- Email: [email protected]
Quantum phase transition of two component dipolar bosons in optical lattices
Physics Letters A 340 (2005) 228–236 www.elsevier.com/locate/pla
Quantum phase transition of two component dipolar bosons in optical lattices Tao Zhang a,b,∗ , Rui-Hong Yue b , W.M. Liu a a Joint Laboratory of Advanced Technology in Measurements, Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China b Institute of Modern Physics, Northwest University, Xi’an 710069, China
Received 14 February 2005; received in revised form 7 April 2005; accepted 7 April 2005 Available online 27 April 2005 Communicated by B. Fricke
Abstract The energy spectrum of two component Bose–Einstein condensates with the dipole–dipole interaction in optical lattice is obtained by Green function technique. Its superfluid and Mott insulator phase transition is analyzed. The results show that the interspecies hopping and dipole interaction can strongly influence the energy spectrum of two component dipolar bosonic system. A paired atom hopping can exist in superfluid phase while single-atom hopping is Mott insulator, a gapless excite, which is essential to super-counter fluid phase. The special case of phase boundary of super-counter phase are discussed. 2005 Elsevier B.V. All rights reserved. PACS: 03.75.Fi; 05.30.Jp; 05.50.+q; 32.80.Pj; 73.43.Nq Keywords: Two-component; Dipolar bosons; Quantum phase transition
1. Introduction Since the theoretical prediction [1] and experimental achievement [2] of superfluid (SF) to Mott insulator (MI) phase transition in optical lattice, a great deal of theorist have turned to the theoretical and experimental study dynamics of quantum phase transition using Bose–Hubbard model [3], which can describe ultracold dilute gas of bosonic atoms in optical lattice, with Green function technique [4] and mean-field approximation [5,6]. Approaching zero temperature, quantum fluctuations drive the bosonic gas from SF to MI phase, the excitation spectra gapless in SF phase whereas having a finite gap in MI phase [2]. * Corresponding author.
E-mail address: [email protected] (T. Zhang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.043
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
229
The single-component bosons without internal degrees of freedom have only two phase in a regular lattice: SF and MI [4], while several bosonic component are in the same optical lattice, because of the presence of interspecies and external species interaction, it may be possible to observe complex quantum phase transition, and many theorist have studied two component phase transition [5,7,8]. In two component BEC, single atom jumps (the hopping drive the same species from one site to another) require high energy, there are properly three phase: (i) MI of sort A and B; (ii) SF of sort A and B; (iii) MI of sort A and SF of sort B and its A ↔ B analog. While exchanging two different component atom (the interspecies hopping) dose not need such energy, Kuklov et al. [8] reveal that there could exist superfluid current, from macroscopical review, the atom numbers do not change in each optical lattice. If two component interaction is repulsive, two-component nonconvertible atoms (such as 23 Na and 87 Rb) exchange or interconvertible atoms convert each other (such as the system compose of hyperfine states of 87 Rb |F = 1, m = −1 and |F = 1, m = +1), the currents of two species are equal values but in opposite directions, this is super-counter-fluid (SCF) [8]. If two component interaction attractive, two species atoms can combine a cooper pair like, this pair hope from one site to another, this is the paired superfluid (PSF) [8]. We can view this paired atom hopping as two single atom hopping convert or exchange between two component. In this Letter, we analytically study the energy spectrum and phase transition of two component BEC with the dipole–dipole interaction in optical lattice. Since the dipole–dipole interaction can be repulsive or attractive which is long-range interaction, quite strong relative to the on-site interaction, these forces significantly modify the ground state and collective excitation of trapped condensation, provide a rich variety of quantum phase transition. It may be crucial to answer some unresolved questions, such as atom–photon coupling or magnetic flied induced polarization, and robust quantum-computing schemers [9]. The results show that the dipole–dipole interaction can be really strongly influence the MI ground state, driving from MI phase to superfluid phase which is agree with the result analyzed in phase diagram. The excited state split into two with interspecies hopping, and have a lower excited state. If the interspecies hopping is big enough, there are phase transition from MI to SF phase, which is essential to SCF and PSF phase. The diagram given by mean-field method indicate the existence of phase transition that the two component united order parameters being nonzero, ψA + ψB = 0, for each component order parameter ψA = ψB = 0, the two atoms exchange (or convert) need a lower energy with a lower excited state which can make the atoms from MI state to superfluid state.
2. Quantum phase transition In general, the Hamiltonian of two component dipolar bosonic system can be written, in the second-quantization natation, as H = Hhop + Hdia , h¯ 2 † 2 +Vopt (r) Ψσ (r), dr Ψσ (r) − Hhop = 2mσ σ,σ =A,B dr Ψσ† (r)VT σ (r)Ψσ (r) + dr dr Ψσ† (r)Ψσ† (r )Vint (r)Ψσ (r )Ψσ (r), Hdia = σ =A,B
(1)
σ,σ =A,B
where Ψσ is the annihilation field operator of a particle at lattice site i of component σ , σ (or σ ) denotes component A or B. When σ and σ are same, the last term in Hdia represents the intra-interaction of same componentbosons; When σ and σ are different, it is mixing interaction between different component bosons. Vopt (r) = 3i=1 V0i sin2 (kxi ) is the optical potential with the wavevector k = 2π/λ and λ is the wavelength of the laser, corresponding to a lattice period a = λ/2, while VT σ is the external magnetic potential trapping two component respectively. Vint (r) is the same description in Ref. [9]. We assume the atoms is in the lowest energy band, which is small compared to excitation energies to the second band, expanding the field operators in the Wannier
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basis and keeping only the lowest vibration states, ΨA† (r) = i Aˆ †i wA (r − ri ), ΨB† (r) = i Bˆ i† wB (r − ri ). Eq. (1) reduces to the Bose–Hubbard Hamiltonian: J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + J12 Aˆ †i Bˆ j + H.c. , Hhop = − i,j
1 µ1i Aˆ †i Aˆ i + µ2i Bˆ i† Bˆ i + U1 Aˆ †i Aˆ †i Aˆ i Aˆ i + U2 Bˆ i† Bˆ i† Bˆ i Bˆ i + 2U12 Aˆ †i Bˆ i† Aˆ i Bˆ i Hdia = 2 i i 1 ˆ† ˆ† ˆ ˆ + ε1 Ai Aj Ai Aj + ε2 Bˆ i† Bˆ j† Bˆ i Bˆ j + 2ε12 Aˆ †i Bˆ j† Aˆ i Bˆ j , 2
(2)
i,j,i=j
where Aˆ †i (Bˆ i† ) and Aˆ i (Bˆ i ) are the creation and annihilation operator of A (B) component atom on site i, they ∗ 2 obey the commutation relations, [Aˆ i , Aˆ †j ] = δij , [Bˆ i , Bˆ j† ] = δij ; the parameters J1,2 = − wA,B (r−ri )[− 2mh¯A,B 2 ∗ 2 +Vopt (r)]wA,B (r − rj ) dr, J12 = − wA (r − ri )[− 2mh¯ AB 2 +Vopt (r)]wB (r − rj ) dr is the strength of the hopping ∗ between sites i and j ; µ1i,2i = wA,B (r − ri )VT A,B wA,B (r − ri ) dr is an energy offset of A (B) component on ith 4π h2 a h¯ 2 aAB site; U1,2 = m¯ A,BA,B |wA,B (r − ri )|4 dr, U12 = 4π2m |wA (r − ri )|2 |wB (r − ri )|2 dr is the repulsion strength AB of two atoms between A and B component, mAB is the reduce mass of atom. The double-time green function at zero temperature is defined as
Aˆ i (t); Aˆ †j (t ) = −iθ (t − t ) Aˆ i (t), Aˆ †j (t )
= −iθ (t − t ) Aˆ i (t)Aˆ †j (t ) − Aˆ †j (t )Aˆ i (t) , (3) where the function θ (t − t ) is the step function, and the green function Aˆ i (t); Aˆ †j (t ) depends only on time difference, we can set t = 0. Fourier transformation of green function is
1 GAij (ω) ≡ Aˆ i (t); Aˆ †j (t ) ω = 2π
∞
dt Aˆ i (t); Aˆ †j (t ) ei(ω+iη) ,
η = +0,
(4)
−∞
using Heisenberg equation and 1 δ(t) = 2π
∞
dω e−iωt ,
(5)
−∞
we can get
ωGAij (ω) = Aˆ i , Aˆ †j + [Aˆ i , H ]; Aˆ †j ω ,
(6)
where · · · denotes the ground-state expectation value. We can solve them in wave vector space. The relation between site space operator Aˆ i (Bˆ i ) and wave-vector-space operator Aˆ k (Bˆ k ) is 1 ikri ˆ ˆ † 1 −ikri ˆ † 1 ikri ˆ ˆ † 1 −ikri ˆ † e Ak , Ai = √ e Bˆ i = √ e Bk , Bi = √ e Aˆ i = √ Ak , Bk , (7) N k N k N k N k and GAij (ω) =
1 ik(ri −rj ) e GAk (ω), N k
where GAk (ω) = Aˆ k ; Aˆ †k denotes the orthogonal Green function in Bloch representation.
(8)
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
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(a)
(b)
(c)
(d)
Fig. 1. The phase diagram of long-range interaction, where the vertical axis is dimensionless chemical potential either µ¯ A or µ¯ B , the horizontal axis is dimensionless dipole interaction strength either ε¯ 1 or ε¯ 2 , respectively, where nA = 1, nB = 2 and ε¯ 12 = 4. (a) U¯ 1 = U¯ 2 = −15, U¯ 12 = −10; (b) U¯ 1 = U¯ 2 = 15, U¯ 12 = −10; (c) U¯ 1 = U¯ 2 = −15, U¯ 12 = 10; and (d) U¯ 1 = U¯ 2 = 15, U¯ 12 = 10. The regions of phase are labelled SF, SM, MS and MI.
We just consider intraspecies hopping at first, the hopping herm in Eq. (2) should be J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + H.c. . Hhop = −
(9)
i,j
We calculate the Green function of two component and get the energy spectrum in wave vector space ωA1 = U12 n20 + ε1 (k) + ε12 (k) + µ1 , (k) + µ2 , ωB1 = U12 n10 + ε2 (k) + ε12
ωA2 = U1 n10 + U12 n20 − J1 (k) + ε1 (k) + ε12 (k) + µ1 ,
ωB2 = U2 n20 + U12 n10 − J2 (k) + ε2 (k) + ε12 (k) + µ2 , (10) † † where n10 = Aˆ i Aˆ i , n20 = Bˆ i Bˆ i are the average occupation number of A (B) component, Jl (k) = a Jl eika (l = 1, 2). ωA1 is the ground state, ωA2 is the excited state. From the energy spectrum we know one component ground state and excited state are related to interspecies interaction, but the energy gap ∆ = Ui ni0 − Ji (k) is the intrinsical property, that is the phase transition independent of other component. We can change the high of optical lattice and transverse confinement of magnetic field, two component phase transition independently. when they are in superfluid, we have the relation Ui ni0 = Ji (k) [5]. In fact, the system is like the single-component, interspecies interaction and long-range interaction just like an additional external potential move the ground state and exited state synchronously. We use mean field method to study phase transition with the grand canonical ensemble and get the same con clusion. We add the chemical potential term, −µA i Aˆ †i Aˆ i − µA i Bˆ i† Bˆ i , into the Hamiltonian, and combine
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(a)
(b)
(c)
(d)
Fig. 2. The phase diagram of on-site interaction, where the vertical axis is dimensionless chemical potential either µ¯ A or µ¯ B , the horizontal axis is dimensionless on-site interaction strength U¯ 1 or U¯ 2 , respectively, other parameter the same as Fig. 1. (a) ε¯ 1 = ε¯ 2 = −5, ε¯ 12 = 4; (b) ε¯ 1 = ε¯ 2 = 5, ε¯ 12 = 4; (c) ε¯ 1 = ε¯ 2 = −5, ε¯ 12 = 2; (d) ε¯ 1 = ε¯ 2 = 5, ε¯ 12 = 2.
the external magnetic potential terms into it. We get the phase transition boundaries of two component dipolar bosons µ¯ A =
1 ¯ 2U1 nA − U¯ 1 + 2U¯ 12 nB + 2z¯ε1 nA + 2z¯ε12 nB − 1 2 ±
µ¯ B =
(U¯ 1 − 1)2 + (z¯ε1 − 1)2 − 4nA (U¯ 1 + z¯ε1 ) + 2zU¯ 1 ε¯ 1 − 1 ,
1 ¯ 2U2 nB − U¯ 2 + 2U¯ 12 nA + 2z¯ε2 nB + 2z¯ε12 nA − 1 2 ±
(U¯ 2 − 1)2 + (z¯ε2 − 1)2 − 4nB (U¯ 2 + z¯ε2 ) + 2zU¯ 2 ε¯ 2 − 1 .
(11)
According to Eq. (11), phase boundary is independent other component intraspecies interaction and atom number, interspecies interaction just move the lobes, the same as before analyzed. Intra-species interaction of long-range and on-site have the same degree, driving atom from SF to MI, show in Figs. 1 and 2. The meaning of SM is that A component is in superfluid phase and B component is in MI phase in this region, vice versa meaning of MS. Interspecies interaction just move the lobes forward or backward. When nA = nB or nA < nB and U12 > 0, the conclusion just like Ref. [5], the difference is phase transition can happen when U1 < 0 (and U2 < 0) because of long-range interaction. Without long-range interaction, the condensates is unstable to local collapse, the long-range attractive interaction can cancel this force, and condensates may be stable. When interspecies interaction is attrac-
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
(a)
233
(b)
Fig. 3. Difference phase diagram of on-site interaction, where nA = 1, nB = 2, and U¯ 12 = −10. (a) The on-site interaction diagram without long-range interaction, two lobes do not intersect; (b) the on-site interaction diagram with long-range interaction, where ε¯ 1 = −10, ε¯ 2 = −10 and ε¯ 12 = 8, at this time two lobes intersect.
tive (here nA < nB and U12 < 0), the lobes A and B can intersect, both two component can be in MI phase (Fig. 3), while two component cannot be in MI phase in Ref. [5]. If there are interspecies hopping, there is convert or exchange atoms between A component and B component, the hopping term is Eq. (2), we also get the energy spectrum = ωA1 , ωA1
ωB1 = ωB1 , 1 2 (k) , ωA2,A3 = ωB2,B3 = M1 + M1 ∓ (M1 − M1 )2 + 4J12 2
(12)
and energy gap 1 2 (k) , M1 − M1 − 2J1 (k) + 2U1 n10 ∓ [M1 − M1 ]2 + 4J12 2 1 2 (k) , ∆B1,B2 = M1 − M1 − 2J2 (k) + 2U2 n20 ∓ [M1 − M1 ]2 + 4J12 2 ∆A1,A2 =
(13)
(k) + µ . where M1 = U1 n10 + U12 n20 − J1 (k) + ε1 (k) + ε12 (k) + µ1 , M1 = U2 n20 + U12 n10 − J2 (k) + ε2 (k) + ε12 2 From the energy spectrum we know the existence of the interspecies hopping, the excited state split into two, a excited state is lower than the excited state without interspecies hopping. The energy gap related to the difference M1 − M1 and interspecies hopping J12 are show in Fig. 4. If M1 − M1 is small and J12 < 2.5, two component is still in MI phase, while M1 − M1 is big, one component is transit from MI to SF phase. When interspecies hopping increase, two component can all transit to SF phase (the black region in Fig. 4). In the proper way, there have SCF and PSF state. If we consider the phase of two component, Eq. (2) rewrite as J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + J12 eiϕ Aˆ †i Bˆ j + H.c. , Hhop = − (14) i,j
where ϕ is the phase difference of two component, we consider each component phase period is a. From the correlation function G12 (ω) =
−J12 (k)eiϕ , (ω − ωA2 )(ω − ωB2 ) − J12 (k)2
G21 (ω) =
−J21 (k)e−iϕ , (ω − ωA2 )(ω − ωB2 ) − J21 (k)2
(15)
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Fig. 4. Two component dipole bosons energy gap relate to difference of M1 − M1 and interspecies hopping J12 , where U1 n10 − J1 (k) = U2 n20 − J2 (k) = 5, that is in MI phase without interspecies hopping. The black is zero plane.
ˆ Bˆ † , G21 (ω) = B; ˆ Aˆ † , and J12 (k) = J21 (k). We can get the current of two fluid density where G12 (ω) = A; ρB→A and ρA→B , because of they have the same energy spectra, |ρA→B | = |ρB→A |, while the current IA + IB = 0,
(16)
this is the condition of SCF phase, we have counter fluid and net current [8]. If the two component have the same phase, ϕ = 0, they have the same density current, and IA = IB , this could be PSF phase. The existence of two component exchange, one component fluctuation relate to the other component fluctuation, each component phase boundary do not simply solved, but we can analysis the united superfluid order parameter ΨA + ΨB (that is total atoms of two component fixed) in special case. Assume J12 = 2J , we have the perturbation term by mean-field method H eff = −2zJ (17) (ΨA + ΨB ) Aˆ †i + Aˆ i + Bˆ i† + Bˆ i − (ΨA + ΨB )2 . i
If two component have the same parameter, we have the phase boundary 1 ¯ ¯ ¯ µ¯ = (U + U12 + z¯ε + z¯ε12 )n − (U + 2) ± (U¯ − 2)2 + (z¯ε − 2)2 − 8n(U¯ + z¯ε ) + 4zU¯ ε¯ − 4 . (18) 2 Fig. 5 shows phase transition of two component dipolar bosons, L2 is the phase boundaries determine by Eq. (11) and L1 is determine by Eq. (18). In MI region, whatever a single atom or a paired atom exchange, their exciting need finite energy. In SCF (or PSF) region, a paired atom exchange needs a lower energy, the interspecies hopping drive two single atoms of two component from MI phase to the paired atom superfluid phase. In 2SF region, whatever a single atom or a pair formed, they are gapless exciting in superfluid phase. For observation of above excitation and phase transition experimentally, we create a polarized atomic dipolar gas by coupling of the atomic ground state to an electrically polarized Rydberg state or using magnetic atomic dipoles, control the trapping geometry changing the sign and value of dipole–dipole interaction with a laser. Two component atoms, such as 87 Rb, created by the π/2 rf pulse out of one component, using absorption imaging technique, adjusting optical lattice potential and the rf pulse (induce two component convert), we can detect transition from MI phase to paired atoms superfluid phase. In the proper way, control the phase of two component, we can detect SCF (or PSF) by spatially selective Ramsey spectroscopy [8].
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Fig. 5. Two component phase diagram with interspecies hopping (L1) and without interspecies hopping (L2), where the vertical axis is dimensionless chemical potential µ, ¯ the horizontal axis is dimensionless on-site interaction strength U¯ , nA = nB = 1, U¯ 12 = 2 and ε1 = ε2 = ε12 = 0.
3. Conclusion In conclusion, we have studied the Bose–Hubbard model of two component bosons energy spectrum and phase transition in optical lattice using Green function and mean-field method. The interspecies hopping is important that it change the energy spectrum of two component with the same excite state, where have a lower excite state, if hopping big enough the paired atom have a gapless exciting from single-atom MI phase to paired superfluid. To some degree we have the counter fluid as the conclusion in literature. The dipole interaction strongly influence the MI phase ground state and phase transition, which is the same as the analysis result in phase diagram. In phase diagram, the SCF (PSF) phase transition boundary is complex, we just consider the special case where ψA + ψB = 0, while to single-atom, ψA = 0 and ψB = 0, this is the paired atom have a lower energy like cooper pair.
Acknowledgements This work is supported by the NSF of China under grant Nos. 60490280, 90403034, 90406017.
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[7] M.Y. Kagan, et al., Phys. Rev. B 65 (2002) 195103. [8] A. Kuklov, et al., Phys. Rev. Lett. 90 (2003) 100401; A. Kuklov, et al., Phys. Rev. Lett. 92 (2004) 030403; A. Kuklov, et al., Phys. Rev. Lett. 92 (2004) 050402; A. Kuklov, et al., Phys. Rev. A 69 (2004) 025601; B. Paredes, et al., Phys. Rev. Lett. 90 (2003) 150402; Z.X. Liang, et al., Phys. Rev. Lett. 94 (2005) 050402. [9] K. Goral, et al., Phys. Rev. Lett. 88 (2002) 170406; W.M. Liu, et al., Phys. Rev. Lett. 88 (2002) 170408; S. Giovanazzi, et al., Phys. Rev. Lett. 89 (2002) 130401.
Quantum phase transition of two component dipolar bosons in optical lattices Tao Zhang a,b,∗ , Rui-Hong Yue b , W.M. Liu a a Joint Laboratory of Advanced Technology in Measurements, Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China b Institute of Modern Physics, Northwest University, Xi’an 710069, China
Received 14 February 2005; received in revised form 7 April 2005; accepted 7 April 2005 Available online 27 April 2005 Communicated by B. Fricke
Abstract The energy spectrum of two component Bose–Einstein condensates with the dipole–dipole interaction in optical lattice is obtained by Green function technique. Its superfluid and Mott insulator phase transition is analyzed. The results show that the interspecies hopping and dipole interaction can strongly influence the energy spectrum of two component dipolar bosonic system. A paired atom hopping can exist in superfluid phase while single-atom hopping is Mott insulator, a gapless excite, which is essential to super-counter fluid phase. The special case of phase boundary of super-counter phase are discussed. 2005 Elsevier B.V. All rights reserved. PACS: 03.75.Fi; 05.30.Jp; 05.50.+q; 32.80.Pj; 73.43.Nq Keywords: Two-component; Dipolar bosons; Quantum phase transition
1. Introduction Since the theoretical prediction [1] and experimental achievement [2] of superfluid (SF) to Mott insulator (MI) phase transition in optical lattice, a great deal of theorist have turned to the theoretical and experimental study dynamics of quantum phase transition using Bose–Hubbard model [3], which can describe ultracold dilute gas of bosonic atoms in optical lattice, with Green function technique [4] and mean-field approximation [5,6]. Approaching zero temperature, quantum fluctuations drive the bosonic gas from SF to MI phase, the excitation spectra gapless in SF phase whereas having a finite gap in MI phase [2]. * Corresponding author.
E-mail address: [email protected] (T. Zhang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.043
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
229
The single-component bosons without internal degrees of freedom have only two phase in a regular lattice: SF and MI [4], while several bosonic component are in the same optical lattice, because of the presence of interspecies and external species interaction, it may be possible to observe complex quantum phase transition, and many theorist have studied two component phase transition [5,7,8]. In two component BEC, single atom jumps (the hopping drive the same species from one site to another) require high energy, there are properly three phase: (i) MI of sort A and B; (ii) SF of sort A and B; (iii) MI of sort A and SF of sort B and its A ↔ B analog. While exchanging two different component atom (the interspecies hopping) dose not need such energy, Kuklov et al. [8] reveal that there could exist superfluid current, from macroscopical review, the atom numbers do not change in each optical lattice. If two component interaction is repulsive, two-component nonconvertible atoms (such as 23 Na and 87 Rb) exchange or interconvertible atoms convert each other (such as the system compose of hyperfine states of 87 Rb |F = 1, m = −1 and |F = 1, m = +1), the currents of two species are equal values but in opposite directions, this is super-counter-fluid (SCF) [8]. If two component interaction attractive, two species atoms can combine a cooper pair like, this pair hope from one site to another, this is the paired superfluid (PSF) [8]. We can view this paired atom hopping as two single atom hopping convert or exchange between two component. In this Letter, we analytically study the energy spectrum and phase transition of two component BEC with the dipole–dipole interaction in optical lattice. Since the dipole–dipole interaction can be repulsive or attractive which is long-range interaction, quite strong relative to the on-site interaction, these forces significantly modify the ground state and collective excitation of trapped condensation, provide a rich variety of quantum phase transition. It may be crucial to answer some unresolved questions, such as atom–photon coupling or magnetic flied induced polarization, and robust quantum-computing schemers [9]. The results show that the dipole–dipole interaction can be really strongly influence the MI ground state, driving from MI phase to superfluid phase which is agree with the result analyzed in phase diagram. The excited state split into two with interspecies hopping, and have a lower excited state. If the interspecies hopping is big enough, there are phase transition from MI to SF phase, which is essential to SCF and PSF phase. The diagram given by mean-field method indicate the existence of phase transition that the two component united order parameters being nonzero, ψA + ψB = 0, for each component order parameter ψA = ψB = 0, the two atoms exchange (or convert) need a lower energy with a lower excited state which can make the atoms from MI state to superfluid state.
2. Quantum phase transition In general, the Hamiltonian of two component dipolar bosonic system can be written, in the second-quantization natation, as H = Hhop + Hdia , h¯ 2 † 2 +Vopt (r) Ψσ (r), dr Ψσ (r) − Hhop = 2mσ σ,σ =A,B dr Ψσ† (r)VT σ (r)Ψσ (r) + dr dr Ψσ† (r)Ψσ† (r )Vint (r)Ψσ (r )Ψσ (r), Hdia = σ =A,B
(1)
σ,σ =A,B
where Ψσ is the annihilation field operator of a particle at lattice site i of component σ , σ (or σ ) denotes component A or B. When σ and σ are same, the last term in Hdia represents the intra-interaction of same componentbosons; When σ and σ are different, it is mixing interaction between different component bosons. Vopt (r) = 3i=1 V0i sin2 (kxi ) is the optical potential with the wavevector k = 2π/λ and λ is the wavelength of the laser, corresponding to a lattice period a = λ/2, while VT σ is the external magnetic potential trapping two component respectively. Vint (r) is the same description in Ref. [9]. We assume the atoms is in the lowest energy band, which is small compared to excitation energies to the second band, expanding the field operators in the Wannier
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basis and keeping only the lowest vibration states, ΨA† (r) = i Aˆ †i wA (r − ri ), ΨB† (r) = i Bˆ i† wB (r − ri ). Eq. (1) reduces to the Bose–Hubbard Hamiltonian: J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + J12 Aˆ †i Bˆ j + H.c. , Hhop = − i,j
1 µ1i Aˆ †i Aˆ i + µ2i Bˆ i† Bˆ i + U1 Aˆ †i Aˆ †i Aˆ i Aˆ i + U2 Bˆ i† Bˆ i† Bˆ i Bˆ i + 2U12 Aˆ †i Bˆ i† Aˆ i Bˆ i Hdia = 2 i i 1 ˆ† ˆ† ˆ ˆ + ε1 Ai Aj Ai Aj + ε2 Bˆ i† Bˆ j† Bˆ i Bˆ j + 2ε12 Aˆ †i Bˆ j† Aˆ i Bˆ j , 2
(2)
i,j,i=j
where Aˆ †i (Bˆ i† ) and Aˆ i (Bˆ i ) are the creation and annihilation operator of A (B) component atom on site i, they ∗ 2 obey the commutation relations, [Aˆ i , Aˆ †j ] = δij , [Bˆ i , Bˆ j† ] = δij ; the parameters J1,2 = − wA,B (r−ri )[− 2mh¯A,B 2 ∗ 2 +Vopt (r)]wA,B (r − rj ) dr, J12 = − wA (r − ri )[− 2mh¯ AB 2 +Vopt (r)]wB (r − rj ) dr is the strength of the hopping ∗ between sites i and j ; µ1i,2i = wA,B (r − ri )VT A,B wA,B (r − ri ) dr is an energy offset of A (B) component on ith 4π h2 a h¯ 2 aAB site; U1,2 = m¯ A,BA,B |wA,B (r − ri )|4 dr, U12 = 4π2m |wA (r − ri )|2 |wB (r − ri )|2 dr is the repulsion strength AB of two atoms between A and B component, mAB is the reduce mass of atom. The double-time green function at zero temperature is defined as
Aˆ i (t); Aˆ †j (t ) = −iθ (t − t ) Aˆ i (t), Aˆ †j (t )
= −iθ (t − t ) Aˆ i (t)Aˆ †j (t ) − Aˆ †j (t )Aˆ i (t) , (3) where the function θ (t − t ) is the step function, and the green function Aˆ i (t); Aˆ †j (t ) depends only on time difference, we can set t = 0. Fourier transformation of green function is
1 GAij (ω) ≡ Aˆ i (t); Aˆ †j (t ) ω = 2π
∞
dt Aˆ i (t); Aˆ †j (t ) ei(ω+iη) ,
η = +0,
(4)
−∞
using Heisenberg equation and 1 δ(t) = 2π
∞
dω e−iωt ,
(5)
−∞
we can get
ωGAij (ω) = Aˆ i , Aˆ †j + [Aˆ i , H ]; Aˆ †j ω ,
(6)
where · · · denotes the ground-state expectation value. We can solve them in wave vector space. The relation between site space operator Aˆ i (Bˆ i ) and wave-vector-space operator Aˆ k (Bˆ k ) is 1 ikri ˆ ˆ † 1 −ikri ˆ † 1 ikri ˆ ˆ † 1 −ikri ˆ † e Ak , Ai = √ e Bˆ i = √ e Bk , Bi = √ e Aˆ i = √ Ak , Bk , (7) N k N k N k N k and GAij (ω) =
1 ik(ri −rj ) e GAk (ω), N k
where GAk (ω) = Aˆ k ; Aˆ †k denotes the orthogonal Green function in Bloch representation.
(8)
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
231
(a)
(b)
(c)
(d)
Fig. 1. The phase diagram of long-range interaction, where the vertical axis is dimensionless chemical potential either µ¯ A or µ¯ B , the horizontal axis is dimensionless dipole interaction strength either ε¯ 1 or ε¯ 2 , respectively, where nA = 1, nB = 2 and ε¯ 12 = 4. (a) U¯ 1 = U¯ 2 = −15, U¯ 12 = −10; (b) U¯ 1 = U¯ 2 = 15, U¯ 12 = −10; (c) U¯ 1 = U¯ 2 = −15, U¯ 12 = 10; and (d) U¯ 1 = U¯ 2 = 15, U¯ 12 = 10. The regions of phase are labelled SF, SM, MS and MI.
We just consider intraspecies hopping at first, the hopping herm in Eq. (2) should be J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + H.c. . Hhop = −
(9)
i,j
We calculate the Green function of two component and get the energy spectrum in wave vector space ωA1 = U12 n20 + ε1 (k) + ε12 (k) + µ1 , (k) + µ2 , ωB1 = U12 n10 + ε2 (k) + ε12
ωA2 = U1 n10 + U12 n20 − J1 (k) + ε1 (k) + ε12 (k) + µ1 ,
ωB2 = U2 n20 + U12 n10 − J2 (k) + ε2 (k) + ε12 (k) + µ2 , (10) † † where n10 = Aˆ i Aˆ i , n20 = Bˆ i Bˆ i are the average occupation number of A (B) component, Jl (k) = a Jl eika (l = 1, 2). ωA1 is the ground state, ωA2 is the excited state. From the energy spectrum we know one component ground state and excited state are related to interspecies interaction, but the energy gap ∆ = Ui ni0 − Ji (k) is the intrinsical property, that is the phase transition independent of other component. We can change the high of optical lattice and transverse confinement of magnetic field, two component phase transition independently. when they are in superfluid, we have the relation Ui ni0 = Ji (k) [5]. In fact, the system is like the single-component, interspecies interaction and long-range interaction just like an additional external potential move the ground state and exited state synchronously. We use mean field method to study phase transition with the grand canonical ensemble and get the same con clusion. We add the chemical potential term, −µA i Aˆ †i Aˆ i − µA i Bˆ i† Bˆ i , into the Hamiltonian, and combine
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(a)
(b)
(c)
(d)
Fig. 2. The phase diagram of on-site interaction, where the vertical axis is dimensionless chemical potential either µ¯ A or µ¯ B , the horizontal axis is dimensionless on-site interaction strength U¯ 1 or U¯ 2 , respectively, other parameter the same as Fig. 1. (a) ε¯ 1 = ε¯ 2 = −5, ε¯ 12 = 4; (b) ε¯ 1 = ε¯ 2 = 5, ε¯ 12 = 4; (c) ε¯ 1 = ε¯ 2 = −5, ε¯ 12 = 2; (d) ε¯ 1 = ε¯ 2 = 5, ε¯ 12 = 2.
the external magnetic potential terms into it. We get the phase transition boundaries of two component dipolar bosons µ¯ A =
1 ¯ 2U1 nA − U¯ 1 + 2U¯ 12 nB + 2z¯ε1 nA + 2z¯ε12 nB − 1 2 ±
µ¯ B =
(U¯ 1 − 1)2 + (z¯ε1 − 1)2 − 4nA (U¯ 1 + z¯ε1 ) + 2zU¯ 1 ε¯ 1 − 1 ,
1 ¯ 2U2 nB − U¯ 2 + 2U¯ 12 nA + 2z¯ε2 nB + 2z¯ε12 nA − 1 2 ±
(U¯ 2 − 1)2 + (z¯ε2 − 1)2 − 4nB (U¯ 2 + z¯ε2 ) + 2zU¯ 2 ε¯ 2 − 1 .
(11)
According to Eq. (11), phase boundary is independent other component intraspecies interaction and atom number, interspecies interaction just move the lobes, the same as before analyzed. Intra-species interaction of long-range and on-site have the same degree, driving atom from SF to MI, show in Figs. 1 and 2. The meaning of SM is that A component is in superfluid phase and B component is in MI phase in this region, vice versa meaning of MS. Interspecies interaction just move the lobes forward or backward. When nA = nB or nA < nB and U12 > 0, the conclusion just like Ref. [5], the difference is phase transition can happen when U1 < 0 (and U2 < 0) because of long-range interaction. Without long-range interaction, the condensates is unstable to local collapse, the long-range attractive interaction can cancel this force, and condensates may be stable. When interspecies interaction is attrac-
T. Zhang et al. / Physics Letters A 340 (2005) 228–236
(a)
233
(b)
Fig. 3. Difference phase diagram of on-site interaction, where nA = 1, nB = 2, and U¯ 12 = −10. (a) The on-site interaction diagram without long-range interaction, two lobes do not intersect; (b) the on-site interaction diagram with long-range interaction, where ε¯ 1 = −10, ε¯ 2 = −10 and ε¯ 12 = 8, at this time two lobes intersect.
tive (here nA < nB and U12 < 0), the lobes A and B can intersect, both two component can be in MI phase (Fig. 3), while two component cannot be in MI phase in Ref. [5]. If there are interspecies hopping, there is convert or exchange atoms between A component and B component, the hopping term is Eq. (2), we also get the energy spectrum = ωA1 , ωA1
ωB1 = ωB1 , 1 2 (k) , ωA2,A3 = ωB2,B3 = M1 + M1 ∓ (M1 − M1 )2 + 4J12 2
(12)
and energy gap 1 2 (k) , M1 − M1 − 2J1 (k) + 2U1 n10 ∓ [M1 − M1 ]2 + 4J12 2 1 2 (k) , ∆B1,B2 = M1 − M1 − 2J2 (k) + 2U2 n20 ∓ [M1 − M1 ]2 + 4J12 2 ∆A1,A2 =
(13)
(k) + µ . where M1 = U1 n10 + U12 n20 − J1 (k) + ε1 (k) + ε12 (k) + µ1 , M1 = U2 n20 + U12 n10 − J2 (k) + ε2 (k) + ε12 2 From the energy spectrum we know the existence of the interspecies hopping, the excited state split into two, a excited state is lower than the excited state without interspecies hopping. The energy gap related to the difference M1 − M1 and interspecies hopping J12 are show in Fig. 4. If M1 − M1 is small and J12 < 2.5, two component is still in MI phase, while M1 − M1 is big, one component is transit from MI to SF phase. When interspecies hopping increase, two component can all transit to SF phase (the black region in Fig. 4). In the proper way, there have SCF and PSF state. If we consider the phase of two component, Eq. (2) rewrite as J1 Aˆ †i Aˆ j + J2 Bˆ i† Bˆ j + J12 eiϕ Aˆ †i Bˆ j + H.c. , Hhop = − (14) i,j
where ϕ is the phase difference of two component, we consider each component phase period is a. From the correlation function G12 (ω) =
−J12 (k)eiϕ , (ω − ωA2 )(ω − ωB2 ) − J12 (k)2
G21 (ω) =
−J21 (k)e−iϕ , (ω − ωA2 )(ω − ωB2 ) − J21 (k)2
(15)
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Fig. 4. Two component dipole bosons energy gap relate to difference of M1 − M1 and interspecies hopping J12 , where U1 n10 − J1 (k) = U2 n20 − J2 (k) = 5, that is in MI phase without interspecies hopping. The black is zero plane.
ˆ Bˆ † , G21 (ω) = B; ˆ Aˆ † , and J12 (k) = J21 (k). We can get the current of two fluid density where G12 (ω) = A; ρB→A and ρA→B , because of they have the same energy spectra, |ρA→B | = |ρB→A |, while the current IA + IB = 0,
(16)
this is the condition of SCF phase, we have counter fluid and net current [8]. If the two component have the same phase, ϕ = 0, they have the same density current, and IA = IB , this could be PSF phase. The existence of two component exchange, one component fluctuation relate to the other component fluctuation, each component phase boundary do not simply solved, but we can analysis the united superfluid order parameter ΨA + ΨB (that is total atoms of two component fixed) in special case. Assume J12 = 2J , we have the perturbation term by mean-field method H eff = −2zJ (17) (ΨA + ΨB ) Aˆ †i + Aˆ i + Bˆ i† + Bˆ i − (ΨA + ΨB )2 . i
If two component have the same parameter, we have the phase boundary 1 ¯ ¯ ¯ µ¯ = (U + U12 + z¯ε + z¯ε12 )n − (U + 2) ± (U¯ − 2)2 + (z¯ε − 2)2 − 8n(U¯ + z¯ε ) + 4zU¯ ε¯ − 4 . (18) 2 Fig. 5 shows phase transition of two component dipolar bosons, L2 is the phase boundaries determine by Eq. (11) and L1 is determine by Eq. (18). In MI region, whatever a single atom or a paired atom exchange, their exciting need finite energy. In SCF (or PSF) region, a paired atom exchange needs a lower energy, the interspecies hopping drive two single atoms of two component from MI phase to the paired atom superfluid phase. In 2SF region, whatever a single atom or a pair formed, they are gapless exciting in superfluid phase. For observation of above excitation and phase transition experimentally, we create a polarized atomic dipolar gas by coupling of the atomic ground state to an electrically polarized Rydberg state or using magnetic atomic dipoles, control the trapping geometry changing the sign and value of dipole–dipole interaction with a laser. Two component atoms, such as 87 Rb, created by the π/2 rf pulse out of one component, using absorption imaging technique, adjusting optical lattice potential and the rf pulse (induce two component convert), we can detect transition from MI phase to paired atoms superfluid phase. In the proper way, control the phase of two component, we can detect SCF (or PSF) by spatially selective Ramsey spectroscopy [8].
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Fig. 5. Two component phase diagram with interspecies hopping (L1) and without interspecies hopping (L2), where the vertical axis is dimensionless chemical potential µ, ¯ the horizontal axis is dimensionless on-site interaction strength U¯ , nA = nB = 1, U¯ 12 = 2 and ε1 = ε2 = ε12 = 0.
3. Conclusion In conclusion, we have studied the Bose–Hubbard model of two component bosons energy spectrum and phase transition in optical lattice using Green function and mean-field method. The interspecies hopping is important that it change the energy spectrum of two component with the same excite state, where have a lower excite state, if hopping big enough the paired atom have a gapless exciting from single-atom MI phase to paired superfluid. To some degree we have the counter fluid as the conclusion in literature. The dipole interaction strongly influence the MI phase ground state and phase transition, which is the same as the analysis result in phase diagram. In phase diagram, the SCF (PSF) phase transition boundary is complex, we just consider the special case where ψA + ψB = 0, while to single-atom, ψA = 0 and ψB = 0, this is the paired atom have a lower energy like cooper pair.
Acknowledgements This work is supported by the NSF of China under grant Nos. 60490280, 90403034, 90406017.
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