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Franck-Condon Physics in A Single Trapped Ion

We propose how to explore the Franck-Condon (FC) physics via a single ion confined in a spin-dependent potential, formed by the combination of a Paul trap and a magnetic field gradient. The correlation between electronic and vibrational degrees of freedom, called as electron-vibron coupling, is induced by a nonzero gradient. For a sufficiently strong electron-vibron coupling, the FC blockade of low-lying vibronic transitions takes place. We analyze the feasibility of observing the FC physics in a single trapped ion, and demonstrate various potential applications of the ionic FC physics in quantum state engineering and quantum information processing.


I. INTRODUCTION
The Franck-Condon (FC) principle is a well-known fundamental law to explain the intensity of vibronic transitions in molecules [1,2], in which the transition intensity is proportional to the FC factor defined by the square of the overlap integral between the vibrational wavefunctions of the two involved states. The FC physics actually exists in various systems of interactions between mechanical and electronic degrees of freedom. In particular, a very small or zero FC factor will cause transition suppression named as the FC blockade and a nonzero FC factor between different vibrational modes may cause vibrational sidebands [3,4]. Besides the conventional experiments of molecules, the electronic transport through quantum dots could be exponentially suppressed in the region of strong electron-vibron coupling [5]. The conspicuous transport property in the strong coupling regime plays an important role in both single-molecule devices [6] and nano-electromechanical systems [7].
To the best of our knowledge, the FC physics has never been clearly demonstrated in a true single-particle system. In recent, an ensemble of atoms confined within a spin-dependent optical lattice [8] has been demonstrated for sideband cooling and coherent operations via FC physics. That experiment can be regarded as an effective single-particle implementation only when the inter-atom interaction can be neglected. However, as the system is cooled to only populating the ground and first-excited vibrational states for each lattice cell, the s-wave scattering between atoms have to be taken into account. For a system of shallow lattices, the atoms can easily tunnel between neighboring sites and therefore the probability of multiple particles in a particular site will be very significant. For a system of deep lattices, if the atomic number per lattice site is larger than one, the on-site interaction between atoms cannot be ignored and the single-atom model is hard to describe the system. Even for a system of no inter-particle interaction, due to the intrinsic nature of a spin-dependent optical lattice, the unavoidable coupling between next-nearest-neighboring sites may destroy the FC physics and induce complex quantum transport along the lattice axis. Moreover, due to the wavelength and intensity limits of the optical lattices, the maximum shift and the total number of vibronic states are both limited.
In this article, we present a proposal for observing the FC physics via a true single-atom system, i.e., a single trapped ion, and discuss its potential applications in quantum state manipulation. The key point is that the electron-vibron coupling in a single trapped ion is induced and controlled by a magnetic field gradient (MFG). Compared to the electron-vibron coupling generated by radiation of non-resonant laser beams on the ion, which is too weak to observe the FC physics, the MFG-induced electron-vibron coupling is controllable and could be strong enough to observe. We may apply this coupling to suppress or even block some undesired transitions, called FC blockade, or to enhance some desirable transitions. Attribute to its clear environment and high controllability, a single trapped ion opens a new area for studying the FC physics at the single-atom level. Beyond the fundamental interests in various fields from quantum spectroscopy to quantum transport, the ionic FC physics is of promising applications in quantum state engineering.

II. MODEL AND FC BLOCKADE
We consider a single ultracold ion confined in a Paul trap [9] and the ion only populates two possible electronic levels |↓ and |↑ of different magnetic dipole moments. In usual ion trap in the absence of MFG, radiation of laser beams on the ion with blue-and red-detuning could yield couplings between the vibrational and electronic degrees of freedom. But this coupling is generally weak for the ultracold ion within Lamb-Dicke limit (LDP). For ensuring FC blockade between low-lying vibrational states, a MFG with a sufficiently large gradient is required to enhance the electron-vibron coupling. Without loss of generality, for a one-dimensional trap and the gradient along the axis, the electron-vibron coupling could be expressed as with the electron-spin g-factor, the Bohr magneton µ B and the phonon creation (annihilation) operators a † (a). δz is the oscillation amplitude of the ion along the z axis. Here, b denotes the magnetic field gradient sensed by the ion, and G = gµ B b /2mω z with the ion mass m and the trap frequency ω z . The spin-dependent potentials can be written as and sketched in Fig. 1. With the electron-vibron coupling given in Eq. (1), it is easy to obtain the shift z 0 = µB b mω 2 z . For a large gradient b, z 0 would be sufficiently large and the overlap between the low-lying vibrational wavefunctions becomes very small. As a result, the transition between the two vibrational ground states, | 0, ↓ ↔| 0, ↑ , would be strongly suppressed. This is the FC blockade. Here, |n with n =0, 1, 2 denotes the vibrational levels. For a sufficiently large distance between the two equilibrium positions, there is no significant overlap between the two low-lying vibrational states so that the FC blockade appears.
Specifically, due to the nonzero gradient, the ionic electron-vibron coupling is governed by a new effective LDP η ′ = η 2 + ε 2 , where η is the original LDP for the case of no gradient, and ε = ∂ω0 ∂z ℏ/2mω 3 z with ∂ω0 ∂z = 2µB b is the additional LDP caused by the gradient [10,11]. This additional LDP has been observed in a recent experiment [12].  [4,[13][14][15] between the vibrational states |0, σ and |n, σ ′ (σ, σ ′ =↓, ↑) could be written as Thus the FC factor is |M 0→n | 2 = e −η ′2 η ′2n /n!, which is exponentially sensitive to the ionic electron-vibron coupling, as plotted in Fig. 2. By adjusting the trap frequency ω z and/or the gradient b, the effective LDP η ′ will suppress the transition between the vibrational ground states of different electronic levels, while allow transitions between the vibrational ground and excited states of different electronic levels. As shown in Fig. 2, the maximum FC factors gradually decrease with the increase of the phonon number n. However, the FC factors between different vibrational states remain non-zero, which may cause the vibrational sidebands useful for cooling the ion.
Eliminating the undesired phase factors on |2 , we obtain a CNOT gate on the electron states conditional on the vibrational states of the ion: |0, ↓ (↑) → |0, ↓ (↑) and |2, ↓ (↑) → |2, ↑ (↓) . The fidelity of the CNOT gate is estimated with respect to the effective LDP η ′ in Fig. 3(b), in which the growth of η ′ improves the fidelity. The FC physics is also observable in the mediate region of 2.6 ≤ η ′ < 3, where the FC factor for |0, ↓ ↔ |0, ↑ is almost zero, but the FC factors for |1, ↓ ↔ |1, ↑ and |2, ↓ ↔ |2, ↑ are still significant. Because |M 2→2 | 2 > |M 1→1 | 2 , we may still encode the control qubit in |0 and |2 . The CNOT gate could still be expressed by Eq. (6) and its fidelity versus η ′ is shown in Fig. 3(b). Compared to the case of larger gradients, an unfavorable effect may appear as a result of the unwanted population on the vibrational state |1 . A simple estimation of the detrimental influence from the undesired population on |1 is obtained in Fig. 3(c) by assuming the initial vibrational state as [α (|0 + |2 ) + β|1 ] / √ 2 with the error factor β. When the gradient is tuned into the region 0 < η ′ ≤ 1, the FC blockade does not happen for the ground vibrational transition but for other high-lying vibrational transitions at some special points, such as the blockade of |1, ↓ ↔ |1, ↑ at η ′ = 1 and the blockade of |2, ↓ ↔ |2, ↑ at η ′ = 0.765. This reminds us of the 'magic' LDP mentioned in [16]. Therefore, tuning the effective LDP to the 'magic' point η ′ = 1, we may employ the vibrational states |0 and |1 to encode the control qubit. The CNOT gate is then implemented in the subspace spanned by |0, ↓ , |0, ↑ , |1, ↓ and |1, ↑ , where the internal states flip only when the vibrational state is |0 . Alternatively, tuning the effective LDP to the other 'magic' point η ′ = 0.765, we may encode the control qubit in the vibrational states |0 and |2 . Different from the schemes for large and mediate gradients, our CNOT gate in the region of small gradients can only be accomplished at some 'magic' points. As a result, the quality of the performance is very sensitive to the effective LDP. We have estimated this sensitivity and the relations of the MFG to the LDP in Fig. 4.
In our CNOT proposal, since two motional states of the ion are encoded as the control qubit, it requires groundstate cooling, which could be made before the MFG is applied or in the presence of a large MFG. As we will briefly discuss later, an appropriately big MFG sometimes helps for laser cooling.

A. Preparation of Fock states
The observable FC physics could be used to prepare motional Fock states in a probabilistic way. We consider the initial state with the population probability P k and the average phonon number P 1 + 2P 2 < 1. To prepare the ground motional state |0 , one has to eliminate the populations in states |1 and |2 . By tuning the gradient to block the transition |1, ↓ ↔ |1, ↑ , the population in |1 could be screened away via a π/2 carrier transition pulse following a measurement on |↑ . Similarly, by tuning the gradient to block the transition |2, ↓ ↔ |2, ↑ , the population in |2 could be screened away via a π/2 carrier transition pulse following a measurement on |↓ . In above operations, the generation of the expected Fock state is probabilistic. So we have to employ the repeat-until-success method [17]. Once the desired internal state is successfully detected, the desired motional state is prepared with a unity fidelity.

B. Modification in single-qubit gate operations
Due to the existence of the FC blockade in the regime of large MFG, some new difficulties for single-qubit operations in a string of trapped ions appear. The proposals [10,11] from Wunderlich's group focused on the regime of small MFG, which corresponds to a small LDP. Under such a small LDP, the first-order expansion works very good and the ground-state cooling is indeed not necessary. In the regime of higher MFG supporting the FC blockade, the large MFG favors a working Ising coupling within a shorter time, which makes the two-qubit conditional operations faster. However, since the large MFG strongly suppresses some vibrational transitions, it becomes more difficult to perform the Hadamard gates via carrier transitions.
We show a specific simulation for a spin flip in the presence and absence of MFG, see Fig. 5. In our simulation, the single trapped ion within the initial internal state | ↓ and thermal motional state (e.g., n = 5 or 0.1) under a MFG (η ′ = 1) evolves according to the dynamical population be determined by interrogative pulse operations with respect to different values of η ′ . This result is also applicable to the refocusing pulses [18] for removing the undesired couplings due to the Ising coupling [19]. As the refocusing pulses are based on carrier transitions, like the Hadamard gate above, stronger pulses are necessary for refocusing under a large MFG. Alternatively, we may switch off the MFG when performing single-qubit gates. To this end, employment of lasers, instead of microwaves as in [10,11], is necessary for individually accomplishing the single-qubit operations, where the FC physics does not work in the carrier transition and cooling to the vibrational ground state is unnecessary.

V. EXPERIMENTAL FEASIBILITY AND CHALLENGE
For a real experimental implementation of the CNOT gate, we may employ the hyperfine levels S 1/2 , F = 0, m F = 0 and S 1/2 , F = 1, m F = 1 of 171 Yb + as |↓ and |↑ , respectively. Here, the transition frequency for |↓ ↔ |↑ is ω 0 ≈ 2π × 12.6 GHz [10,20]. With the z-axis trap frequency ω z = 2π×100 kHz and magnetic dipole coupling strength λ = 2π × 50 kHz, the CNOT gate will be accomplished within 19.2 µs, 8.2 µs, and 6.7 µs for η ′ =3, 1 and 0.765, respectively. To perform the CNOT gates highly coherently, we require the coherence time of the employed hyperfine levels to be at least longer than 192 µs. Fortunately, the latest experiment has shown that this coherence time could be 5 ms [21].
To achieve our scheme, moreover, we may employ the weak region of η ′ , e.g., η ′ = 1. To this end, due to sensitivity to the 'magic' points, a highly stable gradient is required for achieving the high-fidelity operations. We have estimated the sensitivity to the fluctuation of the gradient, see Fig. 4(b) where the fidelities are larger than 0.994 for all schemes if the gradient fluctuation |∆b| ≤ 10 T/m. Compared to [16] with the 'magic' LDP controlled by the wave-vector, phase and intensity of the radiating lasers, the CNOT operation in our scheme is mainly governed by the MFG, whose strength and stability are key to the implementation.
With currently available technologies, a big challenge of our scheme is how to realize a large MFG. Current-carrying coils in an anti-Helmholtz-type arrangement [10,22] and permanent magnets [12] have been applied to attain a gradient up to tens of T/m. To achieve a working Ising type interaction between two ions separated by few micrometers, the MFG is required to be on the order of 100 T/m [23,24]. Although this is still challenging with current techniques, there are some efforts toward this aim using new materials and improved designs [25].

VI. CONCLUSIONS
In summary, we have explored the FC physics in a single trapped ion under an external MFG and discussed the experimental feasibility. Although our discussion above focused on the FC blockade, the strong electron-vibron coupling would probably be useful for quantum simulation of, e.g., Dirac equation [26] or quantum walk [27]. We argue that our study would be useful for further understanding FC physics and its application.
Moreover, the FC physics with strong electron-vibron coupling might also be useful for sideband cooling of the trapped ion, in which vibrational sidebands play the important role and the carrier transitions are excluded. In usual schemes, the cooling does not happen if the cooling laser is orthogonal to the trapping direction [28]. However, for an ion in a spin-dependent potential as shown in Fig. 1, the electron-vibron coupling is caused by the MFG, instead of the cooling laser itself. As a result, the cooling could work even for a red-detuning beam perpendicular to the trapping direction. However, due to FC blockade, the cooling down to the ground motional state is sometimes impossible, but deterministically to some certain motional states, e.g., to n = 1 for b = 208 T/m with ν =100 kHz. Nevertheless, using the conventional laser cooling techniques plus a MFG in parallel, the cooling efficiency should be enhanced. In recent, there appears a work on enhanced cooling via MFG [29], in which the FC physics has not been specifically mentioned.