Decoherence control by quantum decoherence itself

We propose a general approach of protecting a two-level system against decoherence via quantum engineering of non-classical multiple superpositions of coherent states in a non-Markovian reservoir. The scheme surprisingly only uses the system-environment interaction responsible for the decoherence and projective measurements of the two-level system. We demonstrate the method on the example of an excitonic qubit in self-assembled semiconductor quantum dots coupled to the super-Ohmic reservoir of acoustic phonons.

|1〉 and only influences their superposition. In this case, the evolution operator = ⊗ + ⊗ U U  . The entanglement generated by the dephasing can be exploited for the state preparation of the environment. Consider a qubit being prepared initially in the state + = ( + )/ 0 1 2 . After it has undergone interaction with the environment for the duration τ, the projection + + (consisting of the standard π/2 -pulse on the qubit system followed by a projective measurement in the basis of ground/ excited states) is executed on this qubit 23 . The environment E is then projected to the superposition state . The environment is thus engineered in a nonclassical quantum state being a superposition of non-orthogonal states, known as the cat state 24,25 . To test, whether the superposition state α ( ) C E 1 present in the environment can be better for a storage of the qubit, the testing qubit only carrying information in the phase variable φ is interacting during time interval t with the pre-engineered environment by the same type of interaction described by H I . The resulting entangled between the qubit and the environment is still subject to the quantum dephasing. However, the overlap of is now substantially different from D(α). Tracing out the environment, the qubit is then described by the density matrix with the phase damping factor fully characterising the dephasing process after engineering of the environment. The last two terms arise due to interference effects between the state preparation and the subsequent dephasing of the testing qubit. If α = β, then D(α − β)= 1 by definition. On the other hand, since D(α), D(β) and D(α + β) vanish for large α and β, the dephasing factor can interestingly converge to D 1 = 1/2 for large equal interaction times τ = t. This should be contrasted with α ( → ∞) = D 0 for the initially ground state of the environment.
This is a remarkable result, since by a conditional engineering of the environment using the same quantum dephasing process, we are able to protect the subsequent qubit evolution against the very same dephasing mechanism. The protection arises due to a quantum interference term D(α − β) in equation (2) caused by the principal indistinguishability of the state α E being a component in both the states , which is increasing with M. The measurement-induced squeezing of the reservoir momentum P E explains why the interaction κ = H P 1 1 I E causes less dephasing of the qubit, since the variable P E is less fluctuating. As shown numerically in the Methods section the above formula approximates equation (3) very well even for large α's and we find the asymptotic behaviour for sufficiently large α ( , ) This result implies that the dephasing process can be completely stopped by the repeated state engineering based on the system-environment interaction which is itself responsible for the dephasing. From this perspective, some types of the decoherence processes can be more easily corrected, without any external dynamical operations with the environment, comparing to others, more destructive ones. It opens the broad possibility of further investigations, various extensions and refining of operational understanding what the decoherence actually is about. However, it is unclear whether properties of this simplistic case carry over to more realistic situations involving non-degenerate qubits and environments with a large number of finite frequency modes. As we show in detail below, the answer is positive and we identify a whole class of experimentally-relevant solid-state setups where an analogous mechanism of decoherence suppression can be implemented.

Infinite reservoir model & its free dynamics.
The system under study consists of a self-assembled, single level quantum dot under the influence of a reservoir of longitudinal acoustic phonons described by ω = ∑ † H bb k k k k ph , with ω k = vk being the frequency of the phonon mode with the wave vector k (v is the speed of longitudinal sound waves). We consider just two electronic states of the dot forming the qubit: |0〉 when the dot is in its ground state ("empty", i.e. no exciton) and |1〉 indicating the excited QD ("occupied" with an exciton in its ground state) with bare excitation energy ε , i.e., , and "form-factor" ( )≈ , ( )≈ /( ) F x F x x 1 1 1 1 48 2 corresponding to the typical material and spatial parameters for a self-assembled InAs/GaAs structure found in Ref. 28 with anisotropic Gaussian exciton wave functions of 5 nm width in the xy plane and 1 nm along z (for details see the Methods section). The exciton-phonon interaction term in the Hamiltonian is linear in phonon operators and describes a shift of the lattice equilibrium induced by the presence of a charge distribution in the dot associated with the classical energy of the displaced oscillators  describing the decoherence of superposition states between 0 and 1 exhibits non-exponential, i.e., non-Markovian decay ρ ρ The model thus shows features of pure dephasing, i.e., only the coherences, which can be measured by the amplitude of coherent dipole radiation emitted by the dot, decay with time. Moreover, for the super-Ohmic spectral density characteristic of this system, due to the Riemann-Lebesgue lemma the decay saturates at a finite value ( ) for times much longer than the dephasing time τ , thus the pure dephasing is only partial or incomplete 29,32,33 . In the zero-temperature limit β ( → ∞) the asymptotic value of the coherence reads ρ where , ∼ vac vac are the phonon vacua when the QD is empty or occupied, respectively. The overlap of the two mutually displaced vacua is non-zero, which means that despite of the continuous spectrum of phonon modes the orthogonality catastrophe is incomplete -this reflects the asymptotic nature of the couplings f k for small k's (and ω) due to identical phonon coupling to electrons and holes for long phonon wavelengths 34 resulting in the super-Ohmic spectral density of exciton-phonon coupling. Consequently, for small k's the trace left by the exciton in the bath is too weak to be distinguished from the vacuum case and, thus, decoherence is only partial 4 .

Repeated initialisations.
We may study not only the state of the QD considered so far but also the state of the phononic subsystem analogously to the above toy model. The creation of an exciton in the QD perturbs the phonon reservoir state by shifting the coordinates. If the exciton is created in a superposition state, the phonon reservoir will react by following in parallel two different evolutions coherently superposed 35,36 . Now, we may ask again, what is the effect of repeated measurements of the dot state on the degree of the partial pure dephasing. Therefore, we analyse the evolution of the composite system of the dot and the phonon reservoir subject to strong projective measurements 23 performed on the QD subsystem. Each measurement is represented by orthonormal projection operators of the form = ± ± ⊗ ± P  with complementary and orthonormal pure qubit states ± = ( ± ) / φ e 0 1 2 i and the unity in the reservoir subsystem . We consider free evolution of the composite system starting from a factorised initial/re-initialised condition σ = ⊗ init init init init corresponding either to the true initial condition or to an output of previous measurement (see equation (6)  and an arbitrary phonon reservoir density matrix  init . We choose the equal-weight superposition so that neither the dephasing interaction nor the measurement processes, regardless of their outcome, change the occupation factors and therefore, they only influence the coherences.
Under these assumptions the state of the composite system right after the measurement at time τ with the outcome ± is given by Here, R denotes the real part, h.c. the hermitian conjugate, denotes the average of the Weyl operator with respect to this time-evolved phonon density matrix. The respective measurement outcomes are obtained with prob- Note furthermore, that regardless of the measurement outcome, the degree of coherence just after the measurement is fully restored to unity , i.e., the net outcome of the measurement on the state of the qubit is, apart from a possible (controlled) phase shift, just the re-initialisation of the qubit state (compare with equation (2) for β → + 0 ). However, the state of the phonon reservoir does change and this has important consequences for further evolution of the qubit. The scheme outlined above can be iterated to yield results for an arbitrary series of measurements, but it acquires great complexity rapidly with the growing number of measurements. It is therefore convenient to study just the single-measurement scenario, especially since an observable decrease of dephasing can be detected already there. as functions of the delay time t, measurement time τ and the measurement outcome (± ). To this end, we evolve the density matrices from equation (6) for the time span t and then evaluate the coherences ρ τ ( + ) ± t 01 . Calculation follows the line analogous to the free evolution discussed above with the initial thermal density matrix  ph can replaced with those of equation (7)  0 , i.e., for partial dephasing, as is the case of the super-Ohmic bath. We then get for large times τ τ , Obviously, these values oscillate as functions of the delay time τ between the preparation of the qubit and its measurement with the frequency determined by the shifted exciton energy ε (corresponding period is on the order of few femtoseconds) as depicted in the inset of Fig. 1. We also plot there the envelopes of curves (9) on the longer timescale of picoseconds showing the saturation of the initial sub-picosecond transient behaviour. The overall magnitude of the asymptotic degree of coherence decreases with increasing temperature as presented in Fig. 1b). Let us now analyse the formulas (9) in more detail. First, D − is easily obtained from D + by the phase shift φ φ π → + so that it suffices to study the latter one. It always attains a minimum D(1 + D)/2 at φ π = and has a local extremum which is bigger than D since the integrand is never below D (equality happens only at φ = 0,π). Numerical analysis reveals that the maximum difference from the free case is obtained at = .  T ≈ 60 K) with the magnitude roughly 0.019, about 4% of the free case value. These conclusions are consistent with the plots in Fig. 1b).
Several experiments with self-assembled QDs considered here have been recently realised [37][38][39] . We have analysed thus far properties of an idealised model and it is necessary to scrutinise whether our conclusions can be carried over to the experimentally realistic situations. There are several points which might in principle endanger our conclusions. First, we have only considered the Hamiltonian describing the free evolution, which is purely harmonic in the acoustic phonon modes and the excitonic interaction with them is solely of pure dephasing type. In reality there are also optical phonons which cause the relaxation of the exciton occupation and, moreover, there is radiative relaxation channel too -these effects, however, become effective only at much longer timescales on the order of tens or hundreds picoseconds 10 while our asymptotic times are just a few picoseconds. Since the dephasing-suppression mechanism hinges on the creation of "cat states" of the acoustic reservoir modes, their potential dephasing beyond the excitonic interaction by anharmonic terms or by coupling to other (e.g., optical) modes would be detrimental to the predicted effect. While such effects do exist and may be relevant in certain contexts (see, e.g., Ref. 40), the estimated lifetime of the acoustic phonons 41 is on the order of 1 nanosecond, which makes these issues irrelevant for our discussion. Finally, we have assumed an instantaneous projective measurement of the qubit state. This is clearly not realistic as existing projective measurements are achieved by optical pulses whose duration is at least ten(s) femtoseconds during which the freely evolving qubit phase ετ acquires several multiples of 2π's (see the inset of Fig. 1). Thus, one might expect that the effect would be smeared by the phase averaging. However, finite duration of pulses is not necessarily fatal to our predictions. What matters is the short duration of the pulse with respect to the characteristic time scale of the phonons being on the order of 1 ps ( ω ≈ / 1 c ) and the ability to very precisely control the relative phase between the initialisation and measurement pulses. This is currently possible by splitting the initial pulse and using the optical delay line with exquisite sub-cycle tuning of the relative phase as realised in pump-probe and multidimensional optical spectroscopies 10,42 . Thus, the approximation of delta-like pulses is done and justified for the study of phonon dynamics 12 . Even if the experiment is not completely controlled (the relative phase ετ is fluctuating between subsequent runs of the measurement) and/or the measurement outcomes of the qubit state are ignored (e.g., to avoid discarding data), the averaged result described by the quantity D 1 av introduced above and plotted in Fig. 1b) still shows enhancement over the free case, although its magnitude is 3-times less (on the absolute scale) than in the fully controlled case. Altogether, we believe that the predicted effect should be experimentally observable.
To summarise, we have proposed measurement-induced quantum pre-engineering of a non-Markovian environment consisting of a super-Ohmic reservoir of longitudinal acoustic phonons which can be directly exploited to control quantum-dot-based qubit decoherence using only the single type of coupling between the qubit and the environment. A feasible proof-of-principle experimental test of the proposed method with self-assembled semiconductor quantum dots would be a practical test of the quantum nature of dephasing for a solid state system. The method can also be translated to the cavity QED, atomic, or trapped ion experiments.

Environment engineering by many repetitions; derivation of equation (3). After the M-times
identical state preparation with the preparation times τ and subsequent evolution of the qubit during the same time τ, the phase damping factor is defined as an absolute value of the scalar product between states In the limit of small α  1 the constituents of the sums (10) and (11) 2 2 very satisfactorily approximates the exact numerical evaluation of equation (3) for small enough α  1 as we show in Fig. 2.
On the other hand, in the limit of large α, the coherent states |ka〉 E become almost orthogonal for different k's and we can treat them approximately as the basis states. We can therefore approximate the scalar product are the exciton wave functions modelled by anisotropic Gaussians with a = 5 nm width in the xy-plane and c = 1 nm along the z-axis. Therefore, we get for the spectral density (recall that ω =