A minimalistic model for a quantum thermoelectric motor

A minimalistic model for a quantum thermoelectric motor Ulf Bissbort,1 Colin Teo,1 Chu Guo,1 Giulio Casati,2, 3 Giuliano Benenti,2, 4 and Dario Poletti1 1

arXiv:1609.02916v1 [cond-mat.stat-mech] 9 Sep 2016

Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore 2 Center for Nonlinear and Complex Systems, Dipartimento di Scienza e Alta Tecnologia, Universit` a degli Studi dell”Insubria, via Valleggio 11, 22100 Como, Italy 3 International Institute of Physics, Federal University of Rio Grande do Norte, Natal, Brasil 4 Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy We introduce a minimalistic quantum model of coupled heat and particle transport. The system is composed of two spins, each coupled to a different bath and to a particle which moves on a ring consisting of three sites. We show that a spin current can be generated and the particle can be set into motion at a speed and direction which sensitively depends not only on the baths’ temperatures, but also on the various system parameters. This current can withstand dissipative processes, such as dephasing. Furthermore, the particle can perform work against an external driving, thus operating as an efficient quantum motor, whose implementation could be envisaged in trapped ions or solid state systems. A two-particle extension of the model shows that particle interactions can qualitatively change the induced current, even causing current inversion. PACS numbers: 05.70.Ln,05.60.Gg,03.65.Yz

Introduction: A fascinating challenge for future technology is to efficiently convert heat into useful work by means of nanoscale thermal machines. A typical example of energy conversion is coupled (heat and charge) transport as it occurs in a thermoelectric device, where a temperature gradient generates an electric current, or vice versa an electric current can be exploited for refrigeration. The search for optimal nanoscale heat engines and refrigerators has stimulated a large body of activities (for reviews see, e.g., [1–11]). For efficient quantum energy conversion technologies, it is important to investigate what would be optimal design principles. For example, it has been shown that energy filtering [12–14], breaking of time-reversal symmetry [15–17], the presence of conserved quantities (e.g. momentum) [18] or multiterminal setups [19–21] can significantly boost thermoelectric performances. A useful approach, to uncover fundamental principles for improving the performance of energy conversion, is to study minimal models in which different effects can be more easily isolated and analyzed. Minimal models have been used [22–26] to study steady-state heat transfer from thermal reservoirs at different temperatures. In this work, instead, we introduce a minimal model of coupled heat and particle transport. Such a simple, minimalistic set-up simulates two coupled lattices, spin and particles. This set-up allows to study coupled transport both in linear and nonlinear response, as well as in a quantum coherent regime, furthermore allowing for the inclusion of particle interactions. In essence, our model is a minimal quantum simulator for a thermoelectric device. We consider two spins, coupled to two different baths and to one or two (interacting) particles moving on a three-site circuit. In addition to the spin current imposed by the baths, a particle current emerges. We show that particle interactions can significantly alter the transport proper-

ties in the system, even resulting in current inversion. It turns out that the generated particle current is rather robust against a dissipative load (dephasing). We finally show that our model can perform work against an external driving. Therefore, our system can be seen as a minimalistic model of a quantum motor, which moreover can be shown to operate at high efficiency. The model: We consider a system [see Fig. 1(a)] composed of two 1/2-spins and one particle which moves in a minimal circuit consisting of three sites [27]. The evolution of the system is described by a master equation in Lindblad form, acting on the density operator ρˆ : i ˆ dˆ ρ = L(ˆ ρ, t) = − [H(t), ρˆ] + D(ˆ ρ). dt ~

(1)

ˆ is given by The Hamiltonian H ˆ =H ˆσ + H ˆa + H ˆ σa , H ˆ σ = −Jxy (ˆ H σx,1 σ ˆx,2 + σ ˆy,1 σ ˆy,2 ) +

(2) X

hz,l σ ˆz,l ,

l

ˆ a = −J H

X

e−iφ(t) a ˆ†l a ˆl+1 + H.c.,

l

ˆ σa = H

X

κl σ ˆx,l n ˆl,

l=1,2

ˆ σ and H ˆ a are the spin and particle Hamiltonians where H ˆ σa is the coupling between the spins respectively, while H and the particle. Periodic boundary conditions are implicitly understood for the particle. σ ˆi,l is the Pauli spin operator for i = x, y, z, acting on the l−th spin, a ˆl (ˆ a†l ) † annihilates (creates) a particle at site l and n ˆl = a ˆl a ˆl counts the number of particles at site l, φ is a timedependent phase, Jxy is the XY-coupling strength between the spins, 2hz,l sets the Zeeman energy splitting, κl is the spin-particle coupling and J the hopping strength (we set J = 1).

2 (a)

Λ

heat bath 1

1

Jxy

heat bath 2 0.05 (b)

Λ

J

2

0.04

(c)

Jxy =0

0 J =-0.05 xy Jxy =-0.4

Jxy =0.1

-0.05 -0.4 1

0

0

0.4

∆p

(d) Jxy=0.2

-4 (e)

-0.03

Jxy 4

Jxy=1.5

0.3

p2

0.1

0

0.5

0

0

0.5

p1

1

0

0

-0.1

-0.3

0

0.5

p1

1

FIG. 1. (Color online) (a) Minimal transport model: Two spins, each coupled to a different bath and to a particle which can move between three different sites on a ring, generating a current. (b-c) Steady-state particle current J n as a function of the pumping rate ∆p and of the coupling Jxy . (d-e) J n in the p1 –p2 plane for different Jxy . Here the dashed line p1 = p2 of equal driving defines an anti-symmetry axis on which J n = 0. Parameter values: γ = φ = 0, hz = 1, κ = 0.2, ~Λ = 0.4, p = 0.25.

The system undergoes dissipative dynamics, induced by the dissipator D = Dλ + Dγ , where Dλ models the coupling of the spins to the baths and Dγ the particle dephasing. The dissipator Dλ = Dλ,1 + Dλ,2 , with 
Dλ,l (ˆ ρ) = λ + σl+ ρˆσ ˆl− − σ ˆl− σ ˆl+ ρˆ − ρˆσ ˆl− σ ˆl+ l 2ˆ
 +λ− σl− ρˆσ ˆl+ − σ ˆl+ σ ˆl− ρˆ − ρˆσ ˆl+ σ ˆl− , (3) l 2ˆ where the λ± l represent raising or lowering rates for the spins and σ ˆl± = (ˆ σx,l ± iˆ σy,l )/2. The pumping rate for each spin is Λl = λ+ + λ− l l and the relative pumping rate into the upper state is pl = λ+ l /Λl . Henceforth, we focus on the symmetric scenario Λ1 = Λ2 = Λ, κ1 = κ2 = κ, and hz,1 = hz,2 = hz . For uncoupled spins, Jxy = κ = 0, the dissipator Dλ,l fixes the steady-state local density matrix for the l−th spin to be diagonal in the σ ˆz -representation, with occupations pl and 1 − pl for the | ↑i and | ↓i states respectively. Hence, an effective inverse spin temperature βl can be defined by 2hz,l βl = ln [(1 − pl )/pl ]. For non-zero couplings Jxy and κ, the

density matrix is no longer separable and an effective temperature for the spins cannot be defined. Therefore, to characterize the effect of the baths, rather than their temperatures we use the parameters p = (p1 + p2 )/2 and ∆p = p1 − p2 and study the system’s response (e.g. the particle current) to the gradient ∆p. The dissipator Dλ induces a spin current, while the coupling between the spins and the particle generates a particle current. The spin and particle currents are respectively given by J σ = (2Jxy /~)hˆ σy,1 σ ˆx,2 − σ ˆy,2 σ ˆx,1 i and Jln = hˆl i where ˆl = i(J/~)[eiφ(t) a ˆ†l+1 a ˆl − −iφ(t) † e a ˆl a ˆl+1 ] is the particle current operator. These currents are associated with the continuity equation for σ ˆz,l and n ˆ l , respectively. Under steady-state conditions, the particle current J n is independent of the site position l. The dissipator Dγ induces dephasing and tends to suppress the current by destroying quantum coherence. For the sake of simplicity, we consider dephasing only for the third site,
Dγ =γ 2ˆ n3 ρˆn ˆ3 − n ˆ 23 ρˆ − ρˆn ˆ 23 , (4) where γ is the dephasing rate. We now turn to the presentation of our results. We will first consider the case for which persistent currents are generated, that is, there are no loads (γ = φ = 0). Then we will show the robustness of the particle current against dephasing and finally discuss the working of our model as a motor. Persistent currents: Since φ = 0, the Lindbladian L is time-independent. The steady-state behavior of the system is obtained by setting dˆ ρ/dt = L(ˆ ρ) = 0 in Eq. (1). Any density operator lying in the null space of L is timeindependent and can be determined by diagonalization of L. For our system the steady state is unique (except for κ = 0) and is the attractive fixed point under time evolution [28]. In Fig. 1(b-c) we show that the coupling strength Jxy can be used as a sensitive knob to tune the particle current. Indeed, the dependence of J n on Jxy is highly nonlinear and nonmonotonous. While the current is maximum close to Jxy = 0, its sign and magnitude can change significantly as Jxy varies [see Fig. 1(c)]. As shown in Fig. 1(b), nonlinear effects in the dependence of J n on the spin pumping imbalance ∆p become noticeable at large ∆p. In Fig. 1(d-e) we show the steady-state particle current as a function of p1 and p2 . The particle current J n is zero on the diagonal p1 = p2 . This is due to the exchange symmetry between sites 1 and 2. Fig. 1(d-e) shows that J n depends on both p¯ and on ∆p. In particular, for the same ∆p, the particle current may be inverted by changing p¯. As expected from Fig. 1(b-c), the current direction also depends strongly on Jxy . We now briefly discuss the role of interactions. We choose a local contact interaction for bosons by adding

3 0.15

0.12

0.02

0.02

0

0

current inversion

0 -0.15

∆p=0.5

(b)

4 (a) U 2

0.06

-2

0

-0.5

0

∆p

0.5

-2

0

U

2

FIG. 2. (Color online) (a) The effect of a contact interaction on the steady-state particle current for two bosons on the ring. By appropriately tuning the interaction strength U , the particle current can be strongly suppressed or even inverted for arbitrarily strong ∆p. (b) The inversion at small U is more clearly visible along a slice of constant ∆p. Parameter values: γ = φ = 0, h = 1, κ = 0.2, Λ = 0.4, p = 0.25, Jxy = 0.2.

ˆ int = U P n ˆ from Eq. (2). As shown in H nl −1)/2 to H l ˆ l (ˆ Fig. 2, even for the simplest case of two particles, the interaction has a non-trivial effect on the system. Remarkably, the particle current has a strong, non-monotonous dependence on U and can also be inverted. In the limit of large U , the current approaches the value for free fermions. Dissipative and Hamiltonian loads: The currents studied so far are persistent currents. However, it is possible to generate a particle current even in the presence of dissipation or against a driving force. First we consider the effect of dephasing as modeled by the dissipator given in Eq. (4). In Fig. 3 we show the steady-state particle current J n versus ∆p for different values of the dephasing rate γ. The particle current survives the presence of dephasing, and only for large enough values of γ it is suppressed (see inset). We now illustrate the working of the model as a motor. We set γ = 0 henceforth. For simplicity, we consider a driving of the form φ(t) = vt, where v is a phase velocity. The Lindbladian L(t) = L(t + T ) is periodic with period T = 2π/v and the system relaxes to a periodic steady state ρˆps (t), which we found to be unique in all cases studied with κ 6= 0. This state is given by theR null space t0 +T of the Floquet Lindbladian operator Lt0 = T e t0 L(t)dt , where T is the time ordering operator. To explicitly determine it, we numerically propagate a basis of the density operator space over one period from t0 to t0 +T . The eigenvector ρˆps (t0 ) corresponding to eigenvalue 1 yields the periodic steady state at time t0 . The particle current averaged over one period is given by JTn

1 = T

Z

t0 +T

tr [ˆ ρps (t) ˆ] dt,

(5)

t0

which is a quantity independent of the site position l and the initial time t0 . Here ρˆps (t) is obtained by evolving ρˆps (t0 ) for time t−t0 using Eq. (1) [29]. The total energy

-4

0 γ = =0.01 γ γ =0.1

0 γ 10

10

0

-0.02

γ =1

-0.4

-0.2

0

∆p

0.2

0.4

FIG. 3. (Color online) Steady-state particle current as a function of ∆p for various dephasing rates γ. As shown in the inset, the current survives up to relatively large values of γ, and then it is exponentially suppressed. Parameter values: v = 0, hz = 1, κ = 0.2, Jxy = 1.5, ~Λ = 0.4, p = 0.25; in the inset ∆p = 0.4.

exchanged by the system with the baths is given by QT =

XZ i

t0 +T

Q˙ λ,i (t)dt,

(6)

t0

where Q˙ λ,i (t) = tr {ˆ ρps (t)Dλ,i [ˆ ρps (t)]} [30]. Since the steady state is periodic, all observables are periodic, inR t +T dhH(t)i ˆ cluding the total energy. Then t00 dt = QT + dt R t0 +T ∂ Hˆ ˆ h ∂t idt = 0. This equality, after differentiating H t0 n with respect to time, gives QT = 3vJT . Hence, as the current changes direction, the net energy exchanged with the environment changes sign. Due to the driving, a particle current can emerge even for ∆p = 0. As shown in Fig. 4(a), the direction and magnitude of JTn strongly depend on p¯. The dependence of the current on ∆p, for a fixed p¯, is shown in Fig. 4(b). It is possible to control and invert the particle current by tuning the spin current via the spin pumping imbalance ∆p. For ∆p = 0 and Λ > 0, the current JTn < 0 and v > 0 implies that the total energy exchanged with the bath QT < 0. Hence, energy flows from the driving via the system into the baths. However, with increasing imbalance ∆p, the current is inverted JTn > 0, reversing the net flow of energy from the baths “into” the driving load. The current direction can also be inverted by increasing Λ beyond a critical value. By analyzing the energy exchanges over one period, one can define the efficiency η = QT /Qabs , i.e. the ratio of net energy directed into the load QT and the heat absorbed from the heat baths Qabs . In contrast to most simple heat engine models, the sign of heat exchanges with a heat bath may reverse multiple times within one period. The total heat absorbed from the baths in one period is then given by Qabs =

X Z i=1,2

t0 +T

t0

h i Q˙ λ,i (t) Θ Q˙ λ,i (t) dt,

(7)

4 1

0.015

p2

(a) Λ=0.6

1

x 10 (b)

-2

tions. 0.6 Λ=

T

0

0.5

0 Λ=0.1

0

Acknowledgments: We are grateful to J.M. Arrazola, F. Giazotto, J. Gong, M. Governale, A. Roulet and F. Taddei for fruitful discussions. D.P. acknowledges fundings from Singapore MOE Academic Research Fund Tier-2 project (Project No. MOE2014-T2-2-119, with WBS No. R-144-000-350-112), together with U.B., from SUTD-MIT IDC (Project No. IDG21500104), together with C.T. and from AcRF MOE Tier-I (project SUTDT12015005).

Λ=2

Λ=5

-0.015 0

0.5

0.8

η,

p1

1

-1

-0.4 -0.2

0

∆p

0.2

0.4

(c) T

x10

T

0.4

Jxy =-3.1 Jxy =3

0

0

0.5

1

Λ 1.5

2

FIG. 4. (Color online) The working of the system as a motor. The time-averaged particle current JTn in the presence of an external driving is shown as a function of p1 and p2 (a), of ∆p (b) and of ~Λ (c). The efficiency η of the model is also shown in panel (c) (solid curves). Parameter values: ~v = 0.6, in panels (b-c) p = 0.25 [corresponding to the dashed line in panel (a)], in panels (a-b) hz = 3.5, κ = 10, Jxy = 0.3, in panel (c) hz = 1.1, κ = 1.

where Θ(x) denotes the Heaviside function, which is equal to one when heat is absorbed and zero otherwise. In this way, we integrate over one period the absorbed heat rather than the exchanged heat. The efficiency η shown in Fig. 4(c), can reach large values (around 80% in the regimes analyzed) and decreases by increasing the pumping rate Λ. In contrast, the net energy exchanged with the baths QT (proportional to JTn ) vanishes in the Λ → 0 limit and features a maximum at non-zero Λ. It is interesting that the efficiency is still quite high at maximum JTn , i.e. at the maximum delivered power. Conclusions: We have studied a minimal model of quantum coupled transport, in which the particle current can be effectively tuned and also inverted by varying the strength of the coupling between the spins or the particle interaction. It turns out that work can be performed against an external driving, with a remarkably high heat to work conversion efficiency. This system could be implemented in various set-ups with effective spin-1/2 systems made with ultracold ions in microtrap arrays, as well as in solid state systems. In particular, for this last case, it could be possible to engineer this set-up using five quantum dots. Two quantum dots would take on the role of the spins coupled to a bath. Each of these would also be coupled to one of the remaining three (or more) quantum dots which form the circuit [31–34]. Future work could focus further on, for example, the effects of particle statistics, types of interactions, as well as on the emergence of non-equilibrium quantum phase transi-

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