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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 3 — Feb. 10, 2014
  • pp: 3501–3513
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Coherent all-optical control of ultracold atoms arrays in permanent magnetic traps

Ahmed Abdelrahman, Tetsuya Mukai, Hartmut Häffner, and Tim Byrnes  »View Author Affiliations


Optics Express, Vol. 22, Issue 3, pp. 3501-3513 (2014)
http://dx.doi.org/10.1364/OE.22.003501


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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

1. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]

]. 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

2. G. Wilpers, P. See, P. Gill, and A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012). [CrossRef] [PubMed]

] and single atom(s) in optical cavities [3

3. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007). [CrossRef] [PubMed]

6

6. S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012). [CrossRef] [PubMed]

]. The archetypal hybrid quantum system is cavity QED, where strong coupling is produced between light and a single atom [4

4. J. Ye, D. Vernooy, and H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999). [CrossRef]

, 7

7. P. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature 404, 365–368 (2000). [CrossRef] [PubMed]

, 8

8. M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, and E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011). [CrossRef]

]. 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

1. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]

,9

9. A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, and R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004). [CrossRef] [PubMed]

11

11. C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, and A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012). [CrossRef]

], and a Bose-Einstein condensate (BEC) to an optical cavity QED [3

3. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007). [CrossRef] [PubMed]

,12

12. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef] [PubMed]

]. 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

13. P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, and T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006). [CrossRef]

]. In particular, coherent control of two-component (or spinor) BECs on atom chips has been realized [14

14. P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, and P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009). [CrossRef]

], 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

15. M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010). [CrossRef] [PubMed]

], for quantum metrology applications [16

16. A. Sørensen, L. Duan, J. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000). [CrossRef]

]. 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

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

]. The framework has been shown to allow for several types of quantum algorithms and protocols to be possible, such as Deutsch’s algorithm [21

21. T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

] and quantum teleportation [22

22. A. Pyrkov and T. Byrnes, “Quantum teleportation of spin coherent states,” arxiv:1305.2479.

]. In [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

] 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. Sx, Sz and SzSz, where Sx,z 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

18. B. Julsgaard, A. Kozhekin, and E. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001). [CrossRef] [PubMed]

]. On the other hand, while entanglement between a BEC and an atom has been realized [19

19. M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, and G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011). [CrossRef] [PubMed]

], entanglement between two spinor BECs has not been demonstrated yet.

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

8. M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, and E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011). [CrossRef]

] 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

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

] 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
|α,β1N!(αa+βb)N|0
(1)
where 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, mF = −1〉 and |F = 2, mF = 1〉 of the 87Rb atoms, respectively [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

,20

20. A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

]. For a cold atom ensemble, the spin coherent state takes the form
|α,βi=1N(α|i+β|i).
(2)
We use the same ”double-ket” notation for the BEC and spin ensemble case as for the purposes of this paper the manipulations lead to the same results. In order to have a universal description for both the BEC and ensemble cases, we shall call Eqs. (1) and (2) a ”spin coherent qubit” (SC qubit), instead of ”BEC qubit” as introduced in [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

].

Fig. 1 Hybrid quantum processor using permanent magnetic traps and waveguides. (a) Sketch of the proposed device (not to scale) consisting of 1⃞ a substrate of permanent magnetic material, 2⃞ reflective coating on the edges, 3⃞ silica waveguides (vertical) for delivering the control/probe photons, 4⃞ an optical microcavity etched into a 5⃞ silica transparent substrate, 6⃞ a joint silica waveguide (horizontal) for transferring photons between nodes, 7⃞ a thermal phase-shifter and 8⃞ a micropattern into the magnetic material for creating the trapping magnetic fields. (b) Density plot of the simulated magnetic field local minima combined with a cross section of the optical microcavity and the silica waveguide.

The general idea of [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

] is that Eq. (1) can be used in place of standard qubits and controlled in an analogous way. It was shown in [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

] 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
Sx=ab+ba
(3)
Sz=aabb.
(4)
For the ensemble case the corresponding operations are
Sx=i=1Nσix
(5)
Sz=i=1Nσiz,
(6)
where σix,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 SjzSjz, 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
|α,β|0,1.
(7)
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
Pk=|kk|
(8)
with the number state basis as
|k=(a)k(b)Nk|0/k!(Nk)!.
(9)
For an ensemble of atoms, the number state basis reads
|k=i=1N|σi.
(10)
where σi = ↑, ↓ and k is a label running from 1 to 2N denoting the spin configuration. Futher details on the use of SC qubits for quantum information processing may be found in Refs. [17

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

, 20

20. A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

, 21

21. T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

, 23

23. T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013). [CrossRef]

].

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 87Rb is that for a two photon transition where |ΔmF| = 2, this necessarily requires a flip of the nuclear spin [24

24. A. Waxman, “Coherent manipulation of the Rubidium atom ground state,” M.Sc Thesis (Ben-Gurion University of the Negev, 2007).

]. 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
|F=1,mF=1=34|I=3/2,mI=3/2|J=1/2,mJ=1/2+12|I=3/2,mI=1/2|J=1/2,mJ=1/2|F=2,mF=1=12|I=3/2,mI=3/2|J=1/2,mJ=1/2+34|I=3/2,mI=1/2|J=1/2,mJ=1/2,
(11)
thus regardless of any manipulation of the 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 (5P1/2) these are (|F=2,mF=0±|F=1,mF=0)/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
H=ga(J++J)+gb(K++K)+A(L++L)+Δne+Δnf
(12)
where ga,b/h̄ is the Rabi frequency of the laser transitions, Δ is the energy detuning of the laser transition to the atomic transitions, and A is the hyperfine coupling. The operators are defined as for the BEC case as J+ = ea, K+ = fb, L+ = ef, na = aa, nb = bb, ne = ee, nf = ff. For the ensemble case these are defined as J+=i=1N|eiai|, K+=i=1N|fibi|, L+=i=1N|eifi|, na=i=1N|aiai|, nb=i=1N|bibi|, ne=i=1N|eiei|, nf=i=1N|fifi|.

Fig. 2 (a) Single BEC qubit control. Two lasers are applied to the transitions between ground states and the excited states with transition energies ga and gb, and detuned each by an amount Δ. Spontaneous emission from the excited states to the ground states with decay rate Γ is present. (b) Rabi oscillations between levels a and b in the presence of spontaneous emission. The effective decoherence rate exp(−Γefft) is shown as the dotted line. (c) Initialization of SC qubits from various initial conditions: I. |1, 0〉〉, II. |12,12. Parameters used are N = 1000, Δ/A = 1000, Γ/A = 0.1, ga/A = 100, gb/A = 100 in (b) and gb/A = 0 in (c). The timescale is t0 = h̄/A.

By adiabatically eliminating the intermediate e, f states creates an effective Hamiltonian
Hx=h¯Ω1Sx,
(13)
where the effective single SC qubit Rabi frequency is
h¯Ω1=2gagbAΔ2.
(14)
The use of excited states necessarily introduces an additional decoherence channel due to spontaneous decay. The effects of spontaneous emission may be modeled by the master equation
dρdt=ih¯[ρ,H]Γ2[J+Jρ2JρJ++ρJ+J]Γ2[K+Kρ2KρK++ρK+K]
(15)
Figure 2(b) shows the Rabi oscillations induced by the laser configuration starting from an initial state |1, 0〉〉 for the BEC case. The spontaneous emission causes an effective decoherence. The damping envelope of the Rabi oscillations occur at a rate of
Γeff=gagbAΓ(N+1)Δ3.
(16)
This gives a condition for experimentally controllable parameters ga,b, Δ to ensure that the damping rate should be at least as long as other decoherence timescales. For the D1 line in 87Rb, the spontaneous emission rate is Γ = 2π × 6MHz, and the hyperfine coupling is A/h̄ = 400MHz [25

25. D. A. Steck, “Rubidium 87 D Line Data,” Los Alamos National Laboratory (2001).

]. Assuming typical parameters N = 103, ga = gb = 100A, and Δ = 1000A, we obtain Ω1 = 8MHz and Γeff = 2π × 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
Hz=h¯ωzSz,
(17)
where h̄ωz = (EaEb)/2 and Ea,b are the energy levels of the logical states. For example, for 87Rb atoms the energy difference between the F = 1 and F = 2 levels gives ωz/2π = 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

30. D. Press, T. Ladd, B. Zhang, and Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008). [CrossRef] [PubMed]

]. The scheme is again the same as Fig. 2(a), but with gb = 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 〈Sz〉/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 fb. 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 Sz 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 b. Then by setting ga = 0 instead, turning on gb and counting the total number of photons, one obtains the total number of atoms 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

17. T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

]. For universal unitary operations it is sufficient to have a complete single SC qubit control (i.e. Sn, n = x, y, z), and any two BEC qubit operation. We now describe how to implement a SizSjz interaction using the experimental configuration considered in Fig. 1(a). The basic scheme is similar to that described in [20

20. A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

]. Each cavity is off-resonantly coupled to the transition between one of the logical states bi and the excited state ei of the atoms. To initiate the entanglement between two nodes i and j, an off-resonant laser for the transition biei 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 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
HQED=jG(Kj+pj+pjKj)+h¯ω0nje+h¯ωpjpj
(18)
where pi are the photon annihilation operators for each cavity, G is the cavity-atom coupling, h̄ω0 is the energy difference between the exited state ei and the ground state bi, and h̄ω is the resonant mode of the cavity. The photons may hop between the cavities through the waveguides, according to the Hamiltonian
Hc-w=νjpjpj+1eiϕj+H.c.
(19)
where ν is the cavity-waveguide hopping amplitude and ϕj is the combined phase picked up due to the length of the waveguide and the presence of adjustable phase shifters [31

31. J. Matthews, A. Politi, A. Stefanov, and J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]

, 37

37. J. Lepert, M. Trupke, M. Hartmann, M. Plenio, and E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011). [CrossRef]

]. We have assumed a convention that pj with odd j label photons within cavities, while even j label photons in waveguides. Assuming that the coupling strengths νG, and a one-dimensional configuration of cavities and waveguides, we may diagonalize the Hamiltonian Hc-w using
ck=1𝒩kjsin(πkj/2M)pj
(20)
where M is the total number of cavities, and 𝒩k is a normalization factor. For this case there is always zero energy mode 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

20. A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

] may be used to derive an effective Hamiltonian
Hzz=2h¯Ω2cosΦijSizSjz+h¯Ω2[(Siz)2+(Sjz)2]
(21)
where the two SC entangling frequency is
h¯Ω2=G2g24Δ3
(22)
and we have omitted single qubit rotation terms. Numerical estimates for the entangling frequency may be found in [20

20. A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

]. Φij is the total phase that is picked up by the photon when traveling between nodes 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 (Siz)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 thatΦij = π/2, to remove the SizSjz interaction. By noting that the Ω2 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

26. M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008). [CrossRef]

29

29. A. Abdelrahman, M. Vasiliev, K. Alameh, and P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010). [CrossRef]

]. A resulting trapping magnetic field appears at a working distance dmin in space, as shown in Fig. 1(b). The size of the patterns (square holes of size αh in this case) and their separation distance αs determine the value of dmin according to dminαπln(Bref) [29

29. A. Abdelrahman, M. Vasiliev, K. Alameh, and P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010). [CrossRef]

]. In our simulations the dimensions are set to αh = 3μm and αs = 100μm. The reference magnetic field Bref is defined as Bref = B0 (1 − eβτ) where τ is the thin film thickness, B0=μ0Mzπ, Mz is the thin film magnetization along the z-axis, and 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

29. A. Abdelrahman, M. Vasiliev, K. Alameh, and P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010). [CrossRef]

]. 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 Bx-bias = −1G. A vertical displacement is simulated in Fig. 3(b) according to the application of external field along the z-axis of magnitude Bz-bias= −1G.

Fig. 3 Numerically simulated magnetic field local minima of a single trap created at a working distance of dmin ≈ 13.5μm with αh = 3μm, αs = 100μm and τ = 2μm. (a) Density plot of a confining magnetic field with a displaced optical axis of a cavity (small red circule). The magnetic field local minima is created with no external magnetic bias fields applied. (b) For the alignment purpose, the trap is displaced along the positive direction of the x-axis by applying an external magnetic bias field along the x-axis, such that By-bias = Bz-bias = 0 and Bx-bias = −1G. (c) The location of the magnetic trap is below the optical axis of the cavity with no external magnetic bias fields. (d) The magnetic trap is displaced along the z-axis to overlap with the optical axis of the cavity at dmin ≈ 16.0μm with external magnetic bias fields applied along the z-axis at Bx-bias = By-bias = 0 and Bz-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

32. M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, and T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013). [CrossRef] [PubMed]

34

34. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, and T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011). [CrossRef]

]. The microcavities, and hence the magnetically trapped atoms, can all be connected together via UV-written silica waveguides [8

8. M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, and E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011). [CrossRef]

,35

35. S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003). [CrossRef]

,36

36. G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, and P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011). [CrossRef]

] 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).

Fig. 4 (a) Scenarios for implementing the silica microcavity where Bragg mirrors are included in the design (2). (b) Possible implementations for the hybrid quantum device with a scheme to manipulate the traveling photons using the thermal phase shifters and entangling junctions (dotted line square). The connections in (b) are not to scale where the actual physical implementation would be modified accordingly to the experiment. The red circules represent the optical micro-cavities, the solid black lines are the silica waveguides and the yellow squares represent the thermal phase shifters which are used to modify the phase of the propagating photons to exclude particular targeted SC qubit(s) from being entangled.

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

32. M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, and T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013). [CrossRef] [PubMed]

]. We note that a three-dimensional optical confinement can be created by using the other two mirror-free silica waveguides terminals [34

34. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, and T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011). [CrossRef]

].

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 ric, with i = 1, 2 labeling the two cavities. The inner mirrors have an associated reflection coefficient of riwc. Each of the cavities have a round trip phase pickup of ϕic. The silica waveguide is of length Lw and has a phase ϕw. Here ri=Ri where Ri is the reflectivity. The reflected optical fields from the first microcavity, the silica waveguide resonator and the second optical microcavity are written, respectively, as
E1rcEin=r1cr1wcexp[2iϕ1c]1r1cr1wcexp[2iϕ1c]
(23)
EwEin=r1wcr2wcexp[2iϕw]1r1wcr2wcexp[2iϕw]
(24)
E2wcEin=r2wcr2cexp[2iϕ2c]1r2wcr2cexp[2iϕ2c]
(25)
We use these ratios to define the reflection coefficients r˜1wc and eiθ1wc such that [37

37. J. Lepert, M. Trupke, M. Hartmann, M. Plenio, and E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011). [CrossRef]

]
r˜1wc=(r1wc)2+(r˜2c)22r1wcr˜2cη1+(r1wc)2(r˜2c)22r1wcr˜2cη
(26)
θ1wc=tan1(((r1wc)21)r˜2cξr1wc(1+(r˜2c)2)r˜2c((r1wc)2+1)η)
(27)
with η=cos(θ2c+2ϕw+2ϕ1c) and ξ=sin(θ2c+2ϕw+2ϕ1c). The reflection amplitudes of the total composite system can thus be written as
E1rc=r1cr˜1wcexp[i(θ1wc+2ϕ1c)]1r1wcr˜2cexp[i(θ1wc+2ϕ1c)]Ein
(28)
Choosing the length of the two microcavities to be equal L1c(μm)=L2c(μm) and the silica waveguide Lw(μm), we find that the resonance frequency for the two individual cavities is ωresc1=ωresc2=2πc/L1,2c(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 87Rb. For the whole composite system the resonance frequency is ωres = 2π × 0.0021GHz. With a beam waist of roughly (2–4μm) we determine the cavity Bragg mirror radius of curvatures such that
˜1c=˜2c=2π2ω04λ2Lc+Lc285.6μm
with ω0 chosen to be 4μm (the diameter of the silica waveguide).

Fig. 5 The reflected power of the composite cavity system, two micro-cavities connected via a single silica waveguide. The simulation input parameters are R1c=R2c=0.985, R1wc=0.999, R2wc=0.9 with both micro-cavities at equal lengths L1,2c=30μm and the silica waveguide with a length of Lw = 4mm.

We estimate the coupling rate between the two-level N atoms and the composite cavity-waveguide system to be
gcwN3cλ2κπ2ω02(L1c+L2c+Lw)
(29)
which is of the order of ~2πN×3GHz, the coupling rate gcw is much greater than the decay rate κ = 2π × 6MHz, for 87Rb. The amplitude decay rate γc/w for the cavity and the waveguide are calculated independently such that γi=χiti with χi2iRi2 and ti is the time of the photon round trip ti=2Lic with i is the cavity and the waveguide index. For a cavity of length 40μm with R1c=99.97%, R1wc=85.0% we find that the decay rate at cavity (1) is relatively small γc1 ∼ 2π × 0.028GHz which we will also consider to be equal to the decay rate γc2 of the second cavity. For the silica waveguide of optical length nsiLw (nsi is the fused silica refractive index) we calculate the decay rate with R1wc=R2wc, R2c=R1c and Lw = 100μm such that γw ∼ 2π × 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 R1c=R2c=0.985, R1wc=0.999, R2wc=0.9 with both micro-cavities at equal lengths L1,2c=30μm and the silica waveguide with a length of Lw = 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

37. J. Lepert, M. Trupke, M. Hartmann, M. Plenio, and E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011). [CrossRef]

].

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

6. S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012). [CrossRef] [PubMed]

, 38

38. M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

]. 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.

References and links

1.

M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]

2.

G. Wilpers, P. See, P. Gill, and A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012). [CrossRef] [PubMed]

3.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007). [CrossRef] [PubMed]

4.

J. Ye, D. Vernooy, and H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999). [CrossRef]

5.

A. Boozer, A. Boca, R. Miller, T. Northup, and H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006). [CrossRef] [PubMed]

6.

S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012). [CrossRef] [PubMed]

7.

P. Pinkse, T. Fischer, P. Maunz, and G. Rempe, “Trapping an atom with single photons,” Nature 404, 365–368 (2000). [CrossRef] [PubMed]

8.

M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, and E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011). [CrossRef]

9.

A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, and R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004). [CrossRef] [PubMed]

10.

D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, and R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010). [CrossRef]

11.

C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, and A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012). [CrossRef]

12.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef] [PubMed]

13.

P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, and T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006). [CrossRef]

14.

P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, and P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009). [CrossRef]

15.

M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, and P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010). [CrossRef] [PubMed]

16.

A. Sørensen, L. Duan, J. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000). [CrossRef]

17.

T. Byrnes, K. Wen, and Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]

18.

B. Julsgaard, A. Kozhekin, and E. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001). [CrossRef] [PubMed]

19.

M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, and G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011). [CrossRef] [PubMed]

20.

A. Pyrkov and T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]

21.

T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).

22.

A. Pyrkov and T. Byrnes, “Quantum teleportation of spin coherent states,” arxiv:1305.2479.

23.

T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013). [CrossRef]

24.

A. Waxman, “Coherent manipulation of the Rubidium atom ground state,” M.Sc Thesis (Ben-Gurion University of the Negev, 2007).

25.

D. A. Steck, “Rubidium 87 D Line Data,” Los Alamos National Laboratory (2001).

26.

M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, and P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008). [CrossRef]

27.

T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, and R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008). [CrossRef]

28.

S. Whitlock, R. Gerritsma, T. Fernholz, and R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009). [CrossRef]

29.

A. Abdelrahman, M. Vasiliev, K. Alameh, and P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010). [CrossRef]

30.

D. Press, T. Ladd, B. Zhang, and Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008). [CrossRef] [PubMed]

31.

J. Matthews, A. Politi, A. Stefanov, and J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]

32.

M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, and T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013). [CrossRef] [PubMed]

33.

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, and T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012). [CrossRef]

34.

M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, and T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011). [CrossRef]

35.

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003). [CrossRef]

36.

G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, and P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011). [CrossRef]

37.

J. Lepert, M. Trupke, M. Hartmann, M. Plenio, and E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011). [CrossRef]

38.

M. Hijlkema, B. Weber, H. Specht, S. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nature 3, 253–255 (2007).

OCIS Codes
(020.1475) Atomic and molecular physics : Bose-Einstein condensates
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: October 17, 2013
Revised Manuscript: December 9, 2013
Manuscript Accepted: December 10, 2013
Published: February 6, 2014

Citation
Ahmed Abdelrahman, Tetsuya Mukai, Hartmut Häffner, and Tim Byrnes, "Coherent all-optical control of ultracold atoms arrays in permanent magnetic traps," Opt. Express 22, 3501-3513 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3501


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References

  1. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]
  2. G. Wilpers, P. See, P. Gill, A. Sinclair, “A monolithic array of three-dimensional ion traps fabricated with conventional semiconductor technology,” Nat. Nanotechnol. 7, 572–576 (2012). [CrossRef] [PubMed]
  3. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007). [CrossRef] [PubMed]
  4. J. Ye, D. Vernooy, H. Kimble, “Trapping of single atoms in cavity QED,” Phys. Rev. Lett. 83, 4987–4990 (1999). [CrossRef]
  5. A. Boozer, A. Boca, R. Miller, T. Northup, H. Kimble, “Cooling to the ground state of axial motion for one atom strongly coupled to an optical cavity,” Phys. Rev. Lett. 97, 083602 (2006). [CrossRef] [PubMed]
  6. S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, G. Rempe, “An elementary quantum network of single atoms in optical cavities,” Nature 484, 195–200 (2012). [CrossRef] [PubMed]
  7. P. Pinkse, T. Fischer, P. Maunz, G. Rempe, “Trapping an atom with single photons,” Nature 404, 365–368 (2000). [CrossRef] [PubMed]
  8. M. Kohnen, M. Succo, P. Petrov, R. Nyman, M. Trupke, E. Hinds, “An array of integrated atom-photon junctions,” Nat. Photonics 5, 35–38 (2011). [CrossRef]
  9. A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin, R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004). [CrossRef] [PubMed]
  10. D. Schuster, A. Sears, E. Ginossar, L. DiCarlo, L. Frunzio, J. Morton, H. Wu, G. Briggs, B. Buckley, D. Awschalom, R. Schoelkopf, “High-cooperativity coupling of electron-spin ensembles to superconducting cavities,” Phys. Rev. Lett. 105, 140501 (2010). [CrossRef]
  11. C. Eichler, C. Lang, J. Fink, J. Govenius, S. Filipp, A. Wallraff, “Observation of entanglement between itinerant microwave photons and a superconducting qubit,” Phys. Rev. Lett. 109, 240501 (2012). [CrossRef]
  12. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef] [PubMed]
  13. P. Treutlein, T. Steinmetz, Y. Colombe, B. Lev, P. Hommelhoff, J. Reichel, M. Greiner, O. Mandel, A. Widera, T. Rom, I. Bloch, T. Hänsch, “Quantum information processing optical lattices and magnetic microtraps,” Fortschr. Phys. 54, 702 (2006). [CrossRef]
  14. P. Böhi, M. Riedel, J. Hoffrogge, J. Reichel, T. Haensch, P. Treutlein, “Coherent manipulation of Bose-Einstein condensates with state-dependent microwave potentials on an atom chip,” Nat. Phys. 5, 592–597 (2009). [CrossRef]
  15. M. Riedel, P. Böhi, Y. Li, T. Hänsch, A. Sinatra, P. Treutlein, “Atom-chip-based generation of entanglement for quantum metrology,” Nature 464, 1170–1173 (2010). [CrossRef] [PubMed]
  16. A. Sørensen, L. Duan, J. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature 409, 63–66 (2000). [CrossRef]
  17. T. Byrnes, K. Wen, Y. Yamamoto, “Macroscopic quantum computation using Bose-Einstein condensates,” Phys. Rev. A 85, 040306 (2012). [CrossRef]
  18. B. Julsgaard, A. Kozhekin, E. Polzik, “Experimental long-lived entanglement of two macroscopic objects,” Nature 413, 400–403 (2001). [CrossRef] [PubMed]
  19. M. Lettner, M. Mücke, S. Riedl, C. Vo, C. Hahn, S. Baur, J. Bochmann, S. Ritter, S. Dürr, G. Rempe, “Remote entanglement between a single atom and a Bose-Einstein condensate,” Phys. Rev. Lett. 106, 210503 (2011). [CrossRef] [PubMed]
  20. A. Pyrkov, T. Byrnes, “Entanglement generation in quantum networks of Bose-Einstein condensates,” New J. Phys. 15093019 (2013). [CrossRef]
  21. T. Byrnes, “Quantum computation using two component Bose-Einstein condensates,” World Acad. Sci. Eng. Technol. 63, 542 (2012).
  22. A. Pyrkov, T. Byrnes, “Quantum teleportation of spin coherent states,” arxiv:1305.2479.
  23. T. Byrnes, “Fractality and macroscopic entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 88, 023609 (2013). [CrossRef]
  24. A. Waxman, “Coherent manipulation of the Rubidium atom ground state,” M.Sc Thesis (Ben-Gurion University of the Negev, 2007).
  25. D. A. Steck, “Rubidium 87 D Line Data,” Los Alamos National Laboratory (2001).
  26. M. Singh, M. Volk, A. Akulshin, A. Sidorov, R. McLean, P. Hannaford, “One dimensional lattice of permanent magnetic microtraps for ultracold atoms on an atom chip,” J. Phys. B At. Mol. Opt. Phys. 41, 065301 (2008). [CrossRef]
  27. T. Fernholz, R. Gerritsma, S. Whitlock, I. Barb, R. Spreeuw, “Fully permanent magnet atom chip for Bose-Einstein condensation,” Phys. Rev. A 77, 033409 (2008). [CrossRef]
  28. S. Whitlock, R. Gerritsma, T. Fernholz, R. Spreeuw, “Two-dimensional array of microtraps with atomic shift register on a chip,” New J. Phys. 11, 023021 (2009). [CrossRef]
  29. A. Abdelrahman, M. Vasiliev, K. Alameh, P. Hannford, “Asymmetrical two-dimensional magnetic lattices for ultracold atoms,” Phys. Rev. A 82, 012320 (2010). [CrossRef]
  30. D. Press, T. Ladd, B. Zhang, Y. Yamamoto, “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218–221 (2008). [CrossRef] [PubMed]
  31. J. Matthews, A. Politi, A. Stefanov, J. OBrien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nat. Photonics 3, 346–350 (2009). [CrossRef]
  32. M. Malak, N. Gaber, F. Marty, N. Pavy, E. Richalot, T. Bourouina, “Analysis of Fabry-Pérot optical micro-cavities based on coating-free all-Silicon cylindrical Bragg reflectors,” Opt. Express 21, 2378–2392 (2013). [CrossRef] [PubMed]
  33. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Cylindrical surfaces enable wavelength-selective extinction and sub-0.2 nm linewidth in 250μm-gap silicon Fabry-Perot cavities,” J. Microelectromech. Syst. 21(1), 171–180 (2012). [CrossRef]
  34. M. Malak, F. Marty, N. Pavy, Y. Peter, A. Liu, T. Bourouina, “Micromachined Fabry-Perot resonator combining submillimeter cavity length and high quality factor,” Appl. Phys. Lett. 98(21), 211113 (2011). [CrossRef]
  35. S. Nolte, M. Will, J. Burghoff, A. Tuennermann, “Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics,” Appl. Phys. A 77, 109–111 (2003). [CrossRef]
  36. G. Lepert, M. Trupke, E. Hinds, H. Rogers, J. Gates, P. Smith, “Demonstration of UV-written waveguides, Bragg gratings and cavities at 780 nm, and an original experimental measurement of group delay,” Opt. Express 19, 24933–24943 (2011). [CrossRef]
  37. J. Lepert, M. Trupke, M. Hartmann, M. Plenio, E. Hinds, “Arrays of waveguide-coupled optical cavities that interact strongly with atoms,” New J. Phys. 13, 113002 (2011). [CrossRef]
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