"Photonic" cat states from strongly interacting matter waves
““Photonic” cat states from strongly interacting matter waves
Uwe R. Fischer and Myung-Kyun Kang
Seoul National University, Department of Physics and AstronomyCenter for Theoretical Physics, 151-747 Seoul, Korea
We consider ultracold quantum gases of scalar bosons, residing in a coupling strength–densityregime in which they constitute a twofold fragmented condensate trapped in a single well. It isshown that the corresponding quantum states are, in the appropriate Fock space basis, identical tothe photon cat states familiar in quantum optics, which correspond to superpositions of coherentstates of the light field with a phase difference of π . In marked distinction to photon cat states, thevery existence of matter wave cat states however crucially depends on the many-body correlations ofthe constituent particles. We consequently establish that the quadratures of the effective “photons,”expressing the highly nonclassical nature of the macroscopic matter wave superposition state, can beexperimentally accessed by measuring the density-density correlations of the interacting quantumgas. The eponymous cat quantum states have originallybeen constructed by Erwin Schr¨odinger to stimulatedeeper thought on the sometimes surreal aspects ofmacroscopic quantum mechanics [1]. Present-day tech-nology allows for going beyond the pure gedankenexper-iment of Schr¨odinger’s day to create, within the realm ofquantum optics, small cat states, Schr¨odinger “kittens”[2]. They consist typically either in the superposition ofcoherent states of light [3] with a phase difference of π ,coined photon cat states, or in states | N (cid:105) + | N (cid:105) of theNOON type [4]. Highly entangled photonic cat statesconstitute a possible basic building block in quantum in-formation architectures [2, 5–9].We propose in what follows a novel species ofSchr¨odinger cat states, which are in distinction to pho-tonic cat states relying on quantum many-body correla-tions, and hence owe their very existence to strong inter-actions of the constituent bosons. They are generated ina scalar (one component) gas of massive bosons trappedinside a single (harmonic or box) trap. The many-bodystates required to produce the cat states correspond totwofold fragmented condensates, for which the single-particle density matrix has two O ( N ) eigenvalues [10].Condensate fragmentation [11] is by now firmly estab-lished as a many-body effect not describable within amean-field theory of the classic Bogoliubov type [12, 13].Specifically, for sufficiently large and positive contact in-teraction coupling, it derives from the broken transla-tional symmetry and localization in a single trap [14].Therefore, strongly interacting matter wave cat states(termed SIMCAS in what follows) can be experimen-tally accessed, starting from a Bose-Einstein condensate(BEC), by tuning the coupling with a Feshbach resonance[15].Below, we shall demonstrate that SIMCAS, in the ap-propriate Fock space basis, form many-body states es-sentially indistinguishable from photonic cat states, forsufficiently large (but still mesoscopic) number N of gasparticles. The degree of fragmentation of condensate co-herence will be shown to be directly related to the de- gree of macroscopicity of the coherent superposition ofmany-body states, quantified by the superposition sizeof the quantum state [16–19]. By evaluating quadra-tures [5], we show that the macroscopicity of the mat-ter wave quantum superposition is directly measurablethrough density-density correlations after time-of-flightexpansion (TOF).We emphasize that SIMCAS live in a single trap andnot in a double well, and are obtained for a scalar, nota two-component gas, with repulsive interactions, whichdistinguishes them from previously suggested cat stateimplementations in dilute ultracold quantum gases, cf.[20–24]. Furthermore, SIMCAS are the ground states ofthe trapped Bose gas for relatively moderate interactioncouplings, at which the single BEC crosses over to a two-fold fragmented condensate [25] (the gas remaining suf-ficiently dilute to ensure that three-body recombinationrates remain small). This renders SIMCAS different fromthe collapse-and-revival type quantum optics proposalsto generate cat states in a Kerr medium cf., e.g., [26, 27],or involving scattering of matter waves at barriers [28].As a first step, we expand two-mode fragmented statesin the Fock space basis of elementary bosons, | Ψ (cid:105) = N (cid:88) l =0 C l | N − l, l (cid:105) , N (cid:88) l =0 | C l | = 1 , (1)where the Fock space basis vectors | N − l, l (cid:105) = (ˆ a † ) N − l (ˆ a † ) l √ ( N − l )! l ! | vac (cid:105) , with | vac (cid:105) the particle vacuum. Atwo-mode Hamiltonian, with interaction part H int = A ˆ a † ˆ a † ˆ a ˆ a + A ˆ a † ˆ a † ˆ a ˆ a + A (cid:16) ˆ a † ˆ a † ˆ a ˆ a + h . c . (cid:17) + A ˆ a † ˆ a ˆ a † ˆ a (assuming that A ∈ (cid:60) ), valid, e.g., forcontact and dipolar two-body interactions when the twomodes have even and odd parity, respectively, genericallyleads to a distribution of the C l modulus which is, in thecontinuum limit, a Gaussian (of relative width ∝ / √ N ),yielding fragmented condensate states for A > A + A + 2 A − A > distribution center l := (cid:104) ˆ a † ˆ a (cid:105) ∈ [0 , N ] quantifies a r X i v : . [ qu a n t - ph ] M a y the degree of fragmentation F = 1 − | λ − λ | /N , where λ i are eigenvalues of the single-particle density matrix.We note that the exact functional form of the C l distri-bution is not important for the following argument. Theconditions are (a) 0 (cid:28) l (cid:28) N , (b) negligible weight of C l at the boundaries l = 0 , N , and (c) small relative dis-tribution width. The fragmentation degree is F = 2 l /N when l ≤ N/ F = 2(1 − l /N ) when N/ ≤ l ≤ N [14, 31].Self-consistent solutions of the quantum many-bodyproblem have indeed also found such two-mode frag-mented states for specific Hamiltonians, e.g. for quasi-one-dimensional (quasi-1D) BECs at sufficiently largecontact interaction couplings [25, 29]. The large overlapof the modes in a single trap, crucially, leads to maroscop-ically large interaction-induced pair-exchange processes, − (cid:10) ˆ a † ˆ a † ˆ a ˆ a + h . c . (cid:11) ∼ O ( N ) which stabilize (rather thandestabilize) SIMCAS, as shown in [30]. The existenceof negative macroscopic pair coherence due to a pair-exchange coupling A ∼ O ( A , A , A ) is thus a distin-guishing feature of SIMCAS.To proceed, we construct the two ladder operatorsˆ b = lim (cid:15) → (cid:112) ˆ N + (cid:15) ˆ a † ˆ a , ˆ b (cid:48) = lim (cid:15) → (cid:112) ˆ N + (cid:15) ˆ a † ˆ a , (2)where ˆ N i = ˆ a † i ˆ a i . The ladder operators and theirHermitian conjugates ˆ b † , ˆ b (cid:48)† convert particles betweenthe two macroscopically occupied modes, accordingto ˆ b | N − l, l (cid:105) = √ l | N − l + 1 , l − (cid:105) , ˆ b † | N − l, l (cid:105) = √ l + 1 | N − l − , l + 1 (cid:105) , and similarly for ˆ b (cid:48) , ˆ b (cid:48)† . The (cid:15) regularization is introduced for the (finite N ) singular-ity created when ˆ b † (ˆ b (cid:48)† ) acts on | , N (cid:105) ( | N, (cid:105) ), which issingular because there is no particle to be transferred tomode 1 (mode 0) for this state.We demonstrate in the following that a twofold frag-mented state can be written as the superposition of trun-cated coherent states of the ladder operators. These co-herent states live in a finite-dimensional Hilbert space[32, 33] for particle-number-conserved states of the form(1). The ladder operators approximately satisfy bosoniccommutation relations, due to 1 − (cid:104) Ψ | [ˆ b, ˆ b † ] | Ψ (cid:105) = ( N +1) | C N | and 1 − (cid:104) Ψ | [ˆ b (cid:48) , ˆ b (cid:48)† ] | Ψ (cid:105) = ( N + 1) | C | , provided | C | , | C N | (cid:28) / √ N .The truncated coherent states | β (cid:105) of ˆ b are defined byˆ b | β (cid:105) = ˆ b (cid:32) A β N (cid:88) l =0 β l √ l ! | l (cid:105) (cid:33) = β | β (cid:105)− βA β β N √ N ! | N (cid:105) , (3)where | l (cid:105) := | N − l, l (cid:105) . The normalization factor is givenby A β = exp( −| β | / (cid:113) Γ( N +1)Γ( N +1 , | β | ) , with the upper in-complete gamma function Γ( s, x ) = (cid:82) ∞ x t s − e − t d t . Thestate | β (cid:105) equals the usual coherent state in a infinite-dimensional Hilbert space when for N → ∞ the secondterm on the right-hand side of (3) becomes negligible; in fact | β (cid:105) becomes very close to a proper coherent statealready for moderate values of N [34]. Similar considera-tions hold for the truncated coherent states of the pair ˆ b (cid:48) , | β (cid:48) (cid:105) . Hence both | β (cid:105) and | β (cid:48) (cid:105) represent a finite- N gener-alization of the concept of coherent states. The quasipar-ticles created by ˆ b † or, as an equivalent choice (also seebelow), by ˆ b (cid:48)† , will thus assume the role of the “photons.”The fragmented condensate states are transparentlydescribed in terms of phase space states [36, 37], wherethe relevant phase is conjugate to the occupation numberdifference of the modes. For large N , fragmented conden-sates correspond to two phase space states | φ, N, l (cid:105) ∝ (cid:16) √ N − l ˆ a † + e iφ √ l ˆ a † (cid:17) N | vac (cid:105) , separated in phase φ byexactly π [31]. We now show that individual phase-spacebasis elements are well approximated by truncated co-herent states of ˆ b (ˆ b (cid:48) ); we will then conclude that thefragmented ground state can be expressed as a superpo-sition of antipodal coherent states, Eq. (7) below.According to [31], a two-mode many-body state (1)can be expressed in terms of a phase-space basis as | Ψ (cid:105) = (cid:90) π dφ π (cid:88) l N N,l ; l C l e − ilφ | φ, N, l (cid:105) , (4)where the relation of the phase space basis states cen-tered at l to Fock space basis states is given by | l (cid:105) = (cid:82) π dφ π N N,l ; l e − ilφ | φ, N, l (cid:105) , with the normalization fac-tor N N,l ; l = (cid:113) N N ( N − l ) N − l l l (cid:113) ( N − l )! l ! N ! . Inverting the re-lation (4), we have | φ, N, l (cid:105) = (cid:80) Nl =0 1 N N,l l e ilφ | l (cid:105) , andwith (3), defining β = | β | exp[ iφ β ], | φ, N, l (cid:105) = 1 A β (cid:90) π dφ β π (cid:88) l e − il ( φ β − φ ) N N,l ; l √ l ! | β | l | β (cid:105) . (5)Now, by setting | β | = l , that is by choos-ing the mean “photon” number to be equal tothe C l -distribution center, we obtain | φ, N, l (cid:105) = (cid:113) N ! N N A β (cid:82) π dφ β π (cid:16)(cid:80) l (cid:113) ( N − l ) N − l ( N − l )! e − il ( φ β − φ ) (cid:17) | β (cid:105) . Thefactor ( N − l ) N − l / ( N − l )! is proportional to a Pois-son distribution centered around N − l , well approx-imated by a normal distribution with both mean andvariance N − l . Then (cid:80) Nl =0 (cid:113) ( N − l ) N − l ( N − l )! e − il ( φ β − φ ) ∝ (cid:80) Nl =0 e − ( l − l N − l e − il ( φ β − φ ) . This resembles a discreteFourier transform of the normal distribution except that l ≥ | β | = l , that a phases-space basis state maps to a truncated coherent state ofthe ladder operators, i.e., | φ, N, l (cid:105) (cid:39) | β (cid:105) , as long as | β | and hence the fragmentation do not become too small,in which case the details of the C l distribution at theboundary l = 0 matter and the concept of a β -coherentstate breaks down. An analogous argument can be ap-plied to | β (cid:48) (cid:105) (cid:39) | φ, N, l (cid:105) [34]. Below, we restrict ourselvesto the truncated coherent states associated to the ladderoperator ˆ b , with analogous conclusions holding for ˆ b (cid:48) .Using | β (cid:105) (cid:39) | φ, N, l (cid:105) and |− β (cid:105) (cid:39) | φ + π, N, l (cid:105) , wethus make the ansatz | Ψ (cid:105) (cid:39) C β | β (cid:105) + C − β |− β (cid:105) for thefragmented condensate many-body state. To identify thecoefficients, we observe that according to [14, 38], a frag-mented state can equivalently be written as a superposi-tion of two (for large N ) degenerate many-body states | Ψ (cid:105) = c ( | even (cid:105) + u exp[ iθ K ] | odd (cid:105) ) , (6)where | even (cid:105) only contains even l coefficients C l fromthe state (1) and | odd (cid:105) only odd l , and | c | (1 + | u | ) = 1. By (3) and the fact that φ l = arg( C l ) =arg (cid:0) C β + C − β ( − l (cid:1) + lπ (for φ β = π/ | even (cid:105) (cid:39) A β √ e − | β | ) (cid:80) Nl =0 | β | l √ l ! (cid:0) i l (1 + ( − l ) (cid:1) | l (cid:105) ; fur-ther, | odd (cid:105) (cid:39) A β i √ − e − | β | ) (cid:80) Nl =0 | β | l √ l ! (cid:0) i l (1 − ( − l ) (cid:1) | l (cid:105) .The overlap (cid:104)− β | β (cid:105) = A β (cid:80) Nl =0 ( −| β | ) l l ! (cid:39) e − | β | forlarge N , and we obtain even and odd superpositionsof | β (cid:105) and |− β (cid:105) , | even (cid:105) (cid:39) √ e − | β | ) ( | β (cid:105) + |− β (cid:105) ), | odd (cid:105) (cid:39) i √ − e − | β | ) ( | β (cid:105) − |− β (cid:105) ) [34].In summary, we have established that a fragmentedcondensate state can be written as a superposition of“photonic” truncated coherent states of the ladder oper-ators defined in (2), | Ψ (cid:105) = N ( | β (cid:105) + re iθ |− β (cid:105) ) , | β | = l , (7)where N = (cid:0) r + 2 r cos θ exp[ − | β | ] (cid:1) − / . Thephase θ and coefficient r in terms of u, θ K in (6) are foundfrom r exp[ iθ ] = uλ β e i ( θk + π/ − uλ β e i ( θk + π/ , where λ β = (cid:113) e − | β | − e − | β | .In the limit | β | (cid:29) r = (cid:12)(cid:12)(cid:12)(cid:12) (1 − u ) + 2 iu cos θ K (1 + u ) + 2 u sin θ K (cid:12)(cid:12)(cid:12)(cid:12) , θ = tan − (cid:18) u cos θ K − u (cid:19) . (8)We thus find, in the equal-weight case r = 1, that thefragmented two-mode many-body states represent theanalogue of photon cat states [6–9]. For the latter, thetruncated coherent state | β (cid:105) of the massive bosons is re-placed by the coherent state of, e.g., a cavity photon field.It was shown in [30] that a stable variety of frag-mented two-mode many-body states is obtained for r =1, θ = π/ , π/
2. In the quantum optics literature onphoton cat states, due to their potential for creating en-tangled coherent states, cf. the review [39], the states | even (cid:105) , | odd (cid:105) are frequently studied, which correspondto r = 1, θ = 0 , π . They are susceptible to perturba-tions when | β | (cid:29)
1, which has been borne out for thepresently studied ”photonic” case in an interacting gas ofmassive bosons [30]. Finally, r (cid:28) r (cid:29) | β (cid:105) and |− β (cid:105) , respectively, which represent nonfragmentedcondensation. The variances of effective position and momentum vari-ables (quadratures), in ladder space, are obtained from ± ∆(ˆ b ± ˆ b † ) = | β | cos φ β (cid:16) r r (cid:17) + 1 , “ + ” , | β | sin φ β (cid:16) r r (cid:17) + 1 , “ − ” , (9)for | β | (cid:29)
1, and where ∆(ˆ b ± ˆ b † ); = (cid:104) (ˆ b ± ˆ b † ) (cid:105)−(cid:104) ˆ b ± ˆ b † (cid:105) .The superposition (7) is a proper cat state for r → φ β can be considered as a parameterdetermining in which direction of “photonic” quadraturefluctuations are squeezed.The size of the SIMCAS (that is their degree of quan-tum mechanical superposition macroscopicity) is, in ac-cordance with the quadratures (9), determined by | β | .The size of coherent state superpositions such as (7) ismaximal when the overlap (cid:104)− β | β (cid:105) (cid:39) exp[ − | β | ] is min-imal [18, 19], the size M being a simple polynomial func-tion of the overlap, M (cid:39) (1 − exp[ − | β | ]) . Thus, dueto | β | = F N/ M (cid:39) (1 − exp[ −F N ]) ( r = 1) . (10)The exponential dependence of the superposition size M on F highlights the distinctly nonclassical character ofa fragmented condensate in comparison to a BEC (sin-gle condensate). This will be manifest already for rel-atively small values of F , when N is mesoscopic, as inexperimentally realized quantum gases. We remark thatthere exist alternative measures for assessing the macro-scopicity of superpositions, which rely on calculating thequantum Fisher information [40, 41], but obviously alsoinvolve the overlap (cid:104)− β | β (cid:105) in an essential manner.In marked distinction to superpositions of photon co-herent states, the superposition size M strongly dependsnot only on a particle number, N , but also, via F , onthe strength of the microscopic (two-body) interactionsin the constituent system dilute quantum gas; M in factvanishes exponentially fast for weakly interacting systemswhere F →
0. The macroscopic superposition size of theSIMCAS is therefore a genuine many-body effect [42].We note in this regard that the macroscopicity M forour matter wave cat is assessed by the rather straight-forward measurement of density-density correlations (seebelow), and not the elaborate quantum state tomographyrequired for proper photonic cat states.Next we establish the relation of the “photonic”quadratures (9) to the density-density correlations inthe interacting quantum gas. The nonclassical char-acter of the superposition (7), in conjunction with itsmany-body origin, will, by this means, become experi-mentally verifiable. We consider the density-density cor-relations in an effectively one-dimensional (1D) system(also see the below discussion on the experimental im-plementation), ∆ ρ ( z, z (cid:48) ) := (cid:10) ˆ ρ ( z )ˆ ρ ( z (cid:48) ) (cid:11) − ρ ( z ) ρ ( z (cid:48) ),where the density ρ ( z ) = (cid:10) ˆ ψ † ( z ) ˆ ψ ( z ) (cid:11) . We take the ap-proximation that (cid:10) ˆ ρ ( z )ˆ ρ ( z (cid:48) ) (cid:11) (cid:39) (cid:10) ˆ ψ † ( z ) ˆ ψ † ( z (cid:48) ) ˆ ψ ( z (cid:48) ) ˆ ψ ( z ) (cid:11) ,which holds true for (cid:10) ˆ N i (cid:11) (cid:29)
1. For simplicity, we as-sume that the two macroscopically occupied field op-erator modes have even and odd parity, respectively,ˆ ψ ( z ) = ψ ( z )ˆ a + ψ ( z )ˆ a , with ψ ( z ) = ψ ( − z ) and ψ ( z ) = − ψ ( − z ), and are real, ψ i ( z ) ∈ R . Bearing inmind that a fully self-consistent solution of the many-body Schr¨odinger equation will yield the true orbitalscf., e.g. [12, 13, 29, 43], for illustration purposes wetake them to be the ground and first excited states ofthe harmonic oscillator; ψ ( z ) = π − / exp (cid:2) − z / (cid:3) and ψ ( z ) = π − / √ z exp (cid:2) − z / (cid:3) , where the z coordinateis assumed to be scaled by a suitable measure of (halfthe) extension of the cloud.For | β | (cid:29) , r = 1, we obtain the correlation func-tion ∆ ρ ( z, z (cid:48) ) = ψ ( z ) ψ ( z (cid:48) ) ψ ( z (cid:48) ) ψ ( z )[2 (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) + (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) + (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − ( (cid:10) ˆ a † ˆ a + ˆ a † ˆ a (cid:11) ) ]. Convertingthis to “photonic” ladder space, we get∆ ρ ( z, z (cid:48) ) (cid:39) ψ ( z ) ψ ( z (cid:48) ) ψ ( z (cid:48) ) ψ ( z ) × (cid:20)(cid:28)(cid:16) ˆ N / (ˆ b + ˆ b † ) (cid:17) (cid:29) − (cid:16)(cid:68) ˆ N / (ˆ b + ˆ b † ) (cid:69)(cid:17) (cid:21) . (11)The density-density correlations in the strongly interact-ing gas therefore map out the quadrature in the firstline of (9) when the fluctuation of occupation numbers issmall, i.e. for large N at finite F .Before TOF, φ β = π , with no distinct correlation sig-nal present (the negative pair coherence of a fragmentedstate enforces sgn( C l +2 C l ) = − φ β = π + 2 nπ , where n is aninteger [34]). After expansion of the cloud, it has beenshown in [31] that TOF of the cloud in the axial direc-tion effectively performs a rotation φ β → φ β − π/ b in(2) involving ˆ a † ˆ a . This results in φ β = 0, and we con-clude from (9) and (11) that a characteristic correlationsignal develops [44]. We plot the density-density corre-lations according to the state (7) in Fig. 1, which mapsout with increasing accuracy the exact correlations of theunderlying fragmented many-body state (6) when | β | in-creases [34]. One clearly recognizes the emerging strongcorrelation signature of an increasingly macroscopic SIM-CAS for larger “photon” number | β | : The larger the de-gree of fragmentation, the bigger the cat becomes. Toidentify the ‘dead’ and ‘alive’ parts of the superpositionstate, we plot in the bottom part of Fig. 1 the density-density correlator (cid:10) ˆ ρ ( z )ˆ ρ ( z (cid:48) ) (cid:11) , for three limiting cases ofsuperposition weights in (7), r = 0 , , ∞ , yielding | β (cid:105) , √ ( | β (cid:105) + exp[ iθ ] |− β (cid:105) ), and |− β (cid:105) , respectively.For the experimental realization of SIMCAS, quasi-1D systems [45] are favorable, since they possess largerdegrees of fragmentation when compared to higher-dimensional gases [29]. SIMCAS exist in a interaction ∆ ρ ( z, z ) ˆ ρ ( z ) ˆ ρ ( z ) √ ( |− β + e iθ | β ) |− β (‘Dead’) | β (‘Alive’) FIG. 1. Emergence of a “photonic” cat state superposi-tion in an ultracold quantum gas containing N = 100 mas-sive, interacting bosons. Top: 1D density-density correlations∆ ρ ( z, z (cid:48) ), scaled by N /Z , where Z is the extension of thecloud after axial TOF ( Z ∝ t in the long time limit [31]),which is also the scaling unit of z, z (cid:48) . From left to right, thefragmentation degree increases according to F = 0 . , . , . | β | = 5 , ,
20, with r = 1). Bottom: Density-density corre-lator (cid:10) ˆ ρ ( z )ˆ ρ ( z (cid:48) ) (cid:11) (in the same N /Z units) for | β | = 20, and r → r = 1 (middle), r → ∞ (right). The correlatordoes not depend on θ in the large | β | limit. coupling–density region of the phase diagram relativelyclose to the BEC domain and far from the extreme Tonks-Girardeau regime of ultralow density and very large cou-pling [46, 47] (in which the number of fragments equalsthe number of particles [25]). The requirements on gasdensity and contact interaction coupling constant g to re-alize SIMCAS are therefore more moderate than for theTonks-Girardeau gas, and should be readily accessible inexperiment. Finally, the phase θ K and amplitude r in thesuperposition (6) can be engineered by a rapid sweep ofinteraction couplings, as demonstrated for θ K in [38].Our analysis reveals the highly nonclassical characterof fragmented condensate many-body states, as opposedto the essentially classical BEC contained in the sametrap. We therefore anticipate applications of SIMCAS, inter alia , in quantum metrology [48], where the Cram´er-Rao bound furnishes rigorous bounds on the precision towhich parameters in the Hamiltonian can be measured[49]. The quantum superposition macroscopicity of SIM-CAS manifest in the density-density correlations of thebosonic gas thus establishes a potential many-body re-source of parameter estimation theory.We thank Ty Volkoff and Daniel Braun for helpful dis-cussions. This research was supported by the NRF Ko-rea, Grant No. 2014R1A2A2A01006535. [1] E. Schr¨odinger, Die gegenw¨artige Situation in der Quan-tenmechanik, Naturwiss. , 807 (1935).[2] A. Ourjoumtsev, R. 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We here provide a more detailed analysis of both the accuracy of using | β (cid:105) as approximate (truncated) coherentstates, as well as the accuracy of the identification of phase-space basis states with them, | φ, N, l (cid:105) (cid:39) | β (cid:105) . Furthermore,a more extended description of the TOF evolution of density-density correlations is presented, and the accuracy ofthe cat state ansatz | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105) ) for the fragmented condensate many-body state is assessed. Accuracy of Treating | β (cid:105) as a Coherent State In the main text, we introduced in Eq. (2) ladder operators, which represent approximate bosonic annihilationoperators, as follows ˆ b = lim (cid:15) → (cid:112) ˆ N + (cid:15) ˆ a † ˆ a , ˆ b (cid:48) = lim (cid:15) → (cid:112) ˆ N + (cid:15) ˆ a † ˆ a , (S1)and the corresponding truncated coherent states as | β (cid:105) = A β N (cid:88) l =0 β l √ l ! | N − l, l (cid:105) , | β (cid:48) (cid:105) = A β (cid:48) N (cid:88) l =0 β (cid:48) N − l (cid:112) ( N − l )! | N − l, l (cid:105) , (S2)where A β = exp( −| β | / (cid:113) Γ( N +1)Γ( N +1 , | β | ) . Truncated coherent states were previously treated, e.g., by [32]. They arefinite-dimensional-Hilbert-space versions of the bosonic annihilation operators ˆ a and coherent state | α (cid:105) in quantumoptics. Here, we aim at finding a quantity which assesses the accuracy of treating ˆ b and | β (cid:105) ( ˆ b (cid:48) and | β (cid:48) (cid:105) ) as bosonicannihilation operators and their corresponding eigenstates,[ˆ b, ˆ b † ] = 1 , ˆ b | β (cid:105) = β | β (cid:105) , (S3)and which allows us to numerically evaluate the precision to which the above two relations hold.We assume, without loss of generality, (cid:10) ˆ a † ˆ a (cid:11) ≤ N/ | β | ≤ N/ { ˆ b, | β (cid:105)} when (cid:10) ˆ a † ˆ a (cid:11) ≤ N/
2, and { ˆ b (cid:48) , | β (cid:48) (cid:105)} when (cid:10) ˆ a † ˆ a (cid:11) ≥ N/
2. In addition, we note that there is always the freedom to choose ˆ b, | β (cid:105) or ˆ b (cid:48) , | β (cid:48) (cid:105) to describe a given twomode system. With respect to an arbitrary (normalized) two-mode state | Ψ (cid:105)| Ψ (cid:105) = N (cid:88) l =0 C l | N − l, l (cid:105) = N (cid:88) l =0 C l (ˆ a † ) N − l (ˆ a † ) l (cid:112) ( N − l )! l ! | (cid:105) , N (cid:88) l =0 | C l | = 1 , (S4) FIG. S1. Plots of (cid:80) Nl = N − n +1 | β | ll ! (cid:80) Nl =0 | β | ll ! (indicated by loss of | C l | on the vertical axis), with | β | = N/ | β | = N/ N = 25 (left) and N = 50 (right). the accuracy of (S3) can be assessed by evaluating the following two quantities1 − (cid:104) Ψ | [ˆ b, ˆ b † ] | Ψ (cid:105) = ( N + 1) | C N | , (cid:16) (cid:104) β | β ∗ − (cid:104) β | ˆ b † (cid:17) (cid:16) β | β (cid:105) − ˆ b | β (cid:105) (cid:17) | β | (cid:104) β | β (cid:105) = A β | β | N N ! . (S5)It is readily observed that the accuracy of ˆ b, ˆ b † as a proper set of bosonic operators depends on the state | Ψ (cid:105) which isconsidered. In the main text a twofold fragmented state is discussed with the ansatz | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105) ) (also seebelow), and since the | C N | distributions of | β (cid:105) and |− β (cid:105) are identical, it is sufficient to consider | Ψ (cid:105) = | β (cid:105) . Eq. (S5)becomes 1 − (cid:104) β | (ˆ b ˆ b † − ˆ b † ˆ b ) | β (cid:105) = ( N + 1) A β | β | N N ! , (cid:16) (cid:104) β | β ∗ − (cid:104) β | ˆ b † (cid:17) (cid:16) β | β (cid:105) − ˆ b | β (cid:105) (cid:17) | β | (cid:104) β | β (cid:105) = A β | β | N N ! . (S6)Higher order expressions in ˆ b, ˆ b † such as ˆ b ˆ b ˆ b † ˆ b † or the repetitive action of ˆ b against | β (cid:105) , can be similarly assessed.They induce the “loss” of certain amplitudes C N − , · · · , C N − n +1 from the state vector | β (cid:105) , where β n | β (cid:105) − (ˆ b ) n | β (cid:105) = β n A β N (cid:88) l = N − n +1 β l √ l ! | N − l, l (cid:105) . (S7)Therefore the robustness of the coherent state representation can be defined by the degree to which a state of interest | Ψ (cid:105) is not deformed from the loss of the amplitudes C N , · · · , C N − n +1 due to a n -times repeated action of ˆ b on it.Specifically, for | Ψ (cid:105) = | β (cid:105) this robustness can be quantified through (cid:80) Nl = N − n +1 | β | l l ! (cid:80) Nl =0 | β | l l ! = Γ( N + 1 , | β | ) − Γ( N + 1) Γ( N − n +1 , | β | )Γ( N − n +1) Γ( N + 1 , | β | ) (cid:39) exp( −| β | ) N (cid:88) l = N − n +1 | β | l l ! , (S8)which represents the lost fraction of the C l due to repetitive action of ˆ b . We used here Γ( N + 1 , | β | ) (cid:39) Γ( N + 1)which holds within 1% error for N >
10 with given | β | ≤ N/
2. In Fig. S1, we show plots of the quantity definedin (S8) for different
N, n, | β | /N . We see that as N gets larger and | β | /N gets smaller, the robustness significantlyincreases. Specifically, for N = 25 , | β | = 0 . N there is already a negligible loss of C l amplitudes from the state evenupon n = 5 times acting with ˆ b on it. Comparison of | φ, N, l (cid:105) , | β (cid:105) , | β (cid:48) (cid:105) We argue in the main text that | φ, N, l (cid:105) (cid:39) | β (cid:105) for small l = | β | , and | φ, N, l (cid:105) (cid:39) | β (cid:48) (cid:105) for small N − l = | β (cid:48) | with φ = φ β and φ = − φ β (cid:48) , cf. the discussion after Eq. (5) in the main text and Eq. (S9) below. We present in Fig. S2 FIG. S2. Plots of the | C l | distribution of | φ, N, l (cid:105) (red), | β (cid:105) (blue), | β (cid:48) (cid:105) (green) with N = 100 for different l = | β | = N − | β (cid:48) | = 0 . N, . N, . N (Top from left) and l = | β | = N − | β (cid:48) | = 0 . N, . N, . N (Bottom from left). the | C l | distribution for | φ, N, l (cid:105) , | β (cid:105) , | β (cid:48) (cid:105) , respectively, with different l = | β | in the case N = 100. We concludethat | φ, N, l (cid:105) (cid:39) | β (cid:105) is confirmed for small l = | β | , and | φ, N, l (cid:105) (cid:39) | β (cid:48) (cid:105) for small N − l = | β (cid:48) | with φ = φ β and φ = − φ β (cid:48) .Furthermore, we have verified that in the large N limit, the truncated coherent states to increasingly good accuracyrepresent the corresponding phase-space basis vectors even when l → N , that is in the limit of maximal fragmentation, F →
1. In more detail, as | β | approaches | β | = N/ F = 1), the | C l | distribution of the phase state | φ, N, l (cid:105) becomes wider than that of | β (cid:105) or | β (cid:48) (cid:105) . However, | φ, N, l (cid:105) (cid:39) | β (cid:105) still holds. From equation (5) of the main text,which transforms | φ, N, l (cid:105) into the | β (cid:105) basis, we have | φ, N, l (cid:105) = (cid:114) N ! N N A β (cid:90) π dφ β π (cid:32)(cid:88) l (cid:115) ( N − l ) N − l ( N − l )! e − il ( φ β − φ ) (cid:33) | β (cid:105) = (cid:90) π dφ β π C φ β | β (cid:105) . (S9)with | β | = l where C φ β = (cid:113) N ! N N A β (cid:16)(cid:80) l (cid:113) ( N − l ) N − l ( N − l )! e − il ( φ β − φ ) (cid:17) . In Fig. S3, the modulus of the coherent state phasedistribution, | C φ β | , is plotted for l = | β | = N/
2, and with various values of particle number N . It is apparent that,with increasing N , even when N = 25, a rapid convergence of | C φ β | to one peak at φ β = φ occurs. Time-Of-Flight Expansion (TOF) and Phase Rotation
We present here a more detailed analysis of the phase rotation incurred by TOF.After TOF, the assumed even-odd parity of the modes is preserved, ψ ( z, t ) = ψ ( − z, t ) , ψ ( z, t ) = − ψ ( − z, t ).Defining the phase variable ϕ such that e − iϕ ψ ( z, t ) := ¯ ψ ( z, t ) , ψ ∗ ( z, t ) ¯ ψ ( z, t ) ∈ R (S10)the wave functions ψ ( z, t ) and ψ ( z, t ) share a common phase factor [35], keeping (cid:10) ˆ a † i ˆ a j (cid:11) , (cid:10) ˆ a † i ˆ a † j ˆ a k ˆ a l (cid:11) ( i, j, k, l = 0 , t = 0) ϕ = 0 and after TOF (large t ) ϕ = − π/ ϕ could be both ϕ or ϕ + π since e iπ times a real number is again real number. We write FIG. S3. Plots of the | C φ β | distribution of | φ, N, l (cid:105) (cf. Eq. (5) in the main text and the discussion that follows it), for l = | β | = N/ N = 25 , , down the density-density correlation function ∆ ρ ( z, z (cid:48) , t ) as∆ ρ ( z, z (cid:48) , t ) = | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:16)(cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11)(cid:17) + | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:16)(cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11)(cid:17) + (cid:0) | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | + | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:1) (cid:16)(cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11)(cid:17) + (cid:0) | ψ ( z, t ) | ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) + | ψ ( z (cid:48) , t ) | ψ ∗ ( z, t ) ¯ ψ ( z, t ) (cid:1) (cid:104) e iϕ (cid:16)(cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11)(cid:17) + h.c. (cid:105) + (cid:0) | ψ ( z, t ) | ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) + | ψ ( z (cid:48) , t ) | ψ ∗ ( z, t ) ¯ ψ ( z, t ) (cid:1) (cid:104) e iϕ (cid:16)(cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11)(cid:17) + h.c. (cid:105) + ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) (cid:20) (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) + e iϕ (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) + e − iϕ (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) − (cid:16)(cid:10) e iϕ ˆ a † ˆ a + e − iϕ ˆ a † ˆ a (cid:11)(cid:17) (cid:21) . (S11)Furthermore, we can take ¯ ψ ( z, t ) as the new ψ ( z, t ), yielding ψ ( z, t ) ψ ( z, t ) ∈ R , by changing the many-body state | Ψ (cid:105) as follows | Ψ (cid:105) = N (cid:88) l =0 C l | N − l, l (cid:105) → | Ψ ϕ (cid:105) = N (cid:88) l =0 C l e ilϕ | N − l, l (cid:105) . (S12)This rotation of the state in (S12) corresponds to a rotation of the phase φ β of the approximate coherent state | β (cid:105) , | β (cid:105) → (cid:12)(cid:12) βe iϕ (cid:11) . (S13)For a general twofold fragmented state we then get1 √ r + 2 r cos θe − | β | (cid:0) | β (cid:105) + re iθ |− β (cid:105) (cid:1) → √ r + 2 r cos θe − | β | (cid:0)(cid:12)(cid:12) βe iϕ (cid:11) + re iθ (cid:12)(cid:12) − βe iϕ (cid:11)(cid:1) . (S14)Since φ β = π/ φ β = 0 after TOF. We can therefore summarize as follows: TOF evolutionis equivalent to ϕ = 3 π/ − π/ ρ ( z, z (cid:48) ) from (S11) forboth before TOF ( ϕ = 0 or φ β = π/
2) and after TOF ( ϕ = − π/ φ β = 0). Validity of the | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105) ) Ansatz
As a main ingredient of our analysis, we propose an ansatz, | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105) ), to describe a twofold fragmentedstate (Eq. (7) in the main text). To test the validity of the ansatz in more detail, let us first specify the C l distribution0of a general twofold fragmented state. In [14], it was shown that | C l | has a Gaussian distribution of mean l andvariance σ with σ ∼ O ( √ N ) as long as the continuum limit for the C l distribution holds. Furthermore, a generaltwofold fragmented state can also be written as [38] | Ψ (cid:105) = c ( | even (cid:105) + u exp[ iθ K ] | odd (cid:105) ) , | c | (1 + | u | ) = 1 , (S15)which leads to | C l | = √ | c | πσ ) / e − ( l − l σ e iφ l for even l | C l | = √ | c || u | πσ ) / e − ( l − l σ e iφ l for odd l , (S16)where C l = | C l | e iφ l , and φ l obeys φ l +2 = φ l + π, φ l +4 = φ l from the condition sgn( C l C l +2 ) = − (cid:40) φ l +1 = φ l + θ k for even lφ l +1 = φ l + π − θ k for odd l . (S17)We now examine to which extent using | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105) ) is accurate for the calculation of the density-densitycorrelations ∆ ρ ( z, z (cid:48) ) by calculating (S11) with (S16) in terms of σ, l and c, u . We then compare the correspondingresult for the density-density correlations obtained by using the exact large N fragmented state with what we deriveusing Eq. (11) and the quadrature fluctuations of Eq. (9) in the main text. As we will show, we obtain two resultswhose difference depends on the value of the Gaussian width σ .To calculate (S11), we perform the following approximations. When the Gaussian | C l | distribution is well localizedin the interval [0 , N ] and N is large enough to approximate C l to be continuous, then it is permissible to approximatethe expectation value of f ( N − l, l ), which is a polynomial of √ N − l, √ l , as (cid:88) l f ( N − l, l ) | C l | =2 | c | (cid:88) l =0 , , ··· f ( N − l, l ) 1 √ πσ e − ( l − l σ + 2 | c | | u | (cid:88) l =1 , , ··· f ( N − l, l ) 1 √ πσ e − ( l − l σ (cid:39) (cid:90) ∞−∞ f ( N − l, l ) 1 √ πσ e − ( l − l σ dl. (S18)The quantities (cid:10) ˆ a † ˆ a (cid:11) , (cid:10) ˆ a † ˆ a (cid:11) , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) and (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) are directly determined from the above expres-sion. We have (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) = ( N − l ) − ( N − l ) + σ , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) = l − l + σ , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) = ( N − l ) l − σ (S19)with (cid:10) ˆ a † ˆ a (cid:11) = N − l , (cid:10) ˆ a † ˆ a (cid:11) = l .The quantities (cid:10) ˆ a † ˆ a (cid:11) , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) and (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) , in turn, are related to the sum (cid:88) l f ( N − l, l ) C ∗ l C l +1 (cid:39) | c | | u | (cid:88) l f ( N − l, l ) | C l | e i ( φ l +1 − φ l ) , (S20)the last relation holding when | C l | (cid:39) | C l +1 | . Under the proviso that φ l +4 = φ l and | C l | slowly vary so that | C l | (cid:39) | C l +4 | ,we can carry out the summation over e i ( φ l +1 − φ l ) separately from the l summation as (cid:88) l f ( N − l, l ) | C l | e i ( φ l +1 − φ l ) (cid:39) (cid:88) l =0 e i ( φ l +1 − φ l ) | c | | u | (cid:88) l f ( N − l, l ) | C l | . (S21)Therefore we have (cid:88) l f ( N − l, l ) C ∗ l C l +1 (cid:39) (cid:88) l =0 e i ( φ l +1 − φ l ) | c | | u | (cid:90) ∞−∞ f ( N − l, l ) 1 √ πσ e − ( l − l σ dl. (S22)From the following string of phase values, φ l = · · · , , θ k , π, θ k + π, , θ k , π, θ k + π, · · · , where φ = 0 is chosen, weobtain 14 (cid:88) l =0 e i ( φ l +1 − φ l ) = e iθ k + e i ( π − θ k ) + e iθ k + e − i ( π + θ k ) = i sin θ k . (S23)1The function f ( N − l, l ) now includes square roots of N − l or l . It is therefore nontrivial to write down expectationvalues in terms of N, l , σ in closed form, with the exception of (cid:10) ˆ a † ˆ a (cid:11) = 2 | c | ui (cid:112) ( N − l ) l sin θ k . Now, (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) and (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) read (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) | c | ui sin θ k (cid:90) ∞−∞ ( N − l −
12 ) (cid:114) ( N − l + 12 )( l + 12 ) 1 √ πσ e − ( l − l σ dl, (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) | c | ui sin θ k (cid:90) ∞−∞ ( l −
12 ) (cid:114) ( N − l + 12 )( l + 12 ) 1 √ πσ e − ( l − l σ dl. (S24)From the following integrals (cid:90) ∞−∞ l √ πσ e − ( l − l σ dl = l , (cid:90) ∞−∞ l √ πσ e − ( l − l σ dl = l + σ , (S25)we can power expand as (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) | c | ui sin θ k (cid:104) ( N − l ) (cid:112) ( N − l ) l + O ( N − l ) + O ( σ ) (cid:105) , (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) | c | ui sin θ k (cid:104) l (cid:112) ( N − l ) l + O ( N − l ) + O ( σ ) (cid:105) . (S26)The first, highest order, terms are matching (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11) ∝ ( N − l ) (cid:112) ( N − l ) l and (cid:10) ˆ a † ˆ a (cid:11)(cid:10) ˆ a † ˆ a (cid:11) ∝ l (cid:112) ( N − l ) l ;we assume l < N/
2. Finally, for (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) ( (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) is given by the complex conjugate), with slowly varying | C l | , we have (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) (cid:88) l =0 e i ( φ l +2 − φ l ) (cid:90) ∞−∞ (cid:112) ( N − l )( N − l + 1) l ( l + 1) 1 √ πσ e − ( l − l σ dl. (S27)A expansion similar to that leading from (S24) to (S26), and (cid:80) l =0 e i ( φ l +2 − φ l ) = −
1, yields (cid:10) ˆ a † ˆ a † ˆ a ˆ a (cid:11) (cid:39) − ( N − l ) l + O ( N − l ) + O ( σ ) . (S28)Summing up, Eq. (S11) becomes∆ ρ ( z, z (cid:48) , t ) = | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:0) σ − ( N − l ) (cid:1) + | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:0) σ − l (cid:1) + (cid:0) | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | + | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | (cid:1) σ + 2 | c | ui sin θ k (cid:0) | ψ ( z, t ) | ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) + | ψ ( z (cid:48) , t ) | ψ ∗ ( z, t ) ¯ ψ ( z, t ) (cid:1) (cid:2) O ( N − l ) + O ( σ ) (cid:3) + 2 | c | ui sin θ k (cid:0) | ψ ( z, t ) | ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) + | ψ ( z (cid:48) , t ) | ψ ∗ ( z, t ) ¯ ψ ( z, t ) (cid:1) (cid:2) O ( N − l ) + O ( σ ) (cid:3) + 4 ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) (cid:2) ( N − l ) l sin ϕ (cid:0) − (2 | c | u sin θ k ) (cid:1) + O ( N − l ) + O ( σ ) (cid:3) . (S29)Provided the orbitals ψ ( z, t ) and ψ ( z, t ) have large overlap, as to be expected in a single trap, the following fourterms have similar order of magnitude, ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t ) ∼ | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | , | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | , | ψ ( z, t ) | | ψ ( z (cid:48) , t ) | . (S30)This leads to∆ ρ ( z, z (cid:48) , t ) = 4 ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t )( N − l ) l sin ϕ (cid:0) − (2 | c | u sin θ k ) (cid:1) + O ( N − l ) + O ( σ ) . (S31)Therefore, since the first term of (S31) is proportional to ( N − l ) l , and because we assume ( N − l ) l (cid:29) N − l , ( N − l ) l (cid:29) σ , we obtain ∆ ρ ( z, z (cid:48) ) to lowest order as∆ ρ ( z, z (cid:48) ) (cid:39) ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t )( N − l ) l sin ϕ (cid:0) − (2 | c | u sin θ k ) (cid:1) . (S32)To compare with the result in Eq. (11) of the main text, we employ the relation between r, θ and u, θ k specified forlarge | β | by Eq. (8). Then 2 | c | u sin θ k = 2 u/ (1 + u ) sin θ k can be expressed in terms of u, θ k as1 − r r = 2 uλ β u λ β sin θ k = (cid:18) u u sin θ k (cid:19) u λ β ( λ − β + u ) . (S33)2In the λ β = (cid:112) (1 + e − | β | ) / (1 − e − | β | ) → e − | β | → | β (cid:105) and |− β (cid:105) ), thisbecomes 1 − r r = 2 u u sin θ k = 2 | c | u sin θ k . (S34)The result does not depend on θ . Thus we see that when the overlap between | β (cid:105) and |− β (cid:105) gets smaller, the effect of θ on ∆ ρ ( z, z (cid:48) ) becomes negligible. In this limit (S32) in terms of r , with | β | = l , is given as∆ ρ ( z, z (cid:48) ) (cid:39) ψ ∗ ( z, t ) ¯ ψ ( z, t ) ψ ∗ ( z (cid:48) , t ) ¯ ψ ( z (cid:48) , t )( N − l ) l sin ϕ r (1 + r ) , (S35)which agrees with the r = 1 result of the main text in Eq. (11) as long as ( N − l ) l (cid:29) N − l , σ .We conclude that ( N − l ) l (cid:29) N − l , σ , and the large single-trap overlap between the orbitals ψ ( z, t ) and ψ ( z, t )are the conditions to utilize the representation | Ψ (cid:105) = N ( | β (cid:105) + r iθ |− β (cid:105)(cid:105)