Cat states in a driven superfluid: role of signal shape and switching protocol
Jesús Mateos, Gregor Pieplow, Charles Creffield, Fernando Sols
CCat states in a driven superfluid: role of signal shape and switching protocol
Jes´us Mateos, Gregor Pieplow,
1, 2
Charles Creffield, and Fernando Sols ∗ Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain Department of Physics, Humboldt-Universit¨at zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany (Dated: May 12, 2020)We investigate the behavior of a one-dimensional Bose-Hubbard model whose kinetic energy ismade to oscillate with zero time-average. The effective dynamics is governed by an atypical many-body Hamiltonian where only even-order hopping processes are allowed. At a critical value of thedriving, the system passes from a Mott insulator to a superfluid formed by a cat-like superpositionof two quasi-condensates with opposite non-zero momenta. We analyze the robustness of this un-conventional ground state against variations of a number of system parameters. In particular westudy the effect of the waveform and the switching protocol of the driving signal. Knowledge ofthe sensitivity of the system to these parameter variations allows us to gauge the robustness of theexotic physical behavior.
I. INTRODUCTION “Floquet engineering” [1] consists of rapidly oscillating a parameter of a Hamiltonian periodically in time, which,following the elimination of the high-frequency degrees of freedom, produces a time-independent effective Hamil-tonian. The process can both produce new terms in the effective Hamiltonian which do not appear in the un-driven model, or the renormalization of previously existing processes. A well-known example of the latter is theperiodically-shaken Bose-Hubbard model [2, 3], in which the lattice-shaking causes a renormalization of the single-particle tunneling. In principle, any term in the Hamiltonian can be periodically varied, and Floquet theory used toderive the resulting effective model. The most common forms of driving are the variation of an external potential –which includes, for example, the case of lattice-shaking mentioned previously – and, more recently, the oscillationof the interparticle interactions [4, 5].In Ref. [6] we introduced a new form of driving which we term “kinetic driving”. This involves the periodicmodulation of the kinetic component of a system’s Hamiltonian, or equivalently, varying the system’s hoppingparameter. Small modulations of the hopping in the Hubbard model have already been used as a tool to probethe system’s properties [7–10]. In contrast, we considered the hopping parameter to have the time-dependent form J ( t ) = J cos ωt , so that its time-averaged value vanishes . As a consequence, particles can only move via higher-orderprocesses, such as the hopping of particles in pairs (doublons), or by long-range single-particle hopping processesassisted by the presence of another particle elsewhere. As a result instead of condensing at zero momentum, likea standard Bose-Einstein condensate, the system condenses into a macroscopic superposition of two condensateswith non-zero momenta, ± π/
2. In Ref. [11] we showed that this superposition has the form of a Schr¨odinger catstate, and the unusual properties of the system give it remarkable stability.In this work we extend our study of the properties of this exotic superfluid state. After a brief description ofpreceding work in Section II, we consider in Section III non-sinusoidal driving functions and study how the shapeof the waveform affects the response of the system. We then go on to include a temporal phase-shift in the drivingin Section IV, to further characterize the stability of the cat state to perturbations of the driving potential, and inSection V we study the response of the system to an external flux. Finally in Section VI we consider the preparationof the cat state in experiment by adiabatically ramping the driving from zero, thereby evolving the system from aMott state to the superfluid state, and show how the ramp time depends on the choice of the driving function.
II. MODEL
We consider the time-dependent Bose-Hubbard Hamiltonian H ( t ) = − Jf ( t ) L − (cid:88) x =0 (cid:16) a † x a x +1 + a † x +1 a x (cid:17) + U L − (cid:88) x =0 n x ( n x − , (1) ∗ [email protected] a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y where a x ( a † x ) are the usual bosonic annihilation(creation) operators and n x = a † x a x is the number operator. TheHubbard interaction energy is given by U >
0, and f ( t ) is the T -periodic function modulating the hopping amplitude J between nearest-neighbor sites.We may introduce the plane-wave representation a x = 1 √ L L − (cid:88) (cid:96) =0 e ik (cid:96) x a k (cid:96) , a k (cid:96) = 1 √ L L − (cid:88) x =0 e − ik (cid:96) x a x , (2)where k (cid:96) = 2 π(cid:96)/L . In Refs. [6, 11] it was proven that for high frequencies the Hamiltonian (1) can be accuratelyapproximated by an effective time-independent Hamiltonian, obtained by making a unitary transformation to theinteraction picture, and averaging over one period of the driving H eff = U L L − (cid:88) (cid:96),m,n,p =0 δ k (cid:96) + k m ,k n + k p Γ( k (cid:96) , k m , k n , k p ) a † k p a † k n a k m a k (cid:96) , (3)where Γ( k (cid:96) , k m , k n , k p ) = 1 T (cid:90) T dte iF ( t ) κg ( k (cid:96) ,k m ,k n ,k p ) ≡ Γ( κg ) , (4) g ( k (cid:96) , k m , k n , k p ) ≡ cos( k (cid:96) ) + cos( k m ) − cos( k n ) − cos( k p ) . (5)In Eq.(4) we introduce the primitive of the driving potential F ( t ) = ω (cid:82) t f ( t (cid:48) ) dt (cid:48) , and in this work we parameterizethe driving using the dimensionless quantity κ = J/ω , where ω = 2 π/T . For cosenoidal driving, f ( t ) = cos( ωt ), theHamiltonian (3) becomes [11] H eff = U L L − (cid:88) (cid:96),m,n,p =0 δ k (cid:96) + k m ,k n + k p J [2 κg ( k (cid:96) , k m , k n , k p )] a † k p a † k n a k m a k (cid:96) , (6)where J is the zeroth-order Bessel function of the first kind.We note that in this derivation we require κ <
1, to be consistent with the high-frequency approximation.
III. SIGNAL SHAPE
In this Section we will derive the expression of the effective Hamiltonian (3) for driving functions f ( t ) other thanthe cosenoidal signal, but still periodic and with zero time-average over one period. These signals are shown versustime in the insets of Fig. 1. They are taken to have the same period and amplitude.Equation (3) remains the general effective Hamiltonian, but different profiles of the time signal f ( t ) now yielddifferent matrix elements of the interaction between plane waves, i.e., different forms of the function Γ( y ), where y = κg ( k (cid:96) , k m , k n , k p ). We have already seen that for the cosenoidal signal [see Eq. (6)], Γ( y ) = J (2 y ).For the square wave signal f ( t ) = , ωt ∈ [0 , π/ − , ωt ∈ [ π/ , π/ , ωt ∈ [3 π/ , π ) , (7)the matrix element Γ( y ) (4) takes the form Γ( y ) = sin( πy ) πy , (8)while for the triangular signal f ( t ) = 2 π × (cid:26) π/ − ωt, ωt ∈ [0 , π ) ωt − π/ , ωt ∈ [ π, π ) , (9)FIG. 1: The matrix element defining the effective interaction between bosons, Γ, for the various signal shapes,plotted as a function of the variable y = κg ( k (cid:96) , k m , k n , k p ). The analytical expressions for Γ( y ) are given in SectionIII. For the sawtooth case, where Γ cannot be made real (see discussion in Sections III and IV), we plot | Γ( y ) | .the expression becomes Γ( y ) = cos (cid:16) πy (cid:17) C ( √ y ) + sin (cid:16) πy (cid:17) S ( √ y ) √ y , (10)where C and S are the Fresnel cosine and sine integrals [12].Finally, for the sawtooth signal f ( t ) = 1 π × ωt, ωt ∈ [0 , π ) ωt − π, ωt ∈ [ π, π ) , (11)we obtain Γ( y ) = erf (cid:0) √− iπy (cid:1) √− iy , (12)where erf( z ) is the error function.Figure 1 shows the plots of Γ( y ) for the different driving profiles. Since Γ is complex for the sawtooth signal, weplot | Γ | in that case. For all signals we have Γ(0) = 1.FIG. 2: Two particle momentum density, ρ (2) ( k, k (cid:48) ) as a function of κ for each signal shape. For κ = 0 the systemis in the Mott state, and ρ (2) ( k, k (cid:48) ) is peaked along the diagonal k = k (cid:48) . As κ increases, isolated peaks form at ± ( π/ , π/ x, t ) → ( − x, t + T / H eff do not have definite parity [13], and sothe von Neumann-Wigner theorem [14] forbids them making exact crossings as κ varies. Instead, they can only formavoided crossings, which in turn implies that the amplitudes for elementary processes, described by Γ( y ), cannotvanish.From (3), and using the specific expression of Γ( y ) of each signal, we calculate the two-particle momentum density ρ (2) ( k, k (cid:48) ) = 1 N (cid:104) n k n k (cid:48) | n k n k (cid:48) (cid:105) . (13)Results are shown in Fig. 2. All the plots reveal the system is robust against the choice of the driving profile, givingvery similar forms for ρ (2) ( k, k (cid:48) ). The only difference is that for the sawtooth, greater values of κ are needed toproduce peaks with comparable heights to the other cases, which is related to the fact that Γ( y ) for the sawtoothdoes not cross the horizontal axis (see Fig. 1).FIG. 3: Momentum density, ρ ( k ), as a function of κ . (a) Square wave driving. The momentum density is initiallyflat for κ = 0 when the system is in the Mott state. As κ increases the system passes through a phase transitionand ρ ( k ) develops two peaks at k = ± π/
2, indicating the formation of the superfluid cat state. (b) Sawtoothdriving. In contrast to the square wave case, the peaks in ρ ( k ) develop more slowly, and are much less pronounced.We now calculate the one-particle momentum density ρ ( k ) = 1 N (cid:104) n k | n k (cid:105) . (14)for the square and the sawtooth signals, and show the results in Fig. 3. Again, peaks form at k = ± π/ κ grows, indicating the macroscopic occupation of these momentum states as the superfluid state develops. For thesawtooth these peaks in ρ ( k ) develop more slowly as a function of κ , showing that larger values of κ are requiredfor the system to become superfluid.In all the cases considered in this Section we have explicitly checked that the full time-dependent evolution underEq. (1) yields the same results as that obtained from the effective time-independent Hamiltonian H eff . We notethat in the numerical simulation of the time evolution, the system is prepared in the Mott state (i.e. the groundstate of the zero-hopping Bose-Hubbard Hamiltonian). The kinetic driving is then gradually introduced so that thestate evolves adiabatically towards the ground state of the effective time-independent Hamiltonian. IV. INITIAL PHASE
It was noted in Refs. [15–19], and later observed in cold atoms systems [20], that if a phase-shift ϕ is added tothe standard cosenoidal signal f ( t ) = cos( ωt + ϕ ) , (15)then the long-term dynamics can be sensitive to that phase and, in the case of a ring this phase can create the effectof an effective flux threading the ring, enabling the simulation of a synthetic magnetic field. We wish to investigatewhether a similar effect appears in the kinetically-driven system. The effective time-independent Hamiltonianbecomes [21] H eff = U L L − (cid:88) (cid:96),m,n,p =0 δ k (cid:96) + k m ,k n + k p e − i κgF ( ϕ ) Γ( κg ) a † k p a † k n a k m a k (cid:96) , (16)where Γ( y ) is given in the previous Section for the various signal shapes, and the F function reads F ( ϕ ) = sin ϕ (cosine wave) ϕ (square wave) ϕ − ϕ /π (triangular wave) ϕ / π (sawtooth wave) . (17)FIG. 4: Two-particle momentum density ρ (2) ( k, k (cid:48) ) for several values of Φ, with κ = 0 .
6. When Φ = 0 the peaksare centered on ± ( π/ , π/ π/ π/ − π/
4, along the diagonal.Due to the additive structure of the function g [see Eq. (5)], the presence of F ( ϕ ) (cid:54) = 0 amounts to adding a phaseto each boson operator a k → a k e iα k , (18)where α k = 2 κ cos( k ) F ( ϕ ). Addition of a global, k -dependent phase to each one-particle momentum eigenstateof eigenvalue k does not have any physical consequence, from which we conclude that the system properties areexactly independent of ϕ , as can be confirmed numerically.Thus, unlike the single-particle hopping of independent particles, or in the conventional Bose-Hubbard problem[15, 18, 19], it is not possible to create a flux through the ring by tailoring the initial phase of the kinetic driving. V. EFFECTIVE FLUX
An effective external flux Φ threading the ring may result from a controlled or spurious rotation of the ring.We study here how ground state properties of the system change under the effect of this flux. The fundamentaltime-dependent Bose-Hubbard Hamiltonian becomes H ( t ) = − f ( t ) L − (cid:88) x =0 (cid:16) e i Φ a † x +1 a x + e − i Φ a † x a x +1 (cid:17) + U L − (cid:88) x =0 n x ( n x − , (19)where we take the cosenoidal driving without any initial phase, f ( t ) = cos( ωt ). In this case the effective Hamiltonianbecomes H eff (Φ) = U L L − (cid:88) (cid:96),m,n,p =0 δ k (cid:96) + k m ,k n + k p J [2 κg ( k (cid:96) , k m , k n , k p ; Φ)] a † k p a † k n a k m a k (cid:96) (20)where the g -function is now defined by g ( k (cid:96) , k m , k n , k p ; Φ) = cos( k (cid:96) + Φ) + cos( k m + Φ) − cos( k n + Φ) − cos( k p + Φ) . (21)This is the main difference between this case and the previous one; Φ does not appear as a phase factor, butinside the arguments of the cosine functions. Accordingly Φ does have a physically observable effect, shifting themomentum at which the peaks of ρ (2) ( k, k (cid:48) ) occur. In Fig. 4 (left) we shown the two-particle momentum densityfor Φ = 0, which displays narrow peaks centered on ± ( π/ , π/ π/ k = π/
8. However, for the 8-site ring weconsider, this shift is not commensurate with the reciprocal lattice momenta, and so the peaks spread over the twoclosest momenta to the shifted values, as shown in Fig. 4 (center). The smallest non-zero shift of momentum that c Time / T ramp c ramp hold (a)(b)T ramp =320TT ramp =640T p/4 p/2 Driving phase, f c (c) T ramp =640TT ramp =320T FIG. 5: Fidelity of the ramp protocol in preparing the cat state at κ = 0 .
8, as measured by χ . (a) For aramp-time of T ramp = 320 T the fidelity of the final state depends strongly on the phase of the driving.Cosinusoidal driving (black solid line) shows the best performance, and sinusoidal driving (blue dot-dashed line)the worst. An intermediate phase, f ( t ) = cos( ωt − π/
4) (red dashed line) lies between these results. The value of χ in the true ground state of the system is shown by the horizontal dashed line. (b) When the ramp-time isincreased to to T ramp = 640 T the driving phase barely affects the result. (c) Dependence of χ on the phase of thedriving for the two ramp-times. Error bars indicate the amplitude of the oscillations of the final state. Physicalparameters: U = 1, ω = 50.is commensurate with the lattice is ∆ k = 2 π/ π/
4. For Φ = π/
4, the peaks are indeed shifted by this quantity,so they now are located at ( − π/ , − π/
4) and ( π/ , π/ π/ ± π/
2, and a hint of this behavior can already be observed in Fig. 3, wherean enhancement of the occupation is already observed in momenta near ± π/ VI. ADIABATIC STATE PREPARATION
We now consider the feasibility of preparing the cat state in experiment, using the technique of adiabatic ma-nipulation. We initialise the system in the Mott state, with one particle occupying each site of the lattice, andthen gradually increase the value of κ from zero to κ = 0 .
8. If this increase is done sufficiently slowly, so that thesystem remains in its ground-state at all times, the Mott-state will be adiabatically transformed into the exoticsuperfluid state. In particular we consider the protocol in which the amplitude of κ is ramped up linearly over along time-interval T ramp , which may be many hundred of driving periods long, and then held at a constant valuefor a period we term the “hold-time”. To quantify the quality of the state preparation, we measure the quantity χ = (cid:104) n π/ n π/ (cid:105) , that is, the ( π/ , π/
2) component of the two-particle reduced density matrix. This quantity takesa high value in the cat state, and so acts as a good figure of merit [11] to identify a state’s “cattiness”.In Fig. 5a we show the result for sinusoidal driving, f ( t ) = cos ( ωt + ϕ ), for three different values of the drivingphase ϕ , using a ramp-time of T ramp = 320 T . When the driving is cosinusoidal ( ϕ = 0), the value of χ initiallyrises smoothly, but then begins to oscillate towards the end of the ramp, and these oscillations continue during thehold-time. This arises from the form of the quasienergy spectrum of the system. When the system is in the Mottstate, the spectrum is gapped and the ground-state is well-separated from the higher energy states. Consequentlythe adiabatic condition is easily satisfied and the system smoothly tracks the instantaneous ground-state as κ increases. As the system passes through the phase transition to the superfluid state, however, this gap closes,and unless the ramp speed is extremely slow some proportion of the state will be excited out of the instantaneousground state, producing the oscillatory behaviour. Changing the driving-phase to ϕ = π/ χ is smaller, indicating that the final state has been prepared withless fidelity. For a sinusoidal driving ( ϕ = π/
2) the fidelity is reduced even further.The ϕ -dependence of these results may appear surprising, since in Section IV it was shown that the driving-phasehad no effect on the system. However, that lack of ϕ -dependence only applies strictly to the ramping procedurein the adiabatic limit, T ramp → ∞ . In Fig. 5b we show the effect of doubling the ramp-time to T ramp = 640 T .With this slower ramp, the procedure is more adiabatic, and indeed the ϕ -dependence of the results is considerablyreduced. As a consequence the maximum value of χ following the ramp is much closer to the value for the κ = 0 . χ are much reduced. We quantify this further in Fig. 5c, showingthe ϕ dependence of χ for the slow and the fast ramp-times. Clearly as T ramp increases, the ϕ dependence of theresults decreases, and the fidelity of the process improves. From this graph it is also clear that if an experiment islimited to low or moderate values of the ramp-time, the best choice of the driving will be cosenoidal. VII. CONCLUSIONS
We have investigated the sensitivity of a many-boson system to the details of the kinetic driving previouslyexplored in [6, 11], where the hopping energy oscillates periodically in time with zero average. In particular we havestudied the sensitivity of the cat-like structure of the ground state to the shape of the signal profile, phase-shifts ofthe driving, and the presence of an external magnetic flux. We have found that the system is very sensitive to thesymmetry of the time signal. The sawtooth profile exemplifies the case where time-reversal symmetry is absent,and the cat-like properties of the ground state are considerably diminished. The fact that the interaction matrixelements between plane waves cannot vanish makes them weakly dependent on the strength of the driving, thusfavoring less markedly the collision processes that in [11] were shown to underlie the ground-state cat structure.In contrast to the case of the standard driven Bose-Hubbard model, we find that introducing a phase-shift tothe driving signal does not affect the properties of the system at all. Instead of producing hopping phases, asmight be expected, the system is completely blind to this form of perturbation. Introducing hopping phases byrotating the ring to produce an effective flux, only has the effect of trivially displacing the momenta at which thequasi-condensates form. In particular, the cat structure of the ground state remains intact.We have also explored the preparation of the cat-state in experiment, by adiabatically ramping the driving fromzero. In the deep adiabatic limit, the process becomes insensitive to the phase of the driving, while a cosenoidalsignal gives the optimum fidelity for shorter ramp-times.In conclusion, the main properties of the kinetically driven superfluid boson system are rather robust againstvariations in the details of the driving, provided that the signal is time symmetric. In addition, we have shown thatsuitably choosing the phase of the driving allows the adiabatic preparation to be substantially shortened.
ACKNOWLEDGMENTS
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