NOON state of Bose atoms in the double-well potential via an excited state quantum phase transition
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p NOON state of Bose atoms in the double-well potential via an excited state quantumphase transition
A. A. Bychek , , D. N. Maksimov , , and A. R. Kolovsky , Kirensky Institute of Physics, Federal Research Center KSC SB RAS, 660036, Krasnoyarsk, Russia Siberian Federal University, 660041, Krasnoyarsk, Russia Reshetnev Siberian State University of Science and Technology, 660037, Krasnoyarsk, Russia (Dated: September 17, 2018)We suggest a simple scheme for creating a NOON state of repulsively interacting Bose atoms in thedouble-well potential. The protocol consists of two steps. First, by setting atom-atom interactionsto zero, the system is driven to the upper excited state. Second, the interactions is slowly increasedand, simultaneously, the inter-well tunneling is decreased to zero. We analyze fidelity of the finalstate to the NOON state depending on the number of atoms, ramp rate, and fluctuations of thesystem parameters. It is shown that for a given fidelity the ramp rate scales algebraically with thenumber of atoms.
I. INTRODUCTION
Non-classical states of bosonic ensembles play impor-tant role in quantum computing, measurement, and com-munication [1–3]. Among many different implementa-tions [4–10] the two-mode Bose-Hubbard model [11–36]is the most popular playground thanks to its versatility,relative simplicity, and experimental accessibility with ul-tracold atoms in optical potentials [2, 18, 23, 26, 34, 35].In this paper we propose a recipe for generating NOONstates [37], also known as large cat sates [12], in the two-site Bose-Hubbard model. It should be mentioned fromthe very beginning that, due to decoherence processesinevitably present in a laboratory experiment (particlelosses, fluctuations of the optical potential, etc.), theNOON state can be obtained only for relatively smallnumber of atoms. In other words, in the thermodynamiclimit N → ∞ one always gets a statistical mixture oftwo states with all atoms localized in either of two wells– the phenomenon known as spontaneous localization orparity-symmetry breaking phase transition. One of thegoals of this work is to estimate the maximal number ofatoms for which one can create the NOON state with thepresent day experimental facilities.Formally, the NOON state is the ground state of theone-dimensional attractive Bose-Hubbard model [13, 17,20] in the strong interaction regime. In practice, however,this state is hard to reach because the NOON state arefragile to particle losses caused by the collision instability[38]. To avoid this problem we consider repulsive atom-atom interactions where the NOON state appears to bethe upper energy state of the system. In what follows weshow that this state can be reached in the course of adi-abatic passage through an excited state quantum phasetransition (ESQPT) [39–41]. It is generally believed thatsuch an adiabatic passage would require extremely longevolution time, which scales exponentially with the num-ber of particles [21]. Here, by detailed examination ofthe system spectrum in a view of the Landau-Zennertunnelling, we demonstrate that the evolution time ac-tually scales algebraically with the number of bosons N . A pseudo-classical interpretation of the adiabatic passagewith the ESQPT corresponding to a separatrix crossingin the classical phase-space is provided. II. SYSTEM OVERVIEW
We consider N ≫ b H = − J (cid:16) ˆ a † ˆ a + ˆ a ˆ a † (cid:17) + U X l =1 , ˆ n l (ˆ n l − δ (ˆ n − ˆ n ) , (1)where J is the hopping matrix element, U the microscopicinteraction constant, ˆ a l and ˆ a † l the bosonic annihilationand creation operators, ˆ n l the number operator, and δ the difference between the on-site energies. For N bosonsthe Hilbert space of the Hamiltonian (1) of dimension N = N + 1 is spanned by Fock states | N , N i = | N/ − n, N/ n i ≡ | n i , | n | ≤ N/ , (2)where N + N = N . Above we used a symmetric param-eterization to label the Fock states by a single quantumnumber n ( N is assumed to be even). The full spectrumof the system is shown in Fig.1, where we introducedthe macroscopic interaction constant g = U N/ J = 1 − g . Thus, thecase g = 0 corresponds to the system of non-interactingbosons while in the case g = 1 the inter-well tunnel-ing is completely suppressed. It is easy to prove thatthe spectrum is equidistant for g = 0 and quadratic for g = 1, with all energy levels except the ground statebeing twofold degenerate [22]. The spectrum for inter-mediate g possesses the quantum separatrix and can beunderstood by employing the pseudo-classical approachwhich we review in Sec. III.Among the eigenstates | Ψ j i of the Hamiltonian (1) ofparticular interest are the states with minimal and max-imal energy. For g = 0 the ground state of the system is E / N (a) (b) FIG. 1: Energy spectrum of N = 10 (left) and N = 40(right) bosons against the macroscopic interaction constant g = UN/
2. (The other parameters are J = 1 − g and δ = 0.)The quantum separatrix is marked by the red dashed line. a Bose-Einstein condensate with all particles occupyingthe symmetric single-particles state, | Ψ ( g = 0) i = 1 √ N N ! (cid:16) a † + a † (cid:17) N | vac i , (3)while the upper energy state is a Bose-Einstein con-densate with all particles occupying the antisymmetricsingle-particles state, | Ψ N ( g = 0) i = 1 √ N N ! (cid:16) a † − a † (cid:17) N | vac i . (4)Let us follow these states under variation of g . At eachvalue of g eigenfunctions are found as an expansion overthe Fock states (2), | Ψ j ( g ) i = N/ X n = − N/ c ( j ) n ( g ) | n i , (5)For j = 0 and j = N the results are shown in Fig 2. Itis seen that the ground state transforms into the frag-mented condensate [22] | Ψ ( g = 1) i = | N/ , N/ i , (6)while the upper energy state evolves into the NOON state | Ψ N ( g = 1) i = | N OON i ≡ √ | N, i + | , N i ) . (7)Next we consider time evolution of the system accord-ing to the Schr¨odinger equation, i dd t | ψ i = b H ( g ) | ψ i , J = 1 − g , (8) n g (a) −20 0 200.20.40.60.81 n (b) −20 0 200.20.40.60.81 FIG. 2: Squared absolute values of expansion coefficientsEq. (5) of the ground (left) and upper energy (right) statesagainst the macroscopic interaction constant g . (a) -20 0 20 n g (b) -20 0 20 n | c n | (c) -20 0 20 n | c n | FIG. 3: Panel (a): Squared absolute values of expansion co-efficients over the Fock basis as the function of g = νt for theadiabatic passage with ν = 0 .
1. Panels (b) and (c) comparesfinal state of the system for ν = 0 . ν = 0 . with the interaction constant g growing linearly from 0to 1 during the time interval T = 1 /ν . In Fig. 3(a) wepresent the results of numerical simulations of the sys-tem dynamics for | ψ ( t = 0) i = | Ψ N ( g = 0) i and ν = 0 . c n ( t ) = h n | ψ ( t ) i . One can see in Fig. 3(b) that thefinal state | ψ ( t = T ) i does not ideally coincide with thetarget NOON state Eq. (7). With a smaller ramp rate,however, the result is almost perfect, see Fig. 3(c). In thenext sections we analyze the discussed adiabatic passagein more detail and quantify the final state | ψ ( t = T ) i . Topay credits to other works we mention that adiabatic pas-sage for the ground state of the attractive Bose-Hubbardmodel was considered earlier in Ref. [20, 42] and a differ-ent adiabatic passage, which involves the rising potentialbarrier which separates a Bose-Einstein condensate intotwo parts, was analyzed in Ref. [14, 15, 19, 24]. III. PSEUDO-CLASSICAL APPROACH
To get a deeper insight into the discussed adia-batic passage we resort to the pseudo-classical approach.This approach borrows its ideas from the semi-classicalmethod in single-particle quantum mechanics to addressthe spectral and dynamical properties of the systemof N interacting bosons, with 1 /N playing the role ofPlanck’s constant [43–46]. Formally, the creation andannihilation operators are substituted with C -numbersas ˆ a l / √ N → a l and ˆ a † l / √ N → a ∗ l , which also impliesrescaling of the Hamiltonian as b H/N → H . For thetwo-site Bose-Hubbard model this leads to the classicalHamiltonian [11] H = g I − J p − I cos φ , J = 1 − g , (9)where g = U N/ g : 0, 0 .
4, 0 .
7, and 1. By using the re-lation I = sin θ phase portraits of the system can be alsodrawn on sphere of the unit radius. In this representationthe line I = 1 ( I = −
1) reduces to single point – the north(south) pole of the sphere. For g = 0 the phase portraitcontains two elliptic points at ( I, φ ) = (0 ,
0) (minimal en-ergy) and (
I, φ ) = (0 , π ) (maximal energy), see Fig.4(a).As g is increased above g cr = 1 / I, φ ) = ( ± I ∗ , π ),where I ∗ is a function of g . With a further increase of g the island around the point ( I, φ ) = (0 ,
0) vanishes I (a) (b) φ / π I (c) φ / π (d) FIG. 4: Phase portraits of the classical Hamiltonian (9) for(a) g = 0, (b) g = 0 .
4, (c) g = 0 .
7, and (d) g = 1. I g (a) −0.5 0 0.500.20.40.60.81 n (b) −50 0 5000.20.40.60.81 FIG. 5: Comparison between the classical (left panel) andquantum (right panel) dynamics. Parameters are δ = 0, ν =0 .
1, and N = 100. while the islands around ( I, φ ) = ( ± I ∗ , π ) monotonicallygrow, finally leading to the phase-space portrait shownin Fig.4(d).The depicted phase portraits suffice to find the en-ergy spectrum shown in Fig. 1 by using the semiclassi-cal quantization rule, where the phase volume encircledby a trajectory is required to be a multiple of the effec-tive Planck constant h = 1 /N . Then the central islandaround the point ( I, φ ) = (0 ,
0) gives energy levels be-low the quantum separatrix while two symmetric islandsaround (
I, φ ) = ( ± I ∗ , π ) give degenerate levels above thequantum separatrix. The details are given in Ref. [45],where it was demonstrated that the pseudo-classical ap-proach provides an accurate approximation to the exactspectrum even for N = 10.Let us now study dynamics of the classical system(9) when both g and J vary in time as g = νt and J = 1 − g . As the initial condition we take an ensembleof particles with the probability distribution given by thetwo-dimensional Gaussian centered at the elliptic point( I, φ ) = (0 , π ). For comparison with quantum dynamicsthe width of the Gaussian is adjusted to σ = √ N . Theleft panel in Fig. 5 shows the evolution of the classicaldistribution function ρ ( I, t ) for N = 100. (We stress onemore time that the latter parameter determines only thewidth of the initial distribution.) The left panel in Fig. 5should be compared with the right panel showing thequantum evolution. The observed agreement underlinesthe classical phenomenon behind the quantum resultsdiscussed in the previous section. Classically, the parti-cles are captured into the upper and lower islands emerg-ing after bifurcation of the elliptic point ( I, φ ) = (0 , π )and then transported towards I = 1 and I = −
1, respec-tively. The phenomenon of capturing into (and releasingfrom) an elliptic island was considered earlier in Ref. [47] n−50 0 5000.20.40.60.81I g −0.5 0 0.500.20.40.60.81 FIG. 6: Same as in Fig. 5 with δ = 0 . in a different context. It involves the crossing of instanta-neous separatrix that, in turn, was analyzed in Ref. [48].To conclude this section we discuss the effect of non-zero δ . For δ = 0 the emerging islands have differentsize, which makes ρ ( I, t ) asymmetric with respect to I →− I . To characterize this asymmetry we introduce thepopulation imbalance G = Z ρ ( I, T )d I − Z − ρ ( I, T )d I . (10)If δ is increased the population imbalance (10) growsmonotonically, approaching | G | = 1, see Fig. 6. Impor-tantly, the imbalance also grows if ν is decreased and forany finite δ the imbalance is unity in the limit ν → IV. LANDAU-ZENER TUNNELING ANDFIDELITY TO THE NOON STATE
In the previous section we explained the quantum re-sults depicted in Fig. 2 by using the pseudo-classical ap-proach. The quantum-mechanical explanation of theseresults is based on the notion of Landau-Zenner tun-neling. Due to this phenomenon several energy levelsbecome populated as we follow the upper most level inFig. 1 with a finite sweeping rate. This is illustrated inFig. 7(a) which shows the populations of the instanta-neous energy levels P j ( t ), P j ( t ) = |h Ψ j ( g = νt ) | ψ ( t ) i| , (11)for ν = 0 . N = 30. Notice that only even levels arepopulated because of different symmetry of eigenstatesof the Hamiltonian (1) with odd and even index j . Toquantify the effect of Landau-Zener tunneling we intro-duce the fidelity F = |h N OON | ψ ( T ) i| , (12) P j (a) g P j (b) FIG. 7: Populations of the instantaneous energy levels for N = 30, ν = 0 .
1, and δ = 0 (upper panel) and δ = 0 . which characterizes how close the final state is to thetarget NOON state. In the limit ν → F = 1 − exp (cid:18) − π ∆ | α | ν (cid:19) . (13)In Eq. (13) ∆ is the energy gap between the upper mostlevel and the next level of the same symmetry, ν = 1 /T the sweeping rate, and α is determined by the angle atwhich two levels approach each other. Since the energygap ∆ and | α | scales algebraically with 1 /N , we expectthat the evolution time T has to be increased proportion-ally to the number of particles to insure a given fidelity.Direct numerical simulations of the adiabatic passage fordifferent N confirm this hypothesis, see Fig. 8(a). It isinteresting to discuss the depicted result with respect tothe recent laboratory experiment [34] which studies theparity-symmetry-breaking phase transition for N ≈ J/h = 40 Hz and the evolution time ∼ s we get N ≈
30 and this number can be easily increased by re-laxing the fidelity to F = 0 . ν = ν ( t ) that optimizes the adiabatic pas-sage. We stress that the above estimate is obtained underthe assumption of negligible decoherence processes whichwe shall discuss in Sec. V.Next we analyze the effect of non-zero δ ≪ J inthe Hamiltonian (1) from the quantum-mechanical view-point. Non-zero δ breaks the reflection symmetry of thesystem, so that eigenstates of the Hamiltonian (1) at g ≫ J are given by the Fock states | N/ − n, N/ n i and | N/ n, N/ − n i but not their symmetric or anti-symmetric superpositions. (In particular, | Ψ N i ≈ | N, i N T (a) -0.02 -0.01 0 0.01 0.0200.20.40.60.81 F , | G | (b) FIG. 8: Left panel: Minimal evolution time T insuring fidelity F = 0 .
99 versus the number of bosons N . Right panel: Pop-ulation imbalance | G | (dashed line) and fidelity F (solid line)as the function of δ for N = 40 and ν = 0 . and Ψ N − i ≈ | , N i .) This drastically changes Fig. 7(a)– now both odd and even instantaneous energy lev-els become populated during the adiabatic passage, seeFig. 7(b). For the considered extremely small value of δ this difference simply reflects a change of the basis and,physically, both Fig. 7(a) and Fig. 7(b) describe the sameprocess, which results in the NOON state as the finalstate of the system. However, for a larger δ we see con-siderable deviation from the NOON state, see Fig. 8(b).In particular, in full analogy with the classical result, thepopulation imbalance | G | approaches the unity if | δ | isincreased. V. DECOHERENCE EFFECTS
The result depicted in Fig. 8(a) proves that, at leastin principle, one can create arbitrary large cat state bysimply increasing the duration of the adiabatic passageproportionally to the number of particles N . This, how-ever, implicitly assumes the absence of any decoherenceprocess [21] and precision control over the system param-eters, in the first place, over parameter δ . In this sectionwe discuss decoherence caused by fluctuations of δ , whichare unavoidable in a laboratory experiment.In the presence of fluctuations the fidelity(12) shouldbe redefined as F = h N OON |R ( T ) | N OON i , R ( t ) = | ψ ( t ) ih ψ ( t ) | , (14)where the bar denotes the average over fluctuations. Tobe specific, we assume that δ ( t ) is the white noise withvanishing mean value, i.e., δ ( t ) δ ( t ′ ) = δ δ ( t − t ′ ). Thenthe density matrix R ( t ) is easy to show to obey the fol-lowing master equation [49]d R d t = − i [ b H, R ] − δ [ˆ n, [ˆ n, R ]] , (15)where ˆ n = ˆ n − ˆ n . We solve Eq. (15) for the adiabaticpassage discussed above. Fig. 9 shows fidelity (14) asthe function of the noise amplitude δ for three system δ N F −10 0 10−10−50510 −10 0 10−10−50510−10 0 10−10−50510 FIG. 9: Fidelity (14) as the function of the noise amplitudefor N = 10 and ν = 0 . N = 20 and ν = 0 . N = 40 and ν = 0 .
025 (dashed line).Insets show the initial density matrix for N = 20 (lower-leftconner) and final density matrices for δ = 0 and δ = 0 . /N (upper-right conner). sizes N = 10 , ,
40, where we proportionally decreasedthe sweeping rate ν to insure fidelity F ≈
1. One strik-ing feature of the depicted functions is a rapid decay offidelity to F ≈ . < δ < δ ∗ where δ ∗ = δ ∗ ( N ). In this interval the off-diagonal elements ofthe density matrix R ( T ) gradually vanish. On the otherhand, the diagonal elements of the density matrix remainessentially unaffected. Clearly, this result illustrates theusual quantum-to-classical transition due to a decoher-ence process [49–51]. Notice that the larger system is,the more it is sensitive to decoherence. Numerical re-sults depicted in Fig. 9 indicate that δ ∗ decreases with N faster than 1 /N .Next we briefly discuss decoherence due to particlelosses. In the case of not conserved number of particlesthe master equation for the system density matrix readsd R d t = − i [ b H, R ] − γ X l =1 , (ˆ a † l ˆ a l R− a l R ˆ a † l + R ˆ a † l ˆ a l ) , (16)where γ denotes the decay rate (see, for example,Ref. [52]). The value of γ in Eq. (16) crucially dependson the sign of interatomic interactions. For example, inthe already sited experiment [34] with attractively inter-acting atoms the decay rate was γ ≈ . N ∼ γ ∼ − [53]. The negligible decoherence rate due to particlelosses is our main reason for considering the adiabaticpassage for the upper energy state of repulsively inter-acting atoms instead of that for the ground state of at-tractively interacting atoms. In all other aspects thereis no conceptional difference between the adiabatic pas-sages for the ground and upper states. t n (a) (b) FIG. 10: Populations of eigenstates of the Hamiltonian (1) asthe function of time. (Note that for g = 0 the eigenstates of(1) are given by | N − n, n i where n now denotes the numberof particles in the antisymmetric single-particle state.) Pa-rameters are N = 40, g = 0, J = 1, ω = J , δ = 0 .
05 (leftpanel) and N = 40, g = 0, J = 0 . δ = 1 (right panel). VI. PREPARATION OF THE EXCITED STATE
Finally, we discuss a method to excite the system ofnon-interacting bosons ( g = 0) into the highest energystate. A way to do this is to drive the system by pe-riodically changing parameter δ as δ ( t ) = δ sin( ωt ),where the frequency ω coincides with the transition fre-quency between the symmetric and antisymmetric single-particle states uniquely determined by the parameter J .If δ ≪ J (the latter condition justifies the rotating-waveapproximation) the problem can be solved analyticallyand leads to the Rabi oscillations, see Fig. 10(a). Thus,to excite the system in the upper state, we need to driveit for one half of the Rabi period.Another, perhaps even simpler way to obtain the ex-cited state (4) is to quench the system into the parameterregion δ ≫ J by suddenly tilting the double-well. Thenthe time evolution of the expansion coefficients is approx-imately given by c ( j ) n ( t ) = exp( i δnt ) c ( j ) n (0) and after onehalf of the period T B = π/δ (which can be interpreted asthe Bloch period) the state (3) transforms into the state(4), see Fig. 10(b). VII. CONCLUSIONS
We suggested a method for creating the NOON stateof Bose atoms, i.e., coherent superposition of two statesin which all particles are in the same well of the double-well potential. Unlike to previous studies, which almostexclusively focused on the case of attractive interactions[54], we considered the repulsively interacting atoms thatavoids the problem of particle losses. The scheme pro-tocol consists of two steps. First, by setting the inter-atomic interactions to zero we transfer the system fromthe ground state to the upper excited state. Second,adiabatically increasing the interaction strength and si-multaneously decreasing the hopping rate we transformthis excited state to the NOON state. In the Fock spacethe latter stage can be viewed as splitting of the initiallylocalized wave packet into two packets [55]. This pro-cess was shown to have a pseudo-classical counterpartand some of quantum results, for example, the popula-tion imbalance G can be obtained by using pure classi-cal arguments. Of course, the classical approach cannotaddress phase coherence between the packets, which ischaracterized by the fidelity F .Formally, the suggested scheme allows us to create anarbitrary large cat state. However, any experimental re-alization of the scheme protocol imposes fundamentallimitation on the number of atoms due to decoherenceprocesses present in a laboratory experiment. Here, weanalyzed the decoherence caused by fluctuation of theparameter δ (the energy mismatch between the left andright wells of the double-well potential) that appears tobe crucial for the system dynamics. It was shown thatthere is a critical value for the fluctuation amplitude δ ∗ ∼ /N above which the final state of the system be-comes ‘classical NOON state’, i.e., incoherent superposi-tion of two states in which all particles are in the samewell of the double-well potential. Thus to get the NOONstate with a large number of atoms every effort to reducefluctuation of δ should be taken. Acknowledgements.
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