## Abstract

We propose a hybrid architecture for quantum information processing based on magnetically trapped ultracold atoms coupled via optical fields. The ultracold atoms, which can be either Bose-Einstein condensates or ensembles, are trapped in permanent magnetic traps and are placed in microcavities, connected by silica based waveguides on an atom chip structure. At each trapping center, the ultracold atoms form spin coherent states, serving as a quantum memory. An all-optical scheme is used to initialize, measure and perform a universal set of quantum gates on the single and two spin-coherent states where entanglement can be generated addressably between spatially separated trapped ultracold atoms. This allows for universal quantum operations on the spin coherent state quantum memories. We give detailed derivations of the composite cavity system mediated by a silica waveguide as well as the control scheme. Estimates for the necessary experimental conditions for a working hybrid device are given.

© 2014 Optical Society of America

## 1. Introduction

Hybrid quantum devices offer the possibility of creating novel technologies that take advantage of the most attractive aspects of the various quantum systems, by merging them into a single device [1]. For example, photonic based systems can bridge large distances with low decoherence, making them ideal for quantum communication tasks. On the other hand, matter qubits are the natural choice if the quantum information needs to be stored for a given period of time. Various systems have demonstrated the concepts mentioned above, such as ions in electric microtraps [2] and single atom(s) in optical cavities [3–6]. The archetypal hybrid quantum system is cavity QED, where strong coupling is produced between light and a single atom [4, 7, 8]. Recently there has been a large effort to realize strong coupling of different systems, for example superconducting two-level systems/single NV centers to a microwave cavity QED [1,9–11], and a Bose-Einstein condensate (BEC) to an optical cavity QED [3,12]. This allows for the possibility of creating quantum communication channels between macroscopic quantum systems, serving as quantum memories. While steps have been put forward to create different integrated hybrid superconducting platforms, no such architecture has been demonstrated or proposed as yet for BECs.

Ultracold atoms offer the possibility of realizing robust quantum memories due to their low decoherence rates and controllability [13]. In particular, coherent control of two-component (or spinor) BECs on atom chips has been realized [14], allowing for the possibility of producing many quantum memories of BEC type on the same compact device. Such coherent control has been extended to producing spin squeezing [15], for quantum metrology applications [16]. Using such macroscopic ”BEC qubits” or in more general spin coherent state qubits (SC qubits) have been shown to be a potential system for realizing quantum computation [17]. The framework has been shown to allow for several types of quantum algorithms and protocols to be possible, such as Deutsch’s algorithm [21] and quantum teleportation [22]. In [17] it was shown that in a similar way to standard qubits, the minimal requirement for realizing universal quantum operations is the presence of single and two BEC qubit operations, e.g. *S ^{x}*,

*S*and

^{z}*S*, where

^{z}S^{z}*S*denotes the collective spin operators. Entanglement between two atomic ensembles have been realized in the form of two-mode squeezing such as in the experiments of Polzik and co-workers [18]. On the other hand, while entanglement between a BEC and an atom has been realized [19], entanglement between two spinor BECs has not been demonstrated yet.

^{x,z}In this paper we propose an integrated hybrid device to accommodate the control and entanglement of an array of ultracold atoms. The atoms are confined in their BEC states using permanent magnetic traps which are integrated at the vicinity of optical microcavities. The microcavities (the nodes) are connected by silica waveguides for a direct optical access to the atoms [8] and for establishing the optical communication between the trapped ultracold atoms. Entanglement can be initiated among selected nodes whenever a control pulse is delivered to the targeted node(s). We provide concrete methods for producing the single BEC qubit control, initialization, and measurement.

## 2. Proposed device and spin coherent state quantum computation

We first give a brief overview of the device and the types of manipulations that are required in order to realize the quantum processor. The proposed hybrid device sketched in Fig. 1(a) consists of optical microcavities connected via silica waveguides and fabricated on the top of a patterned permanent magnetic thin film. The magnetic thin film is patterned such that at each node shown in the structure there is a trapping center, which allows for a large number (*N* ∼ 1000) of cold atoms to be confined. The atoms may either be an ensemble of cold atoms or a BEC, which may be achieved by standard methods such as laser and evaporative cooling. We use the framework described in [17] to store and manipulate quantum information on the ultracold atoms. Qubit information is stored as a spin coherent state where for a BEC case such state takes the form

*a*and

*b*are the bosonic operators associated with the logical qubit states,

*N*is the number of atoms in the BEC, and

*α*,

*β*are arbitrary coefficients such that |

*α*|

^{2}+ |

*β*|

^{2}= 1. In the following, we asume the logical states are the hyperfine states |

*F*= 1,

*m*= −1〉 and |

_{F}*F*= 2,

*m*= 1〉 of the

_{F}^{87}Rb atoms, respectively [17,20]. For a cold atom ensemble, the spin coherent state takes the form

The general idea of [17] is that Eq. (1) can be used in place of standard qubits and controlled in an analogous way. It was shown in [17] that for universal operations of SC qubits above, single collective spin operations and entangling operations are required. For the BEC case this corresponds to the possibility of performing the Hamiltonians

For the ensemble case the corresponding operations are where ${\sigma}_{i}^{x,z}$ are the Pauli matrices and the*i*index runs over all the atoms in one trapping site. As an entangling gate we propose the operaton ${S}_{j}^{z}{S}_{{j}^{\prime}}^{z}$, where

*j*and

*j′*label two distinct trapping centers. To this end, initialization of the SC qubits is required, which can be considered to irreversibly take any state to a known state. We shall consider the irreversible process If in a quantum algorithm a different initial state is required, a simple unitary rotation of Eq. (7) can then in turn prepare any state. Finally, the readout of the state is required, such as the projective measurement with the number state basis as

*σ*= ↑, ↓ and

_{i}*k*is a label running from 1 to 2

*denoting the spin configuration. Futher details on the use of SC qubits for quantum information processing may be found in Refs. [17, 20, 21, 23].*

^{N}We shall show in the following section that all these operations may be performed optically, using the hybrid architecture shown in Fig. 1(a), thus realizing an architecture for universal quantum computing. Details of the experimental design will be discussed in section 4.

## 3. All-optical control

#### 3.1. Single SC qubit control

All-optical single SC qubit control may be achieved by performing an Raman transition through an excited state. One difficulty with using a standard three level Raman scheme with hyperfine states of ^{87}Rb is that for a two photon transition where |Δ*m _{F}*| = 2, this necessarily requires a flip of the nuclear spin [24]. However, the optical fields only change the state of the electrons. Specifically, the hyperfine states used as the logical states can be written in terms of the electron angular momentum

*J*and nulcear angular momentum

*I*

*J*-states, these states remain orthogonal. In order to complete the transition, the natural hyperfine coupling is required to complete the transition.

The Raman passage that is then relevant to hyperfine ground state manipulation is shown in Fig. 2(a). Two off-resonant lasers detuned from the ground states (denoted as before by annihilation operators *a* and *b*) excite the states *e* and *f*. For the D1 line of Rubidium (5P_{1/2}) these are
$\left(|{F}^{\prime}=2,{m}_{F}=0\u3009\pm |{F}^{\prime}=1,{m}_{F}=0\u3009\right)/\sqrt{2}$ respectively, according to the selection rules of the *σ*^{±} transition. The two intermediate states are connected by a transition element determined by the hyperfine interaction. The Hamiltonian

*g*is the Rabi frequency of the laser transitions, Δ is the energy detuning of the laser transition to the atomic transitions, and

_{a,b}/h̄*A*is the hyperfine coupling. The operators are defined as for the BEC case as

*J*

^{+}=

*e*

^{†}

*a*,

*K*

^{+}=

*f*

^{†}

*b*,

*L*

^{+}=

*e*

^{†}

*f*,

*n*=

^{a}*a*

^{†}

*a*,

*n*=

^{b}*b*

^{†}

*b*,

*n*=

^{e}*e*

^{†}

*e*,

*n*=

^{f}*f*

^{†}

*f*. For the ensemble case these are defined as ${J}^{+}={\sum}_{i=1}^{N}|{e}_{i}\u3009\u3008{a}_{i}|$, ${K}^{+}={\sum}_{i=1}^{N}|{f}_{i}\u3009\u3008{b}_{i}|$, ${L}^{+}={\sum}_{i=1}^{N}|{e}_{i}\u3009\u3008{f}_{i}|$, ${n}^{a}={\sum}_{i=1}^{N}|{a}_{i}\u3009\u3008{a}_{i}|$, ${n}^{b}={\sum}_{i=1}^{N}|{b}_{i}\u3009\u3008{b}_{i}|$, ${n}^{e}={\sum}_{i=1}^{N}|{e}_{i}\u3009\u3008{e}_{i}|$, ${n}^{f}={\sum}_{i=1}^{N}|{f}_{i}\u3009\u3008{f}_{i}|$.

By adiabatically eliminating the intermediate *e*, *f* states creates an effective Hamiltonian

*g*, Δ to ensure that the damping rate should be at least as long as other decoherence timescales. For the D1 line in

_{a,b}^{87}Rb, the spontaneous emission rate is Γ = 2

*π*× 6MHz, and the hyperfine coupling is

*A/h̄*= 400MHz [25]. Assuming typical parameters

*N*= 10

^{3},

*g*=

_{a}*g*= 100

_{b}*A*, and Δ = 1000

*A*, we obtain Ω

_{1}= 8MHz and Γ

_{eff}= 2

*π*× 60kHz, allowing for many coherent oscillations during the effective decoherence.

For full qubit control, rotation around another axis of the Bloch sphere is required. This is realized by the natural energy difference between the states used to hold the logical states, and in terms of the logical operators is

where*h̄ω*= (

^{z}*E*−

_{a}*E*)/2 and

_{b}*E*are the energy levels of the logical states. For example, for

_{a,b}^{87}Rb atoms the energy difference between the

*F*= 1 and

*F*= 2 levels gives

*ω*/2

^{z}*π*= 3.4GHz.

#### 3.2. Initialization and measurement

Initialization can be performed by directly driving one of the transitions and taking advantage of the irreversible spontaneous emission [30]. The scheme is again the same as Fig. 2(a), but with *g _{b}* = 0 and the detuning is Δ = 0. By application of only one branch of the Λ system, this forces all states towards the state |0, 1〉〉, since an atom in level

*a*efficiently transfered to level

*e*via the laser, from which it may decay into level

*b*via spontaneous emission. After decay into level

*b*it is trapped there. In Fig. 2(c) we plot the state from two different initial conditions by evolving (15). We see that in all cases the population evolves towards 〈

*S*〉/

^{z}*N*= −1, corresponding to the state |0, 1〉〉. Measurement is performed by the same process. Spontaneous emission causes an emission of photons due to the decay process between the levels

*f*↔

*b*. Every detected photon arises due to the presence of an atom in level

*a*, thus by counting the number of photons one may obtain a measurement in the

*S*basis of Eq. (9). To obtain expectation values, the total number of atoms is also needed, which would be obtained in an initial calibration step, where initially all the atoms are driven into the level

^{z}*b*. Then by setting

*g*= 0 instead, turning on

_{a}*g*and counting the total number of photons, one obtains the total number of atoms

_{b}*N*.

#### 3.3. Two SC qubit entanglement

An arbitrary unitary operation, as would be necessary for a general quantum algorithm, can be decomposed into single and two qubit gates. The analogous result holds true in the case of SC qubits [17]. For universal unitary operations it is sufficient to have a complete single SC qubit control (i.e. *S ^{n}*,

*n*=

*x*,

*y*,

*z*), and any two BEC qubit operation. We now describe how to implement a ${S}_{i}^{z}{S}_{j}^{z}$ interaction using the experimental configuration considered in Fig. 1(a). The basic scheme is similar to that described in [20]. Each cavity is off-resonantly coupled to the transition between one of the logical states

*b*and the excited state

_{i}*e*of the atoms. To initiate the entanglement between two nodes

_{i}*i*and

*j*, an off-resonant laser for the transition

*b*↔

_{i}*e*is delivered through the control waveguide, labeled by 3⃞ in Fig. 1(a). Entanglement is generated by the process of photon emission from node

_{i}*i*and absorption by node

*j*, by traveling through the silica waveguide labeled by 6⃞, or vice versa. For nodes without the laser illumination, the photon does not get absorbed since they are off-resonant of the transition to the excited state. The Hamiltonian describing the system is given by

*p*are the photon annihilation operators for each cavity,

_{i}*G*is the cavity-atom coupling,

*h̄ω*

_{0}is the energy difference between the exited state

*e*and the ground state

_{i}*b*, and

_{i}*h̄ω*is the resonant mode of the cavity. The photons may hop between the cavities through the waveguides, according to the Hamiltonian where

*ν*is the cavity-waveguide hopping amplitude and

*ϕ*is the combined phase picked up due to the length of the waveguide and the presence of adjustable phase shifters [31, 37]. We have assumed a convention that

_{j}*p*with odd

_{j}*j*label photons within cavities, while even

*j*label photons in waveguides. Assuming that the coupling strengths

*ν*≫

*G*, and a one-dimensional configuration of cavities and waveguides, we may diagonalize the Hamiltonian

*H*

_{c-w}using where

*M*is the total number of cavities, and

*𝒩*is a normalization factor. For this case there is always zero energy mode

_{k}*k*=

*M*which has the same energy as the original cavity resonance. This mode is used as the common mode connecting all the SC qubits to each other, with all other modes being off-resonant and do not contribute to the operation. From here the same derivation as [20] may be used to derive an effective Hamiltonian

*is the total phase that is picked up by the photon when traveling between nodes*

_{ij}*i*and

*j*. Equation (21) shows that the two qubit interactions can be produced. However, as by product we have also created unwanted effective self-interaction terms ${\left({S}_{i}^{z}\right)}^{2}$. These may however be canceled out by implementing a two step process: first, Eq. (21) is applied in order to create the entangling Hamiltonian. Then the phase shifters are adjusted such thatΦ

*=*

_{ij}*π*/2, to remove the ${S}_{i}^{z}{S}_{j}^{z}$ interaction. By noting that the

*h̄*Ω

_{2}has an odd parity with Δ, we may apply a second interaction but with a reverse detuning −Δ, which removes the unwanted self-interaction terms.

## 4. Experimental design

In this section we describe the basic components of the proposed hybrid quantum device. Two different technologies are combined together to facilitate such processing unit; the atomic BEC states (the SC qubits) are created using permanent magnetic traps and the coupling between the atoms and the driving optical fields (write/read/probe lasers) can be enhanced using optical microcavities which are fabricated along side with magnetic traps. The optical communications between the trapped BECs are established via silica waveguides which are fabricated as joint optical wires between the microcavities.

#### 4.1. The permanent magnetic traps

The traps for the atoms are created by milling micropatterns through a thin film of a permanently magnetized material [26–29]. A resulting trapping magnetic field appears at a working distance *d*_{min} in space, as shown in Fig. 1(b). The size of the patterns (square holes of size *α _{h}* in this case) and their separation distance

*α*determine the value of

_{s}*d*

_{min}according to ${d}_{\text{min}}\approx \frac{\alpha}{\pi}\text{ln}\left({B}_{\text{ref}}\right)$ [29]. In our simulations the dimensions are set to

*α*= 3

_{h}*μm*and

*α*= 100

_{s}*μm*. The reference magnetic field

*B*

_{ref}is defined as

*B*

_{ref}=

*B*

_{0}(1 −

*e*

^{−}

*) where*

^{βτ}*τ*is the thin film thickness, ${B}_{0}=\frac{{\mu}_{0}{M}_{z}}{\pi}$,

*M*is the thin film magnetization along the z-axis, and

_{z}*β*=

*π/α*. Due to their spherical quadruple nature, these particular types of magnetic traps produce zero magnetic field minimum where to elevate the minimum value of the trapping magnetic field away from zero external magnetic bias fields are often used, hence avoiding the Majorana spin flip.

Coupling between the atoms and the optical field will occur whenever the positions of the magnetic traps and the optical axes of the cavities are properly aligned. To precisely align the magnetic trap within the center of the cavity an external magnetic bias field must be applied where its source can also be fabricated on chip such as using an independent coil for each trap [29]. The numerical simulation results of Fig. 3 show a displaced magnetic trap along the *x*-axis by applying a bias field along the *x*-axis of *B _{x}*

_{-bias}= −1G. A vertical displacement is simulated in Fig. 3(b) according to the application of external field along the

*z*-axis of magnitude

*B*

_{z}_{-bias}= −1G.

#### 4.2. Atoms-optical fields strong coupling

The proposed hybrid quantum interface assumes strong coupling of the magnetically trapped atomic BECs to optical fields as well as maintaining an efficient optical delivery between the SC qubits whenever an effective quantum bus is established. To accommodate strong coupling optical cavities are required; in this section we describe a convenient method for fabricating the optical micro-cavities which will be combined with the permanent magnetic traps for creating the hybrid system.

For creating the optical microcavity we consider coating-free high-Q Bragg cylindrical reflectors. They are coating free because the mirrors can be fabricated within the silica substrate with no reflective coating process [32–34]. The microcavities, and hence the magnetically trapped atoms, can all be connected together via UV-written silica waveguides [8,35,36] where an efficient connectivity between the optical micro-cavities and the silica waveguides can be achieved by considering one of the two configurations depicted in Fig. 4(a). This configuration allows several microcavities to be connected via waveguides as shown in Fig. 4(b).

In Fig. 4(a)(2), Bragg mirrors are to be fabricated with an air gap included between the end of the waveguide and the mirror so as to avoid any possible surface roughness that may occur during the fabrication process. The number of Bragg mirrors determines the resonator finesse [32]. We note that a three-dimensional optical confinement can be created by using the other two mirror-free silica waveguides terminals [34].

We simulate the case of Fig. 1(a), a system of two microcavities connected via a silica waveguide.The reflection coefficients of the outer mirrors not connected to the waveguide are
${r}_{i}^{c}$, with *i* = 1, 2 labeling the two cavities. The inner mirrors have an associated reflection coefficient of
${r}_{i}^{wc}$. Each of the cavities have a round trip phase pickup of
${\varphi}_{i}^{c}$. The silica waveguide is of length *L ^{w}* and has a phase

*ϕ*. Here ${r}_{i}=\sqrt{{R}_{i}}$ where

^{w}*R*is the reflectivity. The reflected optical fields from the first microcavity, the silica waveguide resonator and the second optical microcavity are written, respectively, as

_{i}*L*(

^{w}*μm*), we find that the resonance frequency for the two individual cavities is ${\omega}_{\mathit{res}}^{c1}={\omega}_{\mathit{res}}^{c2}=2\pi c/{L}_{1,2}^{c}(\text{GHz})$ with

*c*the speed of light (Fig. 5 shows the simulation of the composite cavity, as detailed below), assuming parameters for the D2 line of

^{87}Rb. For the whole composite system the resonance frequency is

*ω*= 2

_{res}*π*× 0.0021GHz. With a beam waist of roughly (2–4

*μm*) we determine the cavity Bragg mirror radius of curvatures such that

*ω*

_{0}chosen to be 4

*μm*(the diameter of the silica waveguide).

We estimate the coupling rate between the two-level *N* atoms and the composite cavity-waveguide system to be

*g*is much greater than the decay rate

^{cw}*κ*= 2

*π*× 6MHz, for

^{87}Rb. The amplitude decay rate

*γ*for the cavity and the waveguide are calculated independently such that ${\gamma}_{i}=\frac{{\chi}_{i}}{{t}_{i}}$ with ${\chi}_{i}\equiv \frac{2-{\sum}_{i}{R}_{i}}{2}$ and

_{c/w}*t*is the time of the photon round trip ${t}_{i}=\frac{2{L}_{i}}{c}$ with

_{i}*i*is the cavity and the waveguide index. For a cavity of length 40

*μm*with ${R}_{1}^{c}=99.97\%$, ${R}_{1}^{wc}=85.0\%$ we find that the decay rate at cavity (1) is relatively small

*γ*

_{c}_{1}∼ 2

*π*× 0.028GHz which we will also consider to be equal to the decay rate

*γ*

_{c}_{2}of the second cavity. For the silica waveguide of optical length

*n*(

_{si}L^{w}*n*is the fused silica refractive index) we calculate the decay rate with ${R}_{1}^{wc}={R}_{2}^{wc}$, ${R}_{2}^{c}={R}_{1}^{c}$ and

_{si}*L*= 100

^{w}*μm*such that

*γ*∼ 2

_{w}*π*× 0.0077GHz. Fig. 5 shows the reflected intensity of a composite cavity system (two micro-cavities mediated by a single silica waveguide). The simulation input parameters are ${R}_{1}^{c}={R}_{2}^{c}=0.985$, ${R}_{1}^{wc}=0.999$, ${R}_{2}^{wc}=0.9$ with both micro-cavities at equal lengths ${L}_{1,2}^{c}=30\mu m$ and the silica waveguide with a length of

*L*= 4mm. Both cavities are at resonance and dips are symmetrically distributed around the zero-resonance with first two dips at the normal modes of the composite cavity system [37].

^{w}## 5. Summary and conclusions

An integrated architecture for quantum information processing was proposed based on the interaction of magnetically trapped ultracold atoms with external optical fields confined in micro-cavity QEDs. The proposed hybrid quantum device can be directly used to store and manipulate quantum information stored on SC qubits. Permanent magnetic traps are proposed here to trap the atoms which have the advantage of negligible technical noise and minimal decoherence rates on the trapped BECs. The hybrid design allows for the efficient delivery of optical fields for control, initialization, and measurement to the magnetically trapped atoms through the optical waveguide. Entanglement between trapped BECs in spatially separated cavities can be created on-demand via a common optical mode induced by the coupled cavities and waveguides. The magnetic traps can be spatially controlled using this architecture, and is also compatible with not only ensembles or BECs of atoms but also single atoms [6, 38]. The controllable nature of the permanent magnetic traps suggests future applications where they are integrated with photonic circuits for control at the single atom and photon level.

## Acknowledgments

This work is supported by the Okawa foundation and the Transdisciplinary Research Integration Center and Center for the Promotion of Integrated Sciences (CPIS) of Sokendai.

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