Dynamical resistivity of a few interacting fermions to the time-dependent potential barrier
DDynamical resistivity of a few interacting fermionsto the time-dependent potential barrier
Dillip K. Nandy & Tomasz Sowi´nski
Institute of Physics, Polish Academy of SciencesAleja Lotnik´ow 32/46, PL-02668 Warsaw, Poland
Abstract.
We study the dynamical response of a harmonically trapped two-component few-fermion system to the external gaussian potential barrier moving across the system. Thesimultaneous role played by inter-particle interactions, rapidity of the barrier, and the fermionicstatistics is explored in systems containing up to four particles. The response is quantifiedin terms of the temporal fidelity of the time-evolved state and the amount of quantumcorrelations between components being dynamically generated. In this way, we show that thedynamical properties of the system crucially depend on non-trivial mutual relations betweentemporal many-body eigenstates, and in consequence, they lead to volatility of the dynamics.Counterintuitively, imbalanced systems manifest much higher resistivity and stability than theirbalanced counterparts.
1. Introduction
Isolated quantum systems described by time-dependent Hamiltonians, due to explicitviolation of the energy conservation, manifest various phenomena that are inaccessible bytheir stationary counterparts. To capture their properties accurately one needs to understandtheir dynamical response to the external driving. This is particularly crucial in the caseof state-of-the-art atomic physics experiments where extremely precise coherent control ofstrongly correlated quantum systems is tremendously rigorous and is treated as the specificdream come true [1].One of the typical directions considered for coherent manipulation of atomic systemsis related to periodic driving [2]. On different levels, it has been demonstrated that suchforcing can be used as a powerful tool for precise controlling of quantum dynamics andit gave a ride to a plethora of fundamentally important and beautiful results. To mentiononly several examples: the dynamical control of the superfluid-Mott insulator transition[3], the AC-induced coherent resonant tunneling [4, 5], expansion dynamics of Bose-Einstein condensates through periodic driving [6], creation of artificial gauge fields [7, 8],the realization of topological insulators via time-periodic potentials [9, 10], the coherentcoupling between different energy bands [11, 12], or frustrated magnetism [13]. On thetheoretical side, the underlying concept which gives an adequate description of a periodicallydriven quantum system is the Floquet theory [14, 15, 16] which gives a nice tool for anappropriate description of the system stroboscopically, i.e. , after each period of the driving. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n ynamical resistivity of a few interacting fermions i.e. , how different few-body results start to resembletheir many-body counterparts when the number of particles is successively increased. Inour work, we make another step in this direction and we focus on transmutations of thesystem’s properties enforced during a single period of external driving. First, we start witha interacting ground state of a system of a few interacting fermions confined in an externalstatic trap. Then the system is non-adiabatically affected by a moving external potentialbarrier that crosses the system back and forth between the edges of the trap. In this way weinvestigate how the system dynamically transits between the two well-explored stationarycases, i.e. , confined in a pure harmonic oscillator [18, 19, 20, 21] and in a double-well[22, 23, 24, 25, 26, 27]. Depending on the rapidity of the barrier we observe how the stateof the system is pulled away from the temporal ground state. By quantifying this distortionwith a well-defined temporal fidelity, we capture the non-trivial interplay between externalforcing, quantum statistics, number of particles, and interaction strength.Since our purely theoretical work is motivated by seminal experiments with two-component mixtures of few fermionic Li atoms confined in quasi-one-dimensional externaltraps [28, 29, 30] we focus on equal-mass components. However, generalization to different-mass atoms (as also conceivable experimentally [31, 32, 33, 34, 35]) is straightforward.Since perfect control and manipulation of these systems were convincingly demonstratedwe believe that our findings may have importance for further experimental and theoreticalprogress (for general reviews see [36, 37, 38]).
2. The Model
In the present study we focus on a few two-component ultra-cold fermionic atoms thatinteract through short-range delta-like potential initially confined in a harmonic trap.In addition to the harmonic potential we introduce a gaussian potential barrier whichperiodically moves across the system. We aim to study a resistivity of the system propertiesto this disturbance depending on its rapidity. This specific situation can be modeled by thefollowing many-body Hamiltonian ˆ H ( x ) = (cid:88) σ (cid:90) d x ˆΨ † σ ( x ) [ H + U ( x )] ˆΨ σ ( x ) + g (cid:90) d x ˆΨ †↑ ( x ) ˆΨ †↓ ( x ) ˆΨ ↓ ( x ) ˆΨ ↑ ( x ) , (1)where the single-particle part of the Hamiltonian contains two parts H and U ( x ) . The firstrepresents the static harmonic oscillator part of the standard form H = − (cid:126) m d d x + m Ω x . (2)The second describes additional gaussian barier of the width β and height U centred around ynamical resistivity of a few interacting fermions x . It can be written as U ( x ) = U √ πβ e − ( x − x ) /β . (3)From now, we will express all quantities in the natural units of the harmonic oscillator. In ourwork we focus on the case of relatively narrow and quite high barrier, i.e. , in these units we set U = 4 and λ = 0 . , respectively. However, generalisation to other barriers is straightforward.In the Hamiltonian (1), the fermionic field operators ˆΨ σ ( x ) annihilate fermion ofcomponent σ ∈ {↓ , ↑} at position x , while g is an effective interaction coupling betweenfermions belonging to opposite components. The field operators obey standard anti-commutation relations { ˆΨ σ ( x ) , ˆΨ † σ (cid:48) ( x (cid:48) ) } = δ σσ (cid:48) δ ( x − x (cid:48) ) and { ˆΨ σ ( x ) , ˆΨ σ (cid:48) ( x (cid:48) ) } = 0 . It isclear that independently on the position of the barrier x and its motion the Hamiltonian(1) commutes with operators of the number of particles in individual components, ˆ N σ = (cid:82) d x ˆΨ † σ ( x ) ˆΨ σ ( x ) . Therefore in the following, we analyze properties of the system in theeigensubspaces of fixed ˆ N ↓ and ˆ N ↑ .In the following, a whole analysis will be carried out numerically in the static, barrier-independent single-particle basis { φ i ( x ) } being the eigenbasis of the pure harmonic oscillatorHamiltonian H . Therefore, after decomposing field operators in this basis as following ˆΨ ↓ ( x ) = (cid:88) i φ i ( x )ˆ a i , ˆΨ ↑ ( x ) = (cid:88) i φ i ( x )ˆ b i , (4)where ˆ a i and ˆ b i are annihilation operators for the ↓ and ↑ fermions at state φ i ( x ) , one rewritesthe many-body Hamiltonian (1) to the form ˆ H ( x ) = (cid:88) i (cid:18) i + 12 (cid:19) (cid:16) ˆ a † i ˆ a i + ˆ b † i ˆ b j (cid:17) + (cid:88) ij U ij ( x ) (cid:104) ˆ a † i ˆ a j + ˆ b † i ˆ b j + h.c. (cid:105) + g (cid:88) ijkl I ijkl ˆ b † i ˆ a † j ˆ a k ˆ b l . (5)where U ij ( x ) = (cid:82) d x φ ∗ i ( x ) U ( x ) φ j ( x ) are matrix elements of the barrier potential in thechosen single-particle basis, while the coefficients I ijkl are determined by the interaction partof the Hamiltonian and they are given by I ijkl = (cid:82) d xφ ∗ i ( x ) φ ∗ j ( x ) φ k ( x ) φ l ( x ) . Due to the mirrorleft-right symmetry of the harmonic oscillator eigenstates these integrals are nonzero onlywhen corresponding sums of the indices ( i + j + k + l ) are even.Since the single-particle basis is independent of the position of the barrier, the wholestatic (but barrier-dependent) Hamiltonian (1) can be simply converted to the time-dependent one by varying the position of the barrier x in time. We will consider the simplestscenario of a periodic motion of the barrier, x ( t ) = A cos( ωt ) , and we will discuss propertiesof the system for different frequencies ω after one period T = 2 π/ω . To make sure thatinitially the system is not influenced by the barrier we set A = 4 . In this case, the ground-state of the system with the barrier in its initial position is in fact the same as the ground-stateof the system in a pure harmonic oscillator. Taking all these initial conditions into account,we investigate properties of the system by solving the time-dependent Schr¨odinger equation, i ddt | ψ ( t ) (cid:105) = ˆ H ( x ( t )) | ψ ( t ) (cid:105) , with the initial initial condition | ψ ( t = 0) (cid:105) = | G (cid:105) , where | G (cid:105) is aground-state of the interacting system confined in a pure harmonic oscillator. ynamical resistivity of a few interacting fermions -4 -2 0 2 40.511.522.533.544.555.5 g = 0.0 -4 -2 0 2 411.522.533.544.55 g = 0.4 -4 -2 0 2 411.522.533.544.55 g = 1.0 -4 -2 0 2 400.511.522.533.544.55 ( + ) g = -1.0 Figure 1.
The many-body spectrum of the Hamiltonian ˆ H ( x ) for N ↑ = N ↓ = 1 particlesas a function of barrier position x and different interaction strengths. For the barrier beingfar away from the center ( x = ± ) the spectrum resembles the corresponding spectrum of apure harmonically trapped system. When the barrier is close to the center, a quasi-degeneracybetween many-body states, characteristic for double-well confinement, is visible. Note, that forstronger interactions the energy gap at x = 0 becomes significantly reduced. The main aim of the present study is to understand the response of a few interactingsystems to a potential barrier moving across a harmonic trap. It is clear that in the case ofadiabatic motion of the barrier ( ω → ), at any moment the system remains in its temporalground state | G ( x ( t )) (cid:105) , i.e. , in the many-body ground state of the Hamiltonian ˆ H ( x ( t )) . Itmeans that after a whole period, T = 2 π/ω → ∞ , the system finally returns to the initialground state | G (cid:105) and thus the final fidelity F = |(cid:104) ψ ( t = T ) | ψ ( t = 0) (cid:105)| is exactly equal tounity. Surely, the situation is different when the barrier makes a loop in a finite period of T since then the system can be partially excited to other many-body states. To capture thispossibility we introduce the temporal fidelity F ω ( t ) = |(cid:104) ψ ( t ) | G ( x ( t )) (cid:105)| , (6)which quantifies the overlap of the transient state of the system | ψ ( t ) (cid:105) with the ground state | G ( x ) (cid:105) of the temporal Hamiltonian ˆ H ( x ( t )) . In this way, we can quite precisely determinemoments when the system becomes excited to higher many-body states and associate theseexcitations with specific properties of the Hamiltonian’s spectrum. We suspect that in theadiabatic limit, ω → , the temporal fidelity F ω ( t ) → for any moment t .
3. Two-particle system
To make our analysis as clear as possible let us start from the simplest system of N ↑ = N ↓ = 1 particles. In Fig. 1 we display the temporal spectrum of the many-body Hamiltonian ˆ H ( x ) as a function of the barrier’s position for different inter-particle interaction strengths. It isclear that whenever the barrier is far from the center of the trap ( x = ± ) the spectrumresembles a well-known spectrum of two particles confined in the harmonic confinement[39] and the ground state is well-isolated from other many-body eigenstates. When thebarrier approaches the center of the trap ( x ≈ ), a characteristic quasi-degeneracy of ynamical resistivity of a few interacting fermions -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = 0.0 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = 0.6 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = 1.2 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = 1.8 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = -0.4 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = -0.8 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = -1.2 -4.0 0.0 4.0 0.0 -4.000.20.40.60.81 g = -1.6 Figure 2.
Time evolution of the temporal fidelity F ω for different interactions g and differentfrequencies ω for the system of N ↑ = N ↓ = 1 . For better comparison, time is expressed in termsof a temporal barrier position x . Note rapid changes of the fidelity when the barrier crossesthe center of the trap. the spectrum is formed. This is a natural consequence of a quasi-degeneracy of the single-particle spectrum in a double-well potential [22, 40]. It is important to note that along withincreasing interactions, the energy gap at x = 0 between the ground-state and excited many-body states becomes smaller. Since this gap has fundamental significance for excitations whena finite-time transition of the barrier is considered, one can suspect that for sufficiently stronginteractions keeping adiabaticity (understood as remaining of the state of the system in thetemporal ground state) may be very challenging.To get a better and quantitative understanding of the dynamical response of the systemdepending on the barrier frequency ω , we calculate the temporal fidelity F ω ( t ) at differenttime frames. These results are shown in Fig. 2 for some chosen values of interactions anddifferent driving frequencies. For convenience, we compare results for different frequencies ω by plotting fidelities as functions of temporal barrier’s position x rather than time. It isclear that in the first stage of the dynamics, i.e. , before the barrier transits through the areaoccupied by particles, the system remains unaffected by the barrier position. Therefore, in theinitial stage, the system is completely described by the ground state of the time-independentHamiltonian and thus the fidelity is close to unity irrespective of the sign of interactionsand the frequency ω . Then, when the barrier is in the vicinity of the center of the trap,the fidelity rapidly decreases for both, repulsive and attractive, interactions. Of course in theadiabatic limit (very small frequency ω ) the fidelity does not change signaling that the system ynamical resistivity of a few interacting fermions F r e qu e n cy ω -2 -1 0 1 2 0 0.002 0.004 0.006 0.008 0.01 Time t=T/2
FidelityEntropy -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1
Time t=T
FidelityEntropy F r e qu e n cy ω -2 -1 0 1 200.0020.0040.0060.0080.01 FidelityEntropyInteraction g -2 -1 0 1 2 0 0.3 0.6 0.9 1.2 1.5 1.8 FidelityEntropy Interaction g Figure 3.
The temporal fidelity F ω (top row) and the inter-component entanglement entropy S (bottom row) calculated for the system of N ↑ = N ↓ = 1 particles at time t = T / (left column)and t = T (right column) as a function of interaction strength and driving frequency ω . Chaoticbehavior of the fidelity at the final instant t = T is clearly visible. It indicates unpredictabilityof the dynamics after the second transition of the barrier through the trap. In contrast, afterthe first transition ( t = T / ) the dynamics is much more regular and predictable. In both cases,departure from the adiabatic limit is much more intensive for attractive forces. remains in the temporal ground state. Note however that the magnitude of a fidelity’s dropdepends significantly on the sign of interactions and is much deeper in the attractive case.This observation can be directly related to the properties of the many-body spectrum. Asclearly seen in Fig. 1, the energy gap between the many-body ground state and other excitedstates closes much differently in the case of attractive interactions and it is more sensitive tothe strength g . Moreover, we checked that for repulsive interactions the lowest excited statein fact does not contribute to the dynamics of the system. Consequently, an effective gapto the first contributing many-body state is essentially larger. Therefore, to get a significantfidelity drop caused by the first transition of the barrier, one needs to consider scenarios beingmuch further from the adiabatic limit (larger frequencies ω ).After the barrier fully transits through the system we observe again that the fidelity doesnot change in time until returning transition. This occurs for all frequencies, irrespectivelyon the sign of interactions. Again, this behavior can be explained by a closer inspectionof the many-body spectrum. After the transition, decomposition amplitudes of a temporalstate of the system to the temporal many-body eigenstates almost does not change since gapsbetween individual states are much larger than the energy scale defined by the frequency ynamical resistivity of a few interacting fermions ω . In consequence, an overlap of the temporal state with the temporal ground state of thesystem remains almost constant.In Fig. 3, we present snapshots of the temporal fidelity at T / and T as a functionof driving frequency and the interaction strength. These plots compare the response ofthe system to single and two transitions of the moving barrier. Of course, in both cases,in the adiabatic limit ( ω → ) the fidelity remains equal to . Nevertheless, for attractiveinteractions achieving adiabaticity requires much smaller frequencies. As can be seen, afterthe first transition of the barrier ( t = T / ), the fidelity is rather a smooth function ofinteractions and frequency. In contrast, at the final moment ( t = T ) the dependence onthese parameters is highly non-trivial and in practice unpredictable. Random behavior ofthe fidelity at time t = T is related to a non-obvious relative dynamics between the internalmotion of the system and the motion of the barrier. Namely, after the first transition of thebarrier, the system evolves almost independently on the barrier’s position. Depending onfrequency ω the second transition effectively happens at quite randomly chosen moment ofthis evolution. This is particularly the case when after the first transition the system becomesessentially excited (far from the adiabatic limit) and its evolution is highly non-trivial.It is very instructive to notice that the situation is substantially different when, instead ofthe fidelity, one considers quantum correlations induced by the moving barrier. For example,let us focus on the inter-component entanglement entropy defined as S ( t ) = − Tr ↑ [ ˆ ρ ↓ ( t ) ln ˆ ρ ↓ ( t )] = − Tr ↓ [ ˆ ρ ↑ ( t ) ln ˆ ρ ↑ ( t )] , (7)where the temporal reduced density matrix of a chosen component ˆ ρ σ ( t ) can be calculatedstraightforwardly by tracing out remaining component σ (cid:48) as ˆ ρ σ ( t ) = Tr σ (cid:48) [ | Φ( t ) (cid:105)(cid:104) Φ( t ) | ] .Analogously as we did for the fidelity, in the bottom row of Fig. 3 we display the entropy S ( t ) as a function of the interaction strengths and driving frequencies at two different moments t = T / and t = T . It is clear that for both cases the inter-component correlations are builtin the system quite smoothly and nearly symmetrically on both sides of the non-interactinglimit. For increasing frequencies, i.e. , when the system rides away from the adiabatic limit,correlations are almost monotonically enhanced. This regularity of the entanglement entropycan be viewed as a direct consequence of the fact that inter-component quantum correlationsare related to period and strength of interactions rather than to a particular decompositionand specific evolution of the quantum state.
4. Spin-balanced system of four particles
After detailed studies of the simplest case of two distinguishable particles, it is worthconsidering slightly larger systems and check roles played by the quantum statistics andparticle imbalance. To make it systematically, let us first focus on the balanced system with N ↑ = N ↓ = 2 . A corresponding many-body spectrum of the Hamiltonian as a functionof the barrier’s position and different interactions is presented in Fig. 4. It is clear thatthe spectrum has substantially different properties than the corresponding spectrum for twoparticles (Fig. 1). The most prominent difference is related to the existence of two barrier ynamical resistivity of a few interacting fermions -4 -2 0 2 445678910 g = 0.0 -4 -2 0 2 444.555.566.577.588.59 g = 0.4 -4 -2 0 2 44.555.566.577.588.59 g = 1.0 -4 -2 0 2 42.533.544.555.566.577.58 ( + ) g = -1.0 Figure 4.
The many-body spectrum of the Hamiltonian ˆ H ( x ) for N ↑ = N ↓ = 2 particlesas a function of barrier position x and different interaction strengths. Note the substantialdifference when compared to the system of two particles in Fig. 1. Here, the spectrum becomesquasi-degenerated for two distant positions of the barrier. The difference significantly changesthe dynamical properties of the system. See the main text for details. F r e qu e n cy ω -2 -1 0 1 2 0 0.002 0.004 0.006 0.008 0.01 Time t=T/2
FidelityEntropy -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1
Time t=T
FidelityEntropy F r e qu e n cy ω -2 -1 0 1 200.0020.0040.0060.0080.01 FidelityEntropyInteraction g -2 -1 0 1 2 0 0.5 1 1.5 2 2.5 3 3.5 FidelityEntropy Interaction g Figure 5.
The temporal fidelity F ω (top row) and the inter-component entanglement entropy S (bottom row) calculated for the system of N ↑ = N ↓ = 2 particles at time t = T / (left column)and t = T (right column) as a function of interaction strength and driving frequency ω . Incontrast to the two particle case (Fig. 3) chaotic behavior of the fidelity is present at the finalinstant ( t = T ) as well as after the first transition of the barrier ( t = T / ). ynamical resistivity of a few interacting fermions F r e qu e n cy ω -2 -1 0 1 2 0 0.002 0.004 0.006 0.008 0.01 0 0.2 0.4 0.6 0.8 1 Fidelity Time t=T/4
Interaction g -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 EntanglementTime t=T/4
Interaction g Figure 6.
The temporal fidelity F ω and the inter-component entanglement entropy S calculatedfor the system of N ↑ = N ↓ = 2 particles when the barrier is at the center of the trap ( t = T / ).As explained in the main text, in this case, the quasi-degeneracy of the spectrum is reachedonly once and therefore the dynamical response of the system is much more regular. positions at which the quasi-degeneracy of the many-body states is present. It may suggestthat the dynamical behavior of the system may be significantly different already after thefirst transition of the barrier. Indeed, as clearly seen in Fig. 5, the temporal fidelity isunpredictable not only at the final moment t = T but also after the first transition t = T / .The phenomenological explanation of this behavior is exactly the same as previously. Afterthe stage the first quasi-degeneracy of the many-body spectrum is present, the dynamicsbecomes almost independent of the barrier position. However, it remains significantlyaffected by particular decomposition (which in fact depends on frequency ω ) to differentexcited states. Therefore, depending on the frequency ω , a moment when the stage of thesecond quasi-degeneracy of the spectrum is activated is highly unpredictable and leads to avery irregular pattern of the fidelity. To make sure that this general picture is correct andargumentation is valid we also check the results for the moment just after the first quasi-degeneracy is deactivated, i.e. , when the barrier is exactly at the center ( x = 0 , t = T / ).Corresponding results are displayed in Fig. 6. It is clear, that at this instant the temporalfidelity is a very smooth function of frequency and interactions. It resembles correspondingresults for N ↑ = N ↓ = 1 at t = T / (see Fig. 3). For completeness and better comparison, inFig. 5 and Fig. 6, we also present entanglement entropy S ( t ) . In this case, similarly as in thecase of two particles, quantum correlations are built in the system much more regularly anddepend mostly on interaction strength and interaction period.
5. Spin-imbalanced systems
Finally, let us discuss the dynamical response of the system having components with anunequal number of particles. These kinds of systems may have substantially differentproperties since, by the construction, they break the symmetry between components. Thedifference is visible already at the initial moment – individual components have different ynamical resistivity of a few interacting fermions -4 -2 0 2 422.533.544.555.566.57 g = 0.0 -4 -2 0 2 42.533.544.555.566.57 g = 0.4 -4 -2 0 2 42.533.544.555.566.57 g = 1.0 -4 -2 0 2 41.522.533.544.555.566.57 ( + ) g = -1.0 -4 -2 0 2 45678910 g = 0.0 -4 -2 0 2 45678910 g = 0.4 -4 -2 0 2 45678910 g = 1.0 -4 -2 0 2 4456789 ( + ) g = -1.0 Figure 7.
The many-body spectrum of the Hamiltonian ˆ H ( x ) as a function of barrier position x for imbalanced systems with N ↓ = 1 and N ↑ = 2 or particles and different interactionstrengths. Note the qualitative difference of these spectra when compared to balanced cases(Fig. 1 and Fig. 4). Here, independently on the interaction strength, the many-body ground-state is always well-isolated from excited states. This causes much better resistivity of thesystem to the external driving (see Fig. 8). density distributions and different spatial sizes. It means that during the evolution theybecome affected by the barrier at different moments and with different intensities. It shouldbe noted however, that although the smaller component becomes directly influenced by thebarrier later, it is affected by its motion earlier due to interactions with the larger componentbeing already affected. Thus naively one can suspect that the dynamics of imbalancedsystems, as being influenced by a larger number of different parameters, should be moreunpredictable. In fact, this phenomenological reasoning is not fully justified. To make itclear let us first analyze the many-body spectra of these systems and their dependence oninteractions and the barrier’s position. We show them in Fig. 7 for two systems with thetotal number of three and four particles. Although their general appearance is analogousto the eigenspectra of the balanced counterparts, one should note a fundamental qualitativedifference – independently of the interaction strength and the barrier position the many-body ground state is always well-isolated from the excited ones. This finite gap leads directlyto a high resistivity of the initial state | G (cid:105) to the driving, provided that the frequency ω is not too high, i.e. , the system remains willingly in its temporal ground state | G ( x ( t )) (cid:105) during the evolution and thus reaching the adiabatic limit is much easier. This effect iswell-captured by the temporal fidelity which remains close to unity for a finite range of ynamical resistivity of a few interacting fermions F r e qu e n cy ω -2 -1 0 1 2 0 0.002 0.004 0.006 0.008 0.01 0.012 Time t=T/2
FidelityInteraction g -2 -1 0 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time t=T
Fidelity Interaction g Figure 8.
The temporal fidelity F ω for the imbalanced system of N ↑ = 2 and N ↓ = 1 particlesat time t = T / and t = T (left and right column, respectively) as a function of interactionstrength and driving frequency ω . It is clear that in both cases, independently of interactions,the adiabatic limit is easily reached for not too high frequencies. It means that the dynamicalresistivity of the system is enhanced. small frequencies independently on interactions (see Fig. 8 for the system with N ↑ = 2 and N ↓ = 1 particles). Of course, for high enough frequencies the system becomes excitedto other many-body eigenstates and the fidelity rapidly drops to 0. This drop is howeverrather smooth and only little irregular deviations (mainly for relatively strong interactions)are visible. Counterintuitively, it means that the imbalanced systems subjected to the externaldriving are much more resistive and predictable when subjected to the moving barrier thantheir balanced counterparts.
6. Conclusions
In this work, we studied the dynamical resistivity of strongly correlated few-fermion systemsto the driving by an external potential barrier. Assuming that the system is initially preparedin its interacting many-body ground state and by solving exactly the time-dependent many-body Schr¨odinger equation we examine how the system drives away from the temporal many-body ground state. It turned out that the dynamics of the system is always regular up to theinstant when a quasi-degeneracy of the many-body spectrum is achieved. For later moments,due to a quite tangled relation between ballistic motion of the system, correlations inducedby interactions, barrier position and speed, and corresponding changes of the many-bodyspectra the dynamics become highly unpredictable and very sensitive to different parameters.Importantly, moments in which the quasi-degeneracy of the spectrum is established highlydepend on the number of particles and may occur more than once during the barriertransition. Moreover, we found that systems with an imbalanced number of particles havea non-vanishing energy gap between the ground and excited states for any interaction andany barrier’s position. Therefore, imbalanced systems are much more resistant to externaldriving. ynamical resistivity of a few interacting fermions
7. Acknowledgments
This work was supported by the (Polish) National Science Center Grant No.2016/22/E/ST2/00555. Numerical calculations were partially carried out in the Interdisci-plinary Centre for Mathematical and Computational Modelling, University of Warsaw (ICM),under Computational Grant No. G75-6.
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