Atomtronic protocol designs for NOON states
Daniel S. Grun, Karin W. Wittmann, Leandro H. Ymai, Jon Links, Angela Foerster
AAtomtronic protocol designs for NOON states
Daniel S. Gr¨un , Karin Wittmann W. , Leandro H. Ymai , ∗ Jon Links & Angela Foerster Instituto de F´ısica da UFRGS, Av. Bento Gon¸calves 9500, Agronomia, Porto Alegre, 91501-970, RS, Brazil Universidade Federal do Pampa, Av. Maria Anuncia¸c˜ao Gomes de Godoy 1650, Malafaia, Bag´e, 96413-170, RS, Brazil School of Mathematics and Physics, The University of Queensland, Brisbane, QLD, 4072, Australia ∗ Corresponding author
Abstract
The ability to reliably prepare non-classical states will play a major role in the realization of quantum technology.NOON states, belonging to the class of Schr¨odinger cat states, have emerged as a leading candidate for severalapplications. Starting from a model of dipolar bosons confined to a closed circuit of four sites, we show how togenerate NOON states. This is achieved by designing protocols to transform initial Fock states to NOON statesthrough use of time evolution, application of an external field, and local projective measurements. By variation ofthe external field strength, we demonstrate how the system can be controlled to encode a phase into a NOON state.We also discuss the physical feasibility, via an optical lattice setup. Our proposal illuminates the benefits of quantumintegrable systems in the design of atomtronic protocols.
Quantum systems are widely considered to be the mostpromising foundation for the next generation of platformsin computing, communication, measurement and simula-tion. This is primarily due to the properties of state su-perposition and entanglement. To realize the potential forprogress, it is necessary to establish protocols that are ca-pable of generating important quantum states.The NOON state is a fundamental example. It is an“all and nothing” superposition of two different modes .For N particles, it has the form | NOON i = 1 √ (cid:0) | N, i + e iϕ | , N i (cid:1) (1)where the phase ϕ typically records information in appli-cations. These include: in the fields of quantum metrologyand sensing, performing precision phase-interferometry atthe Heisenberg limit and overcoming diffraction limits inquantum lithography ; in tests of fundamental physics,NOON states are used to study Bell-type inequalities vi-olation ; they offer promising applications in QuantumCommunication and Quantum Computing , and their uti-lization is expected to extend to areas such as chem-istry and biology . After an early success, using photonpairs and Hong-Ou-Mandel (HOM) interferometry , sev-eral schemes have followed for the production and detec-tion of photonic NOON states . There are also pro-posals using other architectures, such as circuit QED ,trapped ions , and Bose-Einstein condenstates .The atomtronic creation of Bose-atom NOON stateswould enable new tests, using massive states, of the foun-dations for quantum mechanics. One step in this direc- tion is a proposal to demonstrate the matter-wave equiv-alent of the HOM effect . Prospects for creating Bose-atom NOON states using a double-well potential were firstfloated some time ago . This early work considered anattractive system, which is prone to instability. In prin-ciple a more robust repulsive system can be prepared toevolve to a high-fidelity approximation of a NOON state.However, the drawback there is that the process is asso-ciated with an extremely large time scale. Recently, newstudies of the double-well system have been undertaken toreduce the time scale. One example proposes to adiabati-cally vary the system parameters through an excited-statephase transition during the process . Another study em-ploys periodic driving to lower the NOON-state evolutiontime . Nonetheless, the time to generate a NOON statein these examples still, increasingly, scales with the totalnumber of particles.Here we present an alternative to circumvent these is-sues. Our approach adopts a closed-circuit of four sites,with a Fock-state input of M particles in site 1, P parti-cles in site 2, and no particles in sites 3 and 4, denotedas | Ψ i = | M, P, , i . The initial step is to create anuber-NOON state, with the general form | u-NOON i = 12 (cid:0) | M, P, , i + e iϕ | M, , , P i + e iϕ | , P, M, i + e iϕ | , , M, P i (cid:1) for a set of phases { ϕ , ϕ , ϕ } . This state may be viewedas an embedding of NOON states (1) within two-site sub-systems. We then describe two protocols to extract aNOON state from an uber-NOON state, one through dy-1 a r X i v : . [ qu a n t - ph ] F e b igure 1. NOON state generation scheme. The four circles on the left represent the initial state, with white indicatingan empty site, cyan and blue corresponding to M and P particles respectively. The solid lines connecting the circlesdenote tunneling between nearest neighbor sites. Rectangles represent applied external fields to sites 1-3 and 2-4. InProtocol I, the system initially evolves during a time of t m − t µ . Then, an applied field across sites 2-4 is switched on,and a phase is encoded during the time t µ . Finally, the light blue halo portrays a projective measurement process atsite 3, denoted by M , resulting in two possible NOON states across sites 2-4. In Protocol II, the system first evolvesduring a time of t m − t ν . Then, an applied field to sites 1-3 is switched on for time t ν . Next, the system evolves for t m − t µ , after which the applied field to sites 2 − . There existsa choice of the coupling parameters for which this modelis integrable . As in other physically realized integrablesystems , this property facilitates several analytic cal-culations for physical quantities. Here, integrability ex-poses the protocols available for NOON state generation.The execution time is found to be dependent on the dif-ference between the two initially populated sites withinthe four-site system. It is independent of total particlenumber, offering an encouraging prospect for scalability.(ii) The system can be controlled by breaking the inte-grability over small time scales. Encoding of the phaseinto a NOON state only requires breaking of integrabil-ity over an interval that is several orders of magnitudesmaller than the entire execution time. This causes min-imal loss in fidelity. (iii) With currently available tech-nology, the system may be realized and controlled usingdipolar atoms (e.g. dysprosium or erbium) trapped in an optical lattice . In this setup, the evolution times thatwe compute for NOON-state generation are of the orderof seconds.For the four-site configuration, the EBHM Hamiltonianis H = U X i =1 N i ( N i −
1) + X i =1 4 X j =1 ,j = i U ij N i N j − J (cid:2) ( a † + a † )( a + a ) + ( a + a )( a † + a † ) (cid:3) , (2)where a † j , a j are the creation and annihilation operatorsfor site j , and N j = a † j a j are the number operators. Thetotal number operator N = N + N + N + N is con-served. Above, U characterizes the interaction betweenbosons at the same site, U ij = U ji is related to the long-range (e.g. dipole-dipole) interaction between bosons atsites i and j , and J accounts for the tunneling strengthbetween different sites.Below, we describe two protocols that enable the gen-eration of NOON states, with fidelities greater than 0.9.A physical setup to implement them, drawn on currentlyavailable technology, is discussed.2 esults Insights into the physical behaviour of Eq. (2) become ac-cessible at integrable coupling. Setting U = U = U ,the system acquires two additional conserved quantities, Q and Q , such that 2 Q = N + N − a † a − a a † and2 Q = N + N − a † a − a a † . Together with the to-tal number of particles N and the Hamiltonian H , thesystem possesses four independent, conserved quantities.This is equal to the number of degrees of freedom, satisfy-ing the criterion for integrability. Suppose that, initially,there are M atoms in site 1 and P atoms in site 2. Weidentify the resonant tunneling regime as being achievedwhen U | M − P | (cid:29) J (see Methods for details), where U = ( U − U ) /
4. This regime is characterized by setsof bands in the energy spectrum (see Supplementary Note1). In this region, an effective Hamiltonian H eff enablesthe derivation of analytic expressions for several physicalquantities.In the settings discussed above, the system describedby Eq. (2) provides the framework to generate uber-NOON states when N = M + P is odd . To encodephases, however, it is necessary to break the integrabilityin a controllable fashion. Here, we introduce two idealizedprotocols to produce NOON states with general phasesby breaking the system’s integrability with externally ap-plied fields. We call the subsystem containing sites 1 , A , and the one containing sites 2 , B . We denote threetime intervals: t m , t µ and t ν . The first, correspondingto integrable time evolution, is associated with evolutionto a particular uber-NOON state. The others, associatedwith smaller scale non-integrable evolution, produce phaseencoding. Both protocols are built around a general time-evolution operator U ( t, µ, ν ) = exp (cid:18) − it (cid:126) [ H + µ ( N − N ) + ν ( N − N )] (cid:19) , where the applied field strengths µ , ν implement the break-ing of integrability. It is convenient to introduce the phasevariable θ = 2 µt µ / (cid:126) , and to fix t ν = (cid:126) π/ (4 M ν ), with (cid:126) the reduced Planck constant.
Protocol I
In this protocol we employ breaking of integrabilitythrough an applied field to subsystem B and a measure-ment process. The protocol consists of three sequentialsteps, schematically depicted in Fig. 1:(i) | Ψ I1 i = U ( t m − t µ , , | Ψ i ;(ii) | Ψ I2 i = U ( t µ , µ, | Ψ I1 i ;(iii) | Ψ I3 i = M | Ψ I2 i , where t m = (cid:126) π/ (2Ω) (see Methods) and M represents aprojective measurement of the number of bosons at site 3(which could be implemented, in principle, through Fara-day rotation detection ). A measurement outcome of 0or M heralds a high-fidelity NOON state in subsystem B .For other measurement outcomes, the output is discardedand the process repeated (post-selection). Idealized limit
There is an idealized limit for which the above protocolhas perfect success probability and output fidelity. Taking t µ → µ → ∞ such that θ remains finite, and using theeffective Hamiltonian, provides explicit expressions for theuber-NOON states that result at steps (i) and (ii) | Ψ I1 i = 12 (cid:16) β | M, P, , i + | M, , , P i + | , P, M, i − β | , , M, P i (cid:17) | Ψ I2 i = 12 (cid:16) β | M, P, , i + e iP θ | M, , , P i + | , P, M, i − βe iP θ | , , M, P i (cid:17) (3)Note that due to the conservation of N + N and N + N under the effective Hamiltonian, Fock states such as | M, , P, i and | , M, , P i do not appear in the above ex-pression. Next, the two possible states at step (iii) dependon the measurement outcome r at site 3: | Ψ I3 i = √ (cid:0) β | M, P, , i + e iP θ | M, , , P i (cid:1) , r = 0 , √ (cid:0) | , P, M, i − βe iP θ | , , M, P i (cid:1) , r = M, (4)with β = ( − ( N +1) / . These states are recognized asproducts of a NOON state for subsystem B with Fock ba-sis states for subsystem A .In the non-ideal case with non-zero t µ and finite µ ,there is a small probability that the measurement outcome r is neither 0 or M . Numerical benchmarks for the mea-surement probabilities and NOON state output fidelitiesare provided in a later section. Next, we describe a secondprotocol. Protocol II
Now we specify an alternative protocol that does not in-volve measurements, so post-selection is not required. Em-ploying the same initial state | Ψ i , the following sequenceof steps are implemented to arrive at a NOON state insubsystem B (illustrated in Fig. 1):(i) | Ψ II1 i = U ( t m − t ν , , | Ψ i ;(ii) | Ψ II2 i = U ( t ν , , ν ) | Ψ II1 i ;3iii) | Ψ II3 i = U ( t m − t µ , , | Ψ II2 i ;(iv) | Ψ II4 i = U ( t µ , µ, | Ψ II3 i . Idealized limit
Similar to Protocol I, in the limit µ, ν → ∞ , t µ , t ν → U ( t, µ, ν ) with the effective Hamiltonianproduces | Ψ II1 i = 12 (cid:16) β | M, P, , i + | M, , , P i + | , P, M, i − β | , , M, P i (cid:17) ; | Ψ II2 i = 12 (cid:16) β | M, P, , i + | M, , , P i + i | , P, M, i − iβ | , , M, P i (cid:17) ; | Ψ II3 i = 1 √ (cid:16) | M, P, , i + βe − iπ/ | M, , , P i (cid:17) ; | Ψ II4 i = 1 √ (cid:16) | M, P, , i + Υ | M, , , P i (cid:17) (5)where Υ = β exp( i ( P θ − π/ Protocol fidelities
The analytic results provided above are obtained by em-ploying the effective Hamiltonian in an extreme limit, withdivergent applied fields acting for infinitesimally smalltimes. Below we give numerical simulations of the proto-cols to show that, for physically realistic settings where thefields are applied for finite times, high-fidelity outcomes forNOON state production persist.Throughout this section, we use | Ψ i to denote an an-alytic state, obtained in an idealized limit. We adopt | Φ i to denote a numerically calculated state, obtained by timeevolution with the EBHM Hamiltonian (2). Two sets ofparameters are chosen to illustrate the results (expressedin Hz):Set 1: { U/ (cid:126) = 75 . J/ (cid:126) = 24 . µ/ (cid:126) = 20 . } ;Set 2: { U/ (cid:126) = 76 . J/ (cid:126) = 73 . µ/ (cid:126) = 15 . } .For all numerical simulation results presented below, theinitial state is chosen as | Ψ i = | , , , i , i.e. M = 4and P = 11.The fidelities of Protocols I and II are defined as F I = | h Ψ I3 | Φ I3 i | and F II = | h Ψ II4 | Φ II4 i | , respectively. Thisis computed for
P θ ranging from 0 to π , achieved by vary-ing t µ . In the case of Protocol II, we use ν = µ for bothsets of parameters. The systems considered here can, inprinciple, be implemented using existing hardware – seePhysical proposal.The results are presented in Fig. 2, where it is seenthat F II is lower than F I . This can be attributed to twoprimary causes. The first is that, while Protocol I takes τ I ∼ t m to produce the final state, Protocol II requiresdouble the evolution time τ II ∼ t m . The longer evolutiontime contributes to a loss in fidelity. The second reason isthat, the measurement occurring in the final step of Pro-tocol I has the effect of renormalizing the quantum stateafter collapse, which increases the fidelity of the resultingNOON state when a measurement of r = 0 or r = M isobtained. However, there is a finite probability that themeasurement outcome is neither r = 0 nor r = M (seeSupplementary Note 2).In summary, both protocols display high fidelity resultsgreater than 0.9. For Protocol I the outcomes are proba-bilistic (See Supplementary Note 2 for data). By contrast,the slightly lower fidelity results of Protocol II are deter-ministic. F I a Protocol
I :
Set 1: r = r = M b Set 2: r = r = M π /2 π P θ (rad) F II c Protocol
II :
Set 1 0 π /2 π P θ (rad) d Set 2
Figure 2.
Fidelities for Protocols I and II. Numerical cal-culations of the fidelities F I ( a and b ) and F II ( c and d ). To vary P θ , µ is fixed and t µ is varied. a , c Set1: U/ (cid:126) = 75 .
876 Hz, J/ (cid:126) = 24 .
886 Hz, µ/ (cid:126) = 20 . t m ∼ . t ν ∼ . t µ varies from t = 0 to t ’ . P θ ∈ [0 , π ]. b , d Set 2: U/ (cid:126) = 76 . J/ (cid:126) = 73 .
219 Hz, µ/ (cid:126) = 15 .
168 Hz, t m ∼ . t ν ∼ . t µ varies from t = 0 to t ’ . P θ ∈ [0 , π ]. The required times t m , t m to producethe NOON states are comparable with typical lifetimes ofoptical lattice traps, which can be as large as a few min-utes . Readout statistics
A means to test the reliability of the system, through astatistical analysis of local measurement outcomes, is di-rectly built into the design. This results from the system’s4apacity to function as an interferometer . For both pro-tocols, once the output state has been attained we cancontinue to let the system evolve under U ( t m , , | Ψ IRO i , | Ψ IIRO i respec-tively for protocols I and II. In the idealized limits theseare | Ψ IRO i = c ( θ ) √ | M, P, , i + β | M, , , P i )+ is ( θ ) √ β | , P, M, i − | , , M, P i ) , r = 0 ,c ( θ ) √ | M, P, , i − β | M, , , P i ) − is ( θ ) √ β | , P, M, i − | , , M, P i ) , r = M, | Ψ IIRO i = 1 √ s (cid:16) θ − π P (cid:17) ( | M, P, , i + β | M, , , P i ) − i √ c (cid:16) θ − π P (cid:17) ( β | , P, M, i − | , , M, P i ) , where c ( θ ) ≡ cos ( P θ/
2) and s ( θ ) ≡ sin ( P θ/ | Ψ I RO i , the measurement probabilities at site 3 are P (0) =cos ( P θ/
2) and P ( M ) = sin ( P θ/ r = 0 , M in step (iii), we obtainfour possibilities for the total probabilities as P I (0 ,
0) = P I ( M,
0) = 0 . ( P θ/
2) and P I (0 , M ) = P I ( M, M ) =0 . ( P θ/ | Ψ II RO i , the measurementprobabilities at site 3 are P II (0) = sin ( P θ/ − π/
4) and P II ( M ) = cos ( P θ/ − π/ Methods
Resonant tunneling regime
The Hamiltonian (2) has large energy degeneracies when J = 0. Through numerical diagonalization of the inter-gable Hamiltonian for sufficiently small values of J , itis seen that the levels coalesce into well-defined bands,similar to that observed in an analogous integrable three-site model . By examination of second-order tunnelingprocesses (see Supplementary Note 1) In this regime, aneffective Hamiltonian H eff is obtained for this regime. P I a Protocol I P ( M ,0) P (0,0) P ( M , M ) P (0, M ) π /2 π − − π /4 π /2 3 π /4 π P θ (rad) P II b Protocol II P (0) P ( M ) π /2 π − − Figure 3.
Readout probabilities for Protocols I and II.Comparison between analytic and numerically-calculatedprobabilities for parameters of Set 1 (as in Figure 2) fordifferent values of
P θ . a Results for Protocol I. The pinkdot and the blue “x” (green square and the blue “+”) de-pict the probabilities of measuring r = 0 ( r = M ) duringthe readout, having measured r = M or r = 0 in step(iii) respectively. b Results for Protocol II. The probabil-ities of measuring N = 0 ( N = M ) in the readout areshown as green (orange) triangles. The dotted line depictsthe analytic predictions of the probabilities with respectto P θ . The insets show the accordance between predictedand calculated probabilities in semilogarithmic scale.For an initial Fock state | M − l, P − k, l, k i , with totalboson number N = M + P , the effective Hamiltonian is asimple function of the conserved operators with the form H eff = ( N + 1)Ω( Q + Q ) − Q Q , (6)where Ω = J / (4 U (( M − P ) − U = ( U − U ) / J (cid:28) U | M − P | , and it is this inequal-ity that we use to define the resonant tunneling regime.A very significant feature is that, for time evolution un-der H eff , both N + N = M and N + N = P are constant.The respective ( M + 1)-dimensional subspace associatedwith sites 1 and 3 and ( P +1)-dimensional subspace associ-ated with sites 2 and 4 provide the state space for the rele-vant energy band (see Supplementary Note 1). Restrictingto these subspaces and using the effective Hamiltonian (6)yields a robust approximation for (2).5 hysical proposal We propose a physical construction, consisting of dyspro-sium
Dy atoms trapped in an optical lattice, to test thetheoretical results. The trapping is accomplished by em-ploying two sets of counterpropagating laser beams withwavelength λ = 0 . µ m and waist w , with w (cid:29) λ .We consider each set of beams to cross with the other atan angle of 90 ◦ (cyan beams in Fig. 5) , generating asquare, two-dimensional optical lattice, in which the dis-tance between nearest wells is l = λ/ z -direction (blue beam in Fig. 5), with λ = 0 . µ m andwaist w = 1 . µ m, aligned to the center of a four-sitesquare plaquette, to isolate it from the rest of the lattice.Then, to achieve a pancake-shaped trap, it is necessary toinclude a set of two beams with λ = 0 . µ m and waist w ∼ w , whose orientations are disposed at an angle of α = 60 ◦ from each other (orange beams in Fig. 5), in-ducing a trapping aspect ratio of κ ≡ ω z /ω r = 1 . V ( r ): V ( r ) = 12 mω ( x + y ) + 12 mω z z + V sin (cid:20) k (cid:18) x − l (cid:19)(cid:21) + V sin (cid:20) k (cid:18) y − l (cid:19)(cid:21) , (7)where m is the atom’s mass, ω r = s m (cid:18) V k + 2 V w (cid:19) and ω z = s m (cid:18) π V d sw + V R (cid:19) , R k = πw k λ are, respectively, the radial and transverse trapping fre-quencies. Above, ω = p V / ( mw ) arises due to the iso-lation of the four-well system from the optical lattice. Thevalues V = V and V = 9 V are, respectively, the cen-tral beam’s and the x-z crossing beam’s potential depths, V is the 2D lattice potential depth, l is the distance be-tween nearest sites, k = 2 π/λ is the wave number, and d sw = λ/ (2 sin ( α/ z -axis. Since we are considering α = 60°, theminimum distance between the system’s horizontal layerand the next upper (or lower) layer is d sw = 2 l = λ , whichmakes irrelevant the tunneling contributions between dif-ferent horizontal layers.To establish equivalency between V ( r ) and the Hamil-tonian of Eq. (2), we employ the standard second-quantization procedure. From this, we calculate the on-site interaction parameter U as: U = U contact + U dip = κη π (cid:18) g − C dd f ( κ ) (cid:19) , (8) where κ is related to the trapping (pancake) shape as-pect, η ≡ mω r / (2 (cid:126) ), g ≡ π (cid:126) a/m , with a being the s-wave scattering length (tunable via Feshbach Resonance), C dd ≡ µ µ is the coupling constant, where µ is the vac-uum magnetic permeability, µ is the atomic magnetic mo-ment, and f ( κ ) is a function that describes how the dipo-lar interaction behaves for different geometries (encodedin κ ) . Taking site 1 as the “starting point’, the param-eter U , which accounts for the dipole-dipole interactionbetween atoms at sites 1 and j, is expressed as: U = C dd π Z ∞ d r r exp (cid:18) − r η (cid:19) J ( rd ) Z ( r ) , (9) Z ( r ) = r κ ηπ − r exp (cid:18) r κ η (cid:19) erfc r p κ η !! , where J is the Bessel function of first kind, d = l/δ ,if j = 2 ,
4, and d = l √ /δ , if j = 3. Here, the on-sitedipolar interaction is given by U dip = lim d ij → U ∝ f ( κ ).The term δ = 1 + 2 V / ( V k w ) arises when isolating thefour-site region from the rest of the lattice, which causesthe wells to slightly approach each other. Integrability condition
The physical setup above is able to simulate the EBHM.To achieve NOON-state generation, however, relies on theparticular case for which the EBHM is integrable; as ex-plained previously, this can be accomplished by making U = U , which we call the “integrability condition”. Theapproach is to first choose a value for the s-wave scat-tering length via Feshbach Resonance. Then, from thecondition just stated, one has to adjust ω r by varying thelaser beams intensities such that, at some point, U be-comes the same as U . From this point every Hamiltonianparameter is evaluated only after the integrability condi-tion is satisfied, which sets the intensity of the trappingscheme.By considering a = −
21 ( − . a , the system be-comes integrable at ω r ≈ π × .
078 (2 π × . V ≈ . E R (13 . E R ), where E R / (cid:126) = (cid:126) ( kπ ) / (2 ml ) = 26 .
894 kHz is the recoil en-ergy, which characterizes a deep lattice. This allows for ahigher stabilization of the system with a negative value forthe s-wave scattering length . Then, by using this trap-ping frequency to calculate the Hamiltonian parameters,one finds U/ (cid:126) ≈ .
876 (76 . J/ (cid:126) ≈ . . J between diagonal sites (1-3 and2-4), which is not included in the Hamiltonian (2), is verysmall if compared to J . From this, one infers that thetunneling between different horizontal layers of the opti-cal lattice is even smaller, since the distance between these6ayers is bigger than the distance between diagonal sitesby a factor of √ Figure 4.
Fulfillment of integrability condition. The s-wave scattering length value is set, following by a variationof the radial trapping frequency ω r up to the point at which U = U , corresponding to the frequency required for thesystem to be integrable. The long dashed, short dashedand solid lines depict U for a = − a and a = − . a and U , respectively, for different values of ω r . By set-ting a = − a ( − . a ), we find ω r ≈ π × . .
610 kHz ) as the frequency for integrability, whichresults in U / (cid:126) = U / (cid:126) ≈ .
282 Hz (161 .
797 Hz). Thepoints where U = U and the corresponding frequencies ω r are highlighted by the dotted lines. The system is ro-bust for small deviations from the integrable point (seeSupplementary Note 4 for more details). Breaking of integrability
To produce a controllable breaking of integrability, it issufficient to consider a second z -oriented Gaussian beam(green beam in Fig. 5), weaker than the one used for theregion isolation, with waist w b ∼ µ m and wavelength λ = 0 . µ m. This beam is displaced by ∆ x and ∆ y (with | ∆ x | = | ∆ y | ) from the center of the four-well system.When the laser is turned on, it implements the terms ν = 2 V b lw δ (∆ x + ∆ y ) , µ = 2 V b lw δ (∆ x − ∆ y ) , (10)where V b = 5 × − V is the potential depth generatedby the second beam. For | ∆ x | = | ∆ y | = 0 . µ m and thepreviously obtained radial trapping frequency, the param-eters µ and ν can (non-simultaneously) assume the valueof 20 .
870 (15 . M = 4 and P = 11, one should vary t µ from 0 to ∼ .
007 (0 . P θ from 0 to π . Also, from the condition2 νt ν / (cid:126) = π/ (2 M ), t ν ∼ .
009 (0 . λ and 2 λ ) . By changing the relativephase, the breaking of integrability can be simultaneouslycontrolled in all copies of the four-site plaquettes. Discussion
We have offered new techniques to address the highlychallenging problem of designing a framework to facili-tates NOON state creation. Our approach employs dipo-lar atoms confined to four sites of an optical lattice. Thesetup allows for the interactions to be tuned, and to fixthe couplings in such a way that the system is integrable.At these couplings, and for controlled perturbative break-ing of the integrability, the theoretical properties of thesystem become very transparent.The insights gained from integrability allowed usto develop two protocols. Protocol I employs a localmeasurement procedure to produce NOON states withslightly higher fidelities, over a shorter time, than Proto-col II. However Protocol I is probabilistic, requiring post-selection on the measurement outcome. This is in contrastto the deterministic approach of Protocol II. For both pro-tocols, phase-encoding is performed by breaking the sys-tem’s integrability, in a controllable fashion, at specificmoments during the time evolution. And in both protocolsthe output states were shown to have high-fidelity in nu-merical simulations. We also identifed a readout scheme,by converting encoded phases into a population imbalance,that allows verification of NOON state production throughmeasurement statistics.The approaches we have described, that are based onthe formation of an uber-NOON state en route to the fi-nal state, have two significant advantages. One is thatthe evolution time does not scale with the total number ofparticles. Instead, it is only dependent on the difference inparticle number of subsystem A and B in the Fock-stateinput. The other advantage is that all measurements aremade in the local Fock-state basis.We conducted an analysis of the feasibility of a physicalproposal. It was demonstrated that the long-range inter-action between dipolar atoms allows for an integrable cou-pling to be achieved, depending on the interplay betweencontact and dipolar interactions. Through the second-quantization procedure the values for the Hamiltonian pa-rameters were provided, derived by numerical calculations.These are seen to be realistic both in the context of optical-lattice setups and in comparison to literature. We also out-lined a procedure to improve the system’s robustness withrespect to error perturbation (see Supplementary Note 4for a broader description).Besides demonstrating the feasibility of NOON stategeneration, the physical setup we provide can also be em-ployed in the study of thermalization processes and othermany-body features of the EBHM. By establishing a linkbetween integrability and quantum technologies, this workpromotes advances in the field of neutral-atom quantuminformation processing.7 igure 5.
Representation of the trapping scheme. a Trapping scheme of the four-well model. In cyan, the two sets ofcounterpropagating beams are represented, with each set crossing at 90° with the other, providing the two-dimensionallattice trap. In orange, the two beams crossing at an angle of α = 60° are depicted, whose propagation occur inopposite orientations as seen with respect to z -axis, resulting in the pancake-shaped potential. In blue, the single laserbeam is illustrated, whose waist value is at the typical size of the four-site system, isolating it from the rest of the2D lattice. The external beam, used for breaking the system’s integrability, is depicted in green. b Zoom into theregion of the lattice which contains the four-well system. The blue beam represents the single laser beam isolating theregion of interest from the rest of the lattice. The green beam depicts the external beam, used to controllably breakthe system’s integrability, and the four pancakes illustrate the four wells of the system. c The dashed square in the x-y plane illustrates the square plaquette formed by the four-well system. The displacement of the central position ofthe green beam with respect to the center of the four-well system is represented by ∆ x and ∆ y , which implement thebreaking of the system’s integrability. d The light grey background represents the trapping potential in the vicinitiesof the four-well system. The four pancake-shaped wells, at a distance of l between nearest neighbors, are depicted inblue, the cyan spheres illustrate the trapped atoms and the purple arrows represent the aligned dipoles, which inducethe dipole-dipole interaction. Data availability
All relevant data are available on reasonable request fromthe authors.
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D.S.G. and K.W.W. were supported by CNPq (ConselhoNacional de Desenvolvimento Cient´ıfico e Tecnol´ogico),Brazil. A.F. acknowledges support from CNPq - EditalUniversal 430827/2016-4. A.F. and J.L. received fundingfrom the Australian Research Council through DiscoveryProject DP200101339. J.L. acknowledges the traditionalowners of the land on which The University of Queens-land operates, the Turrbal and Jagera people. We thankRicardo R. B. Correia and Bing Yang for helpful discus-sions.
Author contributions
All authors contributed to the conceptualization of theproject, and actively engaged in the writing of themanuscript. D.S.G, K.W.W. and L.H.Y. implemented thetheoretical analyses of the model, detailed the physicalproposal, and processed the numerical computations. J.L.and A.F. designed the research framework, and directedthe program of activities. 10 upplementary Note 1: Energy bands andeffective Hamiltonian
Here we give an overview of the origin for the effectiveHamiltonian. Recall that the integrability condition is U = U = U and U = U = U = U . When J = 0, the Fock state | M − l, P − k, l, k i is eigenstate ofthe Hamiltonian (2) with energy E = C − U ( M − P ) (S.1)where C = ( U + U ) N / − U /
2. The result is indepen-dent of l and k , indicating degeneracies. For small valuesof J , the degeneracies are broken and lead to energy levelsin well-defined bands, each with 2( M + 1)( P + 1) energylevels, except for N even, where the band with the highestenergy, M = P , will have ( M + 1)( P + 1) levels. Thelevel energy structure of the case we are analyzing, with N = 15, is shown in Supplementary Figure 1. In it, wehighlight in cyan the band with M = 4 and P = 11 (andvice versa), while the vertical lines marks the two setsof parameters pointed in the main text (repeated here,expressed in Hz):Set 1: { U/ (cid:126) = 75 . J/ (cid:126) = 24 . µ/ (cid:126) = 20 . } ;Set 2: { U/ (cid:126) = 76 . J/ (cid:126) = 73 . µ/ (cid:126) = 15 . } . U/J -150-100-500 E / J Supplementary Figure 1 . Energy band formation. Di-mensionless energy eigenvalues
E/J as a function of di-mensionless coupling
U/J , where U = ( U − U ) / C = 0 in (S.1). The dashed vertical line marks U/J ∼ U/J ∼
1) (concerning parameter Set 2), whilecyan depicts the band containing the expectation energyof the initial state | Ψ i = | , , , i . The formation ofthe bands is due to the quadratic dependence of ( M − P )in the energy (S.1).An effective Hamiltonian for each band is obtained byconsideration of second-order processes. Associated to la- bels M and P , such that N = M + P , we obtain H eff = J U ( M − P + 1) (cid:16) a a † + a a † (cid:17) (cid:16) a † a + a † a (cid:17) + J U ( M − P + 1) (cid:16) a a † + a a † (cid:17) (cid:16) a † a + a † a (cid:17) − J U ( M − P − (cid:16) a a † + a a † (cid:17) (cid:16) a † a + a † a (cid:17) − J U ( M − P − (cid:16) a a † + a a † (cid:17) (cid:16) a † a + a † a (cid:17) + J U (cid:18) M − P + 1 − M − P − (cid:19) × (cid:16) a † a a a † + a † a † a a + a a † a † a + a a a † a † (cid:17) . For a given initial Fock state, the resonant regime isachieved when the expectation energy lies in a region char-acterized by an energy band. There, the values of theintegrability-breaking parameters µ , ν may be as large asthe band-separation allows, which is depicted in Supple-mentary Figure 2. Supplementary Figure 2 . Energy bands for broken in-tegrability. Four-well model energy distribution for thetwo sets of parameters U and J , as in the main text. a Set 1:
U/J ∼ b Set 2:
U/J ∼
1. The vertical linesindicate the respective integrability-breaking parameters µ/ (cid:126) = 20 .
870 Hz ( a ) and µ/ (cid:126) = 15 .
168 Hz ( b ). The cyanlines represent the energy band associated to the initialstate | Ψ i = | , , , i .1 upplementary Note 2: Probabilities andfidelities Supplementary Table 1 shows the measurement probabil-ities of Protocol I, as well as the fidelity of the resultingstate with the respective NOON state, for M = 4, P = 11and the two aforementioned sets of parameters. The re- sulting NOON state from Protocol I can be either symmet-ric ( r = 0) or antisymmetric ( r = M ). For intermediatevalues for the outcome of measuring N , we calculate thefidelity of the resulting state with the symmetric NOONstate ( r = 0 ,
1) or the antisymmetric state ( r = 2 , , Protocol I:
Set 1
Phase (
P θ )Measurement 0 π/ π/ π/ π/ πr P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I M =4 0.4956 0.9996 0.4957 0.9996 0.4956 0.9996 0.4956 0.9996 0.4957 0.9996 0.4957 0.9996 Set 2
Phase (
P θ )Measurement 0 π/ π/ π/ π/ πr P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I P ( r ) F I M =4 0.4629 0.9886 0.4631 0.9886 0.4632 0.9887 0.4633 0.9887 0.4635 0.9887 0.4640 0.9888 Supplementary Table 1 . Measurement probabilities and NOON state fidelities. Probability of measuring r particlesat site 3 of Protocol I, and fidelity of the resulting state with the symmetric NOON state ( r = 0 ,
1) or the antisymmetricNOON state ( r = 2 , , M = 4 and P = 11. Supplementary Note 3: Readout statistics
For less ideal choices of parameters, it is possible to per-form a fitting on the readout probabilities amplitudes, suchthatProtocol I P (0 ,
0) = P ( M,
0) = c (cid:18) P θ (cid:19) P (0 , M ) = P ( M, M ) = c MM (cid:18) P θ (cid:19) Protocol II P (0) = c sin (cid:18) P θ − π (cid:19) P ( M ) = c M cos (cid:18) P θ − π (cid:19) where c , c MM , c and c M are constants that are ob-tained by fitting the numerically-evaluated data with theanalytic models. By choosing the parameters of Set 2, weobtain the following constants from a least-squares fitting: c = 0 . c MM = 0 . c = 0 .
954 and c M = 0 . .000.250.500.751.00 P I a Protocol I P ( M ,0) P (0,0) P ( M , M ) P (0, M ) π /2 π − − π /4 π /2 3 π /4 π P θ (rad) P II b Protocol II P (0) P ( M ) π /2 π − − Supplementary Figure 3 . Readout probabilities. Com-parison between analytic and numerically-calculated prob-abilities relating to the parameters of Set 2 with µ/ (cid:126) = ν/ (cid:126) = 15 .
168 Hz, for different values of
P θ . a) Probabil-ity distributions for measuring N = 0 ( N = M ) aftertime evolution subsequent to Protocol I. b) Probabilitydistributions of measuring N = 0 ( N = M ) after timeevolution subsequent to Protocol II. In both cases, the dot-ted lines refer to the analytic probabilities adjusted to thenumerical points according to the Eqs. (S.2). The coef-ficients used were c = 0 . c MM = 0 . c = 0 . c = 0 . Supplementary Note 4: Robustness
Here we analyze the system’s robustness in the presenceof a perturbation parameter, and outline a method to en-hance performance. Supposing that the integrability con-dition is subject to an error, denoted by ξ : ξ = U − U . (S.2)We find that the fidelities for the parameters Set 1 areabove 0.9 for an error parameter ξ/J up to ∼ . . ξ/J ∼ . ξ ) and negative ( − ξ ) devia-tions in Eq. (S.2). This can be done, for instance, by con-sidering a sequence of pulses . This is appropriate whenconsidering an error parameter in the physical setup: afterfixing the desired (approximate) s-wave scattering length, the trapping frequency adjustment may not have the re-quired precision, allowing for a minimum-error of ± ξ .Considering perturbations H ± ( µ, ν ) of the Hamilto-nian, with the form H ± ( µ, ν ) = H ( U ± , J ± ) + µ ± ( N − N )+ ν ± ( N − N ) ± ξ ( N N + N N ) , set ¯ U , ¯ J and ¯ µ as the mean values for the two cases + ξ and − ξ . We then calculate the times t m and t µ from thesemean values. Next, running a simulation that alternates N δt times between the extreme coupling values during theintegrable time evolution over t m − t µ leads to an increasein the system’s tolerance to the error, as depicted in Sup-plementary Figure 4. F I a Protocol
I :
Set 1: r = r = M b Set 2: r = r = M ξ/ J F II c Protocol
II :
Set 1 0 0.01 0.02 ξ/ J d Set 2
Supplementary Figure 4 . Robustness. NOON statesfidelities with respect to a perturbation parameter ξ , forProtocols I (panels a and b ) and II (panels c and d ). Pan-els a and c : Set 1. Panels b and d : Set 2. We considered ν = µ in Protocol II evaluations. For every ξ , we evaluatethe fidelities, with P θ = π/
2, for the Hamiltonian param-eters obtained by solving for ω r that corresponds to Eq.(S.2) (cf. Figure 4 of the main text). With N δt = 100 os-cillations between + ξ and − ξ , NOON states are producedwith fidelities higher than 0.9 for ξ/J up to 1 .
2% on theleft, and more than 2 .
0% on the right..
Supplementary Reference [1] Zhou, X., Jin, S. & Schmiedmayer, J. Shortcut load-ing a Bose–Einstein condensate into an optical lattice.
New J. Phys.20