3, 1, y · 2020. 1. 20. · Detecting dynamical quantum phase transition via out-of-time-order correlations in a solid-state quantum simulator Bing Chen,1,2 Xianfei Hou, 1Feifei Zhou,

Detecting dynamical quantum phase transition via out-of-time-order correlations in asolid-state quantum simulator

Bing Chen,1, 2 Xianfei Hou,1 Feifei Zhou,1 Peng Qian,1 Heng Shen,3, ∗ and Nanyang Xu1, †

1School of Electronic Science and Applied Physics,Hefei University of Technology, Hefei, Anhui 230009, China

2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China3Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK

Quantum many-body system in equilibrium can be effectively characterized using the frameworkof quantum statistical mechanics. However, non-equilibrium behaviour of quantum many-bodysystems remains elusive, out of the range of such a well established framework. Experiments inquantum simulators are now opening up a route towards the generation of quantum states be-yond this equilibrium paradigm. As an example in closed quantum many-body systems, dynamicalquantum phase transitions behave as phase transitions in time with physical quantities becomingnonanalytic at critical times, extending important principles such as universality to the nonequi-librium realm. Here, in solid state quantum simulator we develop and experimentally demonstratethat out-of-time-order correlators, a central concept to quantify quantum information scramblingand quantum chaos, can be used to dynamically detect nonoequilibrium phase transitions in thetransverse field Ising model. We also study the multiple quantum spectra, eventually observe thebuildup of quantum correlation. Further applications of this protocol could enable studies otherof exotic phenomena such as many-body localization, and tests of the holographic duality betweenquantum and gravitational systems.

Equilibrium properties of quantum matter can be effec-tively captured with the well-established quantum statis-tical mechanics. However, when closed quantum many-body systems are driven out of equilibrium, a lot ofquestions of how to understand the actual dynamics ofquantum phase transition remain elusive since they arenot accessible within thermodynamic description [1, 2].An exciting perspective arises from quantum simulators,which can mimic natural interacting quantum many-body systems with experimentally controlled quantummatter such as ultracold atoms in optical lattice andtrapped ions. Such analogue system enables the investi-gation of exotic phenomena such as many-body localiza-tion [3, 4], prethermalization [5, 6], particle-antiparticleproduction in the lattice Schwinger model [7], dynamicalquantum phase transitions (DQPT) [8–10] and discretetime crystal [11, 12].

In many of these phenomena, such as the celebratedlogarithmic entanglement growth in many-body localiza-tion [13–15], the propagation of quantum informationplays a central role, opening up new point of view andpossibilities for probing out-of-equilibrium dynamics. Tomeasure the propagation of information beyond quantumcorrelation spreading and characterize quantum scram-bling through quantum many-body systems, the conceptof out-of-time-order correlation (OTOC) is developed re-cently [16–20], leading to new insight into quantum chaos[21, 22] and the black hole information problems [23, 24].

Recent experimental progresses in measuring out-of-time-order correlation (OTOC) [18–20] deliver importantnew insight into a more thorough grasping of how suchquantities characterized complex quantum system. Forinstance, OTOC can be used as entanglement witness

via multiple quantum coherence [25], and in particularbe used to dynamically detect equilibrium as well asnonequilibrium phase transitions [26, 27]. Here, we emu-late the dynamical quantum phase transition of quantummany-body system by a solid-state quantum simulatorbased on nitrogen-vacancy centre in diamond [28]. Fur-thermore, measurement of OTOC is performed to quan-tify the buildup of quantum correlations and coherence,and remarkably to detect the dynamical phase transition.

A very general setting for DQPT is the one emerg-ing from a sudden global quench across an equilibriumquantum critical point [8, 9, 29], and DQPT manifestsitself in discontinuous behaviour of the system at cer-tain critical times. Here, we consider such a protocol.First, the state is initialized in the ground state of the ini-

tial Hamiltonian H0 = −∑Ni σ

xi σ

xi+1 as |Φ〉0 = |+〉

⊗N

where |+〉 = (|↑〉 + |↓〉)/√

2 and N is the number ofspins. At time t=0, the Hamiltonian is suddenly switchedto HTFI = −

∑Ni (σxi σ

xi+1 + gσzi ) and the system state

evolves to |Φ(t)〉 = e−iHt |Φ〉0, realizing a quantumquench. Here σ = (σx, σy, σz) are Pauli spin operators.The rate function f(t) as a function of return probabilityplays a role of a dynamical free energy, signalling the oc-currence of DQPT. To explore DQPT of a spin-chain viaa single solid-state qubit, HTFI is written in momentumspace as HTFI =

∑k Ψ†kHkΨk where Ψk denotes a spinor

with two elements vector composed of fermion opera-tors (See Methods). The associated Bloch Hamiltonianis Hk = d(k) ·σ = [1−cos (k)g]σx+sin (k)gσy with k thequasi-momentum. The bulk dynamics of the system canbe solved since each k-component evolves independently,and after quench the state at each k is given by|Φ(k, t)〉 =

e−iHkt |Φ(k, 0)〉. The rate function is defined as f(t) =

arX

iv:2

001.

0633

3v1

[qu

ant-

ph]

17

Jan

2020

2

ab

c

ms=±1

ms=0MW

ms=±1

ms=03E

3A2

1E1A1

Time

532nmLaser pulse

Microwave pulse

Initialization |↓> Readout

Ry/2

|x>

Preparation Detection

HTFI -HTFIRx(Φ) Ry/2

t tEvolution Backevolution

Rotation

NV sample

Microwave wire

FIG. 1: Measuring out-of-time-order correlation using time reversal in Nitrogen-vacancy centre. a. Illustration of experimentschematics and atomic structure of the Nitrogen-vacancy (NV) centre in diamond. b. Scheme of energy levels of the NVcentre electron spin. Both its ground state (3A2) and excited state (3E) are spin triplets. By applying a laser pulse of 532 nmwavelength with the assistance of intersystem crossing (ISC) transitions, the spin state can be polarized into ms = 0 in theground state (3A2). This process can be utilized to initialize and to read out the spin state of the NV centre. The fluorescencephotons are detected by using the single photon counting module (SPCM). Additionally, a small permanent magnet in thevicinity of the diamond (magnetic field B ≈ 524 G) that is aligned parallel to the symmetry axis of the nitrogen vacancy centresplits the ms = ±1 spin levels. With this magnetic field, the 14N nuclear spin of the NV centre can be also polarized withthe laser pulse, which is enabled by the level anti-crossing in its excited state. c. Laser and microwave pulse sequence for themeasurement of OTOC. The π/2 rotation Ry about the y-axis prepares an initial state with spin pointing along x-axis |x〉. The

state of interest ρ(t) is reached after the first evolution period. The rotation Rx(φ) then imprints a phase mφ on each sectorρm of density matrix. Evolving backward and measuring the overlap with initial state as a function of φ, the coherence Im andmagnetization Am of ρ(t) are retrieved as the Fourier components of this signal.

−1/N∑k log(

⟨Φ(k, 0)

∣∣∣e−iHkt∣∣∣Φ(k, 0)

⟩)2, whose nonan-

alytic behaviour yields DQPT.

In the experiment we use a negatively charged NV cen-tre in type-IIa, single-crystal synthetic diamond sample(Element Six) to simulate the quantum many-body dy-namics in its quasi-momentum representation. As illus-trated in Fig.1, the NV centre has a spin triplet groundstate. We encode ms = −1 and ms = 0 in 3A2 asspin up and down of the electron spin qubit. The stateof the qubit can be manipulated with microwave pulses(ωMW ≈ 2π× 1400 MHz), while the spin level ms = +1

remains idle due to large detuning. By applying a laserpulse of 532 nm wavelength with the assistance of inter-system crossing (ISC) transitions, the spin state can bepolarized into ms = 0 in the ground state. This processcan be utilized to initialize and read out the spin stateof the NV centre. By using a permanent magnet a mag-netic field about 524 G is applied along the NV axis, thenearby nuclear spins are polarized by optical pumping,improving the coherence time of the electron spin.

Experimentally, evolution at different k is performedin independent runs attributed to different rotation axesand speed (See Appendix). As sketched in Fig.2a, we pre-

3

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

t/t0

f(t)

gf=1.2gf=0.8gf=0.5

Time

532nmLaser pulse

Microwave pulse

Initialization |↓> Readout

Ry/2

|x>

Preparation Detection

HTFIRy/2

tEvolution

Rotation

a

b

c

d

1 0.8 0.6 -0.20.4 00.2 -0.4 -0.6 -0.8 -1

ExperimentTheory

gf=0.8

gf=1.2

0

1

20 0.5 1 1.5 2

t/t0

k/

0

1

20 0.5 1 1.5 2

t/t0

0

1

20 0.5 1 1.5 2

t/t0

0

1

20 0.5 1 1.5 2

t/t0

k/

Return Probability

FIG. 2: Rate function dynamics after a quantum quench in an Ising model. a. Pulse sequence for DQPT. b. Rate function aftera quantum quench of HTFI = −

∑Ni (σxi σ

xi+1 + gfσ

zi ) with N=30. The initial state is prepared at |φi〉 = |x〉 = (|↑〉+ |↓〉)/

√2,

as the ground state of H0 = −∑Ni σ

xi σ

xi+1 (i.e. gi = 0 in HTFI). gf is varied in the global quench with the values of 0.5, 0.8

and 1.2. Rate functions are shown in panel b with symbols (square, diamond, circle) and solid lines representing experimentaland theoretical values. c-d. Return probabilities for gf = 1.2 and gf = 0.8, respectively. Theoretical results and experimentaldata are presented in the left and right panels, all sharing the same colorbar. For convince, t is normalized by a period of timet0 = π

|df (k)| (See Appendix).

pare the initial state as |+〉 = (|↑〉+ |↓〉)/√

2, the groundstate of H0, and then switch on the quench Hk by apply-ing a resonant microwave pulse. The return probabilitiesare recorded by projecting the final states on x-basis. Inthe transverse field Ising model, the critical transversefield gc = 1 separates the paramagnetic phase (g > 1)from ferromagnetic phase (g < 1). Since state is initial-ized in ferromagnetic phase (gi = 0), DQPT only canoccur only if gf > 1. It is demonstrated by the experi-mental data of the rate function for different gf values,as shown in Fig.2b, where the sharp peak with nonan-alytic behaviour of rate function indicates the DQPT,being associated with dynamical Fisher zeros [31] (SeeAppendix).

Further signatures of the DQPT are observed bymeasure the OTOC, a quantity probing the spreadof quantum information beyond quantum correlations.OTOC functions of particular interest is defined as Ref.

[18, 32], F (t) =⟨W †(t)V †W (t)V

⟩where W (t) =

e−iHinttWeiHintt with Hint an interacting many-body

Hamiltonian and W and V two commuting unitary op-

erators. Re[F (t)] = 1 −⟨∣∣∣[W (t), V

]∣∣∣2⟩ /2 captures the

degree by which the initially commuting operators failto commute at later times due to the many-body inter-actions Hint, an operational definition of the scramblingrate. In such process the information initially encoded inthe state spread over the other degrees of freedom of thesystem after the interactions, and cannot be retrieved bylocal operations and measurement.

We now outline the protocol to measure the OTOC asillustrated in Fig.1c. In contrast to the pulse sequenceshown in Fig.2 a, we implement the many-body time re-versal by inverting the sign of HTFI which evolves againfor time t to the final state ρf and ideally takes thesystem back to the initial state ρ0. If a state rotation

Rx(φ) = e−iSxφ, i.e. W (0) = Rx(φ), here about the x-axis with Sx = 1/2

∑i σ

ix, is inserted between the two

halves of the time evolution through a variable angle φ,the dependence of the revival probability on this angle

4

0

1

2

3

4

5

6

0.2 0.4 0.6 0.8t/t0

0 1

0

1

2

3

4

5

6

0.2 0.4 0.6 0.8t/t0

0 1

0

1

2

3

4

5

6

m

0.2 0.4 0.6 0.8t/t0

0 1

0

1

2

3

4

5

6

m

0.2 0.4 0.6 0.8t/t0

0 10

0.02

0.04

0.06

0.08

0.12

0.14

0.16

0.18

0.10

0.4 0.6 0.8t/t0

0 0.2 1

g=0.8g=1.2

Magnetization OTOC F (t)gf=0.8a c

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1d e

A1

fMagnetization Fourier components Am

gf=1.2 b

0.4

0.5

0.6

0.7

0.8

0.9

1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(t)

0.4 0.6 0.8t/t0

0 0.2 1

g=0.5g=0.8g=1.2g=1.5

FIG. 3: Probing the DQPT through magnetization OTOC dynamics. The measured magnetization OTOC from time evolutionunder the HTFI and rotations about the x-axis 0− 2π with gf = 1.2 a and gf = 0.8 b. The associated Fourier components Amdynamics for gf = 1.2 d and gf = 0.8 e, respectively. c. The measured magnetization with fixed φ = π by varying gf (0.6, 0.8,1.2, 1.5). f. Fourier components A1 as a function of time where inversed double-well structure at the critical time tc is onlyshown in DQPT (gf > 1).

contains information about ρ(t). At the end of sequence,two different observables can be measured, the collective

magnetization along the x-direction,⟨Sx

⟩= tr[Sxρf ]

and the fidelity Fφ(t) = tr[ρ0ρf ]. In particular, the fi-

delity can be cast as an OTOC by setting V = ρ0, corre-sponding to a many-body Loschmit echo. Measurementof the fidelity is also directly links to the so-called multi-ple quantum intensities Im by the Fourier transformationFφ(t) = tr[ρf ρ0] = tr[ρ(t)ρφ(t)] =

∑Nm=−N Im(t)eimφ

[18, 25, 33].

Similarly, the dynamics of the Fourier amplitude Amof the magnetization Fφ(t) =

∑Nm=−N Am(t)eimφ quan-

tifies the buildup of many-body correlations. However, itis much less sensitive to decoherence compared with thefidelity due to the nature of single-body observables. InFig.3, we show the results of the magnetization OTOCmeasurement sequence and hence a buildup of Fourieramplitudes, Am. Comparison of the data with gf = 0.8to that with gf = 1.2 confirms that the appearanceof double-well-like features in φ − t plane, signals theDQPT. Fig.3c shows the measured magnetization withfixedφ = π by varying gf , in agreement with the con-clusion above. The associated Fourier amplitude Amis extracted and illustrated in Fig.3d-e. Although thenearest-neighbour interaction limits the buildup of highorder components, double-well feature still distinguishes

the DQPT with non-DQPT.In summary, we demonstrate a new approach for inves-

tigating quantum many-body system out of equilibriumby a solid-state quantum simulator based on nitrogen-vacancy centre in diamond. The sharp peak with non-analytic behaviour of rate function indicates the DQPT.Moreover, we measure the magnetization OTOC to quan-tify the buildup of quantum correlations and coherence,and in particular the intriguing feature arising from thedynamical phase transition is characterized to detect theoccurrence of DQPT. Further applications of this proto-col could enable studies of other exotic phenomena suchas many-body localization, and tests of the holographicduality between quantum and gravitational systems.

5

Appendix

Spin chain model We start with a periodically drivenspin chain with transverse-field Ising Hamiltonian as fol-lows,

HTFI = −N∑n

(σxnσxn+1 + gσzn),

with g the transverse field strength. The first term de-scribes the time-independent nearest-neighbour spin-spincoupling.

In order to obtain a single qubit Hamiltonian inmomentum representation, we first apply the Jordan-Wigner transformation to fermionize HTFI , which is of-ten used to solve 1D spin chains with non-local trans-formation [34]. For convenience, Pauli operators areexpressed by spin raising and lowering operators asσ±n = (σxn ± iσyn)/2. By applying Jordan-Wignertransformation, the original Hamiltonian is mapped tothe free-fermion model with the definition of σ+

n =

eiπ∑

j<n f†j fjfn, σ−n = f†ne

−iπ∑

j<n f†j fj and σzn = 1 −

2f†nfn. Here, the fermionic creation and annihilationoperators f†n and fn satisfy the anti-commutation re-lations fm, fn =

f†m, f

†n

= 0 and

fm, f

†n

=

δmn. The fermionized spin chain model reads as H =−∑n(f†nfn+1 + f†nf

†n+1 + h.c.) + g(1− 2f†nfn).

Next, by using Fourier transformation defined as fn =1/√N∑k∈BZ e

iknfk and f†n = 1/√N∑k∈BZ e

−iknf†kwith the quasi-momentum k in the first Brillouin zone(BZ), the associated Hamiltonian can be written as

HTFI =∑k Ψ†k[sin(k)σy + (g − cosk)σz]Ψk in terms of

the spinor basis Ψ†k = (f†k , f−k). And we can denote

Hk = d(k) · σ.

In our experiment, the initial state is prepared at|φi〉 = |x〉 = (|↑〉 + |↓〉)/

√2, one eigenstate of σx. Given

|φi〉 as the ground state of H0 in the protocol, a unitarytransformation U should be used to meet UH0(k)U† =σx, then the applied quench Hamiltonian HTFI is ro-tated to UHTFI U

†. In practice, unitary transformationP and S are introduced to diagonalize σx and H0 asD = P−1σxP and D = S−1H0S respectively. By rewrit-ing σx = PS−1H0SP

−1, one finds U = PS−1.

The spin processes on the Bloch sphere with a periodπ

|df (k)| . In order to explore the full Brillouin zone k

should be varied from 0 to 2π with N + 1 steps (i.e.step size 2π/N) and N is equivalent number of spin.For each k, a unitary rotation operation is applied with

axisdf (k)|df (k)| , i.e. Ω = C |df (k)| with Constant C from

current experimental setting and (θ, φ) = (Ωt, φ) with

φ = arcsin(sin(k)gf√

(1−cos(k)gf )2+(sin(k)gf )2). Since we’d like to

keep the normalized speed for all the k, pulse durationT is varied from 0 to 2π/Ω with NT steps. In the exper-iment, we can fix it as ∼ 100 for instance.

Critical time tc In analogy to the Fisher zeros inthe partition function, which trigger phase transition inequilibrium, dynamical Fisher zeros is introduced [31].At dynamical Fisher zeros the Lochmidt amplitude

G(t) =⟨

Φ0

∣∣∣e−iHkt∣∣∣Φ0

⟩=∏k

[cos(|df (k)| t) + idi(k) · df (k) sin(|df (k)| t)]

goes to zero. It requires the existence of the criticalmomentum k∗ at which the vector df is perpendicularto di, i.e. di(k) · df (k) = 0. and DQPT occurs attc = π

|df (k)| (n+ 1/2), n=0, 1, 2....

Since we employ transverse field Ising model, oneshould satisfy the following condition,

di(k)·df (k) = cos2(k)+sin2(k)−(gi+gf ) cos(k)+gigf = 0

which requires k∗ = ±arccos 1+gigfgi+gfon the condition of

|cos(k)| =∣∣∣ 1+gigfgi+gf

∣∣∣ < 1, causing to sgn[(1 − |gi|)(1 −|gf |)] = −1. This indicates DQPT occurs if and only ifthe initial and final Hamiltonian have to belong to dif-ferent phases.Experiment setup The diamond used in this work is

a 2mm× 2mm× 500µm type-IIa, single-crystal syntheticdiamond sample (Element Six), grown using chemical va-por deposition (CVD) by Element Six, containing lessthan 5 ppb (often below 1 ppb) Nitrogen concentrationand typically has less than 0.03 ppb NV concentration.

Single mode solid-state 532 nm laser is utilized to ini-tialize and readout the electron spin state of the NV cen-ter. We can use an acoustic optical modulator (AOM) tocontrol the laser and create the desired pulse, driven byan amplified signal from a home-built pulse generator.The fluorescence photons emitted from the NV centeris collected by a 1.40 numerical aperture (NA) asphericaplanatic oil condenser (Olympus), passed through a 600nm longpass filter (Thorlabs) and a pinhole with a diam-eter of 50 µm, and detected by the single photon countingmodule (SPCM; Excelitas).

A small permanent magnet creates a bias magneticfield B0 of 524 G along the NV axis, splitting ms = ±1spin levels. With this magnetic field, the 14N nuclearspin of the NV center can be polarized based on opti-cal pumping, improving the coherence time of electronspin. It is observed in the optically detected magneticresonance (ODMR) spectra that the nuclear spin polar-ization is higher than 98% [35].

Fig. shows a schematic of the microwave (MW) setup.A commercial MW source (Rohde&Schwarz) outputs asingle frequency signal, mixed with the output from anarbitrary-waveform generator (AWG610; Tektronix; 2.6GHz sampling rate) via IQ modulator to adjust the MWfrequency and phase. Then the MW signal is ampli-fied (Mini-Circuits ZHL-42W+) before delivery to an

6

impedance-matched copper slotline with 0.1 mm gap, de-posited on a coverslip, and finally coupled to the NVcenter. Note that pulse generator, MW source and AWGare all synchronized by locking to a 10 MHz referencerubidium clock.

∗ Electronic address: [email protected]† Electronic address: [email protected]

[1] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium. Nat. Phys. 11, 124-130(2015).

[2] T. Langen, R. Geiger, and J. Schmiedmayer, Ultracoldatoms out of equilibrium. Annu. Rev. Condens. MatterPhys. 6, 201 (2015).

[3] M. Schreiber, et al. Observation of many-body localiza-tion of interacting fermions in a quasirandom optical lat-tice. Science 349, 842 (2015).

[4] J. Smith, et al. Many-body localization in a quan-tum simulator with programmable random disorder. Nat.Phys. 12, 907-911 (2016).

[5] M. Gring, et al. Relaxation and Prethermalization in anIsolated Quantum System. Science 337, 1318 (2012).

[6] B. Neyenhuis, et al. Observation of Prethermalizationin Long-Range Interacting Spin Chains. Preprint athttps//arxiv.org/abs/1608.00681 (2016).

[7] E. A. Martinez, et al. Real-time dynamics of lattice gaugetheories with a few-qubit quantum computer. Nature534, 516-519 (2016).

[8] P. Jurcevic, et al. Direct Observation of DynamicalQuantum Phase Transitions in an Interacting Many-Body System. Phys. Rev. Lett. 119, 080501 (2017).

[9] J. Zhang, et al. Observation of a many-body dynamicalphase transition with a 53-qubit quantum simulator. Na-ture 551, 601-604 (2017).

[10] N. Flaschner, et al. Observation of dynamical vorticesafter quenches in a system with topology. Nat. Phys. 14,265-268 (2018).

[11] J. Zhang, et al. Observation of a discrete time crystal.Nature 543, 217-220 (2017).

[12] S. Choi, et al. Observation of discrete time-crystallineorder in a disordered dipolar many-body system. Nature543, 221-225 (2017).

[13] M. Znidaric, T. Prosen, and P. Prelovsek, Many-bodylocalization in the Heisenberg XXZ magnet in a randomfield. Phys. Rev. B 77, 064426 (2008).

[14] J. H. Bardarson, F. Pollmann, and J. E. & Moore, Un-bounded Growth of Entanglement in Models of Many-Body Localization. Phys. Rev. Lett. 109, 017202 (2012).

[15] E. Altman, Many-body localization and quantum ther-malization. Nat. Phys. 14, 979983 (2018).

[16] B. Swingle, Unscrambling the physics of out-of-time-order correlators. Nat. Phys. 14, 988-990 (2018).

[17] R. J. Lewis-Swan, A. Safavi-Naini, A. M. Kaufman, andA. M. Rey, A. M. Dynamics of quantum information.Nat. Rev. Phys. 1, 627-634 (2019).

[18] M. Garttner. et al. Measuring out-of-time-order corre-lations and multiple quantum spectra in a trapped-ionquantum magnet. Nat. Phys. 13, 781-786 (2017).

[19] J. Li, et al. Measuring out-of-time-order correlators ona nuclear magnetic resonance quantum simulator. Phys.

Rev. X 7, 031011 (2017).[20] K. Landsman, et al. Verified quantum information scram-

bling. Nature 567, 61-65 (2019).[21] J. Maldacena, S. H. Shenker, and D. J. Stanford, A

bound on chaos. J. High Energy Phys. 2016, 106 (2016).[22] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida,

Chaos in quantum channels. J. High Energy Phys. 2016,4 (2016).

[23] P. Hayden, and J. Preskill, Black holes as mirrors: quan-tum information in raddom subsystems. J. High EnergyPhys. 2007, 120 (2007).

[24] S. H. Shenker, and D. Stanford, Black holes and the but-terfly effect. J. High Energy Phys. 2014, 67 (2014).

[25] M. Garttner, P. Hauke, and A. M. Rey, Relating Out-of-Time-Order Correlations to Entanglement via Multiple-Quantum Coherences. Phys. Rev. Lett. 120, 040402(2018).

[26] H. Heyl, F. Pollmann, and B. Dora, B. Detecting Equilib-rium and Dynamical Quantum Phase Transitions in IsingChains via Out-of-Time-Ordered Correlators. Phys. Rev.Lett. 121, 016801 (2018).

[27] C. B. Dag, K. Sun, and L.-M. Duan, Detection of quan-tum phases via out-of-time-order correlators. Phys. Rev.Lett. 123, 140602 (2019).

[28] B. Chen, et al. Quantum state tomography of a singleelectron spin in diamond with Wigner function recon-struction. Appl. Phys. Lett. 116, 041102 (2019).

[29] H. Heyl, Dynamical Quantum Phase Transitions: a re-view. Rep. Prog. Phys. 81, 054001 (2018).

[30] K. Yang, et al. Floquet dynamical quantum phase tran-sition. Phys. Rev. B 100, 085308 (2019).

[31] S. Vajna, and B. Dara, B. Toplogical classification ofdynamical phase transitions. Phys. Rev. B 91, 155127(2018).

[32] A. I. Larkin, and Y. N. Ovchinnikov, Quasiclassicalmethod in the theory of superconductivity. ZhETF 55,2262 (1969) [Sov. Phys. JETP 28, 1200 (1969)].

[33] N. Y. Yao, et al. Interferometric Approachto Probing Fast Scrambling. Preprint athttps://arxiv.org/abs/1607.01801 (2016).

[34] Franchini, F. An Introduction to Integrable Techniquesfor One-Dimensional Quantum Systems. Lecture Notesin Physics 940 (Springer, Cham. 2017).

[35] Jacques, V. et al. Dynamic Polarization of Single NuclearSpins by Optical Pumping of Nitrogen-Vacancy ColorCenters in Diamond at Room Temperature. Phys. Rev.Lett. 102, 057403 (2009).

Acknowledgements

The authors are grateful to Dayou Yang, Philipp Hauke,Markus Heyl and Xiaojun Jia for fruitful discussions.This work is supported by the National Key R&D Pro-gram of China (Grants No. 2018YFA0306600, andNo. 2018YFF01012500), the National Natural ScienceFoundation of China (Grants No. 11604069 and No.11904070), the Program of State Key Laboratory ofQuantum Optics and Quantum Optics Devices (No.KF201802), the Fundamental Research Funds for theCentral Universities, and the Natural Science Founda-tion of Anhui Province (Grant No. 1708085QA09). H.Shen acknowledges the financial support from the Royal

7

AOM

532nm Laser Lens

Objective

NV sample

Dichroic Mirror

Longpass FilterPinhole

SPCM

AWG

IQModulator

Power Amplifier

Pulse generator

Rb clock

MW generator

Frequency [GHz]

PL [a

.u.]

1.392 1.396 1.400 1.404

1.0

0.9

0.8

ODMR

Copper slotline withΩ-type ring

FIG. 4: Detailed schematic of the experimental setup. The solid colorful lines represent the path of the 532 nm laser andthe fluorescence light. The solid black lines represent electrical connections.The above inset shows the ODMR signal whichthe microwave field frequency is about 1.4 GHz. The lower right corner inset is the copper coplanar waveguide which is themicrowave field delivery setup. The impedance-matched copper coplanar waveguide with gap of 0.1 mm, a open-end Ω-typering in the middle and deposited on a coverslip. The outer diameter and inner diameter of the Ω-type ring are 0.5 mm and 0.3mm.

Society Newton International Fellowship (NF170876) of UK.