Spin-orbit coupled Bose-Einstein condensates
We consider a many-body system of pseudo-spin- bosons with spin-orbit interactions, which couple the momentum and the internal pseudo-spin degree of freedom created by spatially varying laser fields. The corresponding single-particle spectrum is generally anisotropic and contains two degenerate minima at finite momenta. At low temperatures, the many-body system condenses into these minima generating a new type of entangled Bose-Einstein condensate. We show that in the presence of weak density-density interactions the many-body ground state is characterized by a twofold degeneracy. The corresponding many-body wave function describes a condensate of “left-” and “right-moving” bosons. By fine-tuning the parameters of the laser field, one can obtain a bosonic version of the spin-orbit coupled Rashba model. In this symmetric case, the degeneracy of the ground state is very large, which may lead to phases with nontrivial topological properties. We argue that the predicted new type of Bose-Einstein condensates can be observed experimentally via time-of-flight imaging, which will show characteristic multipeak structures in momentum distribution.
pacs:03.75.Ss, 05.30.Jp, 71.70.Ej, 72.25.Rb
Bose-Einstein condensation is an old and thoroughly studied quantum phenomenon, where a many-body system of bosons undergoes a phase transition in which a single-particle state becomes macroscopically occupied. This phenomenon has been observed in condensed matter systems and more recently in experiments on cold atomic gases,Leggett (2001); Dalfovo et al. () which provided a unique avenue to visualize the formation of the Bose-Einstein condensate (BEC). Bose-Einstein condensation is a phase transition driven mostly by the statistics of the underlying bosonic excitations and not by interactions. The statistics of basic particles are determined by the particle spin via the fundamental Pauli spin-statistics theorem:Pauli () The spin must be integer for bosons and half-integer for fermions.
In this paper, we discuss a cold atomic system of multi-level bosons moving in the presence of spatially-varying laser fields, which give rise to an emergent pseudo-spin- degree of freedom for the bosons. We emphasize from the outset that the symmetry operations in the pseudo-spin space are not related to real-space rotations and thus there is no contradiction between the existence of the pseudo-spin- bosons and the fundamental Pauli theorem. To “create” the pseudo-spin- bosons, one can use the experimental setup, proposed in Refs. Juzeliunas and Ohberg, 2004; Jaksch and Zoller, 2003; Ruseckas et al., 2005; Osterloh et al., 2005; Zhu et al., 2006; Satija et al., 2006; Stanescu et al., , in which three degenerate hyperfine ground states , , are coupled to an excited state by spatially varying laser fields. This “tripod scheme” leads to the appearance of a pair of degenerate dark states, spanning a subspace which is well separated in energy from two nondegenerate bright states. The coupling between the dark and the bright states is very weak and will be neglected (adiabatic approximation). The parameter which labels the dark states plays the role of a pseudo-spin index. This emergent pseudo-spin degree of freedom is similar to that studied recently in the context of spinor condensates.Stenger et al. (); Ashhab and Leggett () In particular, various aspects of the pseudo spin- boson physics were addressedSiggia and Ruckenstein (); Fazekas and Entel (); Hall et al. (); Lewandowski et al. (); Moore and Meystre (); Sorensen et al. (); Pu and Meystre (); Helmerson and You (); Raghavan et al. (); Erhard et al. () using two hyperfine states to support the internal degree of freedom associated with the pseudo spin. The key distinctive feature of the systems studied in this paper is that the single-particle Hamiltonian projected onto the sub-space of the degenerate dark states generally possesses a non-Abelian gauge structure. I.e., the kinetic term of the effective Hamiltonian has the form , where is a matrix in the pseudo-spin space. In a recent Letter,Stanescu et al. () we pointed out that under certain conditions this non-Abelian gauge structure is equivalent to a spin-orbit interaction. To understand the nature of this interaction, we note that the dark states are eigenstates of an atom at rest. Once the atom moves in the spatially modulated laser field, the dark state label, i.e., the pseudo-spin index, is not a good quantum number and the pseudo-spin starts to precess about the direction of the momentum. This coupling between the internal degree of freedom associated with the dark state subspace and the orbital movement of the particle represents the spin-orbit interaction. The spin-orbit coupling parameters can be adjusted by changing the properties of the spatially modulated light beams. These conclusions are based entirely on single-particle physics; the particle statistics play no role. Below, we consider a many-particle system of bosons within this tripod scheme. Due to spin-orbit coupling, the degenerate ground states of the system correspond to non-zero momenta, leading to a new type of BEC, the spin-orbit coupled Bose-Einstein condensate (SOBEC).
The article is organized as follows: In Section II we introduce our model and discuss the properties of a non-interacting many body system of bosons with spin-orbit interactions. We find that, in general, the single-particle spectrum is characterized by two degenerate minima at finite momenta and we determine the transition temperature for the bosons condensing into these minima. In Section III we study the effects of density-density interactions using a generalized Bogoliubov transformation (Subsection III.1). We show the quasiparticle excitation spectrum contains an anisotropic free particle component and an anisotropic sound similar to the conventional Bogoliubov phonon. By calculating the energy of the condensate, we find that for a system of bosons the -fold degeneracy of the non-interacting ground state is reduced by the interactions to a two-fold degeneracy corresponding to “left-” or “right-moving” particles. The corresponding many-body wavefunction describes a NOON state,Sanders () suggesting that future studies of the SOBEC state in the context of quantum entanglement and quantum interference are highly relevant. For completeness, we also derive the Gross-Pitaevskii equations for the spin-orbit coupled condensate (Subsection III.2). Linearizing the coupled non-linear equations in the vicinity of a stationary solution leads to a spectrum of excitations that reproduces the generalized Bogoliubov result. A possible experimental signature of the new type of SOBEC is described in Section IV. We argue that a SOBEC can be observed via time-of-flight imaging, which will show a characteristic multi-peak structure of the density profiles. We demonstrate that such a measurement generates distinct outputs for “left-” and “right-moving” condensates and thus can be viewed as a measurement of a qubit. A summary of the paper along with our conclusions are presented in Section V.
Ii Spin-orbit interacting Hamiltonian and single-particle spectrum
We start with the following many-body Hamiltonian describing spin-orbit coupled bosons,
where and are the creation and annihilation operators for bosons in the state with momentum and pseudo-spin , are the Pauli matrices in the pseudo-spin space, and the parameters and characterize the strength and anisotropy of the spin-orbit coupling. We reiterate that this type of spin-orbit-coupled Hamiltonian (1) will appear within the recently proposed tripod schemeJuzeliunas and Ohberg (2004); Jaksch and Zoller (2003); Ruseckas et al. (2005); Osterloh et al. (2005); Zhu et al. (2006); Satija et al. (2006); Stanescu et al. () in which three hyperfine ground states of an atom , , and are coupled to an excited state via spatially modulated laser fields. The underlying laser-atom Hamiltonian is
where is a constant detuning to the excited state and the Rabi frequencies consistent with the realization of an effective spin-orbit interaction can be taken as , , and , with , , and being constants (see, e.g., Refs. Ruseckas et al., 2005 and Stanescu et al., for details). Diagonalizing the atom-laser Hamiltonian (2) via a position-dependent rotation , with and , generates a pair of degenerate dark states
with and , and two non-degenerate bright states The pseudo-spin- structure emerges when the problem is projected onto the subspace spanned by the pair of degenerate dark states.Stanescu et al. () Applying the position-dependent rotation to the kinetic energy term in the Hamiltonian generates a coupling of the pseudo-spin to momentum [see Eq. (1)] with and , in the given parametrization. These coupling constants can be easily adjusted by changing the parameters , and of the laser fields, which provides a knob to tune the strength and form of the spin-orbit interaction.
Now, we concentrate on the generic case characterized by anisotropic spin-orbit interactions and assume for concreteness that . The trap potential and the inter-particle interaction are initially disregarded and discussed in the following sections. The single-particle spectrum of Hamiltonian (1) is (see Fig. 1):
where labels the bands.
The corresponding eigenfunctions are spinors with components
where is the azimuthal angle in the -plane and . The unitary matrix diagonalizes the Hamiltonian (1) (where corresponds to the pseudo-spin index and labels the eigenstates). It is obvious from Eq. (4) that the spectrum of the single particle problem contains two minima at and momenta and (see Fig. 1). Consequently, the single particle ground-state is double-degenerate and the most general expression for the corresponding wave-function is
where and are the fractions of “left-” and “right-moving” states subjected to the constraint , while and are arbitrary phases. Note that by left/right-moving states we mean states with non-zero momentum average, . However, the corresponding average velocity vanishes , so that quasiparticles characterized by these non-zero momentum single-particle states are not actually “moving”, as long as the laser fields generating the spin-orbit coupling are maintained. Note that rotations in the manifold of the double-well ground-states are distinct from rotations in the pseudo-spin Hilbert space, as real-space and pseudo-spin coordinates are mixed up by the spin-orbit interaction. The two-fold degeneracy of the single-particle ground state is preserved if the system is placed in a harmonic trap. For a potential , we can write the Schödinger equation in momentum representation: The trap potential plays the role of “the kinetic energy” and the real kinetic term produces a double-well potential in momentum space, see Fig. 1. The tunnelling processes connect the degenerate vacua in momentum space Polyakov (). However, they do not eliminate the double-degeneracy of the single-particle states, which is protected by the Kramers-like symmetry (see Section III.2).
At low temperatures, the many-body Bose system (1) condenses into the single-particle states corresponding to the double-well minima. The transition temperature of this double-well SOBEC can be calculated using standard text-book procedures.Abrikosov et al. () Let us assume that near and below the transition the band with does not contribute and that we can expand the spectrum near the minima of the band (4). We define the momentum in the vicinity of the left/right minima as follows: , with . Eq. (4) leads to the anisotropic spectrum:
The transition temperature is
We see that if , our approximation is justified and, in particular, the density of particles in the upper band is exponentially small.
In the isotropic limit , the transition temperature formally vanishes. Note that in the isotropic case the spin-orbit term of the Hamiltonian (1) is equivalent to the Rashba model Rashba () and can be reduced to the latter via the rotation in the pseudo-spin space. In this case, the spectrum (4) has minima on a one-dimensional circle (see Fig. 2).
The single-particle ground-state is infinitely degenerate and the most general expression for the corresponding wave-function is
where is the angle-dependent weight of the Bose-condensate on a circle  and is the angle-dependent phase. An especially interesting class of ground states corresponds to not vanishing anywhere on the circle. In this case, the phase must satisfy the constraint , with being an integer winding number. Therefore, there may exist a number of topologically distinct ground states (characterized by the winding number), which can not be deformed into one another via any continuous transformation. We note here that a transition into the ring SOBEC is similar to a “weak-crystallization transition” discussed by Brazovsky Brazovskii (1975) (see also, Refs. Schmalian and Turlakov, ). In this case, the phase volume of fluctuations is very large, which drives the (classical) transition first order. Even though the transition temperature into the ring SOBEC vanishes in the thermodynamic limit, in a finite trapped system, the energy scale for the crossover into this state will be non-zero.Posazhennikova ()
Iii Effects of density-density interaction
The most general ground-state many-body wave-function of a non-interacting “double well BEC” is
where and are the numbers of “left-” and “right-movers,” are the corresponding creation operators, and are arbitrary coefficients satisfying . In the absence of spin-orbit interaction, a two-component bosonic system has a ferromagnetic ground-state with fully polarized pseudo-spin.Kuklov and Svistunov (); Ashhab and Leggett () We emphasize that this is not the case for the double-well many-body ground-state (11) that describes the spin-orbit interacting BEC. All the arguments used for proving the ferromagnetic nature of the ground-state for a two-component systemAshhab and Leggett () are now irrelevant, as the real-space and spin components of the wave-function cannot be factorized due to the spin-orbit coupling. The non-interacting ground-state (11) has an -fold degeneracy. We show bellow that this large degeneracy is partially lifted by interactions and reduced to a two-fold degeneracy. We assume a density-density interaction , where and is the field operator, which is initially defined in terms of the creation/annihilation operators for the original hyperfine states. First, we perform the position-dependent rotation to obtain the effective interaction term, which has the standard form
where is the creation operator for a state with momentum and pseudo-spin in the dark state subspace. We need to perform a second momentum-dependent transformation defined by (5) and (6), which introduces new bosonic operators labelled by the band index (4): and , where and the summation over the spin index is implied. In the limit of relatively weak interactions, (we emphasize that the spin-orbit coupling strength can be tuned to be arbitrarily strong by adjusting the properties of laser fields), the upper band with is irrelevant for the low-energy physics. Thus, it is convenient to express the Hamiltonian in terms of left/right-moving operators, defined as . Correspondingly, we have . This leads to the following interaction Hamiltonian
where the prime sign in the sum over the left and right indices is restricted by the condition , i.e., the numbers of left- and right-movers are conserved, and . We stress that equation (13) is valid in the limit of weak interactions (relative to the spin-orbit coupling) and low temperatures, when only single particle states with momenta in the vicinity of the two minima are occupied.
iii.1 Generalized Bogoliubov transformation
Next, we introduce the projection operators that select the subspace characterized by left-moving and right-moving quasiparticles. The Hamiltonian can be expressed as . An important observation is that the Hamiltonian containing the interaction term (13) preserves the number of left- and right-movers and thus we can consider different “sectors,” , independently. Our goal is to diagonalize each term using a mean-field scheme and reduce the many-body Hamiltonian to the form
where is the contribution of the sector to the condensate energy, while represents the spectrum of quasi-particle excitations. To obtain the mean-field result, we use a Bogoliubov-type approximation in which the operators corresponding to are replaced within each sector by -numbers, and . Next, we notice that at low temperatures, the momenta of uncondensed bosons are . Thus, we can expand the products of -vectors in (13) in terms of the deviations from the minima of the energy bands
with and corrections of order and , respectively. Consequently, contributions to the mean-field Hamiltonian can be expanded in the small parameter . In the zero-order approximation, i.e., neglecting contributions of order and higher, the mean-field Hamiltonian for the sector is
where , is the annihilation operator in a spinor notation, , and is the anisotropic spectrum (8) near the minima. We now introduce new bosonic operators and . The Hamiltonian becomes diagonal for the -particles, which have the “free” spectrum , and has the standard Bogoliubov form Abrikosov et al. () for the -particles. Introducing the new operators and , with , we get
where is the condensate energy Abrikosov et al. () in the zero-order approximation, is the anisotropic free particle quadratic spectrum and is an anisotropic sound similar to the conventional Bogoliubov phonon mode in a BEC. At this level of approximation the condensate energy is n-independent (i.e., it is the same for any particular sector characterized by n left movers and () right movers) and, consequently, the degeneracy of the non-interacting ground state (11) is preserved. In the first order approximation, the mean-field Hamiltonian (16) acquires sector-dependent corrections of order . Following the above recipe, we introduce a set of new operators that diagonalize the term in the Hamiltonian (16) but not the other terms. Next, we diagonalize the full Hamiltonian [up to terms of order ] via a generalized Bogoliubov-type transformation
In the equations (III.1) and (III.1) we already anticipated that some of the terms are corrections of order . The coefficients are determined by requiring that the -operators obey standard commutation relations [to order ] and that the off-diagonal contributions to the Hamiltonian vanish. Assuming for simplicity that we have a point-like interaction, i.e., is momentum-independent for momenta in a range that is relevant for the problem, the groundstate energy in the (n, N-n) sector is
Explicitly evaluating (20) with given by Eq. (21) shows that, at this level of approximation, the energy of the condensate becomes sector-dependent, , and is minimal for and . Thus, the density-density interaction reduces the large -fold degeneracy of the ground state to a two-fold degeneracy. Consequently, in the limit of vanishing interactions , the most general expression for the many-body wave-function is
where represents the fraction of the left/right movers and are arbitrary phases. Notice that Eq. (22) describes a fragmented or entangled BEC, unless . I.e., the many-body state (22) does not correspond to the condensation into one single-particle state. We reiterate that the left- and right-movers in the condensate have non-zero momentum, but zero velocity and do not actually move while the laser fields responsible for the spin-orbit coupling are present. We also note that equation (22) describes a so-called NOON state,Sanders (); Kok et al. () which is quantum correlated state with properties that can be exploited in applications such as quantum sensing and quantum metrology. This suggests that the possibility of using spin-orbit coupled condensates as qubits deserves to be further investigated.
iii.2 Gross-Pitaevskii equations
Let us consider the density-density interaction potential as a contact pseudo-potential, , where and is the inter-atomic scattering length. The full many body Hamiltonian can be written as
in terms of field operators for the original hyperfine states, . In Eq. (III.2) we used the notation for the single particle Hamiltonian in the presence of a trap potential , in addition to the spatially varying laser fields that interact with the atom, . In the adiabatic approximation, after projecting onto the dark state subspace, the first term in Eq. (III.2) becomes , where and are the creation and annihilation operators for bosons with pseudo-spin . The interaction term is given by equation (12). Before writing down the Gross-Pitaevskii equations, let us summarize the three different representations used for describing the system of bosons interacting with the spatially modulated laser fields.
i) Hyperfine states representation: This is the most straightforward way to describe the motion of the bosons and their interaction with the trap potential () and the laser fields (), as well as the density-density interaction (second term in Eq. (III.2)). The field operator that creates a particle in the hyperfine state at point is , while the creation of a free-moving particle with momentum is described by . By performing the position-dependent rotation which diagonalizes the atom-laser Hamiltonian and projecting onto the dark states subspace we switch to the pseudo-spin representation.
ii) Pseudo-spin representation (dark states representation): This is the natural framework for describing the low-energy physics of the atomic system interacting with the laser field. The creation operator for free-moving particles with spin and momentum p is . We can define the corresponding field operator as . Note that the field operators in the hyperfine and pseudo-spin representations are related via the position-dependent rotation, . Diagonalizing the single-particle spin-orbit coupled Hamiltonian, , generates a set of eigenstates described by the spinor eigenfunctions . The quantum number can be viewed as labeling right (left) moving states.
iii) Right/left moving states representation: This is the representation corresponding to the eigenstates of the spin-orbit coupled single particle Hamiltonian. In Section I we have shown that in the absence of a trap potential the single particle spectrum for the generic case is characterized by two minima at non-zero momenta. Here we show explicitly that the double-degeneracy of the single-particle states is a general property of the spin-orbit interacting Hamiltonian, protected by a Kramers-like symmetry. Let us use the following parametrization for the eigenfunctions:
where and n is a set of quantum numbers. The components are the solutions of the following eigenproblem
where is the Hamiltonian in the absence of spin-orbit interaction. More explicitly, satisfy the following system of coupled differential equations:
Taking the complex conjugate of (26) with we obtain an identical set of equations. Consequently we have
and the corresponding energies are degenerate, . Because , the two states are linearly independent. We conclude that the single-particle eigenstates of the spin-orbit coupled Hamiltonian are (at least) double degenerate independent of the symmetries (or lack of symmetry) of the trap potential. Note that this double degeneracy is a consequence of a Kramers-like symmetry of the spin-orbit interacting Hamiltonian, which contains terms that are either quadratic in momentum, or linear in both momentum and spin. The creation operator for a left/right moving particle described by the eigenstate is . The field operators in the pseudo-spin representation can be expressed in terms of operators as
where the terms with and correspond to the right and left moving modes, respectively. Finally, note that in the translation invariant case, , we introduced the eigenfunctions , with given by equations (5) and (6), and the corresponding creation operators, . We then defined the left/right movers for the low energy band and small momenta as . Alternatively, we can directly define the eigenfunctions in the left/right moving representation using the parametrization (24), with no restriction for . The correspondence between the two representations is given by: and . This generalizes our definition of the left/right moving modes to arbitrary energy. Notice however, that a left (right) “moving” state from the high energy band has in fact a positive (negative) momentum.
To write the Gross-Pitaevskii equation in the pseudo-spin representation we use the standard procedure and calculate the commutator , where is the many-body Hamiltonian expressed in terms of pseudo-spin field operators. Using Eq. (III.2) and the relations between representations summarized above we obtain
Relation (III.2), which is a system of two coupled non-linear differential equations, represents the time-dependent Gross-Pitaevskii equation for a spin-orbit coupled Bose-Einstein condensate wave-function. Similar equations can be written in the left/right moving states representation. For simplicity, we will address here only the translation invariant case . The field operator for the left/right moving modes can be written in terms of the corresponding operators as
The non-interacting part of the Hamiltonian is diagonal in terms of left/right moving operators, with eigenvalues that depend on the momentum only. At low-energies, these eigenvalues are given by the anisotropic spectrum with . In general, the interacting Hamiltonian is given by equation (13), but in the low-energy limit we neglect all corrections of order and higher coming from the momentum-dependent matrices . In this limit we obtain
where , . The time-independent Gross-Pitaevskii equations can be obtained by looking for a stationary solution of the form , where is the chemical potential which determined by the condition , with being the total number of bosons. We note that by linearizing with respect to the deviations from the stationary solution we obtain an excitation spectrum consisting in two modes, , identical with those found using the generalized Bogoliubov treatment.
Iv Experimental signature of spin-orbit coupled BEC: measuring a SOBEC qubit
A straightforward way to detect experimentally the new type of BEC would be to probe the momentum distribution of the density of the particles via time-of-flight expansion. After removing the trap and the laser fields, the boson gas represents a system of free particles, each characterized by a certain momentum and a hyperfine state index. In a TOF experiment one determines the momentum distribution by measuring the particle density at various times after the release of the boson cloud. The operator associated with a density measurement is , where is the creation operator for a particle in the hyperfine state positioned at point . Determining the density profile involves a simultaneous measurement of for all the values of corresponding to a ceratin region in space where the boson cloud is located. To insure formal simplicity, we consider a coarse-grained space, i.e., we treat