Dynamical control of the conductivity of an atomic Josephson junction
DDynamical control of the conductivity of an atomic Josephson junction
Beilei Zhu, ∗ Vijay Pal Singh, Junichi Okamoto,
2, 3 and Ludwig Mathey
1, 4 Zentrum für Optische Quantentechnologien and Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany EUCOR Centre for Quantum Science and Quantum Computing,University of Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: September 25, 2020)We propose to dynamically control the conductivity of a Josephson junction composed of twoweakly coupled one dimensional condensates of ultracold atoms. A current is induced by a peri-odically modulated potential difference between the condensates, giving access to the conductivityof the junction. By using parametric driving of the tunneling energy, we demonstrate that thelow-frequency conductivity of the junction can be enhanced or suppressed, depending on the choiceof the driving frequency. The experimental realization of this proposal provides a quantum sim-ulation of optically enhanced superconductivity in pump-probe experiments of high temperaturesuperconductors.
I. INTRODUCTION
Recently, light-induced or enhanced superconductivityhas been discovered in superconducting materials suchas YBa Cu O x [1–5] or K C [6, 7] using pump-probe techniques with mid-infrared lasers. This has trig-gered theoretical investigations of the origin and mech-anism of optical control of superconductivity. Basedon microscopic models, various mechanisms have beenproposed such as enhancement of electron-phonon cou-pling [8–12], control of competing order [13–18], photo-induced η -pairing [19–21] and cooling in multi-band sys-tems [22, 23]. Meanwhile, phenomenological approacheshave been used to understand the effect of supercon-ducting fluctuations in photo-excited systems [24–32]. InRefs. [25–27], we proposed a mechanism based on para-metrically driven Josephson junctions for light-enhancedsuperconductivity. This mechanism is also reflected in aredistribution of current fluctuations, such that the low-frequency part of the system is effectively cooled downleading to an enhancement of the inter-layer tunnelingenergy, see Ref. [25].Given the complexities of light-induced dynamics instrongly correlated solids, it is conceptually instructiveto explore proposed mechanisms in a well-defined phys-ical system, which isolates specific features of the solidstate system. In particular, cold atom systems arehighly tunable model systems, that provide toy modelsfor more complex systems, in the spirit of quantum sim-ulation. In this paper, we will utilize the ability of coldatom technology to design and control Josephson junc-tion systems. Atomic Josephson junctions [33–39] havebeen realized experimentally to study coherent transport[40, 41], atomic conductivity [42], and the dynamics ofJosephson junctions of two dimensional cold atomic gases[43, 44]. This provides an ideal platform to simulate the ∗ [email protected] dynamics of a parametrically driven Josephson junction.In this paper, we propose to perform dynamical controlof the conductivity of a Josephson junction composed oftwo weakly coupled one dimensional (1D) condensates,see Fig. 1(a). This proposal is motivated by the mech-anism of parametrically enhanced conductivity, that weestablished in Refs. [26, 27]. For that purpose, two dy-namical processes have to be introduced in the system ofcoupled condensates: One is the analogue of the probingprocess, and the second one is the analogue of the pump-ing process or optical driving. As shown in Fig. 1(b), weimplement driving and probing via periodical modulationof the tunneling energy and the potential difference be-tween the condensates, respectively. The probe induces acurrent, allowing us to determine the conductivity σ n ( ω ) of the junction of neutral atoms. To serve as a quantumsimulation for a Josephson junction of charged particles,we will determine the relation of σ n ( ω ) to the conductiv-ity σ c ( ω ) of a junction of charged particles below, wherewe demonstrate that σ n ( ω ) is inversely proportional to σ c ( ω ) . This interpretation derives from the difference ofa U(1) symmetry of a system of neutral particles and aU(1) gauge symmetry of a system of charged particles.Using classical field simulations, we show that the densityimbalance between the condensates is suppressed at lowprobe frequencies when the parametric driving frequencyis above the Josephson plasma frequency, and enhancedbelow the plasma frequency, which constitutes dynam-ical control of conductivity. Based on a two-site Bose-Hubbard (BH) model, we derive analytical expressionsfor /σ n ( ω ) for an undriven and a driven system. Thecomparison between the analytical estimates and the sim-ulations shows good agreement. We note that while thekey physics occurs in the motion of the relative phase be-tween the condensates, the 1D geometry of the two sub-systems of the junction serves as an entropy bath, whichslows down the heating of the system. This reduced heat-ing rate of the system allows for a long probing time usedin this proposal. Finally, we relate the dynamical renor- a r X i v : . [ c ond - m a t . qu a n t - g a s ] S e p driving ! d
We consider a Josephson junction composed of two 1Dcondensates, as shown in Fig. 1(a). We numerically sim-ulate the dynamics of this system using the classical fieldmethod of Ref. [52]. For the numerical implementation,we represent the system of two coupled condensates, as alattice model, which takes the form of a BH Hamiltonian ˆ H = − (cid:88) (cid:104) α,β (cid:105) J αβ ˆ b † α ˆ b β + U (cid:88) α ˆ n α (ˆ n α − . (1) ˆ b † α (ˆ b α ) is the bosonic creation (annihilation) operator atsite α . (cid:104) α, β (cid:105) denotes nearest neighbour sites α and β .The lattice dimensions are N x × N z , where we choose N x = 50 and N z = 2 . Each site index α = ( i, j ) en-codes the two coordinates i and j , with i ∈ [1 , and j = 1 , . ˆ n α = ˆ b † α ˆ b α is the number operator at site α .Along the z- direction, the tunneling energy J αβ is givenby J z , which is the tunneling energy of the double-wellpotential. In the undriven system, this tunneling energyis a constant, J z = J . We will use J as the energy scalethroughout this paper. To capture the continuous 1Dcondensates numerically, we discretize the motion alongthe x- direction, with a discretization length l x . This re-sults in an effective tunneling energy J x = (cid:126) / (2 ml x ) ,where m is the atomic mass and (cid:126) is the reduced Planckconstant [53]. In this discretized representation, the on-site repulsive interaction is determined by U = g /l x ,where g = 2 (cid:126) a s / ( ma y a z ) . a s is the s- wave scatteringlength and a y ( a z ) is the oscillator length due to trapconfinement along y- ( z- ) direction. In the following, weset (cid:126) = 1 . We use U = 0 . J throughout this paper. Inour classical field representation, we replace the opera-tors in Hamiltonian (1) and its corresponding equations FIG. 2. Density imbalance ∆ n ( t ) as a function of time t . The system is subjected to the probing term of Eq. (2) with anamplitude V = 0 . J and a frequency ω p = 2 π × . J . Time t is displayed in units of the probing period T p = 500 J − .The dashed lines in (a)-(c) are ∆ n ( t ) , averaged over 500 realizations. The continuous lines are the low-frequency filtered ∆ n ( t ) ,where we use a Gaussian filter with a time scale of . T p , see text. The low-frequency filtered ∆ n ( t ) of (a)-(c) is displayedin (d) as well, for comparison. The probing frequency is significantly smaller than the resonance frequency of the junction,which is ω Jp = 2 . J . In (a) we display ∆ n ( t ) of the undriven system. In (b) and (c) we drive the system parametrically, seeEq. (4). In (b) the driving frequency is ω d /ω Jp = 1 . , and therefore blue-detuned, in (c) it is ω d /ω Jp = 0 . , and thereforered-detuned. The comparison in (d) demonstrates dynamical control of the low-frequency response of the junction. of motion by complex numbers. We sample initial statesfrom a grand canonical ensemble with chemical poten-tial µ and temperature T . We choose J x = 3 . J and T = 0 . J /k B , k B being the Boltzman constant, and ad-just µ such that the density per site is n = 2 . For theprobe, we add the following term ˆ H pr = V ( t ) · ∆ ˆ N , (2)where the number imbalance ∆ ˆ N is ∆ ˆ N = 12 (cid:88) i (cid:0) ˆ n ( i, − ˆ n ( i, (cid:1) (3)with V ( t ) = V cos( ω p t ) . V is the probe amplitude and ω p the probe frequency. The oscillating potential inducesan oscillating current and density motion between thecondensates, which we use to determine the conductivity,as we describe below. To simulate parametric driving, wemodulate the tunneling energy J z as J z ( t ) = J [1 + A cos ( ω d t )] , (4)where A is the driving amplitude and ω d the driving fre-quency. As a key quantity to determine the conductivity σ n ( ω ) , we calculate the density imbalance, averaged overeach 1D condensate, ∆ n ( t ) ≡ (cid:104) ∆ ˆ N ( t ) (cid:105) N x = 2 n N (cid:104) ∆ ˆ N ( t ) (cid:105) (5) where (cid:104) ... (cid:105) denotes the average over the thermal ensembleand N is the total particle number in the system.In Fig. 2 we present an example that demonstrates themain physical effect that we propose to realize experimen-tally. We show the time evolution of the density imbal-ance ∆ n ( t ) , averaged over 500 trajectories, as a functionof time. The system of coupled condensates is subjectedto a probing term with V = 0 . J and a small probingfrequency of ω p = 2 π/T p = 2 π × . J . The probingperiod T p is used as a time scale in Fig. 2. In Fig. 2(a),we show ∆ n ( t ) for an undriven system, which displayshigh frequency fluctuations due to thermal noises. Wefilter these fluctuations using a filter function with Gaus-sian kernel g ( t ) = exp( − t /σ t ) . We choose the time scale σ t = T p / . The low-frequency part of the motion of thedensity imbalance displays a periodic motion at the prob-ing frequency ω p . It is depicted in Fig. 2(a), in additionto the unfiltered data, and also in Fig. 2(d), to be com-pared to the density motion of a parametrically drivensystem, as we describe below.We now calculate ∆ n ( t ) for a driven junction. Wedrive the tunneling energy between the condensates asdescribed by Eq.(4). We use the driving amplitude A = 0 . . In Figs. 2(b) and (c), we show ∆ n ( t ) fora blue-detuned ( ω d /ω Jp = 1 . ) and a red-detuned( ω d /ω Jp = 0 . ) driving frequency, respectively. ω Jp is the Josephson plasma frequency, which we estimate as ω Jp = (cid:112) J ( J + U n ) , as we describe below. For theparameter choice of this example, we have ω Jp = 2 . J ,which we use as a frequency scale for the driving fre-quency. We note that this choice of the driving ampli-tude and driving frequency is outside of the parametricheating regime, which allows for a long driving time.As depicted in Figs. 2 (b) and (c), we also determinethe low-frequency filtered density imbalance, which wecalculate via Gaussian filtering as described above. Wecompare these averaged values in Fig. 2(d). The am-plitude of the oscillation of the density imbalance is in-creased due to parametric driving with a red-detuneddriving frequency, and decreased due to driving with ablue-detuned frequency. This observation exemplifies themain point of parametric control of conductivity, for anatomic Josephson junction. For red-detuned driving, thelow-frequency limit of the response to a potential dif-ference between the two condensates is increased. Toachieve the same response statically, a larger tunnelingenergy would be required. This dynamically induced be-haviour is therefore parametrically enhanced superfluid-ity. For blue-detuned driving, the amplitude of the os-cillation of the density imbalance is reduced, which in-dicates a reduction of superfluidity. This constitutes theessence of control of conductivity via parametric driving.We elaborate on this observation below and relate it tothe conductivity of a parametrically driven junction ofcharged particles. As we demonstrate, the frequency de-pendence is inverted: For blue-detuned parametric driv-ing, the superconducting response is enhanced, for red-detuned driving the response is reduced. III. CONDUCTIVITY
In this section, we derive the conductivity of a Joseph-son junction of neutral particles and of charged particles,at linear order. The resulting expressions are propor-tional to the inverse of each other, as we discuss below.To provide an estimate for the conductivity of anatomic junction, we consider a two-site BH model inphase-density representation, in linearized form: H = (cid:18) J n + U (cid:19) ∆ n + J n θ + V ( t )∆ n. (6) θ is the phase difference of the two condensates. ∆ n =( n − n ) / is the density imbalance. The equations ofmotion are ∆ ˙ n = 2 J n θ, (7) ˙ θ = − (cid:18) J n + U (cid:19) ∆ n − V ( t ) . (8)Eliminating ∆ n , we obtain an equation of motion for θ , ¨ θ + γ ˙ θ + ω θ = − ˙ V ( t ) , (9)where γ is included phenomenologically to describedamping. ω Jp = (cid:112) J ( J + U n ) is the Josephson plasma frequency, as stated in Sec. II. The Fourier trans-form of Eq. (9) can be written as θ ( ω ) = − iωV ( ω ) ω − ω + iγω , (10)which relates the phase to the external probing potential.The particle current is determined by j = − ∆ ˙ n . The mi-nus sign is explicitly included to specify that a positive j refers to a current flowing from condensate to conden-sate , and a negative j to the opposite direction. Theconductivity of a junction of neutral particles is definedas σ n ( ω ) ≡ j ( ω ) /V ( ω ) . Combining Eqs. (7) and (10), weobtain σ n ( ω ) = 2 J n iωω − ω + iγω . (11)So the conductivity of an atomic junction is a Lorentianwith its maximum at the resonance frequency ω Jp , mul-tiplied by the frequency ω .To develop the relation between the transport acrossan atomic junction and a junction of charged particles, wederive the conductivity of the RCSJ model of a junction.The linearized equation of motion for the phase differenceof a Josephson junction is [26] ¨ φ + γ c ˙ φ + ω Jp,c φ = ˜ I, (12)where ˜ I ≡ ω Jp,c
I/J c , with I being the external current. ω Jp,c = (2 e/ (cid:126) ) J c /C is the Josephson plasma frequency,where J c is the bare Josephson coupling energy and C isthe capacity determined by the geometry of the junction.The conductivity is defined as σ c ( ω ) ≡ Id/V c , where d isthe distance between the superconductors. The voltagedifference across the junction is given by the Josephsonrelation, V c = (cid:126) e ˙ φ , where e is the charge of an electron.We then obtain σ c ( ω ) = (cid:126) e Cdiω (cid:0) ω − ω Jp,c + iγ c ω (cid:1) . (13)Comparing Eqs. (11) and (13), we observe that theconductivity σ n ( ω ) and σ c ( ω ) are proportional to eachother’s inverse, i.e., /σ n ( ω ) ∼ σ c ( ω ) . (14)This relation motivates us to display /σ n ( ω ) through-out this paper, for example in Fig. 1(c) and Fig. 4. Thisquantity features a /ω divergence in its imaginary part,and a zero crossing at the resonance frequency, and there-fore directly resembles the conductivity of a junction ofcharged particles.The origin of this relation derives from a comparisonof Eq. (9) and (12). In both cases, the equations havethe form of a driven oscillator. This results in a lin-ear relation between the current and the potential, whenexpressed in frequency space. The phase of the atomicjunction relates to the current, at linear order, and is Ϭ ͘ Ϭ Ϭ Ϭ ͘ Ϭ ϱ Ϭ ͘ ϭ Ϭ Ϭ ͘ ϭ ϱ Ϭ ͘ Ϯ Ϭ ω p /J Ϭ ͘ ϱ ϭ ͘ Ϭ ϭ ͘ ϱ Ϯ ͘ Ϭ Ϯ ͘ ϱ ϯ ͘ Ϭ ω p / ŵ [ / σ n ( ω p ) ] Ƶ Ŷ Ě ƌ ŝ ǀ Ğ Ŷ ω d < ω Jp , A = 0 . ω d > ω Jp , A = 0 . ω d > ω Jp , A = 0 . FIG. 3. Numerical simulation results for ω p Im [1 /σ n ( ω p )] of the undriven system (grey circles), for red-detuned driving(purple circles) with ω d /ω Jp = 0 . , and for blue-detuneddriving with ω d /ω Jp = 1 . and driving amplitudes A = 0 . (diamonds) and A = 0 . (triangles). therefore the quantity that responds to the external per-turbation − ˙ V ( t ) in Eq. (9). However, for the electronicjunction, the phase is related to the external potential,due to the gauge theoretical relation of phase and vec-tor field, whereas the inhomogeneous term in Eq. (12)is the current. Therefore the roles of current and exter-nal potential are reversed between Eq. (9) and Eq. (12),resulting in the inverse response function. IV. NUMERICAL RESULTS
We present how the inverse of the conductivity /σ n ( ω ) is affected by parametric driving at a blue-detuned driving frequency of ω d /ω Jp = 1 . and ared-detuned driving frequency of ω d /ω Jp = 0 . . InFig. 3, we show ω p Im [1 /σ n ( ω p )] in the low probing fre-quency regime. For each ω p , we determine the time evo-lution of ∆ n ( t ) over a time duration of T p , and ex-tract the Fourier component ∆ n ( ω p ) . For the undrivensystem, ω p Im [1 /σ n ( ω p )] approaches a constant value of ∼ ω / (2 J n ) in the limit of ω p → . In the pres-ence of parametric driving, the low frequency response ismodified. When the driving frequency is larger than theJosephson plasma frequency, i.e., ω d > ω Jp , the quan-tity ω p Im [1 /σ n ( ω p )] is enhanced for ω p < ω d − ω Jp , in-dicating a reduced effective tunneling energy across thejunction. The magnitude of enhancement depends on thedriving amplitude, as shown in Fig. 3. For larger drivingamplitude A , ω p Im [1 /σ n ( ω p )] shows a larger enhance-ment. Above the frequency difference, i.e., ω p > ω d − ω Jp ,the quantity ω p Im [1 /σ n ( ω p )] is reduced. This obser-vation that the enhancement of ω p Im [1 /σ n ( ω p )] at lowprobing frequency limit is accompanied by the reduc- Ϭ ͘ Ϭ Ϭ ͘ ϱ ϭ ͘ Ϭ ϭ ͘ ϱ Ϯ ͘ Ϭ Ϯ ͘ ϱ ϯ ͘ Ϭ ω p /J Ϭ Ϯ ϰ ϲ ϴ ϭ Ϭ / ŵ [ / σ n ( ω p ) ] Ƶ Ŷ Ě ƌ ŝ ǀ Ğ Ŷ Ĩ ŝ ƚ Ϯ ͘ ϭ Ϯ ͘ Ϯ Ϯ ͘ ϯ Ϯ ͘ ϰ Ϯ ͘ ϱ Ϭ ͘ ϭ Ϭ ͘ Ϭ Ϭ ͘ ϭ / ŵ [ / σ n ( ω p ) ] FIG. 4. Im [1 /σ n ( ω p )] as a function of the probing fre-quency ω p for an undriven system, obtained from the numer-ical simulation (circles) and analytical prediction of Eq. (15).The probing amplitude is V = 0 . J . We obtain the fit-ting parameters for the density n = 1 . , for the damp-ing γ = 0 . J and for the Josephson plasma frequency ω Jp = 2 . J . Inset: zoom-in near ω Jp . The green ver-tical line indicates the analytical estimate of the Josephsonplasma frequency of ω Jp = 2 . J . tion of ω p Im [1 /σ n ( ω p )] above the frequency difference,i.e., ω p > ω d − ω Jp , is reminiscent of the redistributionof phase fluctuations described in [25]. On the otherhand, for a red-detuned driving frequency, i.e., ω d < ω Jp , ω p Im [1 /σ n ( ω p )] is reduced for ω p < ω d − ω Jp while in-creased for ω p > ω d − ω Jp . We note that the enhancementand suppression effect is most pronounced for ω d close to ω Jp . For ω d far away from ω Jp , the effect of enhancementand suppression is diminished. V. PARAMETRIC CONTROL OFCONDUCTIVITY
Based on the analytical estimate of the conductivitythat we presented in Sec. III, the inverse of the conduc-tivity is /σ n ( ω ) = 12 J n iω (cid:0) ω − ω + iγω (cid:1) . (15)In Fig. 4, we show the numerical results for Im [1 /σ n ( ω p )] .It displays a /ω divergence in the low frequency regime,that is associated with the low-frequency behaviour of theconductivity of a superconductor. We fit the numericaldata with formula (15), which gives for the condensatedensity n = 1 . , for the damping γ = 0 . J andfor the plasma frequency ω Jp = 2 . J . We note thatthe value of n is close to the value of the numericalsimulations, and the value of ω Jp is close to the analyticalestimate ω Jp ≡ (cid:112) J ( J + U n ) ≈ . J . The zerocrossing of Im [1 /σ n ( ω p )] signifies the Josephson plasmafrequency. Again, we find that the analytical estimate isclose to the numerical finding. To indicate the magnitudeof the deviation from the linearized estimate, we displayIm [1 /σ n ( ω p )] in the vicinity of the resonance in the inset.Small deviations are visible around the resonance, wherenonlinear contributions are noticeable, due to the largeamplitudes of the motion near resonance.We now determine how the conductivity σ n ( ω ) is mod-ified by parametric driving. This analysis is closely re-lated to the analysis presented in Ref. [26]. We replace J by J ( t ) in Eq. (6). The equation of motion for ∆ n ( t ) is ∆¨ n = 2 ˙ J ( t ) n θ + 2 J ( t ) n ˙ θ − γ ∆ ˙ n, (16)where we include damping term phenomenologically witha damping parameter γ . We note that time dependenceof J ( t ) contributes an additional term on the right-handside, compared to Ref. [26]. This term is of the form ofa damping term as well, in which the damping rate ismodulated in time. The oscillatory time dependence of J ( t ) couples the mode ∆ n ( ω p ) at the probing frequencyto the modes ∆ n ( ω p ± mω d ) , where m ∈ Z . Using a threemode expansion, we write ∆ n ( t ) = (cid:80) j ∆ n ( ω j ) exp( iω j t ) ,where ω j = ω p + jω d with j = 0 , ± . The full expressionfor /σ n ( ω ) is given in Eq. (A3). In the limit of ω p → ,we obtain lim ω p → Im[ ω p /σ n ( ω p )] = 12 J n A (cid:0) J + ω (cid:1) ( ω d − ω ) (cid:0) J − ω d + ω (cid:1) − ω (cid:104) γ ω d + (cid:0) ω d − ω (cid:1) (cid:105) A ( ω d − ω ) (cid:16) J − ω d + ω (cid:17) − γ ω d − (cid:16) ω d − ω (cid:17) . (17)This modified expression depends on the driving ampli-tude and the driving frequency, and the damping param-eter γ . We note that A appears in the denominator aswell. This is due to the ˙ θ term that couples the probe to J ( t ) , which in turn leads to a probe input of three modesat frequencies ω p , ω p + ω d , ω p − ω d . Using an expansionwith more and more modes, we expect that the contri-bution of A to the denominator to play a lesser role.In Fig.(5), we compare the analytical result based onthe three-mode expansion, Eq. (A3), the numerical resultbased on the Eqs. (7) and (8), and the numerical simu-lation results of the two coupled condensates. We showthe case of blue-detuned driving, ω d /ω Jp = 1 . andthe case of red-detuned driving, ω d /ω Jp = 0 . , bothwith A = 0 . . We use the parameters n = 1 . and γ = 0 . J for the three mode expansion and numericalresult based on Eqs. (7) and (8). The numerical resultof Eqs. (7) and (8) matches the numerical simulation re-sult well. The three mode expansion gives a qualitativeestimate of enhancement and reduction of /σ n ( ω ) . Theoverall shape of the response is that of a resonance polelocated near ω d − ω Jp ≈ . J , broadened by the damp-ing parameter γ , which depends on the temperature andnonlinear terms. VI. MECHANISM
To describe the physical origin of the dynamical con-trol effect that we present in this paper, we consider theequation of motion ¨ φ + γ ˙ φ + ω [1 + A cos ( ω d t )] sin φ = I ( t ) . (18) This is the RCSJ model of a Josephson junction ofcharged particles, see Eq. (12), with an additional para-metric modulation of the Josephson energy, see Ref. [26].Due to the similarity to an atomic Josephson junction,see e.g. Eqs. (9) and (16), this discussion providesan intuition for atomic junctions as well, with the re-interpretation of terms, discussed in Sec. III.Interpreted as a mechanical model, this equation de-scribes a particle moving in a cosine potential, as de-picted in Fig. 6. This is the tilted-washboard poten-tial representation of the RCSJ model [54] with V ( φ ) = − I ( t ) φ − ω [1+ A cos( ω d t )] cos φ . The probe current I ( t ) plays the role of tilting the potential up- and downward.If the external potential oscillates in time, the washboardpotential is modulated with a linear gradient, oscillatingin time. The Josephson plasma frequency ω Jp is the fre-quency of a particle oscillating around a minimum. Theparametric driving term, given by A cos( ω d t ) correspondsto a modulation of the height of the potential in time, asshown in Fig. 6(b).To describe the origin of the renormalization of thelow-frequency response, and its sign change for drivingfrequencies above and below the resonance frequency, wepresent a perturbative argument. This approach providesan estimate of the conductivity that corresponds to theresult of the three-mode expansion, expanded to secondorder. We expand φ = φ (0) + Aφ (1) + A φ (2) + ... by treat-ing the driving amplitude A as the expansion parameter.Assuming small amplitudes of the phase oscillation, welinearize sin φ . Inserting the expansion series into Eq. Ϭ ͘ Ϭ Ϭ ͘ ϭ Ϭ ͘ Ϯ Ϭ ͘ ϯ ω p /J ϭ ͘ Ϯ ϭ ͘ ϯ ϭ ͘ ϰ ϭ ͘ ϱ ω p / ŵ [ / σ n ( ω p ) ] Ϭ ͘ Ϭ Ϭ ͘ ϭ Ϭ ͘ Ϯ Ϭ ͘ ϯ ω p /J ϭ ͘ ϭ ϭ ͘ Ϯ ϭ ͘ ϯ ϭ ͘ ϰ ω p / ŵ [ / σ n ( ω p ) ] ; Ă Ϳ ; ď Ϳ Ƶ Ŷ Ě ƌ ŝ ǀ Ğ Ŷ ω d > ω Jp ω d < ω Jp FIG. 5. Comparison of the numerical simulations (circles,diamonds, triangles) with the three mode expansion (purpledashed lines) of Eq. (A3) and the numerical solution (purplecontinuous lines) of Eqs. (7) and (8) with the parameters n =1 . and γ = 0 . J . The probe amplitude is V = 0 . J .Driving amplitude is A = 0 . and the driving frequency is ω d /ω Jp = 1 . for blue-detuned driving, and ω d /ω Jp = 0 . for red-detuned driving. (18), we the obtain the equations ¨ φ (0) + γ ˙ φ (0) + ω φ (0) = I ( t ) , (19) ¨ φ ( n ) + γ ˙ φ ( n ) + ω φ ( n ) = − ω A cos( ω d t ) φ ( n − (20)in zeroth and n -th order in A , respectively. We notethat the n -th order solution is multiplied with cos( ω d t ) to provide the source term for the ( n + 1) -th order. Witha monochromatic probe current I ( t ) = I e − iω p t , the so-lutions to Eqs. (19) and (20), up to second order contri- FIG. 6. In (a) and (b) we depict the washboard potentialrepresentation of junction dynamics. In (a) we show the in-fluence of an external current, in (b) we show the influenceof parametric modulation of the junction energy. In (c), weshow the response of the phase of the probe, and indicate thefirst and second order contributions to the modified responseat ω p . butions, are φ (0) = I e − iω p t ω − ω p − iγω p , (21) φ (1) = − ω ( A/ φ (0) e − iω d t ω − ( ω p + ω d ) − iγ ( ω p + ω d ) , (22) φ (2) = − ω ( A/ φ (1) e iω d t ω − ω p − iγω p . (23)Each solution is the solution of a driven harmonic oscil-lator, responding to an external driving term. φ (0) os-cillates at frequency ω p determined by the probe current I ( t ) , as indicated in Eq. (19). In the solution for φ (1) , − ω A cos( ω d t ) φ (0) is the source term, and determinesthat φ (1) oscillates at frequency ω d + ω p . If ω d is closeto the resonance frequency ω Jp , the amplitude of the re-sponse is large. As indicated in Fig. 6(c), the motionof φ (1) will pick up an additional phase of π , when thedriving frequency is above the resonance frequency. Thesecond order correction φ (2) oscillates at low frequencydue to the oscillatory factor e iω d t . Therefore φ (2) is thelowest order contribution to the motion at the probingfrequency. The sign change at the resonance translatesinto φ (2) having a positive or negative sign. InsertingEqs. (21) and (22) into Eq. (23), in the limit of ω p → ,we obtain Re [ φ (2) ] = A ω − ω d ( ω d − ω ) + γ ω d . (24)Therefore, when the system is subjected to a blue-detuned driving, i.e., ω d > ω Jp , the combined terms φ (0) ( ω p ) + φ (2) ( ω p ) have a reduced magnitude, resultingin a stabilization of the phase. Similarly, for red-detuneddriving, φ (2) ( ω p ) has the same sign as φ (0) ( ω p ) , thereforethe response of the phase is increased. For σ c ( ω p ) this im-plies that the conductivity is enhanced for blue-detuneddriving and reduced for red-detuned driving, because thephase is proportional to electric field, while the currentis held fixed. A reduction of the motion of the phaseimplies that the same current is induced with a smallerelectric field, indicating an enhanced conductivity. Forthe conductivity of an atomic junction, the phase is pro-portional to the current, at linear order. So a reduction ofthe phase motion implies a reduction of the conductivity,which occurs at blue-detuned driving, while an increasedcurrent occurs at red-detuned driving, resulting in para-metrically enhanced conductivity. VII. CONCLUSIONS
We have demonstrated parametric enhancement andsuppression of the conductivity of an atomic Josephsonjunction, composed of two weakly coupled 1D conden-sates. This is motivated by our proposed mechanismof parametric enhancement of the conductivity of light-driven superconductors [26], which, in its simplest form,manifests itself in a single, parametrically driven Joseph-son junction. To demonstrate the analogous mechanism in a cold atom system, we discuss the relation betweenthe conductivity of a junction of neutral particles and ajunction of charged particles. We demonstrate that theseare proportional to the inverse of each other. Based onthis analogue, we propose to control the inverse of theconductivity of an atomic junction. We implement para-metric control of the junction by periodic driving of themagnitude of the tunneling energy. We show numeri-cally and analytically that the low-frequency limit of theinverse conductivity is enhanced for parametric drivingwith a frequency that is blue-detuned with regard to theresonance frequency of the junction. Similarly, the in-verse of the conductivity is suppressed for parametricdriving with a red-detuned frequency. This effect con-stitutes the central point of parametric enhancement ofconductivity, which we propose to implement and ver-ify in an ultracold atom system, which serves as a well-defined toy model, in the spirit of quantum simulation.
VIII. ACKNOWLEDGEMENT
This work was supported the DFG in the frameworkof SFB 925 and the excellence clusters âĂŸThe Ham-burg Centre for Ultrafast Imaging- EXC 1074 - project ID194651731 and âĂŸAdvanced Imaging of Matter - EXC2056 - project ID 390715994. B.Z. acknowledges sup-port from China Scholarship Council (201206140012) andEqual Opportunity scholarship from University of Ham-burg. J.O. acknowledges support from Research Foun-dation for Opto-Science and Technology and from GeorgH. Endress Foundation.
Appendix A: three mode expansion solution
To solve Eq. (16), we first substitute θ and ˙ θ usingEqs. (7) and (8). A three mode expansion allows us towrite ∆ n ( t ) = ∆ n ( ω p ) e − iω p t + ∆ n ( ω p + ω d ) e − iω p t − iω d t + ∆ n ( ω p − ω d ) e − iω p t + iω d t . (A1)Now Eq. (16) can be written in matrix form as ω Jp − iγ ∆ − d − ∆ d AJ + ω A/ AJ + ( ω − ω d ) A/ ω − iγω p − ω p AJ + ( ω − ω d ) A/