Robust open-loop stabilization of Fock states by time-varying quantum interactions
aa r X i v : . [ qu a n t - ph ] O c t Robust open-loop stabilization of Fockstates by time-varying quantuminteractions ⋆ Alain Sarlette ∗ Pierre Rouchon ∗∗∗
SYSTeMS, Ghent University, Technologiepark Zwijnaarde 914, 9052Zwijnaarde, Belgium (e-mail: [email protected]) ∗∗ Centre Automatique et Syst`emes, Mines ParisTech, 60 boulevardSaint Michel, 75006 Paris, France(e-mail: [email protected])
Abstract:
A quantum harmonic oscillator (spring subsystem) is stabilized towards a target Fockstate by reservoir engineering. This passive and open-loop stabilization works by consecutive andidentical Hamiltonian interactions with auxiliary systems, here three-level atoms (the auxiliaryladder subsystem), followed by a partial trace over these auxiliary atoms. A scalar control inputgoverns the interaction, defining which atomic transition in the ladder subsystem is in resonancewith the spring subsystem. We use it to build a time-varying interaction with individualatoms, that combines three non-commuting steps. We show that the resulting reservoir robustlystabilizes any initial spring state distributed between 0 and 4¯ n + 3 quanta of vibrations towardsa pure target Fock state of vibration number ¯ n . The convergence proof relies on the constructionof a strict Lyapunov function for the Kraus map induced by this reservoir setting on the springsubsystem. Simulations with realistic parameters corresponding to the quantum electrodynamicssetup at Ecole Normale Sup´erieure further illustrate the robustness of the method. Keywords: open-loop control systems, discrete-time, Lyapunov function, stabilization methods,photons, physics, Hilbert spaces.1. INTRODUCTIONThe last decades have seen a surge of developments onthe control of systems that feature essential quantum dy-namics (see e.g. Wiseman and Milburn [2009]). A majormotivation is to harness their peculiar possibilities for ITapplications ranging from fundamentally-secure communi-cation, over hyper-precise measurements, to the quantumcomputer (see e.g. Nielsen and Chuang [2000], Harocheand Raimond [2006]). A basic building block for operationswith quantum dynamics is the ability to produce andstabilize a wealth of different ‘target’ states. The extremefragility, limited measurement and control possibilities inquantum systems make this already a challenging task formany target states with non-trivial (and thus interesting)properties.Since an isolated quantum system fundamentally followspure Hamiltonian dynamics, asymptotic stabilization nec-essarily requires interaction with the external world andrelies on the theory of open quantum systems. Such in-teractions can involve measurement and feedback action,called measurement-based feedback , or also control by tai-lored interaction as broadly advocated in Willems [1995] ⋆ This paper presents research results of the Belgian NetworkDYSCO (Dynamical Systems, Control, and Optimization), fundedby the Interuniversity Attraction Poles Programme, initiated bythe Belgian State, Science Policy Office. The authors were partiallysupported by the ANR, Projet Blanc EMAQS ANR-2011-BS01-017-01 and Projet C-QUID BLAN-3-139579. and called coherent feedback in the quantum context.Measurement-based feedback introduces specific difficul-ties because quantum measurement is fundamentally lim-ited to partial state knowledge and always perturbs themeasured system (“back-action”). Controller design there-fore always needs to follow an interactions-based reasoning(see e.g. Dotsenko et al. [2009], Sayrin and et al. [2011]).Coherent feedback involves a specific structure, based onjoint evolution on a tensor product of the interacting sub-systems that leads to dissipative and/or stochastic quan-tum dynamics for the target subsystem (see e.g. Jamesand Gough [2010], Kerckhoff et al. [2010].
Reservoir engineering is a systematic method related tocoherent feedback for open loop stabilization of quantumsystems through tailored interaction. For discrete-timequantum systems, the target system repeats the sameinteraction with a succession of auxiliary systems, whichare discarded after interaction. The dynamics on the tar-get system resulting from repeating the same interactionprocess, takes the form of a Kraus map, see Kraus [1983].A random Kraus map would generally drive the systemto a mixed quantum state (featuring classical uncertain-ties). For reservoir action, interaction with each individualauxiliary system must be tailored such that the Krausmap has a desired pure state as asymptotically stableequilibrium. General techniques have been proposed totailor Kraus maps, provided any chosen unitary evolutionscan be applied to the target subsystem by controlling itsHamiltonian (kind of ‘complete actuation’ on the sub-ystem Hamiltonian, see e.g. Ticozzi and Viola [2009],Bolognani and Ticozzi [2010]). There are however manysystems where control over the Hamiltonian is instead veryrestricted.This paper considers such a situation, where the tar-get system is an electromagnetic field mode, interactingwith three-level atoms as auxiliary systems. This quan-tum electrodynamics situation is a prototype for so-calleduniversal spring-spin systems (see e.g. Haroche and Rai-mond [2006]). The spin-spring Hamiltonian is controlledby shifting the frequencies of the atomic transitions asa function of time, through the Stark effect. This yieldsa very low-dimensional input signal to tailor the targetsystem’s evolution governed by the resulting Kraus map.Sarlette et al. [2011] propose a method to stabilize so-called ‘Schr¨odinger cat states’ with this setup. It buildsthe overall unitary interaction operator as a symmetricproduct of non-commuting basic operators, by varying theinput signal during the field’s interaction with each atom.We prove here that stabilizing pure photon states (‘Fockstates’) is also possible. Rempe et al. [1990] have proposeda method based on ‘trapping state conditions’, where theauxiliary systems are two-level atoms interacting reso-nantly with the quantized field mode: the target Fock stateis an equilibrium of the Kraus map but it is a saddlepoint with stable manifold in some but not all relevantdirections. The present paper proposes a modification ofthis method that makes the target Fock state a stableequilibrium of the Kraus map in the most relevant Hilbertsubspace. We therefore build an interaction that combinesthe ‘trapping state’ approach of Rempe et al. [1990] withthe construction of symmetric products of non-commutingoperators from Sarlette et al. [2011].We give formal convergence properties for the resultingscheme, both in absence and in presence of a disturbingenvironment (Theorem 1 based on the construction of astrict Lyapunov function, and Proposition 2). In section 2we define the target system, the auxiliary three-levelatoms, the interaction Hamiltonian with its scalar controlinput u and the associated Kraus map. In section 3, theoperators appearing in the Kraus map are computed forour open-loop piecewise constant control (10). Section 4 isdevoted to convergence analysis. Simulations in Section 5briefly explore the influence of parameter uncertainties andillustrate the robustness for realistic QED parameters.2. SYSTEM DESCRIPTIONWe consider a quantum electrodynamics setup as de-scribed e.g. in Haroche and Raimond [2006]. The targetsystem is a field mode, with infinite-dimensional Hilbertspace H = span( | i , . . . , | n i , ... ) and free Hamiltonian H f = + ∞ X n =0 n ω f | n ih n | (1)where ω f > n gives the photon number. This paper denotesby | n i the pure photon number eigenstates, called Fockstates . The photon number operator is defined by N = P + ∞ n =0 n | n ih n | = H f / ω f . We will further use the notation f N = f ( N ) = + ∞ X n =0 f ( n ) | n ih n | = + ∞ X n =0 f n | n ih n | for an operator that is diagonal in the Fock basis, for anyfunction f : N C . We denote by H n n the Hilbertsubspace spanned by Fock states | n i , | n + 1 i , ..., | n i .A stream of identical atoms consecutively interact withthis field mode according to the Jaynes-Cummings model,playing the role of a ‘reservoir’ to stabilize the field in atarget state. We consider three atomic levels | g i , | e i , | m i ,with free Hamiltonian H a = ω m | m ih m | − ω g | g ih g | (2)up to an irrelevant term that is a multiple of the iden-tity operator. The transition frequencies between levels( | g i , | e i ) and ( | e i , | m i ), that is ω g ∈ R and ω m ∈ R respec-tively, depend on an input u ∈ R . The latter represents anenergy shift induced on | e i by an external field through aStark effect, such that ω g ( u ) = ω g + u and ω m ( u ) = ω m − u . (3)The constants ω g and ω m are the transition pulsationsin absence of external field. For simplicity, the followingassumes that ω g ( u ) > ω m ( u ) > | g i correspondsto the lowest atomic level whereas | e i and | m i are thefirst and second excited states (3-level ladder system).The proposed method can be adapted to other energyarrangements, like V-structures, by adding short externalcontrol pulses that switch the atomic state at specificpoints in our scheme.To make atomic transitions and field mode interact, weconsider the following realistic situation, in the spirit ofSantos and Carvalho [2011]: | ω g − ω f | , | ω m − ω f | ≪ ω f ; | u | ∼ | ω g − ω f | , | u | ∼ | ω m − ω f | . Then with the standardrotating wave approximation, the atom-field interaction isdescribed by the Hamiltonian: H c = i Ω2 ( a † ( | g ih e | + | e ih m | ) − a ( | e ih g | + | m ih e | ) ) . (4)Here i = √− g, e ) and( e, m ); photon annihilation operator a is defined by a = P + ∞ n =1 √ n | n -1 ih n | in the Fock basis; and † denotes theadjoint of an operator (complex conjugate transpose of theassociated matrix). We have the fundamental identities a † a = N , as well as a f ( N ) = f ( N + I ) a and its adjoint f ( N ) a † = a † f ( N + I ) for any function f : N C . Equa-tion (4) essentially expresses that atomic state can raisefrom | g i to | e i or from | e i to | m i by absorbing one photonfrom the field, or fall inversely by releasing one photon.The propagator U , expressing the transformation that thejoint atom-field state undergoes during interaction, followsthe Schr¨odinger equation ddt U ( t ) = − i ( H f + H a + H c ) U ( t ) (5)with initial condition U ( t ) = I the identity operator.A standard change of variables | ψ i → e iH f t | ψ i on fieldstate and ( | g i , | e i , | m i ) → ( e − iω f | g i , | e i , e iω f | m i ) onatomic states leads to ‘interaction coordinates’. In thesecoordinates the propagator follows ddt U ( t ) = − i H JC U ( t ) (6) We take the convention, especially useful in quantum mechanics,that N includes 0. here the Jaynes-Cummings
Hamiltonian writes H JC ( u ) = (∆ m − u ) | m ih m | − (∆ g + u ) | g ih g | + H c (7)with ∆ m = ω m − ω f and ∆ g = ω g − ω f . The goalbeing to allow separate interactions of the two atomictransitions with the field, we assume that the | m i and | g i levels are attributed sufficiently different frequenciesin H JC , i.e. | ∆ g + ∆ m | =: ∆ ≫ Ω. Note that H JC actsin parallel on a set of decoupled subspaces spanned by( | g i ⊗ | n + 1 i , | e i ⊗ | n i , | m i ⊗ | n − i ). The associatedmatrix operators are thus block-diagonal with blocks ofsize 3 × T s seconds,to undergo the same interaction with the field. We denotethe field mode density operator just before interacting withthe ( k + 1)th atom by ρ k . The initial state | u at i ∈ C ofthe atoms can be chosen, as well as the Stark detuningsignal u ( t ) during interaction time [0 , T ], with T ≤ T s .Denote U T the solution at time T of (6) with U (0) = I and with the chosen u ( t ), governing a time-varying H JC in (7). Then the atom-field joint state just after the k +1th interaction is given by U T ( ρ k ⊗ | u at ih u at | ) U T . Thisgenerally corresponds to an entangled situation. Since wedo not measure the final atomic state, the expected fieldevolution follows the Kraus map ρ k +1 = Φ( ρ k ) = M g ρ k M † g + M e ρ k M † e + M m ρ k M † m , (8)where the operators M g , M e , M m acting only on the fieldmode are identified from U T and | u at i : ∀| ψ i ∈ H , U T | ψ i| u at i = M g | ψ i | g i + M e | ψ i | e i + M m | ψ i | m i . The goal of open-loop stabilization by reservoir engineer-ing is to select u ( t ) and | u at i such that the dynamics (8)asymptotically stabilize a target pure state ρ k → ρ ∞ = | ψ ∞ ih ψ ∞ | . 3. CONTROL DESIGNOur objective is to stabilize a given Fock state | ψ ∞ i = | ¯ n i .Rempe et al. [1990] have noted that in absence of the | m i level (i.e. | ∆ m − u | ≫ Ω), a single atomic transition( | g i , | e i ) in perfect resonant interaction with the field(i.e. ∆ g + u = 0) allows to “trap” states below | ¯ n i . Thecorresponding propagator (obtained by direct integrationof (6) which is then 2 × U r = cos( θ r √ N ) | g ih g | + cos( θ r √ N + I ) | e ih e | (9) − a sin( θ r √ N ) √ N | e ih g | + sin( θ r √ N ) √ N a † | g ih e | , where θ r = t r Ω is the interaction strength integrated overchosen interaction time t r . Then taking θ r = 2 π/ √ ¯ n + 1,the operator in (9) implies no exchange between jointstate components | g i ⊗ | ¯ n + 1 i and | e i ⊗ | ¯ n i . The sub-space H ¯ n spanned by Fock states | i , | i , ... | ¯ n i then re-mains decoupled from the rest of the field Hilbert spacethroughout reservoir action. Now take | u at i = | e i , thus M g = sin( θ r √ N √ N a † , M e = cos( θ r √ N + I ) and M m = 0.Then ρ k following (8) starting from ρ with support in H ¯ n converges to the trapping state | ¯ n ih ¯ n | [Haroche andRaimond 2006, page 210]. But if the support of ρ isin H n +3¯ n +1 , then ρ k converges to | n + 3 ih n + 3 | as the field gets continuously excited. Therefore any fraction ofdensity that is pushed above | ¯ n i is lost away to highphoton numbers. Since perturbations will always inducetransitions between nearby energy states, this makes themethod of Rempe et al. [1990] unusable in practice as itleaves equilibrium | ¯ n i unstable in important directions.A robust stabilization method should enlarge the basin ofattraction of | ¯ n ih ¯ n | to include at least any ρ with supportin H n n for some n < ¯ n < n .We achieve such stabilization with n = 0 and n = 4¯ n + 3(see Theorem 1) by exploiting the possibility of varying u ( . ) during the interaction. Specifically, we take u ( t ) = − ∆ g for t ∈ [0 , ( T − t s ) / m for t ∈ [( T − t s ) / , ( T + t s ) / − ∆ g for t ∈ [( T + t s ) / , T ] . (10)Fast set-point changes can indeed suitably be experimen-tally implemented. The switching time t s and overall in-teraction time T will be tuned to optimize operation. Westill take | u at i = | e i as initial atomic state.The propagator U T , solution of (5) with u given by(10), readily writes U T = U U U where U =exp[ − iH JC ( − ∆ g ) ( T − t s ) /
2] is the solution at t = ( T − t s ) / t = 0, with constant u = − ∆ g ;and U = exp[ − iH JC (∆ m ) t s ] is the solution at t = t s of(5) starting at t = 0, with constant u = ∆ m . We computethose operators using quantum Hamiltonian perturbationtheory on the decoupled subspaces spanned by ( | g i ⊗ | n +1 i , | e i ⊗ | n i , | m i ⊗ | n − i ) and neglecting terms of orderΩ / ∆ ≪
1. Up to this approximation, the states | m i⊗| n − i (resp. | g i ⊗ | n + 1 i ) remain decoupled from the rest for U (resp. U ). One gets: U T | e i = M g z }| { a † (cid:16) e i ∆ t s + cos θ √ N (cid:17) sin( θ √ N + I )2 √ N + I | g i + M e z }| {(cid:16) cos θ √ N + I cos θ √ N − e i ∆ t s sin θ √ N + I (cid:17) | e i− M m z }| { e − i ∆( T - t s ) / a sin θ √ N √ N cos θ √ N + I | m i (11)with θ = Ω ( T − t s ) / θ = Ω t s . To make thetarget field state | ¯ n i invariant under the Kraus map (8),we make it invariant under U T | e i . This is achieved withthe trapping-like condition:( T − t s ) = 2 π Ω √ ¯ n + 1 ⇔ θ = π √ ¯ n + 1 . (12)For simplicity, we take ∆ t s = 0 nominally . The value of∆( T − t s ) is irrelevant as it drops out of the Kraus mapfor the field evolution. The value of θ remains to be fixed. To this end, the value of ∆ can be slightly tuned by adjusting thetrap that governs field mode frequency ω f . Indeed since ∆ ≫ Ω,a small shift in ∆ allows to sensibly tune ∆ t s for all t s that yieldnon-negligible values of θ = Ω t s . . CONVERGENCE ANALYSISCondition (12) makes the subspace H (4¯ n +3)0 invariant by M g , M e and M m . We denote P (4¯ n +3)0 the orthogonalprojection onto H (4¯ n +3)0 . Consider a candidate Lyapunovfunction of the form V ( ρ ) = trace( f N ρ ), for some function f ( n ) to be determined. Then (8) and (11) lead to V (Φ( ρ )) − V ( ρ ) =trace (cid:16) ρ cos ( α N ) sin ( β N ) ( f ( N − I ) − f ( N ) ) (cid:17) + trace (cid:16) ρ sin ( α N ) cos ( β N ) ( f ( N + I ) − f ( N ) ) (cid:17) with α N = π q N + I ¯ n +1 and β N = θ √ N /
2. Note that the lastline of the above equation vanishes for ρ = | n + 3 ih n + 3 | and any function f N , reflecting the decoupling of H (4¯ n +3)0 from the remainder of the Hilbert space. To formallyrestrict ourselves to H (4¯ n +3)0 , we can take f ( n ) = f (4¯ n +3)for all n > n +3, such that f ( N − I ) − f ( N ) and f ( N + I ) − f ( N ) vanish on H + ∞ (4¯ n +4) . Further take η ∈ (0 , f (¯ n ) = 0, f (¯ n + 1) = f (¯ n −
1) = 1 and set f ( n −
1) = f ( n ) + η sin α n cos β n ( f ( n ) − f ( n +1))for 0 < n < ¯ n ,f ( n +1) = f ( n ) + η sin β n ( f ( n ) − f ( n − n < n < n + 3 . Then V (Φ( ρ )) − V ( ρ ) = trace( q N ρ ) with q n = 0 for n = ¯ n , n > n + 3, and q n = sin α n cos β n ( η sin β n −
1) ( f ( n )- f ( n +1))for 0 ≤ n < ¯ n , (14) q n = sin β n cos α n ( η sin α n cos β n −
1) ( f ( n )- f ( n -1))for ¯ n < n ≤ n + 3 . We then have the following convergence result.
Theorem 1.
Consider the dynamics (8) where M g , M e and M m are defined by (11) with (12), and its restriction todensity operators ρ with support in H (4¯ n +3)0 . Assume that θ = kπ/ √ n for all ( n, k ) ∈ { , ..., n + 3 } × N . Then V ( ρ )built with (13) is a strict Lyapunov function: for any ρ with support in H (4¯ n +3)0 , ρ k converges towards the fixedpoint ρ ∞ = | ¯ n ih ¯ n | . Proof:
Thanks to the assumption on θ , (13) implies f ( n ) > f ( n −
1) for ¯ n < n ≤ n + 3 and f ( n ) > f ( n + 1)for 0 < n < ¯ n . Then the same assumption ensures that q n is strictly negative for all n ≤ n +3 except n = ¯ n . Writing ρ = P j p j | ψ j ih ψ j | (spectral decomposition with p j > P j p j = 1) then yields V (Φ( ρ )) − V ( ρ ) = X j p j h ψ j | q N | ψ j i < P (4¯ n +3)0 | ψ j i = s j | ¯ n i with s j ∈ C for all j , implying P (4¯ n +3)0 ρ P (4¯ n +3)0 = p | ¯ n ih ¯ n | for some p ∈ R . Thus V is astrict Lyapunov function in H (4¯ n +3)0 as stated. (cid:3) Remark:
The above construction of f ( n ) can in fact beextended to n < n + 8. For generic θ , such that sin(2 β n ) never vanishes, the resulting V is then a non-strict Lya-punov function on the invariant Hilbert subspace H (9¯ n +8)0 ,with the Lasalle invariance principle ensuring convergenceof ρ to a ρ ∞ supported on the subspace spanned by | m i , | m + 8 i . Considering even larger Hilbert spaces, oneshows that ρ converges to a state with nonzero populationonly on Fock states | (2 l + 1) (¯ n + 1) − i for l integer.The experiments working in very low temperature envi-ronments ensure that decoherence usually pulls the fieldstate towards vacuum (Fock state | i ) and depopulateshigh-number Fock states.Tailored interaction (10) with the atomic stream thusmakes the otherwise isolated field converge to | ¯ n i forvirtually all θ >
0. This confirms the interest of thissymmetric product operator approach. We next analyzehow the latter can be optimized to strengthen convergencew.r.t. external disturbances. We therefore add to theevolution model a typical relaxation disturbance, alsocalled decoherence: evolution between consecutive atomicsamples becomes ρ k +1 = Φ( ρ k ) − Γ − [ N Φ( ρ k ) + Φ( ρ k ) N − a Φ( ρ k ) a † ] (15) − Γ + [( N + I )Φ( ρ k ) + Φ( ρ k )( N + I ) − a † Φ( ρ k ) a ]with Γ + = κn th T s ≪ Γ − = κ (1+ n th ) T s ≪
1. The terms inΓ − and Γ + model interaction with a thermal environmentthat induces photon annihilation and creation respectively.Parameter κ models coupling strength to this environ-ment, while n th is the average number of thermal pho-tons per mode in the environment. The invariant-subspacestructure no longer rigorously holds, but we can invoke aphysical argument to still truncate our computations at n ≤ n + 8, ensuring that the discretized model remainsvalid and all parameters in the following are well-behaved.Writing the operator ρ k as a matrix with components[ ρ k ] a,b , for a ≥ b ≥
0, the evolution (15) couplescomponent [ ρ k ] a,b only to components [ ρ k ] a + l,b + l i.e. onthe same diagonal of the matrix. To analyze how thefidelity h ¯ n | ρ k | ¯ n i to target | ¯ n ih ¯ n | evolves, we may thusreduce our investigation to the vector r k containing theprincipal diagonal of ρ k , that is [ r k ] a = [ ρ k ] a,a for a =0 , , , ... . Evolution (15) yields r k +1 = B · A · r k with A and B tridiagonal real positive non-symmetricmatrices, representing reservoir and decoherence respec-tively. The upper, lower and principal diagonals of B (respectively A ) have elements b n = Γ − n (resp. d n =sin β n cos α n ) for n ∈ { , , ... } ; c n = Γ + n (resp. e n =sin α n cos β n ) for n ∈ { , , , ... } ; and 1 − c n − b n (resp. 1 − d n − e n ). A and B are thus column-stochastic,reflecting conservation of trace( ρ ) = T r where is thevector of all ones. Therefore B · A has at least one eigen-value 1. The corresponding 1-eigenvector r ∗ characterizesphoton populations for a field pointer state under reservoirand decoherence.We try to approximate it by viewing B · A as a smallperturbation of A . Denote x the r -vector correspondingto | ¯ n ih ¯ n | , let R = A − I and R = ( B − I )( A − I )+ ( B − I ).Then we know that R x = 0 with T x = 1, i.e. x is thesteady-state solution r ∗ when B = I (no perturbation). Inresence of a thermal environment, we get r ∗ = x + x solution of( R + R )( x + x ) = 0 , T x = 0 . (16) Proposition 2.
Approximating R x ≈
0, problem (16)can be explicitly solved. It gives 1+ x n as estimated fidelityof ρ ∞ to our goal, where − x n = b ¯ n e ¯ n − n X m =1 m Y l =2 d ¯ n − l +1 e ¯ n − l + c ¯ n d ¯ n +1 X m> m Y l =2 e ¯ n + l − d ¯ n + l . An approximation error of order ( x n ) is expected. Proof:
Solving (16) approximated (note that d ¯ n = e ¯ n = 0)fixes the components of x , except x n which remains free: x n − = b ¯ n /e ¯ n − ; x n = d n +1 e n x n +1 ∀ n < ¯ n − x n +1 = c ¯ n /d ¯ n +1 ; x n = e n − d n x n − ∀ n > ¯ n + 1 . Then (16) determines x n < (cid:3) Proposition 2 can be used to optimize θ for maximalfidelity in presence of decoherence, see next Section. Notethat θ = 0 (the case of Rempe et al. [1990]) would yield d n = 0 for all n . The small- x approximation would thenlead to an invalid recurrence, suggesting that decoherenceleads to large x i.e. the method of Rempe et al. [1990] ispoorly robust. 5. SIMULATIONSFor the simulations we consider realistic parameters cor-responding to the cavity quantum electrodynamics setupat Ecole Normale Sup´erieure, Paris. See e.g. Haroche andRaimond [2006], Del´eglise and et al. [2008] for detailedexplanations. Vacuum Rabi frequency is given by Ω / π =50 kHz and we take ∆ ≈ u given by(10). For each simulation we slightly adapt ∆ to opti-mize fidelity, see footnote 2. Evolution of ρ k is computedstarting from a vacuum initial state ρ = | ih | . Figure 1illustrates the good working principle of the method, de-spite our approximate reasoning: ρ k +1 = Φ( ρ k ) convergesessentially exactly to ρ ∞ = | ¯ n ih ¯ n | for all ¯ n ∈ { , , ..., } .The simulations are run with an arbitrary θ = 1 / √ ¯ n .We next add decoherence. We take 1 /κ = 0 . s and cryo-genic n th = 0 .
05, corresponding to current high-standardexperiments. This environment-induced decoherence is incompetition with our reservoir strength, that is mainlythe time between consecutive atoms. The latter can re-alistically be set to T s = 60 µ s, but there is only a0 . ρ k to ρ k +1 is thus similar to (a less approximated versionof) (15) except that operator 0 . I + 0 . ρ with the reservoir of Rempe et al. [1990] (top) andwith our symmetric-interaction reservoir (bottom), for a Although the required values of θ and θ allow smaller T , we areexperimentally limited to periods of the order of T s . h ¯ n | ρ k | ¯ n i Fig. 1. Fidelity h ¯ n | ρ k | ¯ n i of ρ k to target state | ¯ n i as afunction of atomic interaction k , for ¯ n = 1 , , ..., − = Γ + = 0i.e. no environmental disturbance.target ¯ n = 3. For the latter, after the | i state has builtup, it progressively drives away towards | i = | n + 3 i and further to higher photon numbers (here accumulatingwhere our Hilbert space is truncated). In contrast ourscheme clearly stabilizes a state close to | i . Achievablesteady-state fidelities are represented on Fig. 3. We alsoshow the values estimated from Proposition 2. They agreewith simulations within the expected approximation error,confirming our theoretical analysis. The selected optimalvalues of θ seem to satisfy θ √ ¯ n ≈ π/ θ appear to give no detectable effect, indicating a rather flatoptimum in θ . Absolute errors of ± π/ t s also barelyaffect fidelity (less than 1%). The fidelities obtained withour reservoir subject to errors in θ (and still in presenceof decoherence) are represented for each ¯ n by the 3rd and4th bars from left on Fig. 3. Relative errors of ±
2% havea detectable but tolerable effect; especially, setting θ inan interval centered slightly below the ideal value seemsto be a good robustness compromise. The same ±
2% erroron θ would be drastically detrimental with the schemeof Rempe et al. [1990]: even in absence of a disturbingenvironment (Γ − = Γ + = 0) the fidelity to | ¯ n i wouldthen quickly decrease towards zero, dropping to about 0 . . .
01 after 0 .
25 s. Overall ourconstruction of atom-field interaction as a product of non-commuting transformations thus leads to a significantlymore robust stabilization scheme.6. CONCLUSIONThis paper proposes an ‘engineered reservoir’ to stabilizeFock states of a quantum harmonic oscillator thanks to itsinteraction with a stream of three-level systems. We use asingle control signal to tailor a time-varying Hamiltonianinteraction. We prove that the resulting propagator yieldsa Kraus map that robustly ‘traps’ the spring state on adesired Fock state. This seems to be a potentially practicalopen-loop alternative to measurement-based feedback forachieving high-fidelity stabilization of Fock states. Theproposed method admits several variations for experimen-tal implementation. Among others, a similar stabilizingeffect is obtained by using a stream of two-level atoms eachundergoing one of two different interactions. Given thepractical possibilities shown in the present and previous
Fig. 2. Evolution of the diagonal elements of ρ k in presenceof a disturbing environment like in (15), with thereservoir of Rempe et al. [1990] (top) and with oursymmetric-interaction reservoir (bottom), for ¯ n = 3.Graduation is given as a function of time = k · µ s. h ¯ n | ρ k | ¯ n i Fig. 3. Fidelity h ¯ n | ρ ∞ | ¯ n i of the steady state ρ ∞ in presenceof a disturbing environment to target states | ¯ n i , for¯ n = 1 , , ...,
8. The left-most bars represent fidelitywith our simulated reservoir, the second are the valuespredicted by Proposition 2, and the right-most onesare obtained after 4 s with the reservoir of Rempeet al. [1990]. The third and fourth bars from leftcorrespond to the same conditions as the first ones,but incorporating a systematic error of −
2% and +2%respectively on θ . papers, future work could further investigate systematicmethods for designing products of interactions that ro-bustly stabilize quantum states.ACKNOWLEDGEMENTSThe authors thank Michel Brune, Igor Dotsenko and Jean-Michel Raimond from Ecole Normale Sup´erieure for usefuldiscussions. REFERENCESS. Bolognani and F. Ticozzi. Engineering stable discrete-time quantum dynamics via a canonical QR decompo-sition. IEEE Trans.Aut.Cont. , 55(12):2721–2734, 2010.S. Del´eglise and I. Dotsenko et al. Reconstruction ofnon-classical cavity field states with snapshots of theirdecoherence.
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