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An atomtronics transistor for quantum gates
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Physics Letters A ••• (••••) •••–•••
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An atomtronics transistor for quantum gates Miroslav Gajdacz a,b , Tomáš Opatrný b,∗ , Kunal K. Das c a b c
Institute for Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark Optics Department, Faculty of Science, Palacký University, 17. Listopadu 12, 77146 Olomouc, Czech Republic Department of Physical Sciences, Kutztown University of Pennsylvania, Kutztown, PA 19530, USA
a r t i c l e
i n f o
Article history: Received 27 September 2013 Received in revised form 28 March 2014 Accepted 22 April 2014 Available online xxxx Communicated by P.R. Holland Keywords: Cold atoms Quantum information Mesoscopic transport
a b s t r a c t We present a mechanism for quantum gates where the qubits are encoded in the population distribution of two-component ultracold atoms trapped in a species-selective triple-well potential. The gate operation is a specific application of a different design for an atomtronics transistor where inter-species interaction is used to control transport, and can be realized with either individual atoms or aggregates like Bose– Einstein condensates (BEC). We demonstrate the operational principle with a static external potential, and show feasible implementation with a smooth dynamical potential. © 2014 Elsevier B.V. All rights reserved.
1. Introduction As research in ultracold atoms shifts more towards practical applications, two of the most promising areas are that of quantum computation [1] and atomtronics or electronics with trapped ultracold atoms [2–4]. Whereas classical computation is intimately tied to electronics, quantum computation in the context of ultracold atoms has so far evolved independently of the relatively new field of atomtronics. Taking a cue from classical computers, it is likely that analogs of standard electronic components like diodes and transistors can be valuable in quantum computation as well. In this paper we propose a different design for an atomtronics transistor which can also be used to implement a two-qubit quantum gate. Our proposal has several distinguishing features: (a) qubit encoding in the spatial coordinates of particles allows implementation with single particle as well as with multi-particle entities such as BEC, (b) a system-independent principle that offers broad choices for physical realization, (c) operation does not require manipulation of internal states, and (d) easily optimizable for high fidelity and speed. In our proposed gate mechanism, qubits are directly encoded in and read out from the spatial distribution of atoms. Spatial mode encoding has been primarily used in optical qubits, in the context of continuous variable quantum computation [5], or in dual-rail schemes [6]. Certain clever proposals for realizing phase gates in double-well potentials [7–9] have also employed vibrational modes of trapped atoms. However, the gate outcome is contained in the
*
Corresponding author.
http://dx.doi.org/10.1016/j.physleta.2014.04.043 0375-9601/© 2014 Elsevier B.V. All rights reserved.
phase of the states and readout was shown to require intermediate encoding on atomic internal states [10]. In contrast, we encode qubits in the population distribution of atoms in a triple-well potential, and readout simply involves determining the presence or absence of atoms in specific wells, possible even for single atoms by direct imaging methods [11]. Likewise, a simpler operational principle underlies our atomtronics transistor, where the atom transport is directly controlled by interspecies interaction. Existing designs are based on manipulating resonant coupling of lattice sites by adjusting the chemical potential or external bias fields [2,3,12]; or on manipulating atomic internal states to transport holes [13], or spin [14]. 2. Quantum gate operation, static case We consider two independently controlled orthogonal triple wells that can be switched between two ‘T-shaped’ configurations, shown in Fig. 1(a, b), containing two mutually-interacting species, each free to move only along one direction. Single qubits are encoded in the spatial degrees of freedom of the two species, with |0 and |1 corresponding to localizations in the respective extreme wells. In any given operation cycle the motion of one species is kept frozen by deep potentials, so only one spatial dimension (1D) needs to be considered at a time. Without loss of generality further we refer to the active species as A and the passive as B as in Fig. 1(a). Along the active direction we label the wells left, central, right, with effective qubit definitions [Fig. 1(d, e)]: Qubit A is in state |0 or |1 if species A is localized in the left or right well respectively; Qubit B is in state |0 or |1 when species B is absent or present in the well that overlaps with the central well of A. A two-qubit CNOT
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Fig. 1. (a)–(b) Interchangeable gate configurations where role of A and B can be swapped; (c) Transistor configuration connected to reservoirs. (d)–(e) Effective qubit definitions in a specific cycle; (f) CNOT quantum gate operation.
gate can be then designed [Fig. 1(f)] such that after a set time T , qubit A is negated if qubit B is in |1, but is unchanged if qubit B is in |0. Notably, such a configuration allows for simple scalability since the roles (‘control’ or ‘controlled’) of the two species can be switched in different cycles.
We first consider a static triple-well potential to describe the gate operation principle, which involves the three lowest eigenstates φ0 , φ1 and φ2 for species A in the triple-well, with eigenenergies E 0 < E 1 < E 2 . The term “static” distinguishes the case in which the potential does not change in time from the dynamic case studied in Section 5. The potential is symmetric about the central well minimum [Fig. 2], so φ1 has its node there while φ0 and φ2 have anti-nodes. Therefore, when species B is present in the central well, the repulsive A–B interaction V AB will shift up the energies E 0 and E 2 , but hardly affect E 1 . A class of potentials exists where the presence of atom B will raise E 0 and E 2 by the same amount, thus leaving E 2 = E 2 − E 0 unchanged while decreasing E 1 = E 1 − E 0 . Species A is prepared in a state |ψ A (t = 0) localized in one of the two extreme wells. Even though we choose this state to be simply a Gaussian with minimized energy, the projection on the three lowest eigenstates is almost complete. The initial phase relations among the eigenstates, shown in Fig. 2(a, b), are such that φ0 (0) and φ2 (0) add up constructively with φ1 (0) in one extreme well and destructively in the other. If present, |ψ B (0) is a minimum energy Gaussian in the tight selective potential for species B, overlapping with the central well of the active potential for atom A. To ensure perfect execution of the gate, one should be able to adjust E 1 and E 2 both with and without atom B. This requires four independent parameters such as separation and height of the barriers, depth of the side wells relative to the middle well and the interaction strength V AB . By simple reparametrization [15], we can find a configuration such that with species B absent, E 2 = 2 × E 1 , and with species B present E 2B = E 2 and E 2B = 4 × E 1B , as seen in Fig. 2. Since all the energy separations are integer multiples of a common energy unit, in both cases (with and without B) species A will undergo periodic dynamics with a revival of the initial state (in the initially occupied extreme well) at
Fig. 2. Static potential: Species B absent (left), present (right): (a, b) The three lowest eigenstates and (c, d) corresponding eigenenergies of species A in the triple-well. The dotted lines show the potential from Eq. (3), which generally matches a lattice potential (solid line in (c), (d)) created with three harmonics. With B absent, E 2 = 2 × E 1 but with B present, E 2B = 4 × E 1B . (e, f) The single-well occupation versus time for species A, with species B absent/present.
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intervals of T = h/ E 1 when B is absent. But, when species B is present, the antisymmetric state φ1 evolves at half the rate than without B, hence after the same time of evolution T it has an opposite phase, or π offset, relative to the symmetric states. This results in localization of the species A in the initially empty extreme well. Thus, the presence or absence of the species B leads to the revival of species A in the original well or transfer to the other extreme well after a set time T , implementing the CNOT gate. It is useful to contrast our mechanism with another interesting proposal [16,17] that uses interspecies interaction to control transport, where the passive species B has two internal states, one that allows passage of species A through its location at a lattice site, and the other that blocks it. The blocking occurs due to destructive interference between two dressed states created by two-photon resonant Raman coupling of the two atoms rather like in Electromagnetically Induced Transparency (a similar mechanism was proposed for realizing collisional blockade in atomic quantum dot structures [18]). Our mechanism is simpler in that there are no internal states, or laser coupling of the atoms, the operation is by modification of the energy levels induced by the presence or absence of atom B which results in switching between atom A making a round-trip or a half-round-trip in a specific period during its natural evolution in a triple well. 3. Transistor operation, static case The gate operation described above is the specific case of a general transistor mechanism in which species B is used to precisely control the flow of species A among the extreme wells. A schematic of operation as an atomtronics transistor is shown in Fig. 1(c), where the left and right wells of the triple-well setup are coupled to two reservoirs of species A, while the central well is coupled transversely to a reservoir for species B. Since V AB directly affects the dynamic evolution of species A it controls the transfer rate from the left well to the right well. A range of interaction strengths can be found for feasible parameters where we can smoothly adjust E 1 which fixes the period for a gate cycle T = h/ E 1 . Therefore, if we instead fix the period of each cycle at T and vary V AB , the transfer per cycle can be controlled; the fraction of species A transferred varies smoothly from zero to complete transfer (inset of Fig. 3(a)). Since V AB depends on the interaction strength g AB and the density |ψ B |2 of species B, for large g AB small variations in |ψ B |2 can be used to control a large flux of species A, creating an amplification effect. As V AB is further increased, transfer of species A is eventually blocked. 4. Physical models: single atoms and BECs During operation the dynamics is kept 1D along the active direction, with transverse motion suppressed by tight harmonic potentials, taken to be cylindrical with angular frequency ω⊥ . We propose implementation in two distinct systems of ultracold atoms: (i) Single atom per species in triple-well optical superlattices, described by an effective Hamiltonian, with 1D hard-core bosonic inter-species interaction strength g AB ,
ˆ = H
−h¯ 2 ∂ 2 + V i (xi , t ) + g AB δ(x A − x B ). 2mi ∂ x2i i= A, B
(1)
This is obtained from the 3D Hamiltonian by integrating out the transverse degrees of freedom [19]. The potential for each species is formed with three harmonics, with the third harmonic generated by counter-propagating beams with wavelength λ and the two lower harmonics obtained using the same light intersecting
3
at angles to increase the spatial periodicity of their interference pattern along the triple-well direction. (ii) Coupled BECs in optical dipole traps described by
ih¯ ∂t ψ A = −
h¯ 2 2m A
∂xx + V A + g AB |ψ B | ψ A 2
h¯ 2 2 2 ih¯ ∂t ψ B = − ∂xx + V B + g B A |ψ A | + g B B |ψ B | ψ B . 2m B
(2)
To ensure linear dynamics, species A has no self-interaction (possible with Feshbach resonance [20]). The triple wells are created with blue-detuned laser barriers in a harmonic well of angular frequency ω :
V A (x, t ) =
1 2
m A ω A x2 + U e
− (x−d2) 2σ
2
+e
− (x+d2) 2σ
2
.
(3)
In simulations we assume the same potential (3) for species B but magnified V B = 20 × V A to keep it localized even when the potential varies (as in the dynamical case described next). In practice [21], V B can be smaller since localization along the active direction can be done by a separate potential that provides the lateral confinement for the passive direction (Fig. 1(a, b)). Both systems have similar outcomes, because the lattice potential in Eq. (1) can be made almost identical to Eq. (3) by suitable choice of parameters as shown in Fig. 2(c, d). The time-evolution is then similar for both models, because due to strong confinement the self-interaction of species B has little effect on the dynamics – which justifies the factorization of the two-component wavefunction |ψAB = |ψ A |ψ B so that Eqs. (2) (apart from the selfinteraction term for B) can be derived as the equations of motion ˆ |ψAB and for the Hamiltonian (1) by taking the projections ψ B | H ψ A | Hˆ |ψAB . Thus here we display simulations [21] that combine features from these almost equivalent scenarios, using potential (3) in the Hamiltonian (1). We time-evolve by a split-operator method applied to a 2D wavefunction of the two species. The evolution of the population of species A (initially in the left well), in the static triple-well during a gate cycle, is shown in Fig. 2(e, f). The populations in the wells are computed by integrating |ψ A (x, t )|2 over the intervals (∞, − D ), (− D , D ), ( D , ∞), where ± D are the coordinates of the barrier maxima. 5. Dynamic gate and transistor The static Hamiltonian demonstrates that both gate and transistor mechanism operate by evolution of dynamical phases. But, to put this in practice, two issues need to be addressed: (i) preparation and readout of the quantum bits and (ii) initiation and termination of each operation cycle. Both goals can be achieved if the initial and final states of species A are the ground state of a single isolated well. The initial state is kept localized by suppressing tunneling by making the barriers high and the central well shallower than the extreme wells. This ensures E 2 E 1 so that φ2 is unoccupied and the state is a superposition of the almost degenerate φ0 and φ1 . A gate cycle is initiated by quickly deepening the central well to set E 2 = 2 × E 1 , whereby φ2 gains population and tunneling ensues. To speed up the tunneling rate, E 1 and E 2 are increased, while maintaining E 2 = 2 × E 1 , by smoothly decreasing U (t ) and d(t ) as shown in Fig. 3(b). The potential energy variation induces a non-adiabatic coupling Akn = ih¯ φk |∂t Hˆ |φn /( E k − E n )2 , that may drive population exchange between the immediate eigenstates. Since the Hamiltonian is kept bilaterally symmetric, such coupling occurs only between states of the same parity. In our simulations, the net population in the three lowest eigenstates is conserved by setting to a low value, the strongest out-coupling A 13 = 0.07, which determines
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Fig. 3. Dynamic potential: (a) Populations in left and right wells after one period t = T , as functions of V AB ; (inset) similar behavior is seen in the static potential. (b) Smooth time evolution of the dynamical parameters: barrier position d and barrier amplitude U ; also shown are the energy separations with species B absent ( E 1B , E 2B , dotted lines) and present ( E 1B , E 2B ). The single-well occupation versus time for species A, when species B is (c) absent and (d) present.
the speed of variation of the potential. The path selecting algorithm stops when the phase difference of the states φ0 and φ1 is π , half of the revival; the second half is just a mirror image in time. The parametric paths, shown in Fig. 3(b), are determined from the properties of the potential with species B absent. We ensure that with species B present, exactly the same path leads to species A localizing in the opposite well, by tuning V AB to maximize gate fidelity, measured by the probability of the projection of the final state at t = T on the desired gate outcome. We stress that during time evolution, the adiabaticity is not assumed among the three relevant lowest eigenstates, but imposed only in their coupling with the higher states. Since the mechanism relies on the natural evolution of the relative dynamical phase of those three states, it can be quite fast. 6. Feasibility analysis In this section we demonstrate feasibility for both individual atoms and BEC by showing that our assumptions can be met by physical systems currently available. We indicate the active direction by coordinate x and assume cylindrical symmetry in transverse directions, denoted by “⊥”. 6.1. Individual atoms in a lattice For lattice implementation we consider 23 Na and 87 Rb as the two species, in orthogonal triple-wells created by λ = 600 nm and 800 nm along their active directions. With D1, D2 lines for 23 Na, 589 nm and D1 = 795 nm, D2 = 780 nm for 87 Rb, this ensures that each species is red-detuned by about 10 nm from its trapping frequency, but far off-resonant from that of the other. Since the species are held by separate lasers, the passive species will be naturally unaffected by the transfer cycle. Transverse trapping for both species can be done with separate lasers at 1064 nm. We use the λ for the active species to set our units, with the recoil energy E R as the energy unit ε (see Table 1 for the definitions). The effective √ = (2π /λ) 2V ⊥ /m; and in these transverse trap frequency is ω⊥ units the oscillator length l⊥ = h¯ /(mω⊥ ) and the confinement energy 2h¯ ω⊥ are fixed by depth of the potential, which we take to be V ⊥ = 80ε . The effective 1D interspecies interaction strength takes the form g AB = a/(π 2l2⊥ A ), where scattering length a = 103a B for the 23 Na–87 Rb [22]. Along the direction of the triple-wells, the well depths never exceed V z = 20ε . The resulting estimates for our units and parameters are summarized in Table 1.
Table 1 Units and parameters for lattice implementation with
87
Rb or
23
Na atoms.
Units and parameters
87
Length unit l = λ
800 nm
600 nm
2.36 × 10−30 J
1.6 × 10−30 J
4.47 × 10−5 s 4.0 × 105 Hz 0.244ε · l 0.053l 36ε 3.24ε
6.7 × 10−6 s 2.7 × 105 Hz 0.325ε · l 0.053l 36ε 4.3ε
2
ε = E R = (2mh λ2 ) A Time unit τ = h¯ /ε Transverse trap freq. ω⊥
Energy unit
Int. strength g AB Transverse osc. length l⊥ Transverse energy 2h¯ ω⊥ Int. energy g 1D /l z
Rb active
23
Na active
These estimates meet the main assumptions of our model and simulations: 1. The values of the interspecies interaction in our estimates are close to that used in our simulation g 1D = 0.26ε · l with the units defined here. 2. Time scale for operation is about 11.9τ which translates to 0.53 × 10−3 s for active 23 Rb and 0.079 × 10−3 s for active 23 Na, implying sub-millisecond operation time scales, competitive with most proposal for quantum gates. 3. Effective 1D dynamics of the active species is justified: The transverse confinement energy 2h¯ ω⊥ = 36ε is much higher than the sum of the longitudinal kinetic energy ∼ ε and interaction energy g AB /l z 4ε . The latter is estimated by the product of the interaction strength and 1D particle density ∼ N / L 1/l z with N ∼ 1 since we consider single particle of each species; the extent of the density overlap of the particles during evolution is estimated by the effective longitudinal os√ cillator length l z = h¯ /(m A ωz ) where ωz = (2π /λ) 2V z /m A and we take V z = 20ε as the upper limit of the longitudinal depth of the wells. 6.2. BECs in a triple-well potential In order to implement the scheme with BECs, the triple well structure can be designed such that the two Gaussian barriers are felt only by the active species (A), possible by keeping their laser frequencies far off-resonant with respect to the passive species (B) which is therefore not affected by the variation of the barriers. Nonlinearity due to interaction can be suppressed for A by Feshbach resonance in optical dipole traps. To demonstrate feasibility, we choose 7 Li and 87 Rb as species A and B. Experiments
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Fig. 4. Snapshots [21] of population density in a gate cycle of period T (time units of τ ) corresponding to Figs. 3(c) and 3(d): upper panels: Density (left axis, filled area) of species A when species B is absent; the potential is shown as solid lines (right axis); lower panels: Joint density of species A (horizontal axis) and species B (vertical axis) when species B is present.
[23] show 7 Li in | F = 1, m F = 1 state has a very weak scattering length a = −1 ∗ a B at 560 G, changes sign and goes up to a = 10a B at 630 G, implying zero scattering length at about 566 G (by linear interpolation). Inter-species s-wave scattering length between 7 Li in | F = 1, m F = 1 and 87 Rb in | F = 1, m F = 1 has been computed [20] to be zero at 438 G and positive above that hitting resonance at 566 G (coincidentally). That coincidence means that the field cannot be at 566 G necessary for a = 0 for lithium, but we can tune the interspecies scattering to about a = +100a B at 560 G by being close but not quite at the resonance while having the species A scattering length of the order of a = −1a B , low enough to have negligible effect on the dynamics or eigenstates of species A. We consider N A = N B = 1000 atoms, large enough to justify using mean field theory [24], but not so large as to create phase fragmentation in elongated traps. Both species are confined in a cigar-shaped optical dipole trap by the same laser of 1064 nm wavelength in the axial (direction of transport), but they experience different axial trap frequencies due to different detuning and masses. For species B the axial (z) and transverse (⊥) trap frequenB cies are chosen to be ωzB = 2π × 10 Hz and ω⊥ = 2π × 100 Hz; A and for species A ω⊥ = 2π × 1000 Hz, and its axial frequency
is fixed by A
ωzA = ωzB /ηz where ηz =
δzA m zA δzB m zB
= 0.36, on using
= 1.6 for the ratio of detuning for the two species in the common axial field. The tight transverse confinement implies a Gaussian shape √ 2 transverse profile for both species : Φi (r i , zi ) = 2βi e −βi r /2 ψi ( zi ) with (i = A, B). Integrating out the transverse direction we get the coupled 1D Gross–Pitaevskii equations in Eq. (2). We define units based on the axial trap frequency ωzA for species A, shown in Taδ δB
ble 2, along with the resulting expressions for some of the relevant parameters like mean field interaction strengths g i j . The mass ratios are denoted by μAB = mAB /m A , μ B = m B /m A with mAB being the reduced mass of species A and B; and ai j are the s-wave scattering lengths. Following Ref. [24], we assume a Thomas–Fermi profile in the axial direction justified because of weak trapping: ψ( z B ) =
3/(4d3B ) d2B − z2B , where z B ∈ [−d B , d B ]. We determine the val-
ues of the physical parameters, summarized in Table 2, by minimizing with respect to parameters β B and d B the resulting energy functional corresponding to GP equation for species B.
E [Φ B ]
ηz N B
d2 (μ B ηz γ B )2 βB + + μ B ηz B 2μ B η z βB 10 1 3N B a B B β B +
=
1
μ B ηz
5d B
(4)
B where ηz = ωzB /ωzA and γ B = ω⊥ /ωzB . Notably, we decoupled the energy functional by neglecting the inter-species interaction assuming it to be small relative to its selfinteraction for species B. But as we see from Table 2, it is quite the opposite. Likewise the cross-interaction for species A is two orders of magnitude higher than is assumed in our simulations. This can be rectified by adding a novel design feature that creates a slight transverse offset of the longitudinal axes to reduce the overlap of the two species. Since the radial profiles are seen to be identical for the two species β A β B 36 = β , the geometric factor due to the reduced overlap is given by the integral:
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0.1% variations in potential parameters [21] leads to a fidelity loss of about 3 × 10−4 .
Table 2 Units and parameters for two-species BEC in triple-well. Units and parameters Energy unit
τ = (ω
8. Discussion and conclusions
1.8 × 10−32 J
ε = h¯ ωzA
Length unit l = Time unit
Numerical values
h¯ m A ωzA
7.2 × 10−6 m
A −1 z )
B Transverse size of B is l⊥ =
Transverse size of A is l⊥ = A
√
5.7 × 10−3 s 1/β B
B β B = 35.85 36 ⇒ l⊥ = 0.17l
1/β A
A β A = 35.97 36 ⇒ l⊥ = 0.17l
√
Longitudinal size of B
d B = 1.58l
Longitudinal size of A
l zA = 1l (by choice of units)
Chemical-potential of B
B 5.7ε (less than 2h¯ ω⊥ = 7.2ε )
Self-interaction strength of B
gBB =
2a B B N B β B
Cross-interaction strength for A
g AB =
aAB N B β B
Cross-interaction strength for B
gB A =
∞ I=
dy e
−∞
=
π β
−β y 2
b/2 ×
Erfc b β/2
dx e
−∞
−β(x−b)2
μB μAB aAB N A β A
μAB
= 28.7 29ε · l
∞ +
= 4.2ε · l
= 28.6 29ε · l
dx e
−β x2
b/2
Snapshots [21] of gate operation shown in Fig. 4 confirm that: (i) species B remains localized unaffected by the evolution of species A, (ii) species A is delocalized during transit (iii) dynamics with Eqs. (1) and (2) is indistinguishable on plot-scale. After a gate cycle, the reduced density matrix ρ A (t ) = Tr B |ψAB (t )ψAB (t )|, gives 1 − Tr ρ 2A ( T ) = 4.4 × 10−4 , indicating a pure state. The von Neumann entropy S ( T ) = −Tr[ρ A ( T ) ln(ρ A ( T )] = 2.3 × 10−3 , with S (t ) < 4.0 × 10−3 during the cycle for the dynamic potential and S (t ) < 12 × 10−3 for the static. These justify the factorization of |ψAB and show absence of significant entanglement between the species. Gate fidelity can be arbitrarily improved to limits ∼ 99.97% set by noise, by using optimal control methods [27] since the mechanism has well-defined initial and target states and a highly optimal initial path. This design can implement a universal set of gates, since single qubit gates, like a Hadamard gate, can be implemented by adjusting E 1 , with species B absent, for desired partial transfer in a cycle.
(5)
which is an error-function and where the line of intersection of the surfaces is found by setting x2 = (x − b)2 ⇒ 2x = b for positive offset b. Using the inter-species interaction strength of 29ε · l for full overlap we solve 29 × I (b) = 0.5 to find that a small offset of b = 0.30l reduces that interaction strength to 0.5ε · l as used in our simulations. We summarize how these estimates satisfy the assumptions of our model: 1. Our use of g 1D = 0.5ε · l for the inter-species interaction can be realized by small transverse offset of the two species. 2. Even though the species B does not “see” the Gaussian barriers creating the triple well, its extent in the dynamical direction is 2d B 3l, localized within the size 2d 4l of the central well measured as the separation between the peaks of the barriers. 3. The dynamics in the active direction is effectively 1D since the chemical potential of even species B (which has selfinteraction) is 5.69ε which is less than the energy required for exciting the 1st transverse excited state 7.2ε . 4. During evolution of species A, species B remains well-localized since it does not feel the time-varying potential and its overlap and interaction with species A remains relatively small. 7. Fidelity and noise We obtained gate fidelities 98% for both static and dynamic potential. The main source of infidelity was projection on φi >3 of the initial Gaussian for the static case and residual non-adiabatic coupling to φi >3 for the dynamic case. Primary noise sources are fluctuations of the potentials, and of T and V AB . High frequency fluctuations > kHz cause heating with negligible impact (∼ 10−4 change in fidelity) in the lattice where cycle time ∼ 10−4 s 10 s durations of stable trapping of single atoms [25]. Heating effects is more significant for BECs with cycle time ∼ 0.25 s, but long lifetimes 10 s can compensate [26]. In the lattice, due to fast cycle time, low frequency noise 100 Hz can be treated as static deviations of the physical parameters. Our mechanism being an interference effect of the lowest three eigenstates, its fidelity goes as sin2 of the eigenstate phase difference θ , which depends linearly on potential variations. So, for small fluctuations, infidelity θ 2 implying a quadratic dependence on fluctuations of the physical parameters. The combined impact of
Acknowledgements T.O. and M.G. acknowledge support of the Czech Science Foundation, Grant No. P205/10/1657. K.K.D. acknowledges support from NSF Grant No. PHY-0970012. References [1] A. Galindo, M.A. Martin-Delgado, Rev. Mod. Phys. 74 (2002) 347. [2] R.A. Pepino, J. Cooper, D.Z. Anderson, M.J. Holland, Phys. Rev. Lett. 103 (2009) 140405. [3] B.T. Seaman, M. Krämer, D.Z. Anderson, M.J. Holland, Phys. Rev. A 75 (2007) 023615. [4] K.K. Das, S. Aubin, Phys. Rev. Lett. 103 (2009) 123007. [5] D.S. Tasca, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, S.P. Walborn, Phys. Rev. A 83 (2011) 052325. [6] M. Miková, H. Fikerová, I. Straka, M. Miˇcuda, J. Fiurášek, M. Ježek, M. Dušek, Phys. Rev. A 85 (2012) 012305. [7] E. Charron, E. Tiesinga, F. Mies, C. Williams, Phys. Rev. Lett. 88 (2002) 077901. [8] J. Mompart, K. Eckert, W. Ertmer, G. Birkl, M. Lewenstein, Phys. Rev. Lett. 90 (2003) 147901. [9] F.W. Strauch, M. Edwards, E. Tiesinga, C. Williams, C.W. Clark, Phys. Rev. A 77 (2008) 050304. [10] E. Charron, M.A. Cirone, A. Negretti, J. Schmiedmayer, T. Calarco, Phys. Rev. A 74 (2006) 012308. [11] J.F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, S. Kuhr, Nature 467 (2010) 68. [12] J.A. Stickney, D.Z. Anderson, A.A. Zozulya, Phys. Rev. A 75 (2007) 013608. [13] A. Benseny, S. Fernández-Vidal, J. Bagudà, R. Corbalán, A. Picón, L. Roso, G. Birkl, J. Mompart, Phys. Rev. A 82 (2010) 013604. ¯ [14] J.Y. Vaishnav, J. Ruseckas, C.W. Clark, G. Juzeliunas, Phys. Rev. Lett. 101 (2008) 265302. [15] M. Gajdacz, T. Opatrný, K.K. Das, Phys. Rev. A 83 (2011) 033623. [16] A. Micheli, A.J. Daley, D. Jaksch, P. Zoller, Phys. Rev. Lett. 93 (2004) 140408. [17] D. Jaksch, P. Zoller, Ann. Phys. 315 (2005) 52. [18] A. Recati, P.O. Fedichev, W. Zwerger, J. von Delft, P. Zoller, Phys. Rev. Lett. 94 (2005) 040404. [19] K.K. Das, M.D. Girardeau, E.M. Wright, Phys. Rev. Lett. 89 (2002) 110402. [20] C. Marzok, B. Deh, C. Zimmermann, P.W. Courteille, E. Tiemann, Y.V. Vanne, A. Saenz, Phys. Rev. A 79 (2009) 012717. [21] Details of feasibility analysis and movies of time evolution are at http:// faculty.kutztown.edu/das/ATQG/ [22] M. Bhattacharya, L. Baksmaty, S. Weiss, N. Bigelow, Eur. Phys. J. D 31 (2004) 301. [23] K.E. Strecker, G.B. Partridge, R.G. Truscott, A.G., Hulet, Nature 417 (2002) 150. [24] K.K. Das, Phys. Rev. A 66 (2002) 053612. [25] M.J. Gibbons, S.Y. Kim, K.M. Fortier, P. Ahmadi, M.S. Chapman, Phys. Rev. A 78 (2008) 043418. [26] A.P. Chikkatur, Y. Shin, A.E. Leanhardt, D. Kielpinski, E. Tsikata, T.L. Gustavson, D.E. Pritchard, W. Ketterle, Science 296 (2002) 2192. [27] S.E. Sklarz, D.J. Tannor, Phys. Rev. A 66 (2002) 053619.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
An atomtronics transistor for quantum gates Miroslav Gajdacz a,b , Tomáš Opatrný b,∗ , Kunal K. Das c a b c
Institute for Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark Optics Department, Faculty of Science, Palacký University, 17. Listopadu 12, 77146 Olomouc, Czech Republic Department of Physical Sciences, Kutztown University of Pennsylvania, Kutztown, PA 19530, USA
a r t i c l e
i n f o
Article history: Received 27 September 2013 Received in revised form 28 March 2014 Accepted 22 April 2014 Available online xxxx Communicated by P.R. Holland Keywords: Cold atoms Quantum information Mesoscopic transport
a b s t r a c t We present a mechanism for quantum gates where the qubits are encoded in the population distribution of two-component ultracold atoms trapped in a species-selective triple-well potential. The gate operation is a specific application of a different design for an atomtronics transistor where inter-species interaction is used to control transport, and can be realized with either individual atoms or aggregates like Bose– Einstein condensates (BEC). We demonstrate the operational principle with a static external potential, and show feasible implementation with a smooth dynamical potential. © 2014 Elsevier B.V. All rights reserved.
1. Introduction As research in ultracold atoms shifts more towards practical applications, two of the most promising areas are that of quantum computation [1] and atomtronics or electronics with trapped ultracold atoms [2–4]. Whereas classical computation is intimately tied to electronics, quantum computation in the context of ultracold atoms has so far evolved independently of the relatively new field of atomtronics. Taking a cue from classical computers, it is likely that analogs of standard electronic components like diodes and transistors can be valuable in quantum computation as well. In this paper we propose a different design for an atomtronics transistor which can also be used to implement a two-qubit quantum gate. Our proposal has several distinguishing features: (a) qubit encoding in the spatial coordinates of particles allows implementation with single particle as well as with multi-particle entities such as BEC, (b) a system-independent principle that offers broad choices for physical realization, (c) operation does not require manipulation of internal states, and (d) easily optimizable for high fidelity and speed. In our proposed gate mechanism, qubits are directly encoded in and read out from the spatial distribution of atoms. Spatial mode encoding has been primarily used in optical qubits, in the context of continuous variable quantum computation [5], or in dual-rail schemes [6]. Certain clever proposals for realizing phase gates in double-well potentials [7–9] have also employed vibrational modes of trapped atoms. However, the gate outcome is contained in the
*
Corresponding author.
http://dx.doi.org/10.1016/j.physleta.2014.04.043 0375-9601/© 2014 Elsevier B.V. All rights reserved.
phase of the states and readout was shown to require intermediate encoding on atomic internal states [10]. In contrast, we encode qubits in the population distribution of atoms in a triple-well potential, and readout simply involves determining the presence or absence of atoms in specific wells, possible even for single atoms by direct imaging methods [11]. Likewise, a simpler operational principle underlies our atomtronics transistor, where the atom transport is directly controlled by interspecies interaction. Existing designs are based on manipulating resonant coupling of lattice sites by adjusting the chemical potential or external bias fields [2,3,12]; or on manipulating atomic internal states to transport holes [13], or spin [14]. 2. Quantum gate operation, static case We consider two independently controlled orthogonal triple wells that can be switched between two ‘T-shaped’ configurations, shown in Fig. 1(a, b), containing two mutually-interacting species, each free to move only along one direction. Single qubits are encoded in the spatial degrees of freedom of the two species, with |0 and |1 corresponding to localizations in the respective extreme wells. In any given operation cycle the motion of one species is kept frozen by deep potentials, so only one spatial dimension (1D) needs to be considered at a time. Without loss of generality further we refer to the active species as A and the passive as B as in Fig. 1(a). Along the active direction we label the wells left, central, right, with effective qubit definitions [Fig. 1(d, e)]: Qubit A is in state |0 or |1 if species A is localized in the left or right well respectively; Qubit B is in state |0 or |1 when species B is absent or present in the well that overlaps with the central well of A. A two-qubit CNOT
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Fig. 1. (a)–(b) Interchangeable gate configurations where role of A and B can be swapped; (c) Transistor configuration connected to reservoirs. (d)–(e) Effective qubit definitions in a specific cycle; (f) CNOT quantum gate operation.
gate can be then designed [Fig. 1(f)] such that after a set time T , qubit A is negated if qubit B is in |1, but is unchanged if qubit B is in |0. Notably, such a configuration allows for simple scalability since the roles (‘control’ or ‘controlled’) of the two species can be switched in different cycles.
We first consider a static triple-well potential to describe the gate operation principle, which involves the three lowest eigenstates φ0 , φ1 and φ2 for species A in the triple-well, with eigenenergies E 0 < E 1 < E 2 . The term “static” distinguishes the case in which the potential does not change in time from the dynamic case studied in Section 5. The potential is symmetric about the central well minimum [Fig. 2], so φ1 has its node there while φ0 and φ2 have anti-nodes. Therefore, when species B is present in the central well, the repulsive A–B interaction V AB will shift up the energies E 0 and E 2 , but hardly affect E 1 . A class of potentials exists where the presence of atom B will raise E 0 and E 2 by the same amount, thus leaving E 2 = E 2 − E 0 unchanged while decreasing E 1 = E 1 − E 0 . Species A is prepared in a state |ψ A (t = 0) localized in one of the two extreme wells. Even though we choose this state to be simply a Gaussian with minimized energy, the projection on the three lowest eigenstates is almost complete. The initial phase relations among the eigenstates, shown in Fig. 2(a, b), are such that φ0 (0) and φ2 (0) add up constructively with φ1 (0) in one extreme well and destructively in the other. If present, |ψ B (0) is a minimum energy Gaussian in the tight selective potential for species B, overlapping with the central well of the active potential for atom A. To ensure perfect execution of the gate, one should be able to adjust E 1 and E 2 both with and without atom B. This requires four independent parameters such as separation and height of the barriers, depth of the side wells relative to the middle well and the interaction strength V AB . By simple reparametrization [15], we can find a configuration such that with species B absent, E 2 = 2 × E 1 , and with species B present E 2B = E 2 and E 2B = 4 × E 1B , as seen in Fig. 2. Since all the energy separations are integer multiples of a common energy unit, in both cases (with and without B) species A will undergo periodic dynamics with a revival of the initial state (in the initially occupied extreme well) at
Fig. 2. Static potential: Species B absent (left), present (right): (a, b) The three lowest eigenstates and (c, d) corresponding eigenenergies of species A in the triple-well. The dotted lines show the potential from Eq. (3), which generally matches a lattice potential (solid line in (c), (d)) created with three harmonics. With B absent, E 2 = 2 × E 1 but with B present, E 2B = 4 × E 1B . (e, f) The single-well occupation versus time for species A, with species B absent/present.
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intervals of T = h/ E 1 when B is absent. But, when species B is present, the antisymmetric state φ1 evolves at half the rate than without B, hence after the same time of evolution T it has an opposite phase, or π offset, relative to the symmetric states. This results in localization of the species A in the initially empty extreme well. Thus, the presence or absence of the species B leads to the revival of species A in the original well or transfer to the other extreme well after a set time T , implementing the CNOT gate. It is useful to contrast our mechanism with another interesting proposal [16,17] that uses interspecies interaction to control transport, where the passive species B has two internal states, one that allows passage of species A through its location at a lattice site, and the other that blocks it. The blocking occurs due to destructive interference between two dressed states created by two-photon resonant Raman coupling of the two atoms rather like in Electromagnetically Induced Transparency (a similar mechanism was proposed for realizing collisional blockade in atomic quantum dot structures [18]). Our mechanism is simpler in that there are no internal states, or laser coupling of the atoms, the operation is by modification of the energy levels induced by the presence or absence of atom B which results in switching between atom A making a round-trip or a half-round-trip in a specific period during its natural evolution in a triple well. 3. Transistor operation, static case The gate operation described above is the specific case of a general transistor mechanism in which species B is used to precisely control the flow of species A among the extreme wells. A schematic of operation as an atomtronics transistor is shown in Fig. 1(c), where the left and right wells of the triple-well setup are coupled to two reservoirs of species A, while the central well is coupled transversely to a reservoir for species B. Since V AB directly affects the dynamic evolution of species A it controls the transfer rate from the left well to the right well. A range of interaction strengths can be found for feasible parameters where we can smoothly adjust E 1 which fixes the period for a gate cycle T = h/ E 1 . Therefore, if we instead fix the period of each cycle at T and vary V AB , the transfer per cycle can be controlled; the fraction of species A transferred varies smoothly from zero to complete transfer (inset of Fig. 3(a)). Since V AB depends on the interaction strength g AB and the density |ψ B |2 of species B, for large g AB small variations in |ψ B |2 can be used to control a large flux of species A, creating an amplification effect. As V AB is further increased, transfer of species A is eventually blocked. 4. Physical models: single atoms and BECs During operation the dynamics is kept 1D along the active direction, with transverse motion suppressed by tight harmonic potentials, taken to be cylindrical with angular frequency ω⊥ . We propose implementation in two distinct systems of ultracold atoms: (i) Single atom per species in triple-well optical superlattices, described by an effective Hamiltonian, with 1D hard-core bosonic inter-species interaction strength g AB ,
ˆ = H
−h¯ 2 ∂ 2 + V i (xi , t ) + g AB δ(x A − x B ). 2mi ∂ x2i i= A, B
(1)
This is obtained from the 3D Hamiltonian by integrating out the transverse degrees of freedom [19]. The potential for each species is formed with three harmonics, with the third harmonic generated by counter-propagating beams with wavelength λ and the two lower harmonics obtained using the same light intersecting
3
at angles to increase the spatial periodicity of their interference pattern along the triple-well direction. (ii) Coupled BECs in optical dipole traps described by
ih¯ ∂t ψ A = −
h¯ 2 2m A
∂xx + V A + g AB |ψ B | ψ A 2
h¯ 2 2 2 ih¯ ∂t ψ B = − ∂xx + V B + g B A |ψ A | + g B B |ψ B | ψ B . 2m B
(2)
To ensure linear dynamics, species A has no self-interaction (possible with Feshbach resonance [20]). The triple wells are created with blue-detuned laser barriers in a harmonic well of angular frequency ω :
V A (x, t ) =
1 2
m A ω A x2 + U e
− (x−d2) 2σ
2
+e
− (x+d2) 2σ
2
.
(3)
In simulations we assume the same potential (3) for species B but magnified V B = 20 × V A to keep it localized even when the potential varies (as in the dynamical case described next). In practice [21], V B can be smaller since localization along the active direction can be done by a separate potential that provides the lateral confinement for the passive direction (Fig. 1(a, b)). Both systems have similar outcomes, because the lattice potential in Eq. (1) can be made almost identical to Eq. (3) by suitable choice of parameters as shown in Fig. 2(c, d). The time-evolution is then similar for both models, because due to strong confinement the self-interaction of species B has little effect on the dynamics – which justifies the factorization of the two-component wavefunction |ψAB = |ψ A |ψ B so that Eqs. (2) (apart from the selfinteraction term for B) can be derived as the equations of motion ˆ |ψAB and for the Hamiltonian (1) by taking the projections ψ B | H ψ A | Hˆ |ψAB . Thus here we display simulations [21] that combine features from these almost equivalent scenarios, using potential (3) in the Hamiltonian (1). We time-evolve by a split-operator method applied to a 2D wavefunction of the two species. The evolution of the population of species A (initially in the left well), in the static triple-well during a gate cycle, is shown in Fig. 2(e, f). The populations in the wells are computed by integrating |ψ A (x, t )|2 over the intervals (∞, − D ), (− D , D ), ( D , ∞), where ± D are the coordinates of the barrier maxima. 5. Dynamic gate and transistor The static Hamiltonian demonstrates that both gate and transistor mechanism operate by evolution of dynamical phases. But, to put this in practice, two issues need to be addressed: (i) preparation and readout of the quantum bits and (ii) initiation and termination of each operation cycle. Both goals can be achieved if the initial and final states of species A are the ground state of a single isolated well. The initial state is kept localized by suppressing tunneling by making the barriers high and the central well shallower than the extreme wells. This ensures E 2 E 1 so that φ2 is unoccupied and the state is a superposition of the almost degenerate φ0 and φ1 . A gate cycle is initiated by quickly deepening the central well to set E 2 = 2 × E 1 , whereby φ2 gains population and tunneling ensues. To speed up the tunneling rate, E 1 and E 2 are increased, while maintaining E 2 = 2 × E 1 , by smoothly decreasing U (t ) and d(t ) as shown in Fig. 3(b). The potential energy variation induces a non-adiabatic coupling Akn = ih¯ φk |∂t Hˆ |φn /( E k − E n )2 , that may drive population exchange between the immediate eigenstates. Since the Hamiltonian is kept bilaterally symmetric, such coupling occurs only between states of the same parity. In our simulations, the net population in the three lowest eigenstates is conserved by setting to a low value, the strongest out-coupling A 13 = 0.07, which determines
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Fig. 3. Dynamic potential: (a) Populations in left and right wells after one period t = T , as functions of V AB ; (inset) similar behavior is seen in the static potential. (b) Smooth time evolution of the dynamical parameters: barrier position d and barrier amplitude U ; also shown are the energy separations with species B absent ( E 1B , E 2B , dotted lines) and present ( E 1B , E 2B ). The single-well occupation versus time for species A, when species B is (c) absent and (d) present.
the speed of variation of the potential. The path selecting algorithm stops when the phase difference of the states φ0 and φ1 is π , half of the revival; the second half is just a mirror image in time. The parametric paths, shown in Fig. 3(b), are determined from the properties of the potential with species B absent. We ensure that with species B present, exactly the same path leads to species A localizing in the opposite well, by tuning V AB to maximize gate fidelity, measured by the probability of the projection of the final state at t = T on the desired gate outcome. We stress that during time evolution, the adiabaticity is not assumed among the three relevant lowest eigenstates, but imposed only in their coupling with the higher states. Since the mechanism relies on the natural evolution of the relative dynamical phase of those three states, it can be quite fast. 6. Feasibility analysis In this section we demonstrate feasibility for both individual atoms and BEC by showing that our assumptions can be met by physical systems currently available. We indicate the active direction by coordinate x and assume cylindrical symmetry in transverse directions, denoted by “⊥”. 6.1. Individual atoms in a lattice For lattice implementation we consider 23 Na and 87 Rb as the two species, in orthogonal triple-wells created by λ = 600 nm and 800 nm along their active directions. With D1, D2 lines for 23 Na, 589 nm and D1 = 795 nm, D2 = 780 nm for 87 Rb, this ensures that each species is red-detuned by about 10 nm from its trapping frequency, but far off-resonant from that of the other. Since the species are held by separate lasers, the passive species will be naturally unaffected by the transfer cycle. Transverse trapping for both species can be done with separate lasers at 1064 nm. We use the λ for the active species to set our units, with the recoil energy E R as the energy unit ε (see Table 1 for the definitions). The effective √ = (2π /λ) 2V ⊥ /m; and in these transverse trap frequency is ω⊥ units the oscillator length l⊥ = h¯ /(mω⊥ ) and the confinement energy 2h¯ ω⊥ are fixed by depth of the potential, which we take to be V ⊥ = 80ε . The effective 1D interspecies interaction strength takes the form g AB = a/(π 2l2⊥ A ), where scattering length a = 103a B for the 23 Na–87 Rb [22]. Along the direction of the triple-wells, the well depths never exceed V z = 20ε . The resulting estimates for our units and parameters are summarized in Table 1.
Table 1 Units and parameters for lattice implementation with
87
Rb or
23
Na atoms.
Units and parameters
87
Length unit l = λ
800 nm
600 nm
2.36 × 10−30 J
1.6 × 10−30 J
4.47 × 10−5 s 4.0 × 105 Hz 0.244ε · l 0.053l 36ε 3.24ε
6.7 × 10−6 s 2.7 × 105 Hz 0.325ε · l 0.053l 36ε 4.3ε
2
ε = E R = (2mh λ2 ) A Time unit τ = h¯ /ε Transverse trap freq. ω⊥
Energy unit
Int. strength g AB Transverse osc. length l⊥ Transverse energy 2h¯ ω⊥ Int. energy g 1D /l z
Rb active
23
Na active
These estimates meet the main assumptions of our model and simulations: 1. The values of the interspecies interaction in our estimates are close to that used in our simulation g 1D = 0.26ε · l with the units defined here. 2. Time scale for operation is about 11.9τ which translates to 0.53 × 10−3 s for active 23 Rb and 0.079 × 10−3 s for active 23 Na, implying sub-millisecond operation time scales, competitive with most proposal for quantum gates. 3. Effective 1D dynamics of the active species is justified: The transverse confinement energy 2h¯ ω⊥ = 36ε is much higher than the sum of the longitudinal kinetic energy ∼ ε and interaction energy g AB /l z 4ε . The latter is estimated by the product of the interaction strength and 1D particle density ∼ N / L 1/l z with N ∼ 1 since we consider single particle of each species; the extent of the density overlap of the particles during evolution is estimated by the effective longitudinal os√ cillator length l z = h¯ /(m A ωz ) where ωz = (2π /λ) 2V z /m A and we take V z = 20ε as the upper limit of the longitudinal depth of the wells. 6.2. BECs in a triple-well potential In order to implement the scheme with BECs, the triple well structure can be designed such that the two Gaussian barriers are felt only by the active species (A), possible by keeping their laser frequencies far off-resonant with respect to the passive species (B) which is therefore not affected by the variation of the barriers. Nonlinearity due to interaction can be suppressed for A by Feshbach resonance in optical dipole traps. To demonstrate feasibility, we choose 7 Li and 87 Rb as species A and B. Experiments
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Fig. 4. Snapshots [21] of population density in a gate cycle of period T (time units of τ ) corresponding to Figs. 3(c) and 3(d): upper panels: Density (left axis, filled area) of species A when species B is absent; the potential is shown as solid lines (right axis); lower panels: Joint density of species A (horizontal axis) and species B (vertical axis) when species B is present.
[23] show 7 Li in | F = 1, m F = 1 state has a very weak scattering length a = −1 ∗ a B at 560 G, changes sign and goes up to a = 10a B at 630 G, implying zero scattering length at about 566 G (by linear interpolation). Inter-species s-wave scattering length between 7 Li in | F = 1, m F = 1 and 87 Rb in | F = 1, m F = 1 has been computed [20] to be zero at 438 G and positive above that hitting resonance at 566 G (coincidentally). That coincidence means that the field cannot be at 566 G necessary for a = 0 for lithium, but we can tune the interspecies scattering to about a = +100a B at 560 G by being close but not quite at the resonance while having the species A scattering length of the order of a = −1a B , low enough to have negligible effect on the dynamics or eigenstates of species A. We consider N A = N B = 1000 atoms, large enough to justify using mean field theory [24], but not so large as to create phase fragmentation in elongated traps. Both species are confined in a cigar-shaped optical dipole trap by the same laser of 1064 nm wavelength in the axial (direction of transport), but they experience different axial trap frequencies due to different detuning and masses. For species B the axial (z) and transverse (⊥) trap frequenB cies are chosen to be ωzB = 2π × 10 Hz and ω⊥ = 2π × 100 Hz; A and for species A ω⊥ = 2π × 1000 Hz, and its axial frequency
is fixed by A
ωzA = ωzB /ηz where ηz =
δzA m zA δzB m zB
= 0.36, on using
= 1.6 for the ratio of detuning for the two species in the common axial field. The tight transverse confinement implies a Gaussian shape √ 2 transverse profile for both species : Φi (r i , zi ) = 2βi e −βi r /2 ψi ( zi ) with (i = A, B). Integrating out the transverse direction we get the coupled 1D Gross–Pitaevskii equations in Eq. (2). We define units based on the axial trap frequency ωzA for species A, shown in Taδ δB
ble 2, along with the resulting expressions for some of the relevant parameters like mean field interaction strengths g i j . The mass ratios are denoted by μAB = mAB /m A , μ B = m B /m A with mAB being the reduced mass of species A and B; and ai j are the s-wave scattering lengths. Following Ref. [24], we assume a Thomas–Fermi profile in the axial direction justified because of weak trapping: ψ( z B ) =
3/(4d3B ) d2B − z2B , where z B ∈ [−d B , d B ]. We determine the val-
ues of the physical parameters, summarized in Table 2, by minimizing with respect to parameters β B and d B the resulting energy functional corresponding to GP equation for species B.
E [Φ B ]
ηz N B
d2 (μ B ηz γ B )2 βB + + μ B ηz B 2μ B η z βB 10 1 3N B a B B β B +
=
1
μ B ηz
5d B
(4)
B where ηz = ωzB /ωzA and γ B = ω⊥ /ωzB . Notably, we decoupled the energy functional by neglecting the inter-species interaction assuming it to be small relative to its selfinteraction for species B. But as we see from Table 2, it is quite the opposite. Likewise the cross-interaction for species A is two orders of magnitude higher than is assumed in our simulations. This can be rectified by adding a novel design feature that creates a slight transverse offset of the longitudinal axes to reduce the overlap of the two species. Since the radial profiles are seen to be identical for the two species β A β B 36 = β , the geometric factor due to the reduced overlap is given by the integral:
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0.1% variations in potential parameters [21] leads to a fidelity loss of about 3 × 10−4 .
Table 2 Units and parameters for two-species BEC in triple-well. Units and parameters Energy unit
τ = (ω
8. Discussion and conclusions
1.8 × 10−32 J
ε = h¯ ωzA
Length unit l = Time unit
Numerical values
h¯ m A ωzA
7.2 × 10−6 m
A −1 z )
B Transverse size of B is l⊥ =
Transverse size of A is l⊥ = A
√
5.7 × 10−3 s 1/β B
B β B = 35.85 36 ⇒ l⊥ = 0.17l
1/β A
A β A = 35.97 36 ⇒ l⊥ = 0.17l
√
Longitudinal size of B
d B = 1.58l
Longitudinal size of A
l zA = 1l (by choice of units)
Chemical-potential of B
B 5.7ε (less than 2h¯ ω⊥ = 7.2ε )
Self-interaction strength of B
gBB =
2a B B N B β B
Cross-interaction strength for A
g AB =
aAB N B β B
Cross-interaction strength for B
gB A =
∞ I=
dy e
−∞
=
π β
−β y 2
b/2 ×
Erfc b β/2
dx e
−∞
−β(x−b)2
μB μAB aAB N A β A
μAB
= 28.7 29ε · l
∞ +
= 4.2ε · l
= 28.6 29ε · l
dx e
−β x2
b/2
Snapshots [21] of gate operation shown in Fig. 4 confirm that: (i) species B remains localized unaffected by the evolution of species A, (ii) species A is delocalized during transit (iii) dynamics with Eqs. (1) and (2) is indistinguishable on plot-scale. After a gate cycle, the reduced density matrix ρ A (t ) = Tr B |ψAB (t )ψAB (t )|, gives 1 − Tr ρ 2A ( T ) = 4.4 × 10−4 , indicating a pure state. The von Neumann entropy S ( T ) = −Tr[ρ A ( T ) ln(ρ A ( T )] = 2.3 × 10−3 , with S (t ) < 4.0 × 10−3 during the cycle for the dynamic potential and S (t ) < 12 × 10−3 for the static. These justify the factorization of |ψAB and show absence of significant entanglement between the species. Gate fidelity can be arbitrarily improved to limits ∼ 99.97% set by noise, by using optimal control methods [27] since the mechanism has well-defined initial and target states and a highly optimal initial path. This design can implement a universal set of gates, since single qubit gates, like a Hadamard gate, can be implemented by adjusting E 1 , with species B absent, for desired partial transfer in a cycle.
(5)
which is an error-function and where the line of intersection of the surfaces is found by setting x2 = (x − b)2 ⇒ 2x = b for positive offset b. Using the inter-species interaction strength of 29ε · l for full overlap we solve 29 × I (b) = 0.5 to find that a small offset of b = 0.30l reduces that interaction strength to 0.5ε · l as used in our simulations. We summarize how these estimates satisfy the assumptions of our model: 1. Our use of g 1D = 0.5ε · l for the inter-species interaction can be realized by small transverse offset of the two species. 2. Even though the species B does not “see” the Gaussian barriers creating the triple well, its extent in the dynamical direction is 2d B 3l, localized within the size 2d 4l of the central well measured as the separation between the peaks of the barriers. 3. The dynamics in the active direction is effectively 1D since the chemical potential of even species B (which has selfinteraction) is 5.69ε which is less than the energy required for exciting the 1st transverse excited state 7.2ε . 4. During evolution of species A, species B remains well-localized since it does not feel the time-varying potential and its overlap and interaction with species A remains relatively small. 7. Fidelity and noise We obtained gate fidelities 98% for both static and dynamic potential. The main source of infidelity was projection on φi >3 of the initial Gaussian for the static case and residual non-adiabatic coupling to φi >3 for the dynamic case. Primary noise sources are fluctuations of the potentials, and of T and V AB . High frequency fluctuations > kHz cause heating with negligible impact (∼ 10−4 change in fidelity) in the lattice where cycle time ∼ 10−4 s 10 s durations of stable trapping of single atoms [25]. Heating effects is more significant for BECs with cycle time ∼ 0.25 s, but long lifetimes 10 s can compensate [26]. In the lattice, due to fast cycle time, low frequency noise 100 Hz can be treated as static deviations of the physical parameters. Our mechanism being an interference effect of the lowest three eigenstates, its fidelity goes as sin2 of the eigenstate phase difference θ , which depends linearly on potential variations. So, for small fluctuations, infidelity θ 2 implying a quadratic dependence on fluctuations of the physical parameters. The combined impact of
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