Orbital excitation blockade and algorithmic cooling in quantum gases
Waseem S. Bakr, Philipp M. Preiss, M. Eric Tai, Ruichao Ma, Jonathan Simon, Markus Greiner
OOrbital excitation blockade and algorithmic cooling in quantum gases
Waseem S. Bakr, Philipp M. Preiss, M. Eric Tai, Ruichao Ma, Jonathan Simon, and Markus Greiner
Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA (Dated: October 30, 2018)Interaction blockade occurs when strong interactions in a confined few-body system prevent aparticle from occupying an otherwise accessible quantum state. Blockade phenomena reveal theunderlying granular nature of quantum systems and allow the detection and manipulation of theconstituent particles, whether they are electrons[1], spins[2], atoms[3–5], or photons[6]. The di-verse applications range from single-electron transistors based on electronic Coulomb blockade[7] toquantum logic gates in Rydberg atoms[8, 9]. We have observed a new kind of interaction blockadein transferring ultracold atoms between orbitals in an optical lattice. In this system, atoms onthe same lattice site undergo coherent collisions described by a contact interaction whose strengthdepends strongly on the orbital wavefunctions of the atoms. We induce coherent orbital excita-tions by modulating the lattice depth and observe a staircase-type excitation behavior as we crossthe interaction-split resonances by tuning the modulation frequency. As an application of orbitalexcitation blockade (OEB), we demonstrate a novel algorithmic route for cooling quantum gases.Our realization of algorithmic cooling[10, 11] utilizes a sequence of reversible OEB-based quantumoperations that isolate the entropy in one part of the system, followed by an irreversible step thatremoves the entropy from the gas. This work opens the door to cooling quantum gases down toultralow entropies, with implications for developing a microscopic understanding of strongly corre-lated electron systems that can be simulated in optical lattices[12, 13]. In addition, the close analogybetween OEB and dipole blockade in Rydberg atoms provides a roadmap for the implementation oftwo-qubit gates[14] in a quantum computing architecture with natural scalability.
An ultracold gas of bosonic atoms in the ground bandof an optical lattice is described by the Bose-Hubbardmodel[15], in which atoms can tunnel between neighbor-ing sites and interact via an onsite repulsive contact in-teraction. In a deep lattice where the interactions domi-nate, the ground state of the system is a Mott insulatorwith a fixed atom number per site that is locally con-stant over a region of the insulator[16]. The energy persite in the absence of tunneling is U gg n ( n − U gg is the interaction energy for two atoms in the groundlattice orbital state and n is the atom number on thesite. The Mott state exhibits a transport blockade phe-nomenon in which the presence of an atom on a siteenergetically prevents tunneling of a neighboring atomonto that site even in the presence of a small bias be-tween the sites. The transport is blocked unless the biasmakes up for the interaction cost, making it possible,for example, to count atoms tunneling across double-wells in a superlattice[3]. In this work, we explore anexcitation blockade phenomenon that does not involvetransport in the lattice. The excitation transfers local-ized atoms between different orbitals on the same sitethrough modulation of the lattice depth at a frequencyclose to a vibrational resonance. Physics in higher opticallattice orbitals has been the focus of much recent exper-imental work including the study of dynamics in higherorbitals[17], multi-orbital corrections to the interactionenergy[18], and unconventional forms of superfluidity in-volving higher orbitals[19, 20].The OEB mechanism can be understood in the sim-plest scenario for two atoms in a single site of a deepthree-dimensional lattice, in which the vibrational fre- δ = U ee + U gg -2 U eg ћ ω z ,0→ m ћ ω z ,0→ m U gg U eg U ee δ FIG. 1. Orbital excitation blockade mechanism in an opticallattice. A single atom on a site is excited to a higher orbitalby resonantly modulating the lattice depth. For two atoms onthe same site, interactions lead to an orbital-dependent energyshift. Modulation at the appropriate frequency excites one ofthe atoms to the higher orbital, but is off-resonant for excitingthe second with a blockade energy δ . quencies in all three directions are taken to be differentto avoid degeneracies. The lattice depth along the z -direction is modulated weakly, which in the presence ofanharmonicity of the lattice potential drives atoms be-tween the ground orbital and a single, specific excited z - a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t p odd abc p odd p odd Modulation Frequency (Hz)0 20 40 60 80 100Modulation Time (ms)0 20 40 60 80 100Modulation Time (ms) iiiiiii iii iii
FIG. 2. Time, frequency and site-resolved coherent transferof atoms in a Mott insulator between orbitals. a , Excitationstransferring a single atom in the n = 1 (orange) or n = 2shell (blue) from the ground to the second excited orbitalare spectroscopically resolved in a two-shell Mott-insulator. b , Rabi oscillations between the two orbitals are observed bydriving at the resonant frequencies for atoms in the n = 1 shelland c , n = 2 shell of a Mott insulator. Bose-enhancementleads to faster oscillations in the n = 2 shell. When theatom number is reduced to obtain one atom per site in theregion previously containing two atoms, the interaction shiftsuppresses oscillations (black). All error bars are one standarderror of the mean. ( i - iii ) Site-resolved snapshots of the Mottinsulator are shown at different points in the Rabi cycles. orbital, subject to a selection rule that only allows cou-pling to orbitals of the same symmetry. For a singleatom, excitation to the m th orbital requires modulationat a frequency ω z, → m which is approximately mω z, → ignoring the anharmonicity of the onsite potential. Withmore than one atom on a site, the interaction introducesan orbital-dependent shift of the energy levels as shownin Fig. 1. In general, the interaction shifts U gg , U ge , and U ee are all different and the differences are a significantfraction of U gg , where g ( e ) denotes atoms in the ground(excited) orbital. If the coupling strength due to themodulation is small compared to these differences and themodulation frequency is tuned to ω z, → m +( U ge − U gg ) / ¯ h ,only a single atom is transferred to the higher orbitaland the transfer of a second atom is off-resonant. In thissense, the first excitation blocks the creation of a secondexcitation.The experimental system has been described in pre-vious work[21]. A two-dimensional Bose-Einstein con-densate of rubidium atoms resides in a single planeof a one-dimensional optical lattice, henceforth referredto as the axial lattice, with a vibrational frequency of a n tot = n + m n -650 Hz -440 Hz -220 Hz Random occupationMott insulatorSuperfluidFilter operations b FIG. 3. Algorithmic cooling in an optical lattice. a , Landau-Zener chirp for transferring entropy from the ground to thefourth band. The lattice modulation frequency is swept acrossthe transition resonances from left to right. The interactionshifts ∆ ω z for excitation of one of n atoms to the fourth or-bital relative to the excitation frequency for a single atom ona site, are shown for different orbital occupations. Excitationprocesses in the same column happen at almost the same fre-quency to within 30 Hz (see Supplementary Table I). b , Astate with random occupation in a deep lattice is far fromthe Mott insulating ground state. Sequential filtering opera-tions followed by reduction of the confinement prepares theground state, which can be adiabatically converted to a ther-malized superfluid in a shallow lattice. Red (blue) spheresdenote atoms in the ground (excited) band. ω z, → = 2 π × . z -axis is perpendicularto the plane and points along the direction of gravity.In addition, we introduce a lattice in the plane with aspacing of a = 680 nm and a depth of 45 E r (trap fre-quency of 17 kHz), where E r = h / ma is the recoilenergy of the effective lattice wavelength, with m themass of Rb. The resulting Mott insulator is at the fo-cus of a high resolution optical imaging system capableof detecting atoms on individual lattice sites through flu-orescence imaging. Light-assisted collisions at the startof the imaging process reduce the occupation of a site toits odd-even parity[21].We start by demonstrating coherent driving of atomsin a Mott insulator between two orbitals. In the presenceof a harmonic trap, the atoms in a 2D Mott insulatorare arranged in concentric rings of fixed atom numberper site, known as shells, with the largest occupation atthe center[21]. We prepare a Mott insulator with twoshells and modulate the axial lattice depth by ± . ω z, → is resonant foratoms in the outer shell with one atom per site ( n = 1).Excitation to the fourth excited orbital is suppressed be-cause of an energy shift of h × | g (cid:105) and | e (cid:105) are detected by loweringthe axial lattice depth at the end of the modulation sothat the excited orbital state becomes unbound and anypopulation in it escapes along the z -axis due to gravity.The Rabi oscillations in that shell, shown in Fig. 2b, havea frequency Ω = 2 π × . n = 2 shell bymodulating at a frequency of ω z, → + ( U ge − U gg ) / ¯ h .For our parameters, U gg , U ge , and U ee are h × , , | g, g (cid:105) and 1 / √ | e, g (cid:105) + | g, e (cid:105) )are detected as an oscillation of the parity between evenand odd after ejecting the atom in the excited orbitaland are shown in Fig. 2c. For the same modulationamplitude as before, the oscillations are expected tooccur √ n = 1 shell owing to Bose-enhancement[5]. Weindeed observe a frequency ratio of 1.42(1) between theoscillation frequencies. A full frequency spectrum inthe two shells is shown in Fig. 2a and the frequencyseparation of the two resonances of 160(10) Hz matcheswell with the theoretically expected value of 165(35) Hzwhen the impact of virtual orbital changing collisions isincluded (see Methods).We next employ OEB to demonstrate a new pathto cooling quantum gases. Evaporative cooling hasbeen the workhorse technique for cooling atomic gasesto nanokelvin temperatures. However, current interestin studying the physics of strongly-correlated materials,such as high- T c cuprates, using ultracold gases[12, 13] hasspurred research into developing new cooling techniquesthat can reach the requisite picokelvin regime[22, 23].The field of quantum information offers an alternativecooling paradigm, wherein a sequence of unitary quan-tum gates purifies a subset of the qubits in a sys-tem by moving entropy and isolating it in another partof the system[10]. One realization of such a coolingscheme, heat-bath algorithmic cooling, has been suc-cessfully demonstrated with solid-state nuclear magneticresonance qubits[11]. We introduce an analogous tech-nique for quantum gases where the unitary operationsare achieved using OEB, building on previous theoreticalproposals in this direction[24–27].A bosonic quantum gas at a finite temperature T adia-batically loaded into the ground band of a optical latticestores its entropy in the form of atom number fluctua-tions in the zero-tunneling limit. Within the local densityapproximation, a lattice site with a local chemical poten-tial µ is described by a density matrix ˆ ρ = (cid:80) n p n | n (cid:105)(cid:104) n | ,where p n , the probability of having n atoms on the site,is given by e − β ( U gg n ( n − − µn ) /Z . Here, β = 1 /k B T and Z is the grand canonical partition function. Cooling tozero temperature is achieved by changing the atom num-ber distribution on each site to obtain p n = δ n, (cid:100) µ/U gg (cid:101) .The crucial ingredient for algorithmic cooling is a uni-tary operation that realizes the transformation | n, m (cid:105) → | n − , m + 1 (cid:105) for each n separately, where | n, m (cid:105) de-notes a Fock state with n atoms in the ground bandand m atoms in an excited band. Resonant lattice mod-ulation in the presence of OEB results in a rotationgate ˆ R nm ( t ) = exp [ i (Ω nm t | n − , m + 1 (cid:105)(cid:104) n, m | + c.c. )],where Ω nm is the transition’s Rabi frequency and the re-quired transformation is achieved for a modulation time t = π/ nm . Entropy is transferred from the groundband to the excited band by performing a sequence of π gates ˆ R N − s,s from s = 0 to s = N − N cho-sen large enough such that p N ≈ µ is readjusted torecover a situation closer to thermal equilibrium by re-ducing the harmonic confinement to a new value ω low sothat µ < U gg throughout the gas. At this point, residualentropy is stored in the resulting n = 1 Mott insulatorin the form of holes that are preferably located near theedge of the cloud. The gas is allowed to rethermalizeby lowering the lattice depth to allow tunneling, and thefinal entropy of the thermalized state would be signifi-cantly reduced compared to the initial state.We start by experimentally demonstrating the algo-rithm on a state with known atom number, namely a fourshell Mott insulator, and reducing the site occupation ev-erywhere in the insulator to a single atom per site. Toincrease the blockade energy for this set of experiments,we transfer atoms to the fourth axial orbital rather thanthe second. We also replace the rotation gates demon-strated in the first part of this work with Landau-Zenertransitions to improve the fidelity of the algorithm. Welinearly sweep the modulation frequency from 20.90 kHzto 21.65 kHz in 250 ms. The chirp realizes a sequenceof quantum operations that transfer atoms to the ex-cited orbital one at a time, until only one atom remainsin the ground state in all shells (Fig. 3a). We probethe ground orbital occupancy at different points in thefrequency chirp by ejecting atoms in the higher orbitalas before and then performing the parity imaging. Theparity of the different shells during the chirp is shown inFig. 4a, and an analogue of the typical Coulomb blockadestaircase is seen in the data. Shell-sensitive manipulationof a Mott insulator had been achieved in previous exper-iments using a microwave transition between hyperfinestates, but the lack of a strong blockade allowed transferof only a small fraction of the population to the targetstate[28].Next, we demonstrate cooling by performing the algo-rithm on a state that is far from the many-body groundstate. To prepare such a state, we non-adiabatically loada condensate into a deep lattice, projecting the wave-function onto a state with Poissonian site occupancy thatrapidly loses coherence between sites. Using the same op-eration sequence as before, we progressively reduce the a b c Dephased cloudRecooled superfluidModulation chirp endpoint (Hz) (i)(ii)
Modulation chirp endpoint (Hz)
FIG. 4. Experimental realization of algorithmic cooling. a , By chirping the modulation towards higher frequencies, atomsin a four-shell Mott insulator are sequentially excited one at a time to the fourth orbital. Population in the higher orbital issubsequently ejected at the end of the chirp. The average parity in the n = 1 (orange), n = 2 (blue), n = 3 (black) and n = 4(green) shell is shown at different points in the chirp, together with single shot images, illustrating the conversion to a three,two and finally one shell insulator. b , The same frequency chirp algorithmically cools a state with random occupancy into an n = 1 Mott insulator, observed as an enhancement in odd occupancy. All error bars are one standard error of the mean. c , ( i )An incoherent cloud does not exhibit an interference pattern in a 5 ms time of flight expansion after adiabatically lowering thelattice depth. ( ii ) Cooling converts the incoherent cloud to a Mott insulator in the deep lattice. After adiabatically loweringthe lattice depth, a superfluid forms and an interference pattern is obtained in the expansion images. randomness of the ground band occupancy, preparing asingle-occupancy Mott insulator (Fig. 3b). We enhancethe odd occupancy from 0.45(1) to 0.76(2) (Fig. 4b),demonstrating significant atom number squeezing limitedby the conversion efficiency of the Landau-Zener transi-tions. To complete the algorithm, we readjust the har-monic confinement to obtain a state close to the many-body ground state. We verify the ground state characterby ramping back adiabatically to a 5 E r lattice in 100 msand releasing the atoms from the lattice. Without cool-ing, we obtain a featureless cloud shown in Fig. 4c(i),indicating an absence of the coherence expected in thesuperfluid ground state. With cooling, the Mott insula-tor is adiabatically converted to a superfluid, giving riseto matter wave interference peaks shown in Fig. 4c(ii).We now discuss the limits on the entropies that canbe achieved with algorithmic cooling. The conversionefficiency to a single-occupancy Mott insulator is tech-nically limited in our system by heating due to sponta-neous emission during the sweep and to a lesser extent, bythe efficiency of the Landau-Zener sweep for clouds withlarge average occupancies (see Methods). While we havedemonstrated cooling of hot clouds, the single-occupancyprobability we have achieved using algorithmic cooling ina two-shell Mott insulator is 0.94(1). This is comparableto what had been previously achieved with evaporativecooling, corresponding to an average entropy of 0 . k B per particle [21]. Nevertheless, lattice heating can bemade negligible by using a further-detuned lattice (e.g.1064 nm), while shaped pulses can improve the Landau-Zener transfer efficiency[25]. More fundamentally, the single-shot cooling algorithm we have implemented is lim-ited by initial holes in the Mott insulator which cannotbe corrected. However, repeated iterations of the algo-rithm can circumvent this problem and bring the cloudto zero entropy with quick convergence[25]. The cyclealternates between (a) using OEB to produce a reducedentropy n = 1 insulator in a harmonic confinement ω low (demonstrated above) and (b) adiabatically increasingthe confinement to ω high in the presence of tunneling tomove hot particles to the center of the cloud where theycan be removed again. Alternatively, the outer edge ofthe cloud containing the holes can be removed using thehigh resolution available in our system[29].In conclusion, we have observed a new blockade phe-nomenon in optical lattices when exciting atoms to higherorbitals, analogous to dipole excitation blockade in Ryd-berg atoms. The blockade permits deterministic manipu-lation of atom number in an optical lattice. We have usedit to convert a multi-shell Mott insulator into a singly-occupied insulator with over a thousand sites, the largestquantum register achieved so far in an addressable sys-tem. The same technique allows initialization of regis-ters in longer wavelength lattices where a Mott insula-tor cannot be prepared [30]. OEB also opens a route toimplementing quantum gates in optical lattices. Single-site addressing[29], possible with our microscope, canperform rotations of individual orbital-encoded qubitsrather than the global rotations demonstrated in thiswork. Controlled-NOT gates can be implemented byconditionally moving the control qubit onto the targetqubit site, and performing an interaction-sensitive ro-tation of the target qubit[14]. 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Our experiments begin with a degenerate 2D Bose gasof Rb atoms prepared in the | F = 1 , m f = − (cid:105) state ina single layer of a 1D optical lattice with spacing 1.5 µ m,in the focal plane of a high resolution imaging system asdescribed in previous work. The atoms are then loadedinto a 2D optical lattice with spacing 680 nm, which isramped up to a depth of 45 E r adiabatically on eithera single- or many- body timescale, depending upon theexperiment to be performed. Higher Band Removal
The orbital blockade is observed through the deter-ministic removal of atoms in higher bands of the 1.5 µ mlattice. For removal of atoms from the second band, thisis achieved by reducing the depth of this lattice to 3.8kHz from an initial depth of 35.8 kHz. Gravity pro-duces a shift of 3.2 kHz per well, thus inducing secondband atoms to Zener tunnel away within a few ms. TheLandau-Zener tunneling rate from the ground state isgiven by Γ LZ = mga π ¯ h e − g c /g ≈
12 Hz. Here g is the grav-itational acceleration and g c = aω z /
4. This effect leadsto a loss of ground state atoms on the percent level, butcan be made negligible by using excitations to the fourthband. ∆ ω z / π [Hz] n = 1 n = 2 n = 3 n = 4 m = 0 0 -217 -434 -651 m = 1 -9 -226 -443 m = 2 -18 -235 m = 3 -27TABLE I. Frequency shifts in Hz for a transition transferringan atom from the ground band to the fourth band, startingwith n and m atoms in each of these bands. The shifts aremeasured relative to an initial state with one atom in theground band and none in the excited band. Band Dependent Energy Shifts
Due to its large spacing, the 1.5 µ m lattice has a recoilenergy of only 250 Hz. Consequently its depth is ∼ m and n may thus be written in terms of the ground band in-teraction energy U as: U nm = U (2 − δ nm ) (cid:82) | ψ m ( x ) | | ψ n ( x ) | dx (cid:82) | ψ ( x ) | dx where ψ m ( x ) is the normalized m th harmonic oscillatorwavefunction. The total interaction shift for M particlesin band m , and N particles in band n is thus: M ( M − U mm + N ( N − U nn + M N U mn . The interactions also produce off-resonant band changingcollisions, with a Rabi frequency:Ω mn ↔ pq = U C mn ↔ pq (cid:82) ψ m ( x ) ψ n ( x ) ψ ∗ p ( x ) ψ ∗ q ( x ) dx (cid:82) | ψ ( x ) | dx up to a combinatoric factor C mn ↔ pq resulting fromBose enhancement. For an energy defect of δ mn ↔ pq (cid:29) Ω mn ↔ pq , this process produces an energy shiftof ∆ mn = −| Ω mn ↔ pq | /δ mn ↔ pq . For our experiment, thedominant band changing collision is | m = 0 , n = 2 (cid:105) →| p = 1 , q = 1 (cid:105) with a Rabi frequency Ω ↔ = 2 π ×
120 Hz. In this case lattice anharmonicity and interactionshifts produce an energy defect of δ ↔ = 2 π ×
330 Hz,resulting in an additional overall shift of the | , (cid:105) stateof ∆ = − π ×
45 Hz.
Limits on entropies achievable with algorithmiccooling