Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits

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1 Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits The Harvard community has made this article oenly available. Please share how this access benefits you. Your story matters. Citation Published Version Accessed Citable Link Terms of Use Rabl, P., D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkof, and P. Zoller Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits. Physical Review Letters 97 (3) (July 21). doi: /hysrevlett doi: /physrevlett Aril 5, :15:22 PM EDT htt://nrs.harvard.edu/urn-3:hul.instreos: This article was downloaded from Harvard University's DASH reository, and is made available under the terms and conditions alicable to Other Posted Material, as set forth at htt://nrs.harvard.edu/urn-3:hul.instreos:dash.current.terms-ofuse#laa (Article begins on next age)

2 Hybrid Quantum Processors: Molecular Ensembles as Quantum Memory for Solid State Circuits P. Rabl, 1 D. DeMille, 2 J. M. Doyle, 3 M. D. Lukin, 3 R. J. Schoelkof, 2,4 and P. Zoller 1 1 Institute for Theoretical Physics, University of Innsbruck, and Institute for Quantum Otics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria 2 Deartment of Physics, Yale University, New Haven, Connecticut 06520, USA 3 Deartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 4 Deartment of Alied Physics, Yale University, New Haven, Connecticut 06520, USA (Received 19 Aril 2006; ublished 21 July 2006) We investigate a hybrid quantum circuit where ensembles of cold olar molecules serve as long-lived quantum memories and otical interfaces for solid state quantum rocessors. The quantum memory realized by collective sin states (ensemble qubit) is couled to a high-q striline cavity via microwave Raman rocesses. We show that, for convenient tra-surface distances of a few m, strong couling between the cavity and ensemble qubit can be achieved. We discuss basic quantum information rotocols, including a swa from the cavity hoton bus to the molecular quantum memory, and a deterministic two qubit gate. Finally, we investigate coherence roerties of molecular ensemble quantum bits. DOI: /PhysRevLett During the last few years we have witnessed remarkable rogress towards the realization of quantum information rocessing in various hysical systems. Highlights include quantum otical systems of traed atoms and ions [1], and cavity QED [2], as well as solid state systems including Cooer air boxes (CPB) [3,4] and quantum dots [5]. In articular, the strong couling regime of circuit CQED [6] was realized using a CPB strongly couled to a striline cavity. In the light of these develoments it is timely to investigate hybrid devices with the goal of combining the advantages of various imlementations, i.e., to build interfaces between, for examle, a quantum otics and solid state qubit with comatible exerimental setus [7]. Such interfaces are articularly imortant in alications where long-term quantum memories or otical interconnects are required [8,9]. In this Letter we study such a scenario by couling a striline cavity to a cloud of cold olar molecules [10]. The cavity may be art of a solid state quantum rocessor involving CPB as charge qubits [3] and microwave hotons as a quantum data bus, while the molecular ensemble serves as a quantum memory with a long coherence time. The condition of strong couling between the cavity and the molecular cloud is achieved via the (large) electric diole moments of olar molecules for rotational excitations in the electronic and vibrational ground state, which are in the tens of GHz regime, and thus rovide an ideal match for resonance frequencies of striline cavities. By adoting a molecular ensemble instead of a single olar molecule [11], we benefit from the enhancement of the coherent couling g N with the number of molecules N and g the single molecule vacuum Rabi frequency. This strong couling between the molecular ensembles and the circuit CQED system also oens the ossibility of a solid state based readout of molecular qubits. In addition, qubits stored in the molecular ensemble can be converted to flying otical qubits, using techniques demonstrated PACS numbers: Ps, Lx, Dv, C for atomic ensembles [12]. This rovides a natural interface between mesoscoic quantum circuits and otical quantum communication. Let us consider the setu of Fig. 1, where two molecular ensembles are couled to a suerconducting cavity. The cavity is assumed to be strongly couled to a CBP reresenting a circuit CQED system, as realized in recent exeriments at Yale [6]. As discussed in detail below, molecular sectroscoy allows us to identify long-lived states, for examle, in the form of a sin qubit j0i, j1i in the ground rotational manifold. Starting with a cloud of N molecules reared in j0i m j N i couling to a microwave or cavity field leads to excitations in the form of symmetric Dicke states, j1i m 1= P N ij i...0 N i m y j0i m, etc. For weak excitation the oerator m obeys aroximate harmonic oscillator commutation relations m; m y 1, and the ensemble excitations are conveniently described as a set of harmonic oscillator states FIG. 1 (color online). (a) Ensembles of olar molecules and a CPB are couled via the quantized field of a striline cavity (see text for more details). (b) Rotational excitation sectrum of molecules with a 2 1=2 ground state where we lot excited states according to the Hamiltonian H H R H SR. For nonzero nuclear sin ~I (here I 1=2) the hyerfine interaction leads to an additional slitting (b MHz) into eigenstates of ~F ~J ~I (hyerfine slitting for the excite states is not shown). Two qubit states in the N 0 manifold, j0i, j1i are couled by a Raman rocess involving a single cavity hoton and an external microwave field, t =06=97(3)=033003(4) The American Physical Society

3 j0i m, j1i m m y j0i m, etc. Our goal below is to use the lowest two of these states as ensemble qubits, which can be maniulated by couling them to the suerconducting cavity and a CPB. The dynamics of the couled system [Fig. 1(a)] can be described in terms of a Hamiltonian H sys H C H M H CM, which is the sum of a Jaynes-Cummings tye Hamiltonian for the circuit CQED system H C, a Hamiltonian for the (sin) excitations of the molecular ensembles H M, and the couling of the molecules to the cavity H CM. In a frame rotating with the cavity frequency! c the CQED Hamiltonian has the form H C c t jeihej g c jeihgj ^c jgihej ^c y : (1) Here jgi and jei denote the ground and the first excited eigenstate of the CPB at the charge degeneracy oint reresenting a charge qubit with a (tunable) transition frequency! cq t and a detuning from the cavity c t! c! cq t. The oerator ^c ( ^c y ) is the cavity annihilation (creation) oerator for microwave hotons. The Hamiltonian describing the internal excitations of the molecular ensembles i 1, 2 and the couling of ensemble states to the striline cavity takes the form H M H CM X m;i t m y i m i X g m;i t m y i ^c H:c: i i Before we enter the details of the derivation of H M and H CM we note the basic structure of the Hamiltonian H sys. The ensemble excitations and the cavity reresent a system of couled harmonic oscillators interacting with a twolevel system (CPB) with controllable coefficients. In general, this rovides the basic ingredients for (i) swa oerations between charge, cavity, and ensemble qubits, (ii) rotations of a single ensemble qubit via the charge qubit, and (iii) 2-qubit entanglement oerations between two ensemble qubits, where the charge qubit lays the role of a nonlinearity. For examle, the CPB can act as a single hoton source ; i.e., we generate a suerosition state of the charge qubit, which by an aroriate control sequence can be swaed over to the cavity, and is finally stored in one of the molecular ensembles, and vice versa: jgi jei j0i c j0i m!jgi j0i c j1i c j0i m!jgij0i c j0i m j1i m. The above discussion has ignored various sources of decoherence. In the Yale exeriment [6], the circuit CQED system realizes the strong couling regime with vacuum Rabi frequency g c & 2 50 MHz. The decoherence of the charge qubit is dominated by the dehasing rate T :5 MHz, while the hoton loss rate is =2 1 to 0.01 MHz; i.e., the charge qubit is the dominant source of decoherence. Below we show that for a cloud of N 10 4 to 10 6 molecules, traed 10 m above the striline cavity, one can reach the regime of strong cavity-ensemble couling g m = MHz, which should be comared with the exected collisional dehasing rates of a few hundred Hz Figure 1(b) shows the rotational sectrum of CaF, which rovides an examle for sectra of alkaline-earth monohalogenides with a 2 1=2 ground state corresonding to a single electron outside a closed shell. The sectrum consists of rotational eigenstates, described by a rigid rotor Hamiltonian H R B ~N 2 with B 2 10 GHz the rotational constant, and ~N the angular momentum of the nuclei. The unaired sin is couled to the molecule rotation according to H SR sr ~S ~N with sr 2 40 MHz and ~S the electron sin (S 1=2). Couled eigenstates are denoted by jn; S; J; M J i with ~J ~N ~S. As seen from Fig. 1, there is a sin rotation slitting ( doubling) for rotationally excited states. In addition, there can be hyerfine interactions, which, in articular, lead to a slitting of the ground state N 0, as in the case of CaF with a nuclear sin I 1=2 which are couled with J 1=2 to F 0 and 1 states. In the following, we denote by j0i, j1i a air of states in the rotational ground state manifold to rovide our sin qubit. Comared to qubits stored in the rotational degrees of freedom this choice of states avoids unfavorable N 1! N 0 collisions while j0i and j1i can still be couled efficiently by a Raman rocess. The cavity mode and microwave fields of aroriate frequency and olarization coule rotational ground states to excited states with electric diole matrix elements ( 5 D). Two microwave driving fields rovide an effective couling Hamiltonian 1 2 eff t j0ih1j H:c: to rotate the single molecule sin qubit, where eff 1 2 =2, with 1;2 the Rabi frequencies and the detuning from the excited state jri, ( * 1;2 ). By similar arguments the couling to the cavity has the form g eff t j1ih0j ^c H:c: with g eff t g t =2, where g E c is the vacuum Rabi frequency, and E c =2 0 d 2 L is the electric field er hoton for a cavity length L and tyical electrode distance d. Tyical values are g= khz for 5Dand d 10 m. The distance d is also an estimate of the traing distance of the molecular cloud from the cavity, which is well in the limit where standard traing techniques work reliably and surface effects are negligible. Rewriting H CM in terms of the collective oerator m we obtain an effective cavity-ensemble couling g m t N geff t. Because of the large wavelength c 1:5 cm a tra volume of V d d c =10 contains N molecules for gas densities of n cm 3 resulting in a couling strength of g m = MHz. The arameters m;i t are Raman detunings, which can be controlled indeendently, e.g., by alying local magnetic and/or electric fields. Thus we obtain the Hamiltonian H M H CM, which allows a SWAP of a cavity and an ensemble state, for examle, by an adiabatic swee of m across the resonance. This corresonds to a read or write oeration c j0i m h0j!j0i c h0j m with c an arbitrary density oerator of the microwave field in the cavity, and m the identical state stored in ensemble excitations. The CPB rovides a nonlinear element in the Hamiltonian H sys. This allows first of all single qubit o-

4 erations of ensemble qubits, e.g., by combining a swa oeration with single qubit rotations of the charge qubit, and second, deterministic entanglement oerations of qubits stored in two molecular ensembles. An examle of such a rotocol, which uses the CPB as a nonlinear hase shifter, is given as follows. We assume that the system is initially reared in the state j i t 0 jgij0i c j i m with the charge qubit far detuned from the cavity resonance, j c 0 j g c, and the two ensemble qubits in an arbitrary state j i m sanned by the basis j 1 2 i m, i 0, 1. In a first ste, in analogy to the single qubit swa, the state j i m is (artially) transferred to the cavity. Assuming symmetric conditions, g m;i g m and m;i t m t, it is convenient to rewrite the ensemble state in terms of the (anti)symmetric oerators m s=a m 1 m 2 = 2 acting on j00im. An adiabatic swee of the Raman detunings then realizes the swa oeration m y s j00i m j0i c!j00i m j1i c and m y s 2 j00i m j0i c! 2 j00im j2i c while the states j00i m j0i c, m y aj00i m j0i c, and m y a 2 j00i m j0i c remain unaffected. In a second ste, the charge qubit is adiabatically tuned close to resonance for a time T, j c T=2 j & g c. During this ulse the nonvacuum states acquire a nonlinear dynamical hase, jni c! e i n jnic, with n R T 0 dt 0 c t 0 2 c t 0 n4g 2 c =2. The ulse form c t and the length T are chosen such that 1 =2 and 2 2 n (see, e.g., Fig. 2). The second condition ensures that after writing the cavity state back into the ensembles states, i.e., reversing ste one, the ensemble states j20i m and j02i m remain unoulated. The total gate sequence corresonds to a SWAP-like gate for two ensembles qubits, with j00i m!j00i m, j10i m! e i =4 j10i m ij01i m = 2, j01im! e i =4 ij10i m j01i m = 2, and j11im!j11i m. A numerical simulation of this gate sequence based on a master equation treatment of the dissiative terms [6] shows that the gate fidelity is only limited by g c T 2 1 ; i.e., the decoherence of the CPB during the time it is tuned close to resonance (see Fig. 2 for more details). We now turn to an analysis of decoherence in the molecular ensemble. In articular, collisional dehasing of the FIG. 2 (color online). (a) Adiabatic energies levels for the states jni c jgi as a function of the charge qubit detuning c. (b) Evolution of the dynamical hases 1 (solid line) and 2 (dashed line) for a ulse c t 0 2t=T 1 2 1, 0 =g c 30, 1 =g c 0:44, and T 44:79=g c. For the same ulse c t the resulting fidelity of the total gate sequence, F G (averaged over all initial states j i m ), is lotted in (c) for different values of the charge qubit dehasing rate, T2 1, and the cavity loss rate ensemble qubit, and a satial variation of the cavitymolecule couling g eff x in combination with the thermal motion of the molecules in the tra, contribute to a finite decoherence time of the molecular quantum memory, and result in imerfections during gate oerations. An oerational definition of the decoherence time of the ensemble qubit can be given in terms of an (idealized) exeriment. A cavity qubit c t 0 j i c h j with j i c j0i c j1i c is written at time t 0 to the molecular memory with all molecules initialized in the state j0i in a (erfect) swa oeration, ket in storage for a time interval, undergoing dehasing collisions. At t, the qubit is transferred back to the cavity mode, resulting in a reduced density matrix c of the cavity with a fidelity F min c ch j c j i c, with the decoherence time of the ensemble memory identified as the decay time of the fidelity. The analysis of this rocess resembles the discussions of clock shifts, and, in articular, studies of collisional dehasing of sins in thermal and quantum degenerate atomic clouds in a Ramsey interferometry setu. A formal theoretical descrition of these henomena is rovided by quantum kinetic theory [13]. We consider a cloud of molecules in the lowest rotational state with external degrees of freedom cooled to a temerature T m & 1mK[10,11] and qubits stored in sin or hyerfine states (Fig. 1). Molecules in the rotational ground state are traed magnetically [14] or by a (sin indeendent) electric rf tra [15]. In a magnetic tra two molecules interact asymtotically according to a V r C 6 =r 6 otential with C 6 2 =4 0 2 =6B. The effective range of this otential is given by R 4 mc 6 =@ 2, which rovides an estimate for the s-wave scattering length a. For examle, for CaCl (which has two magnetically traed hyerfine states) we obtain R 780 a B which is a few times the tyical scattering length encountered for alkali atoms. S-wave scattering dominates for temeratures T m & T 1 K where the thermal energy is below the centrifugal barrier for higher angular momenta, leading to an estimate for the collision rate col 8 a 2 n v Hz for n cm 3 and v the relative thermal velocity. For temeratures T m T higher artial waves contribute, and an estimate of the cross section based on the unitarity limit gives col & Hz for T m 1mK. In electric tras the induced diole moment ind leads to a V r 2 ind =r3 deendence of the asymtotic interaction. Although this long-range behavior significantly changes the low temerature scattering (T m < 1 K) the estimate based on the unitarity limit at T m 1mKstill rovides a valid bound for scattering rates for ind < 1D. We have calculated the decoherence time of an ensemble qubit corresonding to collisional dehasing using quantum kinetic theory [16]. Dehasing of the qubit coherence c 10 ex 10 =2 c 10 0 is associated with sin deendent collisions between the states j0i and j1i, which gives a contribution

5 10 n Z Y m 2 d 3 k i E K K P ~k 1 P ~k 2 i 1::4 jf00 e K fe 01 K j2 jf00 in K j2 jf01 in K j2 : (2) It deends on the difference between f00 e and fe 01, the elastic scattering amlitudes for the internal states j00i and j10i j01i = 2 averaged over the thermal distributions P ~k in the scattering rocess between momenta K ~k 3 ; ~k 4 ~k 1 ; ~k 2, and functions accounting for energy and momentum conservation in the collision. In addition, there may be contributions from inelastic collisions, f00 in and f01 in, which scatter molecules outside the j0i, j1i subsace. While accurate scattering amlitudes for molecular collisions may not be available at resent, we can estimate these contributions in certain limits. For s-wave scattering the above exression simlifies to 10 8 a 00 a 01 2 n v with a 00 and a 01 scattering lengths. If we assume that the scattering length is dominated by a sin exchange otential, the scattering is characterized by a singlet (a S ) and trilet scattering length (a T ). In the simle case of a ure sin qubit fj0i; j1ig fjs 1=2;m s 1=2ig we find a 00 a 01 a T, and the dehasing rate is determined by nonvanishing contributions arising from magnetic diole and sin rotation couling, which are exected to be much smaller. In a similar way, in the resence of hyerfine interactions we can form a qubit j0i jf I 1=2;M F Fi and j1i jf 0 I 1=2;M F F 0 i, where again j00i and j01i j10i contain no sin singlet contribution and the leading dehasing term vanishes. In the worst case the decoherence rate is bounded by the single molecule collision rate 10 col which has been estimated above. Satial variations of the effective single moleculecavity couling, g eff x, result in a dehasing of the qubit during a single swa gate and an incomlete recovery of the state after a redistribution of the molecules between two successive write or read oerations. The inhomogeneity in the couling arises from the variation of the cavity mode function on a scale of the electrode distance, g x g 1 x=d, with a numerical constant, and a osition deendence of the detuning x m! 2 x 2 = 2@. Here! 2! 2 t! 2 r accounts for a difference in the traing otentials for the qubit states (! t ) and the excited state jri (! r ). For an otimal detuning 3 3g 2 N k b T m! 2 2 =! 4 2 the inhomogeneous couling results in a total gate error of 2 k b T m =m! 2 t d 2 k b T m! 2 =@g 2 N! 2 t 2=3 [16]. For g N 2 10 MHz, 2 10 khz and at T m 1mKgate fidelities of F > 0:99 require tra frequencies of! t 2 50 khz and a similar traing otential for the state jri (! 2 0:1! 2 t ). Lower temeratures and an otimized cavity or tra design, e.g.,,! 2! 0, lead to a further significant reduction of gate errors. In conclusion, ensembles of cold olar molecules reresent a good quantum memory that can be strongly couled to striline cavities with a long lifetime, limited essentially only by collisional dehasing. We note that these dehasing channels are virtually eliminated, if the ensemble is reared in a crystalline hase of diolar gases with diole moments induced and aligned by a dc electric field under 2D traing conditions [16]. The resent work oens an exciting avenue towards long-lived molecular quantum memories for solid state quantum rocessors. Work at Innsbruck is suorted by the Austrian Science Foundation, Euroean Networks, and the Institute for Quantum Information. P. R. thanks the Harvard Physics Deartment and ITAMP for hositality. Work at Harvard is suorted by NSF, Harvard-MIT CUA, and Packard and Sloan Foundations. Work at Yale is suorted by NSF Grant No. DMR , the W. M. Keck Foundation, and the Army Research Office. [1] D. Leibfried et al., Nature (London) 438, 639 (2005); H. Häffner et al., Nature (London) 438, 643 (2005). [2] S. Numann et al., Nature Phys. 1, 122 (2005); K. M. Birnbaum et al., Nature (London) 436, 87 (2005). [3] Yu. Makhlin et al., Rev. Mod. Phys. 73, 357 (2001). [4] D. Vion, et al., Science 296, 886 (2002); I. Chiorescu et al., Nature (London) 431, 159 (2004). [5] J. R. Petta et al., Science 309, 2180 (2005); F. H. L. Koens et al., Science 309, 1346 (2005). [6] A. Wallraff et al., Nature (London) 431, 162 (2004); A. Blais et al., Phys. Rev. A 69, (2004). [7] For early ideas involving hybrid imlementations, see A. S. Sorensen et al., Phys. Rev. Lett. 92, (2004); L. Tian et al., Phys. Rev. Lett. 92, (2004). [8] See, e.g., A. M. Steane, Quantum Inf. Comut. 2, 297 (2002). [9] D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information (Sringer, New York, 2000). [10] For a review, see J. Doyle et al., Eur. Phys. J. D 31, 149 (2004), and references cited within. [11] A. Andre et al., quant-h/ [12] M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003); C. W. Chou et al., Nature (London) 438, 828 (2005); T. Chanelièrea et al., ibid. 438, 833 (2005); M. D. Eisaman et al., ibid. 438, 837 (2005); B. Julsgaard et al., ibid. 432, 482 (2004). [13] C. W. Gardiner and P. Zoller, Phys. Rev. A 55, 2902 (1997); A. S. Bradley and C. W. Gardiner, J. Phys. B 35, 4299 (2002). [14] Note that magnetic traing achieved using localized magnetic fields near the nodes of the striline resonator should not significantly degrade the quality of the suerconducting cavity. [15] J. van Veldhoven, H. L. Bethlem, and G. Meijer, Phys. Rev. Lett. 94, (2005). [16] P. Rabl et al. (to be ublished); P. Xue et al. (to be ublished)