Interferometric Approach to Probing Fast Scrambling

N. Y. Yao1, F. Grusdt2, B. Swingle3, M. D. Lukin2, D. M. Stamper-Kurn1, J. E. Moore1, E. Demler2

1Department of Physics, University of California Berkeley, Berkeley, CA 94720, U.S.A.2Department of Physics, Harvard University, Cambridge, MA 02138, U.S.A. and

3Stanford Institute for Theoretical Physics and Department of Physics,Stanford University, Stanford, California 94305, U.S.A.

Out-of-time-order correlation functions provide a proxy for diagnosing chaos in quantum systems.We propose and analyze an interferometric scheme for their measurement, using only local quantumcontrol and no reverse time evolution. Our approach utilizes a combination of Ramsey interferometryand the recently demonstrated ability to directly measure Renyi entropies. To implement ourscheme, we present a pair of cold-atom-based experimental blueprints; moreover, we demonstratethat within these systems, one can naturally realize the transverse-field Sherrington-Kirkpatrick(TFSK) model, which exhibits certain similarities with fast scrambling black holes. We performa detailed numerical study of scrambling in the TFSK model, observing an interesting interplaybetween the fast scrambling bound and the onset of spin-glass order.

PACS numbers: 05.45.Mt, 03.67.-a, 04.60.-m, 37.10.Jk

Much of statistical mechanics rests on the assumptionthat generic systems will, after sufficient time, arrive ata state close to thermal equilibrium. In isolated quan-tum systems, the approach to equilibrium is character-ized by the spreading of entanglement and quantum in-formation. That there might exist fundamental limitson the rate of thermalization has a long history [1–6]. Atone extreme are strongly disordered systems, where ther-malization is absent and quantum information spreadsslowly [7–14]. At the other extreme, certain gauge theo-ries appear to spread quantum information very rapidly[15–17]. However, these gauge theories are special—theirthermal states are “holographically dual” to black holesin Einstein gravity [18]. They are also highly symmetri-cal, display scale invariant physics, and do not order atlow temperatures despite strong interactions.

Thus, a key question is: Where do typical interactingsystems fall between these two extremes? The lack ofgeneral theoretical tools in this context suggests that ex-periments will be essential to explore the nature of infor-mation spreading or “scrambling” in many-body systems[15]. More precisely, scrambling describes the delocaliza-tion of quantum information over all of a system’s de-grees of freedom. The analog of scrambling in a classicalsystem is chaos and is diagnosed by the butterfly effect,which describes the exponential sensitivity of a particle’smotion to small changes in its initial conditions. As anexample, the Poisson bracket of position and momen-tum, x(t), p = ∂x(t)/∂x(0), can be used to quantifythe butterfly effect [1]. The strength of quantum scram-bling can similarly be diagnosed using the commutator,C(t) = 〈[W (t), V (0)]†[W (t), V (0)]〉, where V and W areunitary operators [15, 16, 19–21].

The functional onset of scrambling is particularly in-triguing, with certain systems exhibiting a parametricperiod of exponential growth, C(t) ∼ eλLt [21–24]. Insemi-classical systems, λL can be interpreted as a Lya-

2

(a) /2

/2

i j

ancilla

spin

spin

system A eiHt

V

W

i0 j0

system B

i, i0

j, j0

X, Y

Tr[

A

B]

(b)

FIG. 1. (a) Schematic of our interferometric protocol. Anancilla qubit is coupled to two copies of a quantum system(prepared at inverse temperature β/2) at different locationsi, j and i′, j′. By performing ancilla-state dependent interac-tions both before and after time evolution, one can measureout-of-time-order correlation functions. (b) Circuit diagramillustrating the interferometric protocol. After preparing theancilla in 1/

√2(|↑〉+ |↓〉), controlled unitaries are performed,

first conditioned on |↑〉 (V ) and then conditioned on |↓〉 (W ).Simultaneous measurement of σx on the ancilla and SWAPon the systems results in the correlator F (t).

punov exponent characterizing the strength of chaos. Forgauge theories dual to black holes, there exists a param-eter N , such that the large-N limit is semi-classical andλL = 2πkBT/~, saturating a recently proposed upperbound on the rate of scrambling [17].

While intriguing, the scrambling behavior of systemsdual to black holes is only one extreme of a largely un-explored landscape. For example, although temperatureprovides the only natural energy scale for a black hole,generic quantum many-body systems can exhibit a multi-tude of additional energy scales governed, e.g. by micro-

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scopic couplings and/or low temperature order. Whetherthe presence of these additional scales precludes fastscrambling, at or near the conjectured bound, remainsan open question. Experimental exploration is thus cru-cial to search for new “universality classes” of scramblingbehavior and to develop a comprehensive picture of in-formation scrambling in systems realized in nature.

To directly measure the commutator C(t) amountsto a measurement of an out-of-time-order correlationfunction, F (t) = 〈W †(t)V †(0)W (t)V (0)〉 where C =2 − 2 Re[F ]. Experimentally, the measurement of F (t)is extremely challenging because it effectively involvesreversing the flow of time at various points in the mea-surement, or equivalently, time-evolving with both theHamiltonian, H, and its negative counterpart, −H. Aprotocol using both forward and backward time evolu-tion has been developed to measure F [25]; while thenecessary ingredients can be engineered in a variety ofquantum optical systems, a complementary approach notinvolving backwards time evolution would be advanta-geous.

In this Letter, we introduce an interferometric ap-proach to measure out-of-time-order correlation func-tions which requires only local quantum control and noreverse time evolution (Fig. 1a). Our method utilizes acombination of Ramsey interferometry and the recentlydemonstrated experimental capability to measure Renyientropies [26–28]. Besides not requiring reverse timeevolution, the scheme also enables the measurement ofso-called thermally regulated out-of-time-order correla-tions, e.g. Tr[

√ρW †(t)V †(0)

√ρW (t)V (0)] with ρ the

thermal state. This regulated correlator is believed toexhibit identical early time scrambling signatures andhelps to enable analytical calculations in certain limits[17, 21, 23, 24]. Because our scheme requires the mea-surement of a many-body overlap, it is naturally suitedto study systems with relatively low entropy.

To this end, we perform a numerical study ofscrambling dynamics in the transverse-field Sherrington-Kirkpatrick (TFSK) model with N ∼ 20 spins [29–32],

HTFSK = −1

2

∑

i 6=jJijσ

zi σ

zj − Γ

∑

j

σxj (1)

where ~σ are Pauli operators. The model consists of Nspins in a transverse field Γ, with random all-to-all Isinginteractions, Jij , drawn from a normal distribution of

zero mean and variance σ =√J2/N [32]. The all-to-

all couplings mimic a similar aspect of black hole mod-els and open the possibility of observing fast scrambling[15, 21–24]. The numerics depict a rich variety of scram-bling behaviors even for relatively small system sizes thatshould be readily accessible with our protocol. Intrigu-ingly, we also find a suggestive interplay between thebound on scrambling and the onset of spin-glass order.Finally, we provide a pair of cold-atom-based experimen-

tal blueprints for implementing the TFSK model andmeasuring the scrambling correlation function.General Strategy—Let us begin by considering

a many-body quantum system with HamiltonianH. The correlator F (t) can be written asTr[ρβe

iHtW †(0)e−iHtV †(0)eiHtW (0)e−iHtV (0)], whereρβ is the thermal density matrix of the system andβ = 1/kBT is the inverse temperature. Noting that√ρ ∼ ρβ/2, our strategy will be to measure

F2(t) = Tr[e−iHtV †ρβ/2eiHtW †e−iHtV ρβ/2e

iHtW ], (2)

which is directly proportional to the thermally regulatedcorrelator [33]. Two remarks are in order. First, oneexpects that the magnitude of the experimental signalassociated with F2(t) will decrease with system size asit corresponds to a many-body overlap. Second, F2(t)factorizes as β → ∞, implying that it will serve as anaccurate proxy for F (t) only at finite energy densities[17]. As will be shown in the numerics, for mesoscopicspin ensembles, neither of the above becomes a limitingconcern for intermediate temperatures of interest.

Crucially, the functional form of F2(t) is suggestive ofa 2nd Renyi entropy measurement [34]. Such a measure-ment can be realized by preparing two identical copies ofa quantum system and performing a subsequent sequenceof individual SWAP operations, S, between the copies[26]; indeed, recent experimental progress has harnessedthis approach to explore entanglement-based signaturesof thermalization [28].

With this technique in mind, we devise an interfero-metric approach (Fig. 1b) to measuring out-of-time-ordercorrelation functions by utilizing only a single ancillaqubit (with states |↓〉 , |↑〉) and Hamiltonian evolution.Let us begin with both copies of the system and the an-cilla qubit prepared in the product state, ρ(t = 0) =(ρβ/2)⊗2⊗|+〉〈+|, where (ρβ/2)⊗2 = ρβ/2⊗ρβ/2. Assum-ing that the ancilla is capable of exerting a local state-dependent interaction on the system, time evolution thenyields,

ρ(t) =1

2[|↑〉 〈↑|⊗(U†↑ρβ/2U↑)

⊗2+|↓〉 〈↓|⊗(U†↓ρβ/2U↓)⊗2

+ |↑〉 〈↓| ⊗ (U†↑ρβ/2U↓)⊗2 + |↓〉 〈↑| ⊗ (U†↓ρβ/2U↑)

⊗2] (3)

where U†↑ = W †e−iHt, U†↓ = e−iHtV and V , W are spa-

tially separated at locations i and j. Here, U†↑(↓) are uni-

tary operators that describe the combination of Hamilto-nian time evolution and the ancilla-state-dependent in-teraction (Fig. 1b).

Assuming that V and W are both Hermitian andunitary, the resulting density matrix contains termsof the form (U†↓ρβ/2U↑)

⊗2 = e−iHtV ρβ/2eiHtW ⊗e−iHtV ρβ/2eiHtW , precisely matching the desired formof F2(t) [36]. This suggests that one may be able to ex-tract coherences of the time-evolved density matrix and

3

= 1.1 = 2 = 4

tdt (1/J)

= 1.1

= 1.5 = 1.55 = 1.65 = 1.3

t (1/J)

Nc = 19

(a)

(b)

N

t (1/J)

/2

(d)

/

/2

=

4

T

Spin Glass t (1/J)

T < 0.1

T > 10

(c)

C(t

)C

(t)

R(t)

C(t

)

Cf (t) = 2

1

Ncet

1 + 1Nc

et

!2

C(t

),C

2(t

)

N = 16

FIG. 2. Numerics on the the Sherrington-Kirkpatrick model in a transverse field. For system sizes N = 10, 12, we perform 103

disorder averages, for N = 14, we perform 102 disorder averages and for N = 16, we perform 29 disorder averages. Throughoutthe simulations, J = 1 and Γ = 1.35. (a) Depicts decay of the autocorrelation function R(t) = 〈σzi (t)σzi (0)〉β and growth ofthe scrambling correlator C(t) = −〈[σzi (t), σzj (0)]2〉β at various inverse temperatures β = 1.1, 2, 4. The approximate time scalefor dissipation is indicated as td. (b) Numeric fitting to the phenomenological function Cf (t) where Nc and ∆ are fixed forall curves to extract the Lyapunov exponent, λ, as a function of temperature. (c) Depicts λ as a function of temperature. Atintermediate temperatures, the growth of C(t) seems to be approximately linear. The temperature of the spin glass transitionas determined from Monte Carlo is indicated as the blue dashed line [35]. The red dashed line depicts the scrambling bound;since the exponent controlling the early time growth of Cf (t) is 2∆λ (see footnote), we plot 2πT/2∆ to compare directly withthe extracted λ. The inset shows the differences in C(t) at low and high temperatures. At low temperatures C(t) does notreach its maximally scrambled value even at late times ∼ 102/J . (d) Comparison between C(t) and C2(t) for β = 4. The insetdepicts the ratio of λ as extracted from C(t) and C2(t) as a function of system size (black points) and λβ=4 as a function ofsystem size (blue points).

naturally leads us to consider an interferometric approachbased upon Ramsey spectroscopy.

In particular, by independently measuring σx ⊗ S andσy ⊗ S (where σx,y act on the ancilla qubit and SWAPS acts on the two copies of the system), one is able tomeasure the real and imaginary portions of F2(t). To seethis, let us project ρ(t) onto the eigenstates of σx. Notingthat Tr[ρAρB ] = Tr[SρA ⊗ ρB ], we obtain

Tr [ρ(t)σx ⊗ S] = Tr[(e−iHtV ρβ/2e

iHtW )2]

+ c.c.

= Re[F2(t)]. (4)

and Tr [ρ(t)σy ⊗ S] = Im[F2(t)]. Having demonstratedthe ability to directly measure thermally regulated out-of-time-order correlators, we now turn to a numericalstudy of scrambling dynamics in the TFSK model.

TFSK Numerics—We perform exact diagonalizationand numerical integration of Hamiltonian [Eqn. (1)] evo-lution for systems of up to N = 16 spins. As previ-

ously discussed, while the model features certain simi-larities with fast scrambling gauge theories [15, 21, 22],it remains an open question whether its dynamics (e.g.growth of C(t)) are controlled by the thermal scale orby microscopics; this owes in part to the existence of alow-temperature spin-glass phase for Γ . 1.5 [35, 37–41].

To begin probing thermalization dynamics in HTFSK,we determine the dissipation time, td, which character-izes the decay of local two-point correlation functions.In particular, as shown in Fig. 2a (black lines), wecompute the disorder averaged autocorrelation functionR(t) = 〈σzi (t)σzi (0)〉β ; td is identified as the time at whichcorrelations have dropped to less than 0.05 and is approx-imately independent of temperature for 1 < β < 4.

At the same temperatures, one finds that C(t) =−〈[σzi (t), σzj (0)]2〉 begins to exhibit a sharp rise for t > td,indicative of the onset of scrambling (Fig. 2a). Intuitionfrom the black hole problem suggests that the time to

4

reach a maximally scrambled state is of order ∼ lnNc,where Nc is proportional to the entropy. This suggestsfitting C(t) with a function of the combined variableeλt/Nc; indeed, we observe that the early to interme-diate time data can approximately be captured using aphenomenological function

Cf (t) = 2

(1Nceλt

1 + 1Nceλt

)2∆

, (5)

where ∆ is an operator-dependent fitting parameter (inthe context of black holes, ∆ is the scaling dimension ofthe operator), and λ characterizes the Lyapunov expo-nent. To quantitatively analyze the data, we perform anonlinear regression, keeping Nc and ∆ fixed across alltemperatures and extract λ as a function of β as shown inFig. 2b,c (note that the dotted lines in Fig. 2b correspondto Cf (t) with extracted parameters).

A few remarks are in order. First, at the system sizesstudied, it is hard to distinguish between dissipation andthe onset of scrambling, although the data are consistentwith a small separation in time-scales. Second, Fig. 2cprovides evidence that for T . J , the temperature depen-dence of λ [42] becomes approximately linear, althoughthe slope is more than an order of magnitude smallerthan the conjectured upper bound [17]. At these tem-peratures, λ exhibits a weak decrease as a function ofsystem size (Fig. 2d, inset); intriguingly, this trend isconsistent with a recent holographic calculation of 1/Ncorrections to λL [43].

At even lower temperature, near or below the spin glasstransition, C(t) does not reach its maximally scrambledvalue of 2 at late times (Fig. 2c, inset). Moreover, forthese finite size systems, there is a near collision be-tween the onset of spin-glass order and the temperatureat which the value of λ seems to violate the scramblingbound (Fig. 2c, red dashed line). That the value of λbelow the transition seems to be well above the bound issomewhat surprising as one might have expected scram-bling to slow once the spins freeze.

For high temperatures, T & J , the data show ev-idence of saturation consistent with scrambling beinggoverned by the microscopic scale. Finally, Fig 2d de-picts a direct comparison between C(t) and C2(t) =−Tr[ρβ/2[W (t), V ]ρβ/2[W (t), V ]]. While they agree atshort times, differences begin to accumulate at late times.The ratio of the exponents, λβ and λβ/2, extracted fromC(t) and C2(t) respectively (Fig. 2b, Fig. 2d, inset), in-creases weakly with system size, indicating subtleties inscaling at larger systems. Crucially, extrapolating theratio of C(t) and C2(t) at finite O(1) times suggests thatthe signal to noise observed in current generation exper-iments should allow for exploration of system sizes up toN ∼ 50 spins, well beyond the capability of numerics.

Experimental Implementation—We now propose twoexperimental blueprints for realizing our interferometric

|e ! fii

|g ! fij

i

|f ! eii f

|f ! gij

2

2eiHt

/2

ancilla Spin system i

(b)

(a)

|gii

|eii

|fii

|gis|eis

FIG. 3. (a) Schematic of a one-dimensional lattice of ul-tracold atoms interacting with two impurities. There exists astrong interspecies Feshbach resonance between impurity andatom only when the impurity resides in state |f〉 . By drivingthe left impurity to state |f〉 before time evolution and theright impurity to |f〉 after time evolution (τ is the time neededto accumulate the Feshbach interaction), one can implementcontrolled unitaries V and W at asymmetric positions in timeand space. (b) Schematic of a 1D lattice of Rydberg atomsinteracting with the ancilla via a long-range blockade-basedgate. Each atom consists of two ground state hyperfine lev-els |↑〉, |↓〉 coupled to a Rydberg state |r〉 via an off-resonantlaser field.

protocol, where the underlying system is composed ofan array of ultracold atoms trapped in a 1D optical ar-ray (Fig. 3). First, we will describe an approach basedupon Feshbach interactions and second, we will considera method that utilizes a Rydberg-blockade gate [44–46].Focusing on the latter, we will further demonstrate thatthis system naturally simulates the TFSK model.

Feshbach Interactions—Here, we envision our 1Datomic gas to be interacting with a pair of ancilla qubits(e.g. impurity atoms) at positions i and j (Fig. 3a).Each ancilla consists of three states |g〉, |e〉, |f〉, withonly state |f〉 exhibiting a strong inter-species Feshbachinteraction with the atoms in the 1D chain. The onlyresource required to implement our proposed interfero-metric protocol is that the two ancillae are initialized inan entangled state, |gigj + eiej〉. In order to implementU†e = V e−iHt and U†g = e−iHtW , the Feshbach interac-tion must be applied at different times at the two posi-tions. This can be accomplished via a coherent π-pulseon the ancilla at site i at time t = 0 (before Hamiltonianevolution) and on the ancilla at site j at time t (aftertime evolution), as shown in Fig. 3a.

Rydberg Blockade—In the case where each atom withinthe system is weakly dressed via a Rydberg state, |r〉,a particularly elegant approach emerges. A combina-

5

tion of the Rydberg-blockade [47] and local addressabil-ity [48–50] will then enable the application of ancilla-spin-dependent Hermitian operators (e.g. V = σzi andW = σzj ) at two spatially separated locations i and j.Leveraging the long-range nature of the Rydberg interac-tion allows for the use of only a single ancilla atom sharedbetween the two systems. We note that this general ap-proach can also be implemented in a photonic system,where long optical delay lines play the role of asymmet-ric timing and strong optical nonlinearities play the roleof interactions.

After preparing the systems in a thermal state ρβ/2and the ancilla in (|↑〉+ |↓〉)/

√2, we apply a π-pulse be-

tween the |↑〉c and |r〉c states of the ancilla. Next, a2π-pulse is applied between the |↑〉i and |r〉i states ofatoms i and i′ (Fig. 1 and Fig. 3b). Owing to the block-ade, the |↑〉i |r〉c component of the wavefunction picksup a π phase shift while the phases of all other com-ponents remain unchanged [44–46]. This corresponds tothe application of a Hermitian operator −2Szi (Szc + 1/2)that implements a controlled-V gate (Fig. 1b). A secondπ pulse between |↑〉c and |r〉c restores the state of theancilla. The analogous procedure can be repeated aftertime-evolution to apply W controlled on the ancilla qubitbeing in state |↓〉c, after which the resulting state ρ(t) isprecisely given by Eqn. (3).

In addition to enabling our interferometric protocol, a1D lattice of Rydberg atoms also provides a natural re-alization of the transverse-field Sherrington-Kirkpatrickmodel [Eqn. (1)]. Experimentally, the spin degreesof freedom can be formed from two hyperfine stateswithin each atom, wherein the transverse field Γ = Ωmw

(Fig. 3b) simply corresponds to applying resonant mi-crowave radiation between these states. The Rydbergblockade can again be used to implement all-to-all Isinginteractions within each copy of the system. [51]. Inparticular, as shown in Fig. 3b, an off-resonant laser fieldwith Rabi frequency Ω and detuning ∆i couples each spin|↑〉-state with a highly excited Rydberg state |r〉. Thisinduces an ac Stark shift of |↑〉 that depends on the spinstates of surrounding atoms (e.g. due to van der Waalsinteractions between atoms virtually excited to |r〉) andleads to a long-range interacting Hamiltonian of the form:∑i,j Jijσ

zi σ

zj , where Jij = C6

(ri−rj)6+a6B[52].

The coupling strengths Jij are determined by C6 =C6Ω4/(∆i + ∆j)

4, where C6 is the coefficient of thevan der Waals interaction between Rydberg states |r〉iand |r〉j . In the regime where the blockade radius

aB = (C6/|∆i + ∆j |)1/6 is larger than the system sizeone obtains all-to-all interaction between spins. To avoidstray long-range interactions between the two copies dur-ing evolution, we assume that they are initially spatiallyseparated, while the ancilla qubit is localized betweenthem to allow for simultaneous coupling with each sys-tem. Randomized Ising couplings Jij can be achieved by

introducing a 1D speckle potential [53, 54], which leadsto random detunings ∆j ; if the transverse correlationlength of the speckle is sufficiently large, the two copiesof the system will be subject to identical randomized in-teractions. By introducing a second dressing laser to anattractive Rydberg state |r′〉, one can even obtain a dis-tribution of Jij with zero mean [51].

To summarize, we have shown that concepts originat-ing in the physics of black holes have a rich connection toquantum optical systems. We do so through two results.First, we introduce a general interferometric approach tomeasuring out-of-time-order correlation functions, whichrequires only local quantum control and no reverse timeevolution. Second, we show that the quantum-opticstoolbox can be used to realize the TFSK model and mea-sure C(t); moreover, we demonstrate numerically thatthis model exhibits “pretty fast” scrambling. In partic-ular, a phenomenological fit of C(t) is consistent with atime to reach maximal scrambling proportional to lnN .Because our protocol involves observables that scale withsystem size like Tr(ρ2) = e−S2 , the approach is mostpromising when the entropy is relatively small. Examplesinclude mesoscopic many-body systems at low tempera-ture, systems with an unbounded Hilbert space dimen-sion but finite entropy, and single particle-like modelsthat exhibit quantum chaos.

Looking forward, measurements of C(t) can providequantitative access to important questions related tothermalization and ergodicity in interacting many-bodysystems. For example, scrambling dynamics may give in-sight into the non-local dressing of Heisenberg operatorsunder time evolution. Along these lines, it may be par-ticularly fruitful to study integrable models, which eitherdo not scramble or scramble only very weakly. In par-ticular, by perturbing such models, one can probe howchaos develops and whether out-of-time-order correlationfunctions might exhibit evidence for a quantum analog ofthe celebrated Kolmogorov-Arnold-Moser theorem [55].

It is a pleasure to gratefully acknowledge the insightsof and discussions with D. Stanford, C. Laumann, M.Schleier-Smith, J. McGreevy and H. Pichler. This workwas supported, in part, by NSF-DMR1507141, the Si-mons Foundation, the Miller Institute for Basic Re-search in Science, AFOSR MURI grant FA9550-14-1-0035, Harvard-MIT CUA, NSF-DMR1308435, the Gor-don and Betty Moore foundation, the Humboldt founda-tion, and the Max Planck Institute for Quantum Optics.

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[33] Note that in principle F (t) can also be measured directlyusing our protocol. In this case, one has to replace thesecond copy at ρβ/2 by an infinite temperature system.

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