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Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates
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Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates
Qizhong Zhu1, Chuanwei Zhang2 and Biao Wu1(a)
1 International Center for Quantum Materials, Peking University - 100871, Beijing, China 2 Department of Physics and Astronomy, Washington State University - Pullman, WA, 99164 USA
received 5 July 2012; accepted in final form 20 November 2012 published online 10 December 2012
PACS 03.75.Kk – Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow PACS 03.75.Mn – Multicomponent condensates; spinor condensates PACS 05.30.Jp – Boson systems
Abstract – We study the superfluidity of a spin-orbit coupled Bose-Einstein condensate (BEC) by computing its Bogoliubov excitations, which are found to have two branches: one is gapless and phonon-like at long wavelength; the other is typically gapped. These excitations imply superfluidity that has two new features: i) due to the absence of the Galilean invariance, one can no longer define the critical velocity of superfluidity independent of the reference frame; ii) the superfluidity depends not only on whether the speed of the BEC exceeds a critical value, but also on cross- helicity that is defined as the direction of the cross-product of the spin and the kinetic momentum of the BEC. Copyright c EPLA, 2012
Superfluidity was first discovered in 1938 and has fasci- To put our study into perspective, we briefly review nated physicists ever since. This interesting phenomenon Landau’s theory for a superfluid without SOC. Consider was explained by Landau [1], whose theory has been such a superfluid flowing in a tube. With the Galilean very successful in explaining many important properties transformation, Landau found that the excitation of this of superfluids. However, Landau’s theory of superfluidity flowing superfluid is related to the excitation of a motion- may be facing challenges brought by the recent experi- less superfluid as [1] mental realization of artificial gauge fields for ultracold εv(p)=ε0(p)+p · v, (1) bosonic atoms [2–5]. When the artificial gauge field is non- ε p Abelian [6–8], it is effectively spin-orbit coupling (SOC). where 0( ) is the excitation for a stationary superfluid p SOC has played a crucial role in many exotic phenomena and is the momentum of the excitation. For phonon ε p c|p| ε p such as topological insulators [9]. However, in superfluids, excitation 0( )= , the excitation v( ) can be nega- |v| >c c the SOC is generally absent and its effects have remained tive only when . Therefore, the speed of sound is largely unexplored. Note that this issue is not limited to the critical velocity beyond which the flowing superfluid ultracold atoms since the Bose-Einstein condensation with loses its superfluidity and suffers viscosity. We switch to SOC also exists for excitons in semiconductors [10,11]. a different reference frame, where the superfluid is at rest There have been some theoretical works, where many while the tube is moving. It is apparent to many that these interesting properties of spin-orbit coupled Bose-Einstein two reference frames are equivalent so that the superfluid condensates (BECs) are explored [11–23]. For instance, it will be dragged along only when the tube speed exceeds c was shown in ref. [11] that SOC can lead to unconventional the speed of sound . However, this equivalence is based on Bose-Einstein condensation with the breaking of time- that the superfluid is invariant under the Galilean trans- reversal symmetry. Later, a stripe phase that breaks formation. As SOC breaks the Galilean invariance of the rotational symmetry was found [12,13]. In this letter we system [24], we find that these two reference frames are no concentrate on the superfluidity of the spin-orbit coupled longer equivalent as shown in fig. 1: the critical speed for BEC. scenario (a) is different from the one for scenario (b). For easy reference, the critical speed for (a) is hereafter called the critical flowing speed and the one for (b) the critical (a)E-mail: [email protected] dragging speed.
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We focus on the homogeneous case V (r) = 0 despite that the BEC usually resides in a harmonic trap in experiments. The primary reason is that the superfluidity can be discussed more clearly in the homogeneous case, and be compared directly with the conventional superfluidity of a spinless bosonic system. In addition, the results in the homogeneous case can be adopted to understand the superfluidity in more complicated situations with the local density approximation. We also limit ourselves to the case C1 >C2, where the system is stable against phase separation [12,15]. In the following discussion, for simplicity, we set = m = 1 and ignore the non-essential z-direction, treating the system as two-dimensional. This does not impair the validity of our model. In the following Fig. 1: (a) A superfluid with spin orbit-coupling moves while we assume the BEC moves in the y-direction, and the the tube is at rest. (b) The superfluid is dragged by a tube critical velocity is found to be not influenced by the v v moving at speed . (c) An impurity moves at in the SOC excitation in the z-direction. superfluid. The reference frame is the lab. The two-way arrow The Gross-Pitaevskii equation obtained from the Hamil- indicates the equivalence between different scenarios and the tonian (2) has plane-wave solutions arrow with a bar indicates the non-equivalence. iθ Ψ1 1 e k ik·r−iµ(k)t φk = = √ e , (3) Ψ2 2 −1 Our study of the superfluidity for a BEC with SOC is 2 based on the computation of the elementary excitations where tan θk = kx/ky, µ(k)=|k| /2 − γ|k| +(C1 + C2)/2. using the Bogoliubov equation. We calculate how the The solution φk is the ground state of the system when elementary excitations change with the flow speed and |k| = γ. There are another set of plane-wave solutions, manage to derive from these excitations the critical speeds which have higher energies and are not relevant to our for the two different scenarios shown in fig. 1(a), (b). We discussion. find that there are two branches of elementary excitations The plane-wave solution φk represents a BEC flow for a BEC with SOC: the lower branch is phonon-like with the velocity v = k − γkˆ( kˆ = k/|k|). This velocity at long wavelengths and the upper branch is generally is essentially the kinetic momentum of the BEC, which gapped. Careful analysis of these excitations indicates is different from the conjugate momentum k due to the that the critical flowing velocity for a BEC with SOC presence of SOC. For example, the ground state has a non- (fig. 1(a)) is non-zero while the critical dragging speed is zero conjugate momentum k, but its kinetic momentum zero (fig. 1(b)). This shows that critical velocity depends (or velocity) is zero. Besides its velocity, the BEC flow on the reference frame for a BEC with SOC and, probably, described by φk has another feature, the direction of the for any superfluid that has no Galilean invariance. spin, which has to be included for a complete description In addition, we find that the properties of a flow of of the flow. For example, the flow at k =3γ/2ˆy has the BEC with SOC are also related to its spin direction. We same velocity γ/2ˆy as the flow at k = −γ/2ˆy. However, characterize this spin direction with cross-helicity, which they have different spin directions. With the Rashba- is defined as the cross-product of the spin and the kinetic type SOC, the spin direction of the eigenstate is always momentum of the flow. A BEC flow with Rashba SOC is perpendicular to the velocity and has only two choices. always unstable if its cross-helicity is negative. As a result, it is sufficient and also very convenient to Model: We consider a BEC with pseudospin 1/2and use w = sign[(vˆ × σˆ) · zˆ] to denote the spin direction. This the Rashba SOC. The system can be described by the variable w, which is either 1 or −1, is the cross-helicity Hamiltonian [12,15,25,26] mentioned in the introduction. Note that the usual helicity in literature is defined as the inner product of the spin and 2 2 ∗ ∇ the momentum. H = dr Ψ − + V (r) Ψσ σ 2m Critical velocities: We study first the scenario depicted σ=1,2 in fig. 1(a), where the BEC flows with a given velocity. ∗ ∗ +γ [Ψ1(ipˆx +ˆpy)Ψ2 +Ψ2(−ipˆx +ˆpy)Ψ1] Since the system is not invariant under the Galilean trans- formation, we cannot use eq. (1) to find the excitations for C1 4 4 2 2 + |Ψ1| + |Ψ2| + C2|Ψ1| |Ψ2| , (2) the flowing BEC from the excitation of a stationary BEC. 2 We have to compute the excitations directly. This can be done by computing the elementary excitations of the state where γ is the SOC constant, C1 and C2 are interaction φk with the Bogoliubov equation for different values of k. strengths between the same and different pseudospin Without loss of generality, we choose k = kyˆ with states, respectively. k>0. Following the standard procedure of linearizing the
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Fig. 2: (Color online) Elementary excitations of a BEC flow with the SOC. (a) k =1;(b) k =2.5; (c) k =4. C1 = 10, C2 =4, γ =1.
Gross-Pitaevskii: equation [27,28], we have the following Bogoliubov equation: u1 u1 u2 u2 L = ε , (4) v1 v1 v2 v2 where + 1 1 Hk b12 − C1 − C2 2 2 b H+ − 1 C − 1 C 21 k 2 2 2 1 L = (5) 1 C 1 C H− b 2 1 2 2 k 34 1 1 − Fig. 3: Excitations along the x-axis (the first row) and y-axis C2 C1 b43 Hk 2 2 (the second row) at different values of k. (a1), (a2): k =1;(b1), k k C C γ 2 2 2 (b2): = 3; (c1), (c2): =4. 1 = 10, 2 =4, =1. ± qx+(qy ±k) C1 k with Hk = ± 2 ± A, A = 2 − 2 + γk, b12 = C2 C2 −γ(iqx + qy + k)+ 2 , b21 = γ(iqx − qy − k)+ 2 , b34 = C2 C2 |k| <γ γ(iqx − qy + k) − 2 ,andb43 = −γ(iqx + qy − k) − 2 . imaginary for BEC flows with . This means that As usual, there are two groups of eigenvalues and all the flows with |k| <γ are dynamically unstable and only the ones whose corresponding eigenvectors satisfy therefore do not have superfluidity. For other flows with 2 2 |ui| −|vi| =1 (i =1, 2) are physical. k γ, the excitations are always real and they are plotted For comparison, we consider the case without SOC. This in fig. 2. One immediately notices that the excitations have is to put γ = 0 and reduce our system to a two-component two branches, which are in contact with each other at a BEC system, which is well studied in the literature [29]. single point. A closer examination shows that the upper For this case, L can be diagonalized analytically and there branch is gapped in most of the cases while the lower are two branches of excitations, branch has phonon-like spectrum at large wavelength. These features are more apparent in fig. 3, where only C ± C q4 x y 1 2 2 the excitations along the -axis and -axis are plotted. ε±(q)=qyk + q + . (6) 2 4 In fig. 2(c) and fig. 3(c2), we notice that some of the exci- tations in the upper branch are negative, indicating that These results show that the system at the ground state the underlying BEC flow is thermodynamically unstable k C C / ( = 0) has two different speeds of sound, ( 1 + 2) 2 and has no superfluidity. In fact, our numerical computa- C − C / ε and ( 1 2) 2. Since the excitation − becomes nega- tion shows that there exists a critical value kc: when k>kc tive only when k> (C1 − C2)/2, the critical flowing either part of the upper branch of excitations or part of velocity in this case is (C1 − C2)/2. When C2 = 0, these the lower branch or both become negative. This means two branches of excitations merge into one and the critical that the flows described by the plane-wave solution φk,− velocity is C1/2, which is well known and was confirmed with |k| >kc suffer Landau instability and have no super- in a BEC experiment [30]. fluidity. Combined with the fact that the flows with |k| < In general, there are no simple analytical results. We γ are dynamically unstable, we can conclude that only have numerically diagonalized L to obtain the elementary the flows with γ |k| kc have superfluidity. Physically, excitations. We find that part of the excitations are these super-flows have speeds smaller than vc = kc − γ and
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2.8 3 For the special case γ =0,wehave 2.8 4 2.6 C1 ± C2 q ε±(q)= q2 + . (10) 2.6 0 2 4 c c
v 2.4 v C =11,C =3 2.4 C =14,C =6 1 2 1 2 From eq. (9), we obtain the critical dragging speed 2.2 C −C 2.2 1 2 , which is the same as the critical flowing speed. (a) (b) 2 γ ε q 2 2 When = 0, the excitations 0( ) also share two 0 2 4 6 0 2 4 6 γ γ branches. Along the x-axis, these two branches are q4 ± 2 x 2 4 2 6 Fig. 4: Critical flowing velocity vc (solid line) of a BEC as a ε (qx)= s1 + s2q + ± t1 + t2q + t3q + γ q , 0 x 4 x x x function of the SOC parameter γ.(a)C1 > 3C2;(b)C1 < 3C2. (11) 4 2 2 1 2 where s1 =2γ + γ (C1 − C2), s2 =2γ + C1, t1 = s , cross-helicity w = 1. We have plotted how the critical flow- 2 1 t s s t s γ2 C / 2 y ing velocities vary with the SOC parameter γ in fig. 4. 2 =2 1 2 ,and 3 =2 1 + + 2 2 . Along the -axis, We find two different asymptotic behaviors for the the excitations of the ground state are γ critical velocity when gets large. As shown in fig. 4, 4 C1 C2 qy vc − + 2 the critical flowing velocity becomes a constant, = ε0 (qy)= qy + , (12) (C1 + C2)/2, beyond a threshold value in the parameter 2 4 regime√ C1 > 3C2 while it approaches an asymptotic value C q4 + 2 2 y C1 − C2 in the regime C1 < 3C2. ε0 (qy)=2γqy + 2s1 + s2 − qy + . (13) 2 4 We turn to another reference frame illustrated in fig. 1(b), where the BEC can be viewed as being dragged + When γ>0, the upper branch ε (qx) is always parabolic by a moving tube. To simplify the discussion, we replace √ 0 at small qx with a gap 2s1. When expanded to the second the moving tube with a macroscopic impurity moving order of qx, the lower branch has the following form: inside the BEC as shown in fig. 1(c). Correspondingly, the question as to whether the BEC will be dragged along − 2 C1 + C2 by the moving tube is replaced by an equivalent question: ε0 (qx) ≈ qx . (14) 8γ2 will the impurity experience any viscosity? Suppose that − the moving impurity generates an excitation in the BEC. This shows that ε0 (qx) is parabolic at long wavelengths According to the conservations of both momentum and instead of linear as usually expected for a boson system. energy, we should have This agrees with the results in ref. [17]. This parabolic excitation has a far-reaching consequence: according to m0vi = m0vf + q, (7) eq. (9), the critical dragging velocity vc is zero, very different from the critical flowing velocity for a BEC 2 m v2 moving in a tube. This shows that the critical velocity for a m0vi 0 f = + ε0(q), (8) BEC with SOC is not independent of the reference frame, 2 2 in stark contrast with a homogeneous superfluid without where m0 is the mass of the impurity, vi and vf are the SOC. This surprising finding of course has the root in the initial and final velocities of the impurity, respectively, fact that the BEC described by the SOC Hamiltonian (2) and ε0(q) is the excitation of the BEC at k = γ.Ifthe is not invariant under the Galilean transformation [24]. excitations were purely phonons, i.e., ε0(q)=c|q|, these Rashba and Dresselhaus SOC : We have also investigated two conservations would not be satisfied simultaneously the superfluidity with the general form of SOC, which is a when v ≈|vi|≈|vf |
50003-p4 Exotic superfluidity in spin-orbit coupled Bose-Einstein condensates critical dragging velocity is the smaller of the above two slopes, which are both non-zero. Secondly, the BEC with negative cross-helicity can also be stable; as a result, there is a different critical flowing speed for either cross-helicity. For the Rashba type (α = β), the critical flowing speed for negative cross-helicity is always zero since the BEC is always dynamically unstable for negative cross-helicity. Experimental observation: Spin-orbit coupled BECs have been realized recently by three different groups [3–5] through coupling ultracold 87Rb atoms with laser fields. The strength of the SOC in the experiments can be tuned by changing the directions of the lasers [3–5] or through the fast modulation of the laser intensities [33]. The interaction between atoms can be adjusted by varying the Fig. 5: In (a), (b), (d), the periodic superfluid and the lattice confinement potential, the atom number or through the are locked in and move together with respect to the obstacle. In Feshbach resonance [34]. For the scenario in fig. 1(b), one (c), the lattice is gradually accelerated to speed v. The reference can use a blue-detuned laser to mimic the impurity for frame is the lab. The scenario in (c) is not equivalent to the the measurement of the critical dragging speed similar to scenarios in (a), (b), (d) as there is relative motion between the experiment reported in ref. [30]. For the scenario in the lattice and the fluid in (c). fig. 1(a), there are two possible experimental setups for measuring the critical flowing speed. In the first one, one The presence of a lattice does put a new twist into generates a dipole oscillation similar to the experiment the system. One can gradually accelerate the lattice to in ref. [5] but with a blue-detuned laser in the middle of a certain velocity and see how the superfluid changes the trap. The second one is more complicated: At first, (fig. 5(c)). There exists a critical velocity for the lattice one generates a moving BEC with a gravitomagnetic beyond which the system loses its superfluidity. This trap [35], then uses Bragg spectroscopy [36,37] to measure critical velocity for lattice is different from the critical the excitations of the moving BEC, from which the super- velocity of a moving defect and is called trawler critical fluidity can be inferred. For the typical atomic density of velocity in ref. [41]. This type of critical velocity was 1014–1015 cm−3 achievable in current experiments [30], considered theoretically in ref. [27] and demonstrated and the experimental setup in ref. [3], the critical flowing experimentally in ref. [42]. velocity is 0.2–0.6mm/s, while the critical dragging Note that a more detailed version of the above discus- velocity is still very small, about 10−3–10−2 mm/s. To sion for the superfluidity of a periodic superfluid can be further magnify the difference between the two critical found in ref. [41]. However, an error was made in ref. [41]: velocities, one can use the Feshbach resonance to tune a non-zero critical velocity for the scenarios shown in the s-wave scattering length. fig. 5(a), (b), (d) was predicted. Perspective: This is not the first BEC system, where With BEC, we now have many types of superfluids, there are two different critical velocities of superfluidity. which are different from the conventional spinless and The BEC in an optical lattice [38], a superfluid with its homogeneous superfluid helium. We expect more of density periodically modulated, is another system of two this kind of new superfluids to appear in the future, different critical velocities. Supersolid helium may also be which will surely extend and enrich our understanding of regarded as a periodic superfluid [39]. superfluidity. We consider first the case when the BEC is locked ∗∗∗ with the lattice and they move together as shown in fig. 5(a), (b), (d). Since this system is invariant under We acknowledge helpful discussion with Hongwei the Galilean transformation, the two scenarios depicted Xiong. QZ and BW were supported by the NBRP of in fig. 5(a), (b) are equivalent and Landau’s argument China (2012CB921300, 2013CB921900) and the NSF of is still applicable. The caveat is that the critical velocity China (10825417, 11274024), and the RFDP of China is always zero no matter how the elementary excitation (20110001110091). CZ was supported by DARPA-YFA of the superfluid looks. The key reason is that the (N66001-10-1-4025), ARO (W911NF-09-1-0248), and p momentum of the excitation is not well defined due NSF-PHY (1104546). to the presence of the lattice: two momenta which differ by a reciprocal lattice vector are equivalent. This result REFERENCES has been verified in extensive computations for Cerenkov radiation in a periodic medium [31]. Recently, a similar [1] Lifshitz E. M. and Pitaevskii L. P., Statistical Physics, numerical calculation was done specifically for a moving Part II (Pergamon Press, Oxford.) 1980. defect in a BEC in an optical lattice and zero critical [2] Lin Y. J., Compton R. L., Jimenez-Garcia K., Porto velocity is found [40]. J. V. and Spielman I. B., Nature, 462 (2009) 628.
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