Supplementary Information on “More bang for your buck ... · PDF fileSupplementary Information on “More bang for your buck: Super-adiabatic quantum engines ... allows us to derive

Supplementary Information on “More bang for your buck: Super-adiabatic quantum engines”

A. del Campo,1, 2 J. Goold,3, 4 and M. Paternostro5

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA2Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

3Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom4The Abdus Salam International Centre for Theoretical Physics, 34014, Trieste, Italy

5Centre for Theoretical Atomic, Molecular and Optical Physics,School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN Belfast, United Kingdom

In this Supplementary Information document we provide adetailed analysis of some of the technical points of our inves-tigation.

I. THERMALIZATION PROCESS DURING THEISOCHORIC TRANSFORMATIONS

In this Section we study the thermalisation processes in-herent in the isochoric transformations included in our enginecycle. The starting state of each transformation corresponds toa squeezed thermal state of the working medium (a harmonicoscillator), while the thermalization process itself can be mod-elled as relaxation induced by a bath at a given operating tem-perature. Our goal is to show that thermalisation is achievedwithin finite-time intervals and that the corresponding irre-versible entropy produced across such relaxation can be keptat bay, and considered ineffective for the sake of determiningthe efficiency of the engine that we propose. In what follows,we will make use of the powerful formalism of covariancestates, which is handy given the nature of the states and trans-formations at hand. The covariance matrix σ of a harmonicoscillator is defined as σ i j =

12 (〈{Qi, Q j}〉− 〈Qi〉〈Q j〉) with

Q= (q, p) the vector of quadrature operators q= (a+ a†)/√

2and p = i(a† − a)/

√2 of the oscillator and {·, ·} the anti-

commutator. Here a (a†) is the annihilation (creation) operatorof a harmonic oscillator.

The covariance matrix of a single mode squeezed thermalstate [1] can be straightforwardly obtained from using the re-lation S σthS

T with σth = (2m+1)112 the covariance matrixof a thermal state of mean occupation number m (112 is the2×2 identity matrix) and

S =

(e−r 00 er

)(1)

the linear canonical transformation corresponding to thesqueezing operation e

r2 (a

†2−a2) with r is the squeezing factor.A harmonic oscillator that is in contact with a thermal bath

at inverse temperature βb such that n = (eβbω − 1)−1 evolvesaccording to the master equation

∂tρ =γ

2[(n+1)(2aρ a†−{a†a,ρ})+n(2a†

ρ a−{aa†,ρ})](2)

with ρ the density matrix of the oscillator and γ the oscilla-tor energy damping rate. This equation, which is valid in thelimit of weak-coupling between the oscillator and its environ-ment, can be solved using phase-space methods leading to thefollowing time evolved covariance matrix [2]

σth(t) = (2n+1)(1− e−γt)114 +σthe−γt

=

(e−2r−tγ(2m+1)+(1− e−tγ)(2n+1) 0

0 e2r−tγ(2m+1)+(1− e−tγ)(2n+1)

).

(3)

Our goal here is to estimate the time t needed by the oscil-lator to relax towards a thermal state with a mean phononnumber n. Our figure of merit in this respect is embodied bythe fidelity between the state characterised by the covariancematrix Eq. (3) and the thermal state with covariance matrixσb = (2n+1)114. As we are dealing with Gaussian states andprocesses, we can write such fidelity in terms of the respectivecovariance matrices only as [3]

F(σth(t),σb) =2

√∆+Λ−

√Λ

(4)

with

∆ = det[σth(t)+σb],

Λ = (det[σb]−1)(detσth(t)−1).(5)

The analytic expression for the fidelity enables the calcu-lation of a lower bound to the irreversible entropy ∆Sth

irr pro-duced during the thermalisation process, which is evaluatedby following the formal apparatus presented in Ref. [4] ac-cording to which ∆Sth

irr ≥ B(t) = s[2L (σth(t),σb)/π] with

s[x] = 2x2 +4x4/9+32x6/135+O(x8) (6)

2

(a) (b) (c)

FIG. 1: Fidelity Fσth(t),σb) [panel (a)] and bound to the irreversible entropy B(t) [panel (b)] plotted against the dimensionless system-bathinteraction time γt and the initial phonon occupation number m of the working medium for n = 10 and r = 1. Panel (c) shows a comparisonbetween these two functions (the fidelity [irreversible entropy] being represented by the solid black line [dashed red line]) for m = 5. Thesebehaviours should be taken as typical.

and L (σth(t),σb) = arccos√

F(σth(t),σb) the Bures anglebetween the states defined by their respective covariance ma-trix. Both the fidelity and the bound to the irreversible en-tropy are shown in Fig. 1 (a) and (b) for a choice of the initialdegree of squeezing of the oscillator and temperature of thebath. As we are interested in the time taken by the oscilla-tor to thermalise to the bath it is in contact with, we studysuch functions against the dimensionless oscillator-bath in-teraction time γt and the initial value of m. The behaviorsshown in Fig. 1 should be taken as typical, as verified by ex-ploring these function within a wide range of values of theinvolved parameters. Clearly, a working point exists such thatthe system equilibrates within a time γt∗ ' 1 [such time isdetermined by considering the threshold value of γt at whichF(σth(t∗),σb) & 0.9]. Although B(t) embodies only a lowerbound, we have checked that within the range of parametersused in Figs. 1 (a)-(c), it is faithful to the explicit evalua-tion of the entropy produced in the process according to thegeneral (and much more involved) approach put forward inRef. [5], and involving the quantum Gaussian relative entropybetween the initial squeezed thermal state and the final equi-librium one at the temperature of the bath. As such, in whatfollows we stick to the use of B(t) as providing a reliable andeasily grasped estimate of the irreversible entropy producedin the system, which is kept at quite low levels. A compari-son between F(σth(t),σb) and the bound, here labelled B(t)(we omit its explicit form for simplicity), is given in Fig. 1 (c)for m = n/2 = 5. Needless to say, as this analysis is valid forγ � ω(0),ω(τ) (for the validity of weak-coupling assump-tions), these results should be kept in consideration when eval-uating the power of the engine at hand.

This analysis shows that the running times τ2,4 of the iso-chores needed for the Otto cycle can be kept at finite values,still achieving effective low-entropy thermalisation processesthat would leave Eqs. (1) and (2) of the main paper valid.

II. NONEQUILIBRIUM WORK FLUCTUATIONS

We next present the derivation of Eq. (7) in the main paper.Consider as reference states the fictitious equilibrium state ρ

eqt

and the adiabatic one ρadt . Then, in the adiabatic limit the

average work can be rewritten as

〈Wad(t)〉 = Tr[ρ0(H (t)−H (0)] = ∑n

p0n[εn(t)− εn(0)]

= −1β

∑n

p0nlnpt

n+1β

∑n

p0nlnp0

n−1β

ln(Zt/Z0)

=1β

S(ρadt ||ρ

eqt )+∆F (7)

with Zt = Tr[e−βH (t)] the instantaneous partition function.For a general nonequilibrium process, the average work readsinstead

〈W 〉 = −1β

∑nk

p0n pt

nklnptk+

1β

∑n

p0nlnp0

n−1β

ln(Zt/Z0)

=1β

S(ρt ||ρeqt )+∆F. (8)

As a result, nonequilibrium deviations from the mean adia-batic work take the form

δW =1β[S(ρt ||ρeq

t )−S(ρadt ||ρ

eqt )]. (9)

It is worth considering an an alternative approach, where ρadt

is used as a reference state and the dynamics is restricted to theclass of self-similar processes [6–8], for which conservationof the population in the mode |n(t)〉 as a function of time t issatisfied provided that

βt = βεn(0)/εn(t), (10)

as it is the case for the adiabatic dynamics associated to theshortcuts discussed here. Under such condition the partitionof the instantaneous equilibrium state remains constant Zt =Z0 = Z. Using ρad

t , the average work in the adiabatic limitreads

〈Wad(t)〉 =1β

∑n

p0nlnp0

n−1βt

∑n

p0nlnpt

n−(1βt− 1

β)lnZ

=1βt

S(ρt)−1β

S(ρ0)+∆F, (11)

3

where we have introduced the von Neuman entropy S(ρ) =−Tr[ρlnρ] of an arbitrary state ρ . More generally,

〈W 〉 =1β

∑n

p0nlnp0

n−1βt

∑k,n

ptnk p0

nlnp0k−(

1βt− 1

β)lnZ

=1βt

S(ρt)−1β

S(ρ0)+1βt

S(ρt ||ρadt )+∆F. (12)

This leads to the following compact expression for nonequi-librium work deviations form the adiabatic path,

δW =1βt

S(ρt ||ρadt ). (13)

The two expressions for δW , Eqs. (9) and (13), agree for self-similar processes and vanish at the end of the stroke (either 1or 3 in Fig. 1 of the main paper) both for a shortcut and in theadiabatic limit.

III. UPPER BOUND TO POWER THROUGH THEQUANTUM SPEED LIMIT

The quantum speed limit for a driven quantum system [9]allows us to derive an upper bound for the power of the engine.

For simplicity, we can consider a equal-time shortcuts alongthe two super-adiabats so that τ = τ1 = τ3. Then, it followsthat

P≤−〈Wad,1(τ)〉+ 〈Wad,3(τ)〉

hL(ρ

eqτ ,ρ0

) max{

Eτ ,∆Eτ

}. (14)

where Eτ = τ−1 ∫ τ

0 dtTr[ρtH (t)] with respect to theground state energy, ∆Eτ = τ−1 ∫ τ

0 dt {Tr[ρtH 2(t)] −Tr[ρtH (t)]2}1/2, and the angle in Hilbert space betweeninitial and target states is

L(ρ0,ρ

eqτ

)= arccos

(√F(ρ0,ρ

eqτ

))(15)

in terms of the fidelity F(ρ0,ρ

eqτ

)=[Tr√√

ρ0 ρeqτ

√ρ0

]2. In

a super-adiabatic engine, 〈W 〉ad,1 + 〈W 〉ad,3 equals

∑j=1,3〈Wad,j(τ)〉=

h2(ω0−ωτ)

[coth

βchω(τ)

2− coth

β hω0

2]

(16)where βc is the inverse temperature of the cold bath duringstage 2.

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