Quantum heat engines and refrigerators

George Thomas

Aalto university

March 31, 2021

QTF special seminar course (PHYS-E0541)

George Thomas Quantum heat engines and refrigerators Talk layout

Introduction to quantum heat engines and refrigerators Maser as a heat engine with superconducting quantum circuit Otto cycle Quantum Otto Refrigerator (Bayan) Thermodynamic cost of coherence Supremacy of incoherent sudden cycle Quantum heat valve and rectifier (Bayan)

George Thomas Quantum heat engines and refrigerators Thermodynamics

Historically thermodynamics was developed during industrial revolution, mainly from the aspiration to achieve highly efficient steam engines. Pioneering work in this direction is done by Sadi Carnot.

Tc Maximum achievable efficiency of any engine is ηC = 1 , only Th depends on temperatures −

George Thomas Quantum heat engines and refrigerators Examples of thermal machines

Refrigerator Heat engine

Heat (Qh−W) Working Heat (Qc ) Heat (Qh ) Working Heat (Qc+W) medium medium

Work (W) Work (W) Cold Hot Cold Hot

Heater

(Q ) Heat c Working Heat (Qh ) medium

Work (W) Cold Hot

for refrigerator, COP = Qc for engine, efficiency η = W W Qh

George Thomas Quantum heat engines and refrigerators Motivation

Quantum thermodynamics To extend thermodynamics in quantum domain. To understand thermodynamics in quantum regime where we have non-classical features such as superposition and entanglement. Thermodynamics of a few particle system. Example: A single qubit Is there any quantum supremacy? Heat management in quantum circuits Qubit reset in quantum computation. Heat engine as maser: converting heat into coherent source of radiation

George Thomas Quantum heat engines and refrigerators Laws of thermodynamics

Mean energy

U = Tr(Hρ), where H is the Hamiltonian and ρ is the density matrix.

First law of thermodynamics

dU = Tr(H dρ) + Tr(ρ dH) dQ = Tr(H dρ) dW = Tr(ρ dH)

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

Second law of thermodynamics Work cannot be extracted from a single heat bath in a cycle manner.

George Thomas Quantum heat engines and refrigerators Maxwell’s Demon

It is a thought experiment proposed in 1867 by James Clerk Maxwell to show an apparent violation of Second law of thermodynamics.

Figure: Maxwell’s demon allows the gas molecules with higher velocities to the right and molecules with lower velocities to the left. This creates a temperature gradient between left and right chambers. Further, this temperature gradient can be used to extract work

1

1 H. Leff and A. F. Rex, Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics, Bristol, 2003). George Thomas Quantum heat engines and refrigerators Szilard engine

George Thomas Quantum heat engines and refrigerators Szilard engine1

L R

Experimental realization of a Szilard engine (A) (B) L R with a single electron J. V. Koski, V. F. Maisi, J. P. Pekola, and Dmitri V. Averin,

PNAS, 111 13786 (2014).




(D) (C)









L
























(A) A particle in a box (B) Insertion of the barrier (C) Measurement: to know whether the particle in ‘L’ or ‘R’. (D) Isothermal expansion: this leads to the lifting up of the load.

Landauer’s principle 2

Minimum work needed to erase one bit of information is kB T ln2.

1 L. Szilard, Zeitschrift f¨urPhysik 53, 840 (1929). 2 R. Landauer, IBM J. Res. Dev. 5, 183 (1961)

George Thomas Quantum heat engines and refrigerators Szilard engine

L R (a)

(b)

(A) (B) L R (c)

(D) (C)




Figure: Particle statistics after inserting the




L









barrier: (a) Distinguishable particles, (b)









1,2




Bosons and (c) Fermions .

Classical case: Work required to insert the barrier is zero. Quantum case: Work required to insert the barrier is non-zero2

1G Thomas, D Das, S Ghosh Physical Review E 100, 012123 (2019) 1S.W. Kim et al, Phys. Rev. Lett. 106, 070401 (2011).

George Thomas Quantum heat engines and refrigerators Second law of thermodynamic in quantum regime

Sadi, Carnot (1824), Max. Achievable efficiency for any engine is Tc ηC = 1 Th Is it possible− surpass the Carnot bound? Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence M. O. Scully, M. S. Zubairy, G. S. Agarwal, H. Walthe, Science 299 862 (2003). Energetics of quantum correlations, R. Dillenschneider, E. Lutz EPL 88 50003 (2009). Nanoscale heat engine beyond the Carnot limit, J Roßnagel, O Abah, F Schmidt-Kaler, K Singer, E Lutz Phys. Rev. Lett. 112, 030602 (2014). We cannot ignore the cost for the resources.

George Thomas Quantum heat engines and refrigerators MASER as a heat engine

T2 T1

Filter 3









p ν 3 2 p 2









Filter 2 ν 1

p 1









1

Figure: Three level MASER as a heat engine 2

−hν1 −hν2 k T k T P3 = P1e B 1 , P3 = P2e B 2

population inversion, P2 > P1 = P2 P3 > 1. ⇒ P3 P1 ν1−ν2 T2 = < 1 = ηC (bounded by Carnot efficiency) ⇒ ν1 − T1

2 H. E. D. Scovil and E. O. Schulz-DuBois, Phys. Rev. Lett. 2, 262 (1959). George Thomas Quantum heat engines and refrigerators MASER as a heat engine

In quantum optical set up: M. O. Scully, K. R. Chapin, K. E. Dorfman, M. B. Kim, and A. Svidzinsky, Quantum heat engine power can be increased by noise-induced coherence, Proceedings of the National Academy of Sciences, 108 15097 (2011). Sheng-Wen Li, M. B. Kim, G. S. Agarwal, and M. O. Scully, Quantum statistics of a single-atom Scovil–Schulz-DuBois heat engine, Phys. Rev. A 96, 063806 (2017). A.Ghosh, et. al, Two-level masers as heat-to-work converters, PNAS 115 (40) 9941 (2018). In quantum dots: B. K. Agarwalla, M. Kulkarni, and D. Segal, Photon statistics of a double quantum dot micromaser: Quantum treatment, Phys. Rev. B 100, 035412 (2019)

George Thomas Quantum heat engines and refrigerators Recent experimental realizations∗:

Image credit: Jorden Senior / Aalto University ∗A. Ronzani, B. Karimi, J. Senior , Y-C Chang, J. T. Peltonen, C. Chen,and J. P. Pekola, Nature 14, 991 (2018).

Motivation

To realize thermally pumped MASER in superconducting circuits. Converting heat into coherent source of radiation. Heat management in quantum circuits.

George Thomas Quantum heat engines and refrigerators Motivation

To realize thermally pumped MASER in superconducting circuits. Converting heat into coherent source of radiation. Heat management in quantum circuits. Recent experimental realizations∗:

Image credit: Jorden Senior / Aalto University ∗A. Ronzani, B. Karimi, J. Senior , Y-C Chang, J. T. Peltonen, C. Chen,and J. P. Pekola, Nature 14, 991 (2018).

George Thomas Quantum heat engines and refrigerators Superconducting qubit

LC oscillator

ω = 1/√LC 4 | i 3 L C | i 2 | i Energy 1 ω | i 0 | i Flux

A transmon qubit 3

4 3| i | i 2 ω12 | i

Energy 1 ω01 | i 0 | i ϕ p ω01 ( 8EJ EC EC )/~, ω01 ω12 EC /~ (1) ≈ − − ≈ EJ is the Josephson energy and EC is the charging energy. Current through the junction is I = IC sin ϕ

3 J. Koch et al., Phys. Rev. A 76, 042319 (2007). George Thomas Quantum heat engines and refrigerators Set-up

A three level system with direct 0-2 transitions. Resonators with high quality factors which we need to use as filters. Well defined heat baths. (a)

Tc T ωc ωh h

D Td ωd

Cd′ Z0 D Td

(b) ωd

Cd R C′ ωc ωh C′ R c c Cc Ch h h

Tc Th 1 3 2

Figure: The flux qutrit consists of three Josephson junctions numbered as 1, 2, and 3 arranged in an asymmetric geometry together with superconducting island denoted with dotted lines .

G. Thomas, A. Gubaydullin, D. S. Golubev, and J. P. Pekola, Phys. Rev. B 102, 104503 (2020). George Thomas Quantum heat engines and refrigerators Setup: Qutrit

Thus the potential energy term is

U = EJ [3 cos (ϕx /3) cos ϕ 2 cos ((ϕ ϕx )/2)]. (2) − − −

2  2 3 4  ∂ ϕ¯ ϕx ϕ¯ ϕ¯ H = 4EC +E˜J tan , − ∂ϕ¯2 2 − 3 12 − 32 1.2 U 3E s ϕx J EC tan 3 p 0.6 n 2 ϕˆ n = ϕx n(n 1) h − | | i 3EJ cos 3 6 − 0 -2 0 2 4 6 " # ϕ  4E 1/4 r E C √ C n 1 ϕˆ n = ϕx n 1 + n h − | | i 3EJ cos 3 32EJ

George Thomas Quantum heat engines and refrigerators Thermally pumped maser in superconducting circuits

On-chip model of a thermally pumped maser 4. Coupling constant

2 r ωhCh ~Z0h (a) gh = ϕ02, r = h, c, d 2e π Tc T ωc ωh h Decay rate of the resonator

3 02 (2ωr Z0hRhCh ) κh = π D Td ωd

Cd′ Z0 Transition rate of the qutrit D Td

ωd 2 (b) gh κh γh = 2 2 Cd ∆h + κh/4 R C′ ωc ωh C′ R c c Cc Ch h h

Tc Th Quality factor 1 3 2 π Qh = ωh/κh = 2 02 (2ωhZ0hRhCh ) 4G. Thomas, A. Gubaydullin, D. S. Golubev, and J. P. Pekola, Phys. Rev. B 102, 104503 (2020). George Thomas Quantum heat engines and refrigerators Power and efficiency

On-chip model of a thermally pumped maser 5.

Pout = n κ~ω10. (3) Tc T h i ωc ωh h

For large number of photons in output resonator

D Td γc γh(Nh − Nc ) ωd Pout ≈ ~ω10. (4) γc (2 + 3Nc ) + γh(2 + 3Nh)

Pin = (p0Γ02 − p2Γ20)~ω20 (5) The efficiency of maser is P ω T η = out 10 < 1 c (6) Pin ≈ ω20 − Th

5G. Thomas, A. Gubaydullin, D. S. Golubev, and J. P. Pekola, Phys. Rev. B 102, 104503 (2020). George Thomas Quantum heat engines and refrigerators Wigner function and Poissonian distribution

Tc Th

ωc ωh −hni n Pois e hni Pn = n!

ωd D

(a) (c) (e) (g) 2.5 0.4 0.3 1000 2.0 0.3 ˜ ˜ ˜ x x x 100 1.5 0.2 x/ x/ x/ 0.2 10 1.0 1 0.1 0.1

0.5 Power (aW) 0 0 0 0.1 0.5 0.9 p/p˜ p/p˜ p/p˜ Tc/Th (b) (d) (f) (h) 1 Tc Tc =1.0 0.11 Tc 0.05 =0.13 Th =0.4 Th Th P 0.3 P P n n 0.06 n 0.03 0.5

0.01 Efficiency 0.01 0. 0 0 10 20 0 25 50 0 50 100 0 0.5 1 n n n Tc/Th

G. Thomas, A. Gubaydullin, D. S. Golubev, and J. P. Pekola, Phys. Rev. B 102, 104503 (2020).

George Thomas Quantum heat engines and refrigerators Output resonator is connected to detector bath

Tc Th

ωc ωh

ωd D

From entropy production

~ω01 ~ω12 ~ω02 + 0. (7) kB Td kB Tc − kB Th ≥

The essential condition for population inversion is ~ω12/kB Tc > ~ω02/kB Th. If the lowest levels are under population inversion, the effective temperature estimated from the two levels is negative. This implies that irrespective of the temperature Td , the heat flows to the resistor (Rd ), even if Td > Th.

George Thomas Quantum heat engines and refrigerators Summary

A model of thermally driven on-chip three level maser in a superconducting circuit is discussed which converts heat into coherent radiation. We have also proposed an experimentally feasible method of detecting the population inversion in the artificial atom by thermometry. We have derived a simple analytical expression for the output power of the maser in terms of the circuit parameters. We have shown that powers up to few femtowatts can be achieved for typical parameters of cQED setup . We believe that the proposed way of converting heat into coherent radiation on-chip can be very useful for quantum circuit applications. Our technique is useful to control and detect the populations of the energy levels of the artificial atom.

George Thomas Quantum heat engines and refrigerators Otto cycle

George Thomas Quantum heat engines and refrigerators Classical Otto cycle

P A Nicolaus Otto was the first T1 person to build a working model of classical four-stroke Q engine in 1861. 1 D T’1 Four-stroke gasoline engines T’2 B Q used nowadays are generally 2 called Otto engines. C T2

V V1 2 V Work and Efficiency

0 0 Q1 = Cv (T1 T ), Q2 = Cv (T2 T ) − 2 − 1 W = Q1 + Q2 W T 0 V (γ−1) η = = 1 1 = 1 1 . Q1 − T1 − V2

George Thomas Quantum heat engines and refrigerators Thermodynamic cost of coherence

|e> Fast Slow

|g’> |g’>

|e’> |e’>

George Thomas Quantum heat engines and refrigerators Thermodynamic cost of coherence

Let us consider initial Hamiltonian as

H(0) = Egg g g + Eee e e , (8) | ih | | ih | and the final Hamiltonian is

H(τ) = E 0 g 0 g 0 + E 0 e0 e0 . (9) gg | ih | ee | ih | The general form of the corresponding unitary with time ordering operator is written as is T (−i/ ) R τ H(t0)dt0 U = e ~ 0 . (10) T Let us consider the initial state as a thermal state e−βH(0) ρ(0) = = Pgg g g + Pee e e , (11) Tr[e−βH(0)] | ih | | ih | The initial mean energy is

E ini = Tr[H(0)ρ(0)] = Egg Pgg + Eee Pee (12) h i

George Thomas Quantum heat engines and refrigerators Thermodynamic cost of coherence

† ρ(τ) = Uρ(0)U , E fin = Tr[H(τ)ρ(τ)] (13) h i

E sudden E adiabatic . (14) h ifin ≥ h ifin

We have to do an extra work in a sudden process. This extra work is used to create the coherence in the system. The coherence term is of the 0 0 0 0 form Pg 0e0 = Pgg g g g e + Pee g e e e . When we attach this system to the bath,h | theih system| i decoheresh | ih by| givingi off this energy to the bath. In quantum thermodynamics, this phenomenon is generally known as inner friction 6.

6R. Kosloff and T. Feldmann, Phys. Rev. E 65, 055102(R) (2002). George Thomas Quantum heat engines and refrigerators Sudden Otto refrigeration cycle

J. P. Pekola, B. Karimi, G. Thomas, and D. V. Averin Phys. Rev. B 100, 085405 (2019).

George Thomas Quantum heat engines and refrigerators Thermal machines

Refrigerator Heat engine

Heat (Qh−W) Working Heat (Qc ) Heat (Qh ) Working Heat (Qc+W) medium medium

Work (W) Work (W) Cold Hot Cold Hot

Heater

(Q ) Heat c Working Heat (Qh ) medium

Work (W) Cold Hot

George Thomas Quantum heat engines and refrigerators 1 f = (2δt) → ∞

J. P. Pekola, B. Karimi, G. Thomas, and D. V. Averin Phys. Rev. B 100, 085405 (2019).

Quantum Otto cycle

(a) Otto cycle (b) Short-time cycle protocol.

George Thomas Quantum heat engines and refrigerators Quantum Otto cycle

(a) Otto cycle (b) Short-time cycle protocol.

1 f = (2δt) → ∞

J. P. Pekola, B. Karimi, G. Thomas, and D. V. Averin Phys. Rev. B 100, 085405 (2019).

George Thomas Quantum heat engines and refrigerators Experimental Implementation

Potential experimental realization: the schematic of a superconducting qubit capacitively (Cc ) coupled to coplanar wave resonators, operating at two distinct frequencies, and terminated by resistors RC and RH acting as the heat baths. The energy separation ∆E of the TLS is tuned by applying magnetic flux Φ.

J. P. Pekola, B. Karimi, G. Thomas, and D. V. Averin Phys. Rev. B 100, 085405 (2019).

George Thomas Quantum heat engines and refrigerators Hamiltonian

H = E0(∆σx + qσz ), Q −

The eigenstates in the computational basis + , {| i |−i} r r 1 η(q) 1 + η(q) g = − + + , | i 2 | i 2 |−i r r 1 + η(q) 1 η(q) e = + − , | i 2 | i − 2 |−i

p where η(q) (q/∆)/ 1 + (q/∆)2 ≡ p 2 2 ~ω0 Ee Eg = 2E0 q + ∆ ≡ −

George Thomas Quantum heat engines and refrigerators Cycle

p 2 2 ~ω0 Ee Eg = 2E0 q + ∆ ≡ − Cycle: a b thermalization with cold bath q = 0 → b c q : 0 q (∆EC ∆EH ) → → max → c d thermalization with hot bath q = qmax → 0 d a q : q 0 (∆EH ∆EC ) → max → →

George Thomas Quantum heat engines and refrigerators Thermalization

Our model is based on the fully realistic description of the heat baths. The only assumption that we make about the reservoirs is that they introduce transition rates of the system governed by the detailed balance condition, which is the most general description of an equilibrium heat bath. B −βB∆EB B Γ↑ = e Γ↓ .

George Thomas Quantum heat engines and refrigerators The steady state situation

H = E0(∆σx + qσz ), Q −

 1  2 + i i + i i ρi = D R1 I i i i i R − I 2 − D

George Thomas Quantum heat engines and refrigerators Heat Power

Avearge heat per cycle P = Total cycle time

Heat currents to the cold P = ∆E ( b a)/(2δt) and hot C C D − D P = ∆E ( d c )/(2δt) baths are then given for q /∆ 1 by H H D − D max 

C(H) H(C) Γ Γ (1 e−βC(H)∆EC(H) ) P ∆E ↓ Σ − > 0. C(H) ≈ C(H) C(H) H(C) 4(2ΓΣ + ΓΣ )

In this limit, both baths are heated. Adverse effect of coherence on the performance of a quantum refrigerator.

George Thomas Quantum heat engines and refrigerators Classical (incoherent) cycle

H = E(t)σz , Q −

 1  2 + i 0 ρi = D 1 0 i 2 − D

C C b = a + [Γ Γ a]δt D D ↓ − ΣD c = b, D D H H d = c + [Γ Γ c ]δt D D ↓ − Σ D a = d , D D

George Thomas Quantum heat engines and refrigerators Power

1 ΓCΓH ↓ ↓ −βH(C)∆EH(C) −βC(H)∆EC(H)
PC(H) = C H e e ∆EC(H). 2 ΓΣ + ΓΣ −

For βH∆EH > βC∆EC

PC < 0 and PH > 0,

Cooling Power is independent of δt. Infinite operation frequency is possible. Shows the supremacy of incoherent cycle.

George Thomas Quantum heat engines and refrigerators Experimental Implementation

Taking reasonable parameters, our model can have relaxation rate Γ 100 MHz and the operation frequency f > 100 MHz and can give output∼ power P Γ∆E 10−16 W. B ∼ ∼

George Thomas Quantum heat engines and refrigerators Shortcut to adiabaticity Protocol

H = E0(∆σx + q(t)σz ), Q − The basic shortcuts to adiabaticity involve compensating fields that are proportional to the time derivative of q 7.

~ ∂χ H = σy , CD 2 ∂t where χ = tan−1 [∆/q(t)]. This means infinite fields for sudden cycles, which is infeasible for experimental realization.

7M. G. Bason et al. Nat. Phys. 8, 147 (2012) X. Chen et al. Phys. Rev. Lett. 104, 063002 (2010). George Thomas Quantum heat engines and refrigerators Protocol to avoid coherence

H = E0(∆σx + qσz ), Q −

r r 1 η(q) 1 + η(q) g = − + + , | i 2 | i 2 |−i r r 1 + η(q) 1 η(q) e = + − , (15) | i 2 | i − 2 |−i

p where η(q) (q/∆)/ 1 + (q/∆)2 ≡ p 2 2 ~ω0 Ee Eg = 2E0 q + ∆ ≡ − If q and ∆ are varied in time such that their ratio remains constant throughout the cycle, then the energy eigenstates ( g , e ) become time-independent and hence no coherence will be created,| i | i but varying the parameters q and ∆ changes the energy level spacing and thus restores the refrigeration cycle. George Thomas Quantum heat engines and refrigerators Conclusion

Quantum model of sudden cycle creates coherences which leads to heating of both the baths. Incoherent model of the sudden cycle can act as a refrigerator even in the limit of infinite operation frequency. Supremacy of incoherent sudden cycles We discussed a protocol to avoid coherences during the cycle and to restore cooling in the quantum model. Possible experimental realization of our model using superconducting circuit is discussed.

George Thomas Quantum heat engines and refrigerators Acknowledgment

Jukka P. Pekola Dmitri Golubev Azat Gubaydullin Bayan Karimi Dmitri V. Averin

Thank you!!

George Thomas Quantum heat engines and refrigerators