Theory of Subradiant States of a One-Dimensional Two-Level Atom Chain
TTheory of Subradiant States of a One-Dimensional Two-Level Atom Chain
Yu-Xiang Zhang ∗ and Klaus Mølmer † Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark (Dated: May 23, 2019)Recently, the subradiant states of one-dimensional two-level atom chains coupled to light modeswere found to have decay rates obeying a universal scaling, and an unexpected fermionic characterof the multiply-excited subradiant states was discovered. In this Letter, we theoretically obtainthe singly-excited subradiant states, and by eliminating the superradiant modes, we demonstrate arelation between the multiply-excited subradiant states and the Tonks-Girardeau limit of the Lieb-Liniger model which explains the fermionic behavior. In addition, we identify a new family of stateswith correlations different from the fermionic ansatz.
To achieve controllable and deterministic photon-atominterfaces for applications in quantum information pro-cessing and quantum sensing, large atom ensembles maybe used to enhance the coupling to photons [1]. The pho-tons induce both coherent and dissipative atom-atom in-teractions that yield collective phenomena of super- orsub-radiance [2], wherein a collective excitation of theatom ensemble decays faster or slower than individualatomic excitations. While superradiance has been exten-sively studied since the seminal work of Dicke [3], sub-radiance of a large ensemble was observed only very re-cently in cold atom clouds [4, 5] and metamaterial arrays[6]. Comprehensive theoretical tools for the subradianceare still elusive [7–10] due to the complicated long-rangeinteractions and many-body features of the atomic en-sembles [11–13]. A one-dimensional (1D) chain of equallyspaced two-level atoms offers the simplest geometry togain insight in the collective decay mechanisms, and im-plementation of such chains coupled to nanofibers [14],1D waveguides [15–18], and the full vacuum electromag-netic field in 3D free space [8, 13, 19–21] has attractedconsiderable attention. Super- and subradiance phenom-ena are in these systems supplemented by further inter-esting properties and applications such as atomic mirrors[22], photon Fock state synthesis [23], enhancement ofcooperativity [24] and applications in quantum compu-tation [25].Recently, the subradiant states of such 1D chains of N qubits in 3D free space and coupled to 1D waveg-uide were numerically found to have a series of seem-ingly universal properties [8–10]: In the one-excitationsector where only one of the N atoms is excited, if wesort all eigenstates (to be elaborated) by increasing de-cay rates with integer labels from ξ = 1 to ξ = N ,the most subradiant states ( ξ (cid:28) N ) have decay rates γ ξ ∝ ξ /N . In the multi-excitation sectors, the mostsubradiant states have a fermionic character, e.g., amost subradiant state with two excitations is given by | F , (cid:105) ∝ (cid:80) i 4; andan ( N − H R eff .While the perturbation view is informative, a moredirect approach to the subradiant states applies the fol-lowing exact result for the Bloch states | k (cid:105) ( k (cid:54) = ± k D ), H eff | k (cid:105) = ω k | k (cid:105) − i Γ D g k | k D (cid:105) − h k |− k D (cid:105) ) , (2)where ω k = Γ D (cid:80) (cid:15) = ± cot( k D + (cid:15)k d ), and the “tails” g k = e i ( k − k D ) z − e i ( k − k D ) d and h k = e i ( k + k D ) zN e − i ( k + k D ) d − . It follows that asuperposition of two degenerate states, | k (cid:105) and |− k (cid:105) , isan eigenstate of H eff with eigenvalue ω k and has no tails if k is a solution to the equation g k h − k = g − k h k . Thisequation has only solutions for complex values of k . Inthe regimes k ≈ ± π/d (center or edges of the firstBrillouin zone), supposing k ξ = 0+ δ ξ and k ξ = − π/d + δ ξ respectively, we find to order N − , δ ξ = ξπN d × (cid:40) − i N cot( k D d ) , k ≈ 01 + i N tan( k D d ) , k ≈ − π/d (3)with ξ = 1 , , · · · , ξ (cid:28) N . Note that Eq. (3)amounts to an 1 /N -order imaginary correction to theBloch wavenumber.Next, we substitute Eq. (3) into the expression for ω k ,which is parabolic near k ≈ ± π/d , i.e., ω k ∝ δ ξ .Then the imaginary corrections directly yield the ξ /N -scaling of the decay rates [9]: γ ξ = Γ D π ξ N × (cid:40) cos ( k D d/ ( k D d/ , k ≈ sin ( k D d/ ( k D d/ , k ≈ − π/d. (4)The eigenstates are written as | φ k ξ (cid:105) ∝ g − k ξ | k ξ (cid:105) − g k ξ |− k ξ (cid:105) = 1 √ | k (0) ξ (cid:105) − |− k (0) ξ (cid:105) ) + O ( ξN ) , (5)where k (0) ξ = ξπ/ ( N d ) or − π/d + ξπ/ ( N d ). Universality -The ξ /N -scaling has also been numeri-cally found for 1D atom chains coupled to 3D free-spacemodes [8–10], where the effective Hamiltonian H D, eff isdetermined by the Green’s dyadic tensor (see the Supple-mental Material). Fourier transformation of the Green’stensor reveals a hidden similarity between the couplingto the 1D and 3D quantized radiation fields: H D, eff can be written as weighted integrals of terms resembling H eff with real-valued k D ∈ [0 , k ] and imaginary-valued k D ∈ [ i , + i ∞ ]: H D, eff = − i γ k (cid:90) k d ˜ k π ρ + (˜ k ) N (cid:88) m,n =1 e i ˜ k | z m − z n | σ † m σ n − γ k (cid:90) + ∞ d ˜ k π ρ − (˜ k ) N (cid:88) m,n =1 e − ˜ k | z m − z n | σ † m σ n , (6)where γ is the spontaneous emission rate and k is theresonant wavenumber. If the atoms are polarized parallelto the chain, ρ ± (˜ k ) = 2 π (1 ∓ ˜ k /k ) and the atom-atominteraction is short-range ( ∼ /r ). If the atoms arepolarized transverse to the chain, ρ ± (˜ k ) = π (1 ± ˜ k /k )and the atom-atom interaction is long-range ( ∼ /r ).In combination with the two exact features of our an-alytical results for H eff :1. The leading order solutions of δ ξ and | φ k ξ (cid:105) are in-dependent of the values of k D ,see Eq. (5); 2. The proportionality δ ξ ∝ ξ and the parabolic dis-persion relation ω k ∝ δ ξ hold to order- N − , for allvalues of k D ,this explains the universality of the ξ /N -scaling: Fea-ture 1 implies that the leading order eigenstates of H eff ,shown in Eq. (5), are shared simultaneously by all termsintegrated in H D, eff , and thus by the full H D, eff due tolinearity. Feature 2, hence implies that the correspondingdecay rates, scaling as ξ /N , also apply to the subradi-ant states of H D, eff . The prerequisite is that H I D, eff must have dark states with k ≈ ± π/d . Hence we require k < π/d which implies that the ensemble is only sub-radiant in the 3D field if the atom-atom distance is lessthan half the resonant wave length [8, 13]. Subradiant multiply-excited states - When the numberof atomic excitations n e (cid:28) N , the leading order Holstein-Primakoff (HP) approximation [28] usually applies andone may replace σ † m σ n of Eq. (1) with the bosonic op-erators b † m b n and obtain a quadratic bosonic H eff . Itworks well for the superradiant modes with wavenumber ± k D . But for the subradiant multiply-excited states,the bosonic creation operators prepare exchange symmet-ric combinations of subradiant one-excitation states withdecay rates scaling as N − [8, 9] which is much largerthan the numerically observed N − -scaling [8–10]. In-stead, the numerical results were found to favor fermionicexchange anti-symmetric combinations of the subradiantone-excitation states [8–10].This somewhat surprising result inspires a closerscrutiny of the HP transformation. Including secondorder corrections due to saturation, the HP transfor-mation reads σ m = (1 − b † m b m / b m so that we canwrite H I eff = H SR + Q + Q † with the quadratic term H SR = N Γ D / (cid:80) (cid:15) = ± b † (cid:15)k D b (cid:15)k D and quartic terms Q = − Γ D (cid:88) (cid:15) = ± (cid:88) p,q b † (cid:15)k D b † p + q − (cid:15)k D b p b q . (7)Here, b † k = N − / (cid:80) m e ikz m b † m and the summation overwavenumber is taken over an orthonormal basis {| k (cid:105)} k containing |± k D (cid:105) . The quadratic term H SR has a pref-actor N -times larger than those of Q and Q † , but toassess their influence, we should take account of not onlythe prefactors but also the magnitudes of the operatorterms. For H SR , the magnitude of b † (cid:15)k D b (cid:15)k D can be es-timated by its typical expectation value over the relevantHilbert space, i.e., the subradiant states.Reasonably, one may expect that a subradiant statecontains no excitation of superradiant modes, i.e., typ-ically (cid:104) b † (cid:15)k D b (cid:15)k D (cid:105) ≈ 0. Thus the magnitude of H SR is suppressed. Meanwhile, Eq. (7) shows that Q an-nihilates a two-boson dark state, b † p b † q | ∅ (cid:105) , with respectto H SR , and generates a superradiant two-boson state b † (cid:15)k D b † p + q − (cid:15)k D | ∅ (cid:105) ( | ∅ (cid:105) denotes the boson vacuum). Thisdemonstrates that the saturation correction to the HPapproximation plays a significant role even in the low ex-citation regime ( n e (cid:28) N ), in contrast to its role in manyother applications.An effective theory for how Q couples the dark statesto superradiant states and hereby determines their sub-radiant behavior is illustrated in the right panel of Fig.1. The effect of Q and Q † is distilled by eliminating thesuperradiant states, in a manner similar to the adiabaticelimination of excited state manifolds to restrict the effec-tive dynamics of quantum systems to their ground statemanifold [29]. Note that the subset of superradiant states with onlya single excitation of the superradiant modes and thusthe eigenvalue (decay rate) N Γ D / 4, has the strongestcoupling to the dark/subradiant states. We hence disre-gard the coupling to other superradiant states and theeffective coupling among subradiant states reduces to V sub = N Γ D P DS Q † P SRS QP DS , with projection oper-ators P DS ( SRS ) on the dark and superradiant spaces, re-spectively. To evaluate this expression we use the oper-ator relation that ( b p (cid:48) + q (cid:48) − (cid:15) (cid:48) k D b (cid:15) (cid:48) k D )( b † (cid:15)k D b † p + q − (cid:15)k D ) = δ (cid:15),(cid:15) (cid:48) δ p (cid:48) + q (cid:48) ,p + q , i.e., no population of the superradiant two-boson modes within the dark/subradiant space. Finally,we obtain V sub = 18 N Γ D (cid:88) p,q,k b †− p + q + k b † p b q b k = 18 Γ D N (cid:88) m =1 ( b † m ) ( b m ) . (8)That is, V sub induces decay with rate O (Γ D ) of nom-inally subradiant states having more than a single HPboson excitation at the same site.In the absence of V sub , approximate multiply-excitedstates are created by the operators b † ξ = (cid:80) m (cid:104) e m | φ k ξ (cid:105) b † m ,with | φ k ξ (cid:105) the one-excitation eigenstates Eq. (5) of H eff .As V sub cannot be treated as a perturbation, we studythe effective Hamiltonian H = (cid:80) ξ γ ξ b † ξ b ξ + V sub , whereonly the most subradiant states ( ξ (cid:28) N ) are includedin the sum. In the Supplemental Material we show thatin the continuous limit, H can be written as the second-quantized form of the Hamiltonian H = n e (cid:88) i =1 (cid:20) − ∂ x i m ∗ + V ( x i ) (cid:21) + 2 c LL n e (cid:88) i 0; orthe kinetic energy in a gauge field γ ξ = ( k ξ + π/d ) / (2 m ∗ )when k ≈ − π/d . With the parametrization of themodel, the effective mass in the kinetic energy term reads m ∗ = ξ π / ( N d γ ξ ) ∝ N .We recognize Equation (9) as the Lieb-Liniger model[30] originally proposed for 1D gases of hard-core bosons.As the effective mass m ∗ diverges in the large N limit, thekinetic energy-like part of Eq. (9) vanishes. This impliesthat Eq.(9) reaches the Tonks-Girardeau regime [31, 32]of the Lieb-Liniger model, where the eigenstates of H canbe obtained via a fermion-boson mapping [32, 33]: Fora free fermion model described by (cid:80) ξ γ ξ f † ξ f ξ , where f † ξ = (cid:80) m (cid:104) e m | φ ξ (cid:105) f † m , we write down its eigenstates (e.g.,two-fermion states) f † ξ f † ξ | ∅ (cid:105) , and replace f † m f † n withsign( n − m ) b † m b † n , where sign( n − m ) is necessary to en-sure the consistency with the fermionic commutation re-lation. This yields a fermion-like bosonic state | F ξ ,ξ (cid:105) = (cid:80) m 0) [9]. Thesebehaviors can also be explained from the Lieb-Linigermodel of Eq. (9). The fermion-boson mapping is notexact and the phase factor e iϕ introduced above deviatesfrom unity by a factor in the form of ( k ξ − k ξ ) / ( m ∗ c LL )[30]. Since k ξ − k ξ is O ( N − ) or ≈ π/d in the two casesconsidered, while m ∗ c LL scales as N , their ratio scalesexactly in the same manner as the numerically observedinfidelities [9]. Larger discrepancies with the fermionicansatz are detectable when the decay rates increase. Universality- The mapping to Lieb-Liniger model and mn (a) (b) | ψ ⟩ | ψ ⟩ n Figure 2. (a) The two-excitation eigenstates of system with k D = 0 . π/d and N = 20 are sorted by increasing decayrates. The bars show the maximal fidelity that a fermionicansatz can achieve for each eigenstate. The fermionic ansatzfits a broad range of the most subradiant states while a fewexceptional states (the dips in the fidelity, e.g., state num-ber 7) show distinct non-fermionic behavior. (b) Positiondistributions of the atomic excitations, |(cid:104) ψ | e m , e n (cid:105)| with m ( n ) = 0 , , · · · 19, of a typical fermionic subradiant state, | ψ (cid:105) (upper panel) and the non-fermionic state | ψ (cid:105) (lowerpanel). The lower panel feature at | z m − z n | ≈ d indicatesthat | ψ (cid:105) supports a dimer-like bound excitation. the Tonks-Girardeau gas can also be extended to the 1Datomic chain coupled to 3D free-space modes describedby Eq. (6). Here H I D, eff possesses short-lived eigenstates | ˜ k (cid:105) with ˜ k ∈ [ − k , k ] and different decay rates γ ˜ k . Eachof them will contribute to V sub a term with prefactor γ ˜ k /N . Hence we have V sub ∝ (cid:80) ˜ k γ ˜ k /N = γ /N , sim-ilar to the coefficient in the first line of Eq. (8). Sincethe ξ /N -scaling decay rates apply in the one-excitationsector in 3D free-space, the fermionic ansatz also applieshere. Since only subradiant states with k ≈ ± π/d appearin 3D free-space, the pertaining N − scaling applies tothe infidelities of all states given by the fermionic ansatz.This matches the numerical results [8]. Conclusion and Discussion- In this Letter, we havedeveloped a theory to explain the ξ /N -scaling of subra-diant decay rates and the fermionic behavior of multiply-excited subradiant states identified in numerical calcula-tions on 1D atom chains coupled to both 1D and 3Dradiation reservoirs [8–10]. We find that the universal ξ /N -scaling results from a parabolic dispersion relationof the atomic excitation, and imaginary corrections of theBloch quasi-momentum eigenstates of the non-HermitianHamiltonian. For multiply excited systems quartic cor-rections to the Holstein-Primakoff (HP) expansion of theeffective spin Hamiltonian for the atom chain dominatethe coupling of the sub-radiant states, and lead to aformulation equivalent to the Lieb-Liniger model of a1D bosonic quantum gas in the Tonks-Girardeau regime[31, 32]. The fermionic ansatz solution of that problemexplains the decay rates and the properties of the solu-tions found in Ref. [8, 9]. There is a high current interestand many potential applications of subradiance [22–25]and the analytical findings presented here may inspirefurther study of subradiance in light-matter interactionsof more complex geometries, e.g., systems with higher di-mensional atom arrays [34], chiral waveguides that breakthe parity symmetry [35] and setups with topological ef-fects [36, 37].Let us conclude by discussing a remaining theoreticalissue. We recall our effective separation of the Hamil-tonian into an interaction term, V sub , based on H I eff andan expansion on subradiant eigenmodes b ξ for which H R eff contributes the decay rates γ ξ . Like the numerical cal-culations, a more rigorous analytical approach should in-corporate H R eff and H I eff on an equal footing. The factthat our separate treatment applies may be understoodfrom the perturbation view. The leading order approx-imation of the subradiant states are the dark states of H I eff . They are also approximate eigenstates of H R eff whenrestricted to the most subradiant states. It means thatthe fermionic ansatz, as the leading order approximation,is shared by both H I eff and H eff . Therefore analyzing V sub from the simpler H I eff is sufficient to capture thesalient fermionic behavior. This is also verified by a di-rect construction of the fermionic ansatz without usingthe HP transformation, for both H I eff and H eff (see theSupplemental Material). Interestingly, we find that thefermionic ansatz does not exhaust all the most subradi-ant eigenstates. For a medium-size ensemble of N = 20atoms, we obtained numerical eigenstates of H eff withvery low fermionic state fidelity. The subradiant statesof this different character have well defined “center ofmass” wavenumber, and well defined spatial separation,as illustrated in Fig. 2(b). Further discussion of thesestates is beyond the scope of this manuscript, but maybe of interest for future work possibly together with theinteresting prospects for studying quantum fluctuations[38–40] in the Tonk-Girardeau gas theory by detection ofthe excited state correlations among atoms in a subradi-ant chain. Acknowledgments -This work was supported by theVillum Foundation and by the European Unions Hori-zon 2020 research and innovation program (Grant No.712721, NanOQTech). ∗ [email protected] † [email protected][1] K. Hammerer, A. S. Sørensen, and E. S. Polzik, Rev.Mod. Phys. , 1041 (2010).[2] A. F. Van Loo, A. Fedorov, K. Lalumi`ere, B. C. Sanders,A. Blais, and A. Wallraff, Science , 1494 (2013).[3] R. H. Dicke, Phys. Rev. , 99 (1954).[4] W. Guerin, M. O. Ara´ujo, and R. Kaiser, Phys. Rev.Lett. , 083601 (2016).[5] P. Weiss, M. O. Ara´ujo, R. Kaiser, and W. Guerin, NewJ. Phys. , 063024 (2018).[6] S. D. Jenkins, J. Ruostekoski, N. Papasimakis, S. Savo,and N. I. Zheludev, Phys. Rev. Lett. , 053901 (2017).[7] D. Plankensteiner, L. Ostermann, H. Ritsch, andC. Genes, Scientific Reports , 16231 EP (2015).[8] A. Asenjo-Garcia, M. Moreno-Cardoner, A. Albrecht,H. J. Kimble, and D. E. Chang, Phys. Rev. X , 031024(2017).[9] A. Albrecht, L. Henriet, A. Asenjo-Garcia, P. B. Dieterle,O. Painter, and D. E. Chang, New Journal of Physics , 025003 (2019).[10] L. Henriet, J. S. Douglas, D. E. Chang, and A. Albrecht,Phys. Rev. A , 023802 (2019).[11] P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A.Orozco, and S. L. Rolston, Nature Communications ,1857 (2017).[12] C. Noh and D. G. Angelakis, Rep. Prog. Phys. , 016401(2016).[13] B. Olmos, D. Yu, Y. Singh, F. Schreck, K. Bongs, andI. Lesanovsky, Phys. Rev. Lett. , 143602 (2013).[14] D. F. Kornovan, A. S. Sheremet, and M. I. Petrov, Phys.Rev. B , 245416 (2016).[15] H. R. Haakh, S. Faez, and V. Sandoghdar, Phys. Rev.A , 053840 (2016).[16] J. Ruostekoski and J. Javanainen, Phys. Rev. Lett. ,143602 (2016).[17] J. Ruostekoski and J. Javanainen, Phys. Rev. A ,033857 (2017). [18] H. Zoubi, Phys. Rev. A , 043831 (2014).[19] R. T. Sutherland and F. Robicheaux, Phys. Rev. A ,013847 (2016).[20] R. J. Bettles, S. A. Gardiner, and C. S. Adams, Phys.Rev. A , 043844 (2016).[21] H. H. Jen, M.-S. Chang, and Y.-C. Chen, Phys. Rev. A , 013803 (2016).[22] D. E. Chang, L. Jiang, A. Gorshkov, and H. Kimble,New J. Phys. , 063003 (2012).[23] A. Gonz´alez-Tudela, V. Paulisch, H. J. Kimble, and J. I.Cirac, Phys. Rev. Lett. , 213601 (2017).[24] D. Plankensteiner, C. Sommer, H. Ritsch, and C. Genes,Phys. Rev. Lett. , 093601 (2017).[25] V. Paulisch, H. Kimble, and A. Gonz´alez-Tudela, NewJ. Phys. , 043041 (2016).[26] H. T. Dung, L. Kn¨oll, and D.-G. Welsch, Phys. Rev. A , 063810 (2002).[27] K. Mølmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am.B , 524 (1993).[28] T. Holstein and H. Primakoff, Phys. Rev. , 1098(1940).[29] F. Reiter and A. S. Sørensen, Phys. Rev. A , 032111(2012).[30] E. H. Lieb and W. Liniger, Phys. Rev. , 1605 (1963).[31] L. Tonks, Phys. Rev. , 955 (1936).[32] M. Girardeau, Journal of Mathematical Physics , 516(1960).[33] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, andM. Rigol, Rev. Mod. Phys. , 1405 (2011).[34] G. Facchinetti, S. D. Jenkins, and J. Ruostekoski, Phys.Rev. Lett. , 243601 (2016).[35] T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, Phys.Rev. Lett. , 237203 (2014).[36] J. Perczel, J. Borregaard, D. E. Chang, H. Pichler, S. F.Yelin, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. ,023603 (2017).[37] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi,L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilber-berg, and I. Carusotto, arXiv:1802.04173.[38] B. Paredes, A. Widera, V. Murg, O. Mandel, S. F¨olling,I. Cirac, G. V. Shlyapnikov, T. W. H¨ansch, and I. Bloch,Nature , 277 EP (2004).[39] T. Jacqmin, J. Armijo, T. Berrada, K. V. Kheruntsyan,and I. Bouchoule, Phys. Rev. Lett. , 230405 (2011).[40] M. Budde and K. Mølmer, Phys. Rev. A , 053618(2004). SUPPLEMENTAL MATERIALA. Energy Levels of the Subradiant States We derived the decay rates of the subradiant states inthe one-excitation sector. As a byproduct, their energylevels are given as following: for the subradiant stateswith k ξ ≈ ξπ/ ( N d ) ( ξ (cid:28) N ), we have ω ξ = Γ D k D d ) + Γ D cos( k D d/ ( k D d/ 2) ( ξπ N ) ; (10)for those with k ξ ≈ − π/d + ξπ/ ( N d ) ( ξ (cid:28) N ), we have ω ξ = − Γ D k D d ) − Γ D sin( k D d/ ( k D d/ 2) ( ξπ N ) . (11)Apart from the constant part, the above expressions showthat the subradiant states have Lamb shifts proportionalto ξ /N . The band is parabolic and flat around theextreme point k = 0 or k = ± π/d . B. Hamiltonian of 1D Atom Chain in 3D Free-Space The effective atom-atom coupling Hamiltonian is ex-pressed as H D, eff = − µ ω N (cid:88) i,j =1 d ∗ i · G ( r i , r j , ω ) · d j σ † i σ j , (12)where d i and r i is the dipole moment and the position ofthe i th atom, respectively, and µ is the vacuum perme-ability. For the case of 3D free-space, the dyadic Green’stensor G ( r i , r j , ω ) = G ( r i − r j , ω ) is given as G ( r , ω ) = e ik r πk r (cid:20) ( k r + ik r − I +( − k r − ik r + 3) rr r (cid:21) (13)where k = ω /c . This expression can be transformedto the wave number presentation and yields H D, eff pre-sented as Eq. (6) of the main text. C. Transformation to Continuous Limit Equation (9) of the main text is written in a discretenotation. The continuous expression can be obtainedfrom the discrete notation by the mapping N (cid:88) i =1 → d (cid:90) Nd dx, b i → √ d b x . (14)The bosonic commutation relation changes from [ b i , b † j ] = δ i,j to [ b x , b † y ] = δ ( x − y ). D. Direct construction To see why the fermionic ansatz has the N − decayrate, we introduce the two-excitation state | k , k (cid:105) = (cid:80) m