt-age may be related to the heating of the rotational mode.These factors can be avoided by carefully choosing theexperimental parameters and using the RF voltage sta-bilization method [14]. Other collective modes may heatthe rotational mode. To prevent this effect, further im-plementation of the laser cooling technique to cool allthe collective modes simultaneously [15] may be helpful.Heating of the rotational mode decreases the populationin the motional ground state of the rotational mode, re-ducing the quantum tunneling quality. However, as longas the population of the motional ground state of therotational mode is not zero, the quantum tunneling dy-namics can be deduced from the measurement results.In addition, the coherence time of the quantum tunnel-ing needs to be long. One reason for the degradation ofthe quantum tunneling coherence is the fluctuation of theRF potential. This can be mitigated by implementing RFstabilization [14].
CONCLUSIONS
We have studied the quantum dynamics of a QTRwhen a spin state of one of the ions is flipped. We foundthat symmetry breaking by flipping the spin state of theion of the QTR results in different quantum dynamics.Additionally, we have exploited the quantum dynamicsof the QTR when a magnetic field is present. Our workmay be useful for realizing applications using spin-motioncoupling [5–8]. While quantum tunneling is a fundamen-tal phenomenon, it is difficult to investigate the quantumtunneling dynamics at a single quantum level. Advan-tages such as individual addressability and manipulationof quantum states of the ions provide a well-controllablequantum system. Therefore, a trapped-ion QTR systemcan provide an ideal platform for performing fundamen-tal quantum physics experiments.
ACKNOWLEDGMENTS
R.O. would like to thank Go Tomimasu and AtsushiNoguchi for fruitful discussion. This work was supportedby MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067477.
Appendix: 5-ion QTR
We show the results of the analysis of a QTR usingfive Yb + ions. We assume that the confinement alongthe z direction ( ω z ) is 2 π × ω x /ω z = 1 .
010 are also shown in Fig. 6(a). Thered curve (mode1) is the rotational mode. Fig. 6(b) showsthe effective potential of the 5-ion QTR system for ω x /ω z = 1.010. There are two stable orientations of the ioncrystal, | ψ up (cid:105) and | ψ down (cid:105) . When ω x /ω z = 1.010, | ψ up (cid:105) and | ψ down (cid:105) overlap [Fig. 6(c)]. [1] A. Noguchi, Y. Shikano, K. Toyoda, and S. Urabe, Nat.Commun. , 3868 (2014).[2] Y. Aharonov and D. Bohm, Phys. Rev. , 485 (1959).[3] D. Solenov and L. Fedichkin, Phys. Rev. A , 012313(2006).[4] A. A. Melnikov and L. E. Fedichkin, Sci. Rep. , 34226(2016).[5] M. Lebrat, S. H¨ausler, P. Fabritius, D. Husmann, L. Cor-man, and T. Esslinger, Phys. Rev. Lett. , 193605(2019).[6] J. D. Baltrusch, C. Cormick, G. DeChiara, T. Calarco,and G. Morigi, Phys. Rev. A , 063821 (2011).[7] R. Gerritsma, A. Negretti, H. Doerk, Z. Idziaszek, T.Calarco, and F. Schmidt-Kaler, Phys. Rev. Lett. ,080402 (2012).[8] J. M. Schurer, R. Gerritsma, P. Schmelcher, and A. Ne-gretti, Phys. Rev. A , 063602 (2016). [9] D. Reiss, K. Abich, W. Neuhauser, Ch. Wunderlich, andP. E. Toschek, Phys. Rev. A , 053401 (2002).[10] J. Chen, J. G. Story, J. J. Tollett, and R. G. Hulet, Phys.Rev. Lett. , 1344 (1992).[11] A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C.Spreeuw, and P. S. Jessen, Phys. Rev. Lett. , 1542(1995).[12] G. Poulsen, and M. Drewsen, arXiv, 1210.4309 (2012).[13] W. Gerlach and O. Stern, Z. Phys. , 349 (1922).[14] K. G. Johnson, J. D. Wong-Campos, A. Restelli, K. A.Landsman, B. Neyenhuis, J. Mizrahi, and C. Monroe,Rev. Sci. Instrum. , 053110 (2016).[15] S. Ejtemaee and P. C. Haljan, Phys. Rev. Lett. ,043001 (2017). (a) Mode1 Mode2 Mode3 Mode4 Mode5Mode6 Mode7 Mode8 Mode9 Mode10 (b)(c)
FIG. 6. (a) Normalized ten lowest collective frequencies of the five trapped ions as a function of the ratio of ω x to ω z . The insetshows the eigenvector of each collective mode for ω x /ω z = 1.010. (b) Effective potential for ω x /ω z = 1.010 ( ω z = 2 π × | ψ up (cid:105) and | ψ down (cid:105) .(c) Effective potential and amplitudes of the wavefunctions | ψ up (cid:105) and | ψ down (cid:105) as a function of the angle θθ