Attempts to control the states of quantum systems often provide new insights into the fundamental nature of quantum mechanics and reveal new aspects of the transition from classical to quantum mechanical behaviour. The reason for this is that the concept of quantum control requires us to examine details of the effects causing decoherence which may have been overlooked before. One typical effect causing decoherence is the interaction of excited atoms with the electromagnetic vacuum giving rise to spontaneous emission. It is especially important since coherence is often established by electromagnetic fields, requiring the quantum system to be open to a continuum of modes. The conventional way of dealing with the problem of decoherence in the presence of spontaneous emission is to distinguish no-photon intervals and photon emission events ^1,2,3. However, this is by no means the only way of observing the electromagnetic field propagating away from a quantum system. As pointed out by Ueda⁴, a measurement of the emitted field which is sensitive to the vacuum state as well is logically reversible, as opposed to the sudden transition to the ground state connected with a photon detection event. Therefore it seems preferable to apply measurement schemes different from photon detection if quantum coherence is to be controlled.

In this letter we consider the possibility of observing one quadrature component of the electromagnetic field propagating away from the atomic system by time resolved balanced homodyne detection. The field actually originating from the dipole oscillations of the atomic system on a timescale τ which is much smaller than the lifetime 1/Γ of the excited atomic state is much smaller than the vacuum fluctuations observed on this timescale. Thus it is possible to interpret the fields measured as quantum fluctuations of the electromagnetic field impinging on the system. In this sense the measurement is a measurement of the forces acting on the system and not a measurement of the system state itself. It should be possible to compensate the effect of the observed quadrature component of the electromagnetic field by a coherent field of opposite sign. However, the effect of the unobserved quadrature component must also be compensated if decoherence is to be suppressed. To find out, how this can be achieved as well, it is necessary to investigate the back-action of the homodyne detection on the atomic system.

For the description of the homodyne detection process, we use a non-orthogonal projective measurement base. This type of measurement base for homodyne detection has been derived and applied in a number of publications ^5,6,7,8. Since the observed fields are small, we will only consider that part of the measurement base composed of the zero or one photon contributions. The effective non-orthogonal measurement base is given by

where α is the field amplitude of the coherent field mode emitted by the local oscillator during the time segment τ considered in the measurement and ∆n is the measured photon number difference between the two branches of the homodyne detection setup. Note that α is related to the intensity (or photon rate) I emitted by the local oscillator by I = α ^* α/τ. The relation has been derived using the assumption that α ^* α ≫ 1. Details of the derivation will be given elsewhere ⁹. Within the zero- and one-photon subspace weak coherent fields of amplitude β are approximately given by

The measurement probability of a photon number difference δn of such a coherent field can be calculated from equation (1) by

This is a Gaussian with a variance of 〈∆n ²〉 = α ^* α and a mean value of 〈∆n〉 = α ^* β + β ^* α. If the measured value of ∆n is identified as 2 | α | times the quadrature component of the measured light field in phase with the local oscillator, this result exactly corresponds to the quantum uncertainty of 1/4 and a shift by the component of β in phase with α. This result confirms the interpretation of homodyne detection as a projective measurement of the quadrature component in phase with the local oscillator.

The dynamics of the photon emission process and the interaction of a two level atom with the light field continuum can be analyzed without assuming an optical cavity or using a bath approximation by applying Wigner-Weisskopf theory to the complete system-field Hamiltonian ¹⁰. In the following, however, we will assume fast time-resolved measurements performed on the field long before the emission probability from an excited state approaches unity. During the short time intervals τ with Γτ ≪ 1, the one-photon component of the wavefunction corresponds to a photon in a field mode with a rectangular envelope: zero field amplitude for distances r from the atomic system with r > cτ and a constant probability of finding a photon at distances of 0 < r < cτ. Therefore, the photon possibly emitted during the time interval τ is in a well defined mode. Thus it is possible to write down the wave function which evolves from the light field vacuum and an arbitrary state of the two level atom given by

where | G〉 is the atomic ground state and | Ẽ〉 is the excited state in the interaction picture, i.e. without the phase dynamics at the frequency ω ₀ of the atomic transition. After the time interval τ, the entangled state of the atomic system and the electromagnetic field is

This is the complete quantum mechanical state as it evolves unitarily according to the Hamiltonian of Wigner-Weisskopf theory.

The projective measurement base given in equation (1) may now be applied to determine the change in the state of the atomic system conditioned by a measurement of ∆n in the homodyne detection during the time interval τ. The wavefunction | ψ(τ)〉 of the atomic system after the measurement reads

Since Γτ ≪ 1, the squared length of this state vector which corresponds to the probability of measuring ∆n is approximately independent of the system state and is given by the vacuum distribution,

The major contribution to the change of the state of the atomic system conditioned by the measurement is given by the amplitude proportional to √Γτ. The higher order terms do have some effect on timescales of 1/Γ, corresponding to a large number of measurement intervals τ. These effects will be discussed elsewhere⁹. In the following we will concentrate on the short time fluctuations effective on a timescale of τ.

If the normalized system state is written as | ψ(0)〉 + | δψ(τ)〉, such that | δψ(τ)〉 is the change of the system state orthogonal to | ψ(0)〉, then this change is approximately given by

Since the probability distribution of measurement results ∆n is a Gaussian, this equation describes a diffusion process. Statistically, the diffusion steps cancel on average, causing decoherence because the uncertainty of the actual path chosen by the system dynamics increases with each unknown step. In our scenario however, the length and the direction of each step has been measured by homodyne detection.

 

Fig. 1. Visualization of the diffusion step on the Bloch sphere. The diffusion is represented by lines oriented parallel to the direction of the diffusion with a length proportional to the standard deviation of the step length. a) and b) show the projections into the s_y, s_z and the s_x, s_z plane, respectively.

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We can therefore deduce the evolution of the pure state of the atomic system. In this sense the description of the quantum measurement process is a quantum trajectory description as introduced in ^3,11 and applied to problems of continuous feedback scenarios in ¹². It has not been derived from a master equation of the open system, however, and the field-atom interaction is described using the Schroedingers equation of Wigner-Weisskopf theory, retaining the full atom-field entanglement up to the projective measurement.

In order to visualize the diffusion step, it is useful to describe the state of the atomic system by its Bloch vector s, defined as

where Re(·) and Im(·) denote the real and imaginary part, respectively. s_z is the expectation value of the population inversion and s_x and s_y are the in-phase and the out-of-phase dipole moments of the atomic system, respectively. The change in the Bloch vector of the atomic system δs conditioned by a measurement of ∆n within the time interval τ is then given by

A representation of this diffusion step on the Bloch sphere is shown in Fig. 1.

The linear part of this change in the Bloch vector corresponds to a Rabi rotation around s_y. It is exactly equal to the effects of a coherent field with an amplitude of ∆n/|2α|. The non-linear part is shown in Fig. 2. For positive ∆n, this contribution draws the Bloch vector towards the s_x = +1 pole of the Bloch sphere. For negative ∆n, the Bloch vector moves towards the s_x = - 1 pole.

It is possible to interpret this effect of the quantum fluctuations on the atomic system as an epistemological effect of information on the in-phase dipole component s_x gained in the measurement. Positive values of ∆n make a positive dipole component s_x more likely and negative values of ∆n make a negative dipole component s_x more likely. Although the information obtained in a single measurement is almost negligible, the relative suppression of the amplitude of one dipole eigenstate and the corresponding amplification of the amplitude corresponding to the other dipole eigenstate causes a change in the state of the atomic system unless the system is already in an eigenstate of the dipole component with s_x = ±1.

 

Fig. 2. Non-linear contribution to the diffusion step of the Bloch vector. The representation is in analogy to Fig. 1, with a) and b) showing the projections into the s_y, s_z and into the s_x, s_z planes, respectively.

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The relative smallness of the dipole field compared to the quantum fluctuations makes this measurement a weak measurement in the sense discussed by Aharonov and coworkers in ¹³.

Even though the non-linear dependence of the diffusion step on the previous state of the atomic system prevents a state independent compensation of quantum fluctuations, the measurement is still logically reversible in the sense of ⁴. It can be compensated if the previous state of the system is known with sufficient precision. In the following, we shall focus on atomic system states with s_y = 0. For such states, δs_y is also zero and the whole diffusion process takes place in the s_x, s_z plane. The diffusion steps may then be identified as rotations around the s_y axis. By defining the angle θ such that cos θ = s_z and sin θ = s_x, the diffusion step in the s_x,s_z plane may be written as

This rotation of the Bloch vector conditioned by the measurement of ∆n is equivalent to a Rabi rotation around the s_y axis proportional to the quadrature component measured in the homodyne detection. Despite the quantum mechanical dependence of this Rabi rotation on θ, it is possible to compensate the effects of the quantum fluctuations by simply reversing the rotations corresponding to each measurement. The feedback field f necessary to stabilize a state of the atomic system with θ = $\overline{\theta }$ is given by

where ∆n ₀ is the measurement result associated with the quantum fluctuations which are to be compensated by the feedback term. Each time interval τ is therefore associated with a diffusion step caused by the quantum fluctuations and a time delayed feedback which compensates the diffusion step. The total field interacting with the atomic system is given by a coherent state of field amplitude f(∆n ₀). This corresponds to vacuum-state quantum fluctuations shifted by f(∆n ₀). Consequently the measurement result ∆n_next corresponding to the time interval τ_next during which the feedback field acts on the system will be composed of a stochastic effect of the quantum fluctuations ∆n_qf and a shift δ_next caused by the feedback field

Since the feedback effect itself should not be compensated, only the contribution of the quantum fluctuations ∆n_qf should be applied for the determination of the subsequent feedback field.

Effectively, the atomic system now interacts with a series of light field modes initially in weak coherent states corresponding to vacuum fluctuations shifted by the feedback field. The quadrature component of the field in phase with the local oscillator is measured, revealing the amplitude of this component of the fluctuations. The feedback then correlates the average field of the next light field mode interacting with the system with the measured quadrature component of the fluctuations. In a classical system this compensation would be insufficient since the out-of-phase quadrature component of the fluctuating field is unknown. In the quantum mechanical case the information obtained is complete. Instead of causing uncontrollable changes in the system state, the effect of the quantum fluctuations corresponds to a weak measurement correlating the information gained in the field measurement with the information about the system state. This effect is therefore predictable and can be compensated. The sum of the measured fluctuations and the feedback field reveals which part of the feedback field is necessary to compensate the changes associated with the weak measurement of s_x:

The state dependence of the weak measurement can be illustrated in terms of the three most typical cases:

Dipole eigenstate. For cos $\overline{\theta }$ = 0 the system is in an eigenstate of the in-phase dipole component s_x. No measurements of s_x, whether weak or strong, will change this. Therefore, the compensating field necessary to suppress the effects of quantum fluctuations is equal to the compensation of the classically expected Rabi rotation. Also note that a coherent field along the unknown field quadrature would not affect this state, since the Bloch vector is parallel to the axis of Rabi rotations caused by fields ±π/2 out of phase with the local oscillator.

Ground state. For cos $\overline{\theta }$ = -1 no feedback is necessary for stabilization. This means that the effects of the Rabi rotation and the weak measurement associated with a homodyne detection result ∆n automatically compensate each other. This is a result of the fact that the ground state is polarized by the field in such a way that the dipole emissions interfere destructively with the field. At the same time, the observed field makes a dipole more likely which emits radiation interfering constructively with the fluctuations. This effect may also be understood in terms of energy conservation. The ground state atom absorbs the field by the destructive interference of dipole emission and incoming field, but at the same time it emits radiation associated with the quantum fluctuations of the dipole variables. Both effects cancel and energy conservation is preserved.

Excited state. For cos$\overline{\theta }$ = +1, the feedback necessary to compensate the weak measurement effects is equal to the feedback necessary to compensate the Rabi rotations. The reason for this is that the excited state is polarized by the field in such a way that the dipole emissions interfere constructively with the field. At the same time, the measurement makes such a dipole more likely. Consequently the effect of the quantum fluctuations is doubled. In terms of energy conservation the excited state atom amplifies the field and emits additional radiation associated with the quantum fluctuations of the dipole. The instability of the excited state is thus related to its linear response to the light field which implies gain instead of absorption. The feedback field corrects this property by effectively reversing the sign of the susceptibility, overcompensating the loss in energy associated with the field amplification and establishing a stability equivalent to that of the ground state without feedback.

 

Fig. 3. Animation: Variation of the effective diffusion step with the feedback increasing from (cos$\overline{\Theta }$ = -1) to (cos$\overline{\Theta }$ = +1). (a) Projection into the s_y,s_z plane; (b) Projection into the s_x, s_z plane. [Media 1] [Media 2]

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If the quantum state of the system does not correspond to the state for which the diffusion step is suppressed by the feedback, the total effect of quantum fluctuations and a feedback signal given by cos $\overline{\Theta }$ may be expressed by a general diffusion step on the Bloch sphere. This diffusion step reads

The continuous change of the effective diffusion as a function of feedback is shown in the animation (Fig. 3). The animation visualizes in projection into the s_y, s_z plane (left) and into the s_x, s_z plane (right) the continuous change of the effective diffusion step as the feedback signal is increased from no feedback (cos$\overline{\Theta }$ = -1) to a feedback of twice the observed field (cos$\overline{\Theta }$ = +1). Note that the diffusion step for any state with s_z = cos $\overline{\Theta }$ is perpendicular to the s_z axis, indicating that the feedback stabilizes this value of the inversion expectation value s_z regardless of the phase of the dipole oscillations relative to the local oscillator.

In conclusion, we have shown that homodyne detection of the electromagnetic field propagating from a single two level atom with a known initial quantum state reveals the changes induced in the state of the atom by quantum fluctuations. In the s_x,s_z plane of the Bloch vector representation of the atomic system, randomly fluctuating forces cause rotations around the s_y axis. This effect corresponds to that of a coherent driving field and can consequently be compensated by Rabi rotations of opposite sign induced by a feedback field. The decoherence caused by quantum fluctuations can be suppressed completely with a precision limited only by the time delay between the emission of the field and the measurement by homodyne detection. Even though one quadrature component of the light field remains unobserved, we have demonstrated that quantum control of an arbitrary state of a two-level atomic system is possible by simply applying a coherent feedback field.