31.1 Introduction

Light–atoms quantum interface is an important component of a quantum network. Whereas light is a natural long‐distance information carrier, it is difficult to keep information encoded in light for an extended period of time due to decoherence associated with its propagation. In the best‐case scenario, light at an optimal telecom wavelength propagating in a fiber loses half of its photons in 100 µs. Longer storage times for a quantum state of light require a faithful transfer onto an atomic quantum state where coherence and storage times can be much longer. Even stronger motivation for light–atoms interface is provided by the need to interconnect distant atomic nodes of a quantum network. One example of such connection is long‐distance teleportation of atomic states discussed in this chapter. Another example is a two‐step transfer of a quantum state: First from atomic sample A to light and then from the light onto a distant atomic sample B.

The light–atoms interface considered in this chapter can be characterized as deterministic. That is, the result of it is not conditioned on probabilistic events, such as detecting a photon in a specific mode. The probabilistic type of light–atom interaction, though being another important ingredient of quantum information processing, cannot alone achieve the above‐stated goals for communication and storage.

In this chapter, we shall concentrate on the quantum interface via free‐space interaction of light with an atomic ensemble. This approach is a powerful alternative to the interface of light with a single atom. The latter approach, developed within the framework of cavity quantum‐electrodynamics, requires strong coupling of an atom to a high‐finesse optical cavity. With multiatom ensembles, strong coupling to light can be achieved in the absence of a cavity, due to the fact that the interaction with a collective mode of an ensemble grows as the square root of the number of atoms. As shown in this chapter, the effective “figure of merit” of the light–atomic ensemble quantum interface is the resonant optical density of the atomic sample.

The light–atomic ensembles quantum interface considered in this chapter provides an example of a link between discrete and continuous quantum variables. Although most of the discussion in this chapter is formulated in the language of canonical operators x and p, which are usually associated with continuous variables, the interface we are discussing often works for an arbitrary single‐mode state of light, which means that it also works for a qubit encoded into a state of a single photon.

31.2 Off‐Resonant Interaction of Light with Atomic Ensemble

In this section, we shall describe the basic physics behind the light–matter interface and derive the effective Hamiltonian that governs the evolution of the system (1–3). Consider a polarized light beam propagating along the z‐axis through an ensemble of N _A spin‐ atoms with two degenerate ground states |g, m_z  =   ≡|1〉 and |g, m_z  =  〉 ≡ |2〉 and two excited states |e, m_z  =  〉 ≡ |4〉 and |e, m_z  =  〉 ≡ |3〉. The level structure is depicted in Figure 31.1 and the geometry of the experiment is shown in Figure 31.2.

As imposed by the selection rules, the left‐hand (L) and right‐hand (R) circularly polarized light modes couple to the transitions |2〉 →|4〉 and |1〉 →|3〉, respectively. The light frequency ω _L is strongly detuned from the atomic transition frequency ω _A, and the detuning Δ = ω _L − ω _A satisfies |Δ| ≫ γ where γ is the spontaneous emission decay rate from excited to ground states. Due to the large detuning, only a tiny fraction of atoms gets excited in the course of evolution and most atoms remain in their ground states. The light–atoms interaction becomes dispersive and the light in each circularly polarized mode experiences a refraction index that depends on the number of atoms in state |1〉 or |2〉. If the populations of these two levels are slightly unbalanced, then the atomic medium exhibits a circular birefringence closely resembling a Faraday effect. The back‐action of the light on the atoms results in the Stark shift of the frequency of the atomic transition |1〉 →|3〉 (|2〉 →|4〉) proportional to the intensity of the light beam in mode R (L).

Large detuning of light from atomic resonance helps in several respects. Alkali atoms used in experiments have the total angular momentum F in the ground state higher than 1/2, for example, F = 4 for cesium. Nevertheless, the simplified four‐level model faithfully captures all the essential features of the interaction between atoms and light when the detuning is large compared to the hyperfine detuning of the excited state. Another experimental advantage brought about by the off‐resonant character of the interaction is the insensitivity to the Doppler motion of atoms at detunings much larger than the Doppler width that allows using atomic gases at room temperature.

The effective Hamiltonian for a circularly polarized light beam propagating in a medium with refraction index n is , where a_j , j = L, R, denotes the annihilation operator of the light mode. The unitary transformation corresponding to the beam propagation through the medium is , where k _L = ω _L /c. In our case, the refraction indices n _L and n_R for the two modes L and R may differ and and , where are the collective atomic operators and we assume equal coupling strength of all atoms to the light beam. The resulting unitary where the total effective interaction Hamiltonian reads

31.1

Here, we introduced new coupling constant a = −2k _L β, N _A = Σ₁₁ + Σ₂₂ and . Since the total number or photons and atoms N _L and N _A is constant during the evolution, the term in the Hamiltonian 31.1 proportional to N _A N _L can be dropped.

It is helpful to introduce the components of the collective atomic spin operator J,

31.2

which satisfy the angular‐momentum commutation relations . Similarly, we define the components of the Stokes vector S describing the polarization properties of the light beam,

31.3

It holds that [S_j, S_k ] = iɛ_jklS _l. In terms of the operators J and S, the effective interaction Hamiltonian 31.1 can be rewritten as

31.4

Following closely (2), we shall now outline a more rigorous derivation of the effective interaction 31.4, which provides an explicit expression for the coupling constant a. The light beam is described by the field operators a(z, t) satisfying the equal time commutation relations , j, k = L, R. It is convenient to formally define continuous atomic operators , which satisfy

31.5

Here, A denotes the transverse area of the atomic ensemble and ρ is the number density of atoms, , and L is the length of the ensemble.

The interaction of light with the atoms is governed by the Jaynes–Cummings Hamiltonian,

where the coupling constant and d is the dipole moment of the atomic transition. In the Heisenberg picture, the atomic and field operators evolve according to , and we neglect the spontaneous decay. Since the interaction is off‐resonant, the populations of the excited states are negligible and the atomic coherences σ ₁₃ and σ ₂₄ adiabatically follow the field operators,

Within this approximation, the ground‐state populations σ ₁₁ (z, t) and σ ₂₂ (z, t) as well as the field intensities , are constants of motion and the coupled Maxwell–Bloch equations for a_j and σ_μν simplify to

31.6

where τ = t − z/c is the retarded time and

31.7

The differential equations 31.6 and 31.7 can be solved by a straightforward integration. The resulting transformation for the elements of the total Stokes vector of the whole pulse, , and so on, is

31.8

and similar formulas hold for the elements of the total atomic spin operator J at time T when the light beam passed through the ensemble,

31.9

These expressions coincide with the formulas obtained by solving the Heisenberg equations of motion induced by the effective Hamiltonian 31.4. The physical meaning of Eqs. 31.8 and 31.9 is that the light Stokes vector S is rotated along the z‐axis by the angle proportional to J_z and, simultaneously, J is rotated by an angle proportional to S ₃. The coupling constant

31.10

where λ is the wavelength and we used the relationship between the dipole moment d and the spontaneous decay rate, .

In order to enhance the coupling between atoms and light, the atomic ensemble is prepared in a coherent spin state (CSS) with all atoms oriented along the x‐axis. The ensemble can be polarized by optical pumping with a right‐hand circularly polarized laser propagating along the x‐axis. As a result, the J_x component of the collective atomic spin attains a macroscopic value and the operator can be replaced with a c‐number, J_x ≈ N _A /2. Under these conditions, we can introduce effective quadratures for the atomic system, x _A =  and p _A =  , which satisfy the canonical commutation relations [x _A , p _A] = i. This approximation can be visualized as follows. The atomic ensemble in a CSS, with all atoms polarized along the x‐axis, can be pictured as a vector pointing to the north pole of the Bloch sphere. In the experiment, the state always remains close to the north pole and the Bloch sphere can be locally approximated by a tangent plane whose geometry is that of the phase space of a particle with momentum p _A and position x _A. Analogously, the light beam should contain a strong coherent component linearly polarized along the x‐axis with mean number of photons N _L. The operator S ₁ can be approximated by a c‐number, S ₁ ≈ N _L /2, and we can define the light quadratures and and we have [x _L , p _L] = i. Note that the quadratures x _L and p _L can be interpreted as the quadratures of the optical mode linearly polarized along the y‐axis.

Assuming aS ₃ ≪ 1 and aJ_z  ≪ 1, the transformations 31.8 and 31.9 can be linearized and we obtain the resulting effective linear canonical transformations for the light and atomic quadrature operators,

31.11

The coupling constant can be written as κ ² = α ₀ η, where

31.12

η is the atomic depumping rate due to the absorption of light, α ₀ is the optical density of the atomic sample on resonance, and σ ∝ λ² is the atomic cross‐section on resonance.

For such quantum information protocols as quantum memory (see below), the coupling constant κ should be of the order of unity. Simultaneously, the decoherence and losses that are proportional to the depumping rate η should remain as low as possible. This means that the optical density of the atomic sample should be high, α ₀ ≫ 1. This condition can be marginally satisfied with atomic vapor at room temperature stored in a paraffin‐coated glass cell, where α ≈ 5 has been observed. For other protocols, such as the generation of strongly entangled states, the coupling should be as strong as possible, κ ≥ 1. High optical densities, required for satisfying this condition, can be achieved with cold atomic samples held in a magneto‐optical trap and, in particular, with Bose–Einstein condensates, where α ₀ can be of the order of 10²–10³.

The linear canonical transformation 31.11 is often referred to as the quantum nondemolition (QND) interaction and is well known from the theory of QND measurements. Indeed, the x quadrature of light stores information about the p quadrature of atoms, while the p _A quadrature is a constant of motion, and the noise associated with the measurement of p _A is fed to the conjugate quadrature x _A in the form of the term κp _L. In particular, if κ = 1 then we recover the so‐called continuous‐variable controlled NOT gate (5). In addition to the two‐mode gate 31.11, it is also experimentally feasible to apply arbitrary single‐mode phase‐space rotations. The light quadratures can be rotated by sending the beam through a wave plate and the rotation of atomic quadratures can be accomplished by illuminating the ensemble with strong coherent far‐detuned laser beams. Alternatively, it is also possible to switch between the coupling to the p _A and x _A quadratures by sending the light beam through the atomic sample either along the z‐ or y‐axes, respectively.

The light Stokes vector components (i.e., the x _L or p _L quadratures) can be measured by sending the light through a wave plate and a polarizing beam splitter that spatially separates two linear orthogonally polarized beams; see Figure 31.2. The power of these two beams is measured with the high efficiency linear photodiodes and the two powers are subtracted. The x _L quadrature is proportional to the difference of the power of two linear polarizations oriented at +45 and −45° with respect to the x‐axis. The p _L quadrature is proportional to the difference of the power of two circular polarizations. If the S ₁ component is in a strong coherent state then this scheme becomes equivalent to self‐homodyning where one polarization mode plays the role of the local oscillator while the orthogonal mode is the signal whose quadrature is detected. In quantum information applications, we require that the measurement is shot‐noise limited, and the detected quadrature variance should be proportional to the mean number of photons N _L in the strong beam.

Experimental realization of quantum information protocols described here requires achieving the level of quantum fluctuations for both the light pulse and the atomic collective spin variables. The relative size of the quantum noise (the shot noise for light and the projection noise for atoms) is of the order of , where N is the number of photons per pulse or the number of atoms in the samples, respectively. It is possible to reduce technical noise to the level much lower than this quantum limit with dc detection provided that N _L ≤ 10⁸ in current experiments. In order to achieve sufficiently strong κ for gasses at room temperature, it is necessary to go to a higher number of atoms, and correspondingly to a higher number of photons per pulse. Quantum limited noise for such a high number of particles can be achieved with the help of AC detection at frequency Ω of few hundred of kHz or higher. This approach allows us to suppress technical noise by several orders of magnitude and quantum‐limited measurements can be carried out with up to N _L = 10¹². However, the light sidebands at the frequency ±Ω around the carrier frequency ω _L do not couple to the atoms. This problem is resolved by placing the atoms into a constant magnetic filed B oriented along the x‐axis. The atomic spins precess with Larmor frequency Ω that should coincide with the frequency of the detected light sidebands.

The application of the magnetic field resolves the problems with the technical noise but it significantly alters the light–matter coupling. At each time instant t, the atomic quadrature x _A(t) stores information about S ₃(t). However, after some time the rotation exchanges x _A and p _A, the information about S ₃(t) is fed to the light Stokes operator S ₂(t + Δt) and the QND character of the interaction is lost. The evolution of the collective atomic spin operators is governed by the Heisenberg–Langevin equations

31.13

where Γ is the decay rate of the atomic coherence and ℱ are the quantum Langevin stochastic forces. The input–output relations for the light Stokes vector components at time t can be written as

31.14

The validity of this description has been confirmed in an experiment where a cw polarization squeezed state of light has been sent through the atomic ensemble and the noise spectrum of was measured (4). In this experiment, the pumping and repumping beams were simultaneously applied to the sample that resulted in decay to the CSS, that is, to the vacuum state in the Gaussian approximation. Thus, the decay Γ has to be taken into account. Equation 31.13 can be solved by performing a Fourier transformation and the resulting noise spectrum of normalized to the shot noise can be expressed as

31.15

The first term V _S2 represents the variance in shot‐noise units. The second term in the brackets, 2Γ, represents the quantum noise of the atomic ensemble recorded in the light beam. Most interesting is the first term in the brackets, proportional to the variance of S ₃.This term represents the quantum noise of S ₃ that was recorded in the atomic ensemble and subsequently transferred again back to the light beam to the S ₂ component of the Stokes vector.

With AC detection and a single atomic sample placed in a magnetic field, it is more difficult to recover the QND‐type coupling 31.11, which would be desirable for applications in quantum information processing. Remarkably, the QND coupling can be recovered if two atomic ensembles 1 and 2 polarized in opposite directions and both placed in a magnetic field are used as a single unit and the light passes through both ensembles in series (6,7). This approach has the added bonus that the effective atomic quadratures that couple to the light are nonlocal, that is, balanced superpositions of the quadratures of atomic ensembles 1 and 2. This can be explored to create entanglement of two distant macroscopic atomic clouds as discussed below.

Suppose that by means of optical pumping the atomic spins are aligned along the x‐axis and the two ensembles 1 and 2 are polarized in the opposite directions, 〈J _x1〉 = −〈J _x2〉 = J_x ; see Figure 31.3. The formulas relating the input and output components of the Stokes vector that describes the polarization of the light beam are a direct generalization of Eq. 31.14 and we have

31.16

The Heisenberg equations of motion for the y and z components of the collective atomic spin vectors can be formulated as follows:

31.17

We define atomic operators in the frame rotating with frequency Ω,

It is also helpful to introduce the “nonlocal” operators that are superpositions of the collective spin operators of atomic clouds 1 and 2,

31.18

The solution of the Heisenberg equations of motion 31.17 reads

31.19

Note that the “plus” operators in the rotating frame are constants of motion while the information about the light is fed to the “minus” operators. Using the commutation relations [J _y1 , J _z1] = iJ_x and [J _y2 , J _z2] = −iJ_x one can derive the commutation relations for the nonlocal operators in the rotating frame,

It follows from these relations that the quadrature operators of two effective atomic modes A and B should be defined as follows:

31.20

These operators satisfy the canonical commutation relations [x_j, p_k ] = iδ_jk .

In terms of the nonlocal atomic operators in rotating frame the input–output transformations for the Stokes operators read

31.21

The quadrature operators of light sidebands with modulation cos(Ωt) and sin(Ωt) can be defined as properly normalized Stokes operators,

31.22

where the integration is carried over the whole pulse and S ₁ = ∫S ₁(t)dt. These quadratures satisfy canonical commutation relations provided that the pulse duration is much larger than 2π/Ω. This condition is satisfied in the present experiments where Ω = 330 kHz and the pulse is approximately 1 ms long.

On inserting the definitions of the atomic and light quadrature operators in Eqs. 31.19 and 31.21, we finally obtain two decoupled systems of linear canonical transformations. Coupling of the modes A and L is governed by the transformations 31.11 and the atomic mode B couples to the light mode M according to

31.23

The coupling constant as it would have been for two atomic samples without the magnetic field. We have thus shown that the QND‐type interaction can be recovered if a pair of atomic ensembles with oppositely polarized spins is utilized. In the following sections, we shall illustrate various applications of the QND coupling 31.11 for quantum information processing. For the sake of presentation simplicity, below we will use the term “an atomic ensemble” although depending on the particular implementation the basic unit interacting with light may actually comprise two ensembles polarized in opposite directions.

31.3 Entanglement of Two Atomic Clouds

The basic application of the QND interaction 31.11 is to measure the atomic quadrature in a nondestructive way. This measurement reduces the uncertainty of the quadrature p _A and if the atomic ensemble was initially in a coherent state, then the measurement reduces the fluctuations of p _A below the shot‐noise level and the atomic ensemble is prepared in a squeezed state. The squeezed state is generally not centered on vacuum but is displaced by an amount that is proportional to the value of the measured light quadrature x _L. If the atomic state is displaced in such a way that this off‐set is canceled then the ensemble is unconditionally prepared in a pure squeezed vacuum state. The displacement can be accomplished by a tiny rotation of the atomic spin along the y‐axis, which couples the operators J_x and J_z . For small rotation angles ɛ, we have and the quadrature is displaced by the amount ɛ . In the system consisting of two atomic ensembles, the displacement has to be applied simultaneously and symmetrically to both ensembles so that the appropriate symmetric nonlocal quadrature is displaced.

The optimal classical gain g in the applied displacement can be determined by minimizing the noise of the displaced quadrature

31.24

Assuming that both atoms and light are initially in coherent states, then 〈(Δx _L)²〉 = 〈(Δp _A)²〉 = 1/2 and the optimum gain is given by g _opt = κ/(1 + κ ²). The variance of the atomic quadrature is reduced below the shot‐noise level 1/2,

31.25

and 3 dB squeezing is reached already for κ = 1. The great advantage of the light–atom interaction is that it may be experimentally feasible to achieve κ ≫ 1, which would result in very strong squeezing of the atomic ensemble. For instance, with κ = 5, we would obtain 14 dB of squeezing, which is much higher than the squeezing of light achievable in optical parametric processes. In practice, the maximum amount of squeezing would be mainly limited by the losses, spontaneous emission, and other decoherence effects. As shown in (8), when spontaneous emission is taken into account, the achievable degree of squeezing scales is approximately for large α ₀.

In the setting with two atomic ensembles, the quadrature that is squeezed is a balanced combination of the quadratures of the two atomic ensembles 1 and 2, . In this way, none of the two ensembles is prepared in a squeezed state separately, but the two ensembles are in an entangled Gaussian state. Moreover, in addition to detecting x _L, we can also measure the quadrature x _M and squeeze the atomic mode B in the quadrature . In this way, the two atomic ensembles are prepared in a two‐mode squeezed vacuum state. Such state is an implementation of the Einstein–Podolsky–Rosen entangled state introduced by these authors in 1935 in their famous paper on completeness of quantum mechanics (9).

The entanglement of two distant macroscopic atomic clouds has been demonstrated experimentally (6). The vapor of Cs atoms at room temperature was contained in two glass cells coated from inside with a special paraffin coating to reduce the Cs spin decoherence due to collisions with walls. In order to inhibit the depolarization of spin states, each atomic cell was protected from the external spurious magnetic fields by careful shielding. Initially, the atoms are prepared in a CSS by polarizing along the x‐axis by optical pumping. The close proximity of the prepared state to CSS is checked by observing the linear dependence of the variance of the measured Stokes components on the size of the collective spin J_x . An independent measurement via magneto‐optical resonance yielded the degree of spin polarization better than 99%. The linear dependence combined with experimentally verified, almost perfect spin polarization proves that the ensemble is very close to CSS. Then, the pumping lasers are switched off and the first (entangling) beam is sent through the two atomic cells. At the output, the cos and sin components at frequency Ω of the Stokes operator S ₂ are measured simultaneously, that is, and are detected. This prepares the atomic ensemble in an entangled state. To verify the presence of entanglement after time t, a second strong coherent verifying pulse is sent through the atoms and the two light quadratures and are measured. This provides information about the two squeezed nonlocal quadratures of the entangled atomic ensembles. In this experiment, it is not necessary to physically displace the atomic state after the entangling pulse, instead, one can displace the measured quadratures of the verifying pulse and the atomic squeezing can be inferred from the fluctuations of the difference operators and . In a later experiment, a deterministic entangled state of atoms was achieved by applying to atoms the displacement conditioned on the result of the first measurement.

The entanglement was tested using the Duan criterion (10), which states that the two‐mode state is entangled if the condition

31.26

is satisfied. The variances appearing in the criterion can be inferred from the variances of ΔX and ΔP, respectively. Since the verifying light beam is in a coherent state, we have and . In terms of the measured variances, the entanglement criterion can be thus rephrased as

31.27

In the experiment, the minimum observed EPR variance was , which confirms that an entangled state has been prepared. The entanglement survived for the time 0 .5 ms that was the delay between the entangling and verifying pulses.

31.4 Quantum Memory for Light

One of the major goals of quantum information processing is the development of a reliable deterministic quantum memory for light, where the quantum state of light could be stored for some time period T and retrieved at a later stage. The quantum memory for light is a key element of the envisioned quantum communication networks, where the quantum repeaters should allow distribution of entanglement over arbitrary long distances. The quantum memory is also required for other applications such as scalable quantum computing with linear optics. Note that a conditional atomic state generated upon detection of a photon emitted by an atom is also sometimes referred to as quantum memory. Such protocols usually work in a probabilistic way. As opposed to such approaches, here we discuss the deterministic memory for an unknown, externally provided state of light. The criteria for the quantum memory for light can be summarized as follows:

1. The memory should work for a class of independently prepared quantum states of light.
2. The storage should provide the fidelity higher than the fidelity for a classical storage protocol that involves measurement and repreparation and sends and stores only classical information.
3. The stored state should be readable.

The off‐resonant interaction of light with an atomic ensemble provides a natural interface between light and atoms. The QND interaction entangles the atoms with the light beam and this entanglement can be exploited to transfer the state of the light beam onto the state of the atomic ensembles.

The simplest memory storage protocol consists of sending the light beam through the atomic ensembles, measuring the quadrature of the output light beam, and applying an appropriate feedback to the atoms. This protocol has been already implemented experimentally, and storage of coherent light states with fidelity exceeding the maximum fidelity that can be achieved by measure‐and‐prepare protocols has been demonstrated (7). Consider the light and atomic quadratures after the QND interaction 31.11. If the light quadrature x _L is measured and the atomic quadrature p _A is displaced by an amount −gx _L, then the resulting atomic quadratures read

31.28

In particular, if κ = 1 and g = 1, then the light quadrature x _L is perfectly stored in the atomic quadrature p _A. The conjugate light quadrature p _L was stored in the atomic quadrature x _A during the QND interaction due to the feedback of light on atoms. The storage of p _L is only imperfect due to the noise stemming from the original atomic quadrature . This noise can be suppressed by preparing the atomic ensembles in a squeezed state before the memory protocol is applied. In the limit of infinite squeezing, we obtain in theory an ideal transfer of the light state onto atoms, and .

Note that the above conclusion is reached on the basis of Heisenberg equations of motion that are state independent. This means that an arbitrary single‐mode input state can be perfectly mapped onto an atomic ensemble state. This input state should be in a form of a linearly polarized light pulse. This pulse is then mixed on a polarizing beam splitter with an orthogonally polarized strong pulse in a coherent state. The two pulses must have a common spatio‐temporal mode. Under these conditions, and provided that the atomic sample is initially in a perfectly squeezed state, the quantum memory protocol should work for a qubit state of light, or any other state. The only limitation is that the mean photon number of this state must be much smaller than the mean photon number of the strong coherent pulse that drives the interaction. In the case of less than perfect squeezing of the initial atomic state, or even for the coherent initial state of atoms, the memory protocol can still provide the fidelity of mapping for a light qubit that is better than the fidelity for classical mapping. This subject is beyond the scope of the present article and will be considered in detail elsewhere (11).

The experimental demonstration of quantum memory for light has been carried out for a class of weak coherent states with mean photon number in the range between zero and a few. In the experiment, a pair of atomic ensembles in glass cells placed in external magnetic field and polarized in the opposite directions served as the memory unit; see Figure 31.3. The initial coherent states of light at the Ω sidebands were prepared by an electro‐optical modulator. The weak horizontally polarized coherent state and the strong coherent vertically polarized beam with identical spatio‐temporal profiles were sent through the atoms and the quadrature was detected in a self‐homodyne detector consisting of a polarizing beam splitter, two photodiodes, a lock‐in amplifier, and an integrator. The atoms were then displaced by applying a radio‐frequency magnetic pulse conditioned on the measurement result. The success of the quantum memory storage was verified by the read‐out pulse that was sent through the atoms after a variable delay τ. From the measurement of the read‐out quadrature , we can determine the gain of the storage for the x _L quadrature, , and the variance .

The conjugate quadrature x _A in this scheme does not directly couple to the light. In order to probe this quadrature, in another series of measurements, a magnetic π/2 pulse converting x _A to x _B is applied to the atoms. The quadrature x _B is then measured as the sin(Ωt) component of the signal. The gain g_p as well as the variance are determined similarly as for the x _L quadrature. The fidelity of the mixed Gaussian state stored in the memory with the initial pure coherent state |α〉 is given by

31.29

Note that for nonunit gains g_x and g_p , the fidelity depends on α. In the experiment, coherent states with mean number of photons 0 < |α| ² < 8 were stored in the memory. In this case, the optimal gain is actually slightly less than one and the gains used in the experiment g_x  = 0 .80 and g_p  = 0 .84 were close to the optimum. The mean storage fidelity obtained by averaging 31.29 over the ensemble of the input coherent states was determined from the experimental data as F _exp = 66 .7 ± 1 .7%, which substantially exceeds the maximum benchmark fidelity F _meas = 55 .4% that can be obtained by any classical measurement and repreparation protocol (12). The maximum memory time over which the fidelity was still larger than F _meas was τ _max = 4 ms.

The full retrieval of the quantum memory, that is, the transfer of the quantum state of the atoms onto light can, in principle, be performed in a similar way as the storage. Indeed, the QND interaction is fully symmetric so one simply exchanges the role of the atoms and light. The protocol goes as follows. First, a read‐out pulse is sent through the atoms. Then, a second, measurement, pulse is sent through the atomic ensembles in order to measure the quadrature . This measurement is not perfect since it is partially disturbed by the noise of the measurement pulse. Assuming for simplicity that the coupling strength κ is the same for both light beams, the measured quadrature reads . The p _L quadrature of the first beam is displaced by the amount and the final quadratures of the read‐out beam are

31.30

In contrast to the memory storage protocol, here both quadratures contain some extra noise even if g = κ = 1, since the measurement of the atomic quadrature is indirect and noisy. A perfect memory retrieval is possible only with very strongly squeezed light beams such that . However, even with vacuum light beams and with unity gains the fidelity of transfer of coherent states from the memory to light is F = 2/3, which is much higher than the maximum classical fidelity F = 0 .5. The above memory retrieval protocol implies that the retrieval light pulse does not travel too far away before the second pulse measuring on the atoms has completed its job. This means, in practice, that this retrieval protocol is limited to rather short pulses of light.

31.5 Multiple Passage Protocols

The great practical advantage of the entanglement generation and memory storage protocols is that each light beam has to pass through the atomic ensembles only once and is immediately measured afterward. This is crucial for the experimental feasibility of these schemes, because in current experiments, the duration of each pulse is about 1 ms, and the corresponding length is 300 km, so it is impossible to store the pulse, for example, in a fiber and send it through the atomic samples several times. This problem would complicate the experimental demonstration of the memory read‐out, where one would ideally like to displace the read‐out beam before detection, which would require keeping this beam somewhere while the atomic quadrature is being measured.

The experiments where the light beam traverses through the atomic ensembles several times could provide much more flexibility and allow us to generate entanglement and squeezing and transfer the state of light onto atoms and vice versa in a unitary way, without resorting to measurements and feedback. Note that in the multipass protocols discussed below, it is crucial that the second passage of the light beam through the atoms begins only after the end of the first passage, that is, the head of the pulse could be sent again onto the atoms only when the tail of the pulse already cleared through. Otherwise the various parts of the pulse would couple simultaneously to the ensemble that would invalidate the simple single‐mode description.

Schemes with several passages of light may become experimentally feasible if cold‐trapped atoms are employed, which could allow us to reduce the pulse duration to a few nanoseconds, making the pulse length compatible with table‐top experiments.

The main advantage of schemes with several passages is that it is possible to modify the coupling between the two subsequent passages by applying local phase shifts to atoms and light. For instance, it is possible to switch between the effective QND unitary transformations U_I (κ) = exp(−iκp _L x _A) and U_II (κ) = (−iκx _L p _A) (13). If these two unitaries are applied in sequence, then the resulting unitary will no longer correspond to QND coupling. In addition, it is also in principle possible to modify the coupling strength κ between two passages, for example, by changing the focusing of the light beam, although this would be experimentally challenging. It is insightful to consider the limit of a weak coupling, when κ ≪ 1. In this case, we can write

31.31

and the effective Hamiltonian generating U _tot is a sum of the effective Hamiltonians κ ₁ p _L x _A and κ ₂ x _L p _A. In particular, if κ ₁ = κ ₂ = κ then U _tot describes the two‐mode squeezer with squeezing constant κ while if κ ₁ = −κ ₂ = κ then U _tot represents a beam splitter with mixing angle κ. By repeating the sequence 31.31 many times, the total squeezing constant or mixing angle increases linearly with the number of passages n.

A more realistic evaluation of the schemes with multiple passages requires taking into account the losses and decoherence during the interaction (8). The resulting evolution corresponding to a single passage of light through the ensemble is a Gaussian completely positive map. Let v = (x _A , p _A , x _L , p _L) denote the vector of quadrature operators. The first moments d = 〈v〉 and the covariance matrix γ_jk  = 〈Δv_j Δv_k  + Δv_k Δv_j 〉 that comprises the second moments transform according to

31.32

The symplectic matrix S(κ) describes the QND coupling between atoms and light while the matrix D accounts for the damping due to losses and atomic depumping and G is the noise stemming from losses and decoherence. A simple model predicts that

Here, η is the atomic depumping rate introduced earlier in Section 31.2 and ɛ is the fraction of light lost due to the absorption, reflection from the glass cells, and so on. Note the factor of 2 in the atomic part of the noise matrix G. This additional noise arises because the atoms that decohere are still present in the atomic ensemble and contribute to the noise. Moreover, the damping decreases the coupling constant κ, because , and after n passages, we have κ_n  = [(1 − η)(1 − ɛ)] ^n/2 κ. The net effect of the multiple passages can be evaluated by iterating the map 31.32 with properly chosen S(κ_n ) for each passage. It has been shown that as the total number of passages n increases, the amount of generated entanglement grows and can be arbitrarily high even in the presence of losses and decoherence. To achieve good performance, it is necessary to optimize η, which is connected with coupling strength via κ ² = ηα ₀.The optimal η decreases with increasing n and the value of η can be tuned experimentally, for example, by changing the detuning Δ.

It has been shown that various important two‐mode linear canonical transformations can be implemented with three passages of light through the atoms, provided that κ can be set independently for each passage (14). The resulting effective unitary operation is given by

31.33

The two‐mode squeezing transformation U _TMS = exp[−ir(x _A p _L + p _A x _L)] with squeezing constant r is applied to light and atoms if

31.34

Similarly, it is possible to implement a beam splitter–type interaction by choosing

31.35

In particular, for φ = π/2, we get a beam splitter that swaps the state of atoms and light that can be used for quantum memory storage and retrieval. The advantage of this approach is that it does not require any measurement and feedback and the transfer of the quantum state from light onto atoms is in principle perfect even if the atomic ensemble is not initially squeezed. Importantly, at φ = π/2, the absolute values of all three coupling constants κ_j coincide, |κ_j| = 1, and it is not therefore necessary to change the strength of coupling between the subsequent passages but only apply local phase shifts to atoms and light.

Let us consider the three steps of the unitary quantum‐state swapping in more detail. In the first step, a unitary U ₁ = exp(−ix _A p _L) is applied and we have

31.36

Next follows the unitary U ₂ = exp(ip _A x _L), which results in

31.37

The state transfer is finished by sending the light through the atoms for the third time after local phase shifts such that U ₃ = exp(−ix _A p _L) is effectively applied, and we obtain

31.38

and the states of light and atoms have been mutually exchanged.

It is also possible to squeeze the state of the atoms in a unitary way by sending the light beam through the atomic ensemble several times. If we restrict ourselves to the sequence of unitaries U_I (κ_j ) and U_II (κ_k ), then the single‐mode squeezing of atoms U _SMS = exp requires four passages of light, and the coupling constants depend on r as follows,

31.39

and κ ₁ can be arbitrary. After this sequence of operations, the atomic ensemble is squeezed irrespective of the initial state of light and the scheme is thus robust against noise in the light beam.

31.6 Atoms‐Light Teleportation and Entanglement Swapping

Quantum teleportation is a process for a disembodied transmission of a quantum state between two distant locations via dual quantum and classical channels. The quantum channel consists of an entangled state shared by the sender and receiver. The sender carries out a joint measurement in the basis of maximally entangled states (the so‐called Bell measurement) on her part of the entangled state and on the state she wants to teleport. The measurement result is transmitted to the receiver via a classical channel, and the receiver then applies an appropriate unitary transformation to his part of the entangled state. Under ideal conditions, when the two partners share maximally entangled state and the Bell measurement is perfect, the state is exactly transferred to the receiver.

The quantum teleportation has been originally proposed for finite‐dimensional systems but it has been later extended to the realm of continuous variables (15,16). Here, the entanglement is provided by the two‐mode squeezed vacuum state, which approximates the (unphysical) maximally entangled EPR state. The continuous‐variable Bell measurement on two modes 1 and 2 consists of simultaneously measuring two commuting quadrature operators x ₊ = x ₁ + x ₂ and p₋  = p ₁ − p ₂. For two optical modes, this measurement can be accomplished by mixing the two modes on a balanced beam splitter and measuring the x and p quadratures on the first and second outputs, respectively.

However, the beam splitter is not the only option, and it can be replaced by the QND‐type coupling U = exp(−iκx _A p _L) with properly chosen coupling constant κ = 1. The atom and light quadratures after the interaction, and are exactly the balanced superpositions of the quadratures required for CV Bell measurement. The light quadrature can be measured directly while an auxiliary probe light beam has to be employed to measure the atomic quadrature similarly as in the protocol for atomic memory read‐out discussed in the preceding section.

The Bell measurement can be explored to teleport the quantum state of light beam onto atoms and vice versa (3). Consider first the teleportation of an atomic state onto light. Two light beams L and M are prepared in a two‐mode squeezed vacuum state with reduced fluctuations of the quadratures x ₊ = x _L + x _M and p₋  = p _L − p _M, 〈(Δx ₊)²〉 = 〈(Δp₋ )²〉 = e ^−2r . The horizontally polarized mode L is combined with a strong coherent vertically polarized coherent beam and sent through the atomic ensemble A. The output quadrature is measured and then an auxiliary beam K probes the atomic p‐quadrature, and is measured. The measurement results are communicated to the receiver who possesses the light beam M and displaces the quadratures according to . The resulting quadratures of the mode M read

31.40

This describes the unity‐gain teleportation and the mean values of the quadratures of mode M after teleportation are equal to the mean vales of the initial atomic quadratures. The process of teleportation is imperfect and adds some noise to the two quadratures. In the present case, this noise is unequally distributed since in the Bell measurement one quadrature is detected directly while the other only indirectly using the auxiliary beam K. The quality of the teleportation is often quantified by the fidelity of teleportation of coherent states. Assuming vacuum probe K and pure two‐mode squeezed vacuum in modes L and M, we obtain

31.41

The teleportation of the state of light onto the atoms proceeds in a similar way. The atomic ensembles have to be first prepared in an entangled state following the procedure described in Section 31.3. Then, the light beam is sent through one of the ensembles and the output light quadrature is measured. An auxiliary beam then probes the atomic ensemble and its quadrature is measured. The second ensemble is displaced according to the measurement results by tiny rotations of the collective spin over the y‐ and z‐axes. The relationship between the final atomic quadratures and the initial light quadratures is formally identical to Eq. 31.40, where the role of atoms and light should be interchanged.

One particularly interesting and useful application of quantum teleportation is the entanglement swapping, that is, a teleportation of one part of entangled state. In this way, the quantum entanglement can be distributed over quantum communication network. Consider three nodes, A, B, and C. Suppose that A and B share an entangled state of two atomic ensembles 1 and 2. Simultaneously, nodes B and C share an entangled state of two other atomic ensembles 3 and 4. The ensemble1 is at A, the ensembles 2 and 3 at B, and the ensemble 4 at C. The middle partner B can teleport the state of the atomic ensemble 2 to C by performing the Bell measurement on the pair of ensembles 2 and 3. This can be accomplished using the same procedure that was used to entangle a pair of ensembles. It is advantageous to carry out this experiment with atomic ensembles placed in an external magnetic field since then the Bell measurement can be performed in a single run by detecting the sin and cos modulation at Ω sidebands of the output light beam.

The whole procedure of the entanglement swapping would involve four light beams. First, two beams are used to entangle the pairs of atomic ensembles 1, 2 and 3, 4, as described in Section 31.3. Next, the third beam is sent through ensembles 2 and 3 and measured, which establishes an entanglement between two atomic ensembles 1 and 4 that never directly interacted. Finally, the presence of the entanglement should be verified by sending a probe beam through the ensembles 1 and 4 and measuring it.

31.7 Quantum Cloning into Atomic Memory

Due to the linearity of quantum mechanics, an unknown quantum state cannot be copied, the transformation |ψ〉 →|ψ〉|ψ〉 is forbidden in quantum mechanics. It is, nevertheless, possible to perform approximate copying of quantum states. In the context of continuous variables, particular attention has been paid to copying of coherent states, because the optimal cloning machines can be used as an efficient eavesdropping on quantum key distribution protocols based on coherent states and homodyne detection.

By exploiting the QND interaction between atoms and light, it is possible to combine the optimal Gaussian quantum cloning of coherent states with the storage of the clones into quantum memory and accomplish a direct quantum cloning into atomic memory (17). The cloning requires two passages of the light beam L through the two atomic ensembles A and B. During the first passage, the information about the x quadrature of light is transferred to atoms by engineering the effective interaction U ₁ = exp[−i(p _A + p _B)x _L]. After the first passage of light through the ensembles, we obtain

31.42

In the next step, the information about the quadrature p _L is written to atoms by sending the light beam through the ensembles again. Before this, local phase shifts are applied to atoms and light that change the effective interaction to U ₂ = exp[i(x _A + x _B)p _L]. After the second passage, the quadratures are transformed to

31.43

If the atomic ensembles are initially in a vacuum state (i.e., CSS with all atoms pointing along the x‐axis), then the ensembles A and B contain two optimal Gaussian clones of the coherent state |α〉 of the light beam L (18), each with fidelity F = 2/3.

In current experiments with hot atomic ensembles, it would be impossible to accomplish the second passage of light through the atoms because the light pulse has to be several hundred kilometers long. Luckily, the second passage of the light beam through the atoms can be avoided and replaced by the measurement of the quadrature followed by the displacement of the atomic quadratures conditioned on the measurement outcome,

31.44

The resulting atomic quadratures coincide with those in Eq. 31.43. This renders the cloning experimentally feasible and the whole procedure closely resembles the protocol for the quantum memory storage. The protocol can be also generalized to optimal asymmetric Gaussian quantum cloning where the two clones exhibit different fidelities. The asymmetric cloning is achievable by a suitable preprocessing of the atomic ensembles, by preparing them in a squeezed state with reduced fluctuations of quadratures x _A and p _B. By varying the amount of squeezing, the whole one‐parametric class of optimal Gaussian asymmetric cloning machines for coherent states can be obtained.

31.8 Summary

We have described a quantum interface between a single‐mode light and atomic ensemble(s). The basis of this interface is an off‐resonant dipole interaction that leads to a phase shift (polarization rotation) of light and Stark shift (rotation of the collective Bloch vector) of atoms. Combined with the quantum measurement and feedback, this interaction provides a wide range of operations useful for quantum information processing, such as long‐distance quantum teleportation of atomic states, quantum memory for light, and quantum cloning of light onto atoms.

Acknowledgment

This work was supported by EU under the projects COVAQIAL (FP6‐511004) and Integrated Project QAP, by the Czech Ministry of Education under the project Information and Measurement in Optics (MSM 6198959213) and by Danish National Research Foundation.

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