Reversing the Direction of Heat Flow Using Quantum Correlations

ARTICLE

https://doi.org/10.1038/s41467-019-10333-7 OPEN Reversing the direction of heat ﬂow using quantum correlations

Kaonan Micadei1,2,8, John P.S. Peterson3,8, Alexandre M. Souza 3, Roberto S. Sarthour3, Ivan S. Oliveira3, Gabriel T. Landi4, Tiago B. Batalhão5,6, Roberto M. Serra 1,7 & Eric Lutz2

Heat spontaneously ﬂows from hot to cold in standard thermodynamics. However, the latter theory presupposes the absence of initial correlations between interacting systems. We here

1234567890():,; experimentally demonstrate the reversal of heat ﬂow for two quantum correlated spins-1/2, initially prepared in local thermal states at different effective temperatures, employing a Nuclear Magnetic Resonance setup. We observe a spontaneous energy ﬂow from the cold to the hot system. This process is enabled by a trade off between correlations and entropy that we quantify with information-theoretical quantities. These results highlight the subtle interplay of quantum mechanics, thermodynamics and information theory. They further provide a mechanism to control heat on the microscale.

1 Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Avenida dos Estados 5001, 09210-580 Santo André, São Paulo, Brazil. 2 Institute for Theoretical Physics I, University of Stuttgart, D-70550 Stuttgart, Germany. 3 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Rio de Janeiro, Brazil. 4 Instituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970 São Paulo, SP, Brazil. 5 Singapore University of Technology and Design, 8 Somapah Road, Singapore 487372, Singapore. 6 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore. 7 Department of Physics, University of York, York YO10 5DD, UK. 8These authors contributed equally: Kaonan Micadei, John P.S. Peterson. Correspondence and requests for materials should be addressed to R.M.S. (email: [email protected])

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ccording to Clausius, heat spontaneously flows from a hot approximation. Our aim is to study the heat exchange between Abody to a cold body1. At a phenomenological level, the the 1H (system A) and 13C (system B) nuclear spins under a second law of thermodynamics associates such irrever- partial thermalization process in the presence of initial correla- sible behavior with a nonnegative mean entropy production2.On tions (Fig. 1a). Employing a sequence of transversal rf-field and the other hand, Boltzmann related it to specific initial conditions longitudinal field-gradient pulses, we prepare an initial state of of the microscopic dynamics3–5. Quantitative experimental con- both nuclear spins (A and B) of the form, firmation of this conjecture has recently been obtained for a ρ0 ¼ ρ0 ρ0 þ χ ; ð1Þ driven classical Brownian particle and for an electrical RC cir- AB A B AB 6 7 χ = α 〉〈 + α* 〉〈 cuit , as well as for a driven quantum spin , and a driven where AB |01 10| |10 01| is a correlation term and 8 ρ0 ¼ ðÀβ H Þ=Z β = quantum dot . These experiments have been accompanied by a i exp i i i a thermal state at inverse temperature i surge of theoretical studies on classical and quantum irreversi- 1/(k T ), i = (A, B), with k the Boltzmann constant. The stateji 0 – B i B bility9 19. It has in particular been shown that a preferred (1ji) represents the ground (excited) eigenstate of the Hamilto- H Z ¼ ðÀβ H Þ direction of average behavior may be discerned, irrespective of the nian i, and i Tri exp i i is the partition function. The size of the system20. individual nuclear spin Hamiltonian, in a double-rotating frame fl Initial conditions not only induce irreversible heat ow, they with the nuclear spinsÀÁ (1H and 13C) Larmor frequency, may be also determine the direction of the heat current. The observation H ¼ ν À σi = ν = written as i h 0 1 z 2, with 0 1 kHz effectively of the average positivity of the entropy production in nature is determined by a nuclei rf-fieldÂÃ offset. In Eq. (1), the coupling often explained by the low entropy value of the initial state3. This jαj À ν ðβ þ β Þ= =ðZ Z Þ strength should satisfy exp h 0 A B 2 A B to opens the possibility to control or even reverse the direction of α = fl ensure positivity. We consider two distinct cases: for 0, the heat ow, depending on the initial conditions. In standard ther- spins are initially uncorrelated as assumed in standard thermo- modynamics, systems are assumed to be uncorrelated before dynamics, while for α ≠ 0, the joint state is initially correlated. We thermal contact. As a result, according to the second law, heat will note that since Tr χ ¼ 0, the two spins are locally always in a flow from the hot object to the cold object. However, it has been i AB thermal Gibbs state in both situations. As a result, thermo- theoretically suggested that for quantum-correlated local thermal dynamic quantities, such as temperature, internal energy, heat, states, heat might flow from the cold to the hot system, thus and entropy, are well defined. A partial thermalization between effectively reversing its direction9–12. This phenomenon has been the qubits is described by the effective (Dzyaloshinskii–Moriya) predicted to occur in general multidimensional bipartite 9,10 10 Heff ¼ ðπ= Þ σAσB À σAσB systems , including the limiting case of two simple qubits ,as interaction Hamiltonian, AB i h 2 J x y y x , with well as in multipartite systems11. J = 215.1 Hz28,29, which can be easily realized experimentally. We Here, we report the experimental demonstration of the reversal implementÀÁ the corresponding evolution operator, of heat flow for two initially quantum-correlated qubits (two- U ¼ À τHeff = τ exp i AB h , by combining free evolutions under the spin-1/2 systems) prepared in local thermal states at different natural hydrogen–carbon scalar coupling and rf-field rotations effective temperatures employing Nuclear Magnetic Resonance (Fig. 1c). We further stress that the correlation term should satisfy (NMR) techniques21,22. Allowing thermal contact between the ÂÃ χ ; Heff ≠0 for the heat flow reversal to occur (Supplementary qubits, we track the evolution of the global state with the help of AB AB quantum-state tomography21. We experimentally determine the Information). energy change of each spin and the variation of their mutual information23. For initially correlated systems, we observe a Thermodynamics. In macroscopic thermodynamics, heat is fi spontaneous heat current from the cold to the hot spin and show de ned as the energy exchanged between to large bodies at dif- 2 that this process is made possible by a decrease of their mutual ferent temperatures . This notion has been successfully extended 30 information. The second law for the isolated two-spin system is to small systems initially prepared in thermal Gibbs states , 10 therefore verified. However, the standard second law in its local including qubits . Since the interaction Hamiltonian commutes form apparently fails to apply to this situation with initial ÂÃwith the total Hamiltonian of the two qubits, H þ ; Heff ¼ quantum correlations. We further establish the nonclassicallity of A HB AB 0, the thermalization operation does not the initial correlation by evaluating its nonzero geometric quan- perform any work on the spins31. As a result, the mean energy is tum discord, a measure of quantumness24,25.Wefinally theore- constant in time and the heat absorbed by one qubit is given by = Δ tically derive and experimentally investigate an expression for the its internal energy variation along the dynamics, Qi Ei, where ¼ ρ H heat current that reveals the trade off between information and Ei Tri i i is the z-component of the nuclear spin magnetiza- entropy. tion. For the combined system, the two heat contributions satisfy9–11, β Q þ β Q ΔIðA : BÞ; ð2Þ Results A A B B Experimental system. NMR offers an exceptional degree of where ΔI(A:B) is the change of mutual information between A preparation, control, and measurement of coupled nuclear spin and B. The mutual information, defined as I(A:B) = SA + SB − 21,22 ≥ systems . It has for this reason become a premier tool for the SAB 0, is a measure of the total correlations between two sys- 7,26,27 23 study of quantum thermodynamics . In our investigation, we tems , where Si = −Tri ρi ln ρi is the von Neumann entropy of 13 1 13 ρ consider two nuclear spins-1/2, in the C and H nuclei of a C- state i. Equation (2) follows from the unitarity of the global labeled CHCl3 liquid sample diluted in Acetone-d6 (Fig. 1b). The dynamics and the Gibbs form of the initial spin states. For sample is placed inside a superconducting magnet that produces a initially uncorrelated spins, the initial mutual information is zero. longitudinal static magnetic field (along the positive z-axis) and As a result, it can only increase during thermalization, ΔI(A:B) ≥ + = the system is manipulated by time-modulated transverse radio- 0. Noting that QA QB 0 for the isolated bipartite system, we frequency (rf) fields. We study processes in a time interval of few find9–11, ÀÁ milliseconds, which is much shorter than any relevant deco- Q β À β 0 ðuncorrelatedÞ: ð3Þ herence time of the system (of the order of few seconds)26. The B B A fl ≥ dynamics of the combined spins in the sample is thus effectively Heat hence ows from the hot to the cold spin, QB >0ifTA closed and the total energy is conserved to an excellent TB. This is the standard second law. By contrast, for initially

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a b Sample Initially uncorrelated systems 1H 0 B 13C Interaction rf-field Cl Warm Cold Cl Cl Heat flow

Hot Warm Super conductor magnet conductor Super

rf-coil Initially correlated systems c Interaction time Data rf-pulse acquisition s s x-direction πJ πJ Cold Colder 1 π –π –π π H 2 2 2 2 Initial Locally Heat state thermal flow prep. 13 π –π π –π Hot Hotter C 2 2 2 2

Correlation rf-pulse Free evolution consumption y-direction under the scalar coupling

Fig. 1 Schematic of the experimental setup. a Heat flows from the hot to the cold spin (at thermal contact) when both are initially uncorrelated. This corresponds to standard thermodynamic. For initially quantum-correlated spins, heat is spontaneously transferred from the cold to the hot spin. The direction of heat flow is here reversed. b View of the magnetometer used in our NMR experiment. A superconducting magnet, producing a high-intensity magnetic field (B0) in the longitudinal direction, is immersed in a thermally shielded vessel in liquid He, surrounded by liquid N in another vacuum separated chamber. The sample is placed at the center of the magnet within the radio-frequency coil of the probe head inside a 5-mm glass tube. c Experimental pulse sequence for the partial thermalization process. The blue (black) circle represents x (y) rotations by the indicated angle. The orange HHC ¼ðπh= ÞJσHσC 1 13 connections represents a free evolution under the scalar coupling, J 2 z z , between the H and C nuclear spins during the time indicated above the symbol. We have performed 22 samplings of the interaction time τ in the interval 0 to 2.32 ms correlated qubits, the mutual information may decrease during case α ¼ 0:00 ± 0:01 (α ¼À0:19 ± 0:01) (Supplementary Infor- the thermal contact between the spins. In that situation, we may mation). The value of α was chosen to maximize the current have9–11, reversal. In order to quantify the quantumness of the initial ÀÁ β À β ð Þ: ð Þ correlation in the correlated case, we consider the normalized QB B A 0 correlated 4 2 geometric discord, defined as D ¼ minψ2C2kkρ À ψ , where C is fl g Heat ows in this case from the cold to the hot qubit: the the set of all states classically correlated24,25. The geometric energy current is reversed. We quantitatively characterize the discord has a simple closed-form expression for two qubits that fl occurrence of such reversal by computing the heat ow at any can be directly evaluated from the measured QST data time τ, obtaining (see the Methods section), fi = ÀÁÀÁ (Supplementary Information). We nd the nonzero value Dg Δβ ¼ Δ ð : Þþ ρτ k ρ0 þ ρτ k ρ0 ; ð Þ 0.14 ± 0.01 for the initially correlated state prepared in the QB I A B S A A S B B 5 ÀÁ ÀÁexperiment. Δβ = β − β ≥ ρτ k ρ ¼ ρτ ρτ À ρ where B A 0 and S i i Tri i ln i ln i We experimentally reconstruct the global two-qubit density 23 ρτ ¼ 21 0 denotes the relative entropy between the evolved AðBÞ operator using quantum-state tomography and evaluate the U ρ0 Uy ρ0 changes of internal energies of each qubit, of mutual information, TrBðAÞ τ AB τ and the initial AðBÞ reduced states. The latter fi τ and of geometric quantum discord during thermal contact quanti es the entropic distance between the state at time and (Fig. 2a–f). We observe the standard second law in the absence the initial thermal state. It can be interpreted as the entropy of initial correlations (α ’ 0), i.e., the hot qubit A cools down, production associated with the irreversible heat transfer, or to the Q < 0, while the cold qubit B heats up, Q > 0 (circles symbols in entropy produced during the ensuing relaxation to the initial A B 32,33 Fig. 2a, b). At the same time, the mutual information and the thermal state . According to Eq. (5), the direction of the geometric quantum discord increase, as correlations build up energy current is therefore reversed whenever the decrease of following the thermal interaction (circles symbols in Fig. 2c, d). mutual information compensates the entropy production. The The situation changes dramatically in the presence of initial fact that initial correlations may be used to decrease entropy has quantum correlations (α ≠ 0): the energy current is here reversed fi 34 rst been emphasized by Lloyd and further investigated in in the time interval 0 < τ < 2.1 ms, as heat flows from the cold to 35,36 fl refs. . Heat ow reversal has recently been interpreted as a the hot spin, Q = −Q > 0 (squares symbols in Fig. 2a, b). This refrigeration process driven by the work potential stored in the A B 12 reversal is accompanied by a decrease of mutual information and correlations . In that context, Eq. (5) can be seen as a generalized geometric quantum discord (squares symbols in Fig. 2c, d). In Clausius inequality due to the positivity of the relative entropies. this case, quantum correlations are converted into energy and The coefficient of performance of such a refrigeration is then at fl 12 used to switch the direction of the heat ow, in an apparent most that of Carnot . violation of the second law. Correlations reach their minimum at In our experiment, we prepare the two-qubit system in an around τ ≈ 1.05 ms, after which they build up again. Once they βÀ1 ¼ initial state of the form (1) with effective spin temperatures A have passed their initial value at τ ≈ 2.1 ms, energy is transferred : : βÀ1 ¼ : : βÀ1 ¼ : : 4 66 ± 0 13 peV ( A 4 30 ± 0 11 peV) and B 3 31 ± 0 08 in the expected direction, from hot to cold. In all cases, we obtain βÀ1 ¼ : : peV ( B 3 66 ± 0 09 peV) for the uncorrelated (correlated) good agreement between experimental data (symbols) and

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ac e Uncorrelated Δ ρτ ||ρ0 ρτ ||ρ0 I + S( A A) + S( B B) Uncorrelated 1.8 Heating Correlated 0.05 Δ ρτ ||ρ0 0.3 I + S( A A) 1.6 0.04 ΔI

1.4 Cooling 0.2 0.03

1.2 0.02 Energy (peV) 0.1 1.0 Correlation 0.01 Mutual information (bit) consumption Entropic quantities (bit) Spin A 0.8 0.0 0.00 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Interaction time (ms) Interaction time (ms) Interaction time (ms) bd f 0.15 0.00 1.2 Heating –0.05 1.0 0.10 –0.10 0.8 –0.15 Cooling

Energy (peV) 0.6 0.05 –0.20 Correlation

Entropic quantities (bit) –0.25 0.4 consumption Spin B Geometric quantum discord Correlated 0.00 –0.30 0.2 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Interaction time (ms) Interaction time (ms) Interaction time (ms)

Fig. 2 Dynamics of heat, correlations, and entropic quantities. a Internal energy of qubit A along the partial thermalization process. b Internal energy of qubit B. In the absence of initial correlations, the hot qubit A cools down and the cold qubit B heats up (cyan circles in panel a and b). By contrast, in the presence of initial quantum correlations, the heat current is reversed as the hot qubit A gains and the cold qubit B loses energy (orange squares in panel a and b). This reversal is made possible by a decrease of the mutual information c and of the geometric quantum discord d. Different entropic contributions to the heat current (5) in the uncorrelated e andÀÁ uncorrelated f ÀÁcase. Reversal occurs when the negative variation of the mutual information, ΔI(A:B), S ρτ k ρ S ρτ k ρ compensates the positive entropy productions, A A and B B , of the respective qubits. The symbols represent experimental data and the dashed lines are numerical simulations. Error bars were estimated by a Monte Carlo sampling from the standard deviation of the measured data (Supplementary Information) theoretical simulations (dashed lines). Small discrepancies seen as time is not an absolute but a relative concept that depends on the time increases are mainly due to inhomogeneities in the control choice of initial conditions. We have observed the reversal of the fields. energy current for the case of two spins which never fully ther- The experimental investigation of Eq. (5) as a function of the malize due to their finite size. However, their dynamics is iden- thermalization time is presented in Fig. 2e, f. While the relative tical to that of a thermalization map during the duration of the entropies steadily grow in the absence of initial correlations, they experiment (Methods), a process we have labeled partial ther- exhibit an increase up to 1.05 ms followed by a decrease in malization for this reason. Furthermore, numerical simulations presence of initial correlations. The latter behavior reflects the show that reversals may also occur for a spin interacting with pattern of the qubits already seen in Fig. 2a, b, for the average larger spin environments which induce thermalization (Supple- energies. We note, in addition, a positive variation of the mutual mentary Information). Hence, an anomalous heat current does information in the uncorrelated case and a large negative not seem to be restricted to extremely microscopic systems. The variation in the correlated case. The latter offsets the increase of precise scaling of this effect with the system size is an interesting the relative entropies and enables the reversal of the heat current. subject for future experimental and theoretical investigations. Our These findings provide direct experimental evidence for the results on the reversal of the thermodynamic arrow of time might trading of quantum mutual information and entropy production. also have stimulating consequences on the cosmological arrow of time34. Discussion We have observed the reversal of the energy flow between two Methods quantum-correlated qubits with different effective temperatures, Experimental setup. The liquid sample consist of 50 mg of 99% 13C-labeled fl associated with the respective populations of the two levels. Such CHCl3 (Chloroform) diluted in 0.7 ml of 99.9% deutered Acetone-d6, in a ame sealed Wildmad LabGlass 5 -mm tube. Experiments were carried out in a Varian effect has been predicted to exist in general multidimensional – 500 MHz Spectrometer employing a double-resonance probe head equipped with a systems9 11. By revealing the fundamental influence of initial magnetic field-gradient coil. The sample is very diluted, such that the inter- quantum correlations on the direction of thermodynamic pro- molecular interaction can be neglected, in this way the sample can be regarded as a cesses, which Eddington has called the arrow of time37, our set of identically prepared pairs of spin-1/2 systems. The superconducting magnet (illustrated in Fig. 1b) inside of the magnetometer produces a strong intensity experiment highlights the subtle interplay of quantum mechanics, longitudinal static magnetic field (whose direction is taken to be along the positive ≈ fi 1 13 thermodynamics, and information theory. Initial conditions thus z axes), B0 11.75 T. Under this led, the H and C Larmor frequencies are about not only break the time-reversal symmetry of the otherwise 500 MHz and 125 MHz, respectively. The state of the nuclear spins are controlled by time-modulated rf-field pulses in the transverse (x and y) direction and long- reversible dynamics, they also determine the direction of a pro- fi cess. Our findings further emphasize the limitations of the stan- itudinal eld gradients. Spin-latticeÀÁ relaxation times, measured by the inversion recovery pulse dard local formulation of the second law for initially correlated I H; I C ¼ð : ; : Þ sequence, are 1 1 7 42 11 31 s. Transverse relaxations, obtained by the systems and offers at the same time a mechanism to control heat ÀÁCarr–Purcell–Meiboom–Gill (CPMG) pulse sequence, have characteristic times I ÃH; I ÃC ¼ð : ; : Þ on the microscale. They additionally establish that the arrow of 2 2 1 11 0 30 s. The total experimental running time, to implement

4 NATURE COMMUNICATIONS | (2019) 10:2456 | https://doi.org/10.1038/s41467-019-10333-7 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10333-7 ARTICLE the partial spin thermalization, is about 2.32 ms, which is considerably smaller than 7. Batalhão, T. M. et al. Irreversibility and the arrow of time in a quenched the spin-lattice relaxation and therefore decoherence can be disregarded. quantum system. Phys. Rev. Lett. 115, 190601 (2015). 8. Hofmann, A. et al. Heat dissipation and fluctuations in a driven quantum dot. Heat current between initially correlated systems. We here derive the expres- Phys. Status Solidi B 254, 1600546 (2017). sion (5) for the heat flow between initially correlated systems A and B. For an initial 9. Partovi, M. H. Entanglement versus stosszahlansatz: disappearance of the ρ0 ¼ ðÀβ H Þ=Z = thermodynamic arrow in a high-correlation environment. Phys. Rev. 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Commun. 8, 2180 Δ ¼ ρτ À ρ0 with the variation in the von Neumann entropy, given by Si S i S i (2017). fi Δ ¼ and the internal energy change of the i-th subsystem de ned as Ei 13. Feng, E. H. & Crooks, G. E. The length of time’s arrow. Phys. Rev. Lett. 101, ρτ H À ρ0H Tri i i Tri i i. Energy conservation for the combined isolated system (AB) 090602 (2008). Δ = −Δ = further implies that EA EB QB, for constant interaction energy. As a result, 14. Maccone, L. Quantum solution to the arrow-of-time dilemma. Phys. Rev. Lett. we obtain, ÀÁÀÁ 103, 080401 (2009). Δβ ¼ Δ ð : Þþ ρτ k ρ0 þ ρτ k ρ0 ; ð Þ 15. Parrondo, J. M. R., den Broeck, C. W. & Kawai, R. Entropy production and the QB I A B S A A S B B 8 arrow of time. New J. Phys. 11, 073008 (2009). Δβ = β − β Δ + Δ = Δ where A B is the inverse temperature difference and SA SB I(A: 16. Jarzynski, C. 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scheme (Grant no. NA140436) and the technical support from the Multiuser Experi- Journal peer review information: Nature Communications thanks the anonymous mental Facilities of UFABC. A.M.S. acknowledges support from the Brazilian agency reviewer(s) for their contribution to the peer review of this work. FAPERJ (203.166/2017). K. M. acknowledges CAPES and DAAD for ﬁnancial support. E.L. acknowledges support from the German Science Foundation (DFG) (Grant no. FOR Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in 2724). This research was performed as part of the Brazilian National Institute of Science published maps and institutional afﬁliations. and Technology for Quantum Information (INCT-IQ).

Author contributions Open Access This article is licensed under a Creative Commons K.M., J.P.S.P., T.B.B., R.S.S., and R.M.S. designed the experiment, J.P.S.P., A.M.S., R.S.S., Attribution 4.0 International License, which permits use, sharing, I.S.O., and R.M.S. performed the experiment, K.M., G.T.L., and E.L. contributed to the adaptation, distribution and reproduction in any medium or format, as long as you give theory. All authors contributed to analyzing the data and writing the paper. appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party ’ Additional information material in this article are included in the article s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- article’s Creative Commons license and your intended use is not permitted by statutory 019-10333-7. regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Competing interests: The authors declare no competing interests. licenses/by/4.0/. Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ © The Author(s) 2019

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