Entanglement of neutral-atom chains by spin-exchange Rydberg interaction
Conditions to achieve an unusually strong Rydberg spin-exchange interaction are investigated and proposed as a means to generate pairwise entanglement and realize a SWAP-like quantum gate for neutral atoms. Ground-state entanglement is created by mapping entangled Rydberg states to ground states using optical techniques. A protocol involving SWAP gate and pairwise entanglement operations is predicted to create global entanglement of a chain of atoms in a time that is independent of .
pacs:03.67.Bg, 03.65.Ud, 32.80.Ee, 32.80.Xx
Entanglement is a property of multiparticle quantum states that is essential for implementing quantum information or computation protocols Einstein et al. (1935); Nielsen and Chuang (2000). As a result, schemes for the fast and efficient generation of entanglement among many quantum systems are the subject of intense theoretical and experimental efforts. Atoms offer an ideal arena for the demonstration of quantum protocols given the stability of their ground states and the powerful optical and trapping techniques that have been developed to control their internal and external degrees of freedom Adams and Riis (1997). The excitation of Alkali atoms to energy levels of large principal quantum number, generically named Rydberg states, provides a way to enhance by several orders of magnitude otherwise weak neutral atom interactions. The resulting dipole-dipole or van der Waals interaction between two highly excited Rydberg atoms allows the creation of stable entangled states of atoms in their ground electronic level Møller et al. (2008); Saffman and Mølmer (2009); Zhao et al. (2010); Saffman et al. (2010); Isenhower et al. (2010); Urban et al. (2009); Zhang et al. (2012); Carr and Saffman (2013); Bhaktavatsala Rao and Mø lmer (2013). The methods employed to date Møller et al. (2008); Saffman and Mølmer (2009); Zhao et al. (2010); Saffman et al. (2010); Isenhower et al. (2010); Urban et al. (2009) rely on the Rydberg blockade effect, in which two-atom energy levels with two Rydberg excitations experience a large interaction induced shift, while energy levels with a single Rydberg excitation are unperturbed. As a consequence, under conditions in which a single Rydberg excitation is resonantly excited, the pair excitation probability is strongly suppressed Tong et al. (2004); Heidemann et al. (2007); Vogt et al. (2007); Schwarzkopf et al. (2011); Dudin and Kuzmich (2012); Dudin et al. (2012); Viteau et al. (2012); Barredo et al. (2014).
The excitation of two atoms into Rydberg -orbitals with different principal quantum numbers, and , and opposite electron spin orientation, produces not only the Rydberg blockade shift van Ditzhuijzen et al. (2008); Han and Gallagher (2009); Bariani et al. (2012); Günter et al. (2013); Gorniaczyk et al. (2014); Tiarks et al. (2014); Li et al. (2014); Paredes-Barato and Adams (2014), but also a coupling that exchanges the electron spin states. While the blockade shift is usually the dominant effect, the spin-exchange coupling can be made almost equally strong with the right choice of and . In this regime, when nearby atoms are optically driven, the probability to create double Rydberg excitations can be as large as one, in sharp contrast to the case when . Furthermore, the two-atom Rydberg state created by this mechanism is one of two entangled Bell states, the triplet denoted , in the subspace of the two-atom Rydberg excitations. The orthogonal Rydberg Bell state, the singlet experiences a strong level shift and is effectively decoupled from the excitation. The entangled Rydberg state created in this way is metastable, but the entanglement can be mapped optically to the ground state in a time short compared to the metastable state lifetime. In the following, we explain how to produce ground state entanglement of a pair of atoms by a sequence of three pulses, and then discuss a protocol for global entanglement of a chain of atoms involving a sequence of twelve pulses which minimizes the blockade shifts due to multiple Rydberg excitations along the chain.
The main body of the paper gives an account of the novel interaction mechanism and its applications to quantum computation. Section II introduces the spin-exchange interaction and its potential for two-atom entanglement. Section III discusses an example to achieve pairwise entanglement via optical pumping. In Sec. IV, we investigate the pairwise entanglement efficiency by numerical simulation. In Sec. V, we study a quantum gate that is similar to the SWAP gate, and introduce a protocol of global entanglement of atoms in a chain. Section VI gives a summary. Additional details of the theory are given in the appendices.
Ii Two-atom entanglement
Consider two Rb atoms, denoted and , respectively, which are loaded into two far-off-resonant traps created by tightly focused laser beams. The interatomic axis is defined by the vector . Each of the atoms can be independently driven by laser light, as shown in Fig. 1. We suppose atom is prepared in a hyperfine level of the ground state manifold identified as “spin-up” while atom is prepared in the state “spin-down”: (. These states may be optically coupled via two-photon transitions to the Rydberg states ( of atoms and , respectively. Here we use the hyperfine notation to label ground states and the fine structure notation for Rydberg levels, according to the spectroscopic resolution usually achieved in experiments (see Appendix A).
Consider two Rydberg atoms prepared in the states and , respectively, and separated by a distance large enough that the states are coupled by the van der Waals interaction Saffman et al. (2010); Walker and Saffman (2008). In the case , this coupling is dominant and induces a shift of the doubly excited Rydberg state commonly referred to as the blockade shift. When , and under special conditions, the coupling , which exchanges the electronic spin states, may become equally large. In the two-atom product basis , the total van der Waals interaction is then,
By using the measured results for the relevant quantum defects Li et al. (2003) and a semiclassical expression for the radial matrix elements Kaulakys (1995), we numerically evaluate and for and . We find four pairs of states where the interaction coefficients differ by less than ; see Table 1.
The occurrence of a strong spin-exchange interaction in these cases arises through an interference effect involving a small number of dominant intermediate and states (see Appendix B). We find that the transition matrix elements for spin exchange constructively interfere whereas a partially destructive interference limits the blockade shift. This results in almost equal magnitude of and .
The eigenstates and eigenvalues of Eq. (1) are , and . If the interaction coefficients have equal magnitudes, one of the two energy eigenvalues is unshifted from the non-interacting value. For the four cases shown in Table 1, and have opposite signs, and by choosing an appropriate atomic separation , () can be made much larger (smaller) than the excitation Rabi frequency. As a result, atoms and will be excited to the entangled two-atom Rydberg state . The lifetime of the entanglement created can be enhanced by coupling it to the two-atom ground state, by driving Rabi oscillations and simultaneously on both atoms [Fig. 1(b)], so that the Rydberg triplet state is mapped to the ground level Bell state
Iii Pairwise entanglement protocol
We now describe the complete protocol creating ground state entanglement via a three--pulse sequence.
Pulse 1 on atom : We take the initial state to be . The first pulse acts on atom and excites to via the state, as shown in Fig. 2. Since atom is in its ground state, there is no Rydberg interaction. Thus, by applying a pulse of duration , we generate the product state .
Pulse 2 on atom B: Following pulse 1, apply a two-photon laser pulse to atom with Rabi frequency , as shown in Fig. 2. This pulse excites to via the state as in Fig. 1(b). The evolution of the two-atom wave function is governed by the Hamiltonian
where the basis vectors are , and . In the case of , we can adiabatically eliminate the state . The population on the Rydberg Bell state reaches its maximum
for a pulse of duration . The prefactor indicates that this is a two-atom -pulse, while the correction results from the small shift of .
Pulse 3 on both atoms: The final step is the mapping of the entangled state of Rydberg atoms onto the ground states. In contrast to Pulses 1 and 2, switch on all four Rabi channels simultaneously exciting the two atoms, as shown in Fig. 1(b).
When the Rabi frequencies satisfy only the intermediate states and are coupled to and where and , see Fig. 2. Ordering the states and the Hamiltonian matrix during Pulse 3 reads,
At the end of Pulse 3, of duration , the entangled Rydberg state would be completely mapped onto the entangled ground state with fidelity limited by the residual shift , and the radiative decay rate of the Rydberg level.
Iv Numerical simulation and fidelity
The proposed scheme relies on creation of the Rydberg entangled state, which requires
In order to show that the ground Bell state can be prepared with high fidelity for finite , we numerically study the time evolution of the atomic state following the procedure discussed above. We choose the Rydberg state pair with principal quantum numbers , atomic separation m, and set the single-atom Rabi frequency kHz. We expand the two-atom wavefunction as
and we numerically solve the Schrödinger equation for Pulses 2 and 3 (see Appendices C and D), since pulse 1 is trivial. The achieved ground state fidelity is . To improve the fidelity of the prepared ground Bell state, we numerically optimize by varying Rabi frequencies and pulse durations. Figure 3 shows the state evolution with kHz and kHz: a fidelity of for the ground Bell state is achieved in less than ten microseconds.
The main practical difficulty is to have all four Rabi frequencies for the excitation channels equal to each other. In order to show that the the protocol is robust against dispersion in the Rabi frequencies, we numerically integrate the Schrödinger equation varying and in the interval for Pulse 3, with , , and . By performing such simulations, we find that almost all fidelities are larger than () for (see Appendix D).
V SWAP gate and entanglement of atomic chains
The spin-exchange Rydberg interaction may be used together with Rydberg blockade to implement a quantum logic gate based on a simple combination of three single-atom laser interaction processes: two -pulses applied to atom , are separated by an intermediate -pulse applied to atom A. If both atoms are initially prepared in the same ground state, we obtain the state transformations,
During stages 1 and 3, a single Rydberg excitation is created and removed, while in stage 2, Rydberg blockade of the states and , respectively, prevents any double excitation. Alternatively, when the atoms are initially prepared in opposite spin ground states, there is a crucial difference. As a consequence of the strong spin-exchange interaction, the 2 pulse resonantly couples to the triplet state and flips the spin of both the Rydberg excitation and the atomic ground state; see Fig. 4. In this case the combination of van der Waals interaction and single atom 2 pulse creates a resonant “lambda” transition between two-atom states:
The three stage protocol completes the quantum SWAP gate transformation, with a phase shift of the swapped states.
By choosing m and Rabi frequency kHz for the pulse on atom A, we can demonstrate a gate fidelity for an operation time s.
The combination of quantum SWAP gate with pairwise atom entanglement operations can be used as the basis of a protocol to entangle a chain of an arbitrary number of atoms. The key observation is that the atoms should be entangled sequentially in pairs, allowing gaps between the pairs in order to minimize spurious level shifts due to Rydberg blockade. After the pair-entanglement, a series of SWAP gate operations link all the pairs in a fully entangled state. Such a procedure may be easily sketched for the case of atoms, where the atoms are labeled sequentially: . We first use the two-atom entanglement protocol described above to entangle atoms and , where . This is followed by a similar protocol that entangles atoms and . Because all atoms are now entangled with one of their two neighboring atoms, the two-qubit SWAP gate is used to entangle atoms and , followed by another SWAP operation for entanglement of and . In this way we may entangle all atoms by a twelve-pulse protocol as shown in Fig. 5. In the case of 4 atoms, this protocol generates the entangled state after the application of 9 pulses (see Appendix E).
During the pairwise entanglement and SWAP gate operations, the metastable Rydberg states are populated for a time , thus the fidelity of the entanglement of the -atom chain may be estimated as , where is the lifetime of the Rydberg level. When keeping the leading order term after expanding the exponential of under the condition of , the total error of the prepared many-atom entangled state scales linearly with . Numerical simulations show that pairwise entanglement and SWAP gate operations can each be carried out in around 10s. Thus a 4-atom chain may be entangled in a time s with m. For (a) , ms for and , while for (b) , ms for and Saffman et al. (2010). Decreasing to makes four times larger, and reduces to . Then For , a 4-atom chain can be entangled with , while an 8-atom chain can be entangled with . These fidelities are comparable to the values for 4 and 8 atom entanglement by asymmetric blockade in Ref. Saffman and Mølmer (2009) and dissipation in Ref. Carr and Saffman (2013). The dissipative protocol in Carr and Saffman (2013) does not suffer from the spontaneous emission issue, but in comparison the present coherent process is almost three orders of magnitude faster. Furthermore, the linear scaling of the error with the total number of entangled atoms is similar to the the blockade-based situation of Ref. Saffman and Mølmer (2009): in that case the error scales cubically at low and then saturates to a linear behavior. By contrast to the multi-atom entanglement based on Ryberg interactions and adiabatic passage proposed in Ref. Møller et al. (2008), the duration and the Rabi frequency of our scheme are independent of . The requirements on the Rabi frequencies and principal quantum numbers are well within experimental reach and the individual atomic addressing allows to tailor the distance between the atoms as well as the desired target state. All these considerations show that the proposed spin-exchange mechanism represents a valid candidate to realize fast quantum operations with Rydberg atoms.
Throughout this work we have considered the case of individually trapped atoms addressed by external lasers. An interesting alternative realization of these protocols relies on the creation of a small super-atom in an elongated ensemble. This option has the advantage of solving the probabilistic loading of the individual traps as well as to provide a boost to the single-atom procedure by a factor where is the number of individual atoms in the super-atom. Quantum gates between super-atoms have been recently proposed Paredes-Barato and Adams (2014) and the multiplexing of different qubits in an elongated ensemble has already been realized experimentally Lan et al. (2009).
In conclusion, we have proposed a fast and robust mechanism to entangle neutral atoms. It is based on a variation of the van der Waals interaction between atoms excited to Rydberg states: for different principal quantum numbers, the spin-exchange interaction may be comparable to the Rydberg blockade shift thus induce a resonance between ground state levels and an entangled Rydberg state. This metastable state may then be mapped to a stable, entangled ground state. Furthermore, the entanglement efficiency may be improved by using small ensembles as well as by manipulating the structure of the Rydberg manifolds via external fields.
Based on the spin-exchange interaction, pairwise entanglement along with a SWAP-like gate form the basis of a protocol for the generation of ground-state entanglement of many atoms in a chain configuration. These protocols may be implemented in present experiments leading to quantum manipulation of many-body systems.
XFS and TABK acknowledge support from AFOSR and the Quantum Memories MURI of the Air Force Office of Scientific Research. FB acknowledges support from the DARPA QuASAR program and the US NSF.
Appendix A Single-atom optical excitation
This section shows how to realize the transitions in Fig 1(b) of the main text. We consider an atom optically excited via two linearly polarized light fields, one -polarized, the other -polarized, traveling along and direction, respectively. To have such a pair of light fields, a beam splitter is placed along the axis as shown in Fig. 6. For each incoming light field, the following optical devices are used, (), beam splitter, () and (), mirror, and (), a quarter-wave plate, or a wave plate whose thickness makes it effectively a combination of a half wave-plate and a quarter-wave plate, depending on the specific low-lying intermediate P state. By tuning the positions of the two mirrors so that they are symmetric to each other about the plane of the beam splitter, we have
where is the distance between and , where . Here denotes the position of the atom. Assuming that the light field impinging on the beam splitter is , one can show that the electric field on the atom is:
In order to excite the the Rydberg states and defined in the main text for either atom or atom , one shall use waveplates so that and , respectively. As a result, the transition from the ground level to the Rydberg level via an intermediate level () can be realized by a two-photon Rabi process via a pair of effective left (right)-hand polarized light fields.
The non-interacting Hamiltonian for atom A (the Hamiltonian for atom B is similar) in the dipole approximation and rotating-wave approximation for the atom-field coupling is
where indicates right (left)-hand polarization, with is the energy of a specific atomic manifold, and are the central frequencies of the lasers for the lower and upper transitions, and they satisfy
The Rabi frequencies read,
where or 6, is the electric field of the laser for the lower (upper) transition in Eq. (9), and are the reduced matrix elements of the electric dipole operator obtained via the Wigner-Eckart theorem Rose (1957); Jenkins et al. (2012), with the elementary charge, and a Clebsch-Gordan coefficient Rose (1957).
Here, the column and row indices for are for quantum numbers (from left to right) and (from top to bottom), and the column and row indices for are for quantum numbers , (from left to right) and (from top to bottom).
Below, we derive the effective two-photon Rabi frequency when the two-photon detuning is zero James (2000); Brion et al. (2007); Han et al. (2013). Using the method of Ref. James (2000), we can derive a Hamiltonian for far off resonant optical driving. The method of Ref. James (2000) is essentially an adiabatic approximation. The Hamiltonian in a rotating frame and rotating-wave approximation for a three level system with basis is (the subscripts d and u denote down and up, respectively)
where is the length defined in Eq. (8) for the transition from to (from to ). Here we assume that is real, and the two-photon transition is resonant. We write the wave function as , and starting from state , one can find and
which indicates that Max and Min achieve maximum difference only when . This can guide setting up the condition in experiments. In fact, when the difference of Max and Min reaches its maximum while adjusting one of , the condition is met. When , the Hamiltonian in a rotating frame is
When the lower and upper Rabi transitions have the same Rabi frequency, the effective Rabi frequency between and is . By assuming , we can write the effective Rabi frequency as .
For the system studied in the main text, we shall identify as for the two-photon transition via 5P state of atom , while as for the two-photon transition via 6P state of atom . But since both of these two-photon transitions are resonant, and the ground states or the Rydberg states are degenerate, we immediately find if one sets a common for exciting both of the two Rydberg state and , with or .
Appendix B van der Waals interaction
Since we consider the uncommon situation of the interaction between two Rydberg levels of different principal quantum numbers, we will briefly outline a perturbation calculation here. Consider two Rb Rydberg atoms, one prepared in state , and the other in state , where . We consider the following four channels for the dipole-dipole interaction, each characterized by its energy defect Walker and Saffman (2008),
Here denote the principal quantum numbers of the pair state produced by the scattering process. These four couplings are known to be the dominant ones in our case. In the van der Waals interaction the atoms then go back to the initial levels and the magnetic quantum number of either atom can change up to unit Walker and Saffman (2008). We can separate the angular dependence of the interaction from the principal quantum numbers. Its matrix representation in the basis of is given by
where diag means a diagonal matrix with as diagonal matrix elements. In the basis of ,, , and , Eq. (13) becomes,
The van der Waals interaction strength for each channel is
We can identify two types of coefficients:
with or . Here is the radial part of the atomic wave function, and the integration about can be approximated as in Ref. Kaulakys (1995).
When , we have . Also, the two channels with energy defect and are the same. We report in Table 2 the values of or for different for two atoms at