Coherent all-optical control of ultracold atoms arrays in permanent magnetic traps Ahmed Abdelrahman,1,3,∗ Tetsuya Mukai,2 Hartmut H¨affner,1 and Tim Byrnes3,4 1 Department
of Physics, University of California, Berkeley, California 94720, USA Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan 3 National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 4 The Graduate University for Advanced Studies (Sokendai), Shonan Village, Hayama, Kanagawa 240-0193, Japan 2 NTT
∗
[email protected]
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 OCIS codes: (020.1475) Bose-Einstein condensates; (270.5585) Quantum information and processing.
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#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3501
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#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3502
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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. Sx , Sz and Sz Sz , 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]. 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. 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3503
Bmin(G)
Bmax(G)
Lc(μm)
Lw(μm)
Silica Waveguide
dmin(μm) 8
7
6
5
4
(b)
(a)
Fig. 1. Hybrid quantum processor using permanent magnetic traps and waveguides. (a) Sketch of the proposed device (not to scale) consisting of 1 a substrate of permanent magnetic material, 2 reflective coating on the edges, 3 silica waveguides (vertical) for delivering the control/probe photons, 4 an optical microcavity etched into a 5 silica transparent substrate, 6 a joint silica waveguide (horizontal) for transferring photons between nodes, 7 a thermal phase-shifter and 8 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.
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 N 1 † α a + β b† |0i (1) |α , β ii ≡ √ N!
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 = −1i and |F = 2, mF = 1i of the 87 Rb atoms, respectively [17,20]. For a cold atom ensemble, the spin coherent state takes the form N |α , β ii ≡ ∏ α | ↑ii + β | ↓ii . (2) i=1
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]. 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3504
corresponds to the possibility of performing the Hamiltonians Sx S
= a† b + b† a
z
†
†
= a a − b b.
(3) (4)
For the ensemble case the corresponding operations are Sx Sz
N
=
∑ σix
(5)
∑ σiz ,
(6)
i=1 N
=
i=1
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 Szj Szj′ , 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 ii → |0, 1ii.
(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 = |kihk| (8) with the number state basis as p |ki = (a† )k (b† )N−k |0i/ k!(N − k)!.
(9)
For an ensemble of atoms, the number state basis reads N
|ki = ∏ |σi i.
(10)
i=1
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, 20, 21, 23]. 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 |∆mF | = 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3505
angular momentum J and nulcear angular momentum I r 3 |I = 3/2, mI = −3/2i|J = 1/2, mJ = 1/2i |F = 1, mF = −1i = − 4 1 + |I = 3/2, mI = −1/2i|J = 1/2, mJ = −1/2i 2 1 |I = 3/2, mI = 3/2i|J = 1/2, mJ = −1/2i |F = 2, mF = 1i = 2r 3 |I = 3/2, mI = 1/2i|J = 1/2, mJ = 1/2i, + 4
(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 = 0i ± |F ′ = 1, mF = 0i)/ 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− ) + ∆ne + ∆n f
(12)
where ga,b /¯h is the Rabi frequency of the laser transitions, ∆ 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 + = e† a, K + = f † b, L+ = e† f , na = a† a, nb = b† b, ne = e† e, n f = f † f . For the ensemble case these are defined as J + = ∑Ni=1 |ei ihai |, K + = ∑Ni=1 | fi ihbi |, L+ = ∑Ni=1 |ei ih fi |, na = ∑Ni=1 |ai ihai |, nb = ∑Ni=1 |bi ihbi |, ne = ∑Ni=1 |ei ihei |, n f = ∑Ni=1 | fi ih fi |. By adiabatically eliminating the intermediate e, f states creates an effective Hamiltonian Hx = h¯ Ω1 Sx ,
(13)
where the effective single SC qubit Rabi frequency is h¯ Ω1 =
2ga gb A . ∆2
(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 Γ i dρ Γ = [ρ , H] − J + J − ρ − 2J −ρ J + + ρ J +J − − K + K − ρ − 2K −ρ K + + ρ K +K − (15) dt h¯ 2 2
Figure 2(b) shows the Rabi oscillations induced by the laser configuration starting from an initial state |1, 0ii for the BEC case. The spontaneous emission causes an effective decoherence. The damping envelope of the Rabi oscillations occur at a rate of Γeff =
ga gb AΓ(N + 1) . ∆3
(16)
This gives a condition for experimentally controllable parameters ga,b , ∆ to ensure that the damping rate should be at least as long as other decoherence timescales. For the D1 line in 87 Rb, the spontaneous emission rate is Γ = 2π × 6MHz, and the hyperfine coupling is A/¯h = 400MHz [25]. Assuming typical parameters N = 103 , ga = gb = 100A, and ∆ = 1000A, #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3506
f
∆ ga
gb
Γ
Γ
a
1.0
0.5
0.5
Sz/N
Sz/N
e
A
1.0
0.0
-0.5 -1.0 0
b (a)
I
0.0
-0.5
1
2
3
4 3
5
-1.0 0
II
50 100 150 200
t/t 0 (x10 ) (b)
t/t 0 (c)
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 ∆. 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, 0ii, II. | √1 , √1 ii. 2 2 Parameters used are N = 1000, ∆/A = 1000, h¯ Γ/A = 0.1, ga /A = 100, gb /A = 100 in (b) and gb /A = 0 in (c). The timescale is t0 = h¯ /A.
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¯ ω z Sz , (17) where h¯ ω z = (Ea − Eb )/2 and Ea,b are the energy levels of the logical states. For example, for 87 Rb atoms the energy difference between the F = 1 and F = 2 levels gives ω z /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]. The scheme is again the same as Fig. 2(a), but with gb = 0 and the detuning is ∆ = 0. By application of only one branch of the Λ system, this forces all states towards the state |0, 1ii, 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 hSz i/N = −1, corresponding to the state |0, 1ii. 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 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3507
SC qubits [17]. 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 Siz Szj 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 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 bi ↔ ei is delivered through the control waveguide, labeled by 3 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 6 , 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 = ∑ G(K + ¯ ω0 nej + h¯ ω p†j p j j p j + p jKj ) + h
(18)
j
where pi are the photon annihilation operators for each cavity, G is the cavity-atom coupling, h¯ ω0 is the energy difference between the exited state ei and the ground state bi , and h¯ ω is the resonant mode of the cavity. The photons may hop between the cavities through the waveguides, according to the Hamiltonian Hc-w = ν ∑ p†j p j+1 eiφ j + H.c.
(19)
j
where ν is the cavity-waveguide hopping amplitude and φ j 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 p j with odd 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 Hc-w using 1 ck = √ ∑ sin(π k j/2M)p j Nk j
(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] may be used to derive an effective Hamiltonian i h Hzz = −2¯hΩ2 cos Φi j Siz Szj + h¯ Ω2 (Siz )2 + (Szj )2 (21)
where the two SC entangling frequency is G2 g 2 (22) 4∆3 and we have omitted single qubit rotation terms. Numerical estimates for the entangling frequency may be found in [20]. Φi j 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 that Φi j = π /2, to remove the Siz Szj interaction. By noting that the h¯ Ω2 has an odd parity with ∆, we may apply a second interaction but with a reverse detuning −∆, which removes the unwanted self-interaction terms. h¯ Ω2 =
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3508
(a)
(c)
(b)
(d)
Fig. 3. Numerically simulated magnetic field local minima of a single trap created at a working distance of dmin ≈ 13.5µ m with αh = 3µ m, αs = 100µ m and τ = 2µ m. (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 ≈ 16.0µ m with external magnetic bias fields applied along the z-axis at Bx-bias = By-bias = 0 and Bz-bias = -1G.
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 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 dmin ≈ απ ln(Bref ) [29]. In our simulations the dimensions are set to αh = 3µ m and αs = 100µ m. #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3509
(a) (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.
The reference magnetic field Bref is defined as Bref = B0 1 − e−β τ where τ is the thin film thickness, B0 = µ0πMz , 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]. 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. 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3510
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 ric , with i = 1, 2 labeling the two cavities. The inner mirrors have an associated reflection coefficient of c riwc . Each of the cavities have a round √ trip phase pickup of φi . The silica waveguide is of length w w L and has a phase φ . 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 E1rc Ein Ew Ein E2wc Ein
= = =
r1c − r1wc exp [2iφ1c ] 1 − r1c r1wc exp[2iφ1c ] r1wc − r2wc exp [2iφ w ] 1 − r1wc r2wc exp [2iφ w ] r2wc − r2c exp [2iφ2c ] 1 − r2wc r2c exp[2iφ2c ]
(23) (24) (25)
We use these ratios to define the reflection coefficients r˜1wc and eiθ1 such that [37] s (r1wc )2 + (˜r2c )2 − 2r1wc r˜2c η wc r˜1 = 1 + (r1wc )2 (˜r2c )2 − 2r1wc r˜2c η ! ((r1wc )2 − 1)˜r2c ξ wc −1 θ1 = tan r1wc (1 + (˜r2c )2 ) − r˜2c ((r1wc )2 + 1)η wc
(26)
(27)
with η = cos(θ2c + 2φ w + 2φ1c ) and ξ = sin(θ2c + 2φ w + 2φ1c ). The reflection amplitudes of the total composite system can thus be written as E1rc =
r1c − r˜1wc exp [i(θ1wc + 2φ1c )] Ein 1 − r1wcr˜2c exp[i(θ1wc + 2φ1c )]
(28)
Choosing the length of the two microcavities to be equal Lc1 (µ m) = Lc2 (µ m) and the silica waveguide Lw (µ m), we find that the resonance frequency for the two individual cavities is c1 = ω c2 = 2π c/Lc (GHz) with c the speed of light (Fig. 5 shows the simulation of the ωres res 1,2 composite cavity, as detailed below), assuming parameters for the D2 line of 87 Rb. For the whole composite system the resonance frequency is ωres = 2π × 0.0021GHz. With a beam waist of roughly (2-4µ m) we determine the cavity Bragg mirror radius of curvatures such that 2π 2ω 4 Lc ≈ 85.6µ m R˜ 1c = R˜ 2c = 2 c0 + 2 λ L with ω0 chosen to be 4µ m (the diameter of the silica waveguide). We estimate the coupling rate between the two-level N atoms and the composite cavitywaveguide system to be s √ 3cλ 2 κ cw N≈ (29) g π 2 ω02 (Lc1 + Lc2 + Lw ) #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3511
Fig. 5. The reflected power of the composite cavity system, two micro-cavities connected via a single silica waveguide. The simulation input parameters are Rc1 = Rc2 = 0.985, Rwc 1 c = 0.999, Rwc 2 = 0.9 with both micro-cavities at equal lengths L1,2 = 30µ m and the silica waveguide with a length of Lw = 4mm.
√ which is of the order of ∼ 2π N× 3GHz, the coupling rate gcw is much greater than the decay rate κ = 2π × 6MHz, for 87 Rb. The amplitude decay rate γc/w for the cavity and the waveguide are calculated independently such that γi = χtii with χi ≡ 2−∑2 i Ri and ti is the time
of the photon round trip ti = 2Lc i with i is the cavity and the waveguide index. For a cavity of length 40µ m with Rc1 = 99.97%, Rwc 1 = 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 nsi Lw (nsi is the fused silica wc c c w refractive index) we calculate the decay rate with Rwc 1 = R2 , R2 = R1 and L = 100 µ m 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 wc c are Rc1 = Rc2 = 0.985, Rwc 1 = 0.999, R2 = 0.9 with both micro-cavities at equal lengths L1,2 = w 30µ 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]. 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 microcavity 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3512
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.
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3513
of Physics, University of California, Berkeley, California 94720, USA Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan 3 National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan 4 The Graduate University for Advanced Studies (Sokendai), Shonan Village, Hayama, Kanagawa 240-0193, Japan 2 NTT
∗
[email protected]
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 OCIS codes: (020.1475) Bose-Einstein condensates; (270.5585) Quantum information and processing.
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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. Sx , Sz and Sz Sz , 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]. 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. 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3503
Bmin(G)
Bmax(G)
Lc(μm)
Lw(μm)
Silica Waveguide
dmin(μm) 8
7
6
5
4
(b)
(a)
Fig. 1. Hybrid quantum processor using permanent magnetic traps and waveguides. (a) Sketch of the proposed device (not to scale) consisting of 1 a substrate of permanent magnetic material, 2 reflective coating on the edges, 3 silica waveguides (vertical) for delivering the control/probe photons, 4 an optical microcavity etched into a 5 silica transparent substrate, 6 a joint silica waveguide (horizontal) for transferring photons between nodes, 7 a thermal phase-shifter and 8 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.
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 N 1 † α a + β b† |0i (1) |α , β ii ≡ √ N!
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 = −1i and |F = 2, mF = 1i of the 87 Rb atoms, respectively [17,20]. For a cold atom ensemble, the spin coherent state takes the form N |α , β ii ≡ ∏ α | ↑ii + β | ↓ii . (2) i=1
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]. 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3504
corresponds to the possibility of performing the Hamiltonians Sx S
= a† b + b† a
z
†
†
= a a − b b.
(3) (4)
For the ensemble case the corresponding operations are Sx Sz
N
=
∑ σix
(5)
∑ σiz ,
(6)
i=1 N
=
i=1
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 Szj Szj′ , 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 ii → |0, 1ii.
(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 = |kihk| (8) with the number state basis as p |ki = (a† )k (b† )N−k |0i/ k!(N − k)!.
(9)
For an ensemble of atoms, the number state basis reads N
|ki = ∏ |σi i.
(10)
i=1
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, 20, 21, 23]. 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 |∆mF | = 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3505
angular momentum J and nulcear angular momentum I r 3 |I = 3/2, mI = −3/2i|J = 1/2, mJ = 1/2i |F = 1, mF = −1i = − 4 1 + |I = 3/2, mI = −1/2i|J = 1/2, mJ = −1/2i 2 1 |I = 3/2, mI = 3/2i|J = 1/2, mJ = −1/2i |F = 2, mF = 1i = 2r 3 |I = 3/2, mI = 1/2i|J = 1/2, mJ = 1/2i, + 4
(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 = 0i ± |F ′ = 1, mF = 0i)/ 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− ) + ∆ne + ∆n f
(12)
where ga,b /¯h is the Rabi frequency of the laser transitions, ∆ 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 + = e† a, K + = f † b, L+ = e† f , na = a† a, nb = b† b, ne = e† e, n f = f † f . For the ensemble case these are defined as J + = ∑Ni=1 |ei ihai |, K + = ∑Ni=1 | fi ihbi |, L+ = ∑Ni=1 |ei ih fi |, na = ∑Ni=1 |ai ihai |, nb = ∑Ni=1 |bi ihbi |, ne = ∑Ni=1 |ei ihei |, n f = ∑Ni=1 | fi ih fi |. By adiabatically eliminating the intermediate e, f states creates an effective Hamiltonian Hx = h¯ Ω1 Sx ,
(13)
where the effective single SC qubit Rabi frequency is h¯ Ω1 =
2ga gb A . ∆2
(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 Γ i dρ Γ = [ρ , H] − J + J − ρ − 2J −ρ J + + ρ J +J − − K + K − ρ − 2K −ρ K + + ρ K +K − (15) dt h¯ 2 2
Figure 2(b) shows the Rabi oscillations induced by the laser configuration starting from an initial state |1, 0ii for the BEC case. The spontaneous emission causes an effective decoherence. The damping envelope of the Rabi oscillations occur at a rate of Γeff =
ga gb AΓ(N + 1) . ∆3
(16)
This gives a condition for experimentally controllable parameters ga,b , ∆ to ensure that the damping rate should be at least as long as other decoherence timescales. For the D1 line in 87 Rb, the spontaneous emission rate is Γ = 2π × 6MHz, and the hyperfine coupling is A/¯h = 400MHz [25]. Assuming typical parameters N = 103 , ga = gb = 100A, and ∆ = 1000A, #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3506
f
∆ ga
gb
Γ
Γ
a
1.0
0.5
0.5
Sz/N
Sz/N
e
A
1.0
0.0
-0.5 -1.0 0
b (a)
I
0.0
-0.5
1
2
3
4 3
5
-1.0 0
II
50 100 150 200
t/t 0 (x10 ) (b)
t/t 0 (c)
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 ∆. 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, 0ii, II. | √1 , √1 ii. 2 2 Parameters used are N = 1000, ∆/A = 1000, h¯ Γ/A = 0.1, ga /A = 100, gb /A = 100 in (b) and gb /A = 0 in (c). The timescale is t0 = h¯ /A.
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¯ ω z Sz , (17) where h¯ ω z = (Ea − Eb )/2 and Ea,b are the energy levels of the logical states. For example, for 87 Rb atoms the energy difference between the F = 1 and F = 2 levels gives ω z /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]. The scheme is again the same as Fig. 2(a), but with gb = 0 and the detuning is ∆ = 0. By application of only one branch of the Λ system, this forces all states towards the state |0, 1ii, 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 hSz i/N = −1, corresponding to the state |0, 1ii. 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 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3507
SC qubits [17]. 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 Siz Szj 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 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 bi ↔ ei is delivered through the control waveguide, labeled by 3 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 6 , 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 = ∑ G(K + ¯ ω0 nej + h¯ ω p†j p j j p j + p jKj ) + h
(18)
j
where pi are the photon annihilation operators for each cavity, G is the cavity-atom coupling, h¯ ω0 is the energy difference between the exited state ei and the ground state bi , and h¯ ω is the resonant mode of the cavity. The photons may hop between the cavities through the waveguides, according to the Hamiltonian Hc-w = ν ∑ p†j p j+1 eiφ j + H.c.
(19)
j
where ν is the cavity-waveguide hopping amplitude and φ j 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 p j with odd 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 Hc-w using 1 ck = √ ∑ sin(π k j/2M)p j Nk j
(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] may be used to derive an effective Hamiltonian i h Hzz = −2¯hΩ2 cos Φi j Siz Szj + h¯ Ω2 (Siz )2 + (Szj )2 (21)
where the two SC entangling frequency is G2 g 2 (22) 4∆3 and we have omitted single qubit rotation terms. Numerical estimates for the entangling frequency may be found in [20]. Φi j 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 that Φi j = π /2, to remove the Siz Szj interaction. By noting that the h¯ Ω2 has an odd parity with ∆, we may apply a second interaction but with a reverse detuning −∆, which removes the unwanted self-interaction terms. h¯ Ω2 =
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3508
(a)
(c)
(b)
(d)
Fig. 3. Numerically simulated magnetic field local minima of a single trap created at a working distance of dmin ≈ 13.5µ m with αh = 3µ m, αs = 100µ m and τ = 2µ m. (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 ≈ 16.0µ m with external magnetic bias fields applied along the z-axis at Bx-bias = By-bias = 0 and Bz-bias = -1G.
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 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 dmin ≈ απ ln(Bref ) [29]. In our simulations the dimensions are set to αh = 3µ m and αs = 100µ m. #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3509
(a) (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.
The reference magnetic field Bref is defined as Bref = B0 1 − e−β τ where τ is the thin film thickness, B0 = µ0πMz , 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]. 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. 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 #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3510
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 ric , with i = 1, 2 labeling the two cavities. The inner mirrors have an associated reflection coefficient of c riwc . Each of the cavities have a round √ trip phase pickup of φi . The silica waveguide is of length w w L and has a phase φ . 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 E1rc Ein Ew Ein E2wc Ein
= = =
r1c − r1wc exp [2iφ1c ] 1 − r1c r1wc exp[2iφ1c ] r1wc − r2wc exp [2iφ w ] 1 − r1wc r2wc exp [2iφ w ] r2wc − r2c exp [2iφ2c ] 1 − r2wc r2c exp[2iφ2c ]
(23) (24) (25)
We use these ratios to define the reflection coefficients r˜1wc and eiθ1 such that [37] s (r1wc )2 + (˜r2c )2 − 2r1wc r˜2c η wc r˜1 = 1 + (r1wc )2 (˜r2c )2 − 2r1wc r˜2c η ! ((r1wc )2 − 1)˜r2c ξ wc −1 θ1 = tan r1wc (1 + (˜r2c )2 ) − r˜2c ((r1wc )2 + 1)η wc
(26)
(27)
with η = cos(θ2c + 2φ w + 2φ1c ) and ξ = sin(θ2c + 2φ w + 2φ1c ). The reflection amplitudes of the total composite system can thus be written as E1rc =
r1c − r˜1wc exp [i(θ1wc + 2φ1c )] Ein 1 − r1wcr˜2c exp[i(θ1wc + 2φ1c )]
(28)
Choosing the length of the two microcavities to be equal Lc1 (µ m) = Lc2 (µ m) and the silica waveguide Lw (µ m), we find that the resonance frequency for the two individual cavities is c1 = ω c2 = 2π c/Lc (GHz) with c the speed of light (Fig. 5 shows the simulation of the ωres res 1,2 composite cavity, as detailed below), assuming parameters for the D2 line of 87 Rb. For the whole composite system the resonance frequency is ωres = 2π × 0.0021GHz. With a beam waist of roughly (2-4µ m) we determine the cavity Bragg mirror radius of curvatures such that 2π 2ω 4 Lc ≈ 85.6µ m R˜ 1c = R˜ 2c = 2 c0 + 2 λ L with ω0 chosen to be 4µ m (the diameter of the silica waveguide). We estimate the coupling rate between the two-level N atoms and the composite cavitywaveguide system to be s √ 3cλ 2 κ cw N≈ (29) g π 2 ω02 (Lc1 + Lc2 + Lw ) #199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3511
Fig. 5. The reflected power of the composite cavity system, two micro-cavities connected via a single silica waveguide. The simulation input parameters are Rc1 = Rc2 = 0.985, Rwc 1 c = 0.999, Rwc 2 = 0.9 with both micro-cavities at equal lengths L1,2 = 30µ m and the silica waveguide with a length of Lw = 4mm.
√ which is of the order of ∼ 2π N× 3GHz, the coupling rate gcw is much greater than the decay rate κ = 2π × 6MHz, for 87 Rb. The amplitude decay rate γc/w for the cavity and the waveguide are calculated independently such that γi = χtii with χi ≡ 2−∑2 i Ri and ti is the time
of the photon round trip ti = 2Lc i with i is the cavity and the waveguide index. For a cavity of length 40µ m with Rc1 = 99.97%, Rwc 1 = 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 nsi Lw (nsi is the fused silica wc c c w refractive index) we calculate the decay rate with Rwc 1 = R2 , R2 = R1 and L = 100 µ m 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 wc c are Rc1 = Rc2 = 0.985, Rwc 1 = 0.999, R2 = 0.9 with both micro-cavities at equal lengths L1,2 = w 30µ 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]. 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 microcavity 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
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3512
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.
#199572 - $15.00 USD Received 17 Oct 2013; revised 9 Dec 2013; accepted 10 Dec 2013; published 6 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003501 | OPTICS EXPRESS 3513
