Manipulating Cooper pairs with a controllable momentum in periodically driven degenerate Fermi gases
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Manipulating Cooper pairs with a controllable momentum in periodically drivendegenerate Fermi gases
Zhen Zheng ∗ and Z. D. Wang † Department of Physics and Center of Theoretical and Computational Physics,The University of Hong Kong, Pokfulam Road, Hong Kong, China
We here present an experimentally feasible proposal for manipulating Cooper pairs in degenerateFermi gases trapped by an optical lattice. Upon introducing an in situ periodically driven field,the system may be described by an effective time-independent Hamiltonian, in which the Cooperpairs, generated by the bound molecule state in Feshbach resonance, host a nonzero center-of-mass momentum. The system thus processes a crossover from a Bardeen-Cooper-Schrieffer (BCS)superfluid phase to a Fulde-Ferrell (FF) one. Furthermore, the magnitude and direction of theCooper pairs in the synthetic FF superfluids are both directly controllable via the periodicallydriven field. Our proposal offers a reliable and feasible scenario for manipulating the Cooper pairsin cold atoms, serving as a tunable as well as powerful platform for quantum-emulating and exploringthe FF superfluid phase.
I. INTRODUCTION
Manipulation of cold atoms via optical techniques hasintensively been studied [1], which offers a powerful ex-perimental tool for synthesizing many interesting phasesand models that are hardly accessible in conventionalsolid systems. On one hand, Raman transitions can beimplemented to couple pseudo-spins of cold atoms, and avariety of experiments have been performed for artificialAbelian and non-Abelian gauge fields [2, 3]. On the otherhand, the periodically driven cold-atom system opens analternative window to achieve modulated couplings [4–6]or interactions [7, 8]. This technique leads to a so-calledFloquet engineering [9, 10], and has been employed inquantum simulations, such as the Mott-insulator to su-perfluid transition [11], topological insulator [12–14], fer-romagnetic transition [15], artificial magnetic fields [16],and superlfluid Ising transition [17].Generally, the Floquet engineering is based on enforc-ing periodically time-dependent external fields or me-chanical deformations on the original static system. Byapplying an in situ perturbative driven field, the systemmay be captured by an effective Hamiltonian with variousmodulated parameters [18–21]. This invokes an idea ofcontrolling and manipulating Cooper pairs in cold atomswith these tunable parameters. In particular, we are mo-tivated to search for a reliable and feasible experimentalscheme for Floquet-engineering a superfluid phase withthe fully controllable pairing momentum, which is knownas the Fulde-Ferrell (FF) phase [22, 23].This paper is organized as follows. In Sec.II, we startwith a two-channel model Hamiltonian in Feshbach reso-nance, and address our motivation of this paper for ma-nipulation of Cooper pairs as well as the synthetic FFsuperfluids. In Sec.III, we elaborate that the introduc-tion of the periodically driven field will result in a artifi- ∗ [email protected] † [email protected] cial controllable pairing momentum, which yields the FFsuperfluid phase. Its existence as the ground state of thelattice system will be shown in Sec.IV. We detail howto realize the periodically driven field via current experi-mental techniques in Sec.V. The features of the syntheticFF superfluids in our proposal are discussed in Sec.VI.We make a brief summary the in the last Sec.VII. II. MODEL HAMILTONIAN
We consider a degenerate Fermi gas with two pseudo-spins trapped in a three dimensional (3D) optical lattice.In real experiments, the interaction between two atomswith opposite spins is realized via Feshbach resonance,in which the two atoms collide and bound to a bosonicmolecule state. This system may be described by a twochannel model Hamiltonian composed of three parts [24], H = H c + H b + H bc . (1)The first part H c describes two free atoms in the openchannel of the scattering process, H c = Z d r X σ = ↑↓ ψ † σ ( x )[ − ∇ / m + V L ( r ) − µ ] ψ σ ( x ) , (2)where ψ † σ , ψ σ are the creation and annihilation opera-tors of the fermionic atoms. V L ( r ) = V L sin ( k L x ) + V L sin ( k L y ) + V L sin ( k L z ) is the lattice trap potential. k L = π/a with a as the lattice constant. In the wholepaper we assume ~ = 1. µ is the chemistry potential offermions. The second part H b is the molecule state ofthe close channel, H b = Z d r ϕ † ( x )[ − ∇ / m + V L ( r ) − µ ] ϕ ( x ) , (3)where ϕ † , ϕ are the creation and annihilation operatorsof the bosonic molecule state. 2 µ is introduced due tothe number conservation. The last part H bc correspondsto the coupling of the two channels. For simplicity, wehere consider the contact interaction with strength g , sothat H bc is expressed as H bc = g Z d r ϕ † ( r ) ψ ↑ ( r ) ψ ↓ ( r ) . (4)In the lattice system, we can study the Hamiltonianusing the tight-binding approximation. For details, weexpand ψ σ and ϕ by Wannier wave functions W ( r ), ψ σ ( r ) = X j W ( r − r j ) c jσ , ϕ ( r ) = X j W ( r − r j ) b j . (5)Here we denote c and b as operators of the fermion andmolecule state. Then Hamiltonian (1) is represented asˆ H = ˆ H + ˆ H , (6)where ˆ H is the intra-channel Hamiltonianˆ H = X j,σ ( E b − µ ) b † j b j − µc † jσ c jσ − ( tc † jσ c j +1 σ + H.c. ) (7)and ˆ H describes the inter-channel couplingˆ H = X j U b † j c j ↑ c j ↓ + H.c. (8)
H.c. is the Hermitian conjugate. t is the tunneling mag-nitude stemmed from the kinetic energy in H c . E b isthe bound energy and can be shifted via magnetic fields,which plays the key role for a controllable atomic inter-action via Feshbach resonance. The interaction strength U is given by U = g R d r W ∗ ( r )[ W ( r )] .In this paper, we take the mean field method to studythe lattice system, since it can give a clear physics pic-ture and capture qualitative features of the 3D system.In an ordinary picture, we can replace the molecular fieldby its mean value b j ≈ h b j i = b in Eq.(8). It reveals the s -wave Cooper pairs is dominant whose presence charac-terizes the Bardeen-Cooper-Schrieffer (BCS) superfluidphase. This can be shown by Fourier transforming theHamiltonian (8) into the momentum space,˜ H = X k U b ∗ c k ↑ c − k ↓ + H.c. (9)The reason of using such a mean-field solution is because,for extremely cold atoms, the bosonic molecule state willcondensate on the state with zero momentum. A re-cent study shows that the molecule state can acquire anonzero center-of-mass momentum via the Raman tran-sition to auxiliary levels [25]. The Cooper pairs in thatsystem thus host a nonzero pairing momentum, yieldingthe realization of the FF superfluid phase. It inspiresus with an interesting question: is there a simpler pro-posal for realizing the FF superluids without manipu-lating the molecule state by optical methods? On theother hand, recent investigations on Floquet engineer-ing have addressed how to modulate single-particle fieldswith time-dependent fields. This motivates us to searchfor an alternative proposal with a periodically driven sys-tem.
III. PERIODICALLY DRIVEN ENGINEERING
We consider a perturbative locally and periodicallydriven field as follows, V ( t ) = ∆2 cos( ωt + φ j ) − ω . (10)∆ and ω are the magnitude and frequency of the period-ically driven field. The phase φ j = jηπ depends on thesite index in accompany with a controllable parameter η .For simplicity, in the whole paper, we consider its pro-jection along the x direction φ j = φ j x = j x ηπ instead ( j x is the site index along the x direction). Adding V ( t ) tothe lattice system, its Hamiltonian is given by H = ˆ H + ˆ H + H t , H t = X j,σ V ( t ) c † jσ c jσ . (11)Here ˆ H , have been given in Eqs.(7)-(8). In order to ob-tain a time-independent effective Hamiltonian, we makethe following unitary transformation to eliminate the theperiodically driven term H t : U = exp h i2 X j,σ Ω j ( t ) c † j,σ c j,σ i (12)with Ω j ( t ) = ∆ ω sin( ωt + φ j ) − ωt . In the rotating frame,the effective Hamiltonian is given by H ′ = UHU † − i U ∂ t U † = ˆ H ′ + ˆ H ′ . (13)whereˆ H ′ = E − t X j,σ e i[Ω j ( t ) − Ω j +1 ( t )] / c † jσ c j +1 σ + H.c. (14)ˆ H ′ = U X j e − iΩ j ( t ) b † j c j ↑ c j ↓ + H.c. (15)and E = P j,σ ( E b − µ ) b † j b j − µc † jσ c jσ . Using Besselexpansion e i z sin θ = P n J n ( z )e i nθ ( J n denotes the n -thorder Bessel function), we can getˆ H ′ = E − t X j,n,n ′ J n (∆ /ω ) J n ′ (∆ /ω )e i( n − n ′ ) ωt +i( nφ j − n ′ φ j +1 ) × c † jσ c j +1 σ + H.c. (16)ˆ H ′ = U X j,n J n (∆ /ω )e − i( n − ωt − i nφ j b † j c j ↑ c j ↓ + H.c. (17)In practice, if we tune ω ≫ ∆, we can neglect rapidlyoscillating terms, and obtainˆ H ′ ≈ E − ˜ t X j c † jσ c j +1 σ + H.c. (18)ˆ H ′ ≈ ˜ U X j e − i φ j b † j c j ↑ c j ↓ + H.c. (19)where we have denoted˜ t ≡ t [ J (∆ /ω )] , ˜ U ≡ U J (∆ /ω ) . (20)We submit ˆ H ′ , into Eq.(13) and make the mean-field ap-proximation b j ≈ b . In the momentum space, we obtainthe final time-independent form of the effective Hamilto-nian, H eff = X k ,σ ξ b + ξ k c † k σ c k σ + ˜ U ( b ∗ c k ↑ c Q − k ↓ + H.c. ) (21)with ξ b = ( E b − µ ) | b | , ξ k = − µ − t X i = x,y,z cos( k i a ) . (22)The Cooper pairs host a center-of-mass momentum Q = ηk L ˆ e x , where ˆ e x denoting the lattice primitive vectoralong the x direction. As η can be changed in the period-ically driven field, it indicate that the magnitude as wellas the direction of the pairing momentum are both di-rectly introduced and controllable via optical techniques.By contrast, in a previous Floquet engineering proposal[26], the FF phase emerges due to the orbit band inverseand hence its pairing momentum is fixed. IV. NUMERIC RESULTS
From the effective Hamiltonian (21), in the base Ψ k =( c k ↑ , c † Q − k ↓ ) T , we can write the Bogoliubov-de Gennes(BdG) Hamiltonian, H BdG ( k ) = (cid:18) ξ k ˜ U b ˜ U b ∗ − ξ Q − k (cid:19) . (23)The diagonalization of H BdG ( k ) gives the spectrum of thequasi-particles: E ± k = ξ − k ± q ( ξ + k ) + ˜ U | b | . We denote ξ ± k = ( ξ k ± ξ Q − k ) /
2. The system energy is calculated by E = h H eff i , which is written as E = X k ,γ = ± E γ k Θ( − E γ k ) + ξ k + ξ b . (24)Here Θ( x ) is the Heaviside step function that describesthe Fermi distribution at zero temperature. The gap andnumber equations can be obtained by [27] ∂ E ∂ | b | = 0 , ∂ E ∂µ = − n . (25)Here n is the filling factor and initial determined whenpreparing the degenerate Fermi gas. The superfluid orderparameter | b | and the chemistry potential µ are obtainedsimultaneously by self-consistent solving Eq.(25). When | b | 6 = 0, the ground state is the superfluid phase, mean-while, is the BCS(FF)-type if we set η =( =)0. When | b | vanishes, the system is a normal gas.
20 25 30 35 40 45 50−1.0−0.50.00.51.0 η (a) n=0.8 0.20.30.40.50.6 | b |
20 25 30 35 40 45 50−1.0−0.50.00.51.0 η (b) n=1.0 0.20.30.40.50.6 | b |
20 25 30 35 40 45 50E b /˜t−1.0−0.50.00.51.0 η (c) n=1.2 0.20.30.40.50.6 | b | FIG. 1. Superfluid order parameter | b | as a function in the η - E b plane at different filling factors. The color visualizes thevalue of | b | . We set ˜ U = 20˜ t in the numeric calculations. In Fig. 1, we plot the evolution of the superfluid orderparameter | b | in the η - E b plane. Both two parameters η and E b are experimentally tunable. We can see thesuperfluid phase is still present when η = 0, yieldingthe existence of the FF superfluids as the ground stateof the lattice system. | b | is dominated by E b as well as n , and increase monotonically with the decrease of E b ,which processes a BCS–Bose-Einstein-condensate(BEC)crossover. By contrast, it changes insensitive and slightlywith η and is independent from η ’s sign. It is straight for-ward to understand this phenomenon, because the latticesystem is spin degenerate despite that Q is artificiallyintroduced. The Fermi surfaces of opposite spins do notsplit nor deform, implying the mechanics of the FF-typeCooper pair resembles the BCS-type one’s. V. EXPERIMENTAL REALIZATION
The proposal of the controllable Cooper pairs is readyto be realized in current experimental technique. Thetime-dependent external potential V ( t ), see Eq.(10), iscomposed of two terms. The first term gives the pe-riodically driven field. It can be introduced by add agroup of counter-propagating lasers, whose wavelength is λ L /η along the x direction. Here λ L is the wavelengthof lasers that construct the optical lattice. The counter-propagating lasers can give rise to a perturbative time-dependent lattice potential ∆2 cos( ηk L x + ωt ), where ω satisfies ω = 2 πcη/λ L ( c is the light speed). We shouldnote that laser strength ∆ / ≪ V L , which guarantees thedriven field does not change the lattice configuration.The second term in V ( t ) can be engineered intuitivelyby the AC-Stark shift to the fermions via an additionallaser. As this term is a local static field for the atoms, itcan be recognized as the additional energy level offset be-tween the open and close channels in Feshbach resonance.Therefore, the generation of the the second term in V ( t )is equivalent to shift the bound energy of the moleculestate E b to E b + ω/
2, which can be compensated by themagnetic field used in Feshbach resonance.
VI. DISCUSSIONS
The Cooper pairing momentum Q of the FF super-fluids is originated from the external periodically drivenfield. We emphasize that the synthetic FF superfluids inour proposal bear the following two features: (i) the mag-nitude of Q is proportional to the parameter η , whichstems from the wavelength of lasers that generate thedriven field; (ii) the direction of Q is governed by thesame lasers’ direction. The two features differentiate thisproposal from a piece of earlier Floquet engineering work[28], in which, like many other pieces of cold-atom workson FF superfluids, the pairing momentum is evidencedby the self-consistently solution, and cannot be directlydetermined by the driven field. Our proposal facilitatesthe manipulation of the pairing momentum in the syn-thetic FF phase, providing a simpler method to directlycontrol not only its magnitude but also the direction.Previous cold-atom works use the Zeeman field to break the spin degeneracy, resulting in split the Fermisurfaces of opposite spins. In this way, a nonzero pairingmomentum, i.e. the FF superfluids, is acquired. How-ever, it has been known that the Zeeman field suppressthe superfluid order parameter, leading to a narrow re-gion in the phase diagram [27]. In our proposal, theengineered FF phase does not require the Zeeman field.The lattice system is spin balanced. The absence of Zee-man fields in our proposal will make the FF superfluidsmore robust against fluctuations. It makes our proposala promising candidate to quantum simulate and studythe FF superfluids.It was seen in Sec.III that introduction of the drivenfield does not change the form of onsite and tunnelingHamiltonians, except for the modulated magnitude char-acterized by Bessel functions. This works even in thepresence of the spin-orbital coupling or Zeeman field, be-cause the periodically driven field is spin independent.Therefore in a Rashba spin-orbital coupled Fermi gas, itis possible, in absence of the in-plane Zeeman field, to en-gineer topological nontrivial FF superfluids that supportMajorana fermion states. This is very different from thepicture reported in previous cold-atom works [29, 30], inwhich the in-plane Zeeman field is necessary for emer-gence of the FF phase. VII. SUMMARY
In summary, we have proposed how to manipulateCooper pairs in a periodically driven degenerate Fermigas. Different from the conventional picture, the nonzeroCooper pairing momentum is artificially introduced byoptical techniques. Its magnitude and direction areboth directly designed by the driven field, which makesour proposal more reliable and feasible for manipulatingCooper pairs. Since the breaking of the spin degeneracyis not required, the synthetic FF superfluid phase is morerobust in comparison with the previous proposals basedon spin polarized gases.
VIII. ACKNOWLEDGEMENTS
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