The Journey to the quantum Carnot engine and its quantum signature

Ronnie Kosloff

Institute of Chemistry, Hebrew University Jerusalem, Israel The Fritz Haber Center for Molecular Dynamics

November 12, 2019

Talk content

1 Quantum Thermodynamics: Consistency.

2 Reciprocating heat engines: learning from example.

3 Generalized Canonical Invariance.

4 Thermodynamics of the GKLS Master Equation.

5 The Non Adiabatic Master Equation (NAME).

6 The inertial theorem.

7 Shortcut to Equilibrium (STE).

8 Quantum control of open systems.

9 The Quantum Carnot engine with finite power.

Eitan Geva

Jeff Gordon

Tova Feldmann

Jose Palao

Quantum Thermodynamics

Eitan Geva

Jeff Gordon

Tova Feldmann

Jose Palao

Quantum Thermodynamics

LevAmikamAmikam Lev

Morag Am Shalem

YYairair Rezek RezekRobert Alicki

Quantum ThermodynamicsQuantum Thermodynamics

Finding quantum analogies to:

0) System bath partition:Approach to equilibrium.

1) The first law:Energy change.

2) The second law:Entropy change.

3) The third law: Approaching the absolute zero.

Any theory should be consistent with Thermodynamics

Einstein 1905

ConsistencyEmergence of Thermodynamics from quantum mechanics

Can a Thermodynamical viwpoint be relavant to a single device at the quantum limit

What is the limit of minaturization of a quantum heat engine

What is quantum in a quantum heat engine Is there quantum supremacy

?

?

??

single molecular refrigerator

Learning from example Thermodynamic ideals

?How small a quantum engine can be

What is quantum in quantum heat devices

Power or efficiency?

Efficiency at maximum power Maximum efficiency

How far can we mintuaize?

The quantum Otto refrigeration cycle

Hot Isochore (isomagnetic) A #B $=$h .The working mediumis in contact with the hot bath of temperature Th .

Expansion adiabat (demagnetization) B #CThe field changes from $h to $c

Cold Isochore (isomagnetic) C #D $=$c .The working medium is in contact with the cold bath of temperature Tc .

Compression adiabat (magnetization) D #AThe field changes from $c to $h .

Uh

Uhc

Uc

Uch

Cycle Propagator: Ucycle=UhUhcUcUch

[UhUhc,UcUch] % 0limit cycle

UcycleY=Y

Realizations 2016 Science 352 , 325 (2016)

Reciprocating heat engines

Learning from example

1

2

3

Carnot cycle Hot to cold adiabatic stroke hcCold isotherm cCold to hot adiabatic stroke ch

4 Hot isotherm h

Carnotcycle:cyc=hchchc

Operating conditionsfixed point of CPTP map

<n>

ωωω ω

Compression ratio

ω

<n>Τh

Tc

A quantum-mechanlcal heat engine operating in finite time. A model consisting of spin-1 /2 systems as the working fluid

Eitan Geva and Ronnie Kosloff

Departmenו of Physical Chemistry and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew Uniuersity, Jerusalem 91904, Jsrae/

(Received 28 August 1991; accepted 21 October 1991)

_____ כo _____ ס (ו)

2 2' 3 3'

l3c·,

s2

1. 4 4' Sl ..

S2 .:•: w·-._ W w,,,//Qו,

Qc'°' c S1 ········------ ·._ -......w .Q.5 -------=---

FIG. S. Thc cyclc 1'-2' -3' -4'-1' is of the Curzon-Ahlborn

FIG. 1. The revcrsible Camotcyc]c (so]id line) in וhc ש,S) plane (w is the ficld and S the polariz.ation ). The cyclc is composed of t'י''O revcr$iblc iso­thcrms corresponding 10 the tcmperatures J and /J, (J, > Pג) and of two

adiabatscorrcsponding 10 וhe polari:uוioחs S, and Sג (S 1 <S,; S, ,S0> ג).

Pסsitivc net work production is obtained by going anזiclockwise. The direc­tions ofwork and heat flows along uch branch are indicatd.

J. Chem. Phys., Vol. 96, No. 4, 15 February 1992

Heisenberg picture, reads as follows:

x = i[H,XJ + ax + 2' מ(X>, ar

Vl [X,V0) 0ץ L = (X)ע '2. ] + [V,X]V0 ).

Carnot cycle

No coherence considered

Generalized Gibbs State

Canonical state: ρ =1

Ze−β H

Maximum entropy state subject to the expectation E = 〈H 〉

Generalized Gibbs state:

Aρ = e ∑k λk

ˆk

Maximum entropy state subject to the expectations〈Ak 〉 = trρAk .

and λk are Lagrange multipliers.

How to incorporate coherence

Canonical invarianceConsider the dynamics

d

dtρ = L ρ

Under what conditions on L and ρ the canonical form stays invariant with β = β (t) .

Generalized Canonical invariance:Under what conditions does the gerealized Gibbs statestay invariant to the dynamics.

Canonocal invariance

Generalized Canonocal invariance

Quantum generalized canonical invariance

Consider the closed Lie algebra of operators Ak :

∑k

[Ai ,Aj ] = CijkAk

where C kIJ is the structure factor of the group.

If the dynamics

is generated b y the Hamiltonian

ρ = −i[H ,ρ]:

H =l∑hl Al

Then the equations of motion of the algebra of operators isclosed under the dynamics. In addition the generalized canonical form is invariant to the dynamics.

.

Proof of closed Heisenberg equations

d

dtAi = i [H, Ai ] =∑

l

hl [Al , Ak ] =∑l

hl ∑k

C kli Ak

Proof of invariance of Generalized Canonical form For a closed Lie algebra:

ρ = e∑k λkAk = ∏k

eαkAk

For the dynamics ddt ρ =−i [H, ρ]:

d

dtρ ρ−1=∑

k

fk (~α,~α)Ak =−i∑l

hlAl+∑j

gj(~α)Aji

using the product form

Canonical invariance for open quantum systems

Consider the dynamics

d

dtρ = L ρ

Generalized Canonical invariance:Under what conditions does the gerealized Gibbs state stay invariant to the dynamics.

Lindblads form of the generator L :

ρ =− i

h[H,ρ]+

N2−1

∑i=1

γi

(AiρA

†i −

1

2ρAi

†Ai −1

2A†i Aiρ

).

The canonical invariance for a product form

ρ = eµ1A1eµ2A2...eµk Ak ...eµN AN

where the set A form a closed Lie algebra andµ = µ(t)

d ρ

dtρ−1 = µ1X1 + µ2e

µ1A1A2e−µ1A1 + µ3e

µ1A1eµ1A2A3e−µ1A2e−µ1 A1 + ...

Using the Baker-Housdorff relation.For canonical invariance to prevail during the evolutionL (ρ)ρ−1 also should become a linear combination of

the set of operators AFor a unitary:L (ρ)ρ−1 =−i h[H, ρ]ρ−1 =−i hH− i hρHρ−1

A1

The canonical invariance for a product formFor an open quantum system LD(ρ)ρ−1 also should

forms a linear combination of the set A. Thiscondition becomes:

LD(ρ)ρ−1 = ∑

j

(Fj ρF

†j ρ−1− 1

2(F†

j Fj + ρFj F†j ρ−1)

)Canonical invariance will be obtained If the operatorsFj F

†j and Fj ρF

†j ρ−1 are part of the set A .

Specifically for the harmonic oscillator set P2 , Q2 andD = (QP+ PQ) and the Lindblad operator F = a this

condition is met. On the contrary for F = H , H2 isnot part of the Lie algebra therefore, canonicalinvariance is not met in the case of pure dephasing.

Closed Heisenberg equations of motion for a Lie algebraof operators, leads to generalized canonical invariance for the form:

ρ = e∑k λkAk

Example

The set I, X, P, X2, P2, (XP+ PX) is a closed Lie

The Hamiltonian which is composed of members of the alebra:

H=1

2mP2+

mω(t)2

2X2

will generate a generalized canonical form for the closed algebra.

algebra

An thermodynamically significant time dependent choice of operators

The Hamiltonian H(t) = 12m P2 + 1

2mω(t)2Q2.

The Lagrangian L(t) = 12m P2− 1

2mω(t)2Q2.

The position momentum correlation C(t) = 12ω(t)(QP+ PQ).

The Casimir operator G commutes with all the operators in the algebra:

G=H2− L2− C2

h2ω2

The coherence:C o =

1

hω

√〈L2〉+ 〈C2〉

The generalized Gibbs state in a product form:

ρ =1

Zeγ a2e−β Heγ∗a†2

,

where H = hω

2 (aa† + a†a), C =−i hω

2 (a2− a†2) , L =− hω

2 (a2 + a†2) and

⟨H⟩

=hω(e2β hω −4γγ∗−1)

(eβ hω −1)2−4γγ∗⟨a2⟩=

2γ∗

(eβ hω −1)2−4γγ∗.

γ=

hω

2 (⟨L⟩

+ i⟨C⟩

)⟨L⟩2

+⟨C⟩2− ( hω

2 −⟨H⟩

)2

L2

+ C − hω

2 − H2

Inverese temperature β :

eβ hω =L⟨ ⟩2

+ C⟨ ⟩2

− H⟨ ⟩2

+ h2ω2

4

L⟨ ⟩2

+ C⟨ ⟩2

−(hω

2 − H⟨ ⟩)2

.

The expectation values of the operatorscompletely determine the GGS

squeezed themal state

The Heisenberg equations of motion for the thermodynamical set of operators are expressed as:

d

dt

H

L

C

I

(t) = ω(t)

µ −µ 0 0

−µ µ −2 0

0 2 µ 0

0 0 0 0

H

L

C

I

(t) .

(31)

The generalized Gibbs state can be reconstructed from the observables

µ is the adiabtic parameter: µ= ω/ω2.

P = µω(〈H〉 − 〈L〉

).Power

generating coherence reduces power

The equations of motion for the adiabatsFree Dynamics

The Heisenberg equation of motion for thermalization

d

dt

HLCI

(t) =

−Γ 0 0 Γ〈H〉eq0 −Γ −2ω 00 2ω −Γ 00 0 0 0

HLCI

(t)

where Γ = k↓−k↑ is the heat conductance and

k↑/k↓ = e−hω/kBT obeys detailed balance whereω = ωh/c and T = Th/c are defined for the hot or coldbath respectively.The heat current can be identified as:

Q =−Γ(〈H〉−〈H〉eq)

Otto cycle

Engine cycle [H(t),H(t ′)] 6= 0

Adiabatic efficiency

Quantum friction is the inability to stay on the energy shell

Shortcuts to adiabaticity use coherence to reach frictionless conditions.

At high temperature the efficiency at maximum power becomes:

A

C D

B

Uch

UhUc

Uhc

Canonical Invariance for Otto Cycle

The Problem:Derive a dynamical description for a driven systemcoupled to a bath beyond the adiabatic limit:

H = HS(t) + HB + HSB

Carnot cycle: The isotherms

An open system quantum control problem:State to state control:

ρi → ρf

where we have control only on the system Hamiltonian HS(t):

H = HS(t) + HB + HSB

The system dynamics is governed by:

d

dtρS = LS ρS

where LS(t) depends on the bath implicitly and HS(t).

The theory of open quantum systems

The quantum Markovian Master Equation .

A completely positive map:

Λρ = ∑j

W †j ρWj ,

where ∑j W†j Wj = I

The Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) quantumMaster equation

d

dtρ =− i

h[H , ρ] +

1

2 ∑j

([Vj ρ, V†j ] + [Vj , ρV

†j ])≡− i

h[H , ρ] +L ρ .

1975

Kraus 1971

System and bath are in tensor product form in all times Lindblad 1996

G. Lindblad

Thermodynamical properties.

Davies construction: The weak coupling limit:

H = HS + HB + Hint

The system-bath interaction as Hint = ∑k Sk ⊗ Rk

One obtains the following structure of MME which isin the GKLS form

d

dtρ =−i [H , ρ] +L ρ, L ρ = ∑

k,l∑ω

L ωlk ρ

where

L ωlk ρ =

1

2h2Rkl(ω)

[Sl(ω)ρ, S†

k (ω)] + [Sl(ω), ρ S†k (ω)]

.

Here, the operators Sk (ω) originate from the Fourier decomposition

Davies 1974

Thermodynamical properties.ω- denotes the set of Bohr frequencies of H.

e i/hHt Ske−i/hHt = ∑

ωe−iωt Sk(ω),

Rkl (ω) is the Fourier transform of the bath correlation function 〈 Rk ( t)Rl 〉bathcomputed in the thermodynamic limit

Rkl (ω) = −+

∞

∞ e iωt 〈Rk (t)Rl 〉bathdt.

The derivation of Davis makes sense for a generic stationary state of the bath

and implies two properties:

1) the Hamiltonian part [H , ·] commutes with the dissipative part L ,

2) the diagonal ( in H -basis) matrix elements of ρ evolveindependently of the off-diagonal ones according to the Pauli Master Equationwith transition rates given by the Fermi Golden Rule.

No mixing of energy and coherence

Thermodynamical properties.

If additionaly the bath is a heat bath, i.e. an infinite systemin a KMS state the additional relation implies that:

3) Gibbs state ρβ = Z−1 exp−β H is a stationary solution.

4) Under the condition that only scalar operators

commute with all Sˆk (ω),Sˆ

k†(ω).

Any initial state relaxes asymptotically to the Gibbs state: The 0-Law of Thermodynamics.

The bath is able to "measure" the energy level structure of the systemand transfer heat according to the detailed balance conditionsQuantum conditions of isothermal partition

The derivation can be extended to slowly varying time-dependentHamiltonian within the range of validity of the adiabatic theorem

and an open system coupled to several heat baths at the inversetemperatures βk = 1/kBTk .

The MME in Heisenberg form:

d

dtY (t) =−i [H(t), Y (t)] +L ∗(t)Y (t) +

∂

∂ tY

, L ∗(t) = ∑k

L ∗k(t).

Each L k(t) is derived using a temporal Hamiltonian

H(t), L k(t)ρj(t) = 0 with a temporary Gibbs state

ρj(t) = Z−1j (t)exp−βj H(t).

Thermodynamical properties.

Dynamical I-law of thermodynamics

R. Alicki , J.phys. A Math. Gen. 12 L103 (1979)

Power +Heat currentSpohn, Herbert, and Joel L. Lebowitz. Adv. Chem. Phys 38 (1978): 109-142

Adiabatic limit

The task: Isothermal Dynamics

Starting from a thermal initial state ρi = e−β Hi

Transform as fast and accurate to the state: ρf = e−β Hf

while the system is in contact with a bath of temperatureT = 1/kβ

The protocol: HS (t) with HS (0) = Hi and HS (tf ) = Hf

The ProblemWe can control directly HS (t) but only indirectly the relaxation rate.We need the dissipative equation of motion with a time dependent HS (t) with a time dependent protocol.

Carnot cycle: The isotherms

Non-adiabatic open system dynamics

Time-dependent Markovian Master Eq., R. Dann, A. Levy, and R. Kosloff, Phys. Rev. A 98, 052129 (2018).The Inertial Theorem, R. Dann and R. Kosloff, arXiv:1810.12094 (2018).

How can we obtain the Master equation?

Analytic tools:

Non-Adiabatic Master Equation (NAME) Inertial theorem

NAME - The driven Non-Adiabatic Master Equation.

d

dtρS(t) = Ls ρS

We change H S(t) from H

S(0) to H S(tf ) while coupled to the bath.

Then we expect:d

dtρS(t) =−i [HS(t) + HLS(t), ρS ]

+∑k

ck(t)

(Lk(t)ρS L

†k(t)− 1

2L†

k(t)Lk(t),ρS)

.

Lk are the Lindblad jump operators.

Here HLS (t) is the time dependent Lamb shift Hamiltonian, HLS (t) = ∑k Skk ′ (α(t)F †j (t)Fj (t)).

The Non-Adiabatic Master Equation (NAME)

H(t) = HS(t) + HB + HSB . HSB = ∑k

gkAk⊗ Bk .

Following Davies’s derivation,1) Transformation to the interaction picture:

UB†(t,0)U†

SH(t)(t,0)US(t,0)UB(t,0) = ˆHSB(t) ,

Where the system evolution operatori ddt US(t) = HS(t)US(t) , US(0) = I .

.2) Second order perturbation theory lead to theMarkovian quantum master equation

d

dtρS(t) =−

∫∞

0ds trBHSB(t), [HSB(t− s), [ρS(t)⊗ ρB ]] .

Consistancy with thermodynamics

The Lindblad Jump operators:For free evolution of a static Hamiltonian:The propagator in Heisenberg form is:

U (t) = eith [HS,•]

The eigenoperators of U (t) are:

U (t)|m〉〈n|= e iωnmt |m〉〈n|

where H S|n〉 = εn|n〉 and ωnm = h

1¯(εn − εm) is the Bohr frequency.

.

The H SB can be expanded: with Fk = |m〉〈n|

HSB = ∑k

gkeiωk t Fk ⊗ Bk

leading to Fk ≡ Lk the Lindblad jump operators.

Excitations

The Lindblad Jump operators:For a driven evolution: HS(t)

We choose a time dependent operator base: Xj(t)The propagator in Heisenberg form is:

U (t) = T eih

∫ t0 ([ ˆHS(t′),•]+ ∂

∂ t′ )dt′

We find eigenoperators of U (t):

U (t)Fk = e iθk(t)Fk , U†(t)FkU(t) = e iθk(t)Fk

where Fk are time independent..HSB can be expanded in interaction frame with Fk

HSB = ∑k

gkeiθk(t)Fk ⊗ Bk

leading to Fk ≡ Lk the Lindblad jump operators.

Non Adiabatic Master Equation (NAME)

1. Weak coupling

2. Born- Markov approximation

3. Fast bath dynamics relative to the external driving

R. Dann, A. Levy, and R. Kosloff, Phys. Rev. A 98, 052129 (2018).

Lamb-shift

1. 2. 3.

How to solve the free dynamics with driving ?

Liouville space representation:

Operator basis:

Elements with inner product

The inertial theorem approximates the evolution of a quantum system, driven by anexternal field. The theorem is valid for fast driving provided the acceleration rate is small.

The Inertial Theorem, R. Dann and R. Kosloff, arXiv:1810.12094 (2018).

Inertial Theorem

Non-adiabatic driving

Time-ordering problem We want to solve this!

The Inertial theorem.

For a closed Lie algebra [Ai , Aj ] = ∑k cijk Ak

the Heisenberg equation of motion, for the set A =~v are

d

dt~v (t) =

(i[H (t) ,•

]+

∂

∂ t

)~v (t) , (1)

In a vector notation (1) becomes

d

dt~v (t) =−iM (t)~v (t) , (2)

where M is a N by N matrix with time-dependentelements and ~v is a vector of size N .If we can factor:

M (t) = Ω(t)B (~χ) . (3)

Here, Ω(t) is a time-dependent real function, andB (~χ) is a function of the constant parameters χ.

For this decomposition, the dynamics becomes

d

dθ~v (θ ) =−iB (~χ)~v (θ ) , (4)

θ ≡ θ (t) =∫ t0 dt

′Ω(t ′) is scaled time.The solution

~v (θ ) =N

∑k

ck ~Fk (~χ)e−iλkθ , (5)

where ~Fk and λk are eigenvectors andeigenvalues of B and ck are constantcoefficients. Each eigenvector ~Fk correspondsto the eigenoperator Fk .

Inertial Theorem

R. Dann and R. Kosloff, arXiv preprint arXiv:1810.12094 (2018).

Inertial solution

Geometric phase

Inertial parameter

can very slowly in time Inertial condition

R. Dann and R. Kosloff, arXiv preprint arXiv:1810.12094 (2018).

Protocol: Illustration

Experimental verification of the Inertial Theorem

Inertial solutionExperimentalNumerical

The Inertial Theorem, R. Dann and R. Kosloff, arXiv:1810.12094 (2018).Experimental Verification of the Inertial Theorem, C.K.Hu, R.Dann, et al. , arXiv:1903.00404

Experimental verification of the Inertial Theorem

Experimental Verification of the Inertial Theorem, C.K.Hu, R.Dann, et al. , arXiv:1903.00404

Example Parametric harmonic oscillator

H(t) =p2

2m+

1

2mω

2 (t) q2 , (8)

The setH(t), L(t) = p2

2m −12 ω2 (t) q2, C(t) = ω(t)

2 (qp + pq), K(t) =√

ω (t)q, J(t) = p

m√

ω(t)

is a Lie algebraThe free dynamics in terms of the vector ~v = H, L, C,K, J, IT

d

dθ~v (θ) =−iB~v (θ) (9)

with,

B = i

χ −χ 0 0 0 0−χ χ −2 0 0 00 2 χ 0 0 00 0 0 χ

2 1 00 0 0 −1 − χ

2 00 0 0 0 0 0

. (10)

Here, χ = µ = ω

ω2 , where µ is the adiabatic parameter, θ =∫ t

0 dt ′ω (t)′.

The Non-Adiabatic Master Equation (NAME)for the Harmonic oscillator(in the interaction picture)

d

dtρS(t) =−i

[HLS(t), ρS

]+

k ↑ (t)

(b†

ρS b−1

2

bb†, ρS

)+k ↓ (t)

(bρS b

†− 1

2

b†b, ρS

),

b≡ b(0) =

√mω(0)

2h

(κ + iµ)

κ

(Q +

µ + iκ

2mω(0)P

)[b, b†] = 1

µ = ω

ω2 , κ =√

4−µ2. α(t) = κ

2 ω(t) k ↑ (t)

k ↓ (t)= e− hα(t)

kBT

γna = k ↓ (α(t)) = πmα(t)J(α(t))(N(α(t)) + 1)

Comparing adiabatic to nonadiabatic rates

Nonadiabatic rate:

γna(α(t)) = πmκ

2ω(t)J(

κ

2ω(t))(N(

κ

2ω(t)) + 1)

κ =√

4−µ2

.Adiabatic rare:

γad(ω(t)) = πmω(t)J(ω(t))(N(ω(t)) + 1)

for J(ω) ∝ ω2 and low temperature:

γna(α(t)) =1

8κ

3γad(ω(t))

Slowing down the relaxation rate

Comparing non-adiabatic to adiabatic equation

Adiabatic: Lidar 2012

Both equations have time dependent Lindblad form. Thisguarantee’s complete positivity and consistency withthermodynamics

Doppler like change in frequency:

κ =√

4−µ2 , α(t) =κ

2ω(t)

slowing down the relaxation rate and changing theinstantaneous target of relaxation.

Mixing Energy and coherence: generating squeezing.

Instantaneous attractor

Carnot cycle: The isotherms

Shortcut to Equilibration (STE)

Shortcut to Equilibration of an Open Quantum System, R. Dann, A. Tobalina, and R. Kosloff, PRL 122, 250402 (2019)

Entropy change

Solving NAME for fast isothermal strokes

Change of variable in interaction representation:

d

dtρS(t) = γ ↓

(bρS b

†− 1

2

b†b, ρS

)γ ↑(b†

ρS b−1

2bb†, ρS

)(2)

we can try a solution in a generalized canonical form:

ρS(t) =1

Z (t)eγ(t)b2

eβ (t)b†beγ∗(t)b†2

2Maximum entropy subject to constraints: 〈b†b〉, 〈b† 〉, 〈b2〉

Canonical invariance: Openheim 1964, Alhassid & Levine 1978. Andersen, H. C., Oppenheim, I., Shuler, K. E., & Weiss, G. H., Jour. of Math. Phys. (1964)

Y. Alhassid and R. D. Levine, Phys. Rev. A 18, 89 (1978

Dynamics for any squeezed thermal state.

_β = k↓

(eβ −1

)+k↑

(e−β −1 + 4eβ |γ|2

), (11)

_γ =(k↓+k↑

)γ−2k↓γe

−β ,

We assume that the system is in a thermal state atinitial time, which infers γ(0) = 0. This simplifies to

~ρS (β (t) ,µ (t)) =1

Zeβ b†b(µ) . (12)

The system dynamics are described by

_β = k↓ (t)(eβ −1

)+k↑ (t)

(e−β −1

), (13)

with initial conditions β (0) = hω(0)kBT

and µ (0) = 0.

R. Dann, A. Tobalina, and R. Kosloff, PRL (2019).

For an initial thermal state

Engineering the shortcut to equilibration protocol Guessing a solution in the form of Generalized Canonical form

Control

Shortcuts to Equilibrium (STE)The shortcut protocol H

S (t) → ω(t):

CompressionExpansion

Overshoot

3 fold improvement in time

Compression

Shortcuts to Equilibrium (STE) .The fidelity F and A = − log10(1 −F ):

R. Dann, A. Tobalina, and R. Kosloff, PRL 122, 250402 (2019)

Expansion

Rapid driving costs!

STE- How much does it cost?STE Quench Adiabatic

CompressionExpansion

Efficiency:

Expansion

Compression

STE – Thermodynamic analysis

STE Quench

CompressionExpansion

The cost of shortcuts W , Su

Adiabatic shortcutsStarting on the energy shell:

〈H 〉 6= 0 , 〈L〉 = 0 , 〈C〉 = 0

Nonadiabtic dynamics generates coherence and requires extra work.Shortcuts to Adiabaticity STA retrieve this work cashing on the coherence.The shortcut duration is inversely related to the stored energy.The system entropy remains constant ∆Ssys = 0. Irreversible cost is only in the controller ∆SU ≥ 0.

The process can be classified as catalysis.

The cost of shortcuts W , Su

Shortcuts to Equilibrium (STE)Starting on the energy shell:

〈H 〉 6= 0 , 〈L〉 = 0 , 〈C〉 = 0

Nonadiabtic dynamics generates coherence and requires extra work.The coherence is dissipated generating quantum frictionThe system entropy changes ∆Ssys 6= 0. Irreversibility is inherent ∆SU > 0.

The speedup cost work and entropy production.

Quantum control of open systemsThe task is classified as a state-to-state objectiveρ iS → ρ f

S , governed by open system dynamics:

d

dtρS = L (t)ρS

Controllability: what are the conditions on thedynamics which allow to obtain thestate-to-state objective?

Constructive mechanisms of control, the problemof synthesis.

Optimal control strategies and quantum speedlimits.

Controlability of open systems.Under what conditions can an initial quantum state ρ i

S

be transformed into a final state ρ fS , employing open

systems dynamics?

1 For any initial state ρ iS couple the system to a

bath of temperature T and change theHamiltonian to H f

S until an equilibrium thermal

state ρ fS ,T , with common eigenvalue with ρ f

S .

2 Apply a fast unitary transformation US such that

ρ fS = US ρ f

S ,T U†S .

Quantum controllability of open systemsThe protocol can be achieved under the followingconditions:

1 The isolated quantum system is completelyunitary controllable, i.e. any unitarytransformation US is admissible.

2 There is a complete freedom in modifying theHamiltonian HS (t) of the system, embedded inthe bath of temperature T .

The control of a target state ρ fS requires engineering

the asymptotic invariant of L to become unitarilyequivalent to the target state In the adiabatic limitunder the Davis construction, the thermal state withthe Hamiltonian H f

S becomes the invariant of theGKLS equation

Shortcut to equilibration of a qubit.The Hamiltonian:

HS (t) = ω (t) Sz + ε (t) Sx ,

Define the Liouville vector

~v (t) = HS (t) , L(t) , C (t)T .

L(t) = ε (t) Sz −ω (t) Sxand C (t) = Ω(t) Sy ,

where Ω(t) =√

ω2 (t) + ε2 (t)

Shortcuts to Equilibrium for a qubit.For a protocol satisfying a constant adiabaticparameter

µ ≡ ωε−ωε

Ω3,

the dynamical equation in Liouville space

d~v

dt=

(˙Ω

Ω2I + Ω(t)B (µ)

)~v

, d~wdt = M (t)~w , with M (t) = Ω(t)B (µ). B (µ)

and the diagonalizing matrix V are given by

B (µ) = i

0 µ 0−µ 0 1

0 −1 0

, V =

1 −µ −µ

0 iκ −iκµ 1 1

.

Shortcuts to Equilibrium for a qubit

eigenoperators .

vi (t) =Ω(t)

Ω(0)

3

∑j ,k=1

Vik (µ (t))e−i∫ t

0 λk(t ′)

(d θ(t′)dt′

)dt ′×V −1

kj (µ (t)) vj (0) ,

Fk (µ (t) ,0) = ∑j

V −1kj (µ (t)) vj (0)

and θ (t) =∫ t

0 Ω (t ′)dt ′.

Shortcuts to Equilibrium for a qubit.Next, we utilize the inertial theorem to obtain thedynamics of the eigenoperators of the free propagators

Fk = ξ , σ , σ†

ξ (µ,0) =1

κΩ (0)

(HS (0) + µC (0)

)σ (µ,0) =

1

2κ2Ω (0)

(−µHS (0)− iκ L(0) + C (0)

)and σ† (µ,0), with corresponding eigenvaluesλ1 = 0,λ2 (t) = κ (µ (t)) and λ3 (t) =−κ(µ(t)),where

κ (µ (t)) =√

1 + µ2 (t) .

Shortcuts to Equilibrium for a qubitWe consider a system interacting with an OhmicBoson bath at temperature T , through a single spincomponent

HI = gSy ⊗ B .

The Master equation for the two-level-system:

d

dtρS (t) = k↓ (t)

(σ (µ)ρS σ

† (µ)− 1

2σ† (µ) σ , ρS

)+k↑ (t)

(σ

† (µ) ρS σ − 1

2σ (µ) σ

† (µ) , ρS)

,

k↑ (t) =2α (t)

hcN (α (t)) = k↓ (t)e−α(t)/kBT ,

α (t) = κ (t) Ω(t)

and σ (µ)≡ σ (µ (t) ,0) depends on µ (t).

Solution of the driven open systems.

ρS (t) = Z−1eγ(t)σ(µ)eß(t)ξ (µ)eγ∗(t)σ†(µ)

where Z ≡ Z (t) = tr(ρS (t)) is the partition function,with time-dependent parameters γ (t) and ß(t).The dynamics

ß =γeß

2κ2γ∗−k↓

4κ2(eß + 1

)+ |γ|2eß

16κ4

+k↑

(|γ|2 + e−ß4κ2

)(4(eß + 1

)κ2 + eß|γ|2

)64κ6

γ = k↓γ

8κ2−k↑

γ(2(1 + 2e−ß

)κ2 + |γ|2

)16κ4

,

Design of the control protocol.We assume no initial coherence:

ρS (t) = Z−1eß(t)ξ (µ)

Throughout the evolution, and the equation of motionreduces to a single differential equation

ß (t) =1

4κ2 (µ (t))

[k↑ (t)

(1 + e−ß(t)

)−k↓ (t)

(eß(t) + 1

)].

This equation is the basis for the suggested controlscheme.

Ω Ω

Control dynamics of the qubit

The control strategy goes as follows: We perform a change of variables y (t) = eß(t), and introduce an polynomial solution for y (t) which satisfies the boundary conditions of the control. The initial and target Gibbs states correspond to values

ß(0) = −hΩi /kBT0, ß(tf ) = −hΩf /kBTf , where

i = Ω(0) and ¯f = Ω(tf ) are the generalized Rabi

frequencies of the initial and final Hamiltonians. A fifth order polynomial for y is sufficient to satisfy the boundary condition, leading to

ß(t) = ln

(5

∑k=0

bktk

),

Control tasks.The control scheme allows addressing various controltasks:

1 Transformation between two equilibrium stateswith different Hamiltonians (STE).

2 Transformation between an initial nonequilibrium state to an equilibrium state,accompanied by a change in the Hamiltonian(STE).

3 Transformation to a final state which is colderthen the bath Tf < TB .

Control protocol

Thermodynamic analyasisFrom conservation of energy, heat is then determinedas Q = ∆ES −W , with

∆ES = tr(HS (tf ) ρS (tf )

)− tr

(HS (0) ρS (0)

).

Q =∫ t

0J(t ′)dt ′

J (t ′)≡ tr(HS (t ′) d

dt ′ ρS (t ′))

is the heat current..The work efficiency ηW , for an expansion processηW = W /Wadi , and for compression ηW = Wadi/Wwhere Wadi is the adiabatic work.For fast protocols tf ∼ 6

(2π/Ωref

)the efficiency

obtains values of ηW = 0.71 for expansionand ηW = 0.5 for compression,

At last: Shortcut to four stroke Carnot cycle

Carnot cycle:cyc=hchchc

arXiv:1906.06946Quantum Signatures in the Quantum Carnot CycleRoie Dann, Ronnie Kosloff

Performance of Shortcut to Carnot

minimum cycle time

0.2

0.12

o.04 .o.o4

L 1.0 1.2 1.4 1.6 1.8 2.0

Quantum eqivalence

The propagator: U = eL t

Four stroke cycle propagator:

Ucyc = UcUhcUhUch = eLc teLhc teLhteLcht

In the limit of small action: s = ||L t|| h

Ucyc = eLc t/2eLhc teLhteLchteLc t/2

Ucyc ≈ e(Lc+Lhc+Lh+Lch)t +O(s3)

Raam Uzdin, Amikam Levy, and Ronnie KosloffEquivalence of Quantum Heat Machines, and Quantum-Thermodynamic.(Phys. Rev. X 5, 031044 2015

The Voyage:Seeking for quantum open system description of the Carnot cycle

Non Adiabatic Master Equation NAME.

The inertial theorem.

Shortcuts to non unitary maps with entropy change.

Finite time quantum Carnot cycle.

Thank you

The end

Coherence generated “by” the bath.R. Dann, A. Levy, and R. Kosloff, Phys. Rev. A 98, 052129 (2018).

Protocol

Solution of the NAME for the HO

Slowing down the rate

Solution of the NAME for HOComparison to numerical simulation

R. Dann, A. Levy, and R. Kosloff, Phys. Rev. A 98, 052129 (2018).

Protocol

Solution for the propagatorFor can be solved in terms of

where and

Kosloff, R., & Rezek, Y. (2017 Entropy, 19(4), 136.

The NAME in Heisenberg form:.The Heisenberg picture is given by the equation ofmotion:

d

dtO = V † (t,0)L †(t)O .

For such a case the adjoint propagator has the form:

V †(t, t0) = Texp∫ t

t0dsL †(s)

where T is the anti-chronological time ordering.V †(t, t0) is defines by the adjoint generator L †

by the differential equation

∂

∂ tV †(t, t0) = V †(t, t0)L †(t)

.Breuer, H. P., & Petruccione, F. (2002). The theory of open quantum systems. (pg. 124-125)

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