Feedback-Induced Quantum Phase Transitions Using Weak Measurements

D. A. Ivanov,1 T. Yu. Ivanova,1 S. F. Caballero-Benitez ,2 and I. B. Mekhov 1,3,4,*

1Department of Physics, St. Petersburg State University, 198504 St. Petersburg, Russia2Instituto de Física, Universidad Nacional Autónoma de Mexico, Ciudad de Mexico 04510, Mexico

3Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom4SPEC, CEA, CNRS, Universite Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France

(Received 1 April 2019; revised manuscript received 15 September 2019; published 6 January 2020)

We show that applying feedback and weak measurements to a quantum system induces phase transitionsbeyond the dissipative ones. Feedback enables controlling essentially quantum properties of the transition,i.e., its critical exponent, as it is driven by the fundamental quantum fluctuations due to measurement.Feedback provides the non-Markovianity and nonlinearity to the hybrid quantum-classical system, andenables simulating effects similar to spin-bath problems and Floquet time crystals with tunable long-range(long-memory) interactions.

DOI: 10.1103/PhysRevLett.124.010603

The notion of quantum phase transitions (QPT) [1] playsa key role not only in physics of various systems (e.g.,atomic and solid), but affects complementary disciplines aswell, e.g., quantum information and technologies [2],machine learning [3], and complex networks [4]. Incontrast to thermal transitions, QPT is driven by quantumfluctuations existing even at zero temperature in closedsystems. Studies of open systems advanced the latter case:the dissipation provides fluctuations via the system-bathcoupling, and the dissipative phase transition (DPT) resultsin a nontrivial steady state [5,6].Here we consider an open quantum system, which is

nevertheless not a dissipative one, but is coupled to aclassical measurement device. The notion of fundamentalquantummeasurement is broader than dissipation: the latteris its special case, where the measurement results areignored in quantum evolution [7]. We show that addingthe measurement-based feedback can induce phase tran-sitions. Moreover, this enables controlling quantum proper-ties of the transition by tuning its critical exponent. Such afeedback-induced phase transition (FPT) is driven byfundamentally quantum fluctuations of the measurementprocess, originating from the incapability of any classicaldevice to capture the superpositions and entanglement ofthe quantum world.Feedback is a general idea of modifying system param-

eters depending on the measurement outcomes. It spreadsfrom engineering to contemporary music, including mod-eling the Maxwell demon [8–10] and reinforcement learn-ing [11]. Feedback control has been successfully extendedto the quantum domain [7,12–28] resulting in quantummetrology aiming to stabilize nontrivial quantum states andsqueeze (cool) their noise. The measurement backactiontypically defines the limit of control, thus, playing animportant but negative role [29]. In our work, we shift the

focus of feedback from quantum state control to phasetransition control, where the measurement fluctuationsdrive transition, thus playing an essentially positive rolein the process as a whole.Hybrid systems is an active field of quantum technologies,

where various systems have been already coupled [30]:atomic, photonic, superconducting,mechanical, etc. The goalis to use advantages of various components. In this sense, weaddress a hybrid quantum-classical system, where the quan-tum system can be a simple one providing the quantumcoherence, while all other properties necessary for a tunablephase transition are provided by the classical feedback loop:nonlinear interaction, non-Markovianity, and fluctuations.We show that FPT leads to effects similar to particle-bath

problems (e.g., spin-boson, Kondo, Caldeira-Leggett,quantum Browninan motion, dissipative Dicke models)describing very different physical systems from quantummagnets to cold atoms [31–37]. While tuning quantumbaths in a given system is a challenge, tuning the classical

h(t)

GI(t)

FIG. 1. Setup (details for a BEC system are given in Ref. [62]).Quantum dipoles (possibly, a many-body system) are illuminatedby probe. Scattered light is measured and feedback acts on thesystem, providing non-Markovianity, nonlinearity, and noisenecessary for phase transition. Importantly, the feedback responsehðtÞ can be digitally tuned.

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feedback is straightforward, which opens the way forsimulating various systems in a single setup. This raisesquestions about quantum-classical mapping betweenFloquet time crystals [38,39] and long-range interactingspin chains. Our model is directly applicable to many-bodysystems, and as an example we consider ultracold atoms ina cavity. Such a setup of many-body cavity QED (cf. forreview Refs. [40,41]) was recently marked by experimentaldemonstrations of superradiant Dicke [42], lattice super-solid [43,44], and other phase transitions [45,46], as well astheory proposals [34,35,47–55]. Nevertheless, effects wepredict here require us to go beyond the cavity-inducedautonomous feedback [56].Model.—Consider N two-level systems (spins, atoms,

qubits) coupled to a bosonic (light) mode, which may becavity enhanced (Fig. 1). The Hamiltonian then reads

H ¼ δa†aþ ωRSz þ2ffiffiffiffiN

p Sx½gðaþ a†Þ þGIðtÞ; ð1Þwhich, without the feedback term GIðtÞ, is the standardcavity QED Hamiltonian [57] describing the Dicke (orRabi) model [34,35] in the ultrastrong coupling regime[58–61] (without the rotating-wave approximation). Here ais the annihilation operator of light mode of frequency δ,Sx;y;z are the collective operators of spins of frequency ωR, gis the light-matter coupling constant. The Dicke model wasfirst realized in Ref. [42] using a Bose-Einstein condensate(BEC) in a cavity, and we relate our model to suchexperiments in Ref. [62]. Our approach can be readilyapplied to many-body settings as Sx can represent variousmany-body variables [71–73], not limited to the sum of allspins: e.g., fermion or spin (staggered) magnetization[45,74–76] or combinations of strongly interacting atomsin arrays, as in lattice experiments [43,44].The feedback term GIðtÞ has a form of the time-

dependent operator-valued Rabi frequency rotating thespins [G is the feedback coefficient and IðtÞ is the controlsignal]. We consider detecting the light quadrature xoutθ ðtÞ(θ is the local oscillator phase) and define IðtÞ ¼ffiffiffiffiffi2κ

p Rt0 hðt − zÞF ½xoutθ ðzÞdz. Thus, the classical device con-

tinuously measures xoutθ , calculates the functionF , integratesit over time, and feeds the result back according to the termGIðtÞ. In BEC [62], the quasispin levels correspond to twomotional states of atoms, and coupling of feedback to Sx isachieved by modifying the trapping potential [62]. Variousforms of the feedback response hðtÞwill play the central rolein our work. The input-output relation [77] gives xoutθ ¼ffiffiffiffiffi2κ

pxθ − fθ=

ffiffiffiffiffi2κ

p, where the intracavity quadrature is xθ ¼

ðae−iθ þ a†eiθÞ=2 and κ is the cavity decay rate. Thequadrature noise fθ ¼ ðfae−iθ þ f†aeiθÞ=2 is defined viathe Markovian noise operator fa [hfaðtþ τÞfaðtÞi ¼2κδðτÞ] in the Heisenberg-Langevin equation:

_a ¼ −iδa − i2gffiffiffiffiN

p Sx − κaþ fa: ð2Þ

Effective feedback-induced interaction.—An illustrationthat feedback induces effective nonlinear interaction is usedin quantum metrology [23] for simple cases such as IðtÞ∼xoutθ . One sees this if light can be adiabatically eliminatedfrom Eq. (2), a ∼ Sx. Then the effective Hamiltonian,giving correct Heisenberg equations for spins, containsthe term S2x leading to spin squeezing [23] [cf. Eq. (1) forIðtÞ ∼ xoutθ ∼ Sx]. Note that this is just an illustration and thederivation needs to account for noise as well. Nevertheless,we can proceed in a similar way and expect the interactionas

Rt0 hðzÞSxðtÞF ½Sxðt − zÞdz. For the linear feedback,

F ½Sx ¼ Sx, this term resembles the long-range spin-spininteraction in space; here we have a long-range (i.e., long-memory) “interaction” of spins with themselves in the past.The “interaction length” is determined by hðtÞ.Such a time-space analogy was successfully used in a

spin-boson model [32,33,78,79], describing spins in abosonic bath of nontrivial spectral function: ωs for smallfrequencies [s ¼ 1 for Ohmic, s < 1 (s > 1) for sub-(super-)Ohmic bath, cf. Ref. [62] ]. It was shown that asimilar “time-interaction” term can be generated [78,79].Moreover, an analogy with the spin chain and long-rangeinteraction term in space

Pi;j SiSj=jri − rjjsþ1 was put

forward and the break of the quantum-classical mappingwas discussed [78,80]. For s ¼ 1 a QPT of the Kosterlitz-Thouless type was found [33], while QPTs for the sub-Ohmic baths are still under active research [37,81].In bath problems, such a long-memory interaction can be

obtained only asymptotically [78,79]. Moreover, arbitrarilytuning the spectral properties of quantum baths in a givensystem is challenging (cf. Ref. [82] for quantum simula-tions of the spin-boson model and Refs. [83,84] for acomplex network approach). In contrast, the feedbackresponse hðtÞ can be implemented and varied naturally,as signals are processed digitally, opening paths forsimulating various problems in a single setup. The function

hðtÞ ¼ hð0Þ

t0tþ t0

sþ1

ð3Þ

will correspond to the spatial Ising-type interaction.The instantaneous feedback with hðtÞ ∼ δðtÞ will lead toa “short-range in time” S2x term, as in the Lipkin-Meshkov-Glick (LMG) model [85] originating from nuclear physics.A sequence of amplitude-shaped time delays hðtÞ ∼P

n δðt − nTÞ=nsþ1 will enable studies of discrete timecrystals [38,86–89] and Floquet engineering [39] withlong-range interaction

Pn SxðtÞSxðt − nTÞ=nsþ1, where

the crystal period may be T ¼ 2π=ωR. This is in contrastto standard time crystals, where the parameter modulationis externally prescribed [e.g., periodic gðtÞ]. Here, theparameters are modulated depending on the system state(via Sx), i.e., self-consistently, as it happens in realmaterials, e.g., with phonons. The interaction in time doesnot necessarily require the presence of standard atom-atom

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interaction in space. The global interaction is given byconstant hðtÞ. The Dicke model can be restored even in theadiabatic limit by exponentially decaying and oscillatinghðtÞ mimicking a cavity. All such hðtÞ can be realizedseparately or simultaneously to observe the competitionbetween different interaction types. Our results do not relyon effective Hamiltonians [48]. This discussion motivatesus to use in further simulations hðtÞ, Eq. (3), unusual infeedback control.Feedback-induced phase transition.—We show the

existence of FPT with a controllable critical exponent bylinearizing Eq. (1) and assuming the linear feedbackF ½xoutθ ¼ xoutθ . Using the bosonization by Holstein-Primakoff representation [35] Sz ¼ b†b − N=2, S− ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN − b†b

pb, Sþ ¼ b†

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN − b†b

p, Sx ¼ ðSþ þ S−Þ=2, we

get

H ¼ δa†aþ ωRb†bþ ðb† þ bÞ½gðaþ a†Þ þ GIðtÞ: ð4ÞThe bosonic operator b reflects linearized spin(Sx ≈

ffiffiffiffiN

pX), and the matter quadrature is X ¼ ðb† þ bÞ=2.

Weak measurements constitute a source of competitionwith unitary dynamics [74,75,90,91], which is well seen inquantum trajectories formalism [6,92–97], underlining thedistinction between measurements and dissipation. Thusthey can affect phase transitions, including the many-bodyones [74,98,99]. Feedback was mainly considered forstabilizing interesting states [18–22,100]. Here, we focuson the QPT it induces. In this formalism, the operatorfeedback signal IðtÞ in Eq. (4) takes stochastic valuesIcðtÞ conditioned on a specific set (trajectory) of mea-surement results hxθicðtÞ [7]: IcðtÞ ¼

ffiffiffiffiffi2κ

p Rt0 hðt − zÞ×

½ ffiffiffiffiffi2κ

p hxθicðzÞ þ ξðzÞdz, where ξðtÞ is white noise,hξðtþ τÞξðtÞi ¼ δðτÞ. The evolution of a conditionaldensity matrix ρc is then given by [7] dρc ¼ −i½H; ρcdtþD½aρcdtþH½aρcdW, where D½aρc ¼ 2κ½aρca†−ða†aρc þ ρca†aÞ=2, H½aρc ¼

ffiffiffiffiffi2κ

p ½ae−iθρc þ ρca†eiθ−Trðae−iθρc þ ρca†eiθÞρc, dW ¼ ξdt. In general, averagingsuch a stochastic master equation over trajectories does notnecessarily lead to the master equation for unconditionaldensity matrix ρ used to describe DPTs.Figure 2 compares trajectories for the spin quadrature

hXic at various feedback constants G and s, Eq. (3).Crossing the FPT critical pointGcrit, the oscillatory solutionchanges to exponential growth. For large s (nearly instantfeedback), there is a frequency decrease before FPTand fastgrowth above it. For small s (long memory), before FPTtrajectories become noisier; the growth above it is slow.Note, that even though the trajectories are stochastic, theirfrequencies and growth rates are the same for all exper-imental realizations.To get insight, we proceed with a minimal model

necessary for FPT and adiabatically eliminate the lightmode from Eq. (2): a ¼ ð−2igX þ faÞ=ðκ þ iδÞ. Thiscorresponds well to experiments [42,43,62], where κ

(∼MHz) exceeds other variables (∼kHz). The Heisenbergequations for two matter quadratures then combine to asingle equation describing matter dynamics:

X þω2R −

4ωRg2δκ2 þ δ2

X

−4ωRGgκκ2 þ δ2

Cθ

Zt

0

hðt − zÞXðzÞdz ¼ FðtÞ; ð5Þ

whereCθ ¼ δ cos θ þ κ sin θ. Here the frequency shift is dueto spin-light interaction, the last term originates from thefeedback. The steady state of Eq. (5) is hXi ¼ 0, whichlooses stability if the feedback strength G > Gcrit.Note, that oscillations below Gcrit are only visible at

quantum trajectories for conditional hXic (Fig. 2). They arecompletely masked in the unconditional trivial solutionhXi ¼ 0. Thus, feedback can create macroscopic spincoherence hXic ≠ 0 at each single trajectory (experimentalrun) even below threshold. This is in contrast to dissipativesystems, where the macroscopic coherence is attributed tohXi ≠ 0 above the DPT threshold only.The noise operator is FðtÞ ¼ −ωR½gfa þGðκ −

iδÞe−iθ R t0 hðt − zÞfaðzÞdz=2=ðκ þ iδÞ þ H:c: It has the

following correlation function:

hFðtþ τÞFðtÞi

¼ ω2Rκ

2ðκ2 þ δ2Þ4g2δðτÞ þG2ðκ2 þ δ2Þ

Zt

0

hðzÞhðzþ τÞdz

þ 2gG½ðκ − iδÞe−iθhðτÞ þ ðκþ iδÞeiθhð−τÞ: ð6Þ

FIG. 2. Feedback-induced phase transition at a single trajectory.Conditional quadrature hXic. For long-memory feedback [smalls ¼ 0.5, panel (a)], approaching the transition at G ¼ Gcrit, theoscillatory trajectory becomes noisier and switches to slowgrowth. For fast feedback [large s ¼ 20, panel (b)], the oscillationfrequency decreases (visualizing mode softening), and switchesto fast growth. Even though the trajectories are stochastic, theirfrequencies and growth rates are the same for all experimentalrealizations. g ¼ ωR, κ ¼ 100ωR, δ ¼ ωR, ωRt0 ¼ 1. hð0Þ ¼ sgives the same Gcrit for all hðtÞ. Note the different scales on timeaxes.

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We thus readily see how the feedback leads to the non-Markovian noise in spin dynamics.Performing the Fourier transform of Eq. (5), one gets

DðωÞXðωÞ ¼ FðωÞ, with the characteristic polynomial

DðωÞ ¼ ω2 − ω2R þ 4ωRg2δ

κ2 þ δ2þ 4ωRGgκ

κ2 þ δ2CθHðωÞ; ð7Þ

where X, F, andHðωÞ are transforms of X, F, and hðtÞ. Thespectral noise correlation function is hFðωÞFðω0Þi ¼SðωÞδðωþ ω0Þ with

SðωÞ ¼ πω2Rκ

κ2 þ δ2j2gþ Gðκ − iδÞe−iθHðωÞj2; ð8Þ

whose frequency dependence again reflects the non-Markovian noise due to the feedback.Even a simple feedback acting on spins leads to rich

classical dynamics [101]. Here we focus on the quantumcase, but only for a simple type of phase transitions, wherethe eigenfrequency ω approaches zero [36] (“mode soft-ening,” visualized in quantum trajectories in Fig. 2). Fromthe equation DðωÞ ¼ 0 we find the FPT critical point forthe feedback strength:

GcritHð0Þ ¼ 1

4gκCθ½ωRðκ2 þ δ2Þ − 4g2δ; ð9Þ

where Hð0Þ ¼ R∞0 hðtÞdt. Without feedback (G ¼ 0) this

gives very large gcrit for LMG and Dicke transitions[34,35]. Thus, feedback can enable and control thesetransitions, even if they are unobtainable because of largedecoherence κ or small light-matter coupling g.Quantum fluctuations and critical exponent.—We now

turn to the quantum properties of FPT driven by themeasurement-induced noise FðtÞ, Eq. (5). While themean-field solution is hXi ¼ 0 below the critical point,hX2i ≠ 0 exclusively due to the measurement fluctuationsand can serve as an order parameter. From DðωÞXðωÞ ¼FðωÞ and noise correlations we get hXðtþ τÞXðtÞi ¼R∞−∞ SðωÞeiωτ=jDðωÞj2dω=ð4π2Þ, giving hX2i for τ ¼ 0.To find the FPT critical exponent α we approximate the

behavior near the transition point as hX2i ¼ A=j1 −G=Gcritjα þ B, where A;B ¼ const. Figure 3 demonstrates thatthe feedback can control the quantum phase transitions.Indeed, it does not only define the mean-field critical point(9), but enables tuning the critical exponent as well.Varying the parameter s of feedback response hðtÞ,Eq. (3) allows one changing the critical exponent in abroad range. This corresponds to varying the length of theeffective spin-spin interaction mentioned above. For hðtÞEq. (3), its spectrum is expressed via the exponentialintegral HðωÞ ¼ hð0Þt0e−iωt0Esþ1ðiωt0Þ. At small frequen-cies its imaginary part behaves as ωs for s < 1, resemblingthe spectral function of sub-Ohmic baths. For large s, αapproaches unity, as hðtÞ becomes fast and feedback

becomes nearly instant, such as interactions in openLMG and Dicke models, where α ¼ 1 [34,35,102].Note that a decaying cavity is well known to produce the

autonomous exponential feedback [56] hðtÞ ¼ expð−κ0tÞ[HðωÞ ¼ 1=ðiωþ κ0Þ] crucial in many fields (e.g., lasers,cavity cooling, optomechanics, etc.). Such a simpleHðωÞ isnevertheless insufficient to tune the critical exponent andmeasurement-based feedback is necessary.The linearized model describes FPT near the critical

point, but it does not give a new steady state. The spinnonlinearity can balance the system (cf. Ref. [62]).However, the feedback with nonlinear F ½xoutθ can assurea new steady state even in a simple system of linearquantum dipoles (e.g., for far off-resonant scattering withnegligible upper state population). It is thus the nonlinearityof the full hybrid quantum-classical system that is crucial.Relation to other models.—Feedback control of QPTs

enables simulating models similar to those for particle-bath interactions, e.g., spin-boson (SBM), Kondo, Caldeira-Leggett (CLM), quantum Brownian motion models(cf. Ref. [62]). They were applied to various systems fromquantum magnets to cold atoms with various spectralfunctions [31–37]. Creating a quantum simulator, whichis able to model various baths in a single device, ischallenging, and proposals include, e.g., coupling numerouscavities or creating complex networks simulating multimodebaths [82–84]. In contrast, the feedback approach is moreflexible as tuning hðtÞ of a single classical loop is feasible.E.g., for BEC [62], the typical frequencies are in the kHzrange, which is well below those of modern digital process-ors reaching GHz. Moreover, it can be readily extended forsimulating a broader class of quantum materials and qubitswith nonlinear bath coupling [103] and multiple baths [80].Themulti- (or large-) spin-bosonmodels [34–36,104,105]

are based on Eq. (1) with sum over continuum of bosonicmodes ai of frequencies δi distributed according to thespectral function JðωÞ [62]. The feedback model reproducesexactly the form of bath dynamical equations for Sx;y;z[cf. Eq. (5) for linearized, and Ref. [62] for nonlinearversions] if ℑHðωÞ ∼ JðωÞ − Jð−ωÞ. The noise correla-tion function of linear CLM is hFðω0ÞFðωÞi ¼4πω2

RJðωÞδðωþ ω0Þ, whereas the feedback model containsHðωÞ and additional light-noise term in Eq. (8).

(a) (b)

FIG. 3. Feedback control of critical exponent. (a) Growingfluctuations of unconditional matter quadrature hX2i for variousfeedback exponents s. (b) Dependence of critical exponent α onfeedback exponent s, proving opportunity for QPT control.g ¼ ωR, κ ¼ 100ωR, δ ¼ ωR, hð0Þ ¼ s, ωRt0 ¼ 1.

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In bath models there is a delicate point of the frequencyωR renormalization (“Lamb shift”) [32,34–36]. It may leadto divergences and necessity to repair the model [106]. Thefeedback approach is flexible. The frequency shift inEq. (7) is determined by GHð0Þ ¼ G

R∞0 hðtÞdt and can

be tuned and even made zero, if hðtÞ changes sign.In summary, we have shown that feedback does not only

lead to phase transitions driven by quantum measurementfluctuations, but controls its critical exponent as well. Itinduces effects similar to those of quantum bath problems,allowing their realization in a single setup, and enablesstudies of time crystals and Floquet engineering with long-range (long-memory) interactions. The applications canalso include control schemes for optical informationprocessing [107]. Experiments can be based on quantummany-body gases in a cavity [42–46,100], and circuit QED,where ultrastrong coupling has been obtained [60,61] oreffective spins can be considered [58,59,82]. Feedbackmethods can be extended by, e.g., measuring severaloutputs [15,22,108] (enabling simulations of qubits andmultibath SBMs [80] with nonlinear couplings [103]) orvarious many-body atomic [71,73,75,100] or molecular[109] variables.

We thank Ph. Joyez, A. Murani, and D. Esteve forstimulating discussions. Figures are prepared using MSPowerPoint. Support by RSF (17-19-01097), RFBR (18-02-01095), DGAPA-UNAM (IN109619), CONACYT-Mexico (A1-S-30934), EPSRC (EP/I004394/1), UPSay(d’Alembert Chair).

Note added in the proof.—Recently, the first experiment,where our predictions can be tested was reported inRef. [110].

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010603-7

Feedback-Induced Quantum Phase Transitions Using Weak Measurements -Supplemental Material

D. A. Ivanov,1 T. Yu. Ivanova,1 S. F. Caballero-Benitez,2 and I. B. Mekhov1, 3, 4

1Department of Physics, St. Petersburg State University, St. Petersburg, Russia2Instituto de Fısica, Universidad Nacional Autonoma de Mexico, Ciudad de Mexico, Mexico

3Department of Physics, University of Oxford, Oxford, United Kingdom4SPEC, CEA, CNRS, Universite Paris-Saclay, CEA Saclay, Gif-sur-Yvette, France

I. FEEDBACK CONTROL OF QUANTUM PHASE TRANSITIONS IN ULTRACOLD GASES

The model presented in the paper can be realized using light scattering from the Bose-Einstein condensate (BEC).Two spin levels will correspond to two motional states of ultracold atoms. A very high degree of the light-matterinteraction control has been achieved in several systems, where BEC was trapped in an optical cavity, and the Dickeand other supersolid-like phase transitions were obtained [1–3]. Experiments now include strongly correlated bosonsin an optical lattice inside a cavity [4, 5] and related works without a cavity [6]. Here we propose one realization andunderline that setups can be flexible and extendable for other configurations as well.

We consider a BEC, elongated in the x direction, illuminated by the pump (Fig. S1). The quadrature of scatteredlight is measured and used for the feedback signal. The feedback is provided by the external trapping potential in theform of a standing wave, whose depth is varied according to the feedback signal: V0(t) cos2 k0x (k0 is the wave vectorof laser beam creating the potential). The many-body Hamiltonian has a form (cf. review [7], we work in the units,where h = 1):

H =∑l

ωla†l al +

∫ L

0

Ψ†(x)Ha1Ψ(x)dx+∑l

ζl(a†l + al), (S1)

where al are the annihilation operators of light modes of frequencies ωl interacting with a BEC, ζl are the pumps ofthese modes (if they are shaped by cavities), Ha1 is the single-atom Hamiltonian, Ψ(x) is the atom-field operator,and L is the BEC length. For the far off-resonant interaction, Ha1 is determined by the interference terms betweenthe light fields present [7]:

Ha1 =p2

2ma+ V0(t) cos2 k0x+

1

∆a

∑l,m

glu∗l (x)a†l gmum(x)am, (S2)

where the first term is atomic kinetic energy operator (p2 = d2/dx2, ma is the atom mass), ∆a is the detuning betweenlight modes and atomic transition frequency, gl,m is the light-matter coupling constants. ul,m are the geometricalmode functions of light waves, which can describe pumping and scattering at any angles to the BEC axis [8, 9], whichmaybe convenient depending on the specific experimental realization.

Here, to strongly simplify the consideration, we select the following geometry of light modes (cf. Fig. S1). Thepump is orthogonal to the BEC axis, thus its mode function is constant along x (can be chosen as u(x) = 1); itsamplitude apump is considered as a c-number. A single scattered light mode a1 is non-negligible along x direction, andits mode function is u1(x) = cos k1x, where k1 is the mode wave vector. There is no direct mode pumping, ζ1 = 0.

Feedback

h(t)

GI(t)

pump

diffracted light detector

potential V0(t)

FIG. S1. Setup. A BEC is illuminated by the transverse pump, the scattered (diffracted) light is detected, feedback acts onthe system via the change of the external periodic potential depth V0(t). Feedback provides the non-Markovianity, nonlinearity,and noise, necessary for the controllable quantum phase transition. Importantly, the feedback response h(t) is tunable.

2

The condition k0 = k1/2 assures the maximal scattering of light into the mode a1 (diffraction maximum). Thus, thepump diffracts into the mode a1 from the atomic distribution. Thus, the wavelength of feedback field should be twicethe mode wavelength. The matter-field operator can be decomposed in two modes [10, 11]:

Ψ(x) =1√Lc0 +

√2

Lc1 cos k1x, (S3)

where c0,1 are the annihilation operators of the atomic waves with momenta 0 and k1 (c†0c0 + c†1c1 = N , N is theatom number). Substituting Eqs. (S3) and (S2) in Eq. (S1) and neglecting in a standard way several terms [1, 10–12](which however may appear to be important under specific conditions and thus enrich physics even further), we getthe Hamiltonian (1) in the main text

H = δa†1a1 + ωRSz +2√NSx[g(a1 + a†1) +GI(t)] (S4)

with the following parameters: δ = ω1 − ωpump + Ng21/(2∆a) (the detuning between mode and pump including the

dispersive shift), ωR = k21/2ma is the recoil frequency, g = Ωpumpg1√N/2/∆a with Ωpump = gpumpapump being the

pump Rabi frequency. The feedback signal is GI(t) =√N/8V0(t). The spin operators are Sx = (c†1c0 + c1c

†0)/2,

Sy = (c†1c0− c1c†0)/(2i), Sz = (c†1c1− c

†0c0)/2. The main characteristic frequency of this system is the recoil frequency

that for BEC experiments is ωR = 2π ·4kHz. This makes the feedback control feasible, as the modern digital procossingof the feedback signal can be much faster (up to the GHz values). Other experimental parameters such as δ and g(depending on the pump amplitude) can be tuned in the broad range and, in particular, be close to kHz values. Thecavity decay rate in Ref. [1] is κ = 2π · 1.3 MHz, which is much greater than kHz making the adiabatic eliminationof the light mode to work very well.

There are other configurations relevant to our work. For example, for the BEC in a cavity setup [1], instead ofcreating the external potential V0 with k0 = k1/2, one can inject the feedback signal directly through the cavitymirror as the pump ζ1(t). In this case, an additional laser with doubled wavelength is not necessary. While theHamiltonian Eq. (S4) will be somewhat different, after the adiabatic elimination of light mode, the equation for thematter quadrature operator X [Eq. (5) in the main text] will be the same with ζ1(t) ∼ I(t).

Another possibility is to use atoms tightly trapped in optical lattices [4, 5]. In this case, instead of expanding Ψ(x)in the momentum space, a more appropriate approach is to expand it in the coordinate space using localized Wannierfunctions [7]. The spin operators will then be represented by sums of on-site atom operators Sx ∼

∑iAini, or the

bond operators representing the matter-wave interference between neighboring sites Sx ∼∑iAib

†i bj [13, 14]. For

example, the effective spin can correspond to the atom number difference between odd and even sites [for Ai = (−1)i][15], represent the magnetization [15] or staggered magnetization [16] of fermions, etc. Such a strong flexibilityin choosing the geometrical combination of many-body variables combined in the effective spin operator enablesdefining macroscopic modes of matter fields [17] and assures the competition between the long-range (but structuredin space with a short period comparable to that of the lattice) light-induced interactions and short-range atom-atominteractions and tunneling on a lattice [15]. This will open the opportunities for the competition between the nontrivialfeedback-induced interactions and many-body atomic interactions.

We belive that our proposal will extend the studies in the field of time crystals [18–22] and Floquet engineering[23]. The time crystals is a recently proposed notion, where phenomena studied previously in space (e.g. spin chains,etc.) are now studied in time. Typically, the system is subject to the external periodic modulation of a parameter[e.g. periodic g(t)], which is considered as creating a “lattice in time”. Our approach makes possible introducing theeffective “interaction in time.” This makes the modulation in the system not prescribed, but depending on the stateof the system (via Sx). This resembles a true lattice in space with the interaction between particles. In other words,our model enables not only creating a “lattice in time,” but introducing the tunable “interaction in time” to such alattice (without the necessity of having the standard particle-particle interaction in space).

II. BATH MODELS

The Hamiltonian of the spin-boson model at zero temperature [24] extended for N spins is

H =∑i

δia†iai + ωRSz +

2√NSx∑i

gi(a†i + ai). (S5)

3

It describes the interaction of spins with many (continuum) bosonic modes ai of frequencies δi. The correspondingHeisenberg-Langevin equations are then given by

ai = −iδiai − i2gi√NSx,

Sx = −ωRSy,

Sy = ωRSx −2√NSz∑i

gi(a†i + ai),

Sz =2√NSy∑i

gi(a†i + ai). (S6)

Two equations for Sx and Sy can be joined: Sx = −ω2RSx + 2ωRSz/

√N∑i gi(a

†i + ai). The modes ai can be

formally found from the first Eq. (S6): ai(t) = ai(0)e−iδit − 2igi/√N∫ t0Sx(τ)e−iδi(t−τ)dτ . This can be combined to∑

i gi(a†i + ai) = −4

∫ τ0β(t− τ)Sx(τ)dτ/

√N − Fb(t)/ωR, where

β(t) =∑i

g2i (δi) sin(δit) =1

π

∫ ∞0

J(ω) sinωtdω, (S7)

and the bath spectral function is

J(ω) = π∑i

g2i (δi)δ(ω − δi). (S8)

The noise operator Fb(t) is determined by random initial values of the bosonic mode operators ai(0): Fb(t) =

−ωR∑i gi[ai(0)e−iδit + a†i (0)eiδit].

At the level of Heisenberg-Langevin equations, the linearized system can be obtained by assuming Sz = −N/2,

Sx =√NX, and Sy =

√NY . Alternatively, it can be obtained from the Hamiltonian (S5) using the Holstein-

Primakoff representation as explained in the main text. The bosonized Hamiltonian then reads

H =∑i

δia†iai + ωRb

†b+ (b† + b)∑i

gi(a†i + ai), (S9)

while the linearized equation for Sx is reduced to the equation for the particle quadrature X:

X + ω2RX − 4ωR

∫ t

0

β(t− z)X(z)dz = Fb(t). (S10)

Such Hamiltonian and operator equation correspond to the quadrature-quadrature coupling model and differs fromthe Caldeira-Leggett model only by the renormalized spin frequency ωR [25]. They describe the quantum Brownianmotion as well [25]. Equation (S10) has a structure identical to Eq. (5) obtained for the feedback model in themain text. The additional frequency shift in the case of feedback can be compensated either by modifying the spinfrequency, or by adding the δ(t)-term in the feedback response function h(t).

The bath spectral function J(ω) (S8) is usually approximated as J(ω) = κR(ω/ωc)sPc(ω), where ωc is the cut-off

frequency and Pc(ω) is the cut-off function. For s = 1 the bath is called Ohmic, for s < 1 it is sub-Ohmic, and fors > 1 it is super-Ohmic. This corresponds to the bath response function asymptotically behaving as 1/ts+1 for largetimes [26].

Taking the Fourier transform of the differential equation (S10) one gets(ω2 − ω2

R + 4ωRB(ω))X(ω) = −Fb(ω), (S11)

where B(ω) is the Fourier transform of β(t). The spectral noise correlation function reads

〈Fb(ω′)Fb(ω)〉 = 4πω2RJ(ω)δ(ω + ω′). (S12)

The time noise correlation function is

〈Fb(t+ τ)Fb(t)〉 =ω2R

π

∫ ∞0

J(ω)e−iωτdω =ω2R

π

[∫ ∞0

J(ω) cosωτdω − i∫ ∞0

J(ω) sinωτdω

]. (S13)

A more standard way to write the differential equation is not via β(t), but via γ(t):

X + ω2R

(1− 2γ(0)

ωR

)X + 2ωR

∫ t

0

γ(t− z)X(z)dz = Fb(t), (S14)

4

0 1 2 3(G Gcrit)/Gcrit

0.0

0.1

0.2

0.3

0.4

S x

FIG. S2. Feedback-induced phase transition for a single spin. δ = ωR, g = 0.1ωR, h(0) = s, s = 1, κ = 10ωR, ωRt0 = 1.

where γ(t) = −2β(t),

γ(t) =2

π

∫ ∞0

J(ω)

ωcosωτdω. (S15)

The frequency shift can be incorporated in the renormalized spin frequency. For the Ohmic bath, γ(t) ∼ δ(t), andthe differential equation is reduced to that for a damped harmonic oscillator.

III. FEEDBACK-INDUCED PHASE TRANSITION WITH NONLINEAR SPINS

In the main text, we presented the properties of the phase transition using the linearized model with Hamiltonian(4). Such a linear model cannot give us the value of the stationary state above the critical point. Here we show theresults of numerical simulations for a single nonlinear spin. The original Hamiltonian (1) is

H = δa†a+ ωRSz +2√NSx[g(a+ a†) +GI(t)]. (S16)

The Heisenberg-Langevin equations are then given by

a = −iδa− i 2g√NSx − κa+ fa,

Sx = −ωRSy,

Sy = ωRSx −2√N

[g(a+ a†) +GI(t)]Sz,

Sz =2√N

[g(a+ a†) +GI(t)]Sy. (S17)

Figure 2 shows the results of numerical simulations for the expectation value of the component 〈Sx〉 of a singlespin (N = 1). It shows the phase transition with 〈Sx〉 = 0 below Gcrit and 〈Sx〉 6= 0 above the critical point. This issimilar to the spin-boson model, where 〈Sx〉 6= 0 corresponds to the localization, while 〈Sx〉 = 0 corresponds to thedelocalized phase.

Such solutions can be obtained by calculating the conditional expectation values 〈Sx(t)〉c and then averaging themover multiple quantum trajectories, which gives the stationary unconditional expectation value 〈Sx〉. In general, theclassical measurement and feedback loop produce a single quantum trajectory. Averaging over many trajectoriesreproduces the results of quantum Heisenberg-Langevin equations (the equivalence between the quantum trajectoryand quantum Heisenberg-Langevin approaches is demonstrated for feedback e.g. in Ref. [27]). In turn, similarHeisenberg-Langevin approach describes the interaction of a particle with quantum baths as well.

Similarly to the bath problem, the equations for Sx and Sy can be joined: Sx = −ω2RSx + 2ωRSz/

√N [g(a† + a) +

GI(t)]. The light mode can be adiabatically eliminated from the first equation (S17) to give g(a† + a) + GI(t) =

−[4g2δSx + 4gκGCθ∫ t0h(t − τ)Sx(τ)dτ ]/[

√N(κ2 + δ2)] − F (t)/ωR, where the notations and noise correlations are

5

given in the main text. These expressions have the same structure as those in the bath models. Various methodsto treat the quantum Heisenberg-Langevin equations have been developed in quantum optics [28–30]. The linearized

equation for Sx then reduces to Eq. (5) of the main text for the particle quadrature X (Sx =√NX):

X +

(ω2R −

4ωRg2δ

κ2 + δ2

)X − 4ωRGgκ

κ2 + δ2Cθ

∫ t

0

h(t− z)X(z)dz = F (t), (S18)

which again has the same structure as the equation for X in the bath model. The additional frequency shift in the caseof feedback can be compensated either by modifying the spin frequency, or by adding the δ(t)-term in the feedbackresponse function h(t).

Finally, we would like to comment on the role of limited detector efficiency η < 1. In the Heisenberg-Langevinapproach, it can be taken into account by introducing an additional light noise corresponding to undetected photonsleaked from the cavity, while the noise fa will still correspond to the detected photons (cf. Ref. [31]). The characteristicequation (7) in the main text will then take exactly the same form with G replaced by

√ηG. Therefore, concerning

the position of the critical point, the limited efficiency can be compensated by the increase of the feedback coefficientG. Indeed, when the detector efficiency approaches zero (η = 0) such that it can not be compensated in a real system,the role of feedback (detected photons) vanishes, and the feedback-induced phase transition reduces to a standarddissipative phase transition due to the undetected photons (i.e. the dissipation).

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