Dispersive-dissipative control strategy for quantum coherent feedback *

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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Dispersive-dissipative control strategy Dispersive-dissipative control strategy  Dispersive-dissipative control strategy quantum coherent feedback  quantum coherent feedback quantum coherent feedback ∗ ∗

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Yoshiki Kashiwamura ∗ Naoki Yamamoto ∗ Yoshiki Kashiwamura ∗∗ Naoki Yamamoto ∗∗ Yoshiki Yoshiki Kashiwamura Kashiwamura Naoki Naoki Yamamoto Yamamoto ∗ Department of Applied Physics and Physico-Informatics, ∗ ∗ Department of Applied Physics and Physico-Informatics, Applied Physics and Keio University,of Yokohama, Japan ∗ Department Department ofHiyoshi Applied 3-14-1, PhysicsKohoku, and Physico-Informatics, Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama, Japan Keio University, Hiyoshi 3-14-1, Kohoku, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama, Yokohama, Japan Japan

Abstract: To control a quantum system, recently the coherent feedback method has attracted Abstract: To control quantum system, recently the feedback method has attracted Abstract: To a quantum system, recently coherent feedback has wide attention thanksa its several advantages overcoherent the conventional measurement-based Abstract: To control control a to quantum system, recently the the coherent feedback method method has attracted attracted wide attention thanks to its several advantages over the conventional measurement-based wide attention thanks to its advantages over the conventional measurement-based feedback method. However, for several nonlinear quantum systems, no systematic guiding principle for wide attention thanks to its several advantages over the conventional measurement-based feedback method. However, nonlinear quantum systems, no systematic guiding principle for feedback method. However, for nonlinear quantum systems, no systematic guiding for designing a coherent feedbackfor controller has been known, while the measurement case one can feedback method. However, for nonlinear quantum systems, nofor systematic guiding principle principle for designing a coherent feedback controller has been known, while for the measurement case one can designing a coherent feedback controller has been known, while for the measurement case one can employ the so-called quantum non-demolition (QND) feedback method as a systematic strategy. designing a coherent feedback controller has been known, while for the measurement case one can employ the so-called quantum non-demolition (QND) feedback method as a systematic strategy. employ the quantum non-demolition feedback method as In this paper, we develop a coherent feedback(QND) counterpart of this QND-based control strategy. employ the so-called so-called quantum non-demolition (QND) feedback method as aa systematic systematic strategy. In this paper, we develop a coherent feedback counterpart of this QND-based control strategy. In this paper, we develop a coherent feedback counterpart of this QND-based control strategy. The effectiveness of this proposal is evaluated in some control problems; qubit stabilization and In this paper, we develop a coherent feedback counterpart of this QND-based control strategy. The effectiveness of this proposal is evaluated in some control problems; qubit stabilization and The effectiveness of this proposal is evaluated in some control problems; qubit stabilization single photon production. The of this proposal is evaluated in some control problems; qubit stabilization and and singleeffectiveness photon production. production. single photon single production. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2017,photon IFAC (International Keywords: Quantum control, coherent feedback, single photon Keywords: Quantum Quantum control, coherent coherent feedback, single single photon Keywords: Keywords: Quantum control, control, coherent feedback, feedback, single photon photon 1. INTRODUCTION et al. (2013)]. The basic idea of this control method is 1. INTRODUCTION INTRODUCTION et al. (2013)]. The basic of this control method is 1. et al. (2013)]. The idea of method is as follows. First, QND idea measurement is performed 1. INTRODUCTION et al. (2013)]. Theaa basic basic idea of this this control control method on is as follows. First, QND measurement is performed on as follows. First, a QND measurement is performed on system continuously in time, which probabilistically To control a quantum system via feedback, we have two the as follows. First, a QND measurement is performed on the system system continuously in time, one which probabilistically To control control quantum system via via feedback, we have have two the continuously time, which the system state in towards of probabilistically the eigenstates To aa system feedback, we options in designing a controller; that is, the measurementthe system continuously in time, one which probabilistically To control a quantum quantum system via feedback, we have two two moves moves the system state towards of the eigenstates options in designing a controller; that is, the measurementmoves the system state towards one of the of the measured physical variable (i.e., observable); then options in designing a controller; that is, the measurementbased feedback (MF) scheme and the non-measurement moves the system state towards one of the eigenstates eigenstates options in designing a controller; that is, non-measurement the measurement- of the measured physical variable (i.e., observable); then basedwhich feedback (MF) scheme and the of the measured physical variable (i.e., observable); then the measurement result is fed back to compensate this based feedback (MF) scheme and the non-measurement one is called the coherent feedback (CF). The basic of the measured physical variable (i.e., observable); then based feedback (MF) scheme and the non-measurement the measurement result is fed back to compensate this one which is called the coherent feedback (CF). The basic the measurement result is fed back to compensate this stochastic displacement, which as a result generates the one which is called the coherent feedback (CF). The basic strategy of MF is that, we first measure the target system measurement result is fed back to compensate this one which isMF called the we coherent feedbackthe (CF). Thesystem basic the stochastic displacement, which as a result generates the strategy of is that, first measure target stochastic displacement, which as a result generates target eigenstate deterministically. strategy of MF is that, we first measure the target system and fed the measurement result back to control it. While stochastic displacement, which as a result generates the the strategy of MF is that, we result first measure the target system target eigenstate deterministically. and fed the measurement back to control it. While target eigenstate deterministically. and fed the measurement result back to control it. While the theory MF has been well established and fed the of measurement result back to control[Wiseman it. While target eigenstate deterministically. the theory of MF has been well established [Wiseman the theory MF been well [Wiseman and Milburnof (2009); Jacobs and some important the MF has has been(2014)] well established established [Wiseman and theory Milburnof (2009); Jacobs (2014)] and some important and Milburn (2009); Jacobs (2014)] and some important experiments have been demonstrated [Sayrin et al. (2011); and Milburn (2009); Jacobs (2014)] and some important experiments have beenwedemonstrated demonstrated [Sayrin et al. al. (2011); (2011); experiments have been [Sayrin et Vijay et al. (2012)], need to recall its critical drawexperiments beenwedemonstrated [Sayrin et al. (2011); Vijay which et al. al. have (2012)], need tothe recall its critical critical drawVijay et (2012)], we need to recall its drawback, as a result limits control performance. Vijay et al. (2012)], we need to recall its critical drawback, is, which as aathe result limitsloop theinvolves control some performance. back, which as result limits the control performance. That because feedback classical back, which as a result limits the control performance. That is, is, because because the feedback feedback loop involvesand some classical That the involves some classical components (detectors, signal loop processors, actuators), That is, because the feedback loop involvesand some classical components (detectors, signal processors, actuators), components signal processors, and actuators), the controller(detectors, always suffers from significant signal loss components (detectors, signal processors, and actuators), the controller controller always suffers fromreason significant signal loss loss the always from significant signal and time delays. This issuffers the main why recently the Fig. 1. The schematic of the proposed CF control, for the the controller always fromreason significant signal loss and control time delays. delays. This is issuffers the main why recently the Fig. 1. The schematic of the proposed CF control, for the and time This the main reason why recently the CF has attracted much attention as an alternative 1. of CF and time delays. This is the main reason why recently the Fig. problem of generating squeezed state. CF control has attracted much attention as an alternative 1. The The schematic schematic of the theaa proposed proposed CF control, control, for for the the CF control attracted much attention as an alternative method for has feedback control; that is, because in principle Fig. problem of generating squeezed state. problem of generating a squeezed state. CF control has attracted much attention as an alternative method for feedback control; that is, because in principle problem of generating a squeezed state. method for because in any CF loop does notcontrol; involvethat any is, classical component, it In this paper, we propose a CF version of the above method for feedback feedback that because in principle principle anybasically CF loop loop doesfrom notcontrol; involve any is, classical component, it In this paper, we propose a CF version of the above any CF does not involve any classical component, it is free the above-mentioned drawback. In In this we aa CF of the any CF loop does not involve any classical component, it mentioned QND-based approach forversion controller and is basically basically free from the above-mentioned above-mentioned drawback. In In this paper, paper, we propose propose CF version of design the above above is free from the drawback. In fact recently some important progress both in theory and mentioned QND-based approach for controller design and mentioned QND-based approach for controller design and is basically free from the above-mentioned drawback. In investigate its effectiveness with some examples. The confact recently recentlyhave somebeen important progress both in in theory theory and mentioned QND-based approach for controller design and fact some important progress both and experiment reported in the research field of CF investigate its effectiveness with some examples. The coninvestigate its effectiveness with some examples. The confact recently some important progress both in theory and trol procedure of this method is as follows. Figure 1 is experiment have been reported in the research field of CF investigate its effectiveness with some examples. The conexperiment been reported in the research field of CF [James et al.have (2008); Gough and James (2009); Hamerly trol procedure of this method is as follows. Figure 1 is trol procedure of this method is as follows. Figure 1 experiment have been reported in the research field of CF the phase-space representation illustrating the idea espe[James et al. al. (2012); (2008); Iida Gough and JamesYamamoto (2009); Hamerly Hamerly procedure ofrepresentation this method illustrating is as follows. 1 is is [James et (2008); Gough and James (2009); and Mabuchi et al. (2012); (2014); trol the phase-space phase-space theFigure idea espethe representation illustrating the idea espe[James et al. (2008); Gough and James (2009); Hamerly cially for the case of squeezed state generation. (a) First, and et Mabuchi (2012); Iida Iida et et al. al. (2012); (2012); Yamamoto Yamamoto (2014); (2014); the phase-space representation illustrating the idea espeand Mabuchi (2012); Liu al. (2016)]. cially for the case of squeezed state generation. (a) First, cially for case squeezed state First, and Mabuchi (2012); Iida et al. (2012); Yamamoto (2014); an initial state is of prepared. the generation. figure, it is (a) given by Liu et et al. (2016)]. (2016)]. cially for the the case of squeezedIn state generation. (a) First, Liu anvacuum initial state is prepared. In the figure, it is given by initial state is prepared. In the figure, it is given by Liu et al. al. a(2016)]. state. (b) The system interacts with a probe However, systematic designing method of a CF controller aan an initial state is(b) prepared. In theinteracts figure, itwith is given by a vacuum state. The system a probe However, a systematic designing method of a CF controller aa vacuum state. (b) The system interacts with a probe field through a dispersive coupling Hamiltonian, which is However, a systematic designing method of a CF controller is still lacking, in contrast to the case of MF where parvacuum state. (b) The system interacts with a probe However, a systematic designing method of a CF controller field through coupling Hamiltonian, which is is still still lacking, lacking, inlinear contrast tothe thequantum case of of MF MF whereof parparfield through dispersive coupling Hamiltonian, which is joint represented byaaa adispersive Hermitian operator L1 . Then the is in contrast to the case where ticularly for the case version the field through dispersive coupling Hamiltonian, which is is still lacking, inlinear contrast tothe thequantum case of MF whereof par.. we Then the joint represented by aa Hermitian operator L 1 ticularly for the case version the Then the joint represented by Hermitian operator L system-probe state becomes entangled; if measure the 1 ticularly for the linear case the quantum version of the linear quadratic Gaussian [Nurdin et al. (2009)] and . Then the joint represented by a Hermitian operator L 1 ticularly for the Gaussian linear case the quantum versionand of the state becomes if we measure ∞ linear quadratic [Nurdin et al. al. (2009)] (2009)] the system-probe system-probe state becomes entangled; if the probe field, the system stateentangled; probabilistically changesthe to linear quadratic Gaussian [Nurdin et and H control [James et al. (2008)] established. state becomes entangled; if we we measure measure the linear quadratic Gaussian [Nurdin have et al.been (2009)] and the the system-probe ∞ probe field, the system state probabilistically changes to ∞ control [James et al. (2008)] have been established. H probe field, the system state probabilistically changes to one of the eigenstates of L , which corresponds to the QND 1 H et al. (2008)] have been established. Especially, the MF case, the quantum non-demolition ∞ controlin[James field, the system state probabilistically changes to et case, al. (2008)] have been established. probe H controlin[James one of the eigenstates of L , which corresponds to the QND 1 Especially, the MF the quantum non-demolition one of the eigenstates of L , which corresponds to the QND measurement. In the figure L is given by a momentum 1 Especially, in the MF case, the quantum non-demolition (QND) based control for deterministic state stabilization one of the eigenstates of L11 , which corresponds to the QND Especially, in the MF case, the quantum non-demolition measurement. In the figure L is given by a momentum 1 is given bysystem (QND) based control for state stabilization measurement. the figure L aa momentum p, in In which state is (QND) based control deterministic state stabilization works very effectively, for deterministic a wide class of nonlinear quan- operator measurement. In the case figurethe L11conditional is given bysystem momentum (QND) based control for for deterministic state stabilization operator p, in which case the conditional state is works very effectively, for a wide class of nonlinear quanp, in which case the conditional system state aoperator squeezed state. (c) In the MF case, this probabilistic works very effectively, for a wide class of nonlinear quantum systems [Mirrahimi and van Handel (2007); Inoue operator p, state. in which case the MF conditional system state is is works very effectively, for and a wide class of nonlinear quana squeezed (c) In the case, this probabilistic tum systems [Mirrahimi van Handel (2007); Inoue a squeezed state. (c) In the MF case, this probabilistic changes canstate. be compensated by displacing state ustum systems [Mirrahimi and van Handel (2007); Inoue a squeezed (c) In the MF case, this the probabilistic tum systems [Mirrahimi and van Handel (2007); Inoue changes can be compensated by displacing the state us This work was supported in part by JSPS Grant-in-Aid No. changes can by state using the measurement result. Our idea is tothe replace changes can be be compensated compensated by displacing displacing the state this us This work was supported in part by JSPS Grant-in-Aid No. ing the measurement result. Our idea is to replace this  ing the measurement result. Our idea is to replace this stochastic compensation process by a coherent dissipative 15K06151. This work was supported in part by JSPS Grant-in-Aid No.  This work was supported in part by JSPS Grant-in-Aid No. ing the measurement result. Our idea is to replace this stochastic compensation process by aa coherent dissipative 15K06151. stochastic 15K06151. stochastic compensation compensation process process by by a coherent coherent dissipative dissipative 15K06151.

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Proceedings of the 20th IFAC World Congress Yoshiki Kashiwamura et al. / IFAC PapersOnLine 50-1 (2017) 11760–11763 Toulouse, France, July 9-14, 2017

process as follows; that is, the output probe field associated with L1 is fed back to the system again and interacts with it through a dissipative operator L2 , so that the desired dissipation of the system state toward the target eigenstate occurs. This second system-probe interaction corresponds to the feedback, as will be shown in Fig. 2. Figure 1 (c) illustrates the ideal case where, due to a proper dissipation process, all the conditional squeezed states move to a single target squeezed state. Hence the guideline for designing an effective CF controller is to find a dissipation process L2 , in addition to some system Hamiltonian if necessary, so that the above desirable state transition is realized. The aim of this paper is to demonstrate that this controller synthesis method is in fact effective for some typical control problems. Notation: For a matrix or an operator X, X T and X † denote its transpose and Hermitian conjugate, respectively. [X, Y ] = XY − Y X is the commutator. Tr(X) is a trace of X. ⊗ denotes the Kronecker’s or tensor product. 2. PRELIMINARIES 2.1 Open quantum systems The interaction of an open quantum system with (coherent) probe fields A = (A1 , . . . , Am )T is represented by the unitary time evolution U (t) that obeys the following quantum stochastic differential equation [Gardiner and Zoller (1991); Gough and James (2009); Wiseman and Milburn (2009)]:  dU (t) = Tr[(S − I)dΛ(t)T ] + dA(t)† L − L† SdA(t)  −L† Ldt/2 − iHdt U (t), U (0) = I.

S is a scattering matrix which satisfies S † S = SS † = I, L = (L1 , . . . , Lm )T is a vector of coupling operators, and H is a system Hamiltonian. Also, Λ = (λij ) represents the matrix of gauge processes. Hence the open quantum system is generally characterized by the triplet (S, L, H). Due to this interaction, in the Heisenberg picture, the system variable X evolves to X(t) = U † (t)XU (t), the differential form of which is given by  dX(t) = − i[X(t), H(t)] + L(t)† X(t)L(t)  − L(t)† L(t)X(t)/2 − X(t)L(t)† L(t)/2 dt + dA† (t)S † (t)[X(t), L(t)] + [L† (t), X(t)]S(t)dA(t) + Tr[(S † (t)X(t)S(t) − X(t))dΛ(t)T ]. If no measurement is performed on the probe, or equivalently if all the measurement results are averaged out, the statistics of the system variable can be explicitly calculated in terms of the unconditional state ρ(t) as follows; that is, the mean of X(t) is connected to ρ(t) as X(t) = Tr[Xρ(t)], and ρ(t) obeys the master equation m  ρ˙ = −i [H, ρ] + D [Lj ] ρ, (1) j=1

where

1 1 D [Lj ] ρ = Lj ρL†j − L†j Lj ρ − ρL†j Lj . 2 2

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2.2 The coherent feedback configuration Here we provide the explicit form of the CF scheme introduced in Section 1. This is based on the cascade of two open quantum systems G1 = (S1 , L1 , H1 ) and G2 = (S2 , L2 , H2 ), meaning that the output of G1 is connected to the input of G2 . Then the triplet characterizing this system, G1  G2 , is given by [Gough and James (2009)]  G1  G2 = S2 S1 , L2 + S2 L1 ,  1 H1 + H2 + (L†2 S2 L1 − L†1 S2† L2 ) . (2) 2i Now let us assume that G1 and G2 are the same system; more precisely, (S1 , L1 , H1 ) and (S2 , L2 , H2 ) are the operators living in the same Hilbert space. In this case, as shown in Fig. 2, the cascade G1  G2 represents the CF controlled system where the system’s output field generated through the system-field coupling L1 is fed back to the same system through a different system-field coupling L2 . Then, as described in Section 1, our strategy for synthesizing this CF controlled system is first to take a Hermitian operator L1 = L†1 representing a dispersive coupling and set a superposition of the eigenstates of L1 to be the target state; then design a dissipative operator L2 and other operators if necessary, so that the master equation (1) with triplet (2) dynamically generates the target state.

Fig. 2. The CF controlled system based on the cascaded connection. 3. QUBIT STABILIZATION Let us begin with a simple example, a qubit system, to examine whether in fact the proposed CF control method works effectively. Note that Ref. [Balouchi and Jacobs (2016)] also studied a different CF control problem for a qubit. The control goal is to deterministically stabilize the qubit at the spin-up state |1 = (1, 0)T . Then from the CF strategy described above, it would be reasonable to set L1 and L2 as follows:     √ √ √ √ 1 0 0 0 L1 = κσz = κ , L2 = γσ− = γ . 0 −1 1 0

Actually, L1 represents the dispersive coupling such that the probe light field is scattered depending on the qubit state. Also L2 represents the dissipation process of the qubit to the spin-down state |0 = (0, 1)T , which can be interpreted as a spontaneous emission process for a two energy-levels atom. Note that both |0 and |1 are the eigenstates of L1 . Also as for the other parameters, we

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set S1 = S2 = 1 and H1 = H2 = 0. Then the total CF controlled system obtained by Eq. (2) is given by Stot = 1, √  √ √ κ √ 0 Ltot = κσz + γσ− = √ , γ − κ √ κγ † (σ− σz − σz σ− ) Htot = 2i  √  √ κγ κγ 0 −i σy = − . =− i 0 2 2 The unconditional state of the CF controlled system (Stot , Ltot , Htot ) is obtained by solving the corresponding master equation (1). In particular, the steady state solution in this case is uniquely given by  √  1 2√ κ . (3) ρ(∞) = |φφ|, |φ = √ γ 4κ + γ That is, very interestingly, the steady state is a pure qubit state. Also it is worth noting that the arbitrary pure qubit state (on the xz-plane in the Bloch sphere) can be prepared by suitably choosing the parameters κ and γ. In particular, our control goal can be approximately achieved by setting κ  γ while keeping γ > 0 (note that, if γ = 0, the master equation does not have a unique steady solution). Figure 3 shows the time-evolution of z(t) = Tr[ρ(t)σz ], with parameter values satisfying κ = 20γ and the initial √ state |φ(0) = (1, 1)T / 2. Actually we then observe that the state evolves toward the target spin-up state approximately.

0.8 0.6 0.4

Figure 4 illustrates a schematic of this configuration.

To design H1 , we use the result obtained in [Yamamoto (2005)]; the master equation (1) has a pure steady state |φ if and only if |φ is a common eigenvector of Lj for all  j and iH + j L†j Lj /2. Thus, the condition for |m to be a steady state of the CF controlled system is that

0.2 (t)>

√ √ Gtot = (1, κn, H1 )  (1, γa, 0)  √  √ √ = 1, κn + γa, H1 + κγ(a† n − na)/2i . (4)

Fig. 4. The configuration of the CF controlled system for single photon production. (a) Initially the system is in the vacuum state. (b) A single-photon is produced at the steady state.

1

z

√ L1 = κn, i.e., the number operator, because |m is an eigenstate of L1 ; for the physical implementation of this coupling, see [Geremia (2006)]. In fact, this coupling induces the scattering of the input probe field depending on the number of photons, and thus measuring the output probe field conditionally yields the number state |m. Hence, it is expected that a suitable dissipative process on the system would produce a target number state deterministically. A typical dissipative process of the optical cavity is simply given by an optical loss, √ which is represented by the annihilation operator L2 = γa. Note now that the CF controlled system composed of only these two couplings L1 and L2 (i.e., the case H1 = H2 = 0) merely produces the vacuum state. Hence, here we consider to add a system Hamiltonian H1 , to supply the energy and eventually drive the state into the target state. That is, our CF controlled system is given by

0 -0.2 -0.4 -0.6 -0.8 -1 0

10

20

30

40

50

60

70

80

90

Fig. 3. Time-evolution of z(t) = Tr[ρ(t)σz ]. 4. NUMBER STATE PRODUCTION The next case-study we consider is the problem for producing a number state |m, especially a single photon state |1, via CF control. This control problem was investigated in the MF framework in the theoretical work [Geremia (2006); Yanagisawa (2006); Negretti et al. (2007)] and notably an experimental demonstration was achieved [Sayrin et al. (2011)], but note that no CF-based approach to this control problem has been proposed. The system is a high-Q optical cavity containing a few photons. To attack the control problem for stabilizing the number state |m, our guideline leads us to choose

√ √ √ √ ( κn + γa)|m = m κ|m + γm|m − 1   κ γ √ iH1 + κγa† n + n2 + n |m 2 2  κm2 + γm = iH1 |m + m κγ(m + 1)|m + 1 + |m 2 are both parallel to |m. This requirement can be only satisfied approximately, for the parameters κ  γ and a small number of m. In particular, √ here we take the displacement Hamiltonian H1 = im κγ(a† − a), which compensates  the √ term m κγ(m + 1)|m + 1 though instead produces m κγm|m − 1; that is, we expect that this additional Hamiltonian could reduce the effect of imperfection. Here we describe the result of a numerical simulation, for the case where the single-photon state |1 is the target state. The parameters are chosen so that κ = 200γ, and the initial state is the vacuum. Note that the master equation of the CF controlled system (4) is infinite dimensional, hence we need some approximation for computation. Here we truncate the density operator up to 9 photon states,

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meaning that ρ(t) is a 10 × 10 matrix; we confirmed that this is a good approximation, since Tr[ρ(t)] = 1 is almost satisfied for all t. Figure 5 shows the time-evolution of the fidelity between the system state and the single-photon state, i.e., F (t) = 1|ρ(t)|1; note that 0 ≤ F (t) ≤ 1 for all t and F (t) = 1 implies ρ(t) = |11|. This figure demonstrates that F (t) monotonically increases and reaches the maximum value F ≈ 0.8 at the steady state. This is worse than the value F ≈ 0.9, which was experimentally achieved in the MF-based control setup [Sayrin et al. (2011)]; but in this experiment no steady state was observed, meaning that a near single photon state is produced only at a certain timing and, unfortunately, it must be collapsed immediately. 1 0.9 0.8

Fidelity

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

200

400

600

800

1000

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Fig. 5. Time-evolution of fidelity between the system state and the single-photon state. The parameters are chosen so that κ = 200γ, and the initial state is vacuum. 5. CONCLUSION In this paper, we have proposed a general guideline for designing a CF controller, inspired by the general MF control strategy based on a continuous QND measurement. The method was evaluated by studying the problems of qubit stabilization and single photon production. For the first example we could say the method works nicely, while for the second one a more careful synthesis is necessary to achieve a better performance.

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REFERENCES Balouchi, A. and Jacobs, K. (2016). Coherent vs. measurement-based feedback for controlling a single qubit. arXiv:1606.06507. Gardiner, C.W. and Zoller, P. (1991). Quantum Noise. Springer Berlin. Geremia, J. (2006). Deterministic and nondestructively verifiable preparation of photon number states. Phys. Rev. Lett., 97(7), 073601. Gough, J. and James, M.R. (2009). The series product and its application to quantum feedforward and feedback networks. IEEE Trans. Automat. Cont., 54(11), 2530–2544. Hamerly, R. and Mabuchi, H. (2012). Advantages of coherent feedback for cooling quantum oscillators. Phys. Rev. Lett., 109(17), 173602. 12260