Shortcut to adiabaticity in internal bosonic Josephson junctions
A. Yuste, B. Juliá-Díaz, E. Torrontegui, J. Martorell, J. G. Muga, A. Polls
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p Shortcut to adiabaticity in internal bosonic Josephson junctions
A. Yuste , B. Juli´a-D´ıaz , , E. Torrontegui , J. Martorell , J. G. Muga , , and A. Polls Deptartament d’Estructura i Constituents de la Mat`eria,Facultat de F´ısica, U. Barcelona, 08028 Barcelona, Spain ICFO-Institut de Ci`encies Fot`oniques, Parc Mediterrani de la Tecnologia, 08860 Barcelona, Spain Departamento de Qu´ımica-F´ısica, UPV-EHU, Apartado 644, 48080 Bilbao, Spain and Department of Physics, Shanghai University, 200444 Shanghai, People’s Republic of China
We extend a recent method to shortcut the adiabatic following to internal bosonic Josephson junc-tions in which the control parameter is the linear coupling between the modes. The approach is basedon the mapping between the two-site Bose-Hubbard Hamiltonian and a 1D effective Schr¨odinger-likeequation, valid in the large N (number of particles) limit. Our method can be readily implementedin current internal bosonic Josephson junctions and it improves substantially the production ofspin-squeezing with respect to usually employed linear rampings. PACS numbers: 03.75.Kk, 42.50.Dv, 05.30.Jp, 42.50.LcKeywords: shortcut to adiabaticity, spin squeezing, Bose-Einstein condensates
I. INTRODUCTION
Practical applications of quantum technologies will re-quire the controlled production of many-body correlatedquantum states, in particular ground states (g.s.). Itis thus desirable to find efficient mechanisms for theirfast production. Bosonic Josephson junctions (BJJs)are among the simplest systems whose ground statesalready contain many-body correlations beyond meanfield. Schematically, BJJs are ultracold bosonic vaporsin which, to a good approximation, the atoms populateonly two mutually interacting single-particle levels. Re-cently, BJJs have been studied experimentally by sev-eral groups [1–7]. Current nomenclature calls externalJosephson junctions those in which the two levels are spa-tially separated, usually by means of a potential barrier[1, 4, 7, 8]. In internal Josephson junctions instead, thetwo levels are internal to the same atom [5]. The two-siteBose-Hubbard Hamiltonian provides a suitable theoreti-cal description of both internal and external junctions [8–11]. A notable feature of this simple Hamiltonian is that,within subspaces of fixed number of particles, it can bemapped into an SU(2) spin model. This makes these sys-tems suitable to study very squeezed spin states [12, 13],as proven experimentally in Refs. [2, 4].In previous work [14] we described how a method toshortcut the adiabatic following in elementary quantummechanical systems could be applied to produce of spin-squeezed states in BJJs. In particular, we adapted a sim-ple method developed for harmonic oscillators in whichthe frequency could be varied in time [15, 16]. In [14]we described the most straightforward application, wherethe inter-atomic interaction strength was the control pa-rameter. This is nowadays a parameter that can be var-ied experimentally but it is difficult to control with goodaccuracy on the time scales considered. To overcome thisproblem, here we will extend the earlier protocol by vary-ing instead the linear coupling between the states (atomic levels) in internal junctions . This variation can be donewith fantastic accuracy [5, 17, 18] and we shall focus onthis case.The protocols to shortcut adiabatic evolution are gen-erally designed to drive in a finite time a system fromsome initial state to a final state that could be reachedadiabatically. An important advantage of these protocolsis that they can, in addition, aim at controlling otherproperties during the evolution, e.g. reducing transientenergy excitation, noise sensitivity or optimizing otherquantities of interest [19–21]. In addition, formulas toachieve shortcuts to adiabatic following are analytic forharmonic oscillator Hamiltonians [15]. From the experi-mental point of view the methods are capable to producea stationary eigenstate of the Hamiltonian at the finaltime, making it unnecessary to stop or freeze the dynam-ics.The paper is organized as follows. First in Section II wedescribe the theoretical framework. In Sec. III we presentour numerical results, including a specific subsection withparameter values within reach with current experiments.Finally, in Sec. IV we summarize the results and providesome concluding remarks. II. THEORETICAL FRAMEWORK
The dynamics of a BJJ can be well described by aquantized two-mode model [8, 10, 11], the Bose HubbardHamiltonian H = ¯ h H BH , H BH = − J ˆ J x + U ˆ J z , (1) In external junctions this can be done by increasing the barrierheight between the two wells where the pseudo-angular momentum operator ˆ J ≡{ ˆ J x , ˆ J y , ˆ J z } is defined asˆ J x = 12 (ˆ a † ˆ a + ˆ a † ˆ a ) , ˆ J y = 12 i (ˆ a † ˆ a − ˆ a † ˆ a ) , ˆ J z = 12 (ˆ a † ˆ a − ˆ a † ˆ a ) , (2)and ˆ a † j (ˆ a j ) creates (annihilates) a boson at site j . Forbosons: [ˆ a i , ˆ a † j ] = δ i,j . J is the hopping strength, and U is the non-linear coupling strength proportional to theatom-atom s -wave scattering length. In internal BJJs, U is proportional to a , + a , − a , , with a , and a , theintra-species scattering lengths and a , the inter-speciesone [5]. In this work we consider repulsive interactions, U >
0. For internal BJJs, the inter-species s -wave scat-tering length in Rb atoms can be varied by applyingan external magnetic field thanks to a well characterizedFeshbach resonance at B = 9 . J ( t ), keeping U and N fixed during the time evolution,which should be simpler and more accurate from an ex-perimental point of view.The time dependent Schr¨odinger equation (TDSE) iswritten as ı∂ t | Ψ i = H BH | Ψ i . (3)For a given N , an appropriate many-body basis isthe Fock basis, {| m z = ( N − N ) / i} , with m z = − N/ , . . . , N/
2. A general many-body state, | Ψ i , canthen be written as | Ψ i = N/ X m z = − N/ c m z | m z i . (4)It is useful to define the population imbalance of the stateas z ≡ m z / ( N/ ξ N ( t ) = ∆ ˆ J z / (∆ ˆ J z ) ref , where∆ ˆ J z ≡ h ˆ J z i − h ˆ J z i and (∆ ˆ J z ) ref = N/ h ˆ J y i = h ˆ J z i = 0. The many-bodystate is said to be number-squeezed when ξ N < ξ S = N (∆ ˆ J z ) / h ˆ J x i = ξ N /α , wherethe phase coherence of the many-body state is α ( t ) = h Ψ( t ) | J x /N | Ψ( t ) i . ξ S takes into account the delicatecompromise between improvements in number-squeezingand loss of coherence. States with ξ S < f J ( t ) H z LinearShortcut (a) f F i d e lit y (b) FIG. 1: (a) J ( t ) used in the shortcut protocol compared tothe corresponding linear ramping. The initial and final valuesof γ are γ i =10, γ f =20, and t f = 0 . t ( i )Rabi . In panel (b) wedepict the fidelity (overlap) between the evolved state and theinstantaneous ground state. The number of particles and non-linearity are, N = 100, and U = 1 / (50 t ( i )Rabi ) , respectively. Since we take J as the control parameter we slightly de-tour from the derivation in Refs. [23–25]. Following sim-ilar steps as described in those references, one can obtainin the semiclassical η ≡ /N ≪ iη∂ t ψ ( z, t ) = H N ψ ( z, t ) (5)for the continuous extrapolation of z , where H N ( z ) ψ ( z ) ≡ − η J ∂ z p − z ∂ z ψ ( z ) + V ( z ) ψ ( z ) , (6)and H N ≡ η H BH , V ( z ) = − J √ − z + (1 / N U/ z . ψ ( z ) = p N/ c m z is normalized as R − dz | ψ ( z ) | = 1.For repulsive atom-atom interactions the potential inFock space, V ( z ), is to a very good approximation a har-monic oscillator. Neglecting the z dependence of the ef-fective mass term and expanding √ − z ≃ − z / V ( z ), the Hamiltonian in Eq. (6) reduces to H N ≃ − Jη ∂ z + 12 ( J + N U/ z , (7)A difference with respect to Ref. [14] and to previousapplications of shortcuts-to-adiabaticity to harmonic-oscillator expansions is that now the control parame-ter J ( t ) shows up both as a formal time-dependent (in-verse of) mass and as an additive term in the force con-stant. In the Appendix A we provide the extensionof the shortcut technique for this type of time depen-dence when N U/ ≫ | J | , so that we can approximate( J + N U/ ≃ N U/
2. Defining γ = N U/ (2 J ), this limitcorresponds to γ ≫
1, which is easily attainable in cur-rent experiments. It is also relevant as it corresponds tovery spin-squeezed ground states of the bosonic Joseph-son junction. In Appendix B we verify that the methodis not applicable when | γ | < J ( t ) in the fol-lowing Ermakov equation,¨ b − b ) b = 4 kJ ( t ) b − k η b , (8) Rabi(i) -0.0500.050.10.150.2 J ( t ) SC-PolynomialSC-Non-PolynomialLinear (a)
Rabi(i) α SC-PolynomialSC-adiabaticLinearLinear-adiabatic (b)
Rabi(i) -10-50 ξ N SC-PolynomialSC-adiabaticLinearLinear-adiabatic (c)
Rabi(i) F i d e lit y SC-PolynomialSC-Non-PolynomialLinear (d)
FIG. 2: Panel (a): evolution of J ( t ), (b) coherence of the state, (c) its number squeezing, and (d) the instantaneous fidelity. N = 100 atoms, γ i = 10, γ f = 100 and t f = 0 . t ( i )Rabi . For t > t f we fix γ ( t ) = γ ( t f ). where the dots indicate time derivatives, and k = N U/ b ( t ) must satisfy the boundary conditions b ≡ b (0) = (cid:18) η J (0) N U (cid:19) / , (9) b f ≡ b ( t f ) = (cid:18) η J ( t f ) N U (cid:19) / , ˙ b (0) = ¨ b (0) = ˙ b ( t f ) = ¨ b ( t f ) = 0 . (10)For simplicity we apply the polynomial [15] b poly ( t ) = b + 10( b f − b ) s − b f − b ) s +6( b f − b ) s . (11)with s = t/t f . We also consider a non-polynomial formin some comparisons, b non − poly ( t ) = b (cid:18) b f b (cid:19) s − s +10 s . (12) III. NUMERICAL SIMULATIONS OF THESHORTCUT PROTOCOL
In all cases we will consider the evolution from an ini-tial g.s. corresponding to γ = γ i to a final one with γ = γ f . The control parameter J ( t ), will go from J (0) = J i to J ( t f ) = J f in a time t f with a fixed valueof U . In our first application, we will measure the timein units of the initial Rabi time, t ( i )Rabi = π/J i . Later, wewill consider realistic values of U and t taken from recentexperiments.In Fig. 1 we consider a factor 2 change in γ , from γ i = 10 to γ f = 20 in a time t f = 0 . t ( i )Rabi , with N = 100 and N U/ /t ( i )Rabi . We compare the short-cut protocol using the polynomial ansatz for b ( t ) to alinear ramping: J ( t ) = J i + ( J f − J i )( t/t f ). The short-cut method is shown to work almost perfectly, and weobtain a final fidelity ≃ ≃ . f (ms)11.011.021.031.04 α ( t f ) / α a d i a b a ti c ( t f ) (a)
20 40 60 80 100t f (ms)-0.200.20.40.60.8 ξ Ν ( t f ) / ξ N , a d i a b a ti c ( t f ) ShortcutLinear (b) f (ms)0.60.70.80.9 F i n a l f i d e lit y (c) f (ms)051015 J m a x / ( NU / ) (d) FIG. 3: Properties of the state of the system at t f after a shortcut protocol (black) and a linear ramp (red), for different valuesof t f . (a) depicts the relative coherence, α/α adiabatic . (b) shows the relative number squeezing, ξ N /ξ N, adiabatic . (c) contains thevalue of the final fidelity. (d) shows the maximum value of 1 /γ required for the shortcut process. In all simulations γ i = 10, γ f = 100, N = 100, and U = 0 .
49 Hz.
As it occurred with the harmonic oscillator [15] or inRef. [14], for more stringent processes, i.e. shorter finaltimes or larger changes in γ , the method requires nega-tive values of the control parameter. For instance, if werequire a factor of 10 change, from γ i = 10 to γ f = 100under the same conditions, J ( t ) becomes negative dur-ing part of the evolution. Although for usual tunnelingphenomena the hopping term is always positive, e.g. inexternal Josephson junctions, there are several proposalsto implement negative or even complex hopping terms inoptical lattices [26, 27]. For the internal Josephson junc-tions achieved in Oberthaler’s group negative tunnelingpresents no obstacle as they are able of engineering atunneling term of the form (see Sect. 3.5 of Ref. [17]) J ( t ) h ˆ J x cos φ c ( t )) + ˆ J y sin φ c ( t ) i (13)with φ c ( t ) a phase which can be controlled externally.Our results are shown in Fig. 2. First we see that, forboth polynomial and non-polynomial choices of b ( t ) de-scribed above, J changes its sign at intermediate times,see Fig. 2(a). This implies a transient loss of fidelity (overlap) between the evolved state and the instanta-neous ground state of the system, as shown in Fig. 2(d).With the shortcut protocol both the coherence, Fig. 2(b),and number squeezing, Fig. 2(c), evolve smoothly to-wards their adiabatic value. In contrast, the linear ramp-ing fails to provide the adiabatic values at the final time.The instantaneous ground state coherence, dotted redline in Fig. 2(b), is rather involved as it follows the J ( t )path. As seen in Fig. 2(c) the linear squeezing is ≃ − ≃ −
10 dB. This is a notable featurewhich should be experimentally accessible. The linearramp gives a final fidelity of 0.95, well bellow those of thepolynomial and non-polynomial shortcut protocols whichget final fidelities of nearly 1. It is also worth stress-ing that the many-body state produced by the shortcutmethod at t = t f is almost an eigenstate of the sys-tem, which implies constant coherence and squeezing for t > t f , see Fig. 2(b,c).It is also possible to engineer fidelity-one processeswhere the control J ( t ) is constrained from below and -2-1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.2 J t/t final (100ms) -0.4-0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 z J t/t final (50ms) -0.4-0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 z J t/t final (20ms) -0.4-0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 z J t/t final (10ms) -0.4-0.2 0 0.2 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 z FIG. 4: (left) Evolution of J ( t ) required by the polynomial shortcut protocol for different final times t f = 100 , , ,
10 ms.(right) Spectral decomposition of the many-body state | c z | for the same t f as a function of time. In all simulations γ i = 10, γ f = 100, N = 100, and U = 0 .
49 Hz. above by predetermined values (in particular we couldmake both bounds positive). Prominent examples are thebang-bang protocols, with step-wise constant J , whichsolve the time-minimization variational problem for givenbounds and boundary conditions [16, 19, 28, 29]. A. Simulations using experimental values for theparameters
As explained above, the variation of J with time can bereadily implemented experimentally. In this section, wewill consider realistic values of the parameters. FollowingRefs. [4, 17] we take a value of the non-linearity U = 0 . N = 100 atoms, and make variations of γ duringtypical experimental values of time: t f = 10 , , , t = t f we fix γ ( t ) = γ f and evolve thesystem during an additional small time to check whetherthe state remains close to desired final the ground stateor not.In Fig. 3 we depict the final value of the fidelity (c),number squeezing (b) and coherence of the many-bodystate (a), as a function of the final time imposed t f . The shortcut method (with polynomial ansatz) is comparedto the linear ramping. The first observation is that for t f >
40 ms, the shortcut protocol produces a fidelity ≃ .
95, see (c).Similarly, for t f >
40 ms, the final coherence and num-ber squeezing are essentially those of the correspondingground state (a,b). This is an important finding, as forinstance the linear ramping produces roughly half of thenumber squeezing as compared to the adiabatic or short-cut protocol. For t f <
40 ms, the shortcut protocol isseen to fail, and in particular, the achieved final fideli-ties drop to 0 . t f = 10 ms, smaller than the linearramping ones. As explained above, our shortcut proto-col has been derived assuming the validity of a parabolicapproximation for the potential in Fock space. There-fore we expect the method to fail when the intermediatewave packet spreads far from the central region in Fockspace. In Fig. 4, we have plotted the spectral decompo-sition of the many-body state | c z | as a function of timefor the same final times t f as above. When the finaltime is large, the process is smooth and the wave func-tion does not spread considerably. When we use shorterfinal times J ( t ) takes large values (so γ is small at inter- f -200-1000100200 J ( t ) N=50N=100N=200N=400 (a) F i n a l f i d e lit y t=20mst=10ms (b) FIG. 5: (a) Evolution of J ( t ) required by the polynomialshortcut protocol for different values of N and t f =20 ms. In(b) we depict the final fidelities attained in the process for t f =10 and 20 ms. U = 0 .
49 Hz and γ i = 10, γ f = 100. mediate times), and the effective wave-function spreadsconsiderably in z space. A parameter that affects the J ( t ) functional form is the number of atoms N . Thelarger N , the smoother the J ( t ) path and the better arethe results obtained. This is seen in Fig. 5, where wechoose only two values of t f : 10 and 20 ms, and consider N = 50 , ,
150 and 400 atoms. We also depict J ( t ),which is on average smaller for larger N . IV. SUMMARY AND CONCLUSIONS
We have presented a method to produce ground statesof bosonic Josephson junctions for repulsive atom-atominteractions using protocols to shortcut the adiabatic fol-lowing. We inverse-engineer the accurately controllablelinear coupling J by mapping a Schr¨odinger-like equa-tion for the (imbalance) wavefunction of the Josephsonjunction onto an ordinary harmonic oscillator for whichshortcut protocols can be set easily. The original equa-tion is a priori more involved for that end, as the kinetic-like term includes a time-dependent formal mass. As de-tailed in Appendix A, the mapping requires a reinterpre-tation of kinetic and potential terms, which interchangetheir roles in a representation conjugate to the imbalance.The time dependence of the formal mass of the originalequation (inversely proportional to J ) implies the timedependence of the frequency of the ordinary (constantmass) harmonic oscillator, and J plays finally the roleof the squared frequency. This mapping is different andshould be distinguished from the ones used to treat har-monic systems with a time dependent mass both in thekinetic and the potential terms [30]. From the experi-mental point of view, our protocol should help the pro-duction of spin-squeezed states, increasing the maximumsqueezing attainable in short times. In particular, an im-portant shortcoming of recent experimental setups [17],is that they have sizable particle loss on time scales ofthe order of ≃
50 ms for atom numbers on the order of afew hundreds. For these systems our methods could betargeted at shorter times, as in the examples presented,providing an important improvement with respect to lin-ear rampings.
Acknowledgments
The authors thank M. W. Mitchell for a careful read-ing of the manuscript and useful suggestions. This workhas been supported by FIS2011-24154, 2009-SGR1289,IT472-10, FIS2009-12773-C02-01, and the UPV/EHUunder program UFI 11/55. B. J.-D. is supported by theRam´on y Cajal program.
Appendix A: Shortcut equations for the Josephsonjunction with controllable linear coupling.
In this Appendix we shall transform the Schr¨odinger-like Eq. (5) so that the invariant-based engineering tech-nique for time-dependent harmonic oscillators developedin [15, 16] may be applied. The structure of the Hamil-tonian (7) is peculiar as it involves a time dependent(formal) mass factor in the kinetic-like term. The firststep is to transform this Hamiltonian according to η → ¯ h, J ( t ) → m ( t ) , N U → k , (A1)to rewrite Eq. (7) as H = 12 m ( t ) p + 12 kz , (A2)where p = − i ¯ h∂ z is the “momentum” conjugate to z .These and other transformations performed below areformal so that the dimensions do not necessarily corre-spond to the ones suggested by the symbols and termi-nology used. For example neither p , m ( t ), or z havedimensions of momentum, mass and length, respectively.Multiplying the time-dependent Schr¨odinger equationcorresponding to Eq. (A2) from the left by momentumeigenstates h p | , i ¯ h∂ t Ψ( p, t ) = − ¯ h k ∂ ∂p Ψ( p, t )+ p m ( t ) Ψ( p, t ) . (A3)Finally with the new mapping k → m x , (A4)1 m ( t ) → m x ω x ( t ) ,p → x , the Hamiltonian takes the standard time-dependent har-monic oscillator form H = − ¯ h m x ∂ ∂x + 12 m x ω x ( t ) x . (A5) We shall use the symbol p also for the momentum eigenvaluessince the context makes its meaning clear. t/t Rabi(i) F i d e lit y LinearSC-PolynomialSC-Non-Polynomial
FIG. 6: Instantaneous fidelity as a function of time. γ i = 0 . γ f = 1 with t f = t ( i )Rabi for the shortcut and a linear path. Note that, thanks to the above transformations and basischange, the kinetic-like and potential-like terms in theHamiltonian (7) have interchanged their roles so that thetime dependence of the formal mass has become a timedependence of the formal frequency in Eq. (A5), whereas m x is constant. Fast dynamics between t = 0 and t f ,from ω x (0) to ω x ( t f ) without final excitations for thisHamiltonian may be inverse engineered by solving for ω x ( t ) in the Ermakov equation [15, 16]¨ ρ + ω x ( t ) ρ = ω ρ , (A6) where ω is in principle an arbitrary constant, and ρ ( t ) isa scale factor for the state that we may design, e.g. witha polynomial, so that it satisfies the boundary conditions ρ (0) = (cid:18) ω ω x (0) (cid:19) / , ρ ( t f ) = (cid:18) ω ω x ( t f ) (cid:19) / , ˙ ρ (0) = ˙ ρ ( t f ) = ¨ ρ (0) = ¨ ρ ( t f ) = 0 . (A7)Defining b = ¯ h/ρ , choosing ω = k ¯ h , and undoing thechanges (A4) and (A1) we rewrite the Ermakov equationas Eq. (8) and the boundary conditions become those inEq. (9). Appendix B: Limitations of the method
As explained in the main text, when we have donethe mapping between the results in Appendix A and theBose-Hubbard Hamiltonian, we have assumed
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