Driven-dissipative control of cold atoms in tilted optical lattices
DDriven-dissipative control of cold atoms in tilted optical lattices
Vaibhav Sharma ∗ and Erich J Mueller † Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York (Dated: January 19, 2021)We present a sequence of driven-dissipative protocols for controlling cold atoms in tilted opticallattices. These experimentally accessible examples are templates that demonstrate how dissipationcan be used to manipulate quantum many-body systems. We consider bosonic atoms trapped in atilted optical lattice, immersed in a superfluid bath, and excited by coherent Raman lasers. Withthese ingredients, we are able to controllably transport atoms in the lattice and produce self-healingquantum states: a Mott insulator and the topologically ordered spin-1 AKLT state.
I. INTRODUCTION
While dissipation is often viewed as a hindrance, itcan also be a tool for manipulating quantum states.Here we provide a series of cold atom examples of usingdissipation to prepare quantum states, and control theirbehavior. This approach complements other methodsof state preparation, and avoids some of the challengesthose techniques face. By giving specific protocols fordissipative state preparation and control, we are able toelucidate the underlying principles and directly confrontthe advantages and disadvantages of this approach.In cold atom experiments, dissipation is provided byseveral mechanisms: off-resonant light scattering, three-body atom loss, collisions with background atoms, andeven the conversion of coherent excitations into inco-herent ones through elastic collisions of atoms in thesample. Traditional cooling schemes have long reliedon these processes for equilibration [1], but the ideaof engineering dissipation to target specific many-bodyquantum states is more novel [2]. In the few-particlelimit, there are more established examples, such as op-tical pumping. The driven-dissipative approach to statepreparation can be viewed as a many-body analog of op-tical pumping.This method offers a practical alternative to othertechniques of state preparation, and overcomes some oftheir difficulties. For instance, adiabatic state prepara-tion requires the process to be much slower than themany-body gap [3]. This poses particular challenges ifone needs to traverse a critical region, where the gapvanishes. Another common state preparation techniqueis Hamiltonian engineering where the desired state isthe ground state of some fixed Hamiltonian which canbe reached by cooling. Obstacles to Hamiltonian engi-neering include: (1) The required temperatures may beunachievably low, and (2) the equilibration rates maybecome small as one approaches the state of interest.An example is the Fractional Quantum Hall state, whereparticles avoid one-another, and hence the elastic colli-sion rate is small. In all of these techniques, the key ∗ [email protected] † [email protected] questions are: (1) Can you produce the state of inter-est, and (2) How long does it take? We answer boththese questions for our examples.Dissipative state preparation has the potential tobe fast – the timescale is experimentally controllableand the kinetic bottlenecks can be explicitly removedby properly engineering the environment. The criticalslowing down that plagues adiabatic methods is com-pletely avoided: one is far from equilibrium throughoutthe time evolution. Most importantly, the final state isself-correcting. If one leaves on the dissipation (perhapswith some reduced amplitude) any perturbations canbe healed. This has connections to autonomous errorcorrecting codes: the dissipatively stabilized quantumstate is a protected resource for manipulating quantuminformation.Controlled driven-dissipative state preparation is anactive area of research both theoretically and experi-mentally. For example, in superconducting qubits, dis-sipation has been used to create a stable Mott insu-lator of photons [4] and a long lived two qubit Bellstate [5]. Reservoir engineering has been used in trappedion systems to create a four qubit GHZ state [6]. Ul-tracold atoms can be promising candidates for dissipa-tive state preparation. The Mott-superfluid transitionin a driven-dissipative Bose-Hubbard system has beenrealized experimentally [7]. Some theoretical propos-als include preparation of number and phase squeezedbosonic states [8] and dissipatively prepared topologicalsuperconductors [9].In this work, we analyze a broadly applicable ap-proach to driven-dissipative control in cold atoms, whichboth extends these examples and provides an experi-mentally accessible framework for exploring the generalprinciples. Our setup is schematically shown in Fig. 1.We consider Li atoms in a “tilted” one dimensional op-tical lattice, modelled by the sum of a linear and sinu-soidal potential. Transitions are driven by two-photonRaman processes, which use an electronically excitedstate as an intermediary in changing the spatial modeof an atom. Dissipation is provided by coupling to asuperfluid bath of Na atoms which are not trappedby the lattice. Lithium atoms can decay from excitedvibrational states to lower ones by emitting Bogoliubovexcitations in the superfluid bath. We note that all thedifferent parts of our proposed process have been exper- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Bath
FIG. 1. (color online) Tilted optical lattice immersed in asuperfluid bath. The yellow beams are Raman lasers whichcoherently transfer atoms from the ground state on one siteto an excited state on a neighboring site. The lattice atoms(in blue) irreversibly transition from the higher to lower bandby giving energy to the bath atoms (in green). imentally demonstrated in other works [10–14]. Simi-larly, these ingredients have been studied in a numberof theory works [15, 16]. We explicitly calculate all therelevant energy and time scales and quantitatively showhow the drive and dissipation can produce our targetstates.We give three examples to demonstrate the capabili-ties of our driven-dissipative scheme. First, we show away to control atom transport along a one dimensionaltilted optical lattice potential, both up and down thelattice. This would be relatively simple to implementand an excellent target for initial experiments. Addi-tionally, it provides a way to fully control the speedof atom transport in a lattice. Second, we show howit can be further modified to drive the system into astrongly-correlated Mott insulator state in a tilted lat-tice with a mechanism to self-heal any holes. Finally, wepropose how the iconic AKLT (Affleck-Kennedy-Lieb-Tasaki) state can be achieved with this technique.The AKLT state occurs in a spin-1 chain. It hasunique properties, including symmetry protected topo-logical order and valence bond order. It has emergentspin-1/2 edge modes and a gapped ground state. It isalso a prototypical example of a matrix product state.The AKLT state has never been created experimentallyin cold atom systems. One of our central results is aprotocol to create it in a driven-dissipative ultra coldatomic system.Our paper is organized as follows. In Section II weintroduce the physical system and derive the effectivemodel which we will use to describe it. This includesshowing how the driving and dissipation processes areengineered. In Section III, we describe our protocols andcalculate state preparation time, including the scalingwith system size.
II. MODELA. System and Effective Model
As already introduced, our basic setup is shown inFig. 1. For each of our examples, we consider two popu-lations of bosons, referred to as the lattice and the bathatoms. The lattice atoms are constrained to move inone dimension (1D). They experience a “tilted lattice”potential along that direction, consisting of the super-position of sinusoidal and linear potentials. The bathatoms form a 3D cloud. There are also a series of con-trol lasers that are used to drive Raman transitions.Each of these components are discussed in detail below.In its simplest incarnation, the resulting effectivemodel has the structure of a 1D chain of sites with a lin-ear potential gradient. Each site j contains two states:The ground state | g (cid:105) j and a vibrationally excited state | e (cid:105) j , with an energy gap, (cid:126) ω between them.ˆ H = (cid:88) j − j ∆ | g (cid:105) j (cid:104) g | j + ( − j ∆ + (cid:126) ω ) | e (cid:105) j (cid:104) e | j . (1)Additionally, there is an onsite interaction Hamiltonian,ˆ H int = U gg | gg (cid:105)(cid:104) gg | + U ge | ge (cid:105)(cid:104) ge | + U ee | ee (cid:105)(cid:104) ee | , (2)and coherent drives, H coh = Ω (cid:48) e − iωt | g (cid:105) j (cid:104) e | j +1 + h.c. (3)Our decay terms are modelled by on-site jump operatorsof the form, ˆ L j = √ Γ | g (cid:105) j (cid:104) e | j . (4)Subsections II B through II F derive this effectivemodel, starting with a microscopic description of thetrapped gas. B. 1D Lattice
The system atoms are trapped in a one dimensionaltilted optical lattice. The lattice is generated by inter-fering two counter-propagating laser beams. The tiltcan be generated by using the AC stark shift from alarge waist laser which is incident on the system from adirection perpendicular to the lattice [10]. The resultingpotential is shown in Fig. 2.When the energy difference between neighboringsites, ∆, is large compared to the tunneling ampli-tude, t , then the eigenstates of the tilted lattice becomestrongly localized to individual sites. If we limit our-selves to nearest neighbor hopping, the exact Wannier-Stark eigenstates are [17] ψ m,α = (cid:88) l J l − m ( t α ∆ ) φ αl , (5) ω Δ t gg FIG. 2. One dimensional tilted optical lattice – schemati-cally showing potential energy vs position, with eigenstatesillustrated by horizontal lines. The tilt per site is ∆, tun-neling within the lowest band is t gg and ω is the bandgapbetween motional bands on every site Here, α is the band index, l , m are site indices, φ αl isthe site localized Wannier function for the α ’th band,and t α is the hopping matrix element for that band. J n ( x ) is the n ’th Bessel function: for small arguments, J n ( x ) ∼ x | n | , and we see that the wavefunction fallsoff exponentially. For a sufficiently tight lattice, onecan approximate the Wannier states, φ αm , as harmonicoscillator eigenstates, with energy E m,α = − m (cid:126) ∆ + ( α + 1 / (cid:126) ω , (6)where ω is the small oscillation frequency. Throughoutwe will only consider two bands, labeled by α = g, e –and regardless of the accuracy of the harmonic approx-imation, we can define (cid:126) ω = E m,e − E m,g .For two particles on a site, the on-shell componentsof the on-site interaction are,ˆ H int = U gg | gg (cid:105)(cid:104) gg | + U ge | ge (cid:105)(cid:104) ge | + U ee | ee (cid:105)(cid:104) ee | (7)where | αβ (cid:105) is the state with particles in bands α and β , and the site index has been suppressed. The U ’sscale as a s /d ⊥ , where a s is the scattering length, and d ⊥ is the transverse size of the Wannier state. If oneapproximates the Wannier states as harmonic oscillatoreigenstates, then U gg can be calculated as, U gg = 8 πa s md ⊥ (cid:90) ∞−∞ dx | w g ( x ) | (8)Here, w g is the ground state harmonic oscillator wave-function. Analogous expressions give the interaction en-ergies involving higher bands, and one finds U ge = U gg and U ee = U gg .We model the lattice potential as V ( r ) = V E rec sin (2 πx/d ), where V is the dimensionless lat-tice strength, d is the lattice beam wavelength, and E rec = (cid:126) π md is the recoil energy. This potential hasa lattice spacing of d/ t gg in the lowest motional band in the harmonic approximation can be written as, t gg = (cid:90) dx w ∗ g ( x + d/
2) ˆ Hw g ( x ) (9) ∼ E rec (cid:112) V e − π √ V (10)The tunneling in the higher band can be similarlyfound to be, t ee ∼ t gg √ (1 + π √ V ). The bandgap be-tween the ground and excited motional band is, (cid:126) ω ∼ E rec (cid:112) V . (11)As a rule of thumb, these approximations work wellwhen V (cid:38)
9. For d = 1064nm, a typical laser wave-length, one then has a tunneling amplitude on the orderof tens to hundreds of Hz, while the band gap betweenthe ground and first band is on the order of tens of kHz.Typical background scattering lengths are of the orderof a few nm for Li atoms [18]. These scattering lengthscan be easily tuned via Feshbach resonances, with thecaveat that one may find enhanced inelastic processesif they are made too large. In our system, we envisionthat the lattice atoms are tightly confined in the trans-verse directions with d ⊥ (cid:28) d and thus the interactionenergy is also on the order of tens of kHz.Such one dimenisonal tilted optical lattices have beenrealized in experiments either using the AC stark shiftgradient from a laser [10] or with a magnetic field gra-dient [11]. C. Coherent Drive
Transitions are driven by a two-photon coherentdrive: the lasers are tuned so that absorbing a pho-ton from one beam, and emitting it into the second isresonant with a band-changing hopping event. For ex-ample, as illustrated in Fig. 3, this Raman process coulddrive the transition from the ground band on one siteto the excited band on a neighboring site, in which case ω − ω = ω − ∆. Here ω = c | k | and ω = c | k | arethe frequencies of the lasers. An incoherent scatteringevent will later return the atom to the ground band.The Raman laser frequencies can be adjusted to bringother possible transitions like hopping between groundbands of neighboring sites into resonance.In our concrete scenario, one laser beam drives theatom in the lowest band in site m to a virtual level, cor-responding to an electronically excited state. This sin-gle photon process is detuned by frequency δ , and hastransition rate Ω . The second laser drives the tran-sition from this virtual level to the motionally excitedstate of site m + 1.If δ (cid:29) Ω , the effective rate of this two photon processcan be calculated by adiabatically eliminating the higher 𝛿Ω Ω Γ m m+1 FIG. 3. (color online) Coherent Raman transition betweenground band of site m and excited band of site m + 1. Thetwo Raman beams are depicted by black and red arrows la-beled by rates, Ω . The blue curvy arrow shows spontaneousdecay from excited band to ground band through bath in-teractions with a rate, Γ. The Raman transition is detunedby δ from an intermediate electronic excited state. electronic level and is given by:Ω (cid:48) = Ω δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dx ψ ∗ m,g ( x ) e i ( k − k ) x ψ m +1 ,e ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (12) ∼ Ω δ (cid:18) π √ V (cid:19) e − π √ V . (13)The latter result is derived in Appendix A, in the deeplattice limit, with the angle between the Raman beamschosen to optimize the transition rate.To make unwanted transitions highly off-resonant, wewant to be in the regime where Ω (cid:48) (cid:28) ∆ , ω (which aspreviously explained are on the order of 10 Hz).These conditions would ensure that our drive inducescoherent Rabi oscillations of atom transfer between ad-jacent sites. A similar coherent atom transfer process ina Wannier-Stark ladder has been experimentally demon-strated by Beaufils et al. [12]. They were able to achieveRaman transition rates larger than needed for our pro-posal.
D. Dissipation Medium
Dissipation is provided by scattering from a particlebath of superfluid bosons that do not feel the opticallattice [15]. An atom in the higher band can transitionto the ground band through spontaneous emission of aBoguliobov excitation in the superfluid bath. This spon-taneous decay process is shown schematically in Figs. 1and 3.The temperature of the superfluid bath is muchsmaller than ω , so the bath of Bogoliubov particlescan effectively be taken as T = 0: These excitations canbe created, but are never absorbed.Fermi’s golden rule can be used to calculate the re-sulting decay rate from the motionally excited state tothe ground state. Following the arugment of Griessner et al. [15], we find the decay rate in our system to be,Γ = 8 πρ b a ab (cid:112) E rec. V / m / a e − π mr √ V d ⊥ d f ( m r ) f ( m r ) = (1 + m r ) m r ( √ π erf( √ m r ) − √ m r e − m r )(14)Here, m b and m a are the masses of the bath atoms andlattice atoms respectively, and m r = m b m a is their ratio.The interactions between the lattice and bath atoms areparameterized by the s-wave scattering length, a ab . Thenumber density of bath atoms is ρ b , and erf( x ) is theerror function.The rate can be tuned by changing the superfluid bathdensity, scattering length and atomic mass ratios. Thefirst of these is experimentally the most accessible – butFeshbach resonances can be used to control a ab .The rate strongly depends on the mass ratio, m r . For m r (cid:28) f ( m r ) scales as √ m r while for m r (cid:29)
1, therate exponentially decays as e − π mr √ V d ⊥ d . The rateexponentially decreases at larger mass ratios becausethe energy transfer is poorer in collisions with highermass atoms. The optimal mass ratio occurs for m r ∼ √ V d ⊥ d ].For concreteness, we consider the Li atoms in thelattice immersed in a superfluid bath of Na atoms.For a typical bath atom density of 10 cm − and inter-species scattering length on the order of a few nm, thedecay rate is on the order of a few kHz. This rate setsthe time-scale of the experiment. Our coherent trans-fers rely upon resolving the motional sidebands and thelattice tilts, and hence require Γ (cid:28) Ω (cid:48) , ω , ∆ , U .In this setup, the bath atoms do not feel the opticallattice: this can be arranged due to the different ACpolarizabilities of the atomic species. For example, R.Scelle et al. [13] immerse Lithium atoms in a conden-sate of Sodium atoms where only the Lithium atoms aretrapped in a species-specific optical lattice. A detaileddiscussion of techniques to create species-specific opti-cal lattices for various alkali atom mixtures is given byLeBlanc and Thywissen in [16].This decay process can be viewed as a form of sym-pathetic cooling – and has been experimentally studiedby Chen et al. [14] in that context. E. Additional Potentials
For some of our protocols we also add an additionalpotential, which can be created with an off-resonantlaser. In particular, we wish to be able to create a fi-nite length chain as shown in Fig. 4, by adding barriersat the end of the chain. Such potentials are commonlygenerated in experiments [19].
FIG. 4. A finite length wannier-stark ladder with the addi-tion of a uniform repulsive box potential
F. Limitations of Effective Model
An experiment can only be described by the effectivemodel in Eqs. (2) through (4) if ∆ , ω (cid:29) Ω (cid:48) , Γ.The condition ∆ , ω (cid:29) Ω (cid:48) is required so that thedrive does not produce transitions to unwanted sites.For example, the drive inevitably produces a matrix el-ement connecting | g (cid:105) j and | g (cid:105) j +1 . This transition is off-resonant, though, and the rate is suppressed by a factorof Ω (cid:48) /ω relative to the wanted transition. Similarly, thematrix element connecting | g (cid:105) j and | e (cid:105) j is suppressed byΩ (cid:48) / ∆.The condition ∆ , ω (cid:29) Γ is required so that the levelbroadening does not bring any of those same unwantedtransitions into resonance.Note, these conditions puts constraints on the tech-niques which can be used to introduce the dissipation.For example, it would be challenging to design the pro-tocols so that spontaneous emission of a photon wouldprovide the dissipation. The characteristic scales of op-tical processes are much larger than ∆ and ω . Thescales of the Bogoliubov excitations are better-matched. III. PROPOSALS
As already explained, we propose three scenarios:In section III A, we describe a novel transport schemewhere this driven-dissipative approach controls the mo-tion of a cold gas. In section III B, we show a variant ofthe technique that can be used to heal defects in a Mottstate. Finally, in section III C, we explain how to pumpthe system into the AKLT state – a highly nontrivialexample of state engineering.All of these will be described using variants of themodel introduced in Eqs. (2) through (4).
A. Raman Sideband Elevator
Transport in solid state systems is a driven-dissipativeprocess. A potential gradient provides energy to the sys-tem, while inelastic scattering off of impurities acts as aregulator, controlling the average speed of the electrons.We propose constructing the analogous process in ouratomic system. As in the solid-state system, the atomswill move with constant velocity. By changing the inten-sity and frequency of the Raman lasers, one can controlboth the direction and speed of motion, which leads usto refer to this as a “Raman Sideband Elevator.”As depicted in Fig. 3, the coherent Raman drive res-onantly couples atoms from ground band of one site tothe excited band of its nearest neighbor. The atom inthe higher band interacts with the dissipative bath anddecays to the ground band at that site. This irreversiblytransfers the atom one site down the ladder. The pro-cess can then repeat itself. In steady state, all atoms aremoving at a constant speed. This elevator can transferatoms in either direction, depending on the frequenciesof the Raman lasers: when ω − ω = ω − ∆, the transferis down-hill, while when ω − ω = ω + ∆, the transferis uphill. The analysis is identical, and throughout thissection, we use the notation appropriate for down-hilltransport.We will neglect interactions between atoms, and justmodel the single-particle problem. The weakly inter-acting regime is readily reached by either reducing thestrength of transverse confinement or using a Feshbachresonance.Given that the single-particle eigenstates are localizedon each site, this situation can be modelled via a clas-sical rate equation: the only relevant variable is n i , theexpectation value of the number of particles on site i ,and dn i dt = ˜Γ n i − − ˜Γ n i . (15)The rate ˜Γ is found by modeling the three levels involvedin the transport of atoms by one site. Under the regimewhere the driving rate is greater than the decay rate,that is, Ω (cid:48) (cid:29) Γ, one finds ˜Γ = Γ /
2. The Raman laserscause the atom to execute many Rabi oscillations, soit spends half of its time in the unstable state – whichdecays with rate Γ.These rate equations can be solved exactly to deter-mine the speed at with which the center of mass movesdown-hill and how much the cloud spreads over time.Using Eq. (15), the rate of change of the center of massposition of the atom cloud, X com is given by, dX com dt = 1 (cid:80) i n i d ( (cid:80) i in i ) dt = ˜Γ (16)Similarly, the rate of change of the cloud spread, σ can be derived from Eq. 15 to be, dσ dt = 1 (cid:80) i n i d ( (cid:80) i i n i − ( (cid:80) i in i ) ) dt = ˜Γ (17) mm-1m-2m-3 m+1 m+2 FIG. 5. A partially filled configuration. The driven dissi-pative process leads to incoherent hopping in the down-hilldirection. In the presence of strong interactions, hoppingonto a filled site is detuned and therefore forbidden. This isindicated by the red arrows with x’s through them.
Under this process, the atom cloud’s center of massmoves downhill with a constant speed, controlled by thedissipation rate, ˜Γ. At the same time, there is a linearspread of the square of the cloud size with time. Boththe drift and the spread are readily measured in exper-iments.One can also envision a finite length chain with all theatoms initially confined at the upper end of the ladder,and a potential barrier at the bottom, as described inSec. II E. The final steady state would be reached whenall atoms accumulate in the last site downhill. For achain of length L , the time taken would scale as, t ∼ L/ ˜Γ. B. Mott State
An ideal Mott insulating state contains exactly oneparticle per site. This is the ground state of the BoseHubbard model with very strong interactions [20]. Fi-nite temperature introduces defects, as do atom lossevents. A variant of our Raman Sideband Elevator canpump the system into an ideal Mott state, and heal de-fects which are later created.We consider a finite length chain, with a potentialsimilar to the one in Fig. 4. We require that the atom-atom interaction, U ge (cid:29) Γ , Ω (cid:48) . This large interactionstrength can be engineered by tightening the transverseconfinement, or using a Feshbach resonance.With strong interactions amongst the lattice atoms,the Raman transition between the ground state of site i and the excited state of site i + 1 is only resonant ifthere are no atoms on site i + 1. Effectively, this meansour incoherent hopping only occurs onto an empty site.Starting from a sparsely filled chain where the averageoccupation per site is less than 1, as shown in Fig. 5,the driven-dissipative process enables transport of theatoms down the ladder from filled sites to empty sites.The right-most atom stops when it hits the barrier.Subsequent atoms stop when they encounter the filledstates. The end result is an idealized Mott state, withone atom per site. If holes later develop, they are rapidlyfilled, as all uphill particles shift over by one site.We can estimate the time needed for a partially filled ladder to reach the final Mott state. The model fromSection III A is simply modified to forbid hopping ontoan occupied site. We use a Gillespie algorithm to simu-late the resulting stochastic dynamics [21].We find that the results are very well approximatedby assuming that the probability distributions on dif-ferent sites are uncorrelated and incoherent hopping isonly allowed on empty sites. Thus the dynamics are ef-ficiently simulated using the “mean field” rate equation dn i dt = ˜Γ n i − (1 − n i ) − ˜Γ n i (1 − n i +1 ) (18)As before, n i is the expectation value of the number ofparticles on site i .For a finite length chain, the infinite time solutionof the above equation is the desired Mott state, whereall atoms are jammed up at the right hand side. Forgeneric initial conditions, the time to reach this darkstate scales linearly with the size of the system.Double occupancies are highly disruptive here. Ourprocedure does not allow them to move, and they actas barriers that prevent the motion of other atoms.There are several techniques to remove doubly occupiedsites [22].After the ideal Mott state is formed, an atom lossevent will create a hole. This is healed by the hole hop-ping to the left. The hole needs to hop at most N sites,where N is the number of particles. Thus the charac-teristic time for the repair is τ ∼ N/ ˜Γ. C. AKLT State
Finally, a variant of this set-up can be used to createthe AKLT (Affleck-Kennedy-Lieb-Tasaki) state of thespin-1 chain. As previously explained, the AKLT stateis a symmetry protected topologically ordered state hav-ing topologically protected edge modes.The set-up here is slightly more involved than our pre-vious examples, as we need to manipulate the hyperfinespin degrees of freedom. We consider the constructiondemonstrated in experiments at MIT [23] where theybuild local spin-1 objects by placing two atoms on eachsite. Each atom has two accessible hyperfine states, andis effectively a spin-1/2 object. Because two Bosons inthe same site must be in a spin symmetric state, theseform a spin-1 composite. In the experiments, bosonic Li was used.The physical structure in this experiment parallelsthe “parton” construction often used to understand theAKLT state [24]. In that picture, each spin-1 is brokeninto two (symmetrized) spin-1/2’s as shown in Fig. 6.Each of these spin-1/2’s forms a singlet with a partonfrom a neighboring site. The edge modes are understoodas the leftover spins which do not have singlet partners.The partons are usually a mathematical construct, butin the experiment they represent individual atoms.In the original spin-1 language, the AKLT state isuniquely (up to the boundary modes) defined by the
FIG. 6. AKLT state described in a parton picture, wherethe spin-1’s are represented by two spin-1/2 particles at eachsite. Each blue square corresponds to projection of the twospin-1/2’s into a triplet state while each red double arrowconnecting spin-1/2’s on neighboring sites represents a sin-glet. The green (dashed) spins at the two ends are spin-1/2edge modes. property that if you take any two neighboring spin-1’s– they have zero weight in the spin-2 channel. In otherwords, the AKLT state is annihilated by the operators P ( S T =2) i,j = S i · S j + ( S i · S j ) + , each of which projectsthe spins on neighboring sites i and j into spin-2. Thesimultaneous null-space of these operators is 4-fold de-generate, corresponding to the edge degrees of freedom.The AKLT state is one of the prototypical examples of amatrix product state. It has also been proposed as a po-tential platform for measurement based quantum com-puting [25], and this projector construction has analogswith stabilizer codes [26].Our goal is to engineer a dissipative process which oc-curs only when two neighboring sites are in the spin-2sector. The AKLT state will then be the unique darkstate. Again, uniqueness is only up to the state of theboundary modes. The strategy will be to have the dissi-pation involve an intermediate state with four bosons ona single site – a configuration which can only be reachedif the atoms are in the spin-2 sector.Each site can be in one of the following three tripletstates, which are the different S z projections of spin 1: | ↑↑(cid:105) = | + (cid:105) (19)1 √ | ↑↓ + ↓↑(cid:105) = | (cid:105) (20) | ↓↓(cid:105) = |−(cid:105) (21)As seen in [23], each of the spin-1 states on a site, | + (cid:105) , | (cid:105) and |−(cid:105) have spin-dependent on-site interaction en-ergies, U | σ (cid:105) gg , with σ = − , + ,
0. These interaction en-ergies depend on the magnetic field. Amato-Grill etal. [23] find that there are special values of magneticfields where the three S z projections have equally spacedinteraction energies, that is, U | + (cid:105) gg − U | (cid:105) gg = U | (cid:105) gg − U |−(cid:105) gg =∆ U gg ∼ kHz. Under those conditions, the spin depen-dence of the interactions are equivalent to a magneticfield in the ˆ z direction. In our dynamics, the total S z will be conserved, so this field plays no role. The spinHamiltonian is then effectively rotationally invariant.Note that ∆ U is small compared to U | σ (cid:105) gg ∼ U gg , whichis tens of kHz.The same story works in the excited band, but U | σ (cid:105) ee (cid:54) = U | σ (cid:105) gg . Within our approximations, U | σ (cid:105) ee = (3 / U | σ (cid:105) gg ,which means ∆ U ee = (3 / U gg . (A)(B) FIG. 7. (A) Coherent resonant Raman transfer of twoatoms from the left site to the right site. (B) Coherent res-onant Raman transfer of the two atoms back to the excitedband on the left site. The two atoms then decay to theground band. Due to bosonic symmetry, this sequence isonly possible if the total spin of the four atoms is S T = 2,and hence this process cannot occur in the AKLT state. The protocol is shown in Fig. 7. We switch on a co-herent Raman drive such that it is in resonance withtransitions where two atoms from the ground band of asite are transported to the ground band of the (filled)site next to it down the ladder. Due to Bose statistics,four spin-1/2 bosons on the same site in the same bandmust be in a total spin symmetric state. Thus this Ra-man process can only occur if the atoms on neighboringsites are in a S T = 2 state. In particular, this processcannot occur if we are in the AKLT state.The resulting state with four atoms on a site withinthe ground band would have an interaction energy of6 U gg . The rate of this process would scale as,Ω (cid:48)(cid:48) ∼ Ω (cid:48) | ∆ − U gg | . (22)The energy denominator in this expression comes fromthe intermediate state where one site has a single atomand the second had three.We would operate in the limit where U gg , ∆ (cid:29) Ω (cid:48)(cid:48) (cid:38) ∆ U, Γ. This hierarchy of energy scales ensures that un-wanted processes are far off resonant.Simultaneously, another set of Raman lasers reso-nantly transfers two atoms from the four-atom site backto their original site but in the excited motional bandas shown in Fig. 7. The rate of this process would alsoscale as Ω (cid:48)(cid:48) ∼ Ω (cid:48) / | ∆ − U gg + ω | and the same hier-archy of energy scales makes only this process resonantfor all S z combinations. The two atoms in the higherband can now decay through their interaction with thesuperfluid bath atoms.Spin decoherence is engineered through two mecha-nisms. First, there is a dephasing resulting from timespent in the excited state configuration. For concrete-ness, consider a neighboring pair in the | S T = 2 , S ZT = 1 (cid:105) state, √ ( | + e (cid:105) + | e + (cid:105) ) with atoms on one of thesites in the excited band. The | + e (cid:105) state has energy U | (cid:105) gg + U | + (cid:105) ee , while the | e + (cid:105) state has energy U | + (cid:105) gg + U | (cid:105) ee .Their relative phases wind at a rate ∆ U gg − ∆ U ee =(1 / U . As long as ˜Γ (cid:38) (1 / U , then the phase iseffectively scrambled. Here ˜Γ is the effective decay ratefrom the motionally excited band.A second mechanism for decoherence comes from thedecay process itself. The transition | + e (cid:105) → | + (cid:105) requiresemitting a different frequency Bogoliubov phonon thanthe transition | e (cid:105) → | (cid:105) . The bath learns which-pathinformation, and this is then analogous to a measure-ment. For the energy difference to be resolvable, werequire ˜Γ (cid:38) (1 / U .After the decay, the probability of being in the S T = 2sector has been reduced. These processes would con-tinue until no neighboring pairs are in the S T = 2 chan-nel. The chain is then in the AKLT state, and all dy-namics stop.We used a Lindblad master equation approach tomodel the dynamics. As in our previous treatments, wedo not explicitly model the various intermediate states,and instead work with an effective model, where the in-coherent processes are described by jump operators ofthe form ˆ C i,kk (cid:48) = (cid:112) ˜Γ | kk (cid:48) (cid:105)(cid:104) kk (cid:48) | ˆ P S T =2 i,i +1 (23)Here, k, k (cid:48) ∈ {| + (cid:105) , | (cid:105) , |−(cid:105)} , and ˆ P S T =2 i,i +1 projects thespin-1 objects on neighboring sites to the spin-2 sector.The effective rate is ˜Γ.The master equation for the density matrix ˆ ρ is d ˆ ρdt = i = N − (cid:88) i =1 (cid:88) kk (cid:48) ˆ C i,kk (cid:48) ˆ ρ ˆ C † i,kk (cid:48) − { ˆ C † i,kk (cid:48) ˆ C i,kk (cid:48) , ˆ ρ } (24)By construction, superpositions of the four AKLT statesare the only steady states.We use two approaches for analyzing the behavior:in Sec. III C 1, we numerically calculate the eigenvaluesof the Lindblad super-operator. The real part of thesmallest non-zero eigenvalue gives the time-scale for ap-proaching the AKLT state. The size of the matrix weneed to diagonalize grows exponentially with the size ofthe system, limiting this technique to chains with fewerthan 7 sites.In Sec. III C 2, we instead use a stochastic wavefunc-tion approach which is equivalent to Eq. (24). We writethe wavefunction as a matrix product state, and usetensor network tools to efficiently evolve it in time. Wetake the initial state as a product state of | (cid:105) on all sites.We measure the expectation value of the sum of all thenearest neighbor spin-2 projectors. At long times, this decays exponentially – and we extract the time-scale byfitting this exponential.We find that the time to create the AKLT state scalesas ( N − , where N is the number of sites. We give anintuitive understanding of this result based upon diffu-sion of domain walls.
1. Exact Diagonalization
We vectorize the density matrix by putting all of itselements in a column vector, denoted ˜ ρ . The Lindbladequation then has the structure of a linear differentialequation with constant coefficients and we can use stan-dard linear algebra techniques to find the rate of ap-proaching equilibrium.If we do not take advantage of any symmetries, ourHilbert space has length 3 N , where N is the numberof spins. The density matrix is a 3 N × N matrix, so˜ ρ is a vector of length 3 N . The index α which labelsthe elements of ˜ ρ is associated with a bra (cid:104) ψ | and a ket | φ (cid:105) , and ˜ ρ α = (cid:104) ψ | ˆ ρ | φ (cid:105) α . Here (cid:104) ψ | and | φ (cid:105) are arbitrarystates in the 3 N dimensional Hilbert space.In its vectorized form, the Lindblad equation fromEq. (24) is d ˜ ρ α dt = (cid:88) β ˆ L αβ ˜ ρ β (25)where the the matrix on the right has elementsˆ L αβ = (cid:0) | ψ (cid:105) ⊗ (cid:104) φ | (cid:1) α ˆ L (cid:0) | ψ (cid:105) ⊗ (cid:104) φ | (cid:1) β . (26)The Lindblad superoperator ˆ L isˆ L = i = N − (cid:88) i =1 (cid:88) kk (cid:48) ˆ C i,kk (cid:48) ⊗ ˆ C i,kk (cid:48) −
12 ˆ C † i,kk (cid:48) ˆ C i,kk (cid:48) ⊗ − ⊗ ˆ C † i,kk (cid:48) ˆ C i,kk (cid:48) . (27)The eigenvalues of ˆ L give the rates of decay of variousperturbations. The zero eigenvalues identify the darkstates. The total S z of the chain is conserved in thedynamics, so ˆ L is block diagonal. We restrict ourselvesto the block with S z = 0.We find four zero-eigenvalues, corresponding to twoof the AKLT states, and the coherences between them.These AKLT states have edge modes | ↑↓(cid:105) and | ↓↑(cid:105) .The time taken to reach the AKLT state is controlledby the eigenvalue whose real part has the smallest non-zero magnitude, γ . Figure 6 shows how this slowest ratescales with N for the exact diagonalization calculation.Due to the exponential scaling of the Hilbert space, weare restricted to N ≤ γ ∝ / ( N − .
2. DMRG Calculation
To explore larger systems, we developed another tech-nique to model the time dynamics of this system, andexplore the time required to reach the AKLT state.Rather than working with density matrices, we usethe stochastic wavefunction approach [28], where oneintersperses coherent evolution of wavefunctions withstochastic “quantum jumps.” To efficiently evolve thewavefunction, we make a matrix-product state ansatz,which allows us to model systems with as many as 9 sites(corresponding to a Hilbert space with almost 20,000states, a density matrix with almost 400 million ele-ments, and a super-operator with over 10 elements).We take an initial product state of | (cid:105) on all sites. Wediscretize time, and at all times take a matrix-productstate ansatz for the wavefunction. During each time-step, we evolve the wavefunction as, | ψ ( t + δt ) (cid:105) = ( − i ˆ H eff δt ) | ψ ( t ) (cid:105) , where ˆ H eff = ˆ H − i (cid:80) i,kk (cid:48) ˆ C † i,kk (cid:48) C i,kk (cid:48) isa non-Hermitian effective Hamiltonian. In our modelˆ H = 0 as all dynamics simply comes from the jumpoperators. We construct a matrix product operator,( − i ˆ H eff δt ), and apply it to our wavefunction, usingthe zip-up method [29] to control the bond-dimensionof the resulting state.We then calculate the norm, 1 − p = (cid:104) ψ ( t + δt | ψ ( t + δt ) (cid:105) . We draw a random number x . If x > p , we normal-ize | ψ (cid:105) , then continue with the next time step. If x < p it means a jump has occurred. We use the standardapproach to calculate which jump occurs [28] and applythe jump operator ˆ C i,kk (cid:48) as a matrix product operator.We measured the total spin-2 projection of all nearestneighbor pairs as a function of time and fit the tail witha decaying exponential function. Our resulting estimatefor the slowest decay rate, γ is plotted in Fig. 6 for upto 9 sites. The DMRG simulation reproduces the exactdiagonalization rates and shows the same 1 / ( N − scaling. We use 75 realizations, and error bars in Fig. 6correspond to the statistical uncertainty in γ .This 1 / ( N − scaling can be qualitatively under-stood by analyzing how string order develops in thespin chain. The AKLT state in the spin-1 basis is asuperposition of different arrangements of | + (cid:105) , | (cid:105) and |−(cid:105) – for example, with three sites, an AKLT state is | (cid:105) − | + − (cid:105) − | −(cid:105) + | + 0 −(cid:105) . There is a ”stringorder” here in that if you threw away all of the spin-0sites, each of these terms correspond to an antiferroma-gentic arrangement | + −(cid:105) . This same property occursfor longer chains.Domain walls in the string order can be assigned a“charge” corresponding to the excess local magnetiza-tion: a configuration | − + + −(cid:105) has a positively chargeddomain wall, and | + − − + (cid:105) has a negatively chargeddomain wall. Our stochastic process involves local pro-jections, which conserve the total magnetization, andhence cannot remove a isolated domain wall. Instead,during the stochastic process, the domain walls undergorandom walks – and can be annihilated when two op- ▽▽▽ DMRG Simulation ▽ Exact Diagonalization0.02 0.04 0.06 0.08 0.10 0.12 1 ( N - ) γΓ ˜ FIG. 8. Slowest rate of decay, γ vs 1 / ( N − using exact di-agonalization of the Lindblad super-operator, and a stochas-tic DMRG simulation of the dynamics. N is the number ofsites in the chain, and ˜Γ is the characteristic rate of the dis-sipation. The error bars represent the statistical uncertaintyfrom using 75 realizations in the DMRG calculation. positely charged domain walls touch. For example, toestablish string order in the state | + + 00 − −(cid:105) , the cir-cled spins must either exchange position, or annihilateeach-other.The underlying domain wall dynamics are diffusive,and the slowest processes involve the motion of a domainwall over a distance of order N −
1, where N is thenumber of sites. Thus one expects the time required toscale as ( N − , as seen in the numerics.We conclude that the state preparation time wouldscale quadratically with the number of sites. This scal-ing is quite favorable. In contrast, for an adiabaticpreparation scheme, one expects the smallest gap toscale exponentially in the system size, and hence thepreparation time would also scale in that manner. IV. SUMMARY AND OUTLOOK
We present concrete examples which elucidate howengineered dissipation, in conjunction with an appro-priate drive, can be used to manipulate bosonic atomsin tilted optical lattices. In these examples, the driveis supplied by coherent Raman lasers, and the dissipa-tion comes from band-changing collisions with a super-fluid bath. Using these ingredients, we present proto-cols for controlling transport, forming a Mott insula-tor state, and creating the topologically ordered spin-1 AKLT state. In the latter two cases, the states areautonomously stabilized, and any perturbations can beautomatically healed.We calculate the relevant time scales for state prepa-ration and their dependence on system size. We notethat in all cases, the preparation time scales polyno-mially with system size. By contrast, adiabatic statepreparation techniques typically scale exponentially.All of the ingredients in our protocols have been in-0dividually realized in existing experiments. Moreover,the three examples form a natural progression for anexperimental program: each adding a layer of sophisti-cation to the previous one. While all three examples areimportant, the observation of the AKLT state would beparticularly impactful.As previously stated, these examples are also impor-tant for the way they exemplify general principles of thedriven-dissipative manipulation of quantum states. Indescribing them, we are able to address the interplayof different energy scales, and rates of processes. Thetechniques are readily extended into other systems, andinto other forms of manipulation.In modeling the formation of the AKLT state, we alsodemonstrated how matrix-product states can be incor-porated into a stochastic wavefunction calculation of dy-namics.
ACKNOWLEDGEMENTS
This work was supported by the NSF Grant PHY-1806357. EJM would like to acknowledge discussionswith Wolfgang Ketterle and Andrew Daley during thisresearch’s formative stages.
Appendix A: Raman Rates
Here we derive the expression in Eq. (13) for the Ra-man matrix elements. We begin with Eq. (12),Ω (cid:48) = Ω δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dx ψ ∗ m,g ( x ) e i ( k − k ) x ψ m +1 ,e ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (A1)In the limit where ∆ (cid:29) t α , we can approximate theWannier-Stark eigenstates in Eq. (5) upto leading orderas, ψ m,α ( x ) ∼ φ αm ( x ) (A2) As described in Sec. II B, in the deep lattice limit, φ αm ( x ) ≈ w α ( x − md/ w α ( x − md/
2) are highlylocalized harmonic oscillator eigenstates. The Ramanmatrix element becomes,Ω (cid:48) = Ω δ | I | (A3)where, I = (cid:90) dx w ∗ g ( x − md/ e ikx w e ( x − ( m + 1) d/ . (A4)Here k = k − k where k and k are the wavevectors ofthe two raman lasers. The Gaussian integral is readilycalculated, I = (cid:32) iC − πV / √ (cid:33) e − C − ikd − π √ V (A5)with C = kd/ (cid:16) √ πV / (cid:17) .For a sufficiently deep lattice, the net transition ratescales as,Ω (cid:48) ∼ Ω δ (cid:18) k d π √ V + π √ V (cid:19) e − k d π √ V − π √ V (A6)Many of the coefficients are under experimental con-trol. Ω is the dipole matrix element between theground and excited electronic levels of the lattice atoms,which directly depends on the laser intensity. Thewavevector k is tuned by changing the angle betweenthe two lattice beams, the optimal value is given by k = 8 π √ V /d , in which caseΩ (cid:48) ∼ Ω δ (cid:18) π √ V (cid:19) e − π √ V . (A7) [1] D. McKay and B. DeMarco, Cooling in strongly cor-related optical lattices: prospects and challenges , Rep.Prog. Phys. 74, 054401 (2011)[2] M. M¨uller, S. Diehl, G. Pupillo and P. Zoller,
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