Anti-Zeno cooling of spin baths by quantum probe measurements

Durga Bhaktavatsala Rao Dasari,1 Sen Yang,2 AmitFinkler,3 Gershon Kurizki,3 and Joerg Wrachtrup1

13. Physics Institute, University of Stuttgart,Center for Applied Quantum Technologies, IQST,

MPI for Solid State Research, Stuttgart 70569, Germany.2Department of Physics, The Hong Kong University of Science and Technology,

Clear Water Bay, Hong Kong, China.3Department of Chemical Physics, Weizmann Institute of Science, Rehovot, Israel

(Dated: August 24, 2021)

AbstractWe put forth, theoretically and experimentally, the possibility of drastically cooling down (puri-

fying) thermal ensembles (“baths”) of solid-state spins via a sequence of projective measurements

of a probe spin that couples to the bath in an arbitrary fashion. If the measurement intervals are

chosen to correspond to the anti-Zeno regime of the probe-bath exchange, then a short sequence

of measurements with selected outcomes is found to have an appreciable success probability. Such

a sequence is shown to condition the bath evolution so that it can dramatically enhance the bath-

state purity and yield a low-entropy steady state of the bath. This purified bath state persists after

the measurements, and can be chosen, on demand, to allow for Zeno- or anti-Zeno- like evolution of

quantum systems coupled to the purified bath. The experimental setup for observing these effects

consists of a nitrogen vacancy (NV) center in diamond at low temperature that acts as a probe of

the surrounding nuclear spin bath. The NV sigle-shot measurements are induced by optical fields

at microsecond intervals.

Keywords: quantum measurements, spin-bath cooling, anti-Zeno effect

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I. INTRODUCTION

Quantum system interactions with their environment, alias a bath with many degrees offreedom, are foundationally interesting since they chart the path from quantum coherent dy-namics to behavior described by statistical-mechanics or thermodynamics as the complexityof the system-bath compound increases [1–9]. On the applied side, adequate control oversuch interactions is a prerequisite for all emerging quantum technologies [10–14, 16–32].In particular, fault-tolerant quantum computation [11, 12] requires the ability to protectthe information encoded in the computational qubits from leaking into the environment(bath), thus causing their decoherence. The desirable, often inachievable, goal is to com-pletely decouple the system from the bath [32–39]. More realistically, the adverse effectsof decoherence can be suppressed by control pulses at a rate faster than the inverse mem-ory (correlation) time of the bath [40–46]. Yet, existing system-bath interaction control isseverely constrained by the paradigm whereby the bath is immutable and control is onlyapplicable to the system [37–43]. Correspondingly, theory commonly resorts to master equa-tions that describe only the dynamics of the system (its reduced density matrix) [2, 40, 46].In such a description, the bath can be emulated for most purposes by a classical noisy, fluc-tuating field. The justification is that in typical experimental settings one can only achievea high degree of control and readout precision of the system, but not of the bath.

A major stride beyond this paradigm towards deeper understanding of quantum aspectsof system-bath interactions has been the discovery that the Quantum Zeno effect (QZE) [47–49] and its inverse, the Anti-Zeno (AZE) effect, can profoundly effect open quantum systems[50–54]. These effects occur on time-scales such that system-bath interaction dynamics mayallow for partial revival of the system coherence caused by energy and entropy backflowfrom the bath to the system. The most familiar manifestations of the QZE or the AZE are,respectively, the slowdown or speedup of the open system-state evolution when subjected tomeasurements at appropriate rates [50–52]. Both effects have been extensively investigatedand verified for a variety of systems [46, 55–59].

Not less significant has been the discovery that QZE dynamics can heat up a qubit byequalizing the bath-induced upward and downward transition rates, whereas less frequentmeasurements conforming to the AZE regime can cool the qubit down by predominantlyenhancing downward transitions (relaxation to the ground state) [60, 61]. Such cooling andheating effects have been experimentally verified by some of us for interacting spin-systemsusing nuclear magnetic resonance (NMR) techniques for their quantum simulation [62].

Thus far, the study of the QZE and AZE has been focused only on the evolution of thesystem, and an unaffected bath, a scenario relevant to weak system-bath coupling, for whichthe second- order non-Markovian master equations reveal the dynamics under repeated non-selective (unread) measurements (or phase modulations) of the system [46, 50, 51, 60–62].Here, by contrast, we consider strong system-bath coupling for which a selective (projective)measurement of the system can drastically disturb the bath state depending on its readoutoutcome. This disturbance becomes stronger as the bath gets smaller. In such cases one hasto rely on the exact solution of the Hamiltonian, which is obtainable for only a few system-bath interaction models. We here theoretically investigate such a model of the dephasingof a two-level system (probe spin) by a spin bath with conserved spin, and experimentallyrealize it for a degenerate (non-interacting) spin bath.We show how to employ the coupledsystem-bath evolution under consecutive selective measurements to control (purify) the bathstate. In turn, such a purified bath state is shown to allow, on demand, either dramatically

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(a)

(b)

(c)

(d)

Figure 1. Bath purification by conditional measurements and its posterior (persistent)

effects (a) Schematic description of our scheme: Conditional measurements of the state of the

central probe-spin (S) collapse the spin-bath toward a low entropy state with resolvable spins. (b)

An experimentally obtained measurement sequence of the central spin displaying the outcomes of a

measurement, where 0 corresponds to zero-photon and 1 to one-photon detection. (c) Modification

of the bath spectrum before and after the measurements. (d) Coherence decay of the probe

electron-spin coupled to the original (red) and purified (blue) spin-bath (experiment and theory).

prolonged coherence time of qubits in the QZE regime, or, conversely rapid oscillations ofqubit cohernece in the AZE regime. Measurement induced Quantum State Engineering ofspin and oscillator baths has been studied theoretically to generate exotic entangled states[63] and for ground state cooling/squeezing of cantilever modes [64]. While these studiesonly focus on the preparation of the target state the role of QZE and AZE have not beenexplored in such preparations.

The concrete protocol consists in the preparation of the probe system in a superposition ofenergy states and observation of the decay of this superposition via the probe interaction withthe adjacent spin bath. The idea is to repeatedly check via frequent selective measurementswhether or not the probe is still in its initial state and thereby control the bath state.

II. MODEL

We consider the dipolar interaction of an (effectively) spin-1/2 probe system (S) with Nidentical spin-1/2 systems (the bath – B). The inherent quantization axis of S and the S-Bresonance mismatch (detuning) render their interaction dispersive, which leads to decoher-ence (pure dephasing) of the S-spin. A common variant of this problem involves spin-spininteractions in the bath [44]. The dynamics generated by all such Hamiltonians can be

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exactly solved for any distribution of S-B couplings and any eigenvalue spectrum of B. Herewe assume, for simplicity, that S is coupled to a non-interacting spin bath B in the starconfiguration (Fig. 1a). The Hamiltonian describing the S-B interaction is then given by

H = Sz ⊗ B, (1)

B =∑k

~gk · ~Ik.

Here Sz is the S-spin component along its quantization axis, k labels the B-spins, ~gk arethe dipolar couplings between S and the k-th nuclear spin, and ~Ik its spin (Pauli) operator.The typically random spatial locations of the B-spins result in an inhomogeneous distributionof the couplings ~gk such that B has no preferred spatial symmetries.

In the Sz basis of S spanned by states (|e〉, |g〉), the above Hamiltonian can be rewrittenas

H = Be|e〉〈e|+ Bg|g〉〈g|, Be(g) = ±∑k

~gk(r) · ~Ik (2)

where Be(g) are the spin-bath energy operators associated with the energy states |e(g)〉 inS. The Hamiltonian in Eq. (2) gives rise in the interaction picture to the closed-form time-evolution operator of the “supersystem” S + B

US+B(t) = Ue(t)|e〉〈e| + Ug(t)|g〉〈g| (3)

where Ue(g)(t) = exp (itBe(g)) govern the dynamics of B in a manner conditional on the stateof S. In what follows we review standard dephasing that assumes an unchanging statisticalstate of B and contrast it with the conditional dynamics of B, which is shown to differdramatically.

III. FREE INDUCTION DECAY AND ITS ZENO OR ANTI-ZENO MODIFICA-

TIONS

In standard treatments, the effect of B on S is mimicked by a classical, noisy (random)magnetic field, assuming the initial B-state to be completely depolarized (mixed): the S-Binteraction given above does not lead then to any bath dynamics, i.e.,

ρB(t) = ρB(0), ρB(0) =∑m

Pj(0)|j〉〈j|, Pj(0) =

√π

2Nexp (−j2/2N) (4)

where j labels the collective B-states. As the S − B interaction causes pure dephasing(decoherence) of S, the dynamics only occurs between superposition states of S in the e(g)basis, say, |±〉 = 1√

2[|e〉 ± |g〉]. If we initialize the probe in the |+〉 state, we can measure

its free-induction decay (FID) [39], caused by its mixing with the orthogonal superpositionstate |−〉 via a bath-induced random phase. The ensemble average of their populationscorresponds to the probabilities of measuring these states at time t, given by

P+(t) ≡ 1− P−(t) =1

2[1 + e−(t/T2)2 ]. (5)

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Equivalently, the probe-spin coherence can be measured by the decaying mean value of the〈Sx〉 component of its spin, which displays Gaussian decay, given by

〈Sx〉(t) ≈ e−(t/T2)2 . (6)

It is known that the dephasing time T2 can be modified by dynamical control of S [39]. Acommon (but not always optimal) example of such control are dynamical - decoupling pulsesequences [35, 38, 39, 65, 66]. According to the Kofman-Kurizki (KK) universal formula[40, 46, 50, 51]any dynamically-controlled decoherence of S evolves as

〈Sx〉(t) ∼ exp[−∫G(ω)F (t, ω)dω

]. (7)

Here F (t, ω) is the time-dependent control power-spectrum, which may arise from unitary(pulsed or continuous) operations or from frequent measurements and G(ω) is the bath-coupling spectrum. In principle, one may adapt the F (t, ω) spectrum to the G(ω) spectrumso as to either suppress or enhance the decoherence in Eq. (7) , corresponding to the QZEor AZE regime, respectively, as is done in bath-optimized minimal energy control (BOMEC)[42, 43].

Here we set out to overcome the limitations of any form of conventional dynamical controlof S, be it DD or BOMEC: (i) the extremely high rate of pulses required to effectivelycontrol decoherence in the high- temperature or broad-band (white noise) limit of B, whichare properties that cannot be affected by conventional control; (ii) the need to maintainthe dynamical control during the time that other (quantum-logic or quantum- sensing)operations are performed on the qubits of interest (although this difficulty can be mitigatedby appropriate control [67]. To this end, we resort to unconventional bath control viaconditional measurements of the probe.

IV. BATH DYNAMICS CAUSED BY CONDITIONAL MEASUREMENTS

Let us assume that τ is the interval between consecutive measurements of S, which mustexceed the minimal (resolution) time of probing/ reading out its state. If these intervalsgreatly exceed the bath correlation (memory) time tc � τ , then the regime is Markovian,i.e. the results of each measurement of S are independent of its predecessors. This meansthat S interacts every time with B that is in the same statistical (thermal) state. Then,upon many repetitions of the FID measurement, one can extract the average coherence ata specific time to infer the value of the dephasing time T2. We, in contrast, wish to ventureinto the opposite strongly non-Markovian limit, that corresponds to highly correlated resultsof consecutive measurements. Namely, the readout at the time t = mτ (m is an integer)depends on the outcomes of all its predecessors. In this limit, the B-state evolution isgoverned by the non-unitary evolution operator corresponding to S projections onto eitherof the states in Eq. (5), given by

V±(t) = (Ue(t)± Ug(t))/2, Ue(g) = e±iωjt|j〉〈j|. (8)

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The conditional evolution of B assumes the form

ρ±B(t) ≡ V±(t)ρB(0)V †±(t) =∑j

Pj(t)|j〉〈j| (9)

where |j〉 are the collective eigenstates of the bath operator in Eq. (2), and Pj(t) are con-ditionally modified populations. Repeated application of the conditional evolution operatorin Eqs. (8), (9) can yield a purified state of B that as shown in Fig. (1) can drasticallychange the standard FID of S.

To get an insight into this purification process, let us consider that m-fold measurementsof S are consecutively performed in the |+〉, |−〉 basis. There are 2m measurement sequenceswith different possible outcomes. The projections on the |+〉 or |−〉 state in each of these(single-shot) measurements are designated as ′0′, ′1′ respectively. Each such sequence, de-noted by Mn,m, can be represented as an n-fold string of ‘1’ and m-fold string of ‘0’. Thebath evolution is conditional on a particular sequence or string, which, in parameter space,can be viewed as a trajectory – dubbed henceforth “conditional trajectory” (CT).

For any CT associated to Mn,m, the corresponding non-unitary B-evolution operator is

Vn,m(t) = cosm−n(Bt) sinn(Bt). (10)

The j-distribution that underlies the B-state is progressively narrowed down by the evolutionoperator (10) compared to that of the initial state, Eq. (1), in a manner that depends on therealized CT. This comes about since, at a chosen time t, different j-states will have eithermaximized or minimized probabilities, depending on the arguments of the respective cosineor sine functions.

In what follows we focus on the CT M0,m = {0, 0, · · · 0}, corresponding to an m-foldstring of 0 results that is associated with the bath state

ρB(t,M0,m) =V0,m(t)ρB(0)V †0,m(t)

Tr[V0,m(t)ρB(0)V †0,m(t)], V0,m = cosm(Bt). (11)

The success probability P0,m(t) = Tr[V0,m(t)ρB(0)V †0,m(t)] is maximized for 〈B〉(t) = lπ(l ∈ 1, 2 · · · ). Importantly, as the j-distribution is gradually narrowed down, this probabilitydoes not decrease rapidly with m, unlike the probability of an uncorrelated CT sequence (seeExperiment). Thus, by recording such an m-fold CT with appreciable success probability,one ensures the projection of B onto a specific state ρB. The changes caused by B on S andvice versa along this CT continue until a steady state of B is reached, that has much higherpurity than its initial high-temperature state. The steady states are uncorrelated and aregiven by

ρB(M0,m, t→∞) =1

2N ′

N ′/2∑k=−N ′/2

[|j′〉〈j′|+ | − j′〉〈−j′|], (12)

where | ± j′〉 are the B- eigenstates that have survived the purification by the selected CTand N ′ � N .

Surprisingly, even without optimal feedback that keeps readjusting the time-interval be-tween measurements according to their outcomes (as proposed by some of us [? ]), theconvergence to the steady state turns out to be rather fast and the success probability of

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𝜋 𝜋𝜏

−1 𝑒 +1 𝑒

0 𝑒

𝐸𝑥

CC

I

Four Measurements

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(a) (b) (c)

Figure 2. Probe FID before and after bath purification: (a) Energy-level diagram of the

central probe-spin depicting the basis states and the readout. Shown below is the pulse-sequence

for four measurements. (b) Experimental and theoretical FID of the probe spin before (blue) and

after (red) four measurements with identical (0) results (CT M0,4). The effective FID rate of

the corresponding conditional bath state is 4 times lower than the original (blue) decay rate. (c)

ODMR of the spin-bath spectrum before (blue) and after (red) the conditional measurements, CT

M0,4.

reaching the desired steady state is fairly high. The reason is the sinusoidal j-dependentS-B exchange that allows to select certain j-states by choosing the appropriate interval. Letus choose for the purification the shortest time-interval of resolvable measurement, whichis τ = 1/g, where g =

√∑k g

2k. The steady state of B following such purification is a

statistical mixture of total- magnetization (macroscopic) states

ρB(t→∞) =1

2[|00 · · · 0〉〈00 · · · 0|+ |11 · · · 1〉〈11 · · · 1|]. (13)

We have thus dramatically raised the bath purity from 1/2N to 1/2. Yet, the individualspins are all unpolarized, and the polarization is not local but global (collective).

The steady-state of B following such purification is a persistent imprint left by the corre-sponding CT of this state. This imprint is revealed by initializing the same or another qubitin the |+〉 state and tracking the subsequent evolution of its mean coherence by repeatedmeasurements with selected time-intervals. Remarkably, the QZE or AZE regimes can bestretched on demand over much longer time intervals than before (Fig. 2) because of theextreme non-Markovianity of the steady-state of B in Eqs. (12), (13).

A. Experiment

In the experiment the S-spin is a low strain NV center in diamond. A small magnetic fieldis applied along the NV axis [111] to counter the earth’s magnetic field, thereby eliminatingthe net B-field at the location of the NV center. Hence the | ± 1〉 states in Fig. 2(a) aredegenerate. Only the bright superposition state |+〉 couples to to |0〉, where the two states

|±〉 (|±〉 = [|+1〉±|−1〉]√2

) form the two-level probe-spin (S) subspace described above. For

a projective readout of the probe state, we map the |+〉 state to the ancillary state |0〉

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Figure 3. Zeno and Anti-Zeno effects on the evolution of the probe-spin following bath

purification(a) The experimental pulse diagram depicting the Microwave pulses (blue) interleaved

with repeated initialization (A1) and measurements (Ex), for preparation of the bath state Mn,m.

Following this the FID of the probe spin is measured again and compared with the original FID. (b,

c) The FID of the probe spin interacting with the conditional bath state M0,4 (red), M4,0 (green)

is compared with the average FID obtained from all possible conditional bath state preparations

(blue). The same is repeated is two different time scales τ = 600ns (b) and τ = 1200ns. The

solid-lines are fits to the experimental data.

via a microwave transition. At a low temperature (4K), according to the optical selectionrules, we achieve high-fidelity single-shot readout. We can clearly observe our theoreticalpredictions as the time needed for the purification of the bath and its posterior effects on S,as the nuclear-spin coherence times, reaches minutes at low temperature.

Our protocol starts with the initialization of the electron spin to the state |0〉 by resonantlyexciting it in the A1 transition [69], followed by a microwave transition, |0〉 → |+〉. We letthe prepared state acquire a phase over time τ due to its interaction with the bath and thenmap the population back to the ancillary state |0〉 and perform single shot readout usingthe Ex excitation.

To confirm the actual purification of the bath toward a low-entropy state we perform twodifferent experiments, wherein: (i) we measure and compare the probe-spin FID (Eq. (2))before and after the bath purification by conditional measurements; and (ii) we measurethe bath spectrum via the standard procedure of the Optically Detected Magnetic Reso-nance (ODMR) that records the spin noise spectrum of the probe-spin, before and after theconditional measurements. The results shown in Fig. 2, 3 reveal the prolonged coherencetime of the probe-spin and the corresponding narrowing of its spin noise spectrum, afterfour conditional measurements that purify the bath to a low-entropy state. Having chosen

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Figure 4. Life-time of the purified collective bath-state. (a) The pulse sequence used for

the measurement of the bath life-time. After the conditional bath state preparation M0,4, we

allow a free time t and then measure again the probability for obtaining the result an identical

measurement result. (b) The probability of obtaining the result 0 after a long wait-time t for a bath

state preparations M0,4 (red) is shown for the measurement times τ = 1200ns.These probabilities

are also compared with the case when no apriori conditional bath state preparation. The effective

life-time of the bath extends towards the T e1 life-time of the probe.

the measurement time-scale (τ < T ∗2 ) to be in the Zeno (QZE) regime, the observation ofa measurement string M0,4 projects the bath state to a mixture of the effective zero-fieldstate (j = 0) and the high-j states (Eq. (13)). Since the density of states that correspondto the zero bath-field is exponentially larger, by the factor NCN/2, than the high-j states,the probability of the bath to collapse toward the j = 0 is high.

To change the probe dynamics from the Zeno (QZE) to the Anti-Zeno (AZE) regime,we repeat the experiments with a different CT, M4,0, and also a different measurementinterval τ . In Fig. 3 we show the FID samplings of the probe-spin following differentconditional preparations of the bath by repeated selective measurement samplings of theprobe-spin. We see two dramatically different decay profiles of the probe-spin coherence:either a slowdown of the decay (red-QZE) or an underdamped oscillation (blue-AZE). In theAZE regime, the probe coherence exhibits underdamped oscillation at the mean frequencyof the bath field ωB, reflecting the bath preparation in a non-zero j-state. The best fits tothese AZE-like experimental data take the form C(t) = 1/2 + ae−(t/T )2 cos(ωBt), where ais a fitting parameter. For example in Fig. 3, the bath-field obtained from such fitting formeasurements time scales τ = 600ns and τ = 1200ns is approximately ωB = 200kHz andωB = 80kHz, and a ≈ 1.

We next repeat the same experiment with a much longer time interval between samplingsts ≥ T ∗2 that is well beyond the Zeno or anti-Zeno time scale. Remarkably, we observe similareffects to Fig. 3, but now they are dependent only on the CT, M0,4,M4,0 and not on the

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sampling time-scale ts. Thus, whereas the original QZE and AZE depend on time-resolutionand not on energy resolution, and are caused by non-selective independent measurements,here we observe the opposite: their counter-parts depend on energy resolution rather thanon time-resolution, and are caused by state-selective measurements.

These novel modifications of the QZE and AZE are shown in Fig. 4 where long time delayis introduced between the bath purification by conditional measurements and the subsequentsamplings of the probe-spin coherence. These sequences help to measure the life-time of thepurified collective bath state. One can see that the measurement of the probe-spin result(coherence) is almost unchanged upto a milli second and then slowly decays towards thefully mixed state i.e., C(t) = 0.5. This corresponds to a very long life-time of the collectivebath spin-state prepared by the conditional measurements of the probe-spin.

V. DISCUSSION

We have demonstrated, theoretically and experimentally, the ability to purify the initiallyfully mixed spin-bath state to a low-entropy steady-state by conditional measurements of aspin-probe coupled to it. We have also revealed the remarkable, hitherto unknown, persis-tent posterior effects of such purification on other probe spins. The demonstrated effects arenot observable unless we track a particular conditional measurement sequence. Otherwise,following m projective measurements of the spin-probe S, its mean coherence exhibits decaythat is an average over the 2m possible measurement outcomes which we dub here condi-tional trajectories (CT). One can verify [70] that the corresponding averaged state of thebath B is then the same as the unmeasured state at time t i.e.,

∑N ρB(MN , t) = ρB(t). This

confirms the ergodicity of the interacting S + B (system + bath). Thus, averaging over allpossible FID patterns resulting from all different CT yields the FID of S in the absence ofany measurements. By contrast, we have here searched to unravel individual CTs that cor-respond to specific states of B and hence to particular magnetic (dipolar) field distributionsgenerated by the bath. The FID time T ∗2 , is actually an average of faster and slower decaysof the S-coherence caused by the bath: There are bath states that almost freeze the decay(obeying QZE evolution) and those that , on the contrary, speed it up (under AZE evolu-tion).All such effects become observable upon resolving the CTs of S, as demonstrated here.We wish to highlight the role of conditional measurements that lead to AZE oscillations ofthe probe spin, as they allow to spectroscopically identify strongly-interacting spins. Suchspins are otherwise identified through complex correlation spectroscopy methods [71, 72].For example, from the underdamped (AZE) oscillations in Fig. 3, we identify the presence ofa strongly-interacting spin with coupling strength g ∼ 200kHz. Remarkably, we have shownthe ability to purify an initially unpolarized (high-temperature) bath-state to a low-entropysteady state that is controlled by selecting a CT with fairly high success probability andlimited number of measurement. Due to the rise in the success probability of the selectedresult as the B-distribution narrows down. This favorable scaling of spin-bath purificationwith the number of successful outcomes is akin to that of quantum state engineering byconditional measurements previously introduced by some of us to control the evolution ofquantum observables [68, 73, 74]. One can optimize the present protocol by adopting similarprotocols [68].

The ability, proposed and demonstrated here, to cool down (purify) the bath to a low-entropy steady state, allows us to preserve spin coherence or cause its oscillations overmuch longer times than with a thermal bath. This may substantially benefit quantum

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information processing or sensing, since, unlike existing dynamical control methods (suchas dynamical decoupling (DD) [35, 37–39] or the bath-optimized minimal-energy control(BOMEC) devised by us [42, 43], this measurement-induced steady-state remains imprintedon the bath long after the control has been switched off.

ACKNOWLEDGMENTS

D.D.B.Rao and J.W would like to acknowledge the support from DFG (FOR2724), ERCproject SMeL, DFG SFB/TR21, EU ASTERIQS, QIA, Max Planck Society, the Volkswa-genstiftung, the Baden- Wuerttemberg Foundation. S.Y acknowledges the financial supportfrom Hong Kong RGC (GRF/14304618). A.F would like to acknowledge the historic gen-erosity of the Harold Perlman Family, research grants from the Abramson Family Centerfor Young Scientists, the Ilse Katz Institute for Material Sciences and Magnetic ResonanceResearch and the Willner Family Leadership Institute for the Weizmann Institute of Science.

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