《Quantum Tricriticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates.pdf》由会员分享，可在线阅读，更多相关《Quantum Tricriticality and Phase Transitions in Spin-Orbit Coupled Bose-Einstein Condensates.pdf（5页珍藏版）》请在三一文库上搜索。
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
25、ffi 2?=1 2? q (12) where we have introduced the dimensionless interaction parameter ? G2=G1 g ? gab=g gab. By further increasing ?, the system will enter the phase (III) at the frequency ?IIIII 2k2 0? 2G2: (13) This result, in the limit G2? k2 0, was also discussed in 11. If instead the condition (1
26、0) is not satisfi ed, the transition will occur directly from the phase (I) to (III) at the frequency ?IIII 2k2 0 G1 ? 2k20 G1G1?1=2: (14) In the strong coupling limit G1? k2 0 (14) approaches the constant value k2 0. PRL 108, 225301 (2012) PHYSICALREVIEWLETTERS week ending 1 JUNE 2012 225301-2 The
27、critical point where the phase (II) disappears is fi xed see Eq. (10) by the condition Gc 1 k2 0=4?1 ?, cor- responding to the critical value nc k2 0=2?g (15) for the density. If n nc, only one phase transition (IIII) can take place. In Fig. 1, we plot the momentum k1, the energy per particle E=N, t
28、he transverse and longitudinal spin polar- izations h?xi,andjh?zijasafunctionof?forn nc(right column). In addition to the results for the ground state (open circles), we also show the various quantities for the three phases (colored lines). Figures (a)(d) reveal the emergence of the phase transi- ti
29、ons (III) and (IIIII), while in (e)(h) there is only the transition (IIII). The fi gures also show that the transitions (III) and (IIII) are accompanied by a jump in k1see (a) and (e) and consequently in h?xi see (c) and (g). In particular the jump in k1associated with the transition (IIII) is sizab
30、le and should be easily observable in experi- ments. On the other hand only the transition (III) is accompanied by a jump in the longitudinal spin polariza- tion jh?zij. The transition (IIIII) is instead characterized by a continuous behavior of the relevant physical parame- ters. The experimental c
31、onditions of 3 correspond to values of the average density n much smaller than nc, so the jump in k1could not be detected because it is too small at the transition (III). On the other hand the occur- rence of this phase transition was clearly revealed by the analysis of the spin distribution after t
32、ime of fl ight (see Fig. 2c of 3). InFig.2weshowthephasediagramforthethreedifferent phases. The value of the spin polarization jh?zij and k1are reported in (a) and (b), respectively. The transition lines separating different phases merge at a tricritical point at n nc. The value of jh?zij always van
33、ishes for n nc. However the phase transition (IIII) is well identifi ed by the behavior of the momentum k1. The parameters employed in Fig.2correspondtoratherlargevaluesofthecriticaldensity. More accessible values of nccan be obtained employing smaller values of k0or larger values of ? using differe
34、nt spin states or different atomic species. Reducing the value of k0 would also have the advantage of increasing the spatial separation betweenthe fringes in thestripephase (I), thereby making their experimental detection easier. The description of the quantum phases carried out in the present work
35、is based on the mean-fi eld picture which ignores the role of quantum fl uctuations. In ordinary Bose-Einstein condensed gases the mean-fi eld approach is justifi ed if the gas parameter na3is small. The spin-orbit term in the single-particle Hamiltonian (2) is expected to emphasize the role of quan
36、tum fl uctuations. In particular when the phase (III) approaches the phase (II), quantum fl uctuations are enhanced and, for large values of k0, the usual Bogoliubov ffi ffi ffi ffi ffi ffi ffiffi na3 p dependence of the quantum deple- tionof the condensate is increased bythe factor k2 0=gn1=4. The
37、effect is, however, small for the current values of the spin-orbit parameters. 012 0 0.5 1 k1 / k0 (a) 012 0.5 0 0.5 E / N k0 2 (b) 012 1 0.5 0 x (c) 012 0 0.5 1 | z | / k0 2 (d) n n(c) FIG. 1 (color online).k1, energy per particle E=N, transverse and longitudinal spin polarization h?xi and jh?zij a
38、s a function of ?. Red dashed lines: stripe phase k1? 0 and ? 1=4; blue dotted lines: separated phase k1? 0 and ? 0; green solid lines: zero momentum phase k1 0; open circles: ground state. Theparameters:G1=k2 0 0:2, G2=k2 0 0:05 (a)(d), G2=k2 0 0:16 (e)(h). FIG. 2 (color online).Spin polarization j
39、h?zij (a) and k1=k0(b) as a function of ? and density n=ncin three different phases with G2 0. The white solid lines represent the phase transition (III), (IIIII) and (IIII). The parameters: g 100aB, where aB is the Bohr radius, ? 0:0012, k2 0 2? ? 80 Hz, correspond- ing to nc 4:37 ? 1015cm?3. PRL 1
40、08, 225301 (2012) PHYSICALREVIEWLETTERS week ending 1 JUNE 2012 225301-3 Let us now discuss the effect of the trap. In order to simplify the analysis we have considered harmonic trapping with frequency !0only along the x axis. Without interaction, one can calculate the ground state using a similar v
41、ariation ansatz, replacing the plane waves in (3) by the functions e?ik1xe?!0x 2=2, corresponding, in the absence of the gauge fi eld, to the usual harmonic oscillator Gaussians. The energy per particle is easily calculated and reads: “ !0 2 k2 0? k21 2 ? ? 2k0 ffi ffi ffi ffi ffi ffi ffi ffi ffi ff
42、i ffi ffi ffi ffi ffiffi k2 0? k21 q ? C? 1C2 C?2C1? k2 1 2k2 0 e?k 2 1=!0: (16) The ground state can be found by minimizing “ with respect to k1, C1, and C2with the normalization constraint. The fi rst term in (16) is just the zero point energy due to the presence of the trap. The following two ter
43、ms are the same as for the uniform case without interactions, i.e., (6) with G1 G2 0. The last term shows the effect of the trap, fi xingthe relative phasebetween thecoeffi cients C1andC2 inthegroundstate. Consequentlythedegeneracyoccurring in the uniform case will be lifted even in the absence of i
44、nteractions (where ? 0). Physically this is the conse- quence of the nonorthogonality of the two Gaussian states. According to (16), for k1? 0, the system prefers to stay in the spin mixed phase, and exhibits density modulation in space even without interactions. On the other hand, the interaction i
45、s crucial for the appearance of the phase sepa- rated confi guration. Since the last term of (16) scales exponentially, the effect of the trap is weak for k2 1 ? !0, and becomes more and more important when k2 1is compa- rable to !0. To describe the role of the interaction we implement the mean-fi e
46、ld approximation by solving numerically the Gross-Pitaevskii equation for the condensate wave func- tion using the gradient method in the same 1D trapping conditions. We fi nd that the properties discussed in the fi rst part of the work for the uniform system almost hold in the trapped case. In Fig.
47、 3 we show an example of the numeri- cal calculation. The spin polarization as a function of ?, in the presence of trapping (red solid line), is compared with our analytical results for the uniform case (blue dashed line), using the density in the center of the trap. There is good agreement between
48、the two curves. We have checked that a similar good agreement is ensured also for larger values of the interaction parameter n=nc , confi rm- ing thegeneral validityof the ansatz (3) forthe spinorwave function employed in the fi rst part of the Letter. We fi nally discuss the case ? ? 0 and gaa? gbb
49、, cor- responding to broken spin symmetry. In general one can introduce three interaction parameters: G1 ngaa gbb 2gab=8, G2 ngaa gbb? 2gab=8, and G3 ngaa? gbb=4. In the case of 87Rb atoms, the scattering lengths relative to the spin states jF 1;mF 0i and jF1;mF?1i are usually parameterized as aaa?c0, a