Finite-size effects in a bosonic Josephson junction
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Finite-size effects in a bosonic Josephson junction
Sandro Wimberger,
1, 2
Gabriele Manganelli,
3, 4
Alberto Brollo, and Luca Salasnich
3, 5, 6, 7 Dipartimento di Scienze Matematiche, Fisiche ed Informatiche,Universit`a di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy INFN - Sezione di Milano-Bicocca, gruppo collegato di Parma,Parco Area delle Scienze 7/A, 43124 Parma, Italy Dipartimento di Fisica e Astronomia ’Galileo Galilei’,Universit`a di Padova, via Marzolo 8, 35131 Padova, Italy Scuola Galileiana di Studi Superiori, Universit`a di Padova,via San Massimo 33, 35129 Padova, Italy Padua Quantum Technologies Research Center, Universit`a di Padova,via Gradenigo 6/b, 35131 Padova, Italy INFN - Sezione di Padova, via Marzolo 8, 35131 Padova, Italy CNR-INO, via Nello Carrara 1, 50019 Sesto Fiorentino, Italy
We investigate finite-size quantum effects in the dynamics of N bosonic particles which are tun-neling between two sites adopting the two-site Bose-Hubbard model. By using time-dependentatomic coherent states (ACS) we extend the standard mean-field equations of this bosonic Joseph-son junction, which are based on time-dependent Glauber coherent states. In this way we find 1 /N corrections to familiar mean-field (MF) results: the frequency of macroscopic oscillation betweenthe two sites, the critical parameter for the dynamical macroscopic quantum self trapping (MQST),and the attractive critical interaction strength for the spontaneous symmetry breaking (SSB) of theground state. To validate our analytical results we perform numerical simulations of the quantumdynamics. In the case of Josephson oscillations around a balanced configuration we find that alsofor a few atoms the numerical results are in good agreement with the predictions of time-dependentACS variational approach, provided that the time evolution is not too long. Also the numericalresults of SSB are better reproduced by the ACS approach with respect to the MF one. Insteadthe onset of MQST is correctly reproduced by ACS theory only in the large N regime and, for thisphenomenon, the 1 /N correction to the MF formula is not reliable. PACS numbers: 03.75.Lm; 74.50.+r
I. INTRODUCTION
The Josephson junction is a quantum mechanical de-vice made of two superconductors, or two superfluids,separated by a tunneling barrier [1]. The Josephsonjunction can give rise to the direct-current (DC) Joseph-son effect, where a supercurrent flows indefinitely longacross the barrier, but also to the alternate-current (AC)Josephson effect, where due to an energy difference thesupercurrent oscillates periodically across the barrier[2]. The superconducting quantum interference devices(SQUIDs), which are very sensitive magnetometers basedon superconducting Josephson junctions, are widely usedin science and engineering [3]. Moreover, Josephson junc-tions are now used to realize qubits (see, for instance,[4, 5]).The achievement of Bose-Einstein condensation withultracold and dilute alkali-metal atoms [6] has renewedand increased the interest on macroscopic quantum phe-nomena and, in particular, on the Josephson effect [7].Indeed, contrary to the case of superconducting Joseph-son junctions, with atomic Josephson junctions it is pos-sible to have a large population imbalance with the ap-pearance of the self-trapping phenomenon [8]. A directexperimental observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction was madein 2005 with Rb atoms [11]. More recently, in 2015, Josephson effect has been detected in fermionic superflu-ids across the BEC-BCS crossover with Li atoms [12].The fully quantum behavior of Josephson junctions isusually described by using the phase model [13], whichis based on the quantum commutation rule [14] betweenthe number operator ˆ N and the phase angle operatorˆ φ . Within this model it has been found that quantumfluctuations renormalize the mean-field Josephson oscil-lation [15–17]. However, the phase angle operator ˆ φ isnot Hermitian, the exponential phase operator e i ˆ φ is notunitary, and their naive application can give rise to wrongresults. Despite such problems, the phase model is con-sidered a good starting point in many theoretical studiesof Josephson junctions, because the phase-number com-mutation rule is approximately correct for systems witha large number of condensed electronic Cooper-pairs orbosonic atoms [16].In this paper we study finite-size quantum effects ina Josephson junction avoiding the use of the phase op-erator. The standard mean-field theory is based on theGlauber coherent state | CS i which however is not eigen-state of the total number operator [18]. Here we adoptthe atomic coherent state | ACS i , which is instead eigen-state of the total number operator, and it reduces to theGlauber coherent state only in the limit of a large num-ber N of bosons [19–24]. We prove that the frequency ofmacroscopic oscillation of bosons between the two sites isgiven by p J + N U J (1 − /N ) / ~ , where J is the tun-neling energy, U is the on-site interaction energy. Re-markably, for very large number N of bosons this formulabecomes the familiar mean-field one √ J + N U J/ ~ . Wefind similar corrections for the critical strength of the dy-namical self-trapping and for the critical strength of thepopulation-imbalance symmetry breaking of the groundstate. Once again in these cases the standard mean-fieldresults are retrieved in the limit of a large number N ofbosons. In the last part of the paper we compare the ACStheory with numerical simulations. In the case of Joseph-son oscillations we find a very good agreement betweenACS theory and numerical results also for a small number N of bosons. For the ACS critical interaction strengthof the semiclassical spontaneous symmetry breaking ofthe ground state we obtain a reasonable agreement withthe numerical results. Instead, for the phenomenon ofself-trapping, our numerical quantum simulations suggestthat the 1 /N corrections predicted by the ACS theoryare not reliable. We attribute this discrepancy to the in-creased importance of quantum fluctuations and strongermany-body correlation in the so-called Fock regime, seee.g. [25]. II. TWO-SITE MODEL
The macroscopic quantum tunneling of bosonic parti-cles or Cooper pairs in a Josephson junction made of twosuperfluids or two superconductors separated by a poten-tial barrier can be described within a second-quantizationformalism, see for instance [26]. The simplest quantumHamiltonian of a system made of bosonic particles whichare tunneling between two sites ( j = 1 ,
2) is given byˆ H = − J (cid:0) ˆ a +1 ˆ a + ˆ a +2 ˆ a (cid:1) + U X j =1 , ˆ N j ( ˆ N j − , (1)where ˆ a j and ˆ a + j are the dimensionless ladder operatorswhich, respectively, destroy and create a boson in the j site, ˆ N j = ˆ a + j ˆ a j is the number operator of bosons in the j site. U is the on-site interaction strength of particles and J > ~ . Eq. (1) is the so-calledtwo-site Bose-Hubbard Hamiltonian. We also introducethe total number operatorˆ N = ˆ N + ˆ N . (2)The time evolution of a generic quantum state | ψ ( t ) i of our system described by the Hamiltonian (1) is thengiven by the Schr¨odinger equation i ~ ∂∂t | ψ ( t ) i = ˆ H | ψ ( t ) i . (3)Quite remarkably, this time-evolution equation can bederived by extremizing the following action S = Z dt h ψ ( t ) | (cid:18) i ~ ∂∂t − ˆ H (cid:19) | ψ ( t ) i , (4) characterized by the Lagrangian L = i ~ h ψ ( t ) | ∂∂t | ψ ( t ) i − h ψ ( t ) | ˆ H | ψ ( t ) i . (5)Clearly, Eqs. (3–5) hold for any quantum system. III. STANDARD MEAN-FIELD DYNAMICS
The familiar mean-field dynamics of the bosonicJosephson junction can be obtained with a specific choicefor the quantum state | ψ ( t ) i , namely [27] | ψ ( t ) i = | CS ( t ) i , (6)where | CS ( t ) i = | α ( t ) i ⊗ | α ( t ) i (7)is the tensor product of Glauber coherent states | α j ( t ) i ,defined as | α j ( t ) i = e − | α j ( t ) | e α j ( t )ˆ a + j | i (8)with | i the vacuum state, and such thatˆ a j | α j ( t ) i = α j ( t ) | α j ( t ) i . (9)Thus, | α j ( t ) i is the eigenstate of the annihilation opera-tor ˆ a j with eigenvalue α j ( t ) [18]. The complex eigenvalue α j ( t ) can be written as α j ( t ) = q N j ( t ) e iφ j ( t ) , (10)with N j ( t ) = h α j ( t ) | ˆ N j | α j ( t ) i the average number ofbosons in the site j at time t and φ j ( t ) the correspondingphase angle at the same time t .Adopting the coherent state (7) with Eq. (8) the La-grangian (5) becomes L CS = i ~ h CS ( t ) | ∂∂t | CS ( t ) i − h CS ( t ) | ˆ H | CS ( t ) i = N ~ z ˙ φ − U N z + JN p − z cos ( φ ) , (11)where the dot means the derivative with respect to time t , N = N ( t ) + N ( t ) (12)is the average total number of bosons (that is a constantof motion), φ ( t ) = φ ( t ) − φ ( t ) (13)is the relative phase, and z ( t ) = N ( t ) − N ( t ) N (14)is the population imbalance. The last term in the La-grangian (11) is the one which makes possible the pe-riodic oscillation of a macroscopic number of particlesbetween the two sites.In the Lagrangian L CS ( φ, z ) of Eq. (11) the dynamicalvariables φ ( t ) and z ( t ) are the generalized Lagrangiancoordinates (see, for instance, [28]). The extremizationof the action (4) with the Lagrangian (11) gives rise tothe Euler-Lagrange equations ∂L CS ∂φ − ddt ∂L CS ∂ ˙ φ = 0 , (15) ∂L CS ∂z − ddt ∂L CS ∂ ˙ z = 0 , (16)which, explicitly, become˙ φ = J z √ − z cos ( φ ) + U N z , (17)˙ z = − J p − z sin ( φ ) . (18)These equations describe the mean-field dynamics of themacroscopic quantum tunneling in a Josephson junction,where φ ( t ) is the relative phase angle of the complex fieldof the superfluid (or superconductor) between the twojunctions at time t and z ( t ) is the corresponding relativepopulation imbalance of the Bose condensed particles (orCooper pairs).Assuming that both φ ( t ) and z ( t ) are small, i.e. | φ ( t ) | ≪ | z ( t ) | ≪
1, the Lagrangian (11) can beapproximated as L (2) CS = N ~ z ˙ φ − JN φ − ( JN + U N )2 z , (19)removing a constant term. The Euler-Lagrange equa-tions of this quadratic Lagrangian are the linearizedJosephson-junction equations ~ ˙ φ = ( J + U N ) z , (20) ~ ˙ z = − Jφ , (21)which can be rewritten as a single equation for the har-monic oscillation of φ ( t ) and the harmonic oscillation of z ( t ), given by ¨ φ + Ω φ = 0 , (22)¨ z + Ω z = 0 , (23)both with frequencyΩ = 1 ~ p J + N U J , (24)that is the familiar mean-field frequency of macroscopicquantum oscillation in terms of tunneling energy
J > U , and number N of particles [8].It is straightforward to find that the conserved energyof the mean-field system described by Eqs. (17) and (18)is given by E CS = U N z − JN p − z cos ( φ ) . (25) If the condition E CS ( z (0) , φ (0)) > E CS (0 , π ) (26)is satisfied then h z i 6 = 0 since z ( t ) cannot become zeroduring an oscillation cycle. This situation is known asmacroscopic quantum self trapping (MQST) [8–10]. In-troducing the dimensionless strengthΛ = N UJ , (27)the expression (25) and the trapping condition (26) giveΛ
MQST = 1 + p − z (0) cos( φ (0)) z (0) / > Λ MQST (29)is the familiar mean field condition to achieve MQSTin BECs [8]. We stress that MQST condition cruciallydepend on the specific initial conditions φ (0) and z (0).Let us study the stationary solutions of (11). From thesystem of Eqs. (17) and (18) we obtain the symmetricsolutions (˜ z − , ˜ φ ) = (0 , nπ ) (30)(˜ z + , ˜ φ ) = (0 , (2 n + 1) π ) (31)with n ∈ Z ,respectively with energies ˜ E − = − JN and ˜ E + = JN . Due to the nonlinear interaction thereare degenerate ground-state solutions that break the z-symmetry z ± = ± r − (32) φ n = 2 πn (33)where n ∈ Z . These solutions give a minimum of theenergy with φ = 0 only for Λ = U N/J <
0. Thus, thespontaneous symmetry breaking (SSB) of the balancedground state ( z = 0, φ = 0) appears at the critical di-mensionless strength Λ SSB = − . (34)In other words, for Λ = U N/J < Λ SSB = − z of the ground state of our bosonicsystem becomes different from zero. IV. FINITE-SIZE EFFECTS
Different results are obtained by choosing anotherquantum state | ψ ( t ) i in Eqs. (4) and (5). In this sec-tion, our choice for the quantum state | ψ ( t ) i is | ψ ( t ) i = | ACS ( t ) i , (35)where | ACS ( t ) i = (cid:18)q z ( t )2 ˆ a +1 + q − z ( t )2 e − iφ ( t ) ˆ a +2 (cid:19) N √ N ! | i (36)is the atomic coherent state [19], also called SU(2) coher-ent state or Bloch state or angular momentum coherentstate [20], with | i the vacuum state. This atomic coher-ent state depends on two dynamical variables φ ( t ) and z ( t ) which, as we shall show, can be again interpreted asrelative phase and population imbalance of the Josephsonjunction [21–24, 29, 30]. Contrary to the Glauber coherent state | CS ( t ) i of Eq.(7), the atomic coherent state of Eq. (36) is an eigenstateof the total number operator (2), i.e.ˆ N | ACS ( t ) i = N | ACS ( t ) i . (37)Moreover, the averages calculated with the atomic co-herent state | ACS ( t ) i become equal to the ones per-formed with the Glauber coherent state | CS ( t ) i only inthe regime N ≫ L ACS = i ~ h ACS ( t ) | ∂∂t | ACS ( t ) i − h ACS ( t ) | ˆ H | ACS ( t ) i = N ~ z ˙ φ − U N (cid:18) − N (cid:19) z + JN p − z cos ( φ ) . (38)Comparing this expression with the Lagrangian of theGlauber coherent state, Eq. (11), one immediately ob-serves that the two Lagrangians become equal under thecondition N ≫
1. Moreover, the former is obtained fromthe latter with the formal substitution U → U (1 − /N ).In other words, the term (1 − /N ) takes into account few-body effects, which become negligible only for N ≫ φ = J z √ − z cos ( φ ) + U N (cid:18) − N (cid:19) z , (39)˙ z = − J p − z sin ( φ ) , (40)which are derived as the Euler-Lagrange equations of theLagrangian (38).Assuming that both φ ( t ) and z ( t ) are small, i.e. | φ ( t ) | ≪ | z ( t ) | ≪
1, the Lagrangian (38) can beapproximated as L (2) ACS = N ~ z ˙ φ − JN φ − ( JN + U (cid:0) − N (cid:1) N )2 z , (41)removing a constant term. The Euler-Lagrange equa-tions of this quadratic Lagrangian are the linearizedJosephson-junction equations ~ ˙ φ = (cid:18) J + U N (cid:18) − N (cid:19)(cid:19) z , (42) ~ ˙ z = − Jφ , (43)which can be rewritten as a single equation for the har-monic oscillation of φ ( t ) and the harmonic oscillation of z ( t ), given by ¨ φ + Ω A φ = 0 , (44)¨ z + Ω A z = 0 , (45) both with frequencyΩ A = 1 ~ s J + N U J (cid:18) − N (cid:19) , (46)that is the atomic-coherent-state frequency of macro-scopic quantum oscillation in terms of tunneling energy J , interaction strength U , and number N of particles.Quite remarkably, this frequency is different and smallerwith respect to the standard mean-field one, given by Eq.(24). However, the familiar mean-field result is recoveredin the limit of a large number N of bosonic particles. Inaddition, for N = 1, Eq. (46) gives Ω A = J/ ~ that isthe exact Rabi frequency of the one-particle tunnelingdynamics in a double-well potential.In the same fashion as in the previous section, the con-served energy associated to Eqs. (39) and (40) reads E ACS = U N (cid:18) − N (cid:19) z − JN p − z cos ( φ ) (47)and using the condition (26) we get the inequalityΛ > Λ MQST,A = 1 + p − z (0) cos( φ (0)) z (0) / (cid:0) − N (cid:1) , (48)where Λ MQST,A is the atomic-coherent-state MQST crit-ical parameter in terms of tunneling energy J , interactionstrength U , and number N of particles. Remarkably thisvalue is bigger than the standard mean field one, givenby Eq. (28), which is recovered in the semiclassical ap-proximation of a large number N of bosonic particles.In addition to the usual ground-state stationary solu-tions (30) and (31) we obtain from the system of Eq. (39)and (40) a correction to the symmetry-breaking ones z ACS ± = ± s − (cid:18) − N (cid:19) − (49) φ n = 2 πn (50)with n ∈ Z and Λ = N U/J . It follows that, within theapproach based on the atomic coherent state, the criticalstrength for the SSB of the balanced ground state ( z = 0, φ = 0) reads Λ SSB,A = − (cid:0) − N (cid:1) . (51)This means that for Λ = U N/J < Λ SSB,A = 1 / (1 − /N )the ground state is not balanced. Clearly, for N ≫ N = 1 onefinds Λ SSB,A = −∞ : within the ACS approach with onlyone boson the spontaneous symmetry breaking cannot beobtained. V. NUMERICAL RESULTS
To test our analytical results we compare them withnumerical simulations. The initial many-body state | Ψ(0) i for the time-dependent numerical simulations isthe coherent state | ACS (0) i from Eq. (36), with a givenchoice of z (0) and φ (0). The time evolved many-bodystate is then formally obtained as | Ψ( t ) i = e − i ˆ Ht/ ~ | Ψ(0) i , (52)with ˆ H given by Eq. (1).Knowing | Ψ( t ) i the population imbalance at time t isgiven by z ( t ) = h Ψ( t ) | ˆ N − ˆ N N | Ψ( t ) i . (53)In Fig. 1 we plot the Josephson frequency Ω as a func-tion of the number N of bosons, but with a fixed valueof U N/J = 1. As shown in the figure, the standardmean-field prediction (dashed curve), Eq. (24), predictsan horizontal line. The numerical results (filled circles),which are very far from the standard mean-field predic-tions, are instead reproduced extremely well by Eq. (46),based on atomic coherent states. Indeed, as previouslystessed, for N = 1 Eq. (46) gives the correct Rabi fre-quency. However, this exact result is, in some sense, ac-cidental since, as shown by the figure, for intermediatevalues of N (4 < N <
10) the agreement gets slightlyworse.We investigate numerically also the onset of macro-scopic quantum self trapping (MQST). In Fig. 2 we re-port the numerical time evolution of the population im-balance z ex ( t ) for different values of the number N ofbosons and of the interaction strength N U/J . In the fig-ure the numerical results are obtained with an initial ACS N π Ω / J numerical resultsmean fieldatomic coherent states N U/J = 1
FIG. 1. (Color online). Josephson frequency Ω as a functionof the number N of bosons, with UN/J = 1,
J >
0, and ~ = 1. Filled circles: numerical results. Dashed line: mean-field result, Eq. (24), based on Glauber coherent states. Solidcurve: results of Eq. (46), based on atomic coherent states(ACS). Initial conditions: z (0) = 0 . φ (0) = 0. state | ACS (0) i where z (0) = 0 . φ (0) = 0. In gen-eral, during the time evolution the many-body quantumstate | Ψ( t ) i does not remains close to an atomic coherentstate. This is especially true in the so-called Fock regime,where U/J ≫ N [25]. Unfortunately, this is the regimewhere the MQST can be achieved. Fig. 2 illustrates theproblems in determining a critical value for MQST: for N .
10 interwell oscillations possibly occur with a verylong period even for very large values of Λ, see Fig. 2(a).For larger N = 10 , . . . , N = 20 and various values of U/J for two slightly different initial conditions. We optedfor the definition that just no crossing of zero imbalanceshould happen. This definition typically underestimatesthe values obtained from, e.g., mean-field theory, as seenin the next Fig. 3.In Fig. 3 we show the critical interaction strengthΛ
MQST for the macroscopic quantum self trapping(MQSF) as a function of the number N of bosons. Inthis case neither the mean-field results (dashed line) northe ACS predictions (solid curve) are able to describe ac-curately the numerical findings (filled circles) for a smallnumber N of atoms.Let us conclude this Section by investigating the spon-taneous symmetry breaking (SSB) of the ground stateof the two-site Bose-Hubbard model, which appears for U <
J t / -0.5-0.4-0.3-0.2-0.100.10.20.30.4 popu l a t i on i m ba l an c e N=2, U/J=25N=2, U/J=12.5N=4, U/J=12.5N=4, U/J=25 (a)
J t/ -0.100.10.20.30.40.5 popu l a t i on i m ba l an c e U/J=1, z=0.5U/J=2, z=0.5U/J=2.5, z=0.5U/J=2, z=0.6U/J=2.5, z=0.6 (b)
FIG. 2. (Color online). Time evolution of the numerical pop-ulation imbalance of Eq. (53) for different values of number N = 2 and 4 (a) and N = 20 (b) of bosons and interactionstrength U/J , please see the legends, and
J >
0. The initialquantum state | ACS (0) i is characterized by φ (0) = 0 and z (0) = 0 . z (0) = 0 . N < | GS i = N X j =0 c j | j i ⊗ | N − j i , (54)where | c j | is the probability of finding the ground statewith j bosons in the site 1 and N − j bosons in the site 2.Here | j i is the Fock state with j bosons in the site 1 and | N − j i is the Fock state with N − j bosons in the site2. The amplitude probabilities c j are determined numer-ically by diagonalizing the ( N + 1) × ( N + 1) Hamiltonianmatrix obtained from (1). Clearly these amplitude prob-abilities c j strongly depend on the values of the hoppingparameter J , on-site interaction strength U , and totalnumber N of bosons. For U > P ( | c j | )of the probabilities | c j | is unimodal with its maximumat | c N/ | (if N is even) [31]. However, for U < N Λ M Q S T numerical resultsmean fieldatomic coherent states FIG. 3. (Color online). Critical interaction strength Λ
MQST for the macroscopic quantum self trapping (MQSF) as a func-tion of the number N of bosons. Notice that we take J > z (0) = 0 . φ (0) = 0. distribution P ( | c j | ) becomes bimodal with a local mini-mum at | c N/ | (if N is even) when | U | exceeds a criticalthreshold [31].The semiclassical SSB, described by Eqs. (34) and(51), corresponds in a full quantum mechanical treat-ment to the onset of the bimodal structure in the distri-bution P ( | c j | ) [31]. In Fig. 4 we report the dimension-less interaction strength ( | U | /J ) SSB for the spontaneoussymmetry breaking (SSB) as a function of the number N of bosons. In the figure we compare the numerical re-sults (filled circles) [31] with the semiclassical predictionsbased on Glauber coherent states (dashed curve) andatomic coherent states (solid curve). The figure showsthat the numerical results of SSB are quite well approx-imated by the ACS variational approach, which is moreaccurate with respect to the standard mean-field one. Forlarge N the numerical results end up in the analyticalcurves, which become practically indistinguishable. VI. CONCLUSIONS
In this paper we have adopted a second-quantizationformalism and time-dependent atomic coherent states tostudy finite-size effects in a Josephson junction of N bosons, obtaining experimentally detectable theoreticalpredictions. The experiments with cold atoms in latticesand double wells reported in [32–35], for instance, showedthat atom numbers well below N = 100 can be reachedand successfully detected with an uncertainty of the orderone atom. In particular we have obtained an analyticalformula with 1 /N corrections to the standard mean-field N ( | U | / J ) SS B numerical resultsmean fieldatomic coherent states FIG. 4. (Color online). Dimensionless interaction strength( | U | /J ) SSB for the onset of spontaneous symmetry break-ing (SSB) as a function of the number N of bosons. Noticethat we use J >
0. Filled circles: numerical results obtainedfrom the onset of a bimodal structure in the distribution P ( | c j | ). Dashed line: mean-field result ( | U | /J ) SSB = 1 /N based on Glauber coherent states, see Eq. (34). Solid curve:( | U | /J ) SSB = 1 / ( N − treatment for the frequency of Josephson oscillations. Wehave shown that this formula, based on atomic coherent states, is in very good agreement with numerical simu-lations and it reduces to the familiar mean-field one inthe large N limit. We have also investigated the spon-taneous symmetry breaking of the ground state. At thecritical interaction strength for the spontaneous symme-try breaking the population-balanced configuration is nomore the one with maximal probability. Also in thiscase the agreement between the analytical predictionsof the atomic coherent states and numerical results isgood. Finally, we have studied the critical interactionstrength for the macroscopic quantum self trapping. Herewe have found that the 1 /N corrections to the standardmean-field theory predicted by the atomic coherent statesdo not work. Summarizing, the time-dependent vari-ational ansatz with atomic coherent states is quite re-liable in the description of the short-time dynamics ofthe bosonic Josephson junction both in the Rabi regime,where 0 ≤ | U/J | ≪ /N , and in the Josephson regime,where 1 /N ≪ | U/J | ≪ N [25]. Instead, in the Fockregime, where | U/J | ≫ N , a full many-body quantumtreatment is needed. ACKNOWLEDGEMENTS
The authors thank A. Cappellaro, L. Dell’Anna, A.Notari, V. Penna, and F. Toigo for useful discussions. LSacknowledges the BIRD project “Superfluid propertiesof Fermi gases in optical potentials” of the University ofPadova for financial support. [1] B. D. Josephson, Phys. Lett. , 251 (1962).[2] A. Barone and G. Paterno, Physics and Applications ofthe Josephson effect (Wiley, New York, 1982).[3] E. L. Wolf, G.B. Arnold, M.A. Gurvitch, and JohnF. Zasadzinski,
Josephson Junctions: History, Devices,and Applications (Pan Stanford Publishing, Singapore,2017).[4] D.T. Ladd et al. , Nature , 45 (2010).[5] I. Buluta et al. , Rep. Prog. Phys. , 104401 (2011).[6] M.H. Anderson et al. , Science , 198 (1995); K.B.Davis et al. , Phys. Rev. Lett. , 3969 (1995).[7] I. Bloch, J. Dalibard and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[8] A. Smerzi, S. Fantoni, S. Giovanazzi, and S.R. Shenoy,Phys. Rev. Lett. , 4950 (1997).[9] S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy,Phys. Rev. A , 620 (1999).[10] S. Ashhab and C. Lobo, Phys. Rev. A , 013609 (2002).[11] M. Albiez et al. , Phys. Rev. Lett. , 010402 (2005).[12] G. Valtolina et al. , Science , 1505 (2015).[13] A. Leggett and F. Sols, Found. Phys. , 353 (1991).[14] P. Carruthers and M.M. Nieto, Rev. Mod. Phys. , 411(1968).[15] A. Smerzi and S. Raghavan, Phys. Rev. A , 063601(2000). [16] J. R. Anglin, P. Drummond, and A. Smerzi, Phys. Rev.A , 063605 (2001).[17] G. Ferrini, A. Minguzzi, and F.W.J. Hekking, Phys. Rev.A , 023606 (2008).[18] R. Glauber, Phys. Rev. , 2766 (1963).[19] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas,Phys. Rev. A , 2211 (1972).[20] W-M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod.Phys. , 867 (1990).[21] G. J. Milburn, J. Corney, E.M. Wright, and D.F. Walls,Phys. Rev. A , 4318 (1997).[22] P. Buonsante, V. Penna, and A. Vezzani, Phys. Rev. A , 043620 (2005).[23] P. Buonsante, P. Kevrekidis, V. Penna and A. Vezzani,J. Phys. B: At. Mol. Opt. Phys. , S77 (2006).[24] P. Buonsante and V. Penna, J. Phys. A: Math. Gen. ,175301 (2008).[25] A. J. Leggett, Rev. Mod. Phys. , 307 (2001).[26] M. Lewenstein, A. Sanpera, and V. Ahufinger, UltracoldAtoms in Optical Lattice (Oxford Univ. Press, 2012).[27] L. Amico and V. Penna, Phys. Rev. Lett. , 2189 (1998).[28] R. Franzosi, V. Penna, and R. Zecchina, Int. J. Mod.Physics B , 943 (2000).[29] F. Trimborn, D. Witthaut, and H. J. Korsch, Phys. Rev.A , 043631 (2008). [30] F. Trimborn, D. Witthaut, and H. J. Korsch, Phys. Rev.A , 013608 (2009).[31] G. Mazzarella, L. Salasnich, A. Parola, and F. Toigo,Phys. Rev. A , 053607 (2011).[32] C. Gross, H. Strobel, E. Nicklas, T. Zibold, N. Bar-Gill,G. Kurizki, and M. K. Oberthaler, Nature (London) ,219 (2011). [33] W. Muessel, H. Strobel, M. Joos, E. Nicklas, I. Stroescu,J. Tomkovic, D. Hume, and M. K. Oberthaler, Appl.Phys. B , 69 (2013).[34] D. B. Hume, I. Stroescu, M. Joos, W. Muessel, H. Stro-bel, and M. K. Oberthaler, Phys. Rev. Lett. , 253001(2013).[35] I. Stroescu, D. B. Hume, and M. K. Oberthaler, Phys.Rev. A91