Steady-state entanglement in a double-well Bose-Einstein condensate through coupling to a superconducting resonator
aa r X i v : . [ qu a n t - ph ] J u l Steady-state entanglement in a double-well Bose-Einstein condensate throughcoupling to a superconducting resonator
H. T. Ng and Shih-I Chu , Center for Quantum Science and Engineering, Department of Physics,National Taiwan University, Taipei 10617, Taiwan and Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA (Dated: November 9, 2018)We consider a two-component Bose-Einstein condensate in a double-well potential, where theatoms are magnetically coupled to a single-mode of the microwave field inside a superconductingresonator. We find that the system has the different dark-state subspaces in the strong- and weak-tunneling regimes, respectively. In the limit of weak tunnel coupling, steady-state entanglementbetween the two spatially separated condensates can be generated by evolving to a mixture of darkstates via the dissipation of the photon field. We show that the entanglement can be faithfully indi-cated by an entanglement witness. Long-lived entangled states are useful for quantum informationprocessing with atom-chip devices.
PACS numbers: 03.75.Gg, 03.75.Lm, 42.50.Pq
I. INTRODUCTION
Recently, the realization of a Bose-Einstein condensate(BEC) strongly coupled to the quantized photon field inan optical cavity has been shown [1, 2]. This paves theway to study the interplay of atomic interactions andatom-photon interactions. For example, the novel quan-tum phase transition of a condensate coupled to a cav-ity has been demonstrated [3]. Strong atom-photon cou-pling is useful for quantum communications [4] such asthe light-matter interface [1, 2].Alternatively, strong coupling of ultracold atoms to asuperconducting resonator has been recently proposed[5]. The two long-lived hyperfine states | e i = | F =2 , m F = 1 i and | g i = | F = 1 , m F = − i of Rb [6]are considered to be magnetically coupled to the mi-crowave field via their magnetic dipoles [5, 7]. Since thehigh-Q superconducting resonator can be fabricated to asmall mode volume [8] and the coupling strength can begreatly increased due to the collective enhancement [4].The strong coupling of ultracold atoms in the microwaveregime can be achieved [5].In this paper, we study a two-component BEC in adouble-well potential [9], where all atoms are equally cou-pled to a single-mode of the microwave field inside a su-perconducting resonator. Two weakly linked condensatescan be created in a magnetic double-well potential on anatom-chip [10, 11] or in an optical double-well poten-tial [12]. In fact, the tunneling dynamics between theatoms in two wells has been recently observed [13–15].A double-well BEC coupled to an optical cavity has alsobeen discussed in the literature [16–18]. However, thespontaneous emission rate of excited states used for opti-cal transitions in experiments [1, 2] is much higher thanthe tunneling rate of the atoms between the two wells [13–15]. Here we consider the two hyperfine states | e i and | g i of Rb with the transition frequency 2 π × . | e i and | g i ) are much longer than both the timescales of tunneling and atom-photon interactions. Therefore, thissystem offers possibilities for the study of how the tunnelcouplings between the two spatially separated conden-sates affect the atom-photon dynamics.We focus our investigation on the the system in thelimits of the strong and weak tunnel couplings, respec-tively. We find that the system has the different dark-state subspaces [21] in these two tunneling regimes. Inthe weak-tunneling regime, the system has a family ofdark states which can be used for producing quantumentanglement between the condensates. Here we proposeto efficiently generate steady-state entanglement betweenthe two spatially separated condensates by evolving to amixture of dark states through the dissipation of the pho-ton field [22–24]. Note that our scheme does not requireany adjustment of the tunneling strength. It is differentto other methods [9] which depend on the strength of tun-nel couplings to generate entanglement. In addition, theentanglement generated between the two condensates canbe used for the implementation of quantum state trans-fer [25]. This may be useful for quantum informationprocessing with atom-chip devices [20]. FIG. 1. (Color online) Schematic of a two-component BECcoupled to a single-mode of the photon field inside a supercon-ducting resonator. A two-component condensate is trappedin a double-well potential, and it is placed close to the surfaceof the superconducting resonator. The atoms are coupled tothe magnetic field via their magnetic dipoles. The parame-ters L and w are the length and width of the superconductingresonator, respectively. This paper is organized as follows: In Sec. II, we in-troduce the system of a two-component condensate in adouble-well potential, and the two-level atoms are cou-pled to a superconducting resonator. In Sec. III, we de-rive the two effective Hamiltonians in the strong- andweak-tunneling regimes, respectively. In Sec. IV, we in-vestigate the dark-state subspaces and the atom-photondynamics in the two tunneling limits. In Sec. V, we pro-vide a method to produce the steady-state entanglementbetween the two condensates in a double well. A sum-mary is given in Sec. VI. In Appendix A, we discuss thevalidity of the effective Hamiltonian in the strong tun-neling regime.
II. SYSTEM
We consider a two-component BEC being trapped ina double-well potential [9], and the condensate is placednear the surface of a superconducting resonator as shownin Fig. 1. The atoms, with two internal states | e i and | g i ,are coupled to a single mode of the photon field via theirmagnetic dipoles. A. A two-component condensate trapped in adouble-well potential
We first introduce the system of a two-component con-densate in a one-dimensional (1D) double-well potentialwhich can be described by the Hamiltonian as H = X α Z dx Ψ † α ( x ) h − ~ m α ∂ ∂x + V DW ( x ) + ˜ U α Ψ † α ( x ) × Ψ α ( x ) i Ψ α ( x )+2 ˜ U eg Z dx Ψ † e ( x )Ψ † g ( x )Ψ g ( x )Ψ e ( x ) , (1)where Ψ α ( x ) is the field operator of the atoms for the in-ternal state | α i at the position x , and the indices α = g, e represent the ground and the excited states, respectively.Here m α is the mass of the atom in the state | α i and V DW ( x ) is the 1D double-well potential which is given by[11] V DW ( x ) = V d h − (cid:16) xx (cid:17) i , (2)where V d is the barrier height and x is the distance be-tween the two separate potential wells. The atoms aretransversely confined in the y - and z -directions with thetrap frequencies ω ⊥ . The size of the ground-state wavefunction in the transverse motion is a ⊥ = p ~ /m α ω ⊥ [26, 27], where m e and m g are nearly equal. Since thetransverse frequencies are much larger than the trap fre-quency in the x -direction, the transverse motions of theatoms are frozen out. The parameters ˜ U α and ˜ U eg are the effective 1D interaction strengths between the inter-,and the intra-component condensates, as [26, 27]˜ U α = 2 ~ a α m α a ⊥ (cid:16) − C a α √ a ⊥ (cid:17) − , (3)˜ U eg = 4 ~ m e m g a eg ( m e + m g ) a ⊥ (cid:16) − C a eg √ a ⊥ (cid:17) − , (4)where C ≈ . a α and a eg are thethree-dimensional s-wave scattering lengths for the inter-,and the intra-component condensates.We adopt the two-mode approximation [28] such thatthe field operator Ψ α ( x ) can be expanded in terms of thetwo localized mode functions u α L ( x ) and u α R ( x ) as,Ψ α ( x ) = α L u αL ( x ) + α R u αR ( x ) , (5)where α L and α R are the annihilator operators of theatoms in the state α = e, g for the left and right modes ofthe double-well potential, respectively. The Hamiltonianof the system [9], within the two-mode approximation,can be written as H ′ = ~ E e ( e † L e L + e † R e R ) + ~ E g ( g † R g R + g † L g L ) − ~ J e ( e † L e R + e † R e L ) − ~ J g ( g † L g R + g † R g L )+ ~ U ee [( e † L e L ) + ( e † R e R ) ] + ~ U gg [( g † L g L ) +( g † R g R ) ] + 2 ~ U eg ( e † L e L g † L g L + e † R e R g † R g R ) , (6)where E α = 1 ~ Z dxu ∗ αj ( x ) h − ~ m α ∂ ∂x + V DW ( x ) i u αj ( x ) , (7) J α = − ~ Z dru ∗ αL ( x ) h − ~ m α ∂∂x + V DW ( x ) i u αR ( x ) , (8) U α = ˜ U α ~ Z dx | u αj ( x ) | , (9) U αβ = ˜ U αβ ~ Z dr | u αj ( x ) | | u βj ( x ) | , (10)and j = L, R . The positive parameters E α and J α [29]are the ground-state frequencies of the localized mode α L,R , and the tunneling strengths between the two wellsfor the atoms in the states α . Here U α and U αβ are thetwo positive parameters which describe the inter- andintra-component interaction strengths, respectively. B. Atoms coupled to the photon field in amicrowave cavity
We consider that the atoms are coupled to a single-mode of the photon field via their magnetic dipoles [5].Within the two-mode approximation, the Hamiltonian,describing the system of cavity field, the atoms and theirinteractions, is given by H I = ~ ω a a † a + ~ ω ( e † L e L + e † R e R ) + ~ g [ a ( e † L g L + e † R g R )+H . c . ] , (11)where ω a and a are the frequency and the annihilatoroperator of the single-mode of the photon field, and ω isthe transition frequency of the two internal states. Herewe have assumed that the wavelength of the microwavefield ( ∼ ∼ µ m) [10, 11]. Therefore, all atoms are cou-pled to the photon field with the same coupling strength g = µ B p µ ω a / ~ V [7], where µ B is the Bohr magneton, µ is the vacuum permeability and V is the volume of thesuperconducting resonator. The coupling strength g canattain 1 kHz [7] if the volume V of the superconductingresonator is taken as L × w × t h ∼ × µ m ×
200 nm[7, 8], where L is the length, w is the width, and t h is thethickness of the superconducting resonator. III. EFFECTIVE HAMILTONIANS IN STRONGAND WEAK TUNNELING REGIMES: LOWATOMIC EXCITATIONS
We will derive the effective Hamiltonians of the systemin the limits of strong and weak tunnel couplings, respec-tively, where a few atomic excitations are only involved.Let us first write the total Hamiltonian of the system as H = ~ ω a a † a + ~ ω ( e † L e L + e † R e R ) − ~ J e ( e † L e R + e † R e L ) − ~ J g ( g † L g R + g † R g L ) + ~ U ee [( e † L e L ) + ( e † R e R ) ]+ ~ U gg [( g † L g L ) + ( g † R g R ) ] + 2 ~ U eg ( e † L e L g † L g L + e † R e R g † R g R ) + ~ g [ a ( e † L g L + e † R g R ) + H . c . ] . (12)The total number of atoms N is conserved. We haveomitted the constant term E N for a symmetric doublewell, where E α ≈ E for the two masses m e and m g beingequal. It is convenient to work in the rotating frame byapplying the unitary transformation to the Hamiltonian H in Eq. (12), where the unitary operator U ( t ) is U ( t ) = exp [ − iω a ( a † a + e † L e L + e † R e R ) t ] . (13)The transformed Hamiltonian becomes H ′ = ~ ∆( e † L e L + e † R e R ) − ~ J e ( e † L e R + e † R e L ) − ~ J g ( g † L g R + g † R g L ) + ~ U ee [( e † L e L ) + ( e † R e R ) ] + ~ U gg [( g † L g L ) +( g † R g R ) ] + 2 ~ U eg ( e † L e L g † L g L + e † R e R g † R g R )+ ~ g [ a ( e † L g L + e † R g R ) + H . c . ] (14)where ∆ = ω − ω a is the detuning between the frequen-cies of the photon field and the two internal states.In the strong tunneling regime, the tunnel couplingis dominant and the strength of atom-atom interactionsis relatively weak. On the contrary, in the weak tunnel-ing regime, the atom-atom interactions become dominantand the tunneling strength is negligible. We will showthat these two cases exhibit the different behaviours inthe atom-photon dynamics. We will provide derivationsof the two effective Hamiltonians in the two tunnelinglimits in the following subsections. A. Strong-tunneling regime
In the limit of the strong tunnel coupling, the tunnelingstrengths are much larger than the strengths of the atom-atom interactions, i.e., J e , J g ≫ U e , U g , U eg . The totalHamiltonian of the system can be approximated as H = ~ ∆( e † L e L + e † R e R ) − ~ J e ( e † L e R + e † R e L ) − ~ J g ( g † L g R + g † R g L ) + ~ g [ a ( e † L g L + e † R g R ) + H . c . ] . (15)We have neglected the terms of the atom-atom interac-tions in this Hamiltonian.The symmetric and asymmetric modes g ± and e ± canbe related to the localized modes as g ± = 1 √ g L ± g R ) , (16) e ± = 1 √ e L ± e R ) . (17)The Hamiltonian is then transformed as H ′ = ~ (∆ − J e ) e † + e + + ~ (∆ + J e ) e †− e − − ~ J g ( g † + g + − g †− g − ) + ~ g ( ae † + g + + H . c . ) + ~ g ( ae †− g − + H . c . ) . (18)Here the atoms are in symmetric (asymmetric) mode ifthey are populated in the states g k + | i + or e k + | i + ( g k − | i − or e k − | i − ), where | i + ( | i − ) is the vacuum state of thesymmetric (asymmetric) mode and k is a non-negativeinteger.We consider the system to be initially prepared in theground state in the limit of strong tunnel coupling, i.e.,the ground state of the symmetric mode. The groundstate can be obtained by applying the operator ( g † + ) N tothe vacuum state | i + of the symmetric mode, i.e., | Ψ (0) i = 1 √ N ! ( g † + ) N | i + , (19)where N is the total number of atoms. Note that theatoms in the symmetric and asymmetric modes are inde-pendently coupled to the photon field in Eq. (18). There-fore, all atoms in the symmetric mode are only involvedin the dynamics of the atom-photon interactions if thesystem starts with the state | Ψ (0) i in Eq. (19). In fact,there are only a few excitations in the asymmetric modedue to the atomic interactions. The effect of the exci-tations from the asymmetric mode to the dynamics ofatom-photon interactions is very small. It is becausethe Rabi coupling strength cannot be greatly enhancedwith a small number of atoms in the asymmetric mode.We briefly discuss the validity of this assumption in Ap-pendix A.It is instructive to express the Hamiltonian in terms ofangular momentum operators: S (+)+ = g + e † + , (20) S (+) − = e + g † + , (21) S (+) z = 12 ( e † + e + − g † + g + ) . (22)The Hamiltonian can be rewritten as˜ H ′ = ~ ∆ S (+) z + ~ g ( aS (+)+ + H . c . ) . (23)For simplicity, we have assumed that the tunnelingstrengths J e and J g are equal. We also have omittedthe constant term ~ N ∆ / S (+)+ = b † p N − b † b, (24) S (+) − = b p N − b † b, (25) S (+) z = b † b − N . (26)In the low degree of excitation, the mean excitation num-ber h b † b i are much smaller than the total number ofatoms N . The angular momentum operators can be ap-proximated by the bosonic operators [9, 31]. The effectiveHamiltonian can be obtained as H (1)eff = ~ ∆ b † b + ~ g √ N ( ab † + H . c . ) . (27)Note that the effective Rabi frequency is enhanced bya factor of √ N . This effective Hamiltonian H (1)eff inEq. (27) describes the interactions between the collective-excitation mode and the single mode of the photon field. B. Weak-tunneling regime
Now we investigate the system in the weak tunnel-ing regime, where the atom-atom interaction strengthsare much larger than the tunneling strengths, i.e, U e , U g , U eg ≫ J e , J g . In this limit, we assume that the tun-neling between the two condensates is effectively turnedoff. The total Hamiltonian can be approximated as H = ~ ∆( e † L e L + e † R e R ) + ~ g [ a ( e † L g L + e † R g R ) + H . c . ]+ ~ U ee [( e † L e L ) + ( e † R e R ) ] + ~ U gg [( g † L g L ) +( g † R g R ) ] + 2 ~ U eg ( e † L e L g † L g L + e † R e R g † R g R ) . (28)Here we have ignored the terms of the tunnel couplings.This Hamiltonian can be expressed in terms of the an-gular momentum operators: S j + = g j e † j , (29) S j − = e j g † j , (30) S jz = 12 ( e † j e j − g † j g j ) , (31) where j = L, R . Now the Hamiltonian is rewritten as˜ H = ~ X j = L,R (∆ + δ ) S jz + ~ g ( aS j + + H . c . ) + ~ χS jz , (32)where δ = ( U ee − U gg ) N/ χ = U ee + U gg − U eg .We have omitted the constant term ~ ( U ee + U gg +2 U eg ) N /
16 + ~ N ∆ / | g i are initially pre-pared in the ground state of the Hamiltonian in Eq. (32),which can be described by a product of two number statesas | Ψ (0) i = | N/ i g L | N/ i g R . (33)Without loss of generality, we assume that N is an evennumber.We apply the HPT such that the angular momentumoperators can be mapped onto the harmonic oscillatorsas: S L + = c † q N/ − c † c, S L − = c q N/ − c † c, (34) S Lz = c † c − N , (35) S R + = d † q N/ − d † d, S R − = d q N/ − d † d, (36) S Rz = d † d − N , (37)If the mean numbers of the atomic excitations, h c † c i and h d † d i , are much smaller than the number of atoms N/ H (2)eff = ~ ∆ w ( c † c + d † d ) + ~ g r N a ( c † + d † ) + H . c . ]+ ~ χ [( c † c ) + ( d † d ) ] , (38)where ∆ w = 2∆+ δ − χN/
2. The effective Rabi frequencyis enhanced by a factor of p N/
2. The parameter χ ismuch smaller than the effective Rabi frequency becausethe scattering lengths of the inter- and intra-componentcondensates of Rb are very similar [6]. We will ignorethe terms with the parameter χ in Eq. (38) in our laterdiscussion.The effective Hamiltonian H (2)eff in Eq. (38) describesthe interactions between the single mode of the photonfield and the two modes of the collective excitations of theatoms in the left and right potential wells, respectively.This system can be described by a system of three cou-pled harmonic oscillators. The effective Rabi frequencyfor each atomic mode is proportional to the factor p N/ √ N . IV. DARK STATES AND QUANTUMDYNAMICS OF THE SYSTEM
We now study dark states of the system which hasdifferent dark-state subspaces in the strong- and weak- gt h a † a i gt h b † b i (b)(a) FIG. 2. (Color online) Time evolution of the mean photonnumber (a) and the mean atomic excitations (b) with thedamping rate κ = 100 g and the detuning ∆ = 0. The differentnumber of atoms N are 5 × (black-solid line), 1 × (blue-dashed line) and 2 × (red-dotted line), respectively. tunneling regimes. Let us first introduce the definitionof dark states. Dark states [21] are the eigenstates of theatom-photon interaction operator V , with zero eigenval-ues, i.e., V| Dark i = 0 | Dark i , (39)= 0 . (40)Dark states, in the strong- and weak-tunneling regimes,in this system can be obtained as H ( j )eff | D i j = 0 , (41)where H ( j )eff are the two effective Hamiltonians inEqs. (27) and (38) with zero detunings (∆ = ∆ w = 0)and j = 1 , | D i is the product state of the vacuum state of the pho-ton field and the ground state of the atomic mode b ,which is given by | D i = | i a | i b . (42)This state is the ground state of the coupled system ofthe atoms and the photon field.In the weak-tunneling regime, the system has a familyof dark states. The family of dark states are | D n i = | i a | D an i , (43)where | D an i = 2 − n/ n X j =0 ( − j q C nj | n − j i c | j i d , (44) gt h a † a i gt h c † c i gt h d † d i (a)(b)(c) FIG. 3. (Color online) Dynamics of the mean photon numberand mean atomic excitation numbers with the damping rate κ = 100 g and the detuning ∆ w = 0. (a) the mean photonnumber h a † a i as a function of the time gt . Time evolution ofthe mean atomic excitation numbers of the atomic mode c , in(b), and the atomic mode d , in (c) as shown. The differentnumber of atoms N are 5 × (black-solid line), 1 × (blue-dashed line) and 2 × (red-dotted line), respectively. and C nj is the binomial coefficient. The dark states | D n i are the product state of the vacuum state | i a of thephoton field and the states | D an i are the eigenstates ofthe operator c + d with zero eigenvalues. Note that thestates | D an i in Eq. (44) is a superposition of the states | n − j i c | j i d which have the same degree of atomic excitations.To gain more insight into dark states, let us first inves-tigate the atom-photon dynamics subject to the dissipa-tion of the photon field. For a superconducting resonatorwith the frequency ∼
40 GHz can be cooled down to lowtemperatures ( ∼
25 mK) [32]. This allows us to considerthe cavity field being weakly coupled to the reservoir atthe zero temperature [33]. Note that the relaxation time(several µ s) of the single photon inside the superconduct-ing resonator is much shorter than the coherence time( ∼ ρ j = − i ~ [ H ( j )eff , ρ j ] + κ aρ j a † − a † aρ j − ρ j a † a ) , (45)where ρ j is the density matrix of the total system, and j = 1 ,
2. Obviously, the dark states | D i and | D n i are the steady-state solutions of the master equation inEq. (45). Thus, the dark states are robust against thedissipation of the photon field. In the strong tunnel-ing regime, the steady state is the dark state | D i . Inthe weak tunneling regime, the state of the condensatesevolves as a mixture of dark states | D n i through thedissipation of the photon field.Now we study the dynamics of the system in thestrong-tunneling regime, where the state is prepared as | i a | i b and | i b is a number state. We plot the time ofevolution of the mean photon number and mean atomic-excitation number in Fig. 2. The mean photon numberand mean atomic excitations undergo a few oscillationsand then both of them decay to zero. We also see thatthe faster rate of oscillations can be obtained if a largernumber of atoms N are used.We proceed to investigate the atom-photon dynamicsin the weak-tunneling regime. The system is initiallyprepared as the state | i a | i c | i d , where | i c is a numberstate. In Fig. 3, we plot the mean photon number, andthe mean excitation numbers of the two atomic modesversus the time. When the atom-photon interactions areturned on, the excitation number of the atomic mode c decreases while the mean photon number increases asshown in Fig. 3. Afterwards, the mean excitation numberof the atomic mode d starts to increase. This means thatthe energy of the atomic mode c transfers to the photonfield and the atomic mode d absorbs the energy fromthe photon field. In this way, the two atomic modesexchange the energy via the photon field. The faster rateof exchanging energy between the atoms and the photonfield can be attained if a larger number of atoms N areused. We also note that the mean photon number inFig. 3(a) is about half of the mean photon number inFig. 2(a). It is because the atoms in the atomic mode d ,in the weak tunneling regime, absorbs the energy fromthe photon field.In Fig. 3(a), the mean photon number decays to zeroafter a period of time. However, the mean excitationnumbers of modes c and d remain non-zero as shown inFigs. 3 (b) and (c). It is because the state of the atomsevolves to a mixture of dark states | D i and | D i , and asingle excitation is shared by the atoms in the dark state | D i . This results in the non-zero excitation numbers ofthe two atomic modes. V. GENERATION OF ENTANGLEMENTBETWEEN TWO SPATIALLY SEPARATEDCONDENSATES
We have shown that the system has the different dark-state subspaces in the two tunneling limits. Now westudy the entanglement between the condensates in thetwo different potential wells in the weak tunneling regime.In this regime, the system has a family of dark stateswhich can be used for generating entanglement. Here W gt gt E N ( ρ c d ) (a)(b) FIG. 4. (Color online) Time evolution of the entanglementwitness in (a) and logarithmic negativity in (b), for the damp-ing rate κ = 100 g and the detuning ∆ w = 0. The differentnumber of atoms N are 5 × (black-solid line), 1 × (blue-dashed line) and 2 × (red-dotted line), respectively. we consider the tunneling between the wells to be effec-tively turned off. Therefore, the two independent conden-sates in the two potential wells are initially unentangled.We will show that steady-state entanglement between thetwo condensates can be produced by evolving to a mix-ture of dark states {| D n i } through the dissipation of thephoton field [22–24].To study the quantum entanglement between the twoatomic modes c and d , it is necessary to obtain the den-sity matrix of the atomic condensate. By tracing outthe system of the photon field, we can obtain the densitymatrix ρ cd , ρ cd = Tr a ( ρ ) , (46)where ρ is the density matrix of the total system. Letus first examine the entanglement of a single dark state | D n i . For a dark state | D n i in Eq. (43), the densitymatrix ρ cd is given by ρ cd = | D an ih D an | , (47)where | D an i is the state in Eq. (44). The degree of entan-glement between the two atomic modes can be quantifiedby the von Neumann entropy. It is defined as E F = − Tr( ρ c ln ρ c ) , (48)where ρ c = Tr d ( ρ cd ) is the reduced density matrix. Thevon Neumann entropy is E F = − − n n X j =0 C nj ln | − n C nj | . (49) gt W gt E N ( ρ c d ) (a)(b) FIG. 5. (Color online) Plot of the dynamics of entangle-ment. (a) entanglement witness W and (b) logarithmic neg-ativity E N ( ρ cd ) as a function of the time gt . The initialstate | i a | n i c | i d with the different excitation numbers n areshown, for n = 1 (black-solid line), n = 2 (blue-dashed line)and n = 3 (red-dotted line), respectively. The parameters are κ = 100 g , ∆ w = 0 and N = 5 × . Thus, the state | D an i is an entangled state. The degreeof two-mode entanglement becomes higher for larger n .In general, this density matrix ρ cd is a mixed state. Toquantify the entanglement of a mixed state, the logarith-mic negativity can be used [36]. The definition of thelogarithmic negativity is [36] E N ( ρ cd ) = log k ρ T c cd k , (50)where ρ T c cd is the partial transpose of the density matrix ρ cd and k · k is the trace norm.However, the logarithmic negativity is difficult to beexperimentally determined. It is very useful to study anexperimentally accessible quantity to detect the quantumentanglement between the two bosonic modes [37]. If aninequality |h cd † i| > h n c n d i , (51)is satisfied [37], then the state is an entangled state. Here n c = c † c and n d = d † d are the number operators of theatomic modes c and d , respectively. For convenience, thisquantity W is defined as W = h n c n d i − |h cd † i| . (52)If W is negative, then the state is non-separable. Thisquantity W is called as an entanglement witness [38].We investigate the dynamics of entanglement betweenthe two atomic modes. We consider an initial state as a product state of the three modes, i.e., | i a | i c | i d , where | i c is a number state. We plot the entanglement wit-ness and logarithmic negativity versus time as shown inFig. 4. This figure shows that the entanglement wit-ness decreases and logarithmic negativity increases witha similar rate, and then they saturate after a short time.This shows that the steady-state entanglement can beproduced in a short time via the dissipative photon field.The entanglement can also be produced faster if a largernumber of atoms are used. Besides, we can see that theentanglement witness is consistent with the logarithmicnegativity to indicate the degree of entanglement. Theentanglement witness is a faithful indicator for detectingthe entanglement between the two bosonic modes.Next, we study the generation of entanglement by us-ing an initial state | i a | n i c | i d with a higher degree ofexcitation, where | n i c is a number state and n is largerthan one. In Fig. 5, the entanglement witness and loga-rithmic negativity are plotted versus the time. It showsthat a higher degree of the entanglement can be obtainedif higher excitation numbers n = 2 , VI. SUMMARY
We have studied a two-component condensate in adouble-well potential, where the atoms are magneticallycoupled to a single-mode of the photon field inside a su-perconducting resonator. The system has the differentdark-state subspaces in the strong- and weak-tunnelingregimes, respectively, and it gives rises to the different dy-namics of atomic excitations in the two regimes. Steady-state entanglement between the two spatially separatedcondensates can be produced by evolving to a mixture ofdark states through the dissipative photon field. We haveshown that the entanglement can be faithfully indicatedby an entanglement witness.
ACKNOWLEDGMENTS
H.T.N. thank David Hallwood for his careful readingand helpful comment, and C. K. Law for his useful discus-sion. This work was partially supported by U.S. NationalScience Foundation. We also would like to acknowledgethe partial support of National Science Council of Tai-wan (Grant No. 97-2112-M-002-003-MY3) and NationalTaiwan University (Grant No. 99R80869).
Appendix A: Validity of the effective Hamiltonian inthe strong-tunneling regime
In this appendix, we examine the validity of the effec-tive Hamiltonian H (1)eff in Eq. (27) in the limit of strongtunnel coupling. We express the Hamiltonian in term of FIG. 6. (Color online) Level scheme of the atoms in thedouble-well potential. In the strong-tunneling regime, theatoms, with the two states | e + i and | g + i , in the symmetricmode are coupled to the cavity field. The two ground statesof the symmetric and asymmetric modes ( | g + i and | g − i ) arecoupled to each other via the atom-atom interactions. the symmetric-mode and asymmetric-mode operators as:˜ H = ~ (∆ − J e ) e † + e + + ~ (∆ + J e ) e †− e − − ~ J g ( g † + g + − g †− g − ) + ~ g ( ae † + g + + H . c . ) + ~ g ( ae †− g − + H . c . )+ ~ U ee (cid:2) ( e † + e + + e †− e − ) + ( e † + e − + e †− e + ) (cid:3) + ~ U gg (cid:2) ( g † + g + + g †− g − ) + ( g † + g − + g †− g + ) (cid:3) + ~ U eg (cid:2) ( e † + e + + e †− e − )( g † + g + + g †− g − ) + ( e † + e − + e †− e + )( g † + g − + g †− g + ) (cid:3) . (A1)Let us define F + = g †− g + , F − = g † + g − , (A2) F = 12 ( g †− g − − g † + g + ) . (A3)The commutation relations [ F , F ± ] = ± F ± and[ F + , F − ] = 2 F z are satisfied. The operators F ± and F z ,and S (+) ± and S (+) z in Eqs. (20) to (22) generate a SU(3)algebra [39]. In the limit of large N , we can apply theHPT to the operators: F + = f † p N − f † f , F − = f p N − f † f , (A4) F = f † f − N/ . (A5)Assume that the mean excitation number h f † f i is muchsmaller than N , we can approximate the operators as[39]: f † = 1 √ N F + , f = 1 √ N F − . (A6)In the low-degree-of-excitation regime, the approximatedHamiltonian can be written as˜ H ≈ ~ ω a a † a + ~ ω ′ b † b + ~ g √ N ( ab † + H . c . ) + ˜ H ′ . (A7)The Hamiltonian ˜ H ′ , contains the terms of the operatorsin the asymmetric mode and the terms from the nonlinear J g t h f † f i FIG. 7. (Color online) The expectation value h f † f i as afunction of the time J g t . Different strengths of U gg N areshown: U gg N = J g (red-solid line), U gg N = 5 J g (blue-dashedline), and U gg N = 10 J g (black-dotted line). interactions, and the constant terms are omitted, whichcan be written as˜ H ′ = ~ J g f † f + ~ U gg N f † + f ) + ~ (∆ + J e )( e †− e − − e †− e − ) + ~ g ( ae †− g − + H . c . ) + ~ U ee (cid:2) ( e † + e + + e †− e − ) + ( e † + e − + e †− e + ) (cid:3) + ~ U gg g † + g + + g †− g − ) + ~ U eg (cid:2) ( e † + e + + e †− e − )( g † + g + + g †− g − )+ √ N ( e † + e − + e †− e + )( f † + f ) (cid:3) . (A8)Here we consider the number of atoms in the excitedstates to be very small. We also assume that the strengthof the Rabi coupling g is weak compared to the tunnelingstrength J g and nonlinear strength U gg N but g is muchstronger than U ee , U gg and U eg . Therefore, the Hamil-tonian ˜ H ′ can be approximated by the Hamiltonian H ′′ as ˜ H ′′ = ~ λ f † f + ~ λ ( f † + f ) , (A9)where λ = J g + U gg N, (A10) λ = U gg N . (A11)From Eq. (A9), nonlinear interactions can give rise tothe transitions of the atoms in the symmetric mode to theatoms in the asymmetric mode, and vice versa. The levelscheme is shown in Fig. 6. Note that this Hamiltonian˜ H ′′ is exactly solvable. The time-evolution operator canbe factorized as [35] S ( t ) = exp( − i ˜ H ′′ t/ ~ ) , (A12)= exp (cid:16) Λ f † (cid:17) exp h ln(Λ )4 ( f † f + f f † ) i exp (cid:16) Λ f (cid:17) , (A13)where Λ = (cid:16) cosh β − λ ′ β sinh β (cid:17) − , (A14)Λ = 2 λ ′ sinh β β cosh β − λ ′ sinh β , (A15) β = λ ′ − λ ′ , (A16) λ ′ = − iλ t, λ ′ = − iλ t. (A17)We then apply the time-evolution operator S ( t ) to the vacuum state | i f of the mode f . The state becomes | Ψ s ( t ) i = Λ / ∞ X n =0 p (2 n )! n ! (cid:16) Λ (cid:17) n | n i f . (A18)The mean excitation number h f † f i is h f † f i = (cid:12)(cid:12) Λ / (cid:12)(cid:12) ∞ X n =0 n (2 n )!Λ n n − ( n !) . 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