Interaction effects on PT -symmetry breaking transition in atomic gases
IInteraction effects on PT -symmetry breaking transition in atomic gases Ziheng Zhou and Zhenhua Yu
1, 2, ∗ Laboratory of Quantum Engineering and Quantum Metrology, School of Physics and Astronomy,Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China State Key Laboratory of Optoelectronic Materials and Technologies,Sun Yat-Sen University (Guangzhou Campus), Guangzhou 510275, China (Dated: January 7, 2019)Non-Hermitian systems having parity-time ( PT ) symmetry can undergo a transition, sponta-neously breaking the symmetry. Ultracold atomic gases provide an ideal platform to study interac-tion effects on the transition. We consider a model system of N bosons of two components confinedin a tight trap. Radio frequency and laser fields are coupled to the bosons such that the single par-ticle Non-Hermitian Hamiltonian h P T = − i Γ σ z + Jσ x , which has PT -symmetry, can be simulatedin a passive way. We show that when interatomic interactions are tuned to maintain the symmetry,the PT -symmetry breaking transition is affected only by the SU(2) variant part of the interactionsparameterized by δg . We find that the transition point Γ tr decreases as | δg | or N increases; in thelarge | δg | limit, Γ tr scales as ∼ | δg | − ( N − . We also give signatures of the PT -symmetric and thesymmetry breaking phases for the interacting bosons in experiment. Study of non-Hermitian systems is constantly enrich-ing our knowledge derived from Hermitian ones [1–9]. Ofparticular interest are a class of non-Hermitian systemshaving the parity-time ( PT ) symmetry [10–12]. A repre-sentative model of the class is a two-level system whoseHamiltonian is of the form h P T = − i Γ σ z + Jσ x ; under thecombined transformation of complex conjugate and theswap |↑(cid:105) ↔ |↓(cid:105) , h P T is invariant [13]. Parameter tuningacross the critical point Γ tr = J gives rise to the tran-sition of the two-level system from the PT -symmetricphase to the symmetry breaking phase where exponen-tially growing or decaying modes set in. PT -symmetrybreaking transition has been widely investigated in elec-tromagnetic [4, 5, 14–17], and mechanical systems [18].The transition is the cornerstone of exceptional proper-ties regarding light propagation [19–21], lasing [22–24]and topological energy transfer [25, 26].Recently PT -symmetry breaking transition was suc-cessfully demonstrated in a gas of two component nonin-teracting Li atoms in a passive way [27]; in the exper-iment, the application of a radio-frequency field and alaser inducing loss in one component of the atoms leadsto, apart from kinetic energy, the single particle Hamil-tonian h = − i Γ + h P T , as the term − i Γ gives rise toan overall decay. This approach circumvents the diffi-culty of realizing an atom gain in quantum simulationof PT -symmetric non-Hermitian Hamiltonians in atomicgases [3]. On the other hand, Feshbach resonance enablesunprecedented control of interactions in ultracold atomicgases [28], and deterministic preparation is achievable fora sample of variable N atoms [29–32]. These capabilitiesmake ultracold atoms an ideal platform to probe inter-action effects on PT -symmetry breaking transition [33].In this work, we consider N interacting two compo-nent bosons confined in a tight harmonic trap such thattheir spatial wave-function is frozen to be the groundharmonic state. The bosons are subject to the radio fre- quency field and the laser as in Ref. [27]. Feshbach reso-nance is used to tune the interaction Hamiltonian of the N bosons to maintain the P T -symmetry. We find thatin this interacting system, PT -symmetry breaking tran-sition depends on only the SU(2) variant part of the in-teractions parameterized by δg . The transition point Γ tr decreases as | δg | or N increases. In the large | δg | limit,Γ tr is suppressed as ∼ | δg | − ( N − . Finally we show howthe modification on the transition by the interactions canbe detected experimentally.Figure (1) gives a schematic of the system that we con-sider. Bosonic atoms with two internal states denotedby |↑(cid:105) and |↓(cid:105) are confined in a harmonic trap potential V ( r ) = mω r , where m is the atomic mass. For sim-plicity, we assume the confinement being so tight, i.e., ω is much larger than any other energy scales to beconsidered, that the spatial wave-function of the bosonsis frozen to be the single particle ground state φ ( r ) ofthe harmonic trap. A radio-frequency field of frequencyequal to the internal energy difference E ↑ − E ↓ is used toflip the atoms between the two internal states |↑(cid:105) and |↓(cid:105) with Rabi frequency J . An additional laser is coupled tothe atoms in state |↑(cid:105) and results in a loss rate 4Γ of theatom number in the state. We take (cid:126) = 1 throughout.In the absence of interatomic interactions, the Hamil-tonian for each bosons spanned by |↑(cid:105) and |↓(cid:105) is non-Hermitian and is given by h = − i Γ + h P T , apart fromthe ground harmonic state energy [27]. The interatomicinteraction Hamiltonian of the bosons is given by H int = (cid:88) j (cid:54) = k (cid:34) g ↑↑ σ ( j ) z + 12 σ ( k ) z + 12 + g ↓↓ σ ( j ) z − σ ( k ) z − − g ↑↓ σ ( j ) z + 12 σ ( k ) z − (cid:35) , (1)where g σσ (cid:48) = (4 πa σσ (cid:48) /m ) (cid:82) d r | φ ( r ) | and a σσ (cid:48) is the s-wave scattering length [34, 35], and σ ( j ) are the Pauli a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n matrices for the j th boson. The interaction Hamiltonianis PT -symmetric only if g ↑↑ = g ↓↓ ; in atomic gases, thiscondition can be experimentally fulfilled by the techniqueof Feshbach resonance [28]. We focus on this situation g ≡ g ↑↑ = g ↓↓ in the following discussion.Therefore, the total Hamiltonian of N interactingbosons is given by H = H P T + C (2) H P T = − i Γ S z + 2 JS x + δg ( S z ) (3) C = − iN Γ + g N ( N − − δg N , (4)where S = (cid:80) Nj =1 σ ( j ) /
2, and δg ≡ g − g ↑↓ . Comparedwith the noninteracting case, the PT -symmetry break-ing transition of the interacting boson system is now de-termined by H P T . Since there are two internal states |↑(cid:105) and |↓(cid:105) to accommodate N bosons, the dimensionof the Hilbert space shall be N + 1. It is easy to as-sure oneself that such a space is spanned by the strechedstates | N/ , m (cid:105) with m = − N/ , − N/ , . . . , N/ − , N/ S | N/ , m (cid:105) = N ( N + 2) / | N/ , m (cid:105) and S z | N/ , m (cid:105) = m | N/ , m (cid:105) . In this space, the matrix ele-ment of H P T becomes( H P T ) mm (cid:48) ≡(cid:104) N/ , m | H P T | N/ , m (cid:48) (cid:105) =( − i Γ m + δg m ) δ m,m (cid:48) + J (cid:112) ( N/ N/ − m + m δ m,m (cid:48) +1 + J (cid:112) ( N/ N/ − m − m δ m,m (cid:48) − . (5)The transition occurs when some eigenvalues of H P T co-alesce and turn complex afterwards. Note that when δg = 0, the interactions drop out of H P T ; this is thesituation that the scattering lengths a σσ (cid:48) become all thesame and the interatomic interactions are SU(2) invari-ant. This dependence of the transition on the interactionsis because if the interatomic interactions are SU(2) invari-ant, i.e., [ S , H int ] = 0, the noninteracting PT -symmetricnon-Hermitian Hamiltonian − i Γ S z +2 JS x and H int arecommutable and can be diagonalised simultaneously.The above non-Hermitian Hamiltonian formalism is re-lated to the Lindbald equation describing the bosons sub-ject to pure loss in the following way. In terms of thefield operator b σ ( b † σ ) which annihilates (creates) a bosonof internal state σ in the ground state of the harmonictrap, the Lindbald equation for the density matrix ρ ofthe bosons is given by [36] dρdt = − i [ H s , ρ ] − b †↑ b ↑ ρ + ρb †↑ b ↑ ) + 4Γ b ↑ ρb †↑ , (6)where H s is the Hermitian Hamiltonian of the bosons inthe absence of the external lossy laser coupling. Giventhat in experiment the initial density matrix ρ (0) shall bealways block diagonalized in the number of bosons, i.e., 𝐽 𝑒4𝛤 … FIG. 1: Schematic of the system. Bosons are spatially con-fined in the single particle ground state of a tight harmonicpotential. A resonant radio-frequency field couples two inter-nal states of each bosons, |↑(cid:105) and |↓(cid:105) , with Rabi frequency J . An additional laser couples the internal state |↑(cid:105) to ananother excited state | e (cid:105) and results in a number loss in state |↑(cid:105) of rate 4Γ. ρ αβ ( t = 0) is zero unless α = β , where ρ αβ ( t ) ≡ P α ρ ( t ) P β and P α is the α boson subspace projection operator,so is ρ ( t ). If the last term 4Γ b ↑ ρb †↑ in Eq. (6) werenot there, the time dependent density matrix would begiven by ρ ( t ) = U ( t ) ρ (0) U † ( t ) with U ( t ) = e − i H t and H ≡ H s − i b †↑ b ↑ ; since the projection of H in the N boson subspace is just the non-Hermitian Hamiltonian H in Eq. (2), i.e., H = P N H P N , the properties of H woulddetermine the time evolution of ρ ( t ).To access the importance of the term 4Γ b ↑ ρb †↑ inEq. (6) to a typical experiment starting with N parti-cles, we note that initially only ρ NN ( t = 0) is nonzero.Since the pure loss can only cause the particle numberto decrease, for all the following time, ρ αβ ( t ) = 0 if α > N or β > N . Thus, from Eq. (6) one can firstobtain ρ NN ( t ) = U ( t ) ρ NN (0) U † ( t ); the term 4Γ b ↑ ρb †↑ has no effects on ρ NN ( t ) since the projection of theterm involves only ρ N +1 ,N +1 ( t ) which is identically zero.Note that H commutes with the total particle number.From hereon, one can show ρ N − ,N − ( t ) = (cid:82) t dτ U ( t − τ )[4Γ b ↑ ρ NN ( τ ) b †↑ ] U † ( t − τ ). Likewise, one can solve allthe rest ρ αα ( t ) for α < N in a cascade; the non-diagonalparts are always zero, i.e., ρ αβ = 0 for α (cid:54) = β . The aboveargument justifies one to study the time evolution ofthe purely lossy system by analyzing the non-HermitianHamiltonian H from Eq. (2). Of course, calculations ofobservables should resort to the density matrix ρ ( t ). Thisjustification shall also apply to other similar purely lossysystems.We start with analyzing the non-Hermitian Hamil-tonian H of two interacting bosons. For N = 2,the Hamiltonian H P T in the basis {|↑(cid:105) |↑(cid:105) , ( |↑(cid:105) |↓(cid:105) + g / J t r / J N=2N=3N=5N=10
FIG. 2: The critical value of Γ tr for the PT -symmetry tran-sition versus the interaction parameter δg . As the magnitudeof δg/J increases, Γ tr /J is suppressed from unity; Γ tr /J issymmetric in δg/J . For fixed δg/J , Γ tr /J acquires a smallervalue for larger boson number N . | g / J |10 t r / J N=2N=3N=5N=10
FIG. 3: The asymptotic behavior of Γ tr in the large | δg | limit.The fit to the numerically calculated points gives Γ tr /J ∼| δg/J | − ( N − , agreeing with the argument given in the text. |↓(cid:105) |↑(cid:105) ) / √ , |↓(cid:105) |↓(cid:105)} has the explicit form H P T = − i Γ + δg √ J √ J √ J √ J i Γ + δg . (7)The corresponding characteristic polynomial is f ( λ ) ≡ det[ H P T − λ ]= − λ + 2 δgλ + [4( J − Γ ) − δg ] λ − J δg, (8)whose zeros are the eigenvalues of H P T . Since all the co-efficients of the cubic polynomial f ( λ ) are real, one of thethree zeros of f ( λ ) is real definite. When Γ is increasedfrom zero, the rest two zeros of f ( λ ), which are also real in the first place, coalesce at the the PT -symmetry tran-sition point and become complex afterwards. This coa-lescence occurs when the discriminant of f ( λ )∆( J, Γ , δg ) =16 (cid:2) J − Γ ) + ( J − J Γ − ) δg − Γ δg (cid:3) (9)is zero. When δg = 0, ∆( J, Γ tr ,
0) = 0 retrieves theknown transition point Γ tr /J = 1. For nonzero δg ,Fig. (2) shows that Γ tr /J is suppressed more and moreas | δg/J | increases. By Eq. (9), it is manifest fromthat Γ tr /J is even in δg/J , and one finds (Γ tr /J ) ≈ − | δg/J | / / / for | δg/J | (cid:28)
1, and Γ tr /J ≈ | δg/J | − for | δg/J | (cid:29) tr /J in the limit | δg/J | (cid:29) PT -symmetricHamiltonian, Eq. (7). In such a limit, we recast H P T = H P T,L + V with H P T,L = δg δg ,V = − i Γ √ J √ J √ J √ J i Γ . (10)To the the leading order, H P T,L yields right away thatthe two states |↑(cid:105) |↑(cid:105) and |↓(cid:105) |↓(cid:105) are degenerate and sharethe same eigenvalue δg , and the eigenvalue of the thirdstate ( |↑(cid:105) |↓(cid:105) + |↓(cid:105) |↑(cid:105) ) / √ δg . Since the PT -symmetry transition is expectedto happen at the point where eigenvalues of H P T coa-lesce, to find the effects of Γ and J on the two degenerateeigenvalues originally equal to δg , we use V to carry outa perturbation calculation to derive the effective Hamil-tonian in the subspace spanned by the states |↑(cid:105) |↑(cid:105) and |↓(cid:105) |↓(cid:105) ; we find that to second order of V the effectiveHamiltonian is given by [36] H eff = δg + 2 J /δg + (cid:20) − i Γ 2 J /δg J /δg i Γ (cid:21) . (11)This effective Hamiltonian yields Γ tr /J ≈ | J/δg | − , thesame as from requiring ∆( J, Γ tr , δg ) = 0.For N >
2, we numerically diagonalise H P T and findthat the PT transition is always due to the coalescenceof a pair of eigenvalues of H P T . Figure (2) shows thatthe critical value Γ tr /J is also symmetric in δg/J , whichis because, under the transformation δg → − δg and S →− S , we have H P T → − H P T . We find that for fixed N ,Γ tr /J decreases as | δg/J | increases, while for fixed δg/J ,as N increases, Γ tr /J is more and more suppressed.Figure (3) shows that in the large | δg/J | limit, Γ tr /J ∼| δg/J | − ( N − . This asymptotic behavior can be under-stood by an analysis similar to the one given abovefor N = 2. For arbitrary N , in the large | δg | limit,we separate H P T from Eq. (3) as H P T = H P T,L + V with H P T,L = δg ( S z ) and V = − i Γ S z + 2 JS x . Insuch a limit, the leading order Hamiltonian H P T,L givesrise to a pair of degenerate eigvenvalues in each sub-spaces spanned by | N/ , m (cid:105) and | N/ , − m (cid:105) ; the eigen-values m δg are all well separated from each other.To determine the transition, we use V to derive theeffective Hamiltonian H eff in the each two dimen-sional subspace. It is easy to convince oneself thatto the lowest order of J , the diagonal elements are (cid:104) N/ , ± m | H eff | N/ , ± m (cid:105) = m δg ∓ im Γ, and the off-diagonal elements are generated at order of V | m | , re-sulting in (cid:104) N/ , ± m | H eff | N/ , ∓ m (cid:105) ∼ J | m | /δg | m |− .Thus, by diagonalizing the 2 × H eff , wefind the transition point Γ tr /J ∼ | δg/J | − (2 | m |− foreach two dimensional subspaces. Given that the max-imum value of | m | equals N/
2, overall, the N -body sys-tem enters into the symmetry breaking phase first atΓ tr /J ∼ | δg/J | − ( N − .The relation between the non-Hermitian Hamilto-nian formalism and the Lindbald equation for our sys-tem given above indicates that the signatures of the PT -symmetric and symmetry breaking phases governedby H P T in Eq. (3) can be detected experimentally inthe following way. Let one prepare the experimentinitally with N bosons [29–31] such that ρ NN ( t ) = e − N Γ t e − iH P T t ρ NN (0) e iH †P T t ; the quantity e N Γ t ρ NN ( t )shall have qualitatively different time dependent behav-iors in the symmetric and symmetry breaking phases.For example, by the high accuracy atom number de-tection achieved experimentally [32], one can measurethe rescaled probability of finding N bosons P( N, t ) = e N Γ t ˜P( N, t ) with ˜P(
N, t ) ≡ Tr ρ NN ( t ). In contrast, thetotal number of atoms was measured to distinguish thetwo phases for the noninteracting Li atoms [27]. In ourinteracting case, the total number of atoms ceases to bea good observable for the purpose since the observabledepends on not only ρ NN ( t ) but also ρ αα ( t ) for α < N whose dynamics is not determined by a single Hamil-tonian H for α bosons. Figure (4) plots the rescaledP(2 , t ) in an experiment starting with N = 2 bosons and δg/J = 1 for various values of Γ /J . Note that for N = 2and δg/J = 1, Γ tr /J ≈ . , t ) is bounded in the PT -symmetric phase, and growsexponentially in the symmetry breaking one. Due to theinteractions, the point Γ /J = 3 / tr /J = 1 for the noninteracting case. Acknowledgements.
We thank Jiaming Li for discus-sions. This work is supported by NSFC Grants No.11474179, No. 11722438, and No. 91736103. ∗ [email protected] J t P ( , t ) / J = / J = / J = / J = FIG. 4: The rescaled probability P(2 , t ) of finding two bosonsin an experiment with initially two bosons and δg/J = 1 forvarious values of Γ /J . The cases of Γ /J = 1 / , / PT -symmetric phase, and P(2 , t ) is bounded. The cases ofΓ /J = 3 / , / PT -symmetry breaking phase, andP(2 , t ) grows exponentially.[1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. ,5243 (1998).[2] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, andZ. H. Musslimani, Phys. Rev. Lett. , 103904 (2008).[3] S. Klaiman, U. Gnther, and N. Moiseyev, Phys. Rev.Lett. , 080402 (2008).[4] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M.Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N.Christodoulides, Phys. Rev. Lett. , 093902 (2009).[5] C. E. R¨uter, K. G. Makris, R. El-Ganainy, D. N.Christodoulides, M. Segev, and D. Kip, Nat. Phys. ,192 (2010).[6] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-gashikawa, and M. Ueda, Phys. Rev. X , 031079 (2018).[7] G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman,Y. D. Chong, M. Khajavikhan, D. N. Christodoulides,and M. Segev, Science , eaar4003 (2018);[8] M. A. Bandres, S. Wittek, G. Harari, M. Parto, J. Ren,M. Segev, D. N. Christodoulides, and M. Khajavikhan,Science , eaar4005 (2018).[9] H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. ,146402 (2018).[10] C. M. Bender, S. Boettcher, and P. N. Meisinger, J.Math. Phys. , 2201 (1999).[11] N. Moiseyev, Non-Hermitian Quantum Mechanics ,(Cambridge University Press, Cambridge, 1st edition,2011).[12] C. M Bender, R. Tateo, and P.E. Dorey,