Quantum Tricriticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates.pdf
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1、Quantum Tricriticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates Yun Li,1Lev P. Pitaevskii,1,2and Sandro Stringari1 1Dipartimento di Fisica, Universita di Trento and INO-CNR BEC Center, I-38123 Povo, Italy 2Kapitza Institute for Physical Problems, Kosygina 2, 119334 Mosco
2、w, Russia (Received 14 February 2012; published 29 May 2012) We consider a spin-orbit coupled confi guration of spin-1=2 interacting bosons with equal Rashba and Dresselhaus couplings. The phase diagram of the system at T 0 is discussed with special emphasis on the role of the interaction treated in
3、 the mean-fi eld approximation. For a critical value of the density and of the Raman coupling we predict the occurrence of a characteristic tricritical point separating the spin mixed, the phase separated, and the zero momentum states of the Bose gas. The corresponding quantum phases are investigate
4、d analyzing the momentum distribution, the longitudinal and transverse spin polarization, and the emergence of density fringes. The effect of harmonic trapping as well as the role of the breaking of spin symmetry in the interaction Hamiltonian are also discussed. DOI: 10.1103/PhysRevLett.108.225301P
5、ACS numbers: 67.85.?d, 03.75.Mn, 05.30.Rt, 71.70.Ej A large number of papers have been recently devoted to the theoretical study of artifi cial gauge fi elds in ultracold atomic gases (for a recent review see, for example, 1). First experimental realizations of these novel confi gura- tions have bee
6、n already become available 2,3. This fi eld of research looks very promising from both the theoretical and experimental point of view, due to the possibility of realizing exotic confi gurations of nontrivial topology 4, withtheemergenceofnewquantumphasesinbothbosonic 5 and fermionic 6,7 gases, and t
7、he possibility to simu- late electronic phenomena of solid state physics. In the case of Bosegases a keyfeature of these new systems is the possibility of revealing Bose-Einstein condensation in single-particle states with nonzero momentum. By tuning the Raman coupling between two hyperfi ne states
8、of 87Rb atoms, the authors of 3 have reported the fi rst experimental identifi cation of the new quantum phases exhibited by a spin-orbit coupled Bose-Einstein condensa- tion. Important features of the resulting phases were antici- pated in the paper by Ho and Zhang 8 and discussed in the same exper
9、imental paper3. The purpose of thisLetter isto provide a theoretical description of the phase diagram cor- responding to the spin-orbit coupled Hamiltonian employed in 3. We point out the occurrence of an important density dependence in the phase diagram which shows up in the appearance of a tricrit
10、ical point that, to our knowledge, has never been predicted for such systems. We will consider the mean-fi eld energy functional (for simplicity we set m 1) Eca;cb Z d3r ? c? a c? b ?h 0 ca cb ! gaa 2 jcaj4 gbb 2 jcbj4 gabjcaj2jcbj2 ? (1) describing an interacting spin-1=2 Bose-Einstein conden- sate
11、 at T 0, wherecaandcbare the condensate wave functions relative to the two spin components interacting with the coupling constants gij 4?aij, with aijthe cor- responding s-wave scattering lengths, and h0 1 2 px? k0?z2 p2 ? ? 2 ?x ? 2 ?z Vext(2) is the single-particle Hamiltonian characterized by equ
12、al contributionsofRashba9and Dresselhaus10spin-orbit couplings and a uniform magnetic fi eld in the x-z plane. In Eq. (2) ? is the Raman coupling constant accounting for the transition between the two spin states, k0is the strength associated with the spin-orbit coupling fi xed by the momen- tum tra
13、nsfer of the two Raman lasers, ? fi xes the energy differencebetween the two single-particlespinstates,?iare the usual 2 ? 2 Pauli matrices, while Vextis the external trapping potential. In the fi rst part of the Letter we will consider uniform confi gurations, neglecting the effect of the trapping
14、poten- tial (Vext 0) and assume a spin symmetric interaction with gaa gbb? g and ? 0. The effect of asymmetry will be discussed afterwards. The ground state condensate wave function will be determined using a variational pro- cedure based on the following ansatz for the spinor wave function: ca cb ?
15、 ffi ffi ffi ffi N V s ? C1 cos? ?sin? ? eik1xC2 sin? ?cos? ? e?ik1x ? (3) where N is the total number of atoms, V is the volume of the system. For a given value of the average density n N=V, the variational parameters are then C1, C2, k1, and ?. Their values are determined by minimizing theenergy(1
16、)withthenormalizationconstraint P ia;b R d3rjcij2 N (i.e., jC1j2 jC2j2 1). Mini- mization with respect to ? yields the general relationship ? arccosk1=k0 =2 (0 ? ? ? ?=4), fi xed by the single- particle Hamiltonian (2). Once the other variational pa- rameters are determined, one can calculate key ph
17、ysical quantities like, for example, the momentum distribution PRL 108, 225301 (2012) PHYSICALREVIEWLETTERS week ending 1 JUNE 2012 0031-9007=12=108(22)=225301(5)225301-1? 2012 American Physical Society accounted for by the parameter k1, the longitudinal and transverse spin polarization of the gas h
18、?zi k1 k0 jC1j2? jC2j2;h?xi ? ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi ffiffi k2 0? k21 q k0 (4) and the density nx n 2 6 41 2jC1C2j ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi k2 0? k21 q k0 cos2k1x ? 3 7 5;(5) where ? is the relative phase between C1and C2. The ansatz (3
19、) exactly describes the ground state of the single- particle Hamiltonian h0(ideal Bose gas). In this case, for ? ? 2k2 0, the energy, as a function of k1, exhibits two minima located at the values ?k0 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi
20、 1 ? ?2=4k4 0 q and the ground stateis degenerate,the energybeingindependentof the actual values of C1and C2. For ? 2k2 0 the two minima disappear and all the atoms condense into the zero momentum state k1 0. The same ansatz is well suited to discuss the role of interactions. By inserting (3) into (
21、1), we fi nd that the energy per particle “ E=N takes the form “ k2 0 2 ? ? 2k0 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi k2 0? k21 q ? F? k2 1 2k2 0 G11 2?(6) where we have defi ned the dimensionless parameter ? jC1j2jC2j2(0 ? ? ? 1=4), and the function F? k2 0? 2G2 4G1 2G2? (7
22、) with the interaction parameters G1 ng gab=4, G2 ng ? gab=4. The variational parameters to minimize the energy are then k1and ?. Let us fi rst consider minimization with respect to k1. If ? 2F? the energy (6) is an increasing function of k1 and the minimum takes place at k1 0. If instead ? 0 case,
23、the system will be always in the phase (I) for small values of the Raman coupling constant ?. If the condition k2 0 4G2 4G2 2 G1 (10) is satisfi ed, the systems will exhibit a phase transition (I) to (II) at the frequency ?III 2 ? k2 0 G1k20? 2G2 2G2 G1 2G2 ?1=2 :(11) This generalizes the result der
24、ived in 8, which corre- sponds to the low density (or weak coupling) limit of (11), i.e., G1;G2? k2 0. The transition frequency in this limit approaches the density independent value ?III LD 2k2 0 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi
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