Quantum Thermodynamics applied for Quantum Refrigerators cooling down a qubit Hideaki Okane1Shunsuke Kamimura1 2Shingo Kukita3Yasushi Kondo3and Yuichiro Matsuzaki1 4 1Research Center for Emerging Computing Technologies RCECT

2025-04-29 0 0 661.21KB 10 页 10玖币

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Quantum Thermodynamics applied for Quantum Refrigerators cooling down a qubit

Hideaki Okane,1Shunsuke Kamimura,1, 2 Shingo Kukita,3Yasushi Kondo,3and Yuichiro Matsuzaki1, 4, ∗

1Research Center for Emerging Computing Technologies (RCECT),

National Institute of Advanced Industrial Science and Technology (AIST),

1-1-1, Umezono, Tsukuba, Ibaraki 305-8568, Japan

2Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan

3Department of Physics, Kindai University, Higashi-Osaka 577-8502, Japan and

4NEC-AIST Quantum Technology Cooperative Research Laboratory,

National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan

(Dated: October 7, 2022)

We discuss a quantum refrigerator to increase the ground state probability of a target qubit

whose energy diﬀerence between the ground and excited states is less than the thermal energy of

the environment. We consider two types of quantum refrigerators: (1) one extra qubit with frequent

pulse operations and (2) two extra qubits without them. These two types of refrigerators are

evaluated from the viewpoint of quantum thermodynamics. More speciﬁcally, we calculate the heat

removed from the target qubit, the work done for the system, and the coeﬃcient of performance

(COP), the ratio between the heat ant the work. We show that the COP of the second type

outperforms that of the ﬁrst type. Our results are useful to design a high-performance quantum

refrigerator cooling down a qubit.

I. INTRODUCTION

Thermodynamics has traditionally explained macro-

scopic behavior of classical systems. For example, a re-

frigerator is essential not only for daily life but also for

academic purposes. We need a refrigerator for realizing

interesting phenomena, such as superﬂuidity, supercon-

ductivity, Bose-Einstein Condensation, and so on [1].

On the other hand, many eﬀorts have been devoted to

extending the conventional thermodynamics to the quan-

tum one [2–5]. Such an extension is called quantum ther-

modynamics. Quantum thermodynamics deﬁne the ther-

modynamic properties of microscopic systems: heat and

work. Quantum thermodynamics helps to understand a

quantum refrigerator, which increases the population of

the ground state of a target quantum system. To quan-

tify the performance of a quantum refrigerator, quanti-

ties such as coeﬃcient of performance (COP) and cooling

power are evaluated [6–10].

Electron spin resonance (ESR) is an important tech-

nique to detect target electron spins, which gives informa-

tion of various materials [11]. ESR has recently been per-

formed with several types of quantum detectors such as

a superconducting circuit [12–24] and nitrogen vacancy

(NV) centers [25–31]. Such a kind of detector can also

be utilized as a quantum refrigerator to polarize target

spins [32, 33]. To improve the sensitivity of the ESR,

polarizing the target spins is essential. Therefore, eval-

uation of a quantum refrigerator should contribute to a

further improvement of the ESR sensitivity.

In this work, we evaluate the performance of two

types of quantum refrigerators: (1) one extra qubit

with frequent pulse operations and (2) two extra qubits

∗matsuzaki.yuichiro@aist.go.jp

without them. The former corresponds to the realized

scheme [32, 33]. The latter is a newly proposed method

in this paper, in which one of the extra qubits is for quan-

tum control and the other is for the heat release to the

environment. We calculate the heat removed from the

target qubit, the work done for the system, and the COP

for these two approaches. According to the analysis of

them based on quantum thermodynamics, we ﬁnd that

the latter outperforms the former in terms of the COP.

The rest of this paper is organized as follows. §II

reviews the deﬁnition of heat and work in the bipartite

quantum system, based on Ref. [34]. We present two

models with and without frequent reset in §III and an-

alyze their performance from the viewpoint of quantum

thermodynamics in §IV. §V summarizes our results. In

appendix A, we explain the possible experimental realiza-

tion of our scheme with current technology. In appendix

B, we explain the detail of the conventional protocol. In

appencix C, we calculate the work necessary for a qubit

initialization. We take the natural unit system and thus

we omit ~and kBin this paper.

II. THE DEFINITION OF WORK AND HEAT

BETWEEN INTERACTING BIPARTITE

SYSTEMS

Let us review the deﬁnition of work and heat trans-

ferred between an interacting bipartite quantum system

(a target and extra qubits) based on quantum thermo-

dynamics [34]. In order to quantify the cooling eﬀect, we

need to evaluate a heat transfer from the target qubit to

the extra qubit.

To calculate heat and work in a quantum system, we

need to deﬁne an internal energy. In a single system

with a Hamiltonian Hand a density matrix ρ, the in-

ternal energy is deﬁned as Tr(ρH). By diﬀerentiating

arXiv:2210.02681v1 [quant-ph] 6 Oct 2022

the internal energy with time, we obtain d

dt (Tr(ρH)) =

Tr( ˙ρH) + Tr(ρ˙

H). Here, Tr( ˙ρH) and Tr(ρ˙

H) are deﬁned

as the heat and the work, respectively. However, when

we consider a bipartite system, it is not straightforward

to deﬁne the internal energy for each system. We con-

sider a bipartite system consisting of system A and B.

The Hamiltonian is given as

H=HA⊗IB+IA⊗HB+HAB,(1)

where HA(HB) is the Hamiltonian for system A (B),

HAB is the interaction Hamiltonian, IA(IB) denotes

identity operator for system A (B). Let ρdenote the

density matrix of the total system. The reduced den-

sity matrix of system A (B) is deﬁned as ρA= TrB(ρ)

(ρB= TrA(ρ)). We introduce the correlation between

system A and B as

χ=ρ−ρA⊗ρB.(2)

The naive deﬁnition of the internal energy for sys-

tem Amay be given as TrA(ρAHA). However, the re-

duced density matrix ρAcan be accessible to the inter-

action Hamiltonian, namely, TrA((ρA⊗IB)HAB)6= 0.

So, we should take into account the contribution from

the interaction Hamiltonian to deﬁne the internal en-

ergy for each system. We reconstruct the Hamiltonian

H=H(eﬀ)

A⊗IB+IA⊗H(eﬀ)

B+H(eﬀ)

AB for the reduced

density matrix to be inaccessible to the eﬀective interac-

tion Hamiltonian, satisfying the following conditions

TrA(ρA⊗IB)H(eﬀ)

AB = 0,(3)

TrB(IA⊗ρB)H(eﬀ)

AB = 0.(4)

We adopt such eﬀective Hamiltonians H(eﬀ)

Aand H(eﬀ)

to deﬁne the internal energy for system A and B, respec-

tively. Thus, the heat and the work to system A is given

as TrA˙ρAH(eﬀ)

Aand TrAρA˙

H(eﬀ)

A, respectively. In

the following, we speciﬁcally derive the eﬀective Hamil-

tonian.

We show how to construct the Hamiltonian satisfying

Eqs. (3) and (4). The time evolution of the total density

matrix is written as,

dρ(t)

dt =−i[H(t), ρ(t)] ,(5)

H(t) = HA(t)⊗IB+IA⊗HB+HAB.(6)

By taking the partial trace of system B in Eq. (5), we

obtain the time-evolution equation for system A,

dρA(t)

dt =−i[H0

A(t), ρA(t)] −iTrB([HAB, χ(t)]) ,(7)

A(t) = HA(t) + TrB((IA⊗ρB(t)) HAB).(8)

Similarly, the time-evolution equation for the system B

is given as,

dρB(t)

dt =−i[H0

B(t), ρB(t)] −iTrA([HAB, χ(t)]) ,(9)

B(t) = HB+ TrA((ρA(t)⊗IB)HAB).(10)

By using the new Hamiltonian H0

A(t) and H0

B(t), we can

rewrite the Hamiltonian as follows,

H=H0

A(t)⊗IB+IA⊗H0

B(t) + H0

AB(t),(11)

AB(t) = HAB −TrB((IA⊗ρB(t)) HAB)⊗IB(12)

−IA⊗TrA((ρA(t)⊗IB)HAB),(13)

where H0

AB(t) denotes the new interaction Hamiltonian.

Now, let us check whether the reduced density matrix

ρAis inaccessible to the new interaction Hamiltonian

AB(t),

TrA((ρA(t)⊗IB)H0

AB(t)) = −Tr ((ρA(t)⊗ρB(t)) HAB)IB.

(14)

The reduced density matrix is still accessible

to the interaction Hamiltonian. However, since

−Tr ((ρA(t)⊗ρB(t)) HAB) is a scalar quantity, we can

deﬁne the eﬀective interaction Hamiltonian to extract

the scalar part as follows,

H(eﬀ)

AB (t) = H0

AB(t) + Tr ((ρA(t)⊗ρB(t)) HAB) (IA⊗IB).

(15)

This satisﬁes the inaccessible condition of Eqs. (3) and

(4). By using the eﬀective interaction Hamiltonian, the

total one can be rewritten as,

H(t) = H(eﬀ)

A(t)⊗IB+IA⊗H(eﬀ)

B(t) + H(eﬀ)

AB (t),

(16)

H(eﬀ)

A(t) = H0

A(t)−(1 −α)Tr ((ρA(t)⊗ρB(t)) HAB)IA,

(17)

H(eﬀ)

B(t) = H0

B(t)−αTr ((ρA(t)⊗ρB(t)) HAB)IB,(18)

where an arbitrary parameter α∈Ris introduced for the

general expression of the eﬀective Hamiltonian.

By using the eﬀective Hamiltonian, we deﬁne the in-

ternal energy as follows,

U= Tr (ρ(t)H(t)) = UA+UB+Uχ,(19)

UA= TrAρA(t)H(eﬀ)

A(t),(20)

UB= TrBρB(t)H(eﬀ)

B(t),(21)

Uχ= Tr χ(t)H(eﬀ)

AB (t),(22)

where UA(UB) is the internal energy of the reduced den-

sity matrix ρA(t) (ρB(t)), and Uχis the internal energy

of the correlation χ(t). The heat ﬂux ˙

QA(˙

QB) to system

A (B) is deﬁned as,

QA(t) = TrA˙ρA(t)H(eﬀ)

A(t)

=−iTr ([HAB, χ(t)] (H0

A(t)⊗IB)) ,(23)

QB(t) = TrB˙ρB(t)H(eﬀ)

B(t)

=−iTr ([HAB, χ(t)] (IA⊗H0

B(t))) ,(24)

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QuantumThermodynamicsappliedforQuantumRefrigeratorscoolingdownaqubitHideakiOkane,1ShunsukeKamimura,1,2ShingoKukita,3YasushiKondo,3andYuichiroMatsuzaki1,4,1ResearchCenterforEmergingComputingTechnologies(RCECT),NationalInstituteofAdvancedIndustrialScienceandTechnology(AIST),1-1-1,Umezono,Tsukuba,Ibar...

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Quantum Thermodynamics applied for Quantum Refrigerators cooling down a qubit Hideaki Okane1Shunsuke Kamimura1 2Shingo Kukita3Yasushi Kondo3and Yuichiro Matsuzaki1 4 1Research Center for Emerging Computing Technologies RCECT

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