A theoretical study of quantum memories in ensemble-based media

Karl Bruno Surmacz St. Hugh’s College, Oxford

A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford

Michaelmas Term, 2007

Atomic and Laser Physics, University of Oxford i

A theoretical study of quantum memories in ensemble-based media

Karl Bruno Surmacz, St. Hugh’s College, Oxford Michaelmas Term 2007

Abstract The transfer of information from flying qubits to stationary qubits is a fundamental component of many quantum information processing and quantum communication schemes. The use of photons, which provide a fast and robust platform for encoding qubits, in such schemes relies on a quantum memory in which to store the photons, and retrieve them on-demand. Such a memory can consist of either a single absorber, or an ensemble of absorbers, with a Λ-type level structure, as well as other control fields that affect the transfer of the quantum signal field to a material storage state. Ensembles have the advantage that the coupling of the signal field to the medium scales with the square root of the number of absorbers. In this thesis we theoretically study the use of ensembles of absorbers for a quantum memory. We characterize a general quantum memory in terms of its interaction with the signal and control fields, and propose a figure of merit that measures how well such a memory preserves entanglement. We derive an analytical expression for the entanglement fidelity in terms of fluctuations in the stochastic Hamiltonian parameters, and show how this figure could be measured experimentally. We then examine the operation of an off-resonant Raman quantum memory, in which a single photon and a control field propagate co-linearly and interact with an ensemble of Λ-atoms. We show that, in the Raman limit, a universal mode decom- position allows one to find the conditions, in particular the control pulseshape, under which storage of a given signal photon is optimal. Furthermore, the storage efficiency is specified by a single parameter. Reading out the stored excitation is inefficient unless the coupling for read-out is prohibitively high. We propose a scheme that solves this read-out problem, allowing efficient retrieval of the signal pulse using the same cou- pling as was used to store it. This method also applies to the other absorptive memory schemes, and spatially separates the output signal field from the control fields. We go on to show that the above proposal for efficient read-out can be used to implement a multimode quantum memory, in which multiple photonic qubits encoded in frequency can be stored in a single ensemble. After retrieval the different components of the signal are spatially separated from each other, and from the control fields. Finally, we investigate the possibility of using an optical lattice as a storage medium in a quantum memory. In addition to eliminating motional dephasing, and enabling manipulation of stored qubits, using a lattice allows a reduction of the signal field group velocity due the periodic refractive index structure induced. This leads to an enhanced coupling, and hence a higher memory efficiency than for an atomic vapour with the same optical depth. ii Contents

Abstract ii

Table of Contents v

Chapter 1. Introduction 1

Chapter 2. An introduction to quantum memories 13 2.1 A simple quantum memory ...... 13 2.1.1 Two-level absorber ...... 14 2.1.2 Model for spontaneous emission ...... 15 2.1.3 Three-level absorber ...... 17 2.2 Ensemble-based memories ...... 18 2.3 Spontaneous emission in ensembles of Λ-type absorbers ...... 19 2.4 Figures of merit for a quantum memory ...... 22 2.4.1 Input-output fidelity ...... 22 2.4.2 Entanglement fidelity ...... 24

Chapter 3. Publication: Entanglement fidelity of quantum memories 29

Chapter 4. Entanglement fidelity of quantum memories - methods 43 4.1 Model ...... 43 4.2 Solution ...... 47 4.2.1 Ensemble-photon interaction ...... 48 4.2.2 Entanglement fidelity ...... 51 4.3 Experimental setup ...... 53 4.4 Entanglement fidelity as an entanglement measure ...... 57 iv Contents

Chapter 5. Schemes for ensemble-based quantum memories 63 5.1 Quantum memories using electromagnetically-induced transparency . 65 5.1.1 Model ...... 66 5.1.2 Electromagnetically-induced transparency ...... 71 5.1.3 Quantum memory interaction ...... 75 5.1.4 Quantum state transfer ...... 77 5.1.5 Limitations ...... 79 5.2 Quantum memory using controlled reversible inhomogeneous broadening 82 5.2.1 Scheme ...... 83 5.2.2 Controlling the rephasing artificially ...... 85 5.2.3 Limitations ...... 86 5.3 Feedback quantum memory ...... 88 5.4 Multimode quantum memories ...... 92 5.5 Conclusion ...... 97

Chapter 6. Mapping broadband single-photon wavepackets into an atomic memory 101 6.1 Introduction ...... 101 6.2 Model ...... 103 6.3 Mode decomposition ...... 107 6.4 Solution of the memory interaction ...... 110 6.4.1 Read-in ...... 112 6.4.2 Read-out ...... 114 6.5 Transverse structure ...... 117 6.6 Conclusions and discussion ...... 121 6.6.1 Connecting the optimizations ...... 121 6.6.2 Memory performance ...... 126

Chapter 7. Publication: Efficient spatially-resolved multimode quantum memory 131 Contents v

Chapter 8. Efficient spatially-resolved multimode quantum memory - methods 145

8.1 Introduction ...... 145

8.2 Phasematching a single-mode memory ...... 146

8.3 Phasematching limitations ...... 149

8.4 Phasematching using two control fields ...... 151

8.5 Multimode memory equations ...... 157

8.6 Conclusion ...... 162

Chapter 9. Publication: Quantum memory in an optical lattice 165

9.1 Introduction ...... 166

9.2 Model ...... 168

9.3 Group velocity reduction and reflection ...... 171

9.4 Memory storage ...... 177

9.5 Conclusion ...... 179

Chapter 10. Conclusions 183 Chapter 1

Introduction

The recent advent of quantum information theory has opened up the exciting pos- sibility of exploiting fundamental properties of quantum mechanics to gain significant advantages in information processing and computation. In this theory information is encoded in quantum bits, or qubits, which can then be exploited to perform tasks that would not be possible using a classical system [1, 2]. As well as the huge efforts to build a quantum computer based on this new framework, quantum information the- ory has given rise to quantum communication protocols [3–5], for which entanglement of qubits is a key resource. For example, in the cryptography scheme of Ekert [4], Einstein-Podolsky-Rosen (EPR) states shared between two parties can be used to en- sure that a cryptographic procedure is secure. Inherent in such schemes are periods during which qubits must be stored whilst some other process occurs, and retrieved on demand for further use when the process is completed, using a quantum memory. One example is the classical communication of a measurement result between two parties, the time for which is dependent on the distance between the parties.

To demonstrate the application of quantum memories more fully, consider the prob- lem of distributing entangled qubits. The implementation of quantum communication 2 Introduction and quantum information processing schemes relies on ones ability to distribute en- tanglement across a network. For example, if quantum cryptographic protocols are to be taken seriously as practical methods to securely transmit information, then one would have to demonstrate their use over continental (∼ 500km), and intercontinen- tal (∼ 104km) distances. This requires entanglement to be distributed over similarly large distances. Realizing such long-distance entanglement represents an enormous challenge. As an illustration of this consider two parties Alice and Bob. Alice has an entangled pair of qubits, and she wishes to send one to Bob, for example using a fibre. If Alice and Bob are separated by a distance L, then the probability of the entangled two-qubit state being preserved by this process scales exponentially with L/L0, with

L0 the attenuation length of the communication channel used. This is due to the noise present in the channel between Alice and Bob, which leads to decoherence. The loss rate for light at a wavelength of 800nm in a fibre is 2dB/m [5], so it is not feasible to directly share entangled states over large distances in this way.

This problem can be overcome using a quantum repeater [6–8]. This works by di- viding the distance L between Alice and Bob into segments that are sufficiently short to enable two qubits placed at either end of each segment to be entangled. This entangle- ment can then be propagated across the network using a combination of entanglement swapping and entanglement purification [9, 10]. The result is that Alice and Bob share the entangled state over a large distance, with in principle an arbitrarily high fidelity (depending on the number of resources used [11]) provided the quantum operations can be faithfully performed. Entanglement swapping, which uses quantum teleporta- tion [12], consists of a measurement, followed by classical communication of the result. This means that during the classical communication time the qubits need to be stored, and then retrieved when the result has been communicated, so that the entanglement between the qubits is preserved. 3

It has been shown that photons are a promising candidate for encoding qubits (e.g. in polarization or frequency degrees of freedom) [13], and so a quantum memory that stores photonic qubits, and re-emits them on demand whilst preserving their entanglement with other qubits, is desirable. On the other hand, many potential candidates for the storage unit of such a memory have been proposed and investigated. However, the mechanism by which photonic qubits are to be stored in these media is common to them all. In chapters 2 to 4 of this thesis we investigate a general quantum memory based on this observation. The remainder of the thesis focuses on specific memory implementations.

A general quantum memory consists of one or more absorbers containing a three- level Λ-type internal level structure, with initial state | 1i, excited state | mi, and metastable storage state | 3i [14]. Note that if a memory is set up to store one logical state of a photonic qubit, then the probability of storing the other logical states will in general be low. At this point we assume that this can be overcome, for example, by using two memories. The light pulse to be stored (the signal field) excites the | 1i → | mi transition, and the pulse is stored as a metastable excitation of the medium by a control field that drives the | mi → | 3i transition. After some storage time the stored signal field can be read out using another control field. Typically the quality of such a memory is assessed by their its ability to preserve the input state of the signal field [15]. However, as we have described, the crucial property of a quantum memory used for quantum communication is its ability to store a photonic qubit that is entangled with some other system, and to retrieve it so that the amount of entanglement is preserved. To see why this is not the same as state preservation, consider the storage and retrieval of two qubits, initially in a maximally-entangled state, using quantum memories. Say that the memory interaction results in unitary evolution of the qubits, so that the final two-qubit state is another maximally-entangled state orthogonal to the 4 Introduction

first. This final state could be still be used in the repeater protocol described above, for example, but the input-output fidelity of this memory would be zero. Hence, this figure of merit does not quantify the crucial property of the memory. In the first part of this thesis, we examine this general model of a quantum memory. We characterize a memory in terms of stochastic parameters in the interaction Hamiltonian, and calculate an expression for a figure of merit, based on the notion of entanglement fidelity [16], in terms of these parameters. Our figure of merit measures how well a quantum memory preserves entanglement.

Having characterized a general quantum memory, the second part of the thesis focuses on specific implementations of the memory. Initially we examine the storage of a quantum light pulse, rather than a qubit, in a single memory. The different systems that have been investigated for their suitability as a storage unit each have their advantages and disadvantages. Whilst reversible state transfer from a photonic excitation to a single atom in a cavity has been achieved [17, 18], much recent attention has focussed on using ensembles of absorbers as a storage medium. In such a memory the coupling between the medium and the signal field is enhanced by a factor of the square root of the number of absorbers, so an efficient memory in an ensemble should be more easily realized experimentally. There are different mechanisms with which one can implement an ensemble-based memory. The proposal of Fleischhauer and Lukin [19], which has been demonstrated exerimentally [20–23], uses electromagnetically-induced transparency (EIT) to adiabatically reduce the group velocity of the signal field, whilst transferring the field to an atomic excitation. In EIT the control field induces a window in frequency space over which the medium becomes transparent, and it is the width of this window that limits the bandwidth of the signal field that can be stored. An alternative scheme uses the fact that the excited atomic state | mi can be broadened

(either due to Doppler shift or artificially) to a width such that the ensemble absorbs 5 the signal pulse. This broadening can then be reversed for the read-out of the signal, so that the field re-phases. This scheme is known as CRIB, standing for controlled reversible inhomogeneous broadening [24–26]. The bandwidth limitation in this case comes from the width of the broadened | 1i → | mi transition. The other main memory scheme, which we call a feedback memory [27–29] is qualitatively different from EIT- and CRIB-based memories, in that the signal field is not absorbed by the medium. Instead, the signal and control field interact with the ensemble, and the signal field is measured. The result of this measurement is then fed back onto the ensemble, which results in the state of the initial signal field being transferred to the medium. Since this scheme is based on a scattering-type interaction, if these memories were used in a quantum repeater then a potential source of error would be creation of additional photons. Hence, whilst being the most experimentally advanced of the ensemble-based memories [29, 30], feedback memories are less well-suited to a quantum repeater, since the repeater schemes (e.g. [8]) are not robust against increasing photon number.

Atomic vapours are one of the main candidates for ensemble-based memories [19– 25, 27–29], due to the high number densities that can be achieved. However, as we show using the entanglement fidelity measure described previously, the efficiency of such memories is limited by dephasing and Doppler broadening due to atomic mo- tion. Cooling down the ensemble reduces these problems, but also reduces the number density of the sample, so the coupling strength is compromised. The limitations of atomic vapour memories have led some to consider solid-state implementations [31]. In particular, rare-earth ion-doped metals have been touted as an ideal material for the CRIB memory scheme [32] – the naturally-occurring absorption line is narrow, and it can be broadened artificially using external fields. Arrays of quantum dots have also been considered as a storage medium [33, 34]. However, the density of dots that can be achieved is too low to achieve a sufficient enhancement of the coupling. Also 6 Introduction inhomogeneous broadening due to e.g. variation in the dot size is very large in quan- tum dots, and cannot be mitigated by using off-resonant interaction. The other main candidate for a solid-state memory involves nitrogen-vacancy (NV) centres in diamond [35, 36]. The number density is higher than in quantum dots, but the much lower dipole moment means that again the coupling is very low. This could be increased for NV centre samples using microcavities and waveguides, but processing the diamond for this is technically very difficult. Also, optically pumping the NV-centres into the state | 1i is only about 90% efficient, and splitting of the | 1i and | 3i states is small (2.8GHz), which limits the signal field bandwidth to around a nanosecond.

In the second part of this thesis we consider storage of a photon using a Raman quantum memory, in which the signal and control fields are far-detuned from the excited state | mi. This work was undertaken in collaboration with the group of Prof. Ian Walmsley, which specializes in the generation of ultra-short laser pulses. In a Raman memory the bandwidth limitation arises from the size of the detuning, so by making this large it will be possible to overcome the restrictions of the EIT and CRIB memory schemes, and store ultra-short pulses. The off-resonant nature of the interaction will also reduce the effects of inhomogeneous broadening (Doppler shifts, for example), compared with the other schemes. Because of this, and because of the large optical depths achievable, we first consider an atomic vapour as the storage medium. When analyzing the memory interaction, the temporal shape of the signal is important. It has been shown that in a Raman scattering process, the collective coupling of the atoms to the control field results in the emitted signal field having a certain mode structure [37], which depends on the shape of the control pulse. In the case of a memory, for a given signal pulse there will be some optimal control field pulseshape that stores the signal with the highest possible efficiency. We investigate this optimization, and show that for a moderate coupling strength one can choose the control field pulse shape to 7 give storage of the signal field with near-unit efficiency. Furthermore, the efficiency of the memory is determined by a single overall coupling parameter.

Once the read-in process for an ensemble memory has been optimized, the retrieval of the stored excitation is still a non-trivial problem. Using one of the Raman, EIT, or CRIB schemes for storage results in a collective excitation of the medium, known as a spin wave. This spin wave is in general asymmetric, due to the fact that atoms at the front of the ensemble have a greater probability of absorbing the signal field than atoms at the back. Now, if one wishes to read out the signal field so that it propagates in the same direction as when it was stored, the optimal spin wave to do this will be the mirror-image of the optimally-stored spin wave [38]. The efficiency of the read- out depends on the overlap between these two spin waves, so will be low due to their asymmetry. This overlap can be made unity by considering backwards read-out of the signal field. However, by conservation of momentum the resulting spin-wave will in general have a non-zero momentum (a phase mismatch), unless levels | 1i and | 3i are degenerate. For the purposes of distinguishability of the atomic transitions it would be advantageous for these levels to not be degenerate, but the resultant spin-wave momentum degrades the overlap between the spin wave and the optimal output mode. Furthermore the output signal field propagates in the same direction as the control field used to retrieve it. Hence spectral filtering is required to resolve the two fields, and since the signal field is typically much weaker than the control field (one might, for example, wish to store a single photon), this filtering is difficult to achieve. We solve this problem of reading out the signal field by proposing a modified scheme that stores and retrieves the signal field with high efficiency (i.e. we phasematch the memory), and spatially separates the output signal field from the read-out control field.

Having solved the problem of how to store and retrieve the signal field with high

fidelity, the question of how this can be extended to a memory that stores qubits re- 8 Introduction mains. After all, it is entangled qubits, rather than just light pulses, that are required for quantum repeaters and quantum communication. One could use multiple ensem- bles, with one to store each logical state of the qubit, but this greatly increases the overheads one would require for, say, a quantum repeater. We show that our scheme for phasematching an ensemble-based memory, as discussed above, can be used to imple- ment a multimode memory that stores multiple signal fields of different frequencies in a single ensemble. Crucially, after read-out these multiple modes are spatially separated and hence can be easily distinguished from each other. This allows the storage and retrieval of multiple frequency-encoded qubits without resorting to multiple ensembles.

One of the most desired properties of a multimode quantum memory is the ability to manipulate the stored qubits, i.e. to perform single- and two-qubit operations on them. For a memory using a single atomic ensemble, it is not clear how this could be implemented. Also, any method for manipulating qubits stored in an ensemble would have to be done quickly because of the storage time dephasing due to atomic motion. It has been shown that, if one could store a qubit using atoms trapped in an optical lattice [39], then, due to the tight control one has over the lattice parameters, one would be able to perform operations on the qubit. Furthermore, trapping the atoms in a lattice virtually eliminates dephasing due to atomic motion. In the final part of this thesis we investigate the memory interaction for atoms in a lattice, and show that in addition to these advantages, it is possible to obtain a significantly enhanced efficiency compared to an atomic ensemble with the same optical depth [38]. The signal and control fields induce a refractive index in the atoms, and the periodicity of the lattice results in a photonic crystal structure [40]. Tuning the signal field close to the band edge of the crystal results in a significantly reduced signal field group velocity, which in turn increases the coupling of the signal to the atoms.

The thesis is structured as follows. In chapter 2 we introduce a model for a general BIBLIOGRAPHY 9 quantum memory, and analyze in detail the role spontaneous emission would play in such a memory. We also define the concept of fidelity for such a memory. In chapter 3 we calculate the entanglement fidelity of a general quantum memory, as well as showing how this figure of merit could be measured experimentally. Chapter 4 contains a more detailed description of the methods used in chapter 3, as well as further discussion on the entanglement fidelity measure. In chapter 5 the different schemes for ensemble- based quantum memories are discussed in detail, and in chapter 6 we analyze the modematching problem for such memories. We calculate the optimal control field that enables maximally-efficient storage of the signal field. Our findings are compared to, and put into context with, the work of other groups relating to this problem [38]. We also discuss the problems with reading out the stored excitation, and in chapter 7 we propose a scheme to overcome these problems. This chapter also contains the implementation of a multimode memory. The methods used in chapter 7 are explained more fully in chapter 8, where we also discuss an alternative possibility for overcoming the retrieval problem in ensemble-based memories. In chapter 9 we investigate the proposal of using atoms trapped in an optical lattice for a quantum memory, and demonstrate the increased efficiency that one can achieve in such a setup. The thesis concludes in chapter 10 with a summary of our findings, and a discussion of potential future directions for research in the field of quantum memories and quantum networks.

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An introduction to quantum memories

As discussed in chapter 1, reliable quantum memories are fundamental components of schemes for quantum communication and quantum information processing schemes, in which the storage and on-demand retrieval of entangled qubits is required. In this chapter a basic model for a quantum memory is introduced, and the ways in which the performance of such a memory can be evaluated are discussed.

2.1 A simple quantum memory

Before designing a quantum memory to store qubits, it is necessary to decide how best to encode a qubit. It has been shown that photons are suitable candidate for this - they can be distributed quickly, and there are numerous internal degrees of freedom that can be used for encoding, e.g. polarization or momentum. Hence in this thesis we concentrate on memories to store and retrieve photonic qubits.

A quantum memory should consist of a storage medium, and some form of coupling between the medium and the light pulse to be stored, which we call the signal field.

Let us assume that the signal field occupies a mode with annihilation operatora ˆ, and 14 An introduction to quantum memories that the medium has some excitation with associated annihilation operator Sˆ. An ideal quantum memory could hence be represented by the Hamiltonian

Hˆ = κaˆSˆ† + H.c., (2.1) where κ is the coupling between the signal field and the medium. The storage medium needs to be robust and long-lived, and must contain some degree of freedom with two states - an initial state and a storage state. Hence we begin by considering a simple two-level absorber as our storage medium.

2.1.1 Two-level absorber

The system we consider is illustrated in Fig. 2.1. A single mode light field, with frequency ωs, is incident on a two-level absorber, which has initial state | 1i. The field excites a transition to some excited state | mi, and we assume that the signal field is on resonance, i.e. ωs = ω1m, with ω1m the frequency of the | 1i ↔ | mi transition. Hence, one could naively consider this to be a quantum memory for a single mode of light. However, say that the signal field was absorbed. Then one would like to be able to retrieve the field on demand after some user-specified time. In the two-level atom setup there is no mechanism with which to do this. Also, there will in general be spontaneous emission from state | mi. Here we introduce the formalism used to describe spontaneous emission in a two-level atom, so that we can use it later in the chapter. 2.1 A simple quantum memory 15

Figure 2.1. A possible quantum memory using a two-level atom. A single- mode signal field, with frequency ωs, excites a transition | 1i → | mi in a single atom. The field is assumed to be on-resonance, so that ωs = ω1m.

2.1.2 Model for spontaneous emission

Spontaneous emission can be modelled using a quantum jump approach [1, 2]. First, consider the coherent interaction between the single mode light field, with annihilation operatora ˆ, and the atom, initially in state | 1i. The Hamiltonian for this process in a rotating frame is given by [3]

ˆ H0 = g0aˆσˆm1 + H.c., (2.2) where r ˆ ωs g0 = hm | (d · ²s) | 1i , (2.3) (2~²0V ) ˆ d is the electric dipole operator,σ ˆµν is the transition operator | µi hν |, ²s is the polar- ization vector for the signal field, and V the quantization volume. Writing the general

† state of the system as | ψ(t)i = c1(t) | 1i ⊗ aˆ | vaci + cm(t) | mi ⊗ | vaci, one can easily solve for the evolution of this system. However, we wish to include the fact that, while the atom is in state | mi, spontaneous emission can occur. To model this effect, one must consider interaction of the atom with all quantized field modes. If spontaneous emission is treated as a stochastic process, then one can derive a stochastic Schr¨odinger 16 An introduction to quantum memories equation for the state of the system at time t of the form [1]

· Z ¸ i √ p d | ψ(t)i = − Hˆ dt + Γ dΩ Φ(n)ˆσ e−iω1mn·r/cdB† (t) | ψ(t)i , (2.4) ~ eff n 1m n where iγ Hˆ = Hˆ − | mi hm | , (2.5) eff 0 2 and γ is the spontaneous decay rate of state | mi. The integral term in Eq. (2.4) is called a recycling term, and describes the spontaneous emission of a photon by the atom. This emission results in a recoil of the atom, represented by the phase ω1mn·r/c, in direction n. The integral is over the surface of the unit sphere, with Ωn the small solid angle centred at n. The angular distribution of the emitted light is given by Φ(n), and dBn(t) is the Ito noise increment [1].

Equation (2.4) can be simulated using a Monte Carlo method [4–6] as follows. The system evolves according to Heff , with the γ term in Heff causing a decay in the norm of the wavefunction. A threshold ξ ∈]0, 1[ is chosen at random, so that when the norm of | ψ(t)i falls below ξ, a spontaneous emission occurs. One applies the recycling operator in Hˆ to the wavefunction, and then after renormalization the system continues ˆ to evolve according to Heff , and so on. The spatial distribution of the spontaneously- emitted photons can be determined by choosing a random direction according to the distribution Φ(n).

Since the decay of the norm of the wavefunction is proportional to the population of state | mi, if one were to attempt to use a two-level atom for a quantum memory, then spontaneous emission would occur with a high probability. Hence such a system is unsuitable for a photonic memory. However as we shall see, if one could absorb the signal field, and shelve the atomic excitation into some metastable state, then this would greatly reduce the probability of spontaneous emission. 2.1 A simple quantum memory 17

Figure 2.2. Possible implementation of a quantum memory using a three-level atom.

2.1.3 Three-level absorber

A medium is required that has a metastable storage state,which can be addressed so that the retrieval of the signal field can be performed on demand. To this end we consider an absorber that contains a Λ-type three-level structure, as shown in Fig. 2.2. The absorber is initially in state | 1i, and has an excited state | mi, and a metastable state | 3i. The signal field excites the transition | 1i → | mi, but now an additional laser field, which we call a control field, drives the | 3i ↔ | mi transition, so that the signal excitation is stored in state | 3i. After some storage time another control field is applied, which drives the absorber into state | mi, and the signal field is subsequently re-emitted. Note that whilst state | 3i can be chosen to have a long lifetime, state | mi will still be susceptible to spontaneous emission during the interaction. We shall study this effect in more detail later in this chapter.

There has been a great deal of work studying single-absorber quantum memories [7–9], and such memories have been experimentally realized [10]. However, using a single absorber has disadvantages. Firstly, the strength of the coupling between the signal field and the absorber needed for efficient storage is very high, and difficult to achieve, although the absorbers can be placed in high-Q cavities to alleviate this problem somewhat. A second disadvantage is that single-absorber memories are un- 18 An introduction to quantum memories surprisingly susceptible to loss of the absorber. In the following section we discuss a method to overcome these disadvantages using ensemble-based memories.

2.2 Ensemble-based memories

In light of the disadvantages associated with single-absorber memories, much recent work has focused on instead using an ensemble of three-level absorbers as the storage medium for a memory. Let us consider a signal and control field incident on an ensemble of N Λ-type absorbers. Assume that both fields are detuned from level | mi by ∆, and for now we neglect spontaneous emission from | mi. The Hamiltonian for the interaction between the fields and the absorbers is given by

XN n h io ˆ † (j) (j) (j) (j) H3 = ~ωsaˆ aˆ + ~ω1mσii + ~ω13σee + ~gaσˆ m1 − ~Ωcσm3 + H.c. , (2.6) j=1 where hm | (dˆ · ² ) | 3i Ω = c E , (2.7) c 2~ c is the Rabi frequency of the control field, Ec is the amplitude of the control field (both

Ωc and Ec are assumed to be independent of position r and time t), g is the coupling constant for the signal field as defined in Eq, (2.3), ωµν is the frequency of the transition | νi → | µi, and j labels the absorbers. From Eq. (2.6) one can see that field interacts with the collective Dicke-like states [12] of the ensemble:

| Gi = | 11,..., 1N i XN | Ii = | 11,..., 1j−1, mj, 1j+1,..., 1N i j=1 XN | Ei = | 11,..., 1j−1, 3j, 1j+1,..., 1N i . j=1 2.3 Spontaneous emission in ensembles of Λ-type absorbers 19

ˆ Then the Hamiltonian H3 can be written in a rotating frame in terms of these states as XN h √ i ˆ (j) (j) (j) H3 = ~∆σmm + ~g Naσˆ IG − ~ΩcσIE + H.c. , (2.8) j=1 √ Hence the coupling of the signal field to the ensemble is enhanced by a factor of N. This is the main motivation for using an ensemble of absorbers for a quantum memory.

Also, loss of an absorber during the memory process will have a negligible effect – typical ensembles in atomic quantum memories contain ∼ 108 atoms.

2.3 Spontaneous emission in ensembles of Λ-type

absorbers

We have seen that an ensemble of absorbers with a Λ-type structure form a potentially- useful storage unit for a quantum memory. However, despite the fact that the storage state | 3i is long-lived, spontaneous emission from state | mi could still degrade the performance of the memory. In the group of Prof. Ian Walmsley in Oxford experimental efforts are underway to construct an atomic quantum memory, and one of the steps towards this goal will be to characterize the spontaneous emission from an ensemble of Λ-atoms using four-wave mixing [11, 13]. As we will discuss in more detail in chapters 7 and 8 of this thesis, such a geometry is advantageous in a quantum memory scheme, because any output signal photons (either read out or spontaneously emitted) will propagate in different directions to the control fields. In conventional memory setups the signal and control pulses are colinear, and filtering of some kind is required to resolve the pulses. In this section we analyze the spontaneous emission from a four- wave mixing setup.

The system we consider is shown in Fig. 2.3. A classical signal field, with carrier 20 An introduction to quantum memories

Figure 2.3. Four-wave mixing setup, with a classical signal field (dashed) prop- agating along the z-axis, and two control fields (solid) at angles of ±θc to the z-axis.

frequency ωs, is incident on an ensemble of N absorbers, which contain a level structure as shown in Fig. 2.2, and excites the | 1i ↔ | mi transition. The signal field is assumed to propagate along the z-axis as shown. Two control fields, propagating in the x-z plane at angles of ±θc to the z-axis respectively, are used to drive the | 3i ↔ | mi transition. The control fields both have carrier frequency ωc and Rabi frequency Ωc, and the signal and control fields are assumed to have a common detuning ∆ from state

| mi. The Hamiltonian for the interaction is given by

XN £ ¤ ˆ iksz ikzz HFWM = ∆ˆσmm − Ωse σˆm1 + Ωc cos(kxx)e σˆm3 + H.c. , (2.9) j=1

where kz = ωc cos(θc)/c, kx = ωc sin(θc)/c, and Ωs is the Rabi frequency of the classical signal field. Spontaneous emission from state | mi can be modelled in the same way as for the two-level atoms, but we now have two decay channels | mi → | 1i and

| mi → | 3i, with decay rates Γ1 and Γ3 respectively. With the state of the atomic ensemble denoted by | ψai, one can write the stochastic Schr¨odingerequation for the interaction as before [14]. We simplify matters by noting that a negligible amount of spontaneous emission occurs outside the x-z plane. Then the recycling term becomes 2.3 Spontaneous emission in ensembles of Λ-type absorbers 21

Figure 2.4. Distribution of spontaneously-emitted photons from the four-wave mixing setup using numerical simulation of Eq. (2.10).

an integral over one variable u = cos(θ), with θ defined as shown in Fig. 2.3. This gives

½ · ¸ 2 Z q i i(Γ + Γ ) X p 1 d | ψ (t)i = Hˆ − 1 3 + Γ du N (u)ˆσ a ~ FWM 2 j j jm j=1 −1 ¾ √ 2 −iωjm( 1−u xˆ+uzˆ) † ×e dBu | ψa(t)i , (2.10)

2 2 where N1(u) = 3(1 − u )/4, and N2(u) = 3(1 + u )/8 are the emission distributions, xˆ and zˆ are the unit vectors in the x- and z-directions respectively, and j labels the terms corresponding to decay channel | mi → | ji.

Equation (2.10) can be solved numerically using the Monte-Carlo method described for the two-level atom. For a quantum memory, one spontaneous emission would result in failure of the memory, so once an emission occurs we do not evolve the system further. We average the atomic wavefunction over many trajectories to build up a picture of the distribution of the spontaneously-emitted photons. This distribution is shown in

◦ Fig. 2.4. With θc = 5 , peaks in the intensity distribution appear at θ = ±2nθc (n ∈ N), with the height of the peaks decreasing with increasing n. These findings are consistent with the analytical treatment of a similar problem given in [15]. Since the spontaneous 22 An introduction to quantum memories emissions are in well-defined directions distinct from the control fields, it would be possible to detect such a process by placing photodetectors in the appropriate places. Finally, the number of spontaneous emissions that occur decreases as the detuning ∆ is increased, due to the fact that the state | mi is less populated. In this respect it would therefore be beneficial to perform the memory interaction off-resonance, and we shall return to this later in the thesis.

2.4 Figures of merit for a quantum memory

We have seen a basic model for a generic quantum memory. The obvious first question to ask about such a memory is: how well does it work?. However, one must first decide how best to measure the quality of the memory. This depends on the purpose for which the memory is to be used, and the rest of this section is devoted to discussing this problem.

2.4.1 Input-output fidelity

One way to measure the quality of a quantum memory is to compare the state of the output signal field with the input signal field. More precisely, suppose we wish to store a photonic qubit, assumed initially to be in a pure state | ψi, in a memory. Let us assume that the qubit is encoded in the polarization of the field, with logical states | Hi and | V i. The action of the memory on state | ψi can be represented by a quantum operation Λ (an introduction to quantum operations can be found in [16]). After storage and retrieval the qubit will in general be in some mixed state Λ(| ψi hψ |). To measure how well the memory has stored the state ψ, one can calculate the fidelity of the input and output state, which is given by F = hψ | Λ(| ψi hψ |) | ψi [16]. However, 2.4 Figures of merit for a quantum memory 23

F is a property not only of the memory, but also of the initial state | ψi. To obtain a figure of merit that characterizes only the memory one can take an average, or minimize, for F over a set of input states. In the work of Hammerer et al. [17], a set of input states {ψx}, each of which occurs with probability p(x), is defined. The fidelity is then defined to be X ¯ F = p(x) hψx | Λ(| ψxi hψx |) | ψxi . (2.11) x This measure has been used in experiment as a figure of merit for the quantum memory scheme based on feedback (see [18] and references contained therein).

The measure F¯ measures how well the quantum memory preserves the input states given to the memory. However, this may not be the property that we wish to maximize when designing a memory. In Chapter 1 we identified the distribution of entanglement for quantum communication schemes as one of the most important applications for a quantum memory. In this setting the task for a quantum memory is to preserve entanglement between qubits, rather than preserving the individual states of the qubits. In the example given above for a photonic qubit encoded in polarization, let us now assume that this qubit is maximally entangled with some auxiliary qubit, with logical states | 0i and | 1i. The light field will also have some initial mode structure, which we shall denote by some annihilation operatora ˆ. Then, let us take the initial state of the light field and auxiliary qubit to be

1 † | ψ2i = √ (| H0i + | V 1i) ⊗ aˆ | vaci , (2.12) 2 with | vaci the vacuum state of the field. If the photonic qubit is stored in and retrieved from the quantum memory, and the mode structure of the light field has changed (for example, the field has a different carrier frequency), then the input-output fidelity would be less than 1. However, if the entanglement between the photonic qubit and 24 An introduction to quantum memories the auxiliary qubit is completely preserved, then the memory is still ideal for use in an entanglement distirbution scheme, for example in a quantum repeater. As another √ example, if the two-qubit state after the memory process is (| H0i + | V 1i)/ 2, then although the actual state is now orthogonal to the initial state, it is still maximally entangled. Hence, it would be preferable to have a measure of how well the entangle- ment between the two qubits is preserved. One way that this can be done is to use the entanglement fidelity.

2.4.2 Entanglement fidelity

Entanglement fidelity is a measure that was introduced [19] to characterize how well entanglement of a bipartite system is preserved when a quantum operation acts on one party. To see how this works, consider a quantum system Q, which is in a mixed state ρ. This system is hence entangled with some other system R, such that the joint system RQ is in a pure state [16], which we denote | ψiRQ. If a quantum operation Λ acts on system Q, then the entanglement fidelity, which is a function of Λ and ρ, is defined as

FE = hψRQ | [(IR ⊗ Λ)(| ψi hψ |)] | ψRQi , (2.13)

where IR is the identity operator on system R. The quantity FE gives a measure of how well the entanglement between R and Q is preserved by the process Λ.

The properties of the entanglement fidelity have been discussed in numerous papers

(see for example [20] and [21]). However, FE cannot be used to quantify the preservation of entanglement in the way that we require. To see this, consider the case referred to in the previous subsection, where a memory stores one qubit of the two-qubit state √ √ (| H0i+| V 1i)/ 2, and after the memory the state is (| H0i−| V 1i)/ 2. Both the input and output states are Bell states, and the operation performed is a local unitary, which BIBLIOGRAPHY 25 must preserve entanglement [22]. However, according to Eq. (2.13) the entanglement fidelity of this process is zero. There is a modified definition for entanglement fidelity, which allows for unitary evolution of the system Q, given by

ˆ † ˆ FU = hψRQ | UΛ [(IR ⊗ Λ)(| ψi hψ |)] UΛ | ψRQi , (2.14)

ˆ where UΛ is the unitary operator representing the local unitary evolution of system Q.

To use FU as a measure of the preservation of entanglement of Λ, one would maximize over all possible UΛ, and minimize over all states | ψRQi. This novel approach results in a figure of merit that is a property of only the quantum memory Λ [23].

In the next chapter, we use the modified entanglement fidelity FU to characterize how well a quantum memory preserves entanglement between a photonic qubit and an auxiliary qubit.

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Publication

Entanglement fidelity of quantum memories

K. Surmacz, J. Nunn, F. C. Waldermann, Z. Wang, I. A. Walmsley and Dieter Jaksch

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K.

Physical Review A 74, 050302(R) (2006)

We introduce a figure of merit for a quantum memory which measures the preservation of entanglement between a qubit stored in and retrieved from the memory and an auxiliary qubit. We consider a general quantum memory system consisting of a medium of two level absorbers, with the qubit to be stored encoded in a single photon. We derive an analytic expression for our figure of merit taking into account Gaussian fluctuations in the Hamiltonian parameters, which, for example, model inhomogeneous broadening and storage time dephasing. Finally we specialize to the case of an atomic quantum memory where fluctuations arise predominantly from Doppler broadening and motional dephasing.

The ability to store flying qubits in a quantum memory (QM) is a fundamental com- ponent of many quantum communication schemes [1, 2]. Numerous possible methods 30 Entanglement fidelity of quantum memories for storing and retrieving qubits encoded in light pulses have been proposed [3–5], and some of these proposals have recently been experimentally realized, achieving e.g. stor- age and retrieval of a single photon on demand [6, 7], and entanglement between light and matter [4, 8]. Many promising candidate systems for QMs such as atomic ensem- bles [9], arrays of quantum dots [10] or NV centers in diamond [11] can often effectively be described as ensembles of N two-level absorbers coupling to the incoming qubit. We consider two independent ensembles each storing one of the logical qubit states. The absorbers consist of two meta-stable internal states | 1i and | 3i as shown in Fig. 3.1(a), and a transition | 1i → | 3i is effected by the incoming photon in logical state q via coupling Ωq. The states | 1i and | 3i are usually not directly connected optically, with this transition often being achieved via an intermediate state | mi and additional con- trol fields. For most of this paper details of such additional structure in the absorbing medium are not considered, and we assume that its effects on the properties of the absorbers can be subsumed into stochastic fluctuations of the coupling parameter Ωq.

After a storage time ts another control field is used to retrieve the photonic qubit. Dephasing may take place in the memory during the storage time, which usually leads to different couplings when writing and reading the qubit.

Using these general assumptions and the notion of entanglement fidelity [12] we derive a figure of merit F that measures how well a QM setup can preserve entanglement between a qubit undergoing the memory process (the memory qubit) and an auxiliary qubit. Our figure of merit F is different from commonly-used quality measures such as average fidelity FA for a pre-defined set of input qubit states [13]. This captures the ability of a memory to recreate the initial state of the qubit, and is equal to 1 if and only if the memory stores and retrieves every state perfectly. However, depending on the application of the QM, one might not necessarily be concerned with exactly preserving the quantum state of the qubit. The preservation of entanglement might be 31

Figure 3.1. (a) General level structure of an absorber in a QM. A photon with annihilation operatora ˆq is incident on a medium of N absorbers, and excites one absorber into state | 3i via an intermediate state | mi. (b) Schematic experimental setup. We consider a photonic qubit entangled with an auxiliary qubit produced by an EPR source. The photonic qubit is stored in the memory, and the amount of entanglement that remains after storage is measured.

more desirable in some quantum information processing and quantum communication schemes [14, 15], for example in a quantum repeater [1, 2] or in the cascaded generation of graph states [16]. The entanglement fidelity F also directly relates to the degree of violation of a Bell inequality by an EPR pair of photons, where one photon is stored and subsequently retrieved from the QM while the auxiliary qubit is directly detected as schematically shown in Fig. 3.1(b). The setup shown in Fig. 3.1(b) could thus be used to measure our figure of merit.

In the system outlined above the memory qubit is encoded in a subspace of the overall photon Hilbert space HA. The states of the memory and auxiliary qubits are denoted by | 0Ai and | 1Ai. The Hilbert spaces of the auxiliary qubit and the medium are HB and HC respectively. The system has initial stateρ ˆ0 = | φ0i hφ0 | ⊗ ρˆC , where

| φ0i ∈ HA ⊗ HB andρ ˆC is the initial density operator of the medium. We assume that the absorbers are not correlated initially and characterize a QM as a quantum operation ΛM that acts on the photon as follows

h i ˆ ΛM ⊗ I : | φ0i hφ0 | → trC L(ˆρ0) , (3.1) 32 Entanglement fidelity of quantum memories

ˆ where L is a Liouvillian operating on states in HA ⊗ HC , and I is the identity operator on states in HB. We note that in the work of A. K. Ekert et al. [17] a quantum channel for qubits Λ is characterized by considering the action of the operator Λ ⊗ I on two qubit states. The superoperator ΛM ⊗I preserves entanglement for all two-qubit states only if ΛM is unitary (the converse is well known [18]). This can be seen by using a Kraus decomposition of ΛM . We find that for at least one initial two-qubit state

| φei the application of a non-unitary ΛM will result in a mixed state. Purification of (ΛM ⊗ I)(| φei hφe |) results in the introduction of an extra ancillary system, with which the memory qubit is entangled. By monogamy of entanglement [19–21], the entanglement between the memory and auxiliary qubits decreases.

Motivated by these observations we write a QM entanglement fidelity as follows

n o ˆ † ˆ F(ΛM ) = min hφ0 | UM [(ΛM ⊗ I)(| φ0i hφ0 |)] UM | φ0i . (3.2) | φ0i

The quantity inside the braces is the entanglement fidelity [12] for the process ΛM applied to the state trB[| φ0i hφ0 |] (trB denotes the partial trace over HB). The entan- glement fidelity was introduced as a measure to characterize how well entanglement is preserved by such a process in [12], and detailed discussions of its properties can be found in [12, 22, 23]. Since the standard definition of entanglement fidelity [24] mea- ˆ sures preservation of state as well as entanglement we include a unitary UM , which acts on HA, to allow for evolution of the photon that would not decrease the entanglement present. This unitary is chosen to maximize F, and thus describes an optimized storage process to which ΛM is compared in the same way that gate fidelity [24] measures the success of a quantum gate. We also minimize over all pure two-qubit input states so that F is a property only of the QM that uses the worst-case scenario as a measure of ˆ its success. The QM ΛM (and hence the Liouvillian L) consists of a read-in process, a 33 period of storage, and a read-out process that retrieves the photon on demand a time ˆ ts after read-in. Note that more sophisticated choices for UM conditional on the out- come of measurements on the state of the QM after retrieving the photon might enable further improvement of F. However, such schemes are difficult to realize experimen- tally and are not considered in this paper. Thus if F = 1 we have that ΛM preserves entanglement between the qubits, but the final and initial states of the photon may be deterministically different. The representation of the QM with ΛM illustrates that for F < 1 the memory process will not be unitary.

We now consider the photon and its interaction with the ensemble of absorbers. ¯ ® We define the annihilation operatora ˆ for the photon in state ¯q =a ˆ† | vaci, where q P q | vaci represents the vacuum state and q = 0, 1 denotes the logical state of the qubit

(the underline distinguishes states in HA from memory qubit states). This annihilation operator can be written as Z

aˆq = dkgq(k)ˆak,λq , (3.3)

wherea ˆk,λq destroys a photon with polarization λq and wavevector k. The mode † functions gq(k) are normalized, [ˆaq, aˆq0 ] = δqq0 and for simplicity we have assumed that each logical state has an associated single polarization λq. The absorbers are initially in the collective state | Gi = | 11,..., 1N i, and are assumed to coherently couple to the photon during the whole of the read-in and read-out processes. The Hamiltonian for ¯ ® the read-in interaction between the photon in state ¯q and the jth absorber is given P ˆ (j) (j) (j) by Hq = (Ωq,jaˆq,jσˆ31 + H.c.), whereσ ˆ31 = | 3ij h1 |. During storage each absorber evolves according to the Hamiltonian

ˆ (j) (j) HS,q = sq,j(t)ˆσ33 , (3.4)

with sq,j(t) some time-dependent detuning. The read-out interaction of the photon in 34 Entanglement fidelity of quantum memories

˜ˆ (j) ˜ ˆ (j) logical state q with the absorber is modeled by the Hamiltonian Hq = (Ωq,jbq,jσˆ31 + ˜ ˆ H.c.), with couplings Ωq,j. The dependence of the operatorsa ˆq,j and bq,j on the absorber reflects the fact that due to motion each absorber will in general couple to a slightly different mode. We assume that an appropriate choice of control field can restrict

(a) (a) this effect to a phase δq,j , so thata ˆq,j =a ˆq exp [iδq,j ], and similarly for the output ˆ ˆ (b) photon mode bq,j = bq exp [iδq,j ]. The read-in and read-out processes are assumed to ˜ require a time tp each. In general the couplings Ωq,j and Ωq,j will depend on time t. In the following we assume a simple time dependence where the magnitude of the read-in (read-out) coupling is switched on to a constant value for the time tp that maximizes storage (retrieval), then switched off. For simplicity we also let |Ωq,j| = ˜ |Ωq,j|∀j – the generalization to different couplings is straightforward. Inhomogeneous broadening can furthermore lead to phases linearly increasing with time, and during storage some additional dephasing can occur. As a result of these assumptions we

i(Kq,j t) ˜ i[Mq,j t+fq,j (ts)] write Ωq,j = κq,je and Ωq,j = κq,je , where fq,j(ts) appears as a result ˆ (j) (a) (b) of eliminating HS,q from the dynamics. The parameters κq,j, Kq,j, Mq,j, δq,j , δq,j and fq,j(ts) are all assumed to be real normally-distributed stochastic variables with respect ¯ to the storage medium. For instance Kq,j is broadened around a mean value Kq by a width wK,q and so on.

¯ To obtain an analytical expression for F, we first note that if Kq,j = Kq∀j the system reduces to a two-level problem, and the evolution during read-in can be solved exactly. To this end we rewrite the read-in Hamiltonian

¯ ¯ (K) (j) ˆ (j) iKqt iKqt iδq,j t Hq = κq,j[e + e (e − 1)]ˆaq,jσˆ31 + H.c., (3.5)

(K) and treat the term containing the fluctuation δq,j in Kq perturbatively up to second ˆ (j) order, and similarly for H˜q . Since any mean broadening could be corrected for, we 35

¯ ¯ assume that Kq = Mq = 0 for simplicity. The general initial normalized photon and auxiliary qubit state can be written as | φ0i = α | 0P 0Ai+β | 0P 1Ai+γ | 1P 0Ai+η | 1P 1Ai. P P ˆ ˆ (j) ˜ˆ (j) For each component of | φ0i the evolution operator U according to j Hq and j Hq can be used to calculate the final wavefunction of the system at time tf = ts + 2tp. Averaging over the ensemble similarly to [25] allows us to rewrite Eq. (3.1) as

DD EE ˆ ˆ † ΛM ⊗ I : | φ0i hφ0 | → U(| φ0i hφ0 | ⊗ | Gi hG |)U , (3.6) where hh... ii denotes averaging over the stochastic Hamiltonian variables then tracing out the memory. Since | φ0i is normalized F can be calculated by minimizing over a single parameter X = |α|2 + |β|2 in Eq. (3.2). This results in

© 2­­ 2 ®® ©­­ ∗®®ª F = X0 |b0| + 2X0(1 − X0)Re b0b1

2­­ 2 ®®ª +(1 − X0) |b1| , (3.7)

where X0 is the value of X that achieves the minimization in Eq. (3.2), and bq is the amplitude of the final output photon in logical state q. Differentiating F with respect to X gives a minimum of

­­ ®® ©­­ ®®ª |b |2 − Re b b∗ X = ­­ ®® 1 ©­­ ®®ª0 1 ­­ ®®, (3.8) 0 2 ∗ 2 |b0| − 2Re b0b1 + |b1|

but if this value lies outside [0, 1] then X0 = 0 or 1. Applying second-order perturbation 36 Entanglement fidelity of quantum memories theory to the read-in and read-out processes as previously described gives

· ¸· ­­ ®® Θ(w2 + w2 ) 1 |b |2 = 1 − K,q M,q + q 8N(¯κ2 + w2 ) N q κ,q ¸ (N − 1) 2 2 2 −(wa,q+wb,q+wf,q(ts) ) + 2 2 e , (3.9) N(1 +w ˜κ,q) · −(w2 +w2 +w (t )2)/2 ¸ ©­­ ®®ª Y e a,q b,q f,q s Re b b∗ = 0 1 1 +w ˜2 q=0,1 κ,q · ¸ X (4 + π2)Θ(w2 + w2 ) × 1 − K,q M,q , (3.10) 64N 2(¯κ2 + w2 ) q=0,1 q κ,q

2 4 2 2 wherew ˜κ,q = wκ,q/κq, and Θ = (1+6w ˜κ,q +3w ˜κ,q)/(1+w ˜κ,q) . We see that F decreases

2 2 exponentially in wa,q, wb,q and wf,q(ts), and also decreases as bothw ˜κ,q and wx,q/κq increase (x = K,M). Due to the factors of 1/N appearing in these latter terms, it is the exponential terms that will dominate for large N. Let us also note that to obtain

2 1/2 maximum absorption and emission we set tp = π/2(κqN) , so the terms containing wx,q could alternatively be seen to depend quadratically on tp. Finally, we observe that

2 sufficient conditions for F . 1 are that wa,q, wb,q, wf,q(ts) ¿ 1 and ωK,q, ωM,q ¿ κ , with the latter becoming less important as N → ∞.

The value of X0 represents the class of states that achieve the minimum required in

Eq. (3.2). To illustrate this let us consider some special cases. (i) If the states | 0P i and

| 1P i of the photon are absorbed and emitted in the same way (b0 = b1), then evaluating

X0 gives an indeterminate answer, reflecting the fact that F is minimized by several

­­ 2 ®® choices of | φ0i. Evaluation of F in this situation gives a value F = |b0| . (ii) If state | 1P i is perfectly stored, but state | 0P i is not stored at all, then b0 = 0, X0 = 1, and F = 0. We also compare our measure with the previously-defined fidelity FA. If entanglement is preserved i.e. F = 1 then FA = 1 if and only if the output photon has the same mode function as the input photon. In the case where the photon is stored and emitted with 100% probability, but becomes completely decorrelated with 37

Figure 3.2. Experimental method of measuring F requiring storage of one logical state only. A source S produces a separable pair of photons, so that photon 1 is stored in the QM and photon 2 enters the pulse shaper (PS). The photons interfere at a beam splitter (the PS includes a time delay), and coincidence measurements at detectors D1 and D2 are made.

the auxiliary qubit FA could vary between 0 and 1 depending on the spatial mode function of the output photon, but F = 1/2.

We now describe an experimental setup (shown in Fig. 3.2) that, assuming case (i) above holds, would allow us to measure F. After read-out but before the beam splitter (BS) the state of the photons will be

½ X ¾ † ˆout † ˆout ρˆin =a ˆPS [pm(bm ) | vaci hvac | bm ] + p0 | vaci hvac | aˆPS, (3.11) m

where | vaci denotes the vacuum,a ˆPS is the annihilation operator for the mode of n o ˆout photon 2 after the pulse shaper (PS), and bm with m ≥ 1 is the set of annihilation operators corresponding to the eigenmodes of the state of photon 1. The eigenvalues are in descending order p1 ≥ p2 ≥ ... and p0 is the probability of not retrieving the photon on demand. Noting that most detectors cannot resolve photon number, the probability

P∞ of obtaining a click in one of the detectors Dj (j = 1, 2) is Pj = m=1 pm(1 + Om)/4 + P∞ p0/2, and of a detection in both D1 and D2 is P12 = m=1 pm(1 − Om)/2, where ˆout † 2 Om = | hvac | aˆPS(bm ) | vaci | is the overlap of the field modes after the BS. Both the ˆout minimum value of P12 and the maximum value of P1 +P2 are obtained whena ˆPS = b1 i.e. when the mode of photon 2 is precisely the dominant mode of photon 1 and for this setting p0 + p1 = P1 + P2 − P12. Hence by tuning the PS the dominant mode 38 Entanglement fidelity of quantum memories

Figure 3.3. The entanglement fidelity of a Raman QM with (a) ζ = 4.49 × −14 −13 10 , (b) ζ = 2.25 × 10 . In both cases the photon bandwidth δp = 0.1∆, with ∆ = 1013s−1. The atomic level splittings used are | 1i → | mi = 5×1015s−1 and | 3i → | mi = 3.5 × 1015s−1, and the ensemble consisted of N = 108 atoms.

of the memory photon can be found experimentally. This tuning then corresponds to ­­ ®® ˆ 2 the UM that maximizes F as in Eq. (3.2). We can then deduce p1 = |b0| by removing the beam splitter and measuring the probability of the memory photon not being re-emitted on demand. Therefore F can be deduced.

We conclude our analysis by applying the fidelity measure F to a specific memory setup. We determine F for a QM for one single photon state based on off-resonant stimulated Raman scattering in an ensemble of Λ-atoms [26]. The atoms each have mass M and temperature T , and have the same internal level structure as the general absorbers considered in Fig. 3.1(a). The photon is incident on the ensemble and excites the | 1i → | mi transition. A control field drives | mi ↔ | 3i and stores the photon as a collective excitation in the ensemble. The probe and control fields are assumed to co-propagate with carrier wavevectors of magnitude kp and kc respectively. Retrieval of the photon is achieved by applying another control field a time ts after read-in. We assume that the probe and control fields are both far-detuned (detuning ∆) from level | mi, so this state can be adiabatically eliminated giving a medium consisting effectively of two-level atoms. Therefore the main source of stochastic variation in the coupling of 39 the atoms to the photon arises from the atomic motion, which we treat semiclassically assuming a Boltzmann distribution for atomic velocity components vj in the direction of the field propagation. This leads to Kq,j = Mq,j = vjωc/c, fq,j(ts) = χvj/c and

1/2 1/2 widths given by wK,q = wM,q = ωcζ , wf,q = χζ , where c is the speed of light,

2 ζ = kBT/Mc , χ = (kp − kc)cts, and ωc = ckc. In this scheme fq,j(ts) originates from the motion of the atoms during the storage time and Kq,j and Mq,j arise from the Doppler-shifting of the field frequencies. The parameter κq = κ is defined by the couplings of the photon and control fields to the atoms, and is assumed to be a constant.

The amplitude of the final photon state can be calculated as for the general case, which upon substitution into Eq. (3.7) yields the following expression for F up to N −2,

· 2 ¸ 2 2 3ζ(ω − ω ) F = e−χ ζ/2(1 − χ2ζ) e−χ ζ/2(1 − χ2ζ) − s c . (3.12) 2κ2N 2

We see that the two main contributions to the decrease in F are the Doppler broadening √ terms, which are quadratic in the ratio (ωs − ωc) ζ/κ, and the storage time dephasing √ terms, which depend on χ2ζ. This observation results in the requirement that χ ζ ¿ 1 in order to achieve F . 1 for N À 1. Figure 3.3 shows the entanglement fidelity of the Raman quantum memory for two different values of ζ, and the expected decrease in F with increasing χ is observed.

In summary we have introduced a figure of merit F for a general QM based on gate fidelity and derived an analytical expression for it. Our calculations took into account stochastic fluctuations in the coupling parameters whose origin might vary for different QM schemes. We concluded by applying our formalism to a specific atomic quantum memory. 40 Entanglement fidelity of quantum memories

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and D. Jaksch, e-print quant-ph/0603268 (2006). 42 Entanglement fidelity of quantum memories Chapter 4

Entanglement fidelity of quantum memories - methods

In this chapter we provide additional information, calculations, and discussions relating to the work on entanglement fidelity of quantum memories in chapter 3. First we justify the parameters used to model the memory interaction, and give details on how a solution to the model, and the resulting expression for the entanglement fidelity of the memory, were obtained. We provide more information about the experimental setup proposed in Fig. 3.2. This chapter concludes with an additional discussion about entanglement fidelity, in particular about its suitability for use as an entanglement measure.

4.1 Model

As in chapter 3, we consider a photon, in which a qubit is encoded, entangled with some auxiliary qubit. The initial state of the photon and auxiliary qubit is assumed to be pure, and can be written as

| φ0i = α | 0P i | 0i + β | 0P i | 1i + γ | 1P i | 0i + ν | 1P i | 1i , (4.1) 44 Entanglement fidelity of quantum memories - methods

2 2 2 2 2 where α, β, γ, ν ∈ C so that |α| + |β| + |γ| + |ν| = 1, and | qi ∈ HB = C (q = 0, 1) ¯ ® are the logical states of the auxiliary qubit. The state ¯q =a ˆ† | vaci, with | vaci P q the vacuum state, anda ˆq the annihilation operator defined in Eq. (3.3). The photon interacts with a storage medium consisting of N absorbers, with levels | 1i, | 3i and | mi as depicted in Fig. 3.1(a), via a classical control field. The medium is initially in collective initial state | Gi = | 11,..., 1N i.

The interactions of the photon in logical state q with the storage medium during ˆ ˆ˜ read-in and read-out are modelled by Hamiltonians Hq and Hq respectively, where

XN n o (a) (j) ˆ iKq,j t iδq,j Hq = κq,je aˆqe σˆm1 + H.c. , (4.2) j=1 XN n o (b) (j) ˆ˜ iMq,j tˆ iδq,j Hq = κq,je bqe σˆm1 + H.c. , (4.3) j=1

where j labels the atoms,σ ˆµν = | µi hν |, and for simplicity we have assumed that the magnitude of the coupling to atom j for read-in and read-out are both equal to ˆ κq,j. For read-out the atoms couple to the signal field mode bq. During the storage ˆ (j) time ts each atom evolves according to the Hamiltonian HS,q as previously defined in Eq. (3.4). However, to simplify matters, the evolution of absorber j during storage can R ˆ (j) t 0 0 be denoted by the unitary US,q(t) = exp[i 0 dt sq,j(t )]ˆσee. Defining

Z ts 0 0 fq,j(ts) = dt sq,j(t ), (4.4) 0

ˆ˜ ˆ (j) we can hence absorb the storage time evolution into Hq by multiplying it by US,q(ts).

ˆ ˆ˜ We now discuss the motivation behind the Hamiltonians Hq and Hq, and in partic- ular the set of stochastic variables used to model the interaction. Firstly, the Hamil- tonians depend on the logical state of the memory qubit, which reflects the fact the 4.1 Model 45

Figure 4.1. The atomic level scheme for a general quantum memory. A signal field (dashed) excites the | 1i ↔ | mi transition, whilst a control field (solid) drives the | 3i ↔ | mi transition.

different states will in general interact differently with the medium. To see this consider a memory consisting of a single ensemble of atoms. Transitions between the levels | 1i, | 3i, and | mi will in general have a polarization associated with them. It follows that, for a photonic qubit encoded in polarization, it will not be possible to set up a memory of this kind to store both logical states with maximal efficiency.

The parameters Kq,j and Mq,j are introduced to model broadening during the in- teraction. To see how this might arise, let us consider a single Λ-type three-level atom interacting with a signal and classical control field as shown in Fig. 4.1. The signal and control fields are assumed to be plane waves with frequency ωs and ωc respec- tively, propagating along the z-axis at the speed of light c. The signal field is described by annihilation operatora ˆ and couples to the | 1i ↔ | mi transition with strength g, and the control field couples to the | 3i ↔ | mi transition with Rabi frequency Ωc. The Hamiltonian for this interaction, as for the similar setup in chapter 2 [see e.g. Eq. (2.8)], is given by

£ ¤ ˆ i(ksz−ωst) i(kcz−ωct) H = ω1mσˆmm + ω13σˆ33 + gaˆσˆm1e − Ωcσˆm3e + H.c , (4.5) 46 Entanglement fidelity of quantum memories - methods

where ks = ωs/c, kc = ωc/c, t is the time of the interaction, g is the coupling of the signal field to the atoms as defined in Eq. (2.3), and ωµ,ν is the frequency of the | νi ↔ | µi transition. We further assume that both the signal and control fields are detuned by ∆ from level | mi. Transforming Hˆ into a rotating frame [1] yields the following Hamiltonian:

£ ¤ ˆ iksz i[kcz−(ωc−ωs+ω13)t] Hrot = ∆ˆσmm + gaˆσˆm1e − Ωcσˆm3e + H.c . (4.6)

We further assume that state | mi is negligibly populated, and can hence be adiabat- ically eliminated. This is done by writing the general normalized state of the system as

| ψi = c1(t) | 1i ⊗ aˆ | vaci + cm(t) | mi ⊗ | vaci + c3(t) | 3i ⊗ | vaci , (4.7)

with c1(t), cm(t), and c3(t) the complex amplitudes of the respective states. The ˆ Hamiltonian Hrot and the state | ψi are substituted into Schr¨odinger’sequation to obtain a system of differential equations in terms of the amplitudes. The adiabatic approximation assumes that the population of | mi does not change, so we set dcm/dt =

0. This gives an expression for cm in terms of c3 and c1, and hence allows us to rewrite ˆ Hrot as 2 |Ωc| © ª Hˆ = − σˆ + gΩ∗aσˆ ei[(ks−kc)z−(ωs−ωc−ω13)t] + H.c. . (4.8) ad ∆ mm c 13

We now have a phase term linear in t with coefficient (ωs −ωc −ω13) = 0. However, if we now let the atom have a non-zero velocity, then the signal and control field frequencies will be Doppler-shifted with respect to the atom. As a result this phase term, which we represent with Kq,j and Mq,j in our general model, will be non-zero and different from atom to atom, resulting in a broadening of the transition for non-stationary atoms.

ˆ ˆ˜ Using Eq. (4.8) the other stochastic parameters in the Hamiltonians Hq and Hq 4.2 Solution 47

can be motivated. The couplings κq,j could change from absorber to absorber due to variation in the detuning ∆ from Doppler shifts again arising from the atomic motion. ˆ Also, in Had there is a spatially-dependent phase term exp[i(ks −kc)z]. If the absorbers are moving then after the storage time they will be in different positions to before the storage. Hence for read-out this phase term will have a correction in terms of the ˆ˜ absorber’s velocity and the storage time, which in Hq we represent with the function fq,j(ts).

(a) (b) The parameters δq,j and δq,j arise from fluctuations in the annihilation operatorsa ˆq ˆ and bq respectively. Each of these modes is composed of plane wave modes which have a phase exp(ik · r) associated with them, where k is the wavevector of the plane-wave mode with annihilation operatorsa ˆk, and r is the position. However, for each atom the wavevector k will be Doppler-shifted. Since we assume that the signal field is plane wave in directions perpendicular to the propagation direction, the shift can be written k → k(1 + vj/c), with vj the speed of the atom along the axis of signal propagation, and k = |k|. The signal field modea ˆq has some carrier wavevector with magnitude ks and a spread in frequency space, which is much smaller than the carrier frequency.

Hence we approximate the Doppler shift in wavevector as k → k + vjks/c, resulting in

(a) an extra phase exp(ivjksz/c). This is the origin of the parameter δq,j , and a similar (b) argument follows for δq,j .

4.2 Solution

We now give details of the evolution of the system, and how this is used to calculate the entanglement fidelity of the memory. 48 Entanglement fidelity of quantum memories - methods

4.2.1 Ensemble-photon interaction

ˆ We first solve for the interaction modelled by Hq (the read-out part follows analo- gously). If the term Kq,j is constant for all absorbers, then the system would re- ¯ (K) duce to an effective two-level problem. To this end, we write Kq,j = Kq + δq,j , ˆ and decompose Hq into a mean part and a fluctuation part. To simplify the cal- culation we transform into a rotating frame as follows: for convenience we write ˆ ˆ ˆ | 3ji = | 11,..., 1j−1, 3j, 1j+1,..., 1N i, and let Hq = hq + Hq, where

XN ˆ (K) hq = − δq,j | 3ji h3j | , j=1 and denote the rotating frame Hamiltonian as

ˆ ˆ ˆ ihqt ˆ −ihqt Hq,I = e Hqe , XN nh i o ¯ (a) (j) (K) iKqt iδq,j = κq,je aˆqe σˆ31 + H.c. + δq,j | Eji hEj | , (4.9) j=1

ˆ ˆ ˆ (0) ˆ Solving for Hq,I , we write Hq,I = Hq + Vq, where

XN n o ¯ (a) (j) ˆ (0) iKqt iδq,j Hq = κq,je aˆqe σˆ31 + H.c. , (4.10) j=1 XN ˆ (K) Vq = δq,j | 3ji h3j | . (4.11) j=1

By defining the collective excited state

N X (a) p iδ | Eqi = κq,je q,j | 3ji ΩqN, (4.12) j=1 4.2 Solution 49

P N 2 ˆ (0) with Ωq = j=1 |κq,j|/N, the Hamiltonian Hq can be seen to model a two-level interaction by rewriting it as

p ¯ ˆ (0) iKqt Hq = ΩqNe | Eqi hG | + H.c.. (4.13)

(0) (0) Hence the unitary Uˆq := exp[iHˆq t] can be obtained as

p · ¯ ¸ ˆ iKq ¯ i ΩqN ¯ U (0)(t) = cos(Γ t) + sin(Γ t) e−iKqt/2 | Gi hG | − sin(Γ t)eiKqt/2 q q 2Γ q Γ q pq q i Ω N q −iK¯qt/2 × | Eqi hG | − sin(Γqt)e | Gi hEq | Γq · ¯ ¸ iKq iK¯qt/2 + cos(Γqt) − sin(Γqt) e | Eqi hEq | , 2Γq (4.14)

q ¯ 2 where Γq = ΩqN + Kq /4.

ˆ ˆ ˆ (0)† ˆ ˆ (0) Transforming Vq into the interaction picture according to W = Uq V Uq gives

" # K¯ 2 XN |κ |2 Wˆ (t) = cos2(Γ t) + q sin2(Γ t) q,j δ(K) | E i hE | q 4Γ2 q Ω N q,j q q q j=1 q ½ · ¸ XN |κ |2 iK¯ + q,j sin2(Γ t)δ(K) | Gi hG | + i sin(Γ t) cos(Γ t) − q sin(Γ t) Γ2 q q,j q q 2Γ q j=1 q q ¾ XN 2 |κq,j| (K) × p δq,j | Gi hEq | + H.c. . (4.15) j=1 Γq ΩqN

ˆ (0) † In the interaction picture the wavefunction of the system is | ψI (t)i = Uq (t) | ψ(t)i, and by perturbation theory

· Z Z Z ¸ t t τ2 ˆ ˆ ˆ | ψI (t)i = I − i dτW (τ) − dτ2 dτ1W (τ2)W (τ1) | ψI (0)i , (4.16) 0 0 0 50 Entanglement fidelity of quantum memories - methods

¯ where I is the identity operator. We set Kq = 0 – as we saw from Eq. (4.8), if the mean atomic velocity is zero, then the time-dependent phase disappears. Furthermore, for p t = tp = π/(2 ΩqN), ˆ (0) Uq = −i | Gi hEq | − i | Eqi hG | . (4.17)

Using this the final state after read-in can be obtained

" # N N (K) 2 iπ X X δ |κq,j| | ψ(t )i = −i δ(K)δ(K)|κ |2|κ |2 − q,j | Gi (4.18) p 16(Ω N)3 q,j q,m q,j q,m (Ω N)3/2 q j,m=1 j=1 q " # iπ XN 4 + π2 XN −i 1 − δ(K)|κ |2 − δ(K)δ(K)|κ |2|κ |2 | Ei . 4(Ω N)3/2 q,j q,j 32(Ω N)3 q,j q,m q,j q,m q j=1 q j,m=1

P N 2 2 Recall that Ωq = j=1 |κq,j| /N = |κq| , so one can identify the order of each term in (M) Eq. (4.18) by the number of δq,j terms it contains.

The procedure above can be repeated to calculate the state after the read-out, so we do not detail it here, but there are some additional points to note. Firstly, for the read-out Hamiltonian the collective excited state of the ensemble is defined differently, as N n o ¯ ® X (b) p ˜ 2 i[1δq,j +fq,j (ts)] ¯Eq = |κq,j| e | Eji / ΩqN, (4.19) j=1 where a tilde is used to denote a quantity associated with the read-out. Also, since we are modelling storage of a photon in the ensemble, any part of the photon that is not stored is assumed to be lost. Hence the | Gi component of | ψ(tp)i does not couple to the read-out process, so we shelve it into some loss state | Li. Hence the final state of the system after read-out can be written as

¯ ® ¯ ® ¯ ˜ ˆ† ¯ ˜ † ψ(tf ) = bq | Gi ⊗ bq | vaci + cq E ⊗ | vaci + lq | Li ⊗ aˆq | vaci , (4.20)

where bq, cq, and lq are complex amplitudes, and tf = ts + 2tp. For the purposes 4.2 Solution 51

of calculating the entanglement fidelity, the only necessary amplitude is bq (this will subsequently be shown more explicitly), and this can be shown to be

( ) ½ " # XN 2 XN 2 (K) (M) |κq,j| i[δ(a)−δ(b)−f (t )] iπ |κq,j(δq,j + δq,j ) b = e q,j q,j q,j s − 1 + q Ω N 4 (Ω N)3/2 j=1 q j=1 q · ¸ ¾ XN π2|κ |4 (4 + π2)|κ |2|κ |2 + q,j δ(K)δ(M) + q,j q,m (δ(K)δ(K) + δ(M)δ(M)) . 16(Ω N)3 q,j q,j (Ω N)3 q,j q,m q,j q,m j,m=1 q q (4.21)

4.2.2 Entanglement fidelity

The amplitude bq of the output signal photon can be used to calculate the entanglement fidelity [2]. Recall that the entanglement fidelity of a quantum memory is defined as

n o ˆ † ˆ F(ΛM ) = min hφ0 | UM [(ΛM ⊗ I)(| φ0i hφ0 |)] UM | φ0i , (4.22) | φ0i

where the initial state | φ0i of the photon and auxiliary qubit is as previously defined. The final state of the system, including the ensemble of absorbers, can be written as

· µ ¶ ¯ ® | φf i = α b0 | Gi ¯ 0˜ + l0 | Li | 0i + c0 | Ei | vaci µ ¶¸ ¯ ® +γ b1 | Gi ¯ 1˜ + l1 | Li | 1i + c1 | Ei | vaci ⊗ | 0i · µ ¶ ¯ ® + β b0 | Gi ¯ 0˜ + l0 | Li | 0i + c0 | Ei | vaci µ ¶¸ ¯ ® +µ b1 | Gi ¯ 1˜ + l1 | Li | 0i + c1 | Ei | vaci ⊗ | 1i , (4.23) 52 Entanglement fidelity of quantum memories - methods

¯ ® ¯ ˆ† where q˜ = bq | vaci. To obtain the final density matrix of the two qubits ρf , we trace out the ensemble, and perform a stochastic average over the fluctuation terms [3] i.e.

ρf = h| φf i hφf |ij . (4.24)

The state ρf can be substituted into Eq. (4.22), and only the terms that correspond to ˆ storage and retrieval of the photon need to be kept. We assume that the unitary UM † ˆ† accounts for the change in mode structure of the photon from statea ˆq | vaci → bq | vaci. Then the entanglement fidelity can be written as

· ¡ 2 2¢2 ­ 2® ¡ 2 2¢¡ 2 2¢ F = min |α| + |β| |b0| + 2 |α| + |β| |γ| + |µ| | φ i 0 ¸ ∗ ¡ 2 2¢2 ­ 2® ×Re {hb0b1i} + |γ| + |µ| |b1| . (4.25)

Due to the fact that the initial state is normalized, the coefficients in Eq. (4.25) can be rewritten in terms of X = |α|2 + |β2|, and denoting the value of X that minimizes F with X0 gives the expression in Eq. (2) in chapter 3. The value of X0 given in Eq. (6) in chapter 3 is obtained by minimizing Eq. (4.25) with respect to X.

It remains to calculate the averages in Eq. (4.25). These calculations are longwinded and require only integration and algebraic manipulation, so only the basic method is

2 2 ∗ outlined here. The terms |b0| , |b1| and Re {b0b1} can be obtained from Eq. (4.21). The averages of these expressions are calculated by integrating with respect to each

(M) stochastic parameter over a Gaussian distribution, variable δq,j is assumed to have

M 2 2 probability distribution function exp[−(δq,j) /wMq ] and so on, with the distribution widths as defined in chapter 3. The following result is also useful: for stochastic variable xj, ­ ® hx x i = x2 − δ(j − m) hx i2 , (4.26) j m j j j j j 4.3 Experimental setup 53

4.3 Experimental setup

As we have stated in chapter 3, the problem of storing both logical states of a photonic qubit in a quantum memory that consists of a single storage unit is a nontrivial problem in itself, and one that we shall address in chapters 5 and 7 of this thesis. This makes an experimental measurement of the memory entanglement fidelity F difficult. If, however, it is assumed that each logical state can be stored and retrieved in the same

2 way, i.e. |b0| = |b1|, then F = h|b0| i can be determined using a single storage unit. We now detail this method.

Consider the setup shown in Fig. 3.2 in chapter 3. A separable photon pair is produced so that one photon (the memory photon) is stored in a quantum memory, and the other (the idler photon) passes through a pulseshaper. After the signal and idler photons have passed through the memory and the pulseshaper, respectively, they are incident on a 50-50 beamsplitter. The pulseshaper is assumed to include a delay that ensures that the photons arrive at the beamsplitter at the same time. We as- sume that after the pulseshaper the memory photon is in a pure state ˆb† | vaci. The memory photon state that is incident on the beamsplitter will in general be mixed.

The eigenmodes of this state are defined {aˆj}, with corresponding eigenvalues {pj}, such that p1 ≥ p2 ≥ ... . However there is also a probability that the memory photon is not stored and retrieved, which we call p0. The two-photon state incident on the beamsplitter can be written as à ! X ˆ† † ˆ ρin = b pjaˆj | vaci hvac | aˆq + p0 | vaci hvac | b. (4.27) j

ˆ ˆ P The modes b and {aˆj} can be decomposed into plane waves as b = k ξ(k)ˆa(k), P anda ˆj = k ςj(k)ˆa(k), wherea ˆ(k) is the annihilation operator for the field mode with 54 Entanglement fidelity of quantum memories - methods

wavevector k, and ξ(k) and {ςj(k)} are normalized mode functions. The transforma- tions affected by the beamsplitter are defined as follows:

1 X h i ˆb → √ ξ(k) − ξ˜(k) aˆ(k), (4.28) 2 k 1 X aˆj → √ [ςj(k) +ς ˜j(k)]a ˆ(k), (4.29) 2 k

where ξ(k) and ςj(k) are normalized modefunctions of the transmitted beamsplitter ˆ ˜ modes corresponding to input modes b anda ˆj respectively, and b(k) andς ˜j(k) are modefunctions for the reflected modes. The overall output state of the beamsplitter is hence ( )( ) 1 X X h i X £ ¤ ρ = p ξ∗(k) − ξ˜∗(k) aˆ†(k) ς∗(k) +ς ˜∗(k) aˆ†(k) | vaci hvac | out 4 j j j j (" k # k ) X X h i ˜ × (ςj(k) +ς ˜j(k))a ˆ(k) ξ(k) − ξ(k) aˆ(k) , (4.30) k k X X ¯ ED ¯ 1 ¯ (j) (j) ¯ = p δ 0 0 ¯ Ψ Ψ 0 0 ¯ , (4.31) 4 j B,A,B ,A B,A B ,A j B,A,B0,A0 where " #" # ¯ E X X ¯ (j) ∗ † ∗ † ¯ ΨB,A = B (k)ˆa (k) Aj (k)ˆa (k) | vaci , (4.32) k k and the labels A, A0 can both be either ς orς ˜, and B,B0 can be either ξ or ξ˜. This notation may seem ungainly at first, but in order to compute the detection probabilities ¯ ® ¯ (j) we must consider individual matrix elements of ρout. The introduction of the ΨB,A ’s simplifies future references to these individual matrix elements.

From the density matrix ρout the probabilities of detection in the setup shown in Fig. 3.2 can be calculated (for a more detailed account of photodetection theory see [1] and [4]). The detectors D1 and D2 are assumed to be perfect single-atom detectors, 4.3 Experimental setup 55

centred at positions r1 and r2 respectively. Then, the probability of detecting a photon in D1 and a photon in D2 is given by

Z Z t t1 h i ˆ(−) ˆ(−) ˆ(+) ˆ(+) P12 = dt1 dt2Tr ρoutE (r1, t1)E (r2, t2)E (r2, t2)E (r1, t1) , (4.33) 0 0

and the probability of two photons hitting detector D1 (which, due to the finite time resolution of the detectors would result in a single click) is

Z Z t t1 h i ˆ(−) ˆ(−) ˆ(+) ˆ(+) P11 = dt1 dt2Tr ρoutE (r1, t1)E (r1, t2)E (r1, t2)E (r1, t1) . (4.34) 0 0

Here Eˆ(±)(r, t) are the positive and negative components of the electric field operators. By linearity the components of the density matrix [as defined in Eq. (4.32)] can be considered individually.

We first consider the diagonal terms of ρin. In particular we examine the state ¯ ®­ ¯ ¯ (j) (j)¯ Ψξ,ς˜ Ψξ,ς˜ , which corresponds to both photons entering the upper arm of the inter- ferometer. By expanding the field operators, substituting in the full expression for ¯ ® ¯ (j) Ψξ,ς˜ , and performing the integrals one obtains

Z Z Z Z D ¯ ¯ E 3 3 (j) ¯ ˆ(−) ˆ(−) ˆ(+) ˆ(+) ¯ (j) d r1 d r2 dt1 dt2 Ψξ,ς˜ ¯ E (r1, t1)E (r1, t2)E (r1, t2)E (r1, t1) ¯ Ψξ,ς˜  ¯ ¯  ¯ ¯2 ¯X ¯ = 2 1 + ¯ ξ∗(k)˜ς (k)¯  . (4.35) ¯ j ¯ k

We integrated over the positions r1 and r2 because we assume that any photon incident on the beamsplitter is caught by one of the detectors. The contributing terms from the other diagonal elements of ρin can be similarly computed. The off-diagonal terms of ρin do not contribute to the probabilities of detection. From these calculations one 56 Entanglement fidelity of quantum memories - methods can write the probabilities of a single detector click, and of coincidence clicks, as " # X p X p P + P = j 1 + | ξ˜∗(k)ς (k)|2 + 0 , (4.36) 11 22 4 j 2 j k and " # X p X P = j 1 − | ξ˜∗(k)ς (k)|2 , (4.37) 12 2 j j k respectively.

To see where the maxima and minima of these probabilities occur, we use the set of ˆ P ˜ eigenmodes {ςj} to expand b = j λjaˆj, so that the modefunction ξ(k) can be written ˜ P as ξ(k) = j λjςj(k). Substituting this into Eq. (4.36) gives

X X X 1 pj ∗ ∗ 2 P + P = + | λ 0 ς 0 (k)ς (k)| (4.38) 11 22 2 2 j j j j j0 k 1 X p = + j |λ |2 (4.39) 2 2 j j 1 X p ≤ + 0 |λ |2 (4.40) 2 2 j j 1 = (1 + p ), (4.41) 2 0

where we have used the orthogonality of the modefunctions {aj(k)}. Furthermore, the ˆ upper bound (1 + p0)/2 can be achieved when b =a ˆ1. Hence the probability of a single detector click is maximized (and it follows then that the coincidence probability P12 is minimized) when the pulseshaper is tuned so that ˆb is equal to the eigenmode with the largest eigenvalue. Then, p0 + p1 = P11 + P22 − P12, and since the value of p0 can be evaluated by running the memory many times without the beamsplitter, the value of

2 p1, which is equal to h|b0| i can hence be deduced. This gives the entanglement fidelity of the memory in the case of the two logical states of the memory qubit being stored 4.4 Entanglement fidelity as an entanglement measure 57 and retrieved in the same way.

In section 4.1 we motivated the different fluctuation parameters using the Hamil- tonian in Eq. (4.8). This Hamiltonian can be used to describe the interaction for a Raman quantum memory. Then the fluctuation terms are given by the expressions at the end of chapter 3, and the entanglement fidelity can be calculated using a similar procedure to that given in sections 4.2 and 4.3, resulting in Eq. (3.12).

4.4 Entanglement fidelity as an entanglement mea-

sure

In the work presented in this and the previous chapter, we have used entanglement fidelity as a measure of how well a quantum memory preserves the entanglement be- tween the photon stored in the memory and an auxiliary qubit. One question that has not thus far been addressed, is how suitable entanglement fidelity is for use as an entanglement measure in general. In the remainder of this chapter, we provide a brief discussion of this question.

To illustrate the problem, let us consider the following example. A quantum system consisting of two subsystems R and Q has an initial state | ψRQi. The subsystem Q can be subjected to one of two quantum channels, Λ1 or Λ2. For simplicity let us assume that any unitary evolution in Λ1 and Λ2 is already accounted for, so that we can sensibly use the standard entanglement fidelity definition, as opposed to that in

Eq. (2.14). The entanglement fidelity of the process of Λj acting on | ψRQi, as defined in Eq. (2.13), is denoted by FE(Λj, | ψRQi) for simplicity, although we concede that technically the entanglement fidelity should be a function of TrR {| ψRQi hψRQ |}. The question of whether entanglement fidelity is useful as general entanglement measure 58 Entanglement fidelity of quantum memories - methods

can now be reformulated as follows: if FE(Λ1, | ψRQi) ≥ FE(Λ2, | ψRQi), does this imply that the state Λ1(| ψRQi hψRQ |) is not less entangled than the state Λ2(| ψRQi hψRQ |)? From the suggestively-defined entanglement fidelity, one might naively assume that the answer to this question is ’yes’. However, to our knowledge this has not been proven to be the case, and the reasons why this problem is non-trivial come from the requirements for a function to be an entanglement measure.

An entanglement measure should quantify the amount of entanglement in a given quantum state, so that one can take two states, and determine which of the two is more entangled using the measure. For simplicity we consider bipartite states as above. There are a number of different postulates for the behaviour of an entangle- ment measure, depending on what the entangled state in question will be used for, but a non-exhaustive list of desired properties of an entanglement measure is given as follows:

1: A bipartite entanglement measure E(ρ) is a map from density operators onto pos- itive real numbers defined for arbitrary states of bipartite systems. The measure is usually normalized so that the maximally-entangled state

¯ ® √ ¯ + ψd = (| 0, 0i + | 1, 1i + ··· + | d − 1, d − 1i)/ d, (4.42)

¯ ® ¯ + of two qudits has E( ψd ) = log(d).

2: E(ρ) = 0 if ρ is separable.

3: E cannot increase on average under local operations and classical communication (LOCC), i.e. µ ¶ X A ρA E(ρ) ≥ p E i i , (4.43) i Tr {A ρA } i i i 4.4 Entanglement fidelity as an entanglement measure 59

where the Ai are the Kraus operators that define some LOCC protocol, and the

probability of obtaining outcome i is pi = Tr {AiρAi}.

4: For a bipartite pure state | ψRQi, E reduces to the entropy of entanglement:

E(| ψRQi hψRQ |) = (S ◦ trQ)(| ψRQi hψRQ |). (4.44)

Any condition that satisfies the first three conditions is usually referred to as an entan- glement monotone. There have been numerous proposals for entanglement measures and monotones [5–7], some of which do not satisfy all four conditions (for a review of entanglement measures one can refer to [8]). For the purposes of this discussion, the key condition is number 3. The fact that E cannot increase on average under LOCC imposes an order on the set of all density operators, say of the bipartite system RQ as before. However, this is only a partial order, because one can usually find states ρ and σ of RQ, such that σ cannot be obtained from ρ by LOCC, and vice versa. We call such states unconnected. For two unconnected states, it is meaningless to ask which is more entangled. Different entanglement measures impose a different partial order on the set of density matrices - in fact it has been shown that if two measures impose the same ordering, then they are equivalent [9].

Let us now return to our example involving the quantum channels Λ1 and Λ2.

The first remark to make is that if the states Λ1(| ψRQi hψRQ |) and Λ2(| ψRQi hψRQ |) are unconnected, then it does not make sense to compare the entanglement of these states, even though FE(Λ1, | ψRQi) and FE(Λ2, | ψRQi) can be calculated. So, the question of whether entanglement fidelity can be considered to be a useful entan- glement monotone should be modified to read: given the quantum channels Λ1 and

Λ2 and the state | ψRQi as already defined, if FE(Λ1, | ψRQi) ≥ FE(Λ2, | ψRQi) and

Λ2(| ψRQi hψRQ |) = Λ3[Λ1(| ψRQi hψRQ |)] for some LOCC protocol Λ3, then does this 60 Entanglement fidelity of quantum memories - methods

imply that Λ1(| ψRQi hψRQ |) is more entangled that Λ2(| ψRQi hψRQ |)? This question has recently been addressed [10], but to our knowledge this has neither been proven or disproven. Despite this open question, our use of entanglement fidelity here is justified – it has been investigated in detail [2, 11–13], and is known to be a suitable measure of how well entanglement is preserved between an input state undergoing some noisy operation and a reference system not affected by the operation.

Bibliography

[1] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).

[2] B. Schumacher, Phys. Rev. A 54, 2614 (1996).

[3] L.-M. Duan, J. I. Cirac, and P. Zoller, Phys. Rev. A 66, 023818 (2002).

[4] C. W. Gardiner, Quantum Noise, Springer-Verlag, Berlin (1991).

[5] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002).

[6] M. B. Plenio, Phys. Rev. Lett. 95, 090503 (2005).

[7] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).

[8] M.B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007).

[9] M. B. Plenio and S. Virmani, Phys. Lett. A 268, 31 (2000).

[10] Y. Xiang and S.-J. Xiong, Phys. Rev. A 76, 014301 (2007).

[11] D. Kretschmann and R. Werner, New Journal of Physics 6, 26 (2004).

[12] H. Barnum, M. A. Nielsen, and B. Schumacher, Phys. Rev. A 57, 4153 (1998). BIBLIOGRAPHY 61

[13] M. A. Nielsen, e-print quant-ph/9606012 (1996). 62 Entanglement fidelity of quantum memories - methods Chapter 5

Schemes for ensemble-based quantum memories

We have already seen in chapter 2 that using an ensemble of N Λ-type absorbers as a quantum memory storage medium gives a great advantage over a single absorber. The coupling of the medium with the signal field to be stored is enhanced by a factor √ of N, so using a gas of atoms, for example, with typical atomic density ∼ 1018m−3 could give a potentially huge benefit. Whilst there are different potential ensemble- based media that could be considered for a quantum memory [1–4] (see chapter 1 for a discussion of these), a great deal of recent research in this area has focused on the use of atomic ensembles.

There are different potential schemes for implementing an ensemble-based atomic quantum memory, and the main candidates split into two groups. The first is in which a signal field and a control field are incident on the medium, and the control field facilitates absorption of the signal into the ensemble. After some storage time the material excitation can be read out using another control field. This type of memory will be referred to as an absorptive quantum memory, and it is this class of memories 64 Schemes for ensemble-based quantum memories that is the main subject of this thesis. However, there is another class of quantum memories that merit discussion in this chapter. In this scheme a signal and control field interact with an ensemble of Λ-atoms, but the signal field is not absorbed. Instead, after the interaction the signal field is measured, and the result is fed back onto the ensemble, so that the signal field state is faithfully mapped onto the atoms. We call this type of memory a feedback memory [5–7]. In this chapter we review the ensemble memory schemes based both on absorptive and feedback mechanisms.

In chapter 1 we also cited the quantum repeater as an example of why one might need a quantum memory in the first place. In the wake of the seminal paper of Duan et al. in 2001 [8] there has been a recent interest in the construction of quantum networks and quantum communication protocols using atomic ensemble quantum memories.

In particular, a recent scheme for a quantum repeater was proposed [9], in which multiple multimode quantum memories are used as storage units. This allows for a more realistic scaling of resources if one wanted to build a quantum communication device using atomic ensembles. In this chapter we also provide an overview of the multimode quantum memory scheme proposed, and of its potential application to a scalable quantum network.

The remainder of the chapter is structured as follows. In section 5.1 we give a de- tailed description of a quantum memory based on electromagnetically-induced trans- parency (EIT) [10, 11], and we use this section to introduce the notation and techniques that will be used in chapter 6 of this thesis. Section 5.2 outlines the memory based on controlled reversible inhomogeneous broadening (CRIB) [12–15]. We then give a brief explanation of the feedback-based quantum memory pioneered by Polzik and coworkers, and conclude with a discussion of the recent work on multimode memories. 5.1 Quantum memories using electromagnetically-induced transparency 65

5.1 Quantum memories using electromagnetically-

induced transparency

EIT was predicted theoretically [16], and subsequently observed experimentally [17], by Harris et al.. It was shown that by quantum interference the presence of a strong control field on a system of three-level atoms (see Fig. 5.1) renders the medium effectively transparent to a signal field that propagates through the ensemble. There is a large linear dispersion of the medium associated with this transparency, which can lead to a significant reduction in the group velocity of the signal field [18]. This group velocity reduction is a linear process, and hence the state of the slowed signal pulse is preserved. Hence, one might consider this as a simple storage device for light pulses. However, since a smaller group velocity is accompanied by narrower spectral window of EIT, the ratio of storage time to signal pulse duration is limited by the opacity of the medium.

However, Fleischhauer and Lukin [10, 11] noted that associated with the slow light propagation in EIT is the existence of quasiparticles called dark-state polari- tons (DSPs). A DSP is a mixture of an electromagnetic excitation and a collective spin coherence in the medium, which we call a spin wave. The proportion of each of these contained in the DSP mixture is characterized by a mixing angle that determines the group velocity of the signal field. This mixing angle depends on the atomic density and intensity of the control field. A quantum memory can be realized by rotating the mixing angle, which, along with reducing the propagation velocity, transfers the state of the DSP from a purely photonic state to an atomic excitation. Furthermore, during the rotation the spectrum of the pulse is narrowed, which alleviates the spectral width restriction on a storage unit based purely on slow light. After some storage time the signal field can be regenerated by rotating the mixing angle back to its initial value.

In the rest of this section we present a mathematical description of the interaction 66 Schemes for ensemble-based quantum memories

Figure 5.1. The level scheme for an EIT quantum memory. A signal field, with frequency ωs and detuning ∆s from | mi, excites the | 1i ↔ | mi transition, whilst a strong control field, with frequency ωc and detuning ∆c, drives the | 3i ↔ | mi transition. See text for more details. described above, which will subsequently be used in the remainder of the thesis. We quantitatively analyze the phenomenon of EIT, and using the DSP picture show that, whilst the slowing of light using EIT alone is not enough to implement an efficient quantum memory, combining it with adiabatic passage techniques provide an elegant method for quantum state transfer between a photonic excitation and a collective atomic excitation.

5.1.1 Model

We consider the interaction between a quantum signal field and an ensemble of N Λ- atoms via a classical control field, as depicted in Fig. 5.1. The shape of the interaction region can be measured with the Fresnel number, which is defined as F = A/λsL, with A and L the cross-sectional area and length of the interaction region, respectively, and

λs the wavelength of the signal field. We assume a long, thin medium so that F < 1. The interaction can then be treated in one spatial dimension (which we label z). The 5.1 Quantum memories using electromagnetically-induced transparency 67

signal field, with carrier frequency ωs excites the | 1i ↔ | mi transition, and the electric field operator for it is given by

r Z ∞ ˆ ~ωs iωz/c E(z) = ²s dωaˆ(ω)e + H.c., (5.1) 4π²0cA 0

where ²s is the polarization unit vector, A is the cross-sectional area of the ensemble, and we have a continuum of annihilation operatorsa ˆ(ω) for the field modes with frequency ω that satisfy the commutation relation

[ˆa(ω), aˆ†(ω0)] = δ(ω − ω0). (5.2)

The classical control field, which has carrier frequency ωc and drives the | 3i ↔ | mi transition, is assumed to co-propagate with the signal field, and has the form

i(kcz−ωct) Ec(t, z) = ²cEc(t − z/c)e + c.c., (5.3)

with ²c the polarization unit vector, kc = ωc/c, and E(t − z/c) the control pulse envelope, which is assumed to propagate with a group velocity equal to c. This is a valid approximation since almost all of the atoms will always be in state | 1i, so they will not significantly alter the propagation of the strong control field. The signal and control fields are detuned from state | mi by ∆s and ∆c respectively.

Bloch equations

To model the interaction, we first derive the equations of motion [19, 20] for the atomic

(j) projection-transition operatorsσ ˆµν (t) for atom j, which for time t = 0 are given by

(j) (j) (j) (j) σˆµν (0) = | µi hν |. These operators obey the commutation relation [ˆσik , σˆlp ] =σ ˆip δkl − (j) σˆkl δip. Note that for µ 6= ν,σ ˆµν acts as an atomic raising/lowering operator. Any 68 Schemes for ensemble-based quantum memories atomic operator Aˆ can be represented in terms of the transition operators as follows:

X ˆ ˆ A = hµ | A | νi σˆµν. (5.4) µ,ν

Thus the evolution of theσ ˆµν’s completely specifies the state of the atomic system. The time evolution of these operators is given by the Heisenberg equation

i ∂ σˆ (t) = − [ˆσ (t), Hˆ ], (5.5) t µν ~ µν

ˆ ˆ ˆ ˆ ˆ where the Hamiltonian H, in the dipole approximation, is given by H = H0−d·²E(z, t).

Here H0, which describes the free-field and the self-energies of the atomic levels, is given by Z XN ³ ´ ˆ † (j) (j) H0 = dω~ωaˆ(ω) aˆ(ω) + ~ωm1σˆmm + ~ω31σˆ33 , (5.6) j=1 with m labelleing the state | mi. The second term in Hˆ meanwhile describes the atom- field interaction, with Eˆ(z, t) the total electric field operator, ² the polarization vector, and dˆ the electric dipole operator. This interaction term has the form

" r Z # XN L dˆ · ²Eˆ(t, z) = ~ Ω(t − z /c)σ(j) ei(kczj −ωct) + g dωaˆ(ω)eiωz/cσ(j) + H.c. , j m3 2π m1 j=1 (5.7) where Ω(t − z/c) is the control field Rabi frequency [defined in the same way as in Eq. (2.7)], and the signal field coupling strength is given by

r ˆ ωs g = hm | (d · ²s) | 1i . (5.8) 2~²0AL

Before substituting the constituent parts of Hˆ into Eq. (5.5), we first introduce the 5.1 Quantum memories using electromagnetically-induced transparency 69

slowly-varying operators for atom j with position zj,

ˆ(j) (j) i(kczj −ωct) σ˜m3(t) =σ ˆm3(t)e , (5.9)

(j) (j) ˆ i(kszJ −ωst) σ˜m1(t) =σ ˆm1(t)e , (5.10)

ˆ(j) (j) i[(ks−kc)zj −(ωs−ωc)t] σ˜31 (t) =σ ˆ31 (t)e , (5.11) r Z L Aˆ(t, z ) = e−i(kszj −ωst) dωaˆ(ω, t)eikszj , (5.12) j 2πc wherea ˆ(ω, t) =a ˆ(ω)e−iωt. We also transform Hˆ into a rotating frame with respect to ˆ H0, yielding the interaction picture Hamiltonian

XN n h io ˆ (j) (j) ˆ(j) ˆ ˆ(j) HI = ∆ˆσmm(t) + (∆s − ∆c)ˆσ33 (t) − Ω(t − zj/c)σ˜m3(t) + gA(t, zj)σ˜m1(t) + H.c . j=1 (5.13) Then, Eq. (5.5) gives the following equations for the atomic operator evolution

ˆ(j) ˆ(j) ˆ(j) ˆ ˆ(j) ˆ(j) ∂tσ˜1m(t) = −(γ + i∆s)σ˜1m(t) + iΩ(t − zj/c)σ˜13 (t) + igA(t, zj)[σ˜11 (t) − σ˜mm(t)],

(5.14)

ˆ(j) ˆ(j) ∗ ˆ(j) ˆ ˆ(j) ∂tσ˜13 (t) = −[γs + i(∆s − ∆c)]σ˜13 (t) + iΩ (t − zj/c)σ˜1m(t) − igA(t, zj)σ˜m3(t),

(5.15)

where γ and γs, which represent the decay of the polarization and spin coherence due to dephasing processes respectively, have been introduced phenomenologically. Similar expressions can be obtained for the time derivatives of the other transition operators, but they will not be needed. The reason for this is that if the signal field is not too

(j) strong, most of the atoms will remain in their ground state. Hence we can setσ ˆ11 = 1, (j) (j) (j) ˆ andσ ˆ33 =σ ˆmm =σ ˆ3m = 0. We now define the polarization P (z, t) and the spin-wave 70 Schemes for ensemble-based quantum memories excitation Bˆ(r, t) of the medium as

√ N XNz Pˆ(t, z) = σ˜ˆ(j) (t), (5.16) N 1m z j=1 and √ N XNz Bˆ(t, z) = σ˜ˆ(j)(t), (5.17) N 13 z j=1 respectively. In these definitions the sums are over the Nz atoms contained in a thin slice of the ensemble centred at position z, where the slices are thick enough so that ˆ ˆ Nz À 1 ∀z, but thin enough so that P (z, t) and B(z, t) can be considered to be continuous. Using these operators we arrive at the following equations for the evolution of the ensemble,

√ ˆ ˆ ˆ ˆ ∂tP (t, z) = −(γ + i∆s)P (t, z) + ig NA(t, z) + iΩ(t − z/c)B(t, z), (5.18)

ˆ ˆ ∗ ˆ ∂tB(t, z) = −[γs + i(∆s − ∆c)]B(t, z) + iΩ (t − z/c)P (t, z). (5.19)

We have assumed a uniform distribution of atoms across the ensemble. A non-uniform distribution can be included in this theory [21], but it is not necessary for our purposes here, and we omit this for simplicity.

Wave equation

As well as modelling the interaction between the light fields and the ensemble, it is also necessary to consider the propagation of the signal field through the medium. Neglecting transverse components, the signal pulse obeys the one-dimensional wave equation µ ¶ 2 1 2 ˆ 4π 2 ˆ ∂z − 2 ∂t Es(t, z) = 2 ∂t P(t, z), (5.20) vs c 5.1 Quantum memories using electromagnetically-induced transparency 71

where vs is the signal field phase velocity, and dispersion is neglected. The operator Pˆ(t, z) represents the polarization of the medium and, unlike Pˆ(t, z), is not slowly varying. To simplify the wave equation, we use the slowly-varying (compared to ωs) operators Aˆ(t, z) and Pˆ(t, z), and neglect their second derivatives with respect to z and t. Then Eq. (5.20) becomes

√ ˆ ˆ (vs∂z + ∂t) A(t, z) = ig NP (t, z). (5.21)

This, along with Eqs. (5.18) and (5.19), constitutes the set of Maxwell-Bloch equations for the ensemble-field interaction.

5.1.2 Electromagnetically-induced transparency

Before delving into the memory interaction, we can first use the model above to give a more quantitative description of EIT. To do this we assume that the probe and control

fields are plane waves. Equations (5.18) and (5.19) then become

√ ˆ ˆ ˆ ∂tP (t) = −(γ + i∆s)P (t) + ig N + iΩcB(t), (5.22)

ˆ ˆ ∗ ˆ ∂tB(t) = −[γs + i(∆s − ∆c)]B(t) + iΩc P (t), (5.23)

where Ωc is the peak Rabi frequency of the control field. These equations can be solved using a matrix representation [22]. Writing

      √ ˆ  P (t)   γ + i∆s −iΩc   ig N  R =   , M =   , V =   , (5.24) ˆ ∗ B(t) −iΩc γs + i(∆s − ∆c) 0 72 Schemes for ensemble-based quantum memories allows us to rewrite Eqs. (5.22) and (5.23) as R˙ = −MR + V . This can be solved by integration to give R(t) = M −1V , which yields

√ ˆ ig N[γs + i(∆s − ∆c)] P (t) = 2 . (5.25) [γs + i(∆s − ∆c)](γ + i∆s) + |Ωc|

From this we can calculate the real and imaginary parts of the susceptibility χ ≡ χ0+iχ00 as

√ 2 0 g N {γs[γs∆s + γ(∆s − ∆c)] − (∆s − ∆c)[γγs − ∆s(∆s − ∆c) + |Ωc| ]} χ = 2 2 2 , (5.26) [γγs − ∆s(∆s − ∆c) + |Ωc| ] + [γs∆s + γ(∆s − ∆c)] and

2 00 γs[γγs − ∆s(∆s − ∆c) + |Ωc| ] + (∆s − ∆c)[γs∆s − γ(∆s − ∆c)] χ = 2 2 2 , (5.27) [γγs − ∆s(∆s − ∆c) + |Ωc| ] + [γs∆s + γ(∆s − ∆c)] respectively.

In Figs 5.2(a) and (c) we plot both χ0 and χ00 for a resonant control field, and Figs. 5.2(b) and (d) are for an off-resonant control field, with (c) and (d) having a stronger control field. In all cases χ0 = χ00 = 0 for two-photon resonance (i.e. when

00 ∆s = ∆c). Notice that, for an increased control field, the region for which χ is close to zero is larger than for the weaker control field. This region is called the transparency window.

One can relate the susceptibility of the medium to its dispersion relation, which is given by k2 −v2n2/c2 = 0, with k the magnitude of the wavevector of light propagation √ through the medium, v the group velocity of the light, and n ≡ n0 + in00 = 1 + χ. Then, n0 represents the refractive index of the medium, and n00 is the associated ab-

0 00 sorption coefficient. We plot n and n as functions of ∆s/γ in Fig. 5.3 for reso- nant [Figs. 5.3(a) and (c)] and off-resonant [Figs. 5.3(b) and (d)] control fields, with 5.1 Quantum memories using electromagnetically-induced transparency 73

Figure 5.2. Real and imaginary parts of the susceptibility of an ensemble of −4 three-level atoms as a function of ∆s/γ, with γs = 10 γ, and (a) Ωc = 0.5γ, ∆c = 0, (b) Ωc = 0.5γ, ∆c = γ, (c) Ωc = 2γ, ∆c = 0, and (d) Ωc = 2γ, ∆c = γ. 74 Schemes for ensemble-based quantum memories

Figure 5.3. Refractive index n0 and absorption n00 of an ensemble of three-level −4 atoms as a function of ∆s/γ, with γs = 10 γ, and (a) Ωc = 0.5γ, ∆c = 0, (b) Ωc = 0.5γ, ∆c = γ, (c) Ωc = 2γ, ∆c = 0, and (d) Ωc = 2γ, ∆c = γ.

Figs. 5.3(c) and (d) for larger Ωc. We see that in both cases at ∆s = ∆c the refractive index is unity, and the absorption is zero. Hence the medium is effectively transparent, and we have an example of EIT. However, the group velocity increases with the Raman detuning, even on two-photon resonance [23]. For larger Ωc the transparency extends over a wider range of detuning ∆s. This is particularly important when considering a quantum memory for light pulses that operates in this way – the whole of the signal pulse must be within this transparency window, so the bandwidth that can be stored depends on the Rabi frequency of the control field. We shall return to this later in the chapter. 5.1 Quantum memories using electromagnetically-induced transparency 75

5.1.3 Quantum memory interaction

Having illustrated the principle of EIT, we now examine how this mechanism can be exploited for the implementation of a quantum memory. The explanation here is based on the work of Fleischhauer and Lukin [10, 11], and for more details one should refer to these papers.

Slow light in the adiabatic limit

Again consider the Maxwell-Bloch equations for the ensemble-field interaction, as de- noted in Eqs. (5.18), (5.19), and (5.21). The system is assumed to be on two-photon resonance, so that ∆s = ∆c = ∆, and the spin wave dephasing γs is neglected. Equa- tions (5.18), (5.19), and (5.21) can be analyzed to yield information about the process of optimizing a quantum memory [21]. However, in this thesis we are mainly interested in what is known as the adiabatic limit of the memory interaction. In the adiabatic limit, it is assumed that the frequency at which the system is driven is much larger than the frequency at which the system evolves. As we shall show in subsequent chapters, under this condition the Maxwell-Bloch equations can be solved analytically. How- ever, at this point in the thesis we instead use this approximation to illustrate how an

EIT-based memory works.

In the case of an EIT memory, the Raman detuning satisfies the condition |∆| ¿ dγ, where d = g2NL/(γc2) is the resonant optical depth [24]. Then, the adiabatic condition reduces to dγ À δ, where δ is the signal field bandwidth. Under this condition Eqs. (5.18), (5.19), and (5.21) can be simplified considerably. To see this, we rearrange Eq. (5.19) for P (z, t) and substitute into Eqs. (5.18) and (5.21) to give

√ g N (v ∂ + ∂ ) A(t, z) = ∂ B(t, z), (5.28) s z t Ω∗(t, z) t 76 Schemes for ensemble-based quantum memories

√ · ¸ ig N i i B(t, z) = − A(t, z) − [∂ + (γ + i∆)] − ∂ B(t, z) . (5.29) Ω(t, z) Ω(t, z) t Ω∗(t, z) t

In the adiabatic approximation, the term involving the signal field is the dominant term on the right-hand side of Eq. (5.29). Substituting into Eq. (5.28) then gives

g2N A(t, z) (v ∂ + ∂ ) A(t, z) = − ∂ . (5.30) s z t Ω∗(t, z) t Ω(t, z)

If it is assumed that the control Rabi frequency is constant in time, then the term on the right-hand side of Eq. (5.30) simply modifies the group velocity vg of the signal

−1 field according to vg = c[1 + ng(z)] , where

g2N n (z) = . (5.31) g |Ω(z)|2

The solution to Eq. (5.30) describes propagation of the signal field with a spatially- varying group velocity vg. The temporal profile of the pulse is unaffected by the slowdown, and as a consequence the spectrum of the pulse does not change. However, a spatial change of the group velocity (e.g. due to a spatially-decreasing control pulse) compresses the spatial profile of the pulse. In particular, if the group velocity is stati- cally reduced from c to vg, the, the spatial pulse length is ∆L = vg∆l0/c, with ∆l0 the free-space pulse-length.

Before examining the transfer of the signal excitation to the atomic spin-wave, let us discuss briefly the limitations to the achievable pulse delay td in the EIT medium.

The most obvious limitation is the dephasing γs, which we neglected in the above

−1 analysis. Hence an upper bound for the delay time would be td ≤ γs . However, a much stronger limitation arises from potential violation of the adiabatic condition. As we saw in Figs. 5.2 and 5.3, under EIT the atomic medium behaves as a non-absorbing, dispersive medium only within a certain transparency window around the two-photon 5.1 Quantum memories using electromagnetically-induced transparency 77 resonance. Furthermore, the adiabatic approximation we made essentially assumes that the interaction of the signal field happens within the transparency window. If the pulse becomes too short, or its spectral width larger than this window, then absorption and higher-order effects must be taken into account. Without going into the details here, the frequency width of the transparency window ∆ωtr can be calculated [10] from the √ susceptibility of the medium [as derived in Eqs. (5.26) and (5.27)] as ∆ωtr = α/td, with α the opacity of the medium in the absence of EIT. Hence, large delay times require a long pulse duration. This gives an upper bound (in addition to the bound set by γs) for the ratio of achievable delay time to the initial photon pulse duration T :

t √ d ≤ α. (5.32) T

In practice, the opacities achievable in the laboratory are . 104, limiting the above ratio to td/T ∼ 100. Hence, storing light pulses simply by reducing their group velocity in this way has limited use. However, one can also utilize adiabatic passage techniques which, when combined with EIT, result in an effective method for photon storage and retrieval. This can be seen by reformulating the problem in terms of quasiparticles consisting of superpositions of signal field and atomic excitations. We introduce these quasiparticles – named dark-state polaritons – in the following subsection.

5.1.4 Quantum state transfer

The propagation of a signal field through an EIT medium under stationary conditions (i.e. a time-independent control field) leads to a time-independent Hamiltonian, which meant that quantum state transfer of the signal to the atoms was not possible. However, by using a time-dependent control field it was shown that such one-way state transfer can be realized [10, 11]. Consider a real control field constant in space. We define two 78 Schemes for ensemble-based quantum memories new operators Ψ(t, z) and Φ(t, z) to be mixtures of the signal field and spin wave as follows

Ψ(t, z) = A(t, z) cos θ(t) − B(t, z) sin θ(t), (5.33)

Φ(t, z) = A(t, z) sin θ(t) + B(t, z) cos θ(t), (5.34) with the angle θ(t) given by g2N tan2 θ(t) = . (5.35) Ω2(t)

The operators Ψ(t, z) and Φ(t, z) are the so-called bright-state and dark-state polari- tons, respectively, and the mixtures of signal and spin wave in them can be controlled by changing the strength of the control field. In the linear limit Ψ and Φ obey the usual bosonic commutation relations [10]. Equations (5.28) and (5.29) can be rewritten in terms of Ψ(t, z) and Φ(t, z). Then making the adiabatic approximation in a similar way as before (recalling that for this we require T dγ ¿ 1, with T the duration of the signal pulse) gives Φ(t, z) ' 0, which results in the following equation of motion

£ 2 ¤ ∂t + vs cos θ(t)∂z Ψ(t, z) = 0. (5.36)

The polariton is now defined in terms of the signal field and spin-wave as

A(t, z) = Ψ(t, z) cos θ(t), (5.37)

B(t, z) = −Ψ(t, z) sin θ(t). (5.38)

The solution to Eq. (5.36) can be written

µ Z t ¶ Ψ(t, z) = Ψ z − c dt0 cos2 θ(t0), 0 , (5.39) 0 5.1 Quantum memories using electromagnetically-induced transparency 79 which describes state- and shape-preserving propagation of the polariton with velocity

2 vg(t) = c cos θ(t). One can see that for a strong control field [θ(t) → 0] the polariton has a purely photonic character, and for a weak control [θ(t) → π/2] the polariton is purely an atomic excitation. Furthermore the EIT mechanism means that in the weak control field limit the group velocity of the polariton approaches zero. Hence a mapping from a photonic signal field to an atomic spin-wave excitation, whilst slowing down the polariton effectively to a standstill, can be achieved by adiabatically rotating the angle

θ(t) from 0 to π/2 (i.e. by turning off the initially strong control field). Similarly, to re-accelerate the polariton and retrieve the signal field after some storage time, one can turn back on the control field and rotate θ → 0.

To conclude our description of the EIT memory scheme, we return to the problem of ensuring that the signal pulse is contained completely within the transparency win- dow. During the process of reading the signal field into the ensemble, the width of the transparency window decreases. However, the temporal profile of the signal pulse increases in length due to the group velocity reduction, resulting in a decrease of the pulse’s spectral width. Furthermore one can show that, for realistic parameters, the ratio of the signal spectral width and the transparency window width stays approxi- mately constant during the interaction. Hence absorption can be prevented during the dynamic state transfer process by ensuring that the initial pulse spectrum lies within the initial transparency window.

5.1.5 Limitations

As already discussed, the width of the EIT transparency window puts an upper limit on the bandwidth of signal field that can be stored. This limit is determined by the maximum control field Rabi frequency, a typical value for which is 108s−1. This gives 80 Schemes for ensemble-based quantum memories an upper bound of roughly 1ns for the duration of a signal pulse.

As well as this there are other limitations to an EIT quantum memory. Because the signal and control fields are on single-photon resonance, then spontaneous emission (represented by γ) from state | mi is significant. The single-photon resonance also means that the system is sensitive to any deviations from the intended signal and control frequencies. This could either come from experimental inaccuracy in the laser frequencies, or from motion of the atoms. The latter effect is due to the atoms all seeing different Doppler-shifted signal and control frequencies. One way to overcome this problem is to use an atomic configuration such that the Stokes shift ω13 ' 0. Then during read-in no net momentum is transferred to the ensemble and the two-photon transition experiences no Doppler shift. However, it is favourable to have a significantly non-zero Stokes shift for two reasons. Firstly, one needs to ensure that no control

field photons drive the | 1i ↔ | mi transition, and since this cannot be completely guaranteed by selection rules, one needs a Stokes shift greater than the bandwidths of the fields. Secondly, a zero Stokes shift means that the signal and control fields have the same carrier frequency. In this case after read-out it would be very difficult to distinguish the weak signal field from the much stronger control field. The on-resonant nature of the EIT memory also means that it would be difficult to adapt the scheme to semiconductor systems, such as quantum dots or NV-centers, where the absorber transition frequencies are not well-defined.

In addition to the above limitations, the motion of the atoms during the memory storage time will lead to dephasing of the atomic excitation. All of these effects can be taken into account using the entanglement fidelity measure [25] proposed in Chapter 3 of this thesis. To calculate this fidelity, we require the Hamiltonian of the system to be in the form of the two-level system considered in Chapter 3. However, as in Eq. (5.18) we must consider spontaneous emission in an EIT memory, since the fields are on- 5.1 Quantum memories using electromagnetically-induced transparency 81 resonance with state | mi. The terms that represent spontaneous emission cannot be included in a Hamiltonian for the system, as they give non-Hermitian terms. However, we can define a complex detuning ∆ − iγ. Then, as with our treatment of spontaneous emission in Eq. (2.5), the effective interaction can be split up into a Hermitian part and a decay term representing the spontaneous emission. For the EIT case in the adiabatic limit we write the effective Hamiltonian as (" # ) XN ³ ´ ˆ gΩ(t, z) (j) (j) −1 γ Heff = −p A(t, z)ˆσ + H.c. − iˆσ tan , (5.40) 2 2 31 33 ∆ j=1 ∆ + γ where we have neglected the Stark shift of atomic level | 3i. Recall that this Hamilto- nian is in a rotating frame. If atomic motion is taken into account, then each atom, with mass M, will see different Doppler-shifted field frequencies (and hence different wavevectors and detunings). Hence, when one removes the quickly-varying components of the fields, there will be a small phase left over from atom to atom. Since the Stokes shift ω13 will in general be much smaller than the splitting of the levels | 1i and | mi, we can assume that despite the Doppler shift the signal and control fields remain on Raman resonance. As in chapters 3 and 4, we simplify the interaction by assuming a constant control field switched on for a time tp, and that the signal field occupies a single mode (with annihilation operator Aˆ) that couples to the atoms. Then the ˆ Hermitian part of Heff can be rewritten as

XN h i ˆ ˆ −ivj ω13(t−z/c)/c HI = |κj|Ae + H.c. , (5.41) j=1

∗ 2 where vj is the velocity (assumed to be along the z-axis) of atom j, and κj = gΩ (∆j +

2 −1/2 γ ) is the magnitude of the coupling. The detuning ∆j of atom j is given by

∆j = ∆ − ωsvj/c. The atomic motion during storage time ts can be incorporated into 82 Schemes for ensemble-based quantum memories the read-out Hamiltonian as in chapters 3 and 4. In exactly the same way as in those chapters, we use Eqs. (3.9) and (4.21) to calculate the entanglement fidelity of an EIT memory, which gives

µ 2 ¶ 2 −(χ ζ+γtp) Θ(ωs − ωc) ζ F = e 1 − 2 2 , (5.42) 4Nκ¯ + wκ

2 where ζ = kbT/Mc , χ = (ks − kc)cts, with T the temperature of the ensemble, and Θ is as defined in Chapter 3. The mean coupling isκ ¯ = gΩ∗/γ, and the width of the coupling distribution wκ comes from the distribution of the detuning ∆j, which has √ width ωs ζ. As with the Raman memory entanglement fidelity (see chapter 3), we see that motional dephasing is a source of degradation of F. However the spontaneous emission also affects the memory fidelity. In agreement with [21], if the coupling is large and the Stokes shift of the atoms is small, then spontaneous emission is the limiting process. Also, because here the mean detuning is zero, the width of the coupling parameter κj relative to its mean is larger than for a Raman quantum memory. This results in the inhomogeneous broadening term having a more detrimental effect on the memory in the EIT regime. The broadening is still small, however, when the atomic Stokes shift is small.

5.2 Quantum memory using controlled reversible

inhomogeneous broadening

The above EIT-based memory and the forthcoming Raman quantum memory model, whilst being quantitatively different, could essentially be thought of as similar schemes in two different parameter regimes. However the proposal outlined in this section, known as CRIB, uses a qualitatively different mechanism for photon storage and re- 5.2 Quantum memory using controlled reversible inhomogeneous broadening 83 trieval, and is based on reversible broadening of the signal field absorption line [12–14]. In the remainder of this section we describe this method in detail.

5.2.1 Scheme

Consider an ensemble of N Λ-type atoms as before. A signal pulse, on resonant with the | 1i ↔ | mi transition, is sent into the medium, and is completely absorbed. Note that in order to facilitate absorption of the whole pulse, the ensemble has some inhomogeneous

(e.g. Doppler) width δi, such that δi ≥ δ (recall δ is the signal bandwidth), as shown in Fig. 5.4(a). The state of the ensemble after absorption of the signal (assuming complete absorption) can be written as

XN | φatomsi = cj | 11,..., 1j−1, mj, 1j + 1,..., 1N i , (5.43) j=1

2 PN 2 where |cj| is the probability that atom j absorbed the signal (with j=1 |cj| = 1), and a subscript j labels the state of atom j. After absorption of the signal field the atoms will begin to dephase relative to each other. Neglecting spatial evolution, the state of the ensemble a time t after the signal pulse is

XN −iω(j+)t | φatomsi = cje 1m | 11,..., 1j−1, mj, 1j + 1,..., 1N i , (5.44) j=1

(j+) where ω1m is the Doppler-shifted (with respect to atom j) frequency of the transition

| 1i ↔ | mi. A time tc after absorption a π pulse, resonant with the | 3i ↔ | mi transition and propagating in the same direction as the signal field, is applied to the ensemble [see Fig. 5.4(b) for an illustration of the sequence of pulses]. Due to the large number of atoms in the ensemble this pulse preserves its shape throughout the 84 Schemes for ensemble-based quantum memories

Figure 5.4. An illustration of the CRIB memory scheme. (a) Signal and con- trol fields drive the | 1i ↔ | mi and | 3i ↔ | mi transitions respectively. The absorption line is broadened to a width δi, which must be larger than the signal field bandwidth δ. (b) The sequence of pulses applied for read-in (top), and read-out (bottom). medium. The resulting atomic state is

N X (j+) 0 −iω1m tc | φatomsi = cje | 11,..., 1j−1, 3j, 1j + 1,..., 1N i . (5.45) j=1

Hence the signal field is now stored as a spin wave in the ensemble. One can already see that this method is different to the EIT and Raman memories, in that a CRIB memory absorbs the signal first, and then a control pulse is applied resulting in state

| mi becoming populated. In the Raman and EIT memories the aim is to apply both fields at roughly the same times1 to implement the coherent transfer of the atomic state from | 1i to | mi, whilst involving state | mi as little as possible.

After a storage time ts the state of the ensemble is

N X (j+) (j+) 0 −i[ω1m tc+ω13 ts] | φatomsi = cje | 11,..., 1j−1, 3j, 1j + 1,..., 1N i , (5.46) j=1

1As with stimulated Raman adiabatic passage techniques, the control field is in fact usually slightly ahead of the signal field. 5.2 Quantum memory using controlled reversible inhomogeneous broadening 85

(j+) with ω13 the inhomogeneous broadened (with respect to atom j) Stokes shift. Let us (j) assume that the broadening of this frequency is small, so that ω13 ' ω13 ∀j. Then, the stored spin wave is read out by applying a second π-pulse resonant with the | 3i ↔ | mi transition, but which is now counter-propagating with respect to the first control pulse

[see Fig. 5.4(b)]. Because of this, the Doppler shifts of the atomic transitions will effectively be inverted, and the evolution of the atoms reverses the relative dephasing experienced during read-in. The atomic state a time t after the second control pulse is

N X (j+) (j−) 00 −iω1m tc −iω1m t | φatomsi = cj e e | 11,..., 1j−1, mj, 1j + 1,..., 1N i , (5.47) j=1

(j−) with ω1m the Doppler-shifted transition frequency for atom j during read-out. Writing (j±) ω1m = ω1m(1 ± vj/c), with vj the velocity of atom j, one can see that at a time tc after the application of the second control pulse, the Doppler shifts in the exponents of Eq. (5.47) cancel out. Then the atomic state is identical to that immediately after absorption of the signal pulse. The system will now evolve in the opposite way as for the signal absorption – the atomic state will return to | 11,..., 1N i, and a time-reversed signal field will be emitted in the opposite direction to its initial direction.

5.2.2 Controlling the rephasing artificially

Although the rephasing of the atoms due to Doppler broadening is a clever mecha- nism for efficient memory operation, in practice it would be advantageous to obtain this rephasing by controlling some artificial broadening, whilst suppressing the natural inhomogeneous broadening. For example, in rare-earth-metal-ion-doped crystals [13], which are a potential solid-state implementation for a CRIB memory, inhomogeneous broadening is caused by the ions being located in slightly different surroundings. This kind of level shift would be difficult to reverse in a similar way to above. However, one 86 Schemes for ensemble-based quantum memories can suppress this natural broadening by optical pumping [14], to give a single atomic absorption line that can be used for the state | mi. This line can then be broadened artificially, for example using an electric or magnetic field gradient. For these materials, the group state is split into long-lived hyperfine levels that are suitable for use as the storage state | 3i. The level splittings range from a few MHz to several GHz, so once the solid is optically pumped, the memory can then be operated as explained earlier in the section. The broadening can be reversed for read-out simply by inverting the field used to produce it.

5.2.3 Limitations

In a CRIB-based memory, the signal field bandwidth is limited by the width to which the absorption line can be broadened. In rare-earth-ion-doped crystals, this width is limited to the order of 1GHz by other nearby absorption lines, and currently to ∼10MHz by the magnitude of the Stark shift that can be induced.

To calculate the entanglement fidelity of the CRIB memory, a slightly different approach must be used to that for the EIT and Raman memories. In particular, since the signal and control fields are applied separately in CRIB, we must consider the transitions | 1i → | mi and | mi → | 3i separately, rather than an overall transition from | 1i to | mi. However, the same principle can apply. The signal field is assumed to be a single mode, with carrier frequency ωs, which couples to the | 1i ↔ | mi transition. State | mi has a spontaneous emission rate γ, which we include in this calculation phenomenologically. After absorption of the signal, the strong control field is applied to drive the atoms into state | 3i. We assume that the control field is applied immediately after maximal absorption of the signal, and that the state transfer affected by the control is instantaneous. Recall that in our formulation of a general quantum 5.2 Quantum memory using controlled reversible inhomogeneous broadening 87 memory [25] in chapters 3 and 4, we included broadening terms in the read-in and read-out Hamiltonians. However, in CRIB the broadening splits into a controlled part, which will not affect the entanglement fidelity, and a natural part, which one would like to make as small as possible. The natural broadening is represented by the stochastic parameter Kj (as in chapter 4), and is assumed to be the same for storage and retrieval. We assume that the absorbers have velocity along the z-axis during the storage time ts, with the velocities obeying a Boltzmann distribution defined the absorber mass M and temperature T . We also assume that the control field is strong enough so that natural broadening of the | 3i ↔ | mi transition can be neglected – as we showed in chapter 4 the fidelity depends on the ratio of the broadening to the coupling strength. Taking these approximations into account, we find that the entanglement fidelity of a CRIB memory is 2 2 ω F ∼ e−χ ζ e−γtp (1 − K ), (5.48) 4g2N where g is the atom-signal field coupling, ωK is the width of the (Gaussian) distribution

2 of Kj, and as in chapter 4 χ = (ks − kc)cts and ζ = kBT/Mc . We see that the effect of spontaneous emission on the entanglement fidelity depends on the interaction time √ √ tp ' π/(2g N), so a high fidelity requires a large optical depth so that g N ¿ γ. Otherwise we see that, as for a Raman memory, the motion of the absorbers and the natural inhomogeneous broadening cause a reduction in the fidelity. However, the move towards using rare-earth ion-doped metals as a storage medium for CRIB would eliminate motional dephasing. Any natural broadening would then come not from Doppler shifts, but from other sources e.g. inhomogeneity in the medium. The 1/N scaling in the broadening term however means that this effect does not have to be kept too low to get a good entanglement fidelity. 88 Schemes for ensemble-based quantum memories

5.3 Feedback quantum memory

Having examined the different proposed schemes for absorptive quantum memories, we now turn to the feedback memory mentioned at the start of this chapter. This approach involves interacting the quantum signal field with an atomic ensemble, and projecting the state of the ensemble into the initial signal field state by measuring the signal. This scheme has been pioneered by Polzik and co-workers, and has been recently demonstrated experimentally [26]. In this section we give a toy-model explanation of the memory protocol, and show how it is realized using atomic ensembles and photons. Since this memory scheme is qualitatively different to that which we consider, we only give a brief overview, and instead direct the reader to [5–7, 27–29] for further details.

To illustrate how measurement can be used to implement a quantum memory, let us first consider two qubits A and B. Qubit A is initially in some unknown normalized state | ψi = α | 0Ai+β | 1Ai, with α, β ∈ C, and | 0Ai and | 1Ai the logical states of qubit

A. Qubit B meanwhile, with logical states | 0Bi and | 1iB, is initially in state | 0Bi. Applying a controlled-NOT gate [30] to qubits A and B results in the transformation

| ψi | 0Bi → α | 0A0Bi + β | 1A1Bi . (5.49)

Now measure qubit A in the σx basis. A result of ±1 leaves the state of qubit B as α | 0i ± β | 1i, respectively. In the case of the −1 outcome the state can be corrected to α | 0i + β | 1i by applying a phase gate. Hence, by using a feedback mechanism, the final state of qubit B is now the initial state of qubit A. Note that the state | ψi was an arbitrary single-qubit state.

Hence, instead of absorbing a light field into an atomic ensemble, a quantum mem- ory can be operated using a similar principle to the one above, where via some interac- 5.3 Feedback quantum memory 89

Figure 5.5. Schematic description of the feedback quantum memory scheme. 0 (a) The level scheme. The 6S1/2 (| 1i and | 3i) and 6P3/2 (| mi and | m i) are split by a magnetic field, and are coupled by a weak signal (dashed) and strong control (solid) field. (b) Experimental setup, with the signal polarized in the y-direction, and the control in the x-direction. After interaction a measurement is made on the signal field, and the result is fed back onto the ensemble.

tion and feedback, an atomic ensemble is prepared in a state that represents some initial photonic state. However, light fields and atomic ensembles are infinite-dimensional sys- tems, so it is more convenient to work in the Heisenberg picture as with the EIT and

CRIB memories. In particular, we define the canonical operators positionx ˆA and mo- mentump ˆA for an ensemble A, such that [ˆxA, pˆA] = i. Similarly we definex ˆB andp ˆB for an ensemble mode B. Any observable of the light field can be written as a power series ofx ˆA andp ˆA (and similarly for the ensemble mode), hence the evolution ofx ˆA,B andp ˆA,B completely determines the interaction of the light and the atoms.

One can model interactions between the light and atomic modes using a Hamilto- nian in terms of the canonical operators above. Consider the following Hamiltonian:

ˆ H =p ˆApˆB, (5.50) the significance of which will become clear shortly. The evolution of the operators 90 Schemes for ensemble-based quantum memories according to Eq. (5.50) is given by

out in in out in in out in out in xˆA =x ˆA + κpˆB ,x ˆB =x ˆB + κpˆA ,p ˆA =p ˆA ,p ˆB =p ˆB , (5.51) where κ is the coupling of the atoms to the signal field. After this evolution, we can measure the operatorx ˆB and, as our feedback step, displace the operatorp ˆA by the result of the measurement. Then, setting κ = 1 the solutions read

out in in out in xˆA =x ˆA +p ˆB ,p ˆA = −xˆB . (5.52)

If the initial state of the atoms is an infinitely-squeezed state, then we see that the expressions in Eq. (5.52) represent a process by which the momentum of the quantum light field is transferred to the position operator of the atoms, and the position operator of the field is exchanged with the momentum operator of the atoms, i.e. we have a complete mapping from the light mode to the atomic mode.

A system that to a good approximation realizes the Hamiltonian in Eq. (5.50), and hence has solutions of the form of the expressions in Eq. (5.52), can be constructed using atomic ensembles and photons. We depict such a setup in Fig. 5.5(a). We consider an ensemble consisting of Caesium atoms, and the memory interaction utilizes the 6S1/2 to 6P3/2 transition [see Fig. 5.5(a)]. Furthermore, these two levels are split into Zeeman sublevels, m, m0 = ±1/2, by an external magnetic field. The ensemble ˆ ˆ ˆ ˆ is hence characterized by its total angular momentum operator J = (Jx, Jy, Jz). To ˆ simplify matters the atoms are polarized in the x-direction, so that the Jx operator can be treated as a c-number. A signal and control pulse are incident on the ensemble, propagating along the z-axis as shown in Fig. 5.5(b). The weak signal is polarized in the x-direction, and the much stronger control is polarized in the y-direction. The light field is characterized in terms of its polarization (Stokes) operators, and due to 5.3 Feedback quantum memory 91 the relative strengths of the fields, one assumes that the x-mode operators are also c-numbers. The choice of polarizations ensures that the signal field excites both the | 1i ↔ | e0i and | 3i ↔ | ei transitions [see Fig. 5.5(a)], whilst the control pulse drives the | 1i ↔ | ei and | 3i ↔ | e0i transitions. Both fields are detuned by ∆.

If the dynamics of the above interaction are solved, one sees that the angular momentum operators oscillate at the Larmor frequency ΩL of the magnetic field. The application of the magnetic field is important so that the signal field is imprinted on a sideband, rather than at its carrier frequency, enabling distinguishability of the signal and control pulses on read-out. However, this oscillation leads to unwanted noise in the memory scheme. To overcome this, two ensembles are used, where the second ensemble ˆ ˆ ˆ is polarized such that Jx,1 = −Jx,2 = Jx, where Jx,j is the operator for the x-component of angular momentum for ensemble j (j = 1, 2). Then, whilst the angular momentum ˆ ˆ operators for the individual ensembles will still be oscillating, the sums Jy,1 + Jy,2, and so on, will not oscillate. Then one can construct appropriate canonical operators from the total angular momentum operators of the ensembles, and the polarization operators of the fields, such that the canonical operators satisfy the expressions in Eq. (5.51), where κ is now the coupling strength. Hence, for appropriate coupling and for a sufficiently-squeezed atomic ensemble, one can achieve faithful mapping of the quantum state of the signal field to the atomic ensembles by appropriately measuring the signal photon after the interaction, and feeding the result back onto the ensembles.

There has been great progress on both realizing and improving this experimental setup [6]. In particular, the improvements have focused on reducing the storage unit to a single ensemble by using multiple passes of the light fields through the ensemble. However, as stated this type of memory interaction is not the main topic of this thesis, so we again refer the reader to the references given at the start of the section for further details. The main disadvantage that this kind of memory has over the absorptive 92 Schemes for ensemble-based quantum memories memories is that repeater schemes are robust to photon loss, and not photon number increase. In a feedback memory, since photons can be created during the interaction processes, these memories are less-suited for use in a quantum repeater protocol.

5.4 Multimode quantum memories

We have given a detailed introduction to the various schemes for storage and retrieval of photonic excitations in a quantum memory. However, as discussed in chapter 1, one would hope to be able to use such a memory in some quantum communication or quantum information processing protocol [31–33], in which preservation of entangle- ment between qubits is essential. As mentioned in earlier chapters, one obvious solution to the problem of storing and retrieving qubits on demand is to produce a storage unit for each logical state of each qubit. This however would lead to an undesirable in- crease in the resources required for any non-trivial communication scheme. To this end there has been a recent interest in devising schemes for multi-mode memories, in which multiple modes of light can be stored in and retrieved from a single storage unit. Furthermore these modes should be distinguishable on read-out of the memory. This would allow one to consider, for example, scalable entanglement distribution protocols using experimentally-feasible setups. In the remainder of this chapter we outline the recent development of a repeater scheme that uses multimode memories [9] based on the CRIB protocol. A description of our own multimode memory will follow in chapter 7.

Simon et al. noticed that, using their CRIB memory scheme, one can store multiple signal field modes, where the different modes correspond to time-bins in a train of pulses. This is illustrated in Fig. 5.6(a). A source S generates photon pairs with probability p/2, and is activated successively at time intervals of T . The source is 5.4 Multimode quantum memories 93

Figure 5.6. Entanglement generation using multimode quantum memories (a) A source S emits pairs of photons, with probability p/2 at intervals of time T . One train is stored in a CRIB-based memory M, and the other is coupled into a fibre. (b) Two sources SA and SB each emit photon pairs, and detection of a single photon behind the beamsplitter BS projects the memories MA and MB into an entangled state.

arranged so that one of the photon pair is incident on a quantum memory M, and the other on a detector. Since the generation of more than one photon pair introduces errors into the repeater protocol (as with the feedback memory), p is kept small. The memory consists of an ensemble of absorbers, and operates in the same way as the CRIB protocol described in section 5.2. A narrow absorption line inside a wide spectral hole is prepared, and the line is artificially broadened. The train of pulses incident on the memory can then be stored in this broadening window, and the atomic excited-state population is transferred to a metastable collective state using a π-pulse, propagating in the same direction as the signal pulse train. One can determine how many modes were stored in the memory by observing the number of clicks in the detector. In fact, it is important that the time-resolution of the detector is less than T , since one must know which temporal modes were stored in the memory. After some storage time the pulse train can be read out of the memory by applying another π pulse in the opposite direction to the first. The train is emitted in inverted order, and the efficiency is not limited by reabsorption. 94 Schemes for ensemble-based quantum memories

Using this multimode memory scheme one can construct a quantum repeater proto- col for generating short-range entanglement, and propagating it over larger distances.

Consider two remote locations A and B, with a memory MA(B) and photon pair source

SA(B) at each [see Fig. 5.6(b)]. The sources are repeatedly coherently excited in the same way as for a single memory above, with the mth excitation (m = 1,...,N) pro- ducing the state

· r ¸ p | ψ i = 1 + (ˆa† aˆ0† + ˆb† ˆb0† ) + O(p) | vaci , (5.53) 0 2 m m m m

0 ˆ ˆ0 where | vaci is the vacuum state,a ˆm anda ˆm (bm and bm) are annihilation operators

th for the output modes corresponding to the m excitation of SA (SB). The photons ˆ 0 ˆ0 in modesa ˆm and bm are stored in the corresponding memories, and modesa ˆm and bm are coupled into optical fibres and combined on a beamsplitter. The modes after the beamsplitter are

1 ³ ´ 0 −iφA ˆ0 −iφB a˜ˆm = √ aˆ e + b e , (5.54) 2 1 ³ ´ ˜ˆ 0 −iφA ˆ0 −iφB bm = √ aˆ e − b e , (5.55) 2

where φA,B are phases acquired by the photons on their way to the beamsplitter.

Neglecting O(p) corrections, detection of a single photon in mode a˜ˆm creates the state ¯ E (m) ¯ † iφA ˆ† iφB ˆ ¯ ψAB = (ˆame + bme ) | vaci, where the photons in modesa ˆ and b are now stored in the memories. Hence | ψmi can be rewritten as an entangled state of memories MA and MB, ¯ E 1 ¯ (m) i(φB −φA) ¯ ψ = √ [| 1m,Ai | 0Bi + e | 0Ai | 1m,Bi], (5.56) AB 2 where | 0Ai denotes the empty state of MA, | 1m,Ai is the state of MA storing the single photon in modea ˆm, and similarly for MB. 5.4 Multimode quantum memories 95

This entanglement between remote memories can be extended by entanglement swapping. We introduce two more sites C and D [as shown in Fig. 5.7(a)], both containing memories that are entangled using the same method as for MA and MB.

th Say that the mode corresponding to the n excitation of sources SC and SD, producing ˆ photons in modes with annihilation operatorsc ˆn and dn, resulted in the creation of the entangled state

¯ E 1 ¡ ¢ ¯ (n) i(φD−φC ) ¯ ψ = √ | 1n,C i | 0Di + e | 0C i | 1n,Di . (5.57) CD 2

The states | 0C i and | 1n,C i, and the phase φC (and similarly for site D) are defined analogously with | 0Ai, | 1m,Ai, and φA respectively. Since we know which time bin m and n correspond to from the detectors, read-out can then be performed on memories ˆ MB and MC , and modes bm andc ˆn can be combined on a beamsplitter. Detection of a single photon after this beamsplitter leads to an entangled state

¯ E h ´ ¯ (m,n) 1 ¯ ψ = √ aˆ† ei(φA−φC ) + dˆ† ei(φB −φD) ] | vaci , (5.58) AD 2 m n

th th between the m mode stored in MA and the n mode stored in MD.

Now, suppose that locations A and D both contain a pair of memories, and en- tanglement has been established between MA1 and MD1, and between MA2 and MD2 [see Fig. 5.7(b)]. Then, by post-selecting cases where there is one memory excitation in each of A and D, then one can create a state of the form

³ ´ 1 † ˆ† † ˆ† | ΨADi = √ aˆ d +a ˆ d | vaci , (5.59) 2 1 2 2 1

wherea ˆ1 is the annihilation operator for the photon mode stored in memory MA1 ˆ ˆ (similarly fora ˆ2, d1, and d2), and we have dropped the labelling of the modes (m and 96 Schemes for ensemble-based quantum memories

Figure 5.7. Entanglement swapping using multimode memories. (a) The mth th time-bins of MA and MB are entangled, as are the n time bins of MC and MD. Reading out of memories MB and MC , and combining these time-bins on a beamsplitter, results in entanglement between MA and MD. (b) Useful entan- glement is created between two distant memories by combining the appropriate time-bins on a beamsplitter. 5.5 Conclusion 97

n) for simplification. State | ΨADi is analogous to a conventional entangled state, which can then be used for quantum communication by converting the memory modes back to photonic modes.

The reason that this scheme is advantageous to previous repeater schemes based on photons and atomic ensembles [8] is due to the speedup in entanglement generation. The multimode capability of the memory enables one to have multiple attempts at generating the lowest-level entangled states. If the distance from a source to a detector is L0 [see Fig. 5.6(a)], then in schemes that employ single mode memories, one has to wait a time L0/c to determine whether entanglement generation was successful. In the multimode case [9], one can have multiple attempts per time L0/c, the number of which is determined by both the repetition rate of the source (typically much higher than

−1 c/L0 s ) and the photodetector resolution. This greatly increases the success rate of the repeater. Note that the fact that the time-bin modes are emitted from the memory separately in a train, rather than at the same time, is crucial to this scheme. One would lose information about the entanglement if the modes were indistinguishable after read-out.

5.5 Conclusion

We have described the different quantum memory schemes based on EIT, CRIB, and feedback. We have seen that the resonant nature of the EIT and CRIB schemes makes them more susceptible to spontaneous emission and inhomogeneous broadening, al- though the latter can be controlled in CRIB depending on the implementation. These absorptive memories are also limited in the signal field bandwidth that can be stored. We have also seen that, if one can implement a multimode quantum memory, then sig- nificant improvements to the existing repeater protocols can be achieved. In the next 98 Schemes for ensemble-based quantum memories chapter we describe in detail a proposal for an off-resonant Raman quantum memory, which practically eliminates spontaneous emission, reduces the Doppler broadening, and gives a potentially larger bandwidth window for the signal field. Then in chapter 7 we show how this memory could be used to construct a multimode memory.

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Mapping broadband single-photon wavepackets into an atomic memory

6.1 Introduction

In the previous chapter we discussed the quantum memory schemes for photons based on electromagnetically induced transparency (EIT), and controlled reversible inhomo- geneous broadening (CRIB) in atomic ensembles, at some length. In this chapter 1 we now turn to a related proposal based on the properties of stimulated Raman scattering

[3–5] and stimulated Raman adiabatic passage (STIRAP) [6, 7] in atomic ensembles. Stimulated Raman scattering is the process by which the | 1i → | mi transition in an ensemble of Λ-atoms [see Fig. 6.1(a)] is driven by a strong control field far from res- onance, and a quantum light field (known as the Stokes field) is emitted as a result of the atoms decaying to state | 3i. Similarly, STIRAP involves two classical fields, one driving the | 1i → | mi transition, and one driving the | mi → | 3i transition. The pulses can be tailored so that the atomic state is completely transferred to state | 3i.

1The work in this chapter originally appeared as [1], and was later published in a shorter form [2]. However, I was not the principal author, and we rewrite it here for clarity and consistency. 102 Mapping broadband single-photon wavepackets into an atomic memory

It will become clear that one can construct a quantum memory consisting of a Raman scattering process in reverse, where a quantum Stokes pulse is stored in a STIRAP-like process.

More precisely, a Raman quantum memory operates in a similar way to EIT-based memories, with Raman-resonant signal and control fields propagating through an en- semble of three-level atoms. The main difference is that the signal and control fields are now far off one-photon resonance. Because of this there is no reduction of the signal field group velocity, so intuitively Raman memories are better suited to short signal wavepackets. The possibility of a Raman quantum memory was in fact previ- ously investigated by Kozhekin et al. [8] for narrowband pulses, and it was shown that there were problems with matching the control and signal pulses for optimal absorp- tion. However, it is this broadband capability that provides the main motivation for re-examining such a memory. Recall that in the EIT scheme the signal bandwidth was limited by the width of the transparency window. In the Raman case the only bandwidth limitations come from the size of the detuning ∆ and the size of the Stokes shift of the atoms. So given a suitable atom, one can increase the detuning in order to increase the bandwidth capability of the memory, although this also requires an increase in coupling strength. Having the interaction off-resonant also makes the mem- ory more robust to Doppler broadening – a Doppler shifting of the signal and control fields has less of an effect when they are already far from resonance than if they are on resonance. Raman memories are also less susceptible to collision-induced fluorescence than on-resonant memories, and also allow a greater versatility in the frequencies to which the signal and control lasers are tuned.

The remainder of this chapter is set out as follows: we establish a model for the Raman quantum memory, including a discussion of the ways in which it differs from the EIT-memory model. We show that for broadband signal pulses, proper shaping 6.2 Model 103 of the control field allows the mapping of a signal field with arbitrary temporal profile onto a long-lived collective atomic excitation, or spin wave. We show that, although the memory dynamics is similar to those investigated in stimulated Raman scattering [3–5], and in proposals for entanglement generation by spontaneous scattering [9], the photon storage process differs from these, in that it exhibits an explicit time-reversal symmetry due to its fundamental mode structure. The use of a one-dimensional model for the mode-matching is justified by considering the transverse structure of the in- teraction. We then discuss the possibility of retrieving the stored signal field by both forwards and backwards read-out, and illustrate the problems associated with each of these approaches. We conclude the chapter with a discussion of our results, in partic- ular comparing our results with recent work undertaken by other groups on EIT- and Raman-based memories [10].

6.2 Model

We begin from the model derived in Chapter 5 for an EIT-based memory2, and the situation is outlined in Fig. 6.1. The signal and control field frequencies are Raman resonant (detuning ∆) with carrier frequencies ωs and ωc, and wavevectors of magnitude ks and kc oriented along the z-axis, respectively. The fields are incident on an ensemble, with length L, of N Λ-type three-level atoms as shown in Fig. 6.1(a), with Stokes shift

ω13. The signal field is represented at position z and time t by the slowly-varying operator A(t, z) as defined in Eq. (5.12), and the control field is represented by the slowly-varying pulse envelope Ω(t, z) [see Eqs. (5.3) and (2.7)]. The polarization and spin wave are denoted by collective coherences of the ensemble as P (t, z) and B(t, z), as in Eqs. (5.16) and (5.17), respectively [5]. The atoms are assumed to all have initial

2All variables, unless specified in this chapter, are as defined in chapter 5 104 Mapping broadband single-photon wavepackets into an atomic memory

Figure 6.1. Schematic setup for an atomic ensemble quantum memory. (a) The Λ-type level structure used for the memory interaction. The atoms are initially in state | 1i, and are excited by the signal field (dotted) and control field (solid). (b) The propagation of the signal and control pulses through the ensemble parallel to the z-axis.

state | 1i and the population of the metastable state | 3i is assumed to be negligible for each atom, so a linear theory can be used. The interaction between the ensemble and the signal and control fields can be modelled by the same Maxwell-Bloch equations as the EIT memory discussed in the previous chapter, and we repeat them here for convenience:

√ ˆ ˆ ˆ ˆ ∂tP (t, z) = −(γ + i∆)P (t, z) + ig NA(t, z) + iΩ(z, t)B(t, z), (6.1)

ˆ ∗ ˆ ∂tB(t, z) = iΩ (z, t)P (t, z), (6.2) √ ˆ ˆ (vs∂z + ∂t) A(t, z) = ig NP (t, z), (6.3) where N is the number of atoms, γ is the polarization decay rate, and g is the signal field coupling. Note that we again assume a uniform distribution of atoms for simplicity, but a nonuniform distribution can be included into the model. We also neglect the decay of the spin wave coherence, as the time-scale for this can be assumed to be much longer than the storage times considered. 6.2 Model 105

As with the EIT model, one can simplify Eqs. (6.1)-(6.3) by considering the adi- abatic limit. However, the adiabatic condition is different for Raman memories com- pared to EIT-based memories. In a Raman memory we simply require that the excited state | mi is never significantly populated, which can be ensured by setting ∆ much larger than the signal field bandwidth δ and the control field Rabi frequency. In the adiabatic limit, the change in polarization over time is assumed to be negligible, and Eqs. (6.1)-(6.3) reduce to

µ ¶ √ i|Ω(t, z)|2 ig N ∂ − B(t, z) = Ω∗(t, z)A(t, z), (6.4) t Γ Γ µ ¶ √ ig2N ig N v ∂ + ∂ − A(t, z) = Ω(t, z)B(t, z), (6.5) s z t Γ Γ where Γ = ∆ − iγ := |Γ| exp(iθ) is the complex detuning. We introduce the local time

τ = t − z/vs, and transform into a comoving frame (with respect to the signal field).

−1 −1 Then the control field, with group velocity vc, can be written as Ω[τ − (vc − vs )z].

In the general case when vs 6= vc, Eqs. (6.4) and (6.5) can be solved numerically [9]. In this chapter however we focus on the dispersionless case. Setting vs = vc = c results in the following equations

µ ¶ |Ω(z, τ)|2 κ ∂ − i B(z, τ) = i Ω∗(τ)A(τ, z), (6.6) τ Γ Γ µ ¶ κ2 κ ∂ − i A(τ, z) = i Ω(τ)B(τ, z), (6.7) z Γ Γ

√ p where κ = g N/c ≡ dγ/L, with d the resonant optical depth [10]. Note that the control pulse envelope is now a function of τ only. The control field can be eliminated from the dynamics by transforming to a new set of coordinates. We define the function

Z τ ω(τ) = |Ω(τ 0)|2 dτ 0, (6.8) 0 106 Mapping broadband single-photon wavepackets into an atomic memory and introduce the memory time and effective distance as ²(τ) = ω(τ)/ω(T ) and ζ(z) = z/L, respectively, with T the signal pulse duration. Note that both ζ and ² run from 0 to 1. Finally, we rescale the signal field and spin wave as

r ω(T ) α(², ζ) = e−iχ(τ,z)A(τ, z)/Ω(τ), (6.9) r C L β(², ζ) = e−iχ(τ,z)B(τ, z), (6.10) C

p where C = κ Lω(T )/|Γ|, we have assumed κ to be real, and

χ(τ, z) = [ω(τ) + |κ|2 z]/Γ. (6.11)

The first term in χ(τ, z) describes a Stark shift due to the control field, and the second represents a modification of the signal group velocity vs. Under these new variables, the Maxwell-Bloch equations given in Eqs. (6.6) and (6.7) reduce to a much simpler form:

iθ ∂ζ α(², ζ) = Ce β(², ζ), (6.12)

iθ ∂²β(², ζ) = −Ce β(², ζ). (6.13)

The solution of these equations holds for all control pulseshapes and arbitrary initial conditions. The variation of the control pulse has been eliminated from the equations, so the problem is reduced to that of a CW control field coupled to the ensemble. The parameter C defines the efficiency of the interaction, and can be rewritten as

r πα ~NN f C = f c , (6.14) me |Γ|A where f is the geometric mean of the oscillator strengths for the signal and control tran- 6.3 Mode decomposition 107

sitions, αf is the fine structure constant, me is the electron mass, and Nc is the number of photons comprising the control field. These equations can be solved straightfor- wardly, but they, along with the more general case where vs 6= vc, also lend themselves to a decomposition into input and output modes. Hence before solving Eqs. (6.12) and (6.13) explicitly, we first discuss this decomposition.

6.3 Mode decomposition

To illustrate the properties of the memory dynamics that allow a mode decomposition, let us return to the potentially dispersive case (i.e. we do not assume that vs = vc) modelled by Eqs. (6.6) and (6.7). However for now we neglect spontaneous emission γ – then the interaction can be decomposed as follows. The solutions to these equations can be written as [9]:

Z T Z L 0 0 0 0 0 A(τ, L) = CA(τ, τ )A(τ, 0)dτ + SA(τ, z )B(0, z )dz , (6.15) 0 0 Z L Z T 0 0 0 0 0 B(T, z) = CB(z, z )B(0, z)dz − SB(z, τ )A(τ , 0)dτ , (6.16) 0 0

where the integral kernels CA,B, SA,B are Green’s functions that propagate the initial conditions A(τ, 0) and B(0, z). The coordinates τ and z can be discretized over an arbitrarily fine mesh, so that the interaction can be written (up to any chosen accuracy) as the following matrix equations

AL = CAA0 + SAB0, (6.17)

BT = CBB0 − SBA0, (6.18) 108 Mapping broadband single-photon wavepackets into an atomic memory

with column vectors A0 and B0 replacing the initial signal and spin-wave amplitudes, and AL and BT replacing the final respective amplitudes. The integral kernels are replaced by the matrices CA,B and SA,B. This transformation describes the evolution of quantum mechanical operators, and hence should be unitary. This can be seen to be the case by considering the continuity relation implied by the evolution equations

† † ∂zA (τ, z)A(τ, z) + ∂τ B (τ, z)B(τ, z) = 0. (6.19)

Integrating this expression over a rectangle in (τ, z) space, and successively setting (τ, z) = (0, 0), (0,L), (T, 0), (T,L) gives the flux-excitation conservation condition

† † † † AL · AL + BT · BT = A0 · A0 + B0 · B0, (6.20)

which must hold for arbitrary A0 and B0. Alternatively Eq. (6.20) can be written as

† † X · X = X0 · X0, where    A0  X0 =   , (6.21) B0      AL   CA SA  X =   = UX0 =   X0. (6.22) BT −SB CB

Condition (6.20) implies that the transformation X0 → X is norm-preserving, and hence U must be unitary. The evolution of the operators describing the memory interaction is therefore canonical as one would expect.

This canonical evolution can be exploited as follows: multiplying out the relation 6.3 Mode decomposition 109

UT∗U = UUT∗ = I, with I the identity matrix, yields the following conditions

T∗ T∗ CA CA + SB SB = I, (6.23)

T∗ T∗ CB CB + SA SA = I, (6.24) along with a pair of antinormally-ordered counterparts. Note that we write the Hermi- tian conjugate of the matrix U as UT∗ to avoid confusion with the operator Hermitian conjugate. The Bloch-Messiah reduction [11] can now be applied to Eq. (6.23). This

T∗ involves spectrally decomposing [12] the positive Hermitian matrix products CA CA

T∗ and SB SB. These products must commute with each other, and are hence both ren- dered diagonal in the same orthonormal basis. Similar conclusions can be reached from Eq. (6.24), and from the antinormally-ordered versions of Eqs. (6.23) and (6.24). From this one can obtain [9] simple relationships between the singular-valued decompositions of the integral kernels as follows

X∞ 0 0 CA(τ, τ ) = φi(τ)µiψi(τ ), (6.25) i=1 X∞ 0 0 SB(τ , z) = φi(z)λiψi(τ ), (6.26) i=1 X∞ 0 0 CB(z, z ) = ϕi(z)µiςi(z ), (6.27) i=1 X∞ 0 0 SA(τ, z ) = ϕi(τ)λiςi(z ), (6.28) i=1

where each set of functions {φi}, {ψi}, {ϕi}, {ςi} forms a complete orthonormal basis,

2 2 and λi, µi are real positive singular values for which λi + µi = 1 ∀i. When substituted into Eqs. (6.6) and (6.7), Eqs. (6.25)-(6.28) imply a set of independent beamsplitter transformations of ensemble modes and light field modes. 110 Mapping broadband single-photon wavepackets into an atomic memory

Figure 6.2. The five largest eigenvalues of the kernel G0, plotted as a function of the read-in coupling C.

6.4 Solution of the memory interaction

We now return to Eqs. (6.12) and (6.13). The theory presented in the previous section can be applied to show that α(², ζ) and β(², ζ) evolve canonically provided that θ = 0.

Note that here the flux-excitation condition becomes

Nα(C) + Nβ(C) = Nα(0) + Nβ(0), (6.29)

where the number operators Nα(ζ) and Nβ(²) count the number of signal photons at effective distance ζ and the number of spin-wave excitations at memory time ², respectively. Hence the dynamics of Eqs. (6.12) and (6.13) can be decomposed into a set of independent transformations between light-field and spin-wave modes. With θ = 0 the solution of Eqs. (6.12) and (6.13) can be easily obtained using Fourier transforms (in section 6.5 we detail this calculation explicitly, including the transverse 6.4 Solution of the memory interaction 111 mode structure), and is given by [5]

Z 1 α(1, ², ) = [G1(² − x, 1)α(0, x) + G0(1 − x, ²)β(x, 0)] dx, (6.30) 0 Z 1 β(ζ, 1) = [G1(ζ − x, 1)β(x, 0) − G0(1 − x, ζ)α(0, x)] dx, (6.31) 0 where the integral kernels are

√ G0(p, q) = J0(2 pq), (6.32) √ G1(p, q) = δ(p) − Θ(p)J1(2 pq), (6.33)

th Jn denotes the n Bessel function of the first kind, and the Heaviside step function Θ ensures that the convolutions respect causality. In a similar spirit to Eqs. (6.17) and (6.18), we could write the above solution in terms of matrix equations. The matrices representing G0(p, q) and G1(p, q) would then be symmetric about their main antidiagonal. To see this note that G1 as it appears in Eq. (6.30) varies only as as function of ²−x, i.e. all of its contours are parallel to the line ² = x, which corresponds to the main diagonal of the matrix representing G1. The corresponding symmetry for

G0 follows from the Hermiticity of G0(p, q) in its arguments. These symmetries allow decomposition of the kernels using input and output modes related by time reversal, or equivalently space reversal:

X∞ G0(1 − ², ζ) = φi(ζ)λiφi(1 − ²), (6.34) i=1 X∞ G1(ζ − ², 1) = φi(ζ)µiφi(1 − ²), (6.35) i=1 where the λi’s and µi’s are defined as before, but there is now a single complete or- thonormal set of modefunctions {φi}. 112 Mapping broadband single-photon wavepackets into an atomic memory

6.4.1 Read-in

We use the solutions in Eqs. (6.30) and (6.31), and Eqs. (6.34) and (6.35), to examine the process of storing the signal pulse in the ensemble, and in particular to modematch the control pulse to optimally store a given signal pulseshape. The output mode operators can be defined [5] as

X α(², C) = φi(²)Ai, (6.36) Xi β(C, ζ) = φi(ζ)Bi, (6.37) i and for the input operators we use the time-reversed expansions

X α(², 0) = φi(C − ²)ai, (6.38) Xi β(0, ζ) = φi(C − ζ)bi, (6.39) i

The solutions in Eqs. (6.30) and (6.31) can now be written as

Ai = µiai + λibi, (6.40)

Bi = µibi − λiai, (6.41)

† with [Xi,Xj ] = δij,(X = A, B, a, b). The interaction can be viewed as a beamsplitter transformation on a mode-by mode basis. Also, since the atoms are all initially in state | 1i, we replace β(0, ζ) by its expectation value β(0, ζ) = 0. Hence Eq. (6.31) describes a mapping of the optical input mode φi(C − ²) to the spin wave output mode φi(ζ) with transfer amplitude −λi. Transforming back into local time τ, the input modes 6.4 Solution of the memory interaction 113 are written s C Φ (τ) = eiχ(τ,0)Ω(τ)φ [C − ²(τ)]. (6.42) i ω(T ) i

The efficiency of the read-in process can be quantified using the expectation value of the spin-wave number operator hNβ(C)i after storage. Expressing the signal pulse R P 2 2 T ∗ Es(τ) in terms of the Φi’s gives hNβ(C)i = i λi |ξi| , where ξi = 0 ξi (τ)Φi(τ)dτ. In Fig. 6.2 the first five transfer amplitudes are plotted as a function of the coupling

C. These are given by the eigenvalues of the kernel G0, which come from numerically solving the eigenvalue equation

Z C √ J0(2 xy)φi(y)dy = λiφi(x). (6.43) 0

It is desirable to limit the energy of the control pulse, so we look for the minimum C for which complete storage of the signal is permitted. We see that for small C the interaction is dominated by the lowest mode, and as C increases the higher modes are significantly coupled. For C ' 2, λ1 ' 1, but the higher modes are poorly coupled. So for optimally-efficient storage one should use a coupling strength C ≥ 2, and set ξ1 = 1,

ξj = 0 (j > 1), which implies that hNβ(C)i = λ1 ' 1. If Gaussian optics (A ∼ cL/ωs) are used to illuminate a region of length ∼ 1cm of a typical atomic vapour (f ∼ 1) with density ∼ 1020m−1, with 100 nJ control pulses, then a 1 ps signal field wavepacket can be stored optimally with C ' 2.

In experiment the control field could be matched to the signal field using a mea- surement and feedback scheme, where the control pulse profile is augmented until the transmission of the signal is minimized [13]. The expression in Eq. (6.42) that relates the transmitted signal pulse to the control pulse is non-invertible, so the optimal con- trol field for a given signal pulse needs to be found numerically. The results of such an optimization for a Gaussian signal pulse are shown in Fig. 6.3. 114 Mapping broadband single-photon wavepackets into an atomic memory

­ ® † Figure 6.3. Left: intensity A© (τ, z)(τ, z) of a Gaussianª signal photon, with 2 wavepacket amplitude ∝ exp −2 ln 2[(τ − τ0)/σ] , as it propagates through the atomic ensemble, with C = 2. Here σ = T/8, τ0 = 2T/3. Right: the op- timized control field intensity plotted alongside the initial signal field intensity (scaled for clarity).

6.4.2 Read-out

Having performed the modematched storage of the signal photon as described above, the ensemble is left in output mode φ1(ζ) with amplitude λ1. Let us consider the possibility of reading out the photon by sending a second control pulse, propagating in the same direction as the initial control pulse, into the ensemble. The parameters associated with the read-out control pulse (e.g. bandwidth, centre frequency, intensity) may differ to those of the initial control field, so we use a superscript r to denote quantities associated with the read-out. The equations used to model the read-in can also be used for the read-out, with appropriate new boundary conditions. It is assumed that the signal field begins in the vacuum state at the start of read-out, so

r that hNα(0)i = 0. Decoherence and dephasing of the spin wave during the storage time are neglected. Furthermore, any changes in the group velocities of the control and signal pulses are assumed to be negligible, and the change in detuning is assumed to be small. Then, since the atomic Stokes shift is fixed, the phases χ and χr of the read-in and read-out spin waves [see Eq. (6.10)] cancel, so that Br(0, z) = B(T, z), i.e. the read-out spin wave is modematched to the read-in spin wave. 6.4 Solution of the memory interaction 115

Figure 6.4. Probability of retrieving a photon N , for forward read-out, plotted as a function of the read-in and read-out coupling parameters C and Cr.

After the read-out some of the spin wave will be transferred back to the optical

field. The efficiency of this process depends upon how much the stored spin wave

1/2 iχ(T,z) mode ψ1(z) = (C/L) e φi(ζ) overlaps with the input modes for the read-out process, which have the form

r r C r Ψr(z) = eiχ (0,z)φr[Cr(1 − z/L)]. (6.44) i L i

r The functions {φi } solve the eigenvalue equation

Z Cr √ r r r J0(2 xy)φi (y)dy = λi φi (x). (6.45) 0

r r Each read-out input mode φi [C (1 − z/L)] is transferred to the optical field with

r amplitude λi . A measure of the efficiency of the memory is the expectation value of 116 Mapping broadband single-photon wavepackets into an atomic memory the output photon number operator N , where

Z r T ­ ® N = Ar†(τ, L)Ar(τ, L) dτ, 0 1 X∞ D E = Ar†Ar , Cr i i i=1 X∞ 2 r 2 = λ1 λi 2fi , (6.46) i=1 where the ith read-out overlap is defined as

Z L ∗ fi = ψ1(z)Ψi(z)dz. (6.47) 0

The variation in N is plotted in Fig. 6.4 as a function of C and Cr. As one would expect, the retrieval probability is low if either C or Cr is small. However, even if C ≥ 2, i.e. the signal pulse is stored with maximal efficiency, the probability of retrieval only approaches unity slowly as Cr is increased. To see the reason for this, consider the case where C = Cr. Then, the lowest read-out mode is the mirror image of the spin wave mode, due to the time-reversal symmetry. For small C the spin wave mode

φ1(Cz/L) is monotonic and relatively flat, so f1 is larger. However, the amplitudes λ1

r and λ1 are small, resulting in low N . Increasing C (and hence λ1) results in a more asymmetric stored spin wave mode, and hence f1 falls. Intuitively one can see this because a higher coupling means that the atoms near the front of the ensemble will absorb the signal photon with a higher probability. To achieve efficient read-out it is necessary to increase Cr well above C so that higher modes, with which the spin wave mode overlaps significantly, are coupled to the optical field. Along the line C = 2 the retrieval probability is maximized, but a read-out coupling of Cr > 10 is required to achieve N = 0.95. Note also that matching the control field to the lowest mode at read-in is the best strategy for maximizing the memory efficiency. Matching to a higher 6.5 Transverse structure 117 mode, or some combination of modes, simply increases the optimal read-in coupling above C ' 2.

A natural suggestion for solving the above read-out problem would be to use a control field for read-out that propagates in the opposite direction to the read-in control pulse. This sends φi(Cz/L) → φi[C(1 − z/L)] so that the spin-wave mode overlaps exactly with the lowest read-out mode with C = Cr. However, after storage the spin wave has a residual momentum given by the phase eiω13z/c due to the Stokes shift of the atoms. For backwards read-out this phase does not cancel, and leads to a decrease

r in the overlap integrals fi. Note that in this case the functions χ and χ also do not cancel, but far off-resonance they can be assumed to be negligibly small. It is clear that one could eliminate this residual momentum by using atoms for which ω13 = 0. However, for experimental purposes one would like the atomic Stokes shift to be at least as large as the signal and control bandwidths, so that control photons do not excite atoms in state | 1i (selection rules are rarely stringent enough to ensure this given the large number of control photons). A more rigorous treatment of this problem, as well as a solution, can be found in chapter 7.

6.5 Transverse structure

In the above analysis, we have used a one-dimensional model for the atom-field inter- action. Whilst it is clear that this is a valid approximation when considering plane wave fields, for example, it is not obvious that a 1D model is also suitable when mode- matching the signal and control pulses. In this section we show that the modes {φi} do indeed also describe the transverse profile of the signal field.

Let us allow the signal field and spin wave both to depend of the transverse position p coordinates (x, y), which we rescale as ρ := (X,Y ) = (x, y) ω1m/2Lc, with ω1m the 118 Mapping broadband single-photon wavepackets into an atomic memory frequency of the | 1i ↔ | mi transition. Then the Maxwell-Bloch equations in the paraxial approximation [4] can be derived in a near-identical manner to Eqs. (6.12) and (6.13) to give

µ ¶ 1 ∇2 + i∂ α(², ζ, ρ) = −Cβ(², ζ, ρ), (6.48) 4q ρ ζ

∂²β(², ζ, ρ) = Cα(², ζ, ρ), (6.49)

where q = ωs/ω1m. Note that here we still assume that the control field has no dependence on x or y. The Fresnel number of the interacting region is given by [14]

ω A f = s = |ρ |2, (6.50) 2πcL max

so, since the limit of the paraxial approximation is usually f = 1, we set |ρmax| = 1.

Equations (6.48) and (6.49) can be solved by alternately Fourier-transforming over the variables (ρ, ζ) and (ρ, ²). For example, for the read-in interaction transforming

(ρ, ²) → (k⊥, ω) (where k⊥ = (kX , kY )) gives

µ ¶ |k |2 − ⊥ + i∂ α˜(ω, ζ, k ) = −Cβ˜(ω, ζ, k ), (6.51) 4q ζ ⊥ ⊥ ˜ iωβ(ω, ζ, k⊥) = Cα˜(ω, ζ, k⊥), (6.52) where the tildes denote fourier transformed variables. One can eliminate β˜ to get an equation just in terms of the transformed signal field:

µ ¶ C2 |k |2 ∂ α˜(ω, ζ, k ) = i + ⊥ α˜(ω, ζ, k ), (6.53) ζ ⊥ ω 4q ⊥

which has solution µ ¶ 2 2 C |k⊥| i ω + 4q ζ α˜(ω, ζ, k⊥) =α ˜(ω, 0, k⊥)e . (6.54) 6.5 Transverse structure 119

Equation (6.54) can be Fourier transformed to give an expression for the signal field after read-in. The spin wave can similarly be found, as can the solutions for read-out. The general solution for the interaction can then be written as

Z Z 1 2 0 £ 0 0 α(², 1, ρ) = d ρ dx R(ρ − ρ , 1)G1(² − x, 1)α(x, 0, ρ ) A 0 0 0 ¤ +R(ρ − ρ , 1 − x)G0(1 − x, ²)β(0, x, ρ ) , (6.55) Z Z 1 2 0 £ 0 0 α(1, ζ, ρ) = d ρ dx R(ρ − ρ , ζ − x)G1(ζ − x, 1)β(0, x, ρ ) A 0 0 0 ¤ −R(ρ − ρ , ζ)G0(1 − x, ζ)α(x, 0, ρ ) , (6.56) where the diffraction kernel is given up to some normalization constant by

R(ρ, x) ∼ e−i|ρ|2/x/x. (6.57)

This kernel is singular for x = 0, since in the near-field the paraxial approximation breaks down. However, in the limit that the interaction is negligible close to the exit face of the ensemble, we can make the approximation R(ρ, x) ' R(ρ, 1) [4]. The diffraction kernel then factorizes out in Eqs. (6.55) and (6.56). We introduce the spectral decomposition

X∞ 0 0 R(ρ − ρ ,C) = ψj(ρ)σjψj(ρ ), (6.58) j=0

where the σj’s are real eigenvalues and {ψj} is a complete orthonormal set of paraxial modes satisfying Z ∗ 2 ψj(ρ)ψk(ρ)d ρ = δjk (6.59) A for all j, k. We assume that the transverse spatial profile of the signal depends only on the radial coordinate |ρ| ≡ ρ, which allows us to neglect modes with any depen- 120 Mapping broadband single-photon wavepackets into an atomic memory dence on azimuthal angle. These cylindrically symmetric modes are then given by

2 (1) 2 (1) ψj(ρ) = exp(iρ )φj (ρ ), where the superscript 1 indicates that the function φj solves Eq. (6.43) with C = 1. As in Eqs. (6.36)-(6.39) we define input and output mode operators as

X∞ α(², 0, ρ) = φi(1 − ²)ψj(ρ)aij, (6.60) i,j=0 X∞ β(0, ζ, ρ) = φi(1 − ζ)ψj(ρ)bij, (6.61) i,j=0 X∞ α(², 1, ρ) = φi(²)ψj(ρ)Aij, (6.62) i,j=0 X∞ β(1, ζ, ρ) = φi(ζ)ψj(ρ)Bij, (6.63) i,j=0 with

† † † † [Aij,Akl] = [Bij,Bkl] = [aij, akl] = [bij, bkl] = δikδjl. (6.64)

The following beamsplitter transformations can be obtained:

Aij = σj(µiaij + λibij), (6.65)

Bij = σj(µibij − λiaij). (6.66)

The lowest mode is dominantly coupled, with σ1 = 0.995, and its transverse spatial profile is well approximated by a Gaussian with a waist of rs = 1.45 (in units of the normalized transverse coordinates). The control can also be considered to have a

Gaussian transverse profile, but should be less focussed, with a waist of rc & 3rs, so that the approximation that the control field consists of plane waves is valid. For such signal and control fields only the lowest paraxial mode is involved in the interaction, so the index j can be dropped from Eqs. (6.65) and (6.66). Setting σj ' 1 recovers 6.6 Conclusions and discussion 121

Eqs. (6.40) and (6.41), and hence the one-dimensional treatment is justified.

6.6 Conclusions and discussion

In this chapter we have presented a scheme showing that an off-resonant Raman con-

figuration for a Λ-type ensemble quantum memory can be used to implement deter- ministic, controllable, unitary transfer of the temporal structure of broadband single photons to a stationary spin wave, in the adiabatic regime. We optimized the dynamics using a universal mode decomposition that is valid for all control pulses and arbitrary input states. Furthermore, the optimal efficiency of the memory depends on a single parameter C. We have also shown the difficulties that arise for both forwards and backwards read-out of the stored excitation, in particular when the Stokes shift ω13 of the atoms is non-zero. However it is desirable to have ω13 6= 0 for addressability of the individual transitions, and we provide a solution to this read-out problem in Chapter 7.

6.6.1 Connecting the optimizations

Independently of our work, Gorshkov et al. carried out an analysis of the optimization of quantum memory schemes in both the EIT and Raman regimes [10, 16]. Although these works use the same model as are used in this chapter, in particular starting from the Maxwell-Bloch equations (6.1)-(6.3), the conclusions reached in his work seemed, on first appearance, to conflict with our own. In [10] it was shown that the retrieval efficiency of a given stored spin wave depends only on the optical depth d, as opposed to C in our work. Since d is independent of the detuning and the control field Rabi frequency, this represents a significant difference. Also, in our work we neglected the 122 Mapping broadband single-photon wavepackets into an atomic memory spontaneous emission γ, thus making the interaction unitary. The fact that this can be done turns out to not be obvious [15], and this point bears itself out in the fact that in [16], the Raman condition is given as dγ ¿ ∆, whereas in our work the Raman limit was taken to be γ ¿ ∆. Hence even when treating the Raman limit, Gorshkov et al. do not neglect the spontaneous emission. These discrepancies, among other things, lead to quantitatively different solutions for the optimal spin waves. However, we will now show that these differences can be resolved by observing that our analysis and the off-resonant analysis in [16] are in fact different limiting cases of the same problem, and it is these different limits that lead to the disagreements described above.

In the work of Gorshkov et al. [10, 16], it is shown that the storage process is simply the time-reverse of the retrieval process, so it is instructive to consider the retrieval

first. We can denote the stored spin wave βr(0, z) and retrieved signal field αr(², 0) as column vectors b and a respectively, so that the entries in the vectors are the spin wave modes and output optical modes. Then the retrieval process can be written as a map of the form a = Mb, (6.67) where M is a matrix that represents the mapping of the memory. Due to spontaneous emission M is not necessarily unitary. The aim of optimizing the memory is to find the spin wave vector b that maximizes the norm of a. This can be achieved by applying a singular valued decomposition (SVD) on M, such that M = UDV †, with U and V unitary and D a diagonal matrix. The columns of V are the right singular vectors, and form an orthonormal basis spanning the input space of M. Similarly the columns of U are the left singular vectors, which span the output space of M. So, to maximize the norm of a, b is chosen so that it coincides with the right singular vector associated with the largest singular value of M. The corresponding left singular vector then determines 6.6 Conclusions and discussion 123 a.

Alternatively, the above SVD problem can be turned into an eigenvalue problem, and this can then be solved, as is done in both the work in this chapter and in [16]. This can be seen by noting that M †M = VD2V †, and MM † = UD2U †. So, one can solve for the optimal spin wave by finding the eigenvector of the normally-ordered product M †M with the largest eigenvalue. Recall our definition for the normalized signal field in Eq. (6.9). We define the ratio R as R2 = ω(T )/(γd), and rewrite the phase χ(τ, z) as

χ(τ, z) = Ce−iθ [²(τ)R + ζ/R] , (6.68)

= C(cos θ − i sin θ)[²(τ)R + ζ/R] . (6.69)

p The ratio R determines the balance of the interaction, and goes as R ' Nc/N. Hence for large R the interaction can be said to be light-biased, and matter biased for small R. We have split χ(τ, z) into real and imaginary parts because, whilst the real part can be incorporated into the normalized signal fields and spin waves, the imaginary parts correspond to non-unitary evolution, and hence must be included in the kernel as they will affect the SVD. Note that in our earlier analysis of the Raman memory, θ → 0, and so Im {χ} = 0. Define a new normalized signal field and spin wave as

r ω(T ) −iC cos(θ)[²(τ)R+ζ/R] αG(², ζ) = e A(τ, z)/Ω(τ), (6.70) r C L β (², ζ) = e−iC cos(θ)[²(τ)R+ζ/R]B(τ, z). (6.71) G C

Then the memory equations can be formulated in terms of αG and βG, and analogously 124 Mapping broadband single-photon wavepackets into an atomic memory to Eq. (6.30), the output signal field after read-out can be written as

Z 1 ³ √ ´ iθ −C sin(θ)[²R+ζ/R] iθ ²ζ αG(², 1) = Ce dζe J0 2Ce βG(0, 1 − ζ), (6.72) 0 and a similar equation can be obtained for the spin wave after read-in. One can see that, if the coordinates ² and ζ are discretized into {²j} and {ζk} respectively, then the interaction in Eq. (6.72) can be written as a map of the form in Eq. (6.67), where M has matrix elements

³ √ ´ iθ −C sin(θ)[²j R+ζk/R] iθ ²j ζk mj,k = Ce e J0 2Ce ∆step, (6.73)

and ∆step is the size of the discretization step. The optimal spin wave mode for retrieval can be found using the SVD method described previously. The storage process can be treated analogously, and the optimal signal mode can also be determined from the SVD of the matrix in Eq. (6.73).

From Eq. (6.73), one can see that the optimal solutions are characterized by three parameters: R, C, and θ; equivalently one can use d, ω(T ) and ∆/γ. However, in the limits considered in both our work and the work of Gorshkov et al. the optimization reduces to an eigenvalue problem that depends only on one number.

Our model - off resonant limit

In the limit θ → 0 (or alternatively ∆ À γ) the map M = [mjk] becomes Hermitian, and the SVD of M reduces to an ordinary spectral decomposition. Hence it is not nec- essary to square up M to get the eigenvalue equation, we just solve for the eigenvalues of M, as we did in Eq. (6.43). Note that this limit also requires a roughly balanced equation, because for the problem to reduce to the eigenvalue equation in Eq. (6.43), 6.6 Conclusions and discussion 125

Figure 6.5. A comparison of the optimal spin waves calculated from a general SVD, the solutions presented in this chapter (labelled C), and the solutions given in [16] (labelled d). For both figures θ = 0.01 and C = 2. (a) The off-resonant, balanced interaction, with Nc/N = 1 and d = 200. (b) The off- resonant, light-biased interaction, with Nc/N = 50, and d = 4. the exponential terms in Eq. (6.73) must be small. As well as θ → 0, this also requires γ/∆ ¿ R ¿ ∆γ, which for off-resonant fields is satisfied when R ∼ 1. In this case we saw that the optimization depends only on the parameter C. This is illustrated in Fig. 6.5(a). In the off-resonant balanced limit, the optimal spin wave calculated using Eq. (6.43) overlaps with that calculated by taking a numerical SVD of M with no approximations, and is different to that calculated using the method in [16].

The light-biased limit

In [16] the SVD problem is turned into an eigenvalue problem as described previously by calculating M †M, which gives the kernel whose eigenfunctions we need to calculate. Switching to continuous coordinates, this product is equivalent to an integral over the time coordinate. In [16] this integral is taken with an upper limit tending to ∞, and results in a integral kernel that depends only on d. However, such an upper limit 126 Mapping broadband single-photon wavepackets into an atomic memory

Figure 6.6. Comparison of the loss of the off-resonant quantum memory (solid) with the minimum possible loss given a coupling C. We set Nc/Na = 1, and θ = 0.01 so that d = 100C. corresponds to having a large control pulse energy, or equivalently, being in the light- biased limit (large R). In Fig. 6.5(b) we show that in this limit the optimal spin wave calculated directly using a numerical SVD coincides with that calculated using the light-biased approximation [16], whereas the spin wave determined from Eq. (6.43) is different. Hence we see that the apparent quantitative discrepancies between the results are due to the different approaches actually solving the same equation in two different parameter regimes.

6.6.2 Memory performance

To conclude this chapter we compare the performance of the off-resonant quantum memory in the balanced limit to both the EIT-based memories and the light-biased Raman memory (which have essentially been shown to be equivalent). Gorshkov et al. calculated [10] the optimal memory efficiency given a certain optical depth d, which we compare to the off-resonant balanced memory in Fig. 6.6. We see that for coupling 6.6 Conclusions and discussion 127 strengths of C > 3 the off-resonant balanced memory approaches the optimal memory, but below this the performance is significantly lower than optimal. Note that if we increase the coupling by increasing the control field strength, then eventually we leave the off-resonant balanced regime and the interaction becomes light-biased, with the efficiency depending only on d. This illustrates the discrepancy between our work and that of [10] regarding the neglecting of spontaneous emission. If the control pulse energy is limited then we have seen that C is the only important parameter. If C is increased by increasing the control pulse energy [represented by ω(T )], then eventually the efficiency becomes independent of ω(T ), and the fundamental limit is governed by d. Hence for arbitrary control fields of arbitrary strength, the spontaneous emission always limits the efficiency.

The transparency window, and hence the signal field bandwidth, is limited by the maximum control Rabi frequency in an EIT quantum memory. Our original motivation for studying a Raman quantum memory was that one might be able to store greater bandwidths than in an EIT memory. It turns out that this is true to some extent. If we are in the off-resonant, balanced regime, then increasing the detuning increases the bandwidth that can be stored, but decreases C. This decrease must be compensated for by increasing either ω(T ), N, or g (e.g. by using either a cavity [17] or a better choice of atom), to give the same efficiency for an increased bandwidth. In the EIT regime, one must increase the control pulse energy to increase the bandwidth capacity of the memory, but the efficiency is independent of this. Hence for a limited control pulse energy, greater bandwidths can be stored in a Raman memory. 128 Mapping broadband single-photon wavepackets into an atomic memory

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[17] A. V. Gorshkov, A. Andr´e,M. D. Lukin, and A. S. Sørensen, e-print quant- ph/0612084 (2006). 130 Mapping broadband single-photon wavepackets into an atomic memory Chapter 7

Publication

Efficient spatially-resolved multimode quantum memory

K. Surmacz, J. Nunn, F. C. Waldermann, K. C. Lee, Z. Wang, I. A. Walmsley and

D. Jaksch

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K.

Submitted to Physical Review Letters

We propose a method that enables efficient storage and retrieval of a photonic exci- tation stored in an ensemble quantum memory consisting of Lambda-type absorbers with non-zero Stokes shift. We show that this can be used to implement a multimode quantum memory storing multiple frequency-encoded qubits in a single ensemble. The read-out scheme applies to memory setups based on both electromagnetically-induced transparency and stimulated Raman scattering, and spatially separates the output signal field from the control fields.

The distribution of entangled states over large distances is a basic requirement for realizing quantum networks and numerous quantum communication protocols [1, 2]. 132 Efficient spatially-resolved multimode quantum memory

It is also of importance for fundamental experimental tests of quantum theory [3]. As recently pointed out [4] current entanglement distribution protocols using quan- tum repeater stations between two parties [5, 6] will only work over large distances if significantly improved quantum memories become available. Ideally these memories should also be able to store multiple qubits and allow their selective retrieval [7]. Here we propose a scheme to achieve these goals for photonic qubits stored in a quantum memory (QM) consisting of an ensemble of three level Λ-type absorbers as shown in

Fig. 7.1(a).

Our scheme is based on the approach [8–14] where the qubit is encoded in a signal light pulse, and stored in the ensemble via a classical control field [see Fig. 7.1(a)] prop- agating colinearly with the signal in the positive z-direction [as defined in Fig. 7.1(b)]. Storage of the signal produces a highly-asymmetric spin wave in the medium [15, 16].

After some time the pulse can be retrieved by applying another control field. If this field also propagates in the positive z-direction, the spin wave of the medium has a low overlap with the read-out mode, resulting in a low memory efficiency unless one resorts to a much larger coupling strength for read-out. Instead one can reverse the read-out control field, which maximizes this overlap. However for non-zero Stokes shift

ω13 [see Fig. 7.1(a)] the memory efficiency suffers due to momentum conservation. It is desirable to have ω13 6= 0 for independent addressability of the transitions | 1i ↔ | mi and | 3i ↔ | mi, as selection rules are rarely stringent enough to ensure this. Further- more, in colinear quantum memories one also has to spectrally separate the retrieved signal field from the stronger control field. For a weak signal this filtering is difficult, and another method for resolving the fields, such as orienting the control field slightly off-axis [17, 18], would be advantageous.

The aim of this paper is to overcome these limitations in the read-out. Our pro- posal applies to QMs based on both stimulated Raman scattering [15, 19, 20] and 133

Figure 7.1. Scheme for efficient storage and phasematched spatially-resolved read-out in an ensemble QM. (a) Absorber level configuration, and (b) schematic experimental setup. A signal pulse (frequency ωs, wavevector ks) is stored by a control field (frequency ωc, wavevector kc). The configuration is such that initial state | 1i is energetically higher than storage state | 3i. The control field is angled so that the resulting spin wave has zero z-momentum, allowing efficient retrieval of the signal in a different direction to the read-out control field.

electromagnetically-induced transparency (EIT) [8, 11, 13, 14]. We show that using our method, a multimode QM can be realized whereby frequency-encoded qubits are stored in and retrieved from a single ensemble. This would allow manipulation of stored qubits, for example in an optical lattice QM [21], without needing to use multiple en- sembles for the different logical states. The number of modes that can be stored in this way is limited experimentally by angular resolution and the achievable frequency of signal and control fields. In a conventional single-mode memory the signal and control pulses are colinear and the levels are chosen so that state | 1i is energetically lower than

| 3i. In this case ω13 6= 0, and the stored spin wave has a momentum ∆k = ks − kc (we call this a phase mismatch), rendering backwards read-out of the signal inefficient. To overcome this (i.e. to phasematch the system) we choose the level configuration shown

−1 in Fig. 7.1(a), and orient the read-in control field at an angle of θc = cos (|ks|/|kc|) to the signal propagation direction [see Fig. 7.1(b)]. This eliminates the longitudinal phase mismatch and allows efficient retrieval of the signal field. Furthermore the output signal field direction, determined by momentum conservation, is spatially distinct from 134 Efficient spatially-resolved multimode quantum memory

Figure 7.2. Demonstration of the increase in read-out efficiency for an ensemble QM with reverse read-out. Retrieval probability ν is plotted against the read-in and read-out couplings Cin and Cout (defined in text). In (a) we use colinear signal and control fields. The phase mismatch introduced by non-zero ω13 degrades the efficiency, with ν = 0.23 for Cin = Cout = 2, which is sufficient for storage with near-unit efficiency [15]. In (b) the control field is angled appropriately, and Cin = Cout = 2 gives ν = 0.95, which is also much higher than the efficiency achieved using colinear read-in and colinear forward readout (ν = 0.52) [15].

the read-out control field. The scheme allows high-fidelity retrieval of signal pulses with duration T provided that L ¿ T c, and T ω13 À 1, with L the length of the ensemble and c the speed of light. In the following we detail the application of this method to a general QM, including a discussion of the constraints that arise and the storage times that can be achieved. We then show how our scheme could be used to implement a multimode Raman QM consisting of a single ensemble, and derive an expression for the maximum number of modes it is possible to store.

We consider the setup shown in Fig. 7.1. The classical control field for read-in is oriented in the x-z plane at an angle of θc to the signal. If θc is small, the walk- off between the two fields can be neglected. The control field envelope can then be represented at time t and longitudinal position z by the slowly varying Rabi frequency

Ω(τ), where τ = t − z/c. The frequency of the transition | 1i ↔ | mi is ω1m. We introduce dimensionless coordinates ζ(z) ≡ z/L for the longitudinal position; ²(τ) ≡ R τ 0 2 0 ω(τ)/ω(T ), with ω(τ) ≡ 0 |Ω(τ )| dτ the integrated Rabi frequency, for the time; and 135

p ρ = (X,Y ), where X = x ω1m/2Lc (analogously for Y ) for the transverse position. p We define the slowly-varying operators α(², ζ, ρ) = ω(T )/CA(r, τ)e−iχ(τ,z)/Ω(τ) for p the signal field and β(², ζ, ρ) = L/CB(r, τ)e−iχ(τ,z) for the spin wave, where A(r, τ) and B(r, τ) are the slowly-varying signal and spin-wave amplitudes respectively, and χ(τ, z) ≡ [ω(τ) + |κ|2z]/Γ, which describes a Stark shift due to the control field and a modification of the signal group velocity. We decompose the phase mismatch as

∆k = ∆k⊥ + ∆kzzˆ, with zˆ the unit vector oriented along the z-axis. The coupling of the signal field to the ensemble is |κ|2 = dγ/L, with d the resonant optical depth [16] and Γ = ∆ − iγ, where γ accounts for dephasing and loss of the optical polarization.

We assume that either dγ or ∆ are much larger than the signal bandwidth, the maximum control Rabi frequency, and γ, and adiabatically eliminate the excited state | mi. Within this limit our theory describes the interaction for arbitrary signal and control pulse shapes in both the EIT and Raman regimes. The absorbers are all initially optically pumped into state | 1i, and the population of state | 3i is assumed to remain negligible for each absorber. In the slowly-varying and paraxial approximations the Maxwell-Bloch equations are

−ipX ∂²β = −Ce α, (7.1)

³ 2 ´ ∇ρ ipX 4q + i∂ζ α = Ce β, (7.2)

p −1/2 where q = |ks|c/ω1m, p = |kc| sin(θc)(ω1m/2Lc) , and the coupling C ≡ |κ| Lω(T )/|Γ|. Note that these equations are similar to those in [15], except we also include the trans-

2 2 2 verse Laplacian ∇ρ = ∂X + ∂Y [22].

The dynamics of Eqs. (7.1) and (7.2) decomposes into a linear mapping between signal and spin wave modes. The optimally-efficient mapping excites an asymmetric spin wave β1 approximated by exponential decay in ζ. The symmetry of the interaction 136 Efficient spatially-resolved multimode quantum memory

r [15, 16] dictates that the optimal spin wave for forwards retrieval, β0, is the mirror image

r in ζ of this, i.e. β0(ζ) = β1(1 − ζ), where a superscript r identifies a quantity associated with the read-out. The efficiency of forward readout then depends on the overlap of

r β1 with β0, which is generally low. Switching the propagation direction of the control

r field, to readout in the backward direction, flips β0 around, so that its overlap with

β1 is perfect. However, due to the residual momentum ∆k of the spin-wave coherence

r after storage, we then have β0 (ζ, ρ) = β1 (ζ, ρ) exp [iϕ (ζ, ρ)], where

½r ω ϕ(ζ, ρ) = 1m (∆k + ∆kr ) .ρ + (∆k + ∆kr ) 2Lc ⊥ ⊥ z z ¾ ×Lζ + χ[T, Lζ] − χr[0,L(1 − ζ)] .

The linear ζ-dependence of the real terms in ϕ reduces the overlap of the stored spin wave and the output mode, leading to a reduction in the memory efficiency. This is shown in Fig. 7.2(a) for a caesium QM, where levels | 1i and | 3i are the F = 3 and

F = 4 hyperfine levels of the 6S1/2 state respectively (i.e. | 1i is energetically lower than | 3i). Our strategy is to find a configuration for which <{ϕ} is independent of ζ. To do

r −1 this we choose | 1i to be energetically higher than | 3i and set θc = θc = cos (ks/kc),

r where the angle of the read-out control field θc is as defined in Fig. 7.1(b). This phasematches the memory in the Raman and EIT configurations – far from resonance the χ functions in ϕ are real but negligibly small, and on resonance they are imaginary, so do not contribute to the mismatch. Figure 7.2(b) shows that this results in a high- fidelity read-out in 133Cs, where now | 1i and | 3i are the F = 4 and F = 3 hyperfine levels respectively. Note that a transverse phase mismatch does not degrade the readout efficiency; any residual ∆k⊥ simply modifies the angle at which the retrieved signal field is emitted.

We have assumed that the walk-off along the z-axis between the signal and control 137

p fields is negligible, i.e. small compared to the pulse durations, so that θc ¿ T c/L.A more stringent condition is that the transverse walk-off of the beams is small compared √ to their beam waists, which requires θc ¿ A/L, with A the cross-sectional area of the control pulse. As well as setting the angle θc, the Stokes shift imposes an upper limit to the control field bandwidth, since there should be no control photons at the signal frequency. Diffraction limits the length of the ensemble according to the condition A ∼ λL, where λ is the control wavelength. These conditions can be combined into

√ T c À L À A, (7.3)

with the maximum storage bandwidth given by T ω13 À 1. The upper bound on L is relaxed if the control field focussing is loosened, but the control pulse energy should then be increased. However, in general, the above considerations imply that the number density required for high-efficiency (>90%) phasematched operation of a Λ-ensemble QM is given approximately by n = ΘT −2, with Θ ' 1m−3s2, in both EIT and Raman configurations.

133 In a Cs QM the 6S1/2 state is split by the hyperfine interaction with the nuclear spin (I = 7/2), to give the F = 3 (our | 3i), and F = 4 states (our | 1i), with

ω13 = 9.2GHz . The 6P3/2 state (D2 line), which lies 351.7THz (852 nm) above state | 1i, would be used for the excited state | mi. If ∆ = 10GHz, then the phasematching

◦ angle is θc ' 0.5 . This would enable high-efficiency storage and retrieval of a single photon with T = 250ps in an ensemble with L ' 2 cm. This bandwidth requires a number density of around 1019m−3, which can be achieved in a caesium vapour at a temperature of Te ' 360K. For a thermal vapour we must consider motional dephasing of the memory, as the phase introduced by the read-in control field will average out

[23, 24]. The stored spin wave oscillates at the frequency c|kc| sin θc ' c|kc|θc, and so 138 Efficient spatially-resolved multimode quantum memory

Figure 7.3. Storage and retrieval of two signal field components with different frequencies: (a) the stored spin wave intensity Ib, and (b) the retrieved signal field intensity Ia. Both are Fourier-transformed in the x-coordinate to illustrate the different spectral components. The two signal components require control ◦ fields angled at θc = ±3 , thus creating two components to the spin wave at either side of kx = 0. Read-out is performed with a control field angled at r ◦ θc = 3 , and hence one output photon propagates along the z-axis, and the other propagates at an angle of 6◦ to the z-axis.

2 2 2 the entanglement fidelity of the memory [25] is given by F' exp[−|kc| θc tskBTe/M], with ts the storage time of the memory, kB Boltzmann’s constant, and M the atomic mass. For the above parameters the maximum storage time with F > 0.9 is ts ' 200ns, i.e. three orders of magnitude larger than the signal pulse duration. Longer storage times could be achieved by using a storage unit consisting of an ultra-cold gas, but this requires a larger sample than has currently been achieved experimentally.

Using the off-axis QM scheme described above, it is possible to store multiple signal fields of different frequencies in a single ensemble, and retrieve them as spatially separated, as demonstrated in Fig. 7.3. Hence one can implement a multimode QM [7] in which frequency-encoded qubits are stored, or frequency entanglement can be converted into spatial entanglement. To see this we consider the same setup as in Fig. 7.1, but now the signal field consists of two components, labeled 1 and 2, of frequency ωs,1 and ωs,2 respectively. Signal component j (j = 1, 2) is Raman resonant with control field j, which has frequency ωc,j, detuning ∆j, and is angled at θc,j to the signal for phasematching. Both signal components can be stored in the ensemble, and 139

the resulting spin wave has two components separated in kx (where kx is the momentum component in the x-direction), due to the different transverse momenta of the control fields [see Fig. 7.3(a)]. Backwards read-out is performed using a single control field, resulting in two spatially-separated output signal components [as shown in Fig. 7.3(b)].

We ensure that |∆1 − ∆2| À (δ1 + δ2), with δj the bandwidth of signal component j, so that signal field j cannot initially interact with control field i (i 6= j). This condition requires that max[∆1, ∆2] is large, i.e. in the Raman regime. However, once a material excitation is created by the absorption of signal j, this excitation may interact with control field i.

To calculate the form of the stored spin wave and the retrieved signal field, we again assume that the angles relative to the signal field of both control fields are small.

Control field j has wavevector in the x-direction with rescaled magnitude pj. We assume that control field j has Rabi frequency Ωj(τ) = Ωjf(τ), with Ωj the maximum

2 Rabi frequency, f(τ) an envelope function normalized so that ωj(T ) = |Ωj| T , and R τ 0 2 0 the integrated Rabi frequency ωj(τ) = 0 |Ωj(τ )| dτ . Also due to the large number of absorbers in our ensemble, the probability of a single absorber being excited by both signal components is assumed to be negligible. The coupling constant is now √ P2 2 Cm = κ LW , where W = j=1 ωj(T )/|Γj| . The dimensionless operator for signal p −iχj (τ,z) component j is written as aj(², ζ, ρ) = T/Cme Aj(r, τ)/f(τ), and the spin p −iχb(τ,z) wave operator is now b(², ζ, ρ) = L/Cme B(r, τ), where Aj(r, t) is the slowly- varying amplitude of signal field j, and the spin-wave momentum mismatch is now

P2 2 ∆k = zˆω13/c. The complex exponents are χj(τ, z) = i=1 ωi(τ)/Γi + |κ| z/Γj, and

P2 2 χB(τ, z) = i=1 ωi(τ)/Γi respectively. The Maxwell-Bloch equations are thus

¡ ¢ 1 2 −i(pj X+ζ/Rj ) 4q ∇ρ + i∂ζ aj = Cmcje b, (7.4) £ ¤ ∗ i(p1X+ζ/R) i(p2X+ζ/R2) ∂²b = −Cm e c¯1a1 + e c¯2a2 , (7.5) 140 Efficient spatially-resolved multimode quantum memory

p p √ √ ∗ with cj = T/W (Ωj/Γj),c ¯j = T/W (Ωj /Γj) and Rj = Γj W/|κ| L; subscripts indicate the signal components to which the quantities refer. The solution for the two- component signal field, which can be achieved in the same way as for the single-field case, is the convolution of the integral kernels for the single-field interaction [15] with terms that arise due to the mixing of the two components. This mixing occurs, for example, when during storage signal component j interacts with both control fields by a second-order process, and could potentially degrade the read-in. However, our calculations have shown that the mixing only becomes significant when the read-in coupling is Cm & 15 – much larger than the coupling sufficient for signal storage with near-unit efficiency (C = 2). For Cm = 2 we see that the storage process for the two- component signal field is almost identical to the results obtained when the two signal components are assumed to interact independently of each other. Also, since we use a single control field to retrieve the signal, the read-out obeys Eqs. (7.1) and (7.2), and the high efficiency of our proposed multimode QM follows from the efficiency of the single-mode memory discussed earlier. Let us note that if the cj’s have a relative phase of π between them, then the two absorption processes interfere destructively, and the storage efficiency is low. For effective storage of the signal components this phase must be zero, which can be ensured either by changing the sign of the ∆j’s, or by phase-shifting the Ωj’s.

Since one can in principle have arbitrarily-far blue-detuned fields (avoiding any higher levels), the main limitations on the number of modes that can be stored are experimental – the achievable optical frequencies and the possible angular resolution √ ∆θ. The maximum control field angle allowed is given by θmax = A/L and the number of different control fields allowed is bθmax/∆θc. The smaller angles considered here correspond to the fields being far blue-detuned, which may be outside the optical regime. Taking this into account, the number of modes that can be stored is given by BIBLIOGRAPHY 141

1/2 nm = b{θmax −[2ω13/(ω1m +∆max)] }/∆θc, with ∆max the largest allowed blue detun- ing from | mi. For the 133Cs QM discussed previously, taking the angular resolution to √ −7 2 be ∆θ = λ/ A, setting A = 10 m , and restricting ∆max so that no other transitions are excited, this corresponds to nm ∼ 100 modes. As with the single-mode case the phase introduced by control field j is ωc,jθc,j/c = ω13/(cθc,j). Hence the limitation on storage time of the memory due to dephasing, as calculated earlier for a single mode, is given by the entanglement fidelity of the mode that requires the smallest control field angle for phasematching.

In summary, we have demonstrated a QM scheme using an ensemble of three-level Λ-type absorbers that allows efficient storage and phase-matched spatially-resolved read-out of the signal field. We use this method to demonstrate how a multimode QM could be implemented using a single atomic ensemble. Multimode memories will enable the storage of multiple qubits in one memory setup, significantly decreasing the resources required for entanglement distribution protocols. Furthermore, storing multiple qubits in a single ensemble opens up new possibilities for scalable entanglement manipulation.

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[25] K. Surmacz, J. Nunn, F. C. Waldermann, Z. Wang, I. A. Walmsley, and D. Jaksch, Phys. Rev. A 74, 050302(R) (2006). 144 Efficient spatially-resolved multimode quantum memory Chapter 8

Efficient spatially-resolved multimode quantum memory - methods

8.1 Introduction

In chapter 7, we proposed a modified memory setup that resolves the problems with reading out of an ensemble-based quantum memory. The strategy was to angle the control field with respect to the signal field, so that the longitudinal phase mismatch of the resulting spin wave is zero [1]. In this chapter we provide supplementary details of the analysis of this method. We also analyse another possible method for phasematch- ing the memory read-out. This scheme uses two control fields for read-in, angled so that they have the same longitudinal wavevector, but the opposite transverse wavevec- tor (see Fig. 8.1). We show that using such a read-in scheme creates two spin-wave excitations in the atomic medium, and that reading out using one or two control fields leads to the signal field being emitted in a superposition of different directions, with the weightings of each superposition determined by the strengths of the control fields used. To conclude this chapter we describe the multimode quantum memory scheme 146 Efficient spatially-resolved multimode quantum memory - methods proposed in chapter 7 in more detail, including an explicit solution of the memory evolution equations.

8.2 Phasematching a single-mode memory

Consider the setup shown in Fig. 7.1. A signal and a control field are incident on an ensemble of Λ-type atoms, with the signal exciting the | 1i ↔ | mi transition, and the control driving the | 3i ↔ | mi transition. The signal field has frequency

ωs and carrier wavevector in the z-direction with magnitude ks, whilst the control

field is angled in the x-z plane at θc to the z-axis, and hence has carrier wavevector kc := kc[sin(θc)xˆ + cos(θc)zˆ], with xˆ and zˆ the unit vectors in the x- and z-directions respectively. Both the signal and control fields are detuned by ∆, and we write the complex detuning as Γ = ∆ − iγ, with γ the polarization dephasing rate. The control

field Ec(t, r) at position r = (x, y, z) and time t can be written in terms of a slowly- varying envelope Ec(t, z), as in Eq. (5.3), with the transverse component also included to give

i(kcz cos θ−ωct) ikcx sin θc Ec(t, r) = ²cEc(t, r)e e + H.c., (8.1)

where ²c is the polarization unit vector. Note that we have assumed that the transverse component of the control field is plane wave. Since the angle θc is small, a similar argument to that given in section 6.5 can be used to show that a modematched control field will have a much larger transverse width than the signal field, so approximation to a plane wave is valid. We define the control Rabi frequency as in Eq. (2.7). The slowly- varying signal field operator A(t, r), is defined as in Eq. (5.12), with the exception that the integral is now a sum over modes in all three dimensions. The slowly-varying 8.2 Phasematching a single-mode memory 147 spin-wave operator B(t, r) is defined as

√ N XNr B(t, r) = σ˜ˆ(j)(t), (8.2) N 31 r j=1

ˆ(j) where σ˜31 (t) is the slowly-varying spin coherence as defined in Eq. (5.11), N is the number of atoms, and the sum is over the Nr atoms contained in a small volume of the ensemble centred at r.

By defining the rescaled coordinates ², ζ, and ρ, and the rescaled variables α(², ζ, ρ) and β(², ζ, ρ) as in chapter 7, one can arrive at Eqs. (7.1) and (7.2) using the same method as for Eqs. (6.12) and (6.13) in chapter 6. As with Eq. (6.48) however we keep the transverse Laplacian from Maxwell’s equation. The exponential in X comes from the transverse component of the control field, and by writing βX (², ζ, ρ) = β(², ζ, ρ) exp(−ipX) one recovers the memory equations for the linear case, which give solutions for α and βX that are identical to those for α and β for the co-linear case

(θc = 0, see section 6.4). Hence one can solve for the optimally-stored spin wave

βX (1, ζ, ρ) (recall that ζ and ² are normalized to run from 0 to 1), and modematch the memory, in the same wave as in chapter 6. Note that p is simply the transverse wavenumber kc sin θc normalized such that pX = kcx sin θc.

As described in subsection 6.4.2, forwards read-out of the stored excitation leads to low efficiency of the memory due to the asymmetry of the spin wave and the time- reversal symmetry of the interaction [2, 3]. Hence the logical next step is to consider backwards read-out, as we do here and in chapter 7. We denote the quantities asso- ciated with the read-out process by a superscript r. Having solved for the optimally stored spin wave one can calculate the initial condition for read-out to which this 148 Efficient spatially-resolved multimode quantum memory - methods corresponds, as

βr (0, 1 − ζ, ρ) = βr(0, 1 − ζ, ρ)eiprX , (8.3) X r L r r = Br(τ = 0,L − z, x, y)e−iχ (0,L−z)eip X , (8.4) Cr r √ Nr L N X r r = σˆ(j)(t)ei[(ks−kc cos θc)(L−z)−(ωs−ωc)t−χ (0,L−z)+p X], Cr N 13 r j=1 (8.5) r r r L i[p X−χ (0,L−z)+(ks−kc cos θc)(L−2z)] = r B(τ = T, r)e , (8.6) rC C r r = β (1, ζ, ρ)ei[(p −p)X+χ(T,z)−χ (0,L−z)+(ks−kc cos θc)(L−2z)], (8.7) Cr X where χ(τ, z) is as defined in Eq. (6.11), L is the length of the ensemble, and the read- out control field is angled so that the normalized transverse wavenumber is pr. The phase in Eq. (8.7) is as defined in Eq. (7.3):

r r ϕ(ζ, ρ) = (p − p)X + χ(T, ζ) − χ [0,L(1 − ζ)] + (ks − kc cos θc)L(1 − 2ζ). (8.8)

We recall from Eq. (6.30) that the solution for the signal field after read-out can be written as

Z Z 1 p r 2 0 0 r r α (1, ², ρ) = d ρ R(ρ − ρ ,C ) dxJ0[2 (1 − x)²]βX (0, x, ρ), (8.9) A 0

th where J0 is the 0 order Bessel function of the first kind, and R(ρ, x) is the diffraction kernel in Eq. (6.57). It is now clear that the z-dependent phase in Eq. (8.7) will adversely affect the amplitude of the emitted signal field, and hence the efficiency [as defined in Eq. (6.46)], since it reduces the overlap of the spin wave mode with the input modes for the read-out process. Hence, to phasematch the memory, we wish to make ϕ(ζ, ρ) independent of z. As stated, in chapter 7, the χ terms can be neglected 8.3 Phasematching limitations 149 in the Raman and EIT memory regimes, since far off-resonance they are negligibly small, and on-resonance they are purely imaginary so do not contribute to the phase.

Hence phasematching means setting ks = kc cos θc, which can be done by appropriately angling the read-in control field. Note that, since the transverse diffraction kernel R is Gaussian, the X-dependence in ϕ(ζ, ρ) from (pr − p) does not affect the efficiency, but results in a momentum shift in the x-direction of the signal field.

8.3 Phasematching limitations

We now describe the limitations outlined in chapter 7 for the phasematching scheme in more detail. One main restriction is that the longitudinal and transverse walk-offs between the signal and control fields must be small compared to the duration of the signal pulse T . The times taken for the signal and control fields to travel a distance

L in the z-direction are L/c and L cos(θc)/c, respectively. Since the difference in these times must be small compared to the pulse durations, this leads to

2 L Lθc (1 − cos θc) ' ¿ T , (8.10) c r 2c T c ⇒ θ ¿ , (8.11) c L

where we have used the fact that θc is small, and c is the speed of light. We also require that the transverse walk-off is small compared to the size of the beam waists. In the time taken for the signal pulse to travel a length L through the ensemble, the control pulse will move a distance L sin θc in the x-direction. The beam waist is given roughly √ by A, with A the cross-sectional area of the control pulse, and using θc ¿ 1 gives √ the condition θc ¿ A/L. For propagation of the signal field through the ensemble, the Fresnel number of the interaction region should be around 1. Assuming that the 150 Efficient spatially-resolved multimode quantum memory - methods cross-sectional area of this region is defined by the transverse control pulse area, the Fresnel number can be written F = A/(λL), with λ the control pulse wavelength. This gives the condition that A ∼ λL.

To obtain the condition in Eq. (7.3), we use the fact that

ωs = ωc cos(θ) = ωc − δ, (8.12)

2 which gives δ = ωcθc /2, assuming that θc is small. Substituting the condition arising from F ∼ 1 into this expression for δ gives

Aδ θ2 ' . (8.13) c πLc

Putting this into the longitudinal and trasnverse walk-off conditions results in

√ T c ¿ A, (8.14) and T c ¿ L, (8.15) respectively. For a long, thin pencil-shaped interaction region, Eq. (8.15) gives the √ more stringent condition, and the fact that such an ensemble will require L ¿ A gives Eq. (7.3).

In chapter 7 we also gave an expression for the number density n of the medium one requires to store a signal pulse of duration T . This expression was calculated as follows: firstly C2 = T dγΩ2/∆2, with d the resonant optical depth [3]. One can rewrite d as πα~Nf dγ = 1m , (8.16) meA 8.4 Phasematching using two control fields 151

Figure 8.1. An alternative phasematching scheme using two control fields. (a) Storage: two control fields (solid lines), angled at ±θc to the z-axis, transfer the signal field excitation (dashed line) to an atomic excitation. (b) Retrieval: r two control fields, angled at ±θc to the z-axis, affect reverse read-out of the r signal field. In this chapter we also consider the special case θc = 0, i.e. colinear read-out.

where me is the electron mass, f1m is the oscillator strength for the | 1i → | mi tran- sition, α is the fine structure constant, and N is the number of atoms, which can be written N = nAL. We set f1m ' 1, and since ∆ > Ω for the adiabatic condition to hold, we let ∆ ' µ1Ω, where µ1 is roughly equal to 3. We also showed that T c ¿ L is required for the phasematching scheme to work, so we set T c ' µ2L, with µ2 ∼ 10. Substituting all of this into the expression for C2 above gives

C2m µ2 n = e 1 T −2, (8.17) πα~cµ2 where the prefactor of T −2 is roughly equal to 1m−3s2.

8.4 Phasematching using two control fields

Instead of using a single control field at an angle of θc to the signal field, one could alternatively use two control fields on Raman resonance and angled at ±θc to the signal. This four-wave mixing geometry (shown in Fig. 8.1) is similar to that considered in 152 Efficient spatially-resolved multimode quantum memory - methods

chapter 2 for the calculation of spontaneous emission, but now the angle θc is chosen to eliminate the longitudinal momentum of the resulting spin wave as for the single mode case illustrated in this chapter. Maxwell-Bloch equations can be derived exactly as for the single mode case, except that we define two control field Rabi frequencies Ω1(τ) and

Ω2(τ), and we assume that the two control pulses have the same temporal profile f(τ), so that Ωj(τ) = Ωjf(τ)(j = 1, 2). Again θc must be small to avoid any significant

2 2 2 walk-off of the pulses. Defining the coupling parameter C2 = T dγ(Ω1 + Ω2)/Γ , the Maxwell-Bloch equations are

−ipX ipX ∂²β = −C2(Ω1e + Ω2e )α, (8.18) µ ¶ ∇2 ρ + i∂ α = C (Ω∗eipX + Ω∗e−ipX )β, (8.19) 4q ζ 2 1 2 where the normalized signal field and spin wave operators α and β are defined exactly as in chapter 7. These equations can be transformed into Fourier space and solved as before. Transforming (ρ, ²) → (kX , kY , ω), and defining k⊥ = (kX , kY ), results in

µ ¶ |k |2 h i − ⊥ + i∂ α˜(k ) = C Ω β˜(k + p) + Ω β˜(k − p) , (8.20) 4q ζ X 2 1 X 2 X iC β˜(k ) = 2 [Ω∗α˜(k − p) + Ω∗α˜(k + p)] , (8.21) X ω 1 X 2 X where the tildes correspond to Fourier-transformed variables, and we have dropped the other function arguments ofα ˜ and β˜ for notational convenience. Hence we see that due to the two control fields, the signal field couples to two different sidebands of the spin wave, and vice versa. One can eliminate β˜ from Eqs. (8.20) and (8.21) to give

i|k |2 £ ¤ ∂ α˜ = − n − iC2 Ω Ω∗α˜ + Ω∗Ω α˜ + (|Ω |2 + |Ω |2)˜α , (8.22) ζ n 4q 2 1 2 n+1 1 2 n−1 1 2 n 8.4 Phasematching using two control fields 153

where we have usedα ˜n ≡ α˜(kX + 2np) and kn = (kX + 2np, kY ). Equation (8.22) is in general not analytically solvable. However, we can find an approximate solution by assuming that the nth sideband is coupled to itself with a strength that does not depend on n. This solution can then be transformed back into real space, and the general form of the final signal field and spin wave is then

Z Z 1 ½ 2 0 0 0 α(², 1, ρ) = d ρ dx α(x, 0, ρ )G1[² − x, C2(sin(2ψ) cos(2pX ) + 1)/2] A 0 √ · ³ ´ 2 0 ×R(ρ − ρ0, 1) + cos(ψ) eipX + e−ip(X−2X ) + sin(ψ) 4 µ ¶¸ −ipX ip(X−2X0) 0 × e + e β(0, ζ, ρ)G0[1 − x, ²(sin(2ψ) cos(2pX ) + 1)/2] ¾ ×R(ρ − ρ0, 1 − x) , (8.23)

Z Z 1 ½ 2 0 0 0 β(1, ζ, ρ) = d ρ dx β(0, x, ρ )G1[ζ − x, C2(sin(2ψ) cos(2pX ) + 1)/2] A 0 √ · ³ ´ 2 0 ×R(ρ − ρ0, ζ − x) − cos(ψ) eipX + e−ip(X−2X ) + sin(ψ) 4 ³ ´ ¸ −ipX ip(X−2X0) 0 × e + e × α(², 0, ρ)G0[1 − x, ζ(sin(2ψ) cos(2pX ) + 1)/2] ¾ ×R(ρ − ρ0, ζ) , (8.24)

0 where G0(x, y), G1(x, y), and R(ρ−ρ , x) are the integral kernels defined in Eqs. (6.32),

−1 (6.33), and (6.57), and ψ = tan (Ω1/Ω2) defines the relative control pulse strengths. Note that these solutions are very similar in form to those for the colinear case in Eqs. (6.55) and (6.56), with the extra terms coming from the added complication of having two control fields.

Numerical evaluations of Eqs. (8.23) and (8.24) for different parameters are plotted in Fig. 8.2. Firstly, we see in Figs. 8.2(a) and (b) that using two angled control

fields results in a stored spin wave that has two components separated in kX -space at kX = ±p. This corresponds to the fact that the signal field could have been stored by 154 Efficient spatially-resolved multimode quantum memory - methods

Figure 8.2. Single mode quantum memory using two angled control fields (θc = ◦ ±0.6 ) for read-in, and one control field for read-out. Coupling C2 = 2 for storage and retrieval. (a) and (b) show a cross-section in kX through the resulting signal field intensity Iβ, with the read-in control fields (a) equally weighted (ψ = 1), and (b) unequally weighted (ψ = 2). (c) and (d) plot cross- sections through the retrieved signal field intensities Iα for the parameters used in (a) and (b), respectively. 8.4 Phasematching using two control fields 155 either one of the control pulses. Furthermore, the relative amplitudes of these two spin- wave depends on the relative strengths of the control fields, given by ψ. Then, if one reads out using a single control field (recall that to ensure modematching the read-out is in the reverse direction), the resulting output signal field is a superposition of two different directions corresponding to the two spin wave components [see Figs. 8.2(c) and (d)]. If a single photon were stored in the memory, then the output state would be of the form µ | 01i + ν | 10i (the ket entries correspond to the different directions), where the ratio |µ|/|ν| could be controlled by the strengths of the control fields. Instead, one could use two angled control pulses for read-out (see Fig. 8.3). This splits each stored spin- wave component into two output signal components. In Fig. 8.3(a) the read-out control

fields are also angled at ±θc to the z-axis. We then see that there are now three output components, at kX = 0, ±2p. If one sets the read-out control fields at different angles to the read-in fields, then one can observer four output signal components. Again, the relative weightings of the signal components can be controlled by the relative strengths of the read-in and read-out control fields [Figs. 8.3(c) and (d)]. Hence one could also generate tunable single-photon states of the form µ | 100i + ν | 010i + η | 001i. We see that by using this kind of angular multiplexing with multiple control fields, it may be possible to read-in a photonic state, and generate an entangled output state1, which could be tuned by changing the control field intensities. We finally point out that the solutions plotted in Figs. 8.2 and 8.3 are not properly modematched, i.e. the control fields are not shaped for optimal storage of the signal pulse. This modematching for the four-wave mixing geometry remains an open problem.

√ 1It is a matter of opinion whether one refers to the photonic state (| 10i + | 01i)/ 2 as entangled. For reasons why I do so here refer to [4], 156 Efficient spatially-resolved multimode quantum memory - methods

Figure 8.3. Cross-sections of output signal field intensities for a memory using ◦ angled control pulses for both storage and retrieval. In all cases θc = 0.6 (a) r ◦ r r ◦ r r ◦ θc = 0.6 , ψ = ψ = 1. (b) θc = 0.3 , ψ = ψ = 1, (c) θc = 0.6 , ψ = 2, r r ◦ r r ψ = 1, and (d) θc = 0.6 , ψ = ψ = 1/2. The read-out control field angle θc is as defined in Fig. 7.1. 8.5 Multimode memory equations 157

8.5 Multimode memory equations

We now turn to our proposal for a multimode quantum memory. The setup we consider demonstrates the principle of the memory for two modes, and is shown in Fig. 8.4. A signal field consists of two components, which we refer to as component 1 and component 2, with carrier frequencies ωs,1 ωs,2 respectively. The signal pulse propagates in the z-direction, and component j has a wavevector with magnitude ks,j (j = 1, 2). Both components excite the | 1i ↔ | mi transition, and component j is detuned from

| mi by ∆j. Two control fields drive the | 3i ↔ | mi transition, such that control field j is Raman resonant with signal field j. Furthermore, control field j is angled at

θj = ωs,j/(ωs,j + ω13) such that it would phasematch the read-in of signal pulse j, with

ω13 the Stokes shift of the atoms. For the scheme to work, we require that initially each control field couples to only one signal field. More precisely, with all the atoms initially in state | 1i, the initial transition | 1i → | 3i can only be affected by signal component j and then control pulse j. This requires that |∆1 − ∆2| À δ1 + δ2, with

δj the bandwidth of signal component j. However, as stated previously, once such a transition has occurred, an atom could then subsequently interact with control pulse i (i 6= j). These higher-order processes lead to mixing of the signal components, but as we shall see they do not significantly degrade the performance of the memory for reasonable couplings.

In the same way as for the linear quantum memories in chapters 5 and 7 we can define slowly-varying operators. For example, for signal field j we have

r Z L A (t, r) = e−i(ks,j z−ωs,j t) dωaˆ(ω, t)eiks,j z, (8.25) j 2πc wherea ˆ(ω, t) are the slowly-varying annihilation operators as defined in chapter 5. We can similarly define the slowly-varying Rabi frequency Ωj(t, r) for control field j, and 158 Efficient spatially-resolved multimode quantum memory - methods

Figure 8.4. Scheme for a multimode quantum memory. (a) The atomic tran- sitions used, and (b) A schematic experimental setup.

the slowly-varying spin wave B(t, r), analogously with the definitions in Eqs. (5.3), (5.8), (5.11) and (5.17). Note that we can define a single slowly-varying spin wave because ωs,j − ωc,j = ω13 for j = 1, 2. We assume that signal field Aj(t, r) couples to a slowly-varying polarization operator Pj(t, r), due to the fact that the signal compo- nents are sufficiently separated in frequency. Then, in the same way as we obtained Eqs. (5.18) and (5.19), the Bloch equations are derived to be

√ ∂τ Pj(τ, r) = −(γ + i∆j)Pj(τ, r) + ig NA(τ, r) + iΩj(τ)B(τ, r), (8.26) and X2 ∗ ∂τ B(τ, r) = i Ω (τ)jPj(τ, r), (8.27) j=1 where we have used the fact that the walk-off between the signal and control pulses is small, and moved into a co-moving frame τ = t − z/c. We have also assumed that

|ωs,1 − ωs,2| ¿ min(ωs,1, ωs,2), so that the atom-signal coupling, given by g, is the same for both components. The propagation equation for each component of the signal

field can be used, as with Eqs. (6.7) and (6.48) [5], to obtain the third Maxwell-Bloch 8.5 Multimode memory equations 159

Figure 8.5. The errors present for the storage of two photons in a single en- semble. (a)-(c) show a cross-section in kX of the intensity Ia of one photon ◦ ◦ ◦ after read-in for (a) C = 2, θc,1 = 3 , θc,2 = −3 ; (b) C = 15, θc,1 = 3 , ◦ ◦ ◦ θc,2 = −3 , and (c) C = 15, θc,1 = 5 , θc,2 = −5 . In (a) the transverse pro- file remains Gaussian, but in (b) and (c) there is significant perturbation at non-zero kX due to the photon being stored with one control field and read out by the other. Increasing the angle between the two control fields causes the perturbation to have a higher transverse momentum, showing that these perturbations are due to coupling to other transverse modes. (d) quantifies ˜ the difference between the spin wave b(1, ζ, k⊥) for the two-photon absorption, ˜ and the spin wave bid(1, k⊥) if the two absorption process are assumed to be ˜ ˜ 2 ˜ 2 completely decoupled. Here δb = |b − bid| /|bid| , with the function arguments omitted for convenience. equation, µ 2 2 ¶ ∇T |κ| κΩj(τ) + i∂z + Aj(τ, r) = B(τ, r), (8.28) 2ks Γj Γj √ 2 2 2 where ∇T = ∂x + ∂y , and κ = g N/c as in chapter 7. The polarizations can be adia- batically eliminated in the same way as we obtained Eqs. (6.4) and (6.5). Substitution of the parameters defined in chapter 7 results into the resulting equation, and some straightforward but tedious rearranging, results in Eqs. (7.4) and (7.5).

Equations (7.4) and (7.5) can be solved by Fourier transforming as demonstrated 160 Efficient spatially-resolved multimode quantum memory - methods earlier in the chapter for the single mode memory with two control fields. For the read- in we transform (², X, Y ) → (ω, kX , kY ), and eliminate the transformed spin wave. The resulting equations for thea ˜j can be written as

· ¸ 2 2 X2 |k⊥| i|C| − + i∂ a˜ (k ) = c eiζ/Rj eiζ/Ri c¯ a˜ (k + p − p ), (8.29) 4q ζ j X ω j i i X i j i=1 where we again drop the other function arguments ofα ˜j for convenience. If we define

· ¸ |k |2 −i ⊥ +|C|2c c¯ /ω ζ ˜ 4q j j Aj(ω, ζ, k⊥) =a ˜j(ω, ζ, k⊥)e , (8.30)

and let m = p1 − p2, then Eq. (8.29) for the signal field components reduces to

2 |C| c1c¯2 iζ[R−1−R−1−(m2−2k m)/q] ∂ A˜ (k ) = e 1 2 X ζ 1 X ω ˜ ×A2(kX + m), (8.31) 2 |C| c¯1c2 −iζ[R−1−R−1−(m2−2k m)/q] ∂ A˜ (k + m) = e 1 2 X ζ 2 X ω ˜ ×A1(kX ), (8.32)

These equations can be solved to give

· µ ¶ ζ p A˜ (ω, ζ, k ) = = e−iξζ/2 a˜ (ω, 0, k ) cos K2 + ξ2 1 ⊥ 1 ⊥ 2 ξa˜ (ω, 0, k ) − Ke−iφc a˜ (², 0, k − m, k ) + 1 ⊥ p 2 X Y K2 + ξ2 µ ¶ ¸ ζ p ×i sin K2 + ξ2 , (8.33) 2 · µ ¶ ζ p A˜ (ω, ζ, k − m, k ) = eiξζ/2 a˜0(ω, 0, k − m, k ) cos K2 + ξ2 2 X Y 2 X Y 2 ξa˜ (ω, 0, k − m, k ) + Keiφc a˜ (ω, 0, k ) + 2 X p Y 1 ⊥ K2 + ξ2 µ ¶ ¸ ζ p ×i sin K2 + ξ2 , (8.34) 2 8.5 Multimode memory equations 161

−1 −1 2 2 where ξ = R1 − R2 − (m − 2kX m)s/q, K = |C| |c1c¯2|/ω, and φc = arg(c1c¯2). Note ˜ that we have used thea ˜j’s as the initial conditions because Aj(ω, 0, k⊥) =a ˜j(ω, 0, k⊥).

From Eqs. (8.33) and (8.34) one can obtain the final transformed signal fields a˜j(ω, 1, kX , kY ), and from these calculate the signal and spin wave operators in real space. The Fourier transform back into real space must be done numerically. Let us ˜ first make the observation that, in the definition of the Aj(ω, ζ, k⊥), the exponential factor has the form as the solution obtained for the case of a single mode interaction ˜ [see Eq. (6.54)]. The Aj’s hence describe the mixing of the signal field components – if they do not change significantly then the interaction can be approximated by two uncoupled single mode processes. In Fig. 8.5(a)-(c) we see that these higher-order mixing processes do not affect the interaction for a coupling of C = 2, which is suffi- cient for high-efficiency storage and retrieval of the modes. When C reaches around 15 however then the mixing is significant, and there is evidence of higher-order processes. Furthermore in Fig. 8.5(d) it can be seen that for C ¿ 10 the read-in process for two signal components is essentially the sum of the two single-photon processes. Hence, the operation of the multimode memory follows from the fact that we have phasematched and optimized a single mode memory.

We conclude this section with a note on our derivation of the number of modes that can be stored in the multimode memory. The angle required for phasematching a signal field detuned by ∆ is θc, where

ωs ω1m − ∆ cos θc = = , ωc ω1m − ∆ + ω13 ω ω ⇒ 1 − θ2/2 = (1 + 13 )−1 ' 1 − 13 , c ω − ∆ ω − ∆ r 1m 1m 2ω13 ⇒ θc ' , (8.35) ω1m − ∆

where ω1m is the frequency of the | 1i ↔ | mi transition. The maximum allowed 162 Efficient spatially-resolved multimode quantum memory - methods

√ phasematching angle is given by θc ¿ A/L, as previously calculated, and the mini- mum allowed is limited by the maximum possible detuning, and the angular resolution achievable in experiment. From this, one can arrive at the condition for the number of modes given in chapter 7.

8.6 Conclusion

In this chapter we have provided additional information regarding the operation of the phasematched single mode quantum memory scheme introduced in chapter 7. We have also shown that one can phasematch using two read-in control fields instead of one, and that this leads to some interesting output signal pulse states. Whether one can modematch such a memory to give near-unit efficiency retrieval remains open. We then derive and solve the Maxwell-Bloch equations for the multimode memory scheme, also proposed in chapter 7. We give a detailed explanation of the limitations of the multimode memory, and demonstrate that for the coupling regime in which we are interested, the coupling between the different modes is negligible, thus ensuring the high-efficiency operation of such a memory.

Bibliography

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[2] J. Nunn, I. A. Walmsley, M. G. Raymer, K. Surmacz, F. C. Waldermann, Z. Wang,

and D. Jaksch, Phys. Rev. A 75, 011401 (2007). BIBLIOGRAPHY 163

[3] A. V. Gorshkov, A. Andr´e,M. Fleischhauer, A. S. Sørensen, and M. D. Lukin, Phys. Rev. Lett. 98, 123601 (2007).

[4] S. J. van Enk, Phys. Rev. A 72, 064306 (2005).

[5] M. G. Raymer, I. A. Walmsley, J. Mostowski, and B. Sobolewska, Phys. Rev. A, 32, 332 (1985). 164 Efficient spatially-resolved multimode quantum memory - methods Chapter 9

Publication

Quantum memory in an optical lattice

K. Surmacz, J. Nunn, F. C. Waldermann, K. C. Lee, Z. Wang, V. Lorenz, I. A. Walmsley

and D. Jaksch

Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, U.K.

To be submitted to Physical Review A

We investigate the possibility of using Lambda-type atoms trapped in an optical lattice as a storage medium for a quantum memory. We show that a periodic refractive index induced in the medium allows one to significantly reduce the group velocity of the signal pulse. This is achieved by tuning the lattice lasers so that the signal frequency is close to the band edge of the photonic crystal dispersion relation, and reflection of the pulse is avoided. This leads to an enhanced coupling of the signal field to the atoms, and hence a significantly higher memory efficiency than for an atomic ensemble with the same optical depth. 166 Quantum memory in an optical lattice

9.1 Introduction

The need for robust, long-lived storage units for photonic qubits in quantum commu- nication protocols [1, 2] that involve distributing entanglement [3, 4] has led to a great deal of recent research into quantum memories. It is well known that atomic ensembles are a promising candidate for such memories [5–8], as the enhanced coupling due to the large number of atoms means that less laser power is required to achieve the required coupling strengths than for a single atom. Indeed setups have been realized experimen- tally that store and retrieve pulses of light on demand, but the optical depths required to implement efficient memories are currently difficult to achieve experimentally. Also, such memories are subject to dephasing, for example due to atomic motion [9, 10] and collisions [11]. Once efficient memories for light pulses have been achieved, one would ideally want to generalize the light-matter interface to be able to manipulate the stored qubits. Whilst there have been proposals for ensemble-based multimode quantum memories capable of storing qubits [12–14], it is still not clear how qubit manipulation could be performed in such memories.

One possible approach to the above problems is to use Λ-type atoms trapped in an optical lattice as a storage unit. If the lattice is in a Mott-insulating phase, then the atomic motion will be almost completely eliminated, leading to greatly increased storage times. It has also been shown that the high degree of control one can exert over an optical lattice allows manipulation of stored qubits [15]. In this paper we study in detail the quantum memory interaction between light and an optical lattice, and reveal another beneficial feature of such a memory. We show that, when the memory operates in the electromagnetically-induced transparency (EIT) regime [6], the periodic refractive index induced in the lattice gives a photonic crystal-like structure [18]. If the wavevector of the signal field to be stored is close to the band edge of the crystal, then 9.1 Introduction 167

Figure 9.1. Scheme for a quantum memory using atoms trapped in an optical lattice. (a) Schematic experimental setup. A signal field is incident on one face of an optical lattice (site separation a), and is written into the atoms using a control field. (b) The level structure of the trapped atoms. The lasers used to create the lattice drive the same transition as the signal field.

the field propagates through the medium with a reduced group velocity. This results in an increased coupling of the signal to the medium, leading to an increase in the efficiency of the memory, compared to an atomic ensemble with the same optical depth.

We demonstrate that for an fixed optical depth that results in read-out probability of 80% with an ensemble, using a lattice in this way can give up to around 95%. Alternatively, our scheme reduces the optical depth required to achieve a given fidelity, thus alleviating the experimental requirements. Furthermore, the lattice lasers can be finely tuned so that reflection at the boundary of the medium is avoided [19].

The remainder of the paper is structured as follows. We first outline the model used to describe an optical lattice memory in section 9.2. In section 9.3 this model is then used to calculate the group velocity of the signal field in the medium. We also determine the reflectivity of the medium, and work out the maximum signal field bandwidth that can be used, such that group velocity and reflectivity do not change significantly across the width of the pulse. In section 9.4 we then use the memory equations to demonstrate that the reduction in group velocity enables an increase in 168 Quantum memory in an optical lattice the memory efficiency.

9.2 Model

We consider an ensemble of N Λ-type atoms trapped in an optical lattice. The lattice is generated by six lasers, propagating in the ±x, ±y, and ±z directions respectively, which drive the | 1i ↔ | mi transition, with detuning ∆latt, Rabi frequency Ωlatt and wavelength λlatt. The lattice is assumed to be in a Mott-insulating phase with one atom per lattice site, and the lattice lasers are oriented so that the lattice has a simple cubic structure, with lattice constant a = λlatt/2 [20–22]. The signal field to be stored

Es(r, t) propagates along the z-axis [as defined in Fig. 9.1(a)] with carrier frequency

ωs and excites the | 1i ↔ | mi transition. The storage is mediated by a strong classical control field Ec(r, t) with carrier frequency ωc, which drives the | 3i ↔ | mi transition. Both the signal and control fields are detuned from state | mi by ∆. The signal field has bandwidth δs (which corresponds to a pulse duration Ts), and to ensure that the control field does not interact with the lattice laser we assume that |∆ − ∆latt| > δs. The signal field propagates through the lattice according to Maxwell’s equation for the operators, which is µ ¶ 1 ∇2 − ∂2 E (r, t) = µ ∂2P (r, t), (9.1) c2 t s 0 t M where µ0 is the permeability of free space, c is the speed of light, and PM (r, t) is the macroscopic polarization of the lattice of atoms at position r = (x, y, z) and time t. The signal field and macroscopic polarization operators can be written as follows:

r X ωs ik·r Es(r, t) = ²sEs(r, t) = ²si ake , (9.2) 2²0V k 1 X p P = d σ(β) := d n(r)P , (9.3) M δV s m1 s β 9.2 Model 169

with ²s the signal field polarization vector, V the volume of the interacting region,

²0 the permittivity of free space, and ak the annihilation operator for the mode with wavevector k. The polarization PM is formed by summing up the coherences σm1 = | mi h1 | of the atoms (labelled β) located in a small volume δV centered at position r, with n(r) the number density of atoms and ds the signal field dipole matrix element (divided by ~).

We define slowly-varying operators for the signal field A(r, t), polarization P(r, t), control pulse Ω(r, t), and spin wave B(r, t), as

X √ ik·r ∗ iωst A(r, t) = ake φs(r)e / 2π, k ∗ iωst P(r, t) = P (r, t)φs(r)e ,

−i(kc·r−ωct) Ω(r, t) = Ec(r, t)e , X 1 (β) ∗ −ik·r i(ωs−ωc)t B(r, t) = p σ13 φs(r)e e , [δV n(r)] β

respectively, with φs(r) the quickly-varying signal field modefunction, which we will shortly define, and B(r, t) the slowly-varying spin wave associated with the atomic medium. The atomic equations of motion can be written in terms of these variables as

∗ ∂tB(r, t) = iΩ(r, t) P(r, t), (9.4)

∂tP(r, t) = −(γ + i∆)P(r, t) + iΩ(r, t)B(r, t) r πωs p +ids A(r, t) n(r), (9.5) ²0V where γ is the polarization dephasing rate due to e.g. spontaneous emission, and we have assumed that the number of atoms N in the lattice is sufficiently large so that the 170 Quantum memory in an optical lattice atoms remain in state | 1i with near-unit probability. We assume that the adiabatic condition holds, i.e. either dγ or ∆ are much larger than the signal bandwidth, the √ √ maximum control Rabi frequency, and γ, with d = g N/ γc the resonant optical depth [17]. Then P(r, t) can be adiabatically eliminated, and the resulting expression for P(r, t) can be substituted into Eq. (9.1). We see that if we define the modefunction

φs(r) to be the solution of

· µ √ ¶¸ ω2 2c2d2µ πn(r) ∇2 + s 1 + s 0 φ (r) = 0, (9.6) c2 Γ s then by using the slowly-varying envelope and paraxial approximations, and projecting onto the subspace of φs(r), Eq. (9.1) becomes

√ 2dsα ωsV (vs∂z + ∂t)A(r, t) = − √ Ω(r, t)B(r, t). (9.7) Γ ²0

The extra term in the coupling is given by

Z p 2 3 α = |φs(r)| n(r)d r, (9.8) cell

(which we note has dimensions of number density), where the integral is over the unit cell of the lattice. We discuss this parameter further in section 9.4. The signal group velocity vs = vszˆ (by symmetry) is

2 Z c ∗ 3 vs = φs(r)[−i∇φs(r)]d r, (9.9) ωs cell where zˆ is the unit vector in the z-direction. Substituting P(r, t) into Eq. (9.4), and 9.3 Group velocity reduction and reflection 171

transforming into a comoving frame τ = t − z/vs yields

√ 2 2gαV ∂zA(r, τ) = − Ω(r, τ)B(r, τ), (9.10) vsΓ · ¸ r i|Ω|2 π gα ∂ − B(r, τ) = i Ω∗(r, τ)A(r, τ), (9.11) τ Γ 2 Γ p where we have defined g = ds ωs/2~²0V . Let us note that in the case of an atomic vapour quantum memory, the modefunction φs(r) is a plane wave. Here however, the periodicity of the lattice structure can result in a mode different to a plane wave, which leads to a modification of the group velocity vs. We discuss this in the following section.

9.3 Group velocity reduction and reflection

The interaction of the signal field with the memory induces a refractive index in the atoms, which is represented by the modulation term containing n(r) in Eq. (9.6). Hence the lattice behaves as a photonic crystal [18] – a regular array of dielectric media. Depending on the size of the modulation term this can affect both the reflectivity of the medium, and the group velocity of the transmitted part of the signal field. Before we examine these effects in detail, let us first note that for a Raman quantum memory, where ∆ ∼ 1012s−1 dominates the spontaneous emission γ (∼ 10−8s−1), the modulation term is too small to affect either the reflectivity or the velocity vs. Then the mode φs(r) will be approximately equal to a plane wave, vs = c, and Eqs. (9.10) and (9.11) reduce to the Maxwell-Bloch equations for an ensemble-based memory [17, 24]. Thus a Raman memory in an optical lattice is in principle as efficient as in an atomic vapour, with the added benefits of reduced atomic motion, and potential for qubit manipulation [15].

For the remainder of this paper we consider the EIT regime. Setting ~ds ∼ 10−29Cm, n(r) ∼ 1020m−3, ∆ = 0 and γ ∼ 108s−1, we see that the modulation term 172 Quantum memory in an optical lattice

2 2 √ in Eq. (9.6) gives 2c dsµ0 πn(r)/Γ ∼ 0.1. This, as shown in Figs. 9.2(b)-(d), is suffi- ciently large to induce a reduction in vs when the lattice lasers are tuned so that the signal field frequency is near the band edge. Here vs is calculated by solving Eq. (9.6) for

φs(r) using the plane-wave method (described in [23]), and substituting into Eq. (9.9). The lattice is assumed to be deep enough so that the optical potential at a site can be approximated by a harmonic potential [21]. Then the ground state wavefunction of the atom in the z-direction is well approximated by

1 −z2/2a2 ψ(z) = √ e 0 , (9.12) rA pi

√ 1/2 2 with rA = (~/MωT ) the size of the ground state, and ωT = Ωlattωlatt/ 2Mc ∆latt the trapping frequency of the lattice. We also see that if the number density distribution of an atom changes [see Fig. 9.2(c)], which could be achieved by reducing the Rabi frequency of the lattice lasers, then the group velocity reduction can be made more, or less, pronounced.

It is in fact preferable experimentally to have the signal frequency near the band edge. If the lattice laser wavelength is λlatt, then a = λlatt/2. The band edge occurs at a wavevector with magnitude Kmax = π/a. In an experiment it would be easier to use the same atomic transition to generate the lattice as for the memory interaction, hence the wavelength of the signal field will be close to that of the lattice laser.

The reduction in vs for signal fields close to the band edge of the induced crystal illustrates that the mode φs(r) differs significantly from a plane wave, and hence that the interaction is different to that for the case of an optically-thick atomic gas. How- ever, it is possible that the periodic refractive index induced in the medium results in reflection of the signal field. To this end we calculate the reflectivity R of the medium.

We do so by assuming that the signal field is plane-wave in the x-y plane [as defined in 9.3 Group velocity reduction and reflection 173

Figure 9.2. A demonstration of the signal group velocity reduction that can be achieved using an optical lattice, and the corresponding reflectivity of the medium. (a) The magnitude of the signal field wavevector ks outside the medium plotted as a function of the signal wavevector magnitude K inside the lattice. This illustrates the dispersion relation of the medium. (b)-(d) Plots of the velocity vs/c (dashed) and the reflectivity R (solid) against K for different atomic radii rA and numbers of atomic layers in the signal propaga- tion direction Nz, with (b) Nz = 100, rA = 0.1a; (c) Nz = 200, rA = 0.1a; (d) Nz = 100, rA = 0.2a. For (a)-(d) the memory operates in the EIT regime, as described in the text. 174 Quantum memory in an optical lattice

Fig. 9.1(b)], and treating the lattice as a one-dimensional periodic stratified medium, with rA the radius of the atoms. Defining kA = ωsnA/c, with nA the refractive index of the medium, the reflectivity of the medium can be calculated [19, 25] to be

|C|2 R = , (9.13) 2 2 2 |C| + sin (Ka)/ sin (NzKa) where K = cos−1[(A + D)/2]/a,

½ µ ¶ i ks kA A = e−2ikAR cos[k (a − 2r )] − − s A 2 k k ¾ A s

× sin[ks(a − 2r)] , (9.14) ½ µ ¶ ¾ i k k −i2kAR s A C = e − sin[ks(a − 2rA)] , (9.15) 2 kA ks

∗ D = A , and Nz is the number of atomic layers in the signal propagation direction

(Nz = Lz/a, with Lz the length of the lattice in the z-direction). The parameters A, C and D are elements of the unit cell translation matrix, which relates the am- plitudes of the incident and reflected plane wave in one layer of a unit cell to those amplitudes in the next unit cell (for more details see [19]). The wavenumber K rep- resents the magnitude of the wavevector of the signal field inside the medium, given that the wavevector outside the medium has magnitude ks. The relationship between

K and ks hence illustrates the dispersion relation of the medium, and this is plotted in Fig. 9.2(a). Note here that the calculation of R assumes ‘square’ atoms, rather than the Gaussian distribution used for the group velocity calculation. However, our calcu- lations show that this does not significantly affect either vs or R (see also [26]), and so this approximation is justified. One sees that in order to calculate R, the refractive index of the medium is required. To derive this, we begin with Eq. (9.1), and make 9.3 Group velocity reduction and reflection 175 the paraxial approximation for the signal field as before. Rearranging Eq. (9.4) gives

i P(r, t) = − ∂ B(r, t), (9.16) Ω∗(r, t) t and substituting into Eq. (9.5), and keeping only the lowest-order terms in time, gives

r d πω p B(r, t) = − s s n(r)A(r, t). (9.17) Ω(r, t) ²0V

Using Eqs. (9.6), (9.16), and (9.17) in the simplified Maxwell equation results in a modified equation for the propagation of the signal field through the medium

· µ √ 2 ¶ √ 2 ¸ πdsα2ωs i πdsα2ωs vs∂z + ∂t 1 + 2 + A(r, t) = 0, (9.18) |Ω| ²0 Γ²0

R 2 3 where α2 = cell |φs(r)| n(r)d r. Note that there are two terms in Eq. (9.18) that modify the group velocity of the signal field. The term containing Γ is due to the periodicity of the lattice, and originates from the term in Eq. (9.6) that changes the signal mode function. The term dependent on Ω meanwhile is due to the EIT effect of the interaction, and for the purposes of the memory evolution this is contained in Eqs. (9.7), (9.10) and (9.11).

In Figs. 9.2(b)-(d), along with the velocity vs, the reflectivity R is plotted for different K. First note that at the band edge there is a forbidden region where the reflectivity approaches 1. However, whilst the overall shape of the curve for R follows the trend of 1 − vs, the reflectivity oscillates so that there are nodes where R = 0.

In fact, the number of these nodes is equal to Nz − 1 [19]. This is illustrated by

Fig. 9.2(b), where Nz is larger than in Fig. 9.2(a), and hence has a more quickly- oscillating reflectivity. If the signal field wavevector is near the band edge, and is chosen so that it sits at one of these nodes, then one can achieve a reduction in vs 176 Quantum memory in an optical lattice without the pulse being reflected. The troughs in R at higher values of K are narrower in width, and so there is a trade-off between the reduction in vs and the maximum bandwidth of the signal field that does not result in reflection. A further limitation on the reduction of vs comes from the fact that vs should be constant across the width of the pulse. In Fig. 9.3(a) we demonstrate the minimum vs that can be achieved for different signal field bandwidths. The jumps in the curves correspond to the fact that when the signal pulse becomes too broadband, either R or vs deviate significantly across the width of the pulse, and hence the pulse must be positioned in the next- nearest trough to the band edge. We see that one can achieve vs . 0.5c if the signal pulse duration is around 1ns, and for more broadband photons the reduction will be less. Also, as Nz is decreased, then the trough in reflectivity closest to the band edge occurs at higher vs, but is actually wider. Hence, for example, if one requires the lowest possible value of vs given a signal of duration 10ps, then one would choose Nz = 100 as opposed to the other values shown. Figure 9.3(b) shows the minimum group velocity that can be achieved for a range of values of Nz, and the corresponding duration of the shortest pulse that can be transmitted without dispersion. This illustrates the trade-off between bandwidth and velocity reduction. Finally let us note that, whilst R has been calculated for a finite lattice, the plane-wave calculations used to obtain vs assumed a lattice of infinite extent. Since the oscillations in R are due to finite-size effects, one may also ask whether similar behaviour would be observed for the group velocity in a finite-size calculation. However, the gradient of the dispersion relation (and hence vs) plotted in Fig. 9.2(a) is monotonically decreasing, in agreement with our earlier calculations of vs. 9.4 Memory storage 177

Figure 9.3. (a) A plot of the minimum value of vs that can be achieved for a given signal pulse duration Ts without reflection and dispersion, with Nz = 100 (solid), 250 (dashed), 350 (dotted), and 450 (dot-dashed). (b) The minimum group velocity that can be achieved without reflection and the corresponding minimum pulse duration possible for transmission without dispersion. In both (a) and (b) the conditions for transmission and dispersionless propagation are that over the FWHM of the signal pulse, the variation in R and vs must be < 1% and < 2%, respectively.

9.4 Memory storage

Having analyzed the group velocity and reflectivity properties of the signal pulse, we now show that tuning the signal close to the band edge can lead to more efficient operation of the quantum memory. Let us return to Eqs. (9.10) and (9.11). The fact that the modefunction φs(r) is not necessarily a plane wave results in the appearance of the parameter α in the overall light-atom coupling. In Fig. 9.4(a) we plot α over the first Brillouin zone, and see that the coupling of the interaction is enhanced as the signal wavevector K approaches the band edge, in addition to the accompanying decrease in vs.

However, when vs is significantly less than the speed of light, we have to take into account the fact that there will be a walk-off between the signal and control pulses. When the signal and control field group velocities are approximately equal, Eqs. (9.10) and (9.11) can be solved by making a coordinate transformation to eliminate the control pulse envelope from the equations [24]. However, even when the fields have different 178 Quantum memory in an optical lattice

Figure 9.4. The improvement in performance of the memory as a result of using an optical lattice. (a) The enhancement α of the coupling plotted as a function of the signal field wavenumber inside the medium K, where α0 is the value of α for an atomic ensemble with the same number density of atoms. (b) The probability F of storing the signal pulse in the memory as a function of K relative to the band edge. Recall that the maximum possible value of K that does not result in reflection depends on Nz. In both figures the atom width was taken to be rA = 0.1a. In (b) the optical depth is d ' 5. group velocities, Eqs. (9.10) and (9.11) can be solved numerically [27]. We optimize the read-in process in this way for different values of vs, keeping the resonant optical √ √ depth d = g N/ γc constant. The results of this are illustrated in Fig. 9.4(b). We see that as the signal field approaches the band edge (decreasing vs), the optimal read-in efficiency increases significantly – the lattice memory reaches an efficiency of 0.95 using our method, where an ensemble with the same optical depth would have an efficiency of 0.8. Let us stress once again that for a given setup one cannot use any value of K due to reflection, but the possible values, and the corresponding bandwidths and velocities, can be determined using the earlier calculations, and Fig. 9.4(b) shows the resulting benefit.

Having solved for the read-in process, the read-out interaction can be easily solved since it is the time-reverse of the storage [17]. Furthermore, using angled control fields to phasematch the read-in ensures that the retrieval is in principle as efficient as the storage [12]. Furthermore, the memory efficiency will not be degraded by motional 9.5 Conclusion 179 dephasing or collisional broadening, since the atoms are trapped in the lattice. Hence the high-fidelity read-out follows from the efficient storage.

9.5 Conclusion

In summary, we have shown that using atoms trapped in an optical lattice as a storage device in a quantum memory can result in a significantly higher memory efficiency than for an atomic ensemble with the same optical depth. The control pulse induces a periodic refractive index in the medium, and by tuning the lattice appropriately one can reduce the group velocity vs of the signal without causing reflection. This, along with the enhancement of the parameter α, results in a larger coupling between the signal field and the atoms, and despite the walk-off between the signal and control fields, this gives the improvement in efficiency. The position of the signal field wavevector relative to the band edge can be controlled by the wavelength of the lattice lasers, and so the regions of zero reflection can be accurately found. In addition, if the memory was part of some larger quantum communication scheme, the forbidden region at the band edge could be used to reflect the signal pulse completely if one wanted to prevent the memory interaction from taking place.

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Conclusions

We conclude this thesis with a summary of our findings, along with a discussion of the potential further directions of the field of quantum memories.

After introducing the basic idea of a quantum memory, and its potential applica- tions, we calculated the entanglement fidelity of a general quantum memory [1]. This figure of merit captures the essential feature of a quantum memory necessary for many quantum communication and quantum information processing schemes, namely how well the memory preserves entanglement. We modeled a general quantum memory in terms of stochastic parameters, whose fluctuations lead to degradation of the fidelity, and an expression for the entanglement fidelity was derived in terms of the means and variances of these parameters. A possible experimental setup to measure this entangle- ment fidelity was put forward. We then introduced the most relevant atomic ensemble quantum memory schemes and gave an expression for the entanglement fidelity of each of them.

We then went on to analyze the storage of a quantum signal field in a Raman quantum memory consisting of an atomic ensemble [2]. We showed that in the Raman 184 Conclusions limit the interaction can be decomposed into a set of beamsplitter relations. We used this result to find the shape of the control field that optimally stores a given signal field pulseshape. This optimization illustrated that the signal field can be stored with high efficiency, even with a moderate coupling strength, and that this coupling strength can be characterized in terms of a single parameter. These results are qualitatively similar, but at first glance quantitatively different, to those obtained by other workers in the field [3], and we resolved the apparent conflict between the two sets of findings, showing that they are due to considering different parameter regimes.

Having optimized the read-in process for a Raman memory, we then analyze the read-out interaction. We proposed a storage and retrieval protocol [4] that enables efficient read-out of the signal photon from Raman, EIT, and CRIB memories. Our method solves the problem encountered in previous versions of these memories, where for non-degenerate initial and storage states, efficient read-out could not be performed without resorting to very high coupling. This scheme should give significant experimen- tal benefit, since it is desirable to have non-degenerate atomic levels for the purposes of addressing the individual memory transitions.

Our solution for the memory read-out problem can be used to implement a mul- timode quantum memory, by which different frequency modes of a signal pulse are stored in transverse modes of the atomic ensemble [4]. We showed that this multimode memory could be realized, storing on the order of 100 modes for sensible coupling pa- rameters. A functioning multimode memory would allow one to store multiple qubits in a single ensemble, thus reducing the resources required to implement a quantum repeater, for example. Multimode memories also open up the possibility of performing single- and two-qubit gates on the stored qubits.

In the final part of this thesis we considered the possibility of implementing a quantum memory using atoms trapped in an optical lattice [5]. Optical lattices are good 185 candidates for a storage medium because of the lack of atomic motion and collisions, and also because storage in an optical lattice makes the manipulation of a stored qubit a distinct possibility [6]. We revealed a new feature of such a memory, namely that the periodic refractive index induced by the control field leads to a reduction in the group velocity of the signal pulse through the medium, and hence an increase in the coupling strength. This allows one to achieve a significantly higher memory efficiency for the same optical depth as in an atomic ensemble memory.

The duration of this research has seen numerous quantum memory schemes compre- hensively studied theoretically. In, particular it has been shown what can be achieved using the different schemes, and how one would go about obtaining the optimal memory efficiency. However, it remains to be shown experimentally that a quantum memory can be implemented with the kind of fidelity (∼ 90%) required, for example, to im- plement a quantum repeater. The original protocols for repeaters [10, 11], and other quantum communication schemes, usually assumed that perfect, or at least very high- fidelity memories are available, and there has been recent pessimistic speculation that building memories with sufficiently high efficiency is going to be difficult [12]. This observation that a quantum memory is somehow ‘expensive’ has led people to consid- erations of how one can minimize the ‘cost’ of a given setup, or what one can achieve given a certain amount of resources. As discussed earlier in this thesis, the recent papers on multimode memories [4, 13] address this problem by showing that multiple qubits can be stored in a single memory. However, other works have focussed away from the memories themselves, and instead on how repeater strategies can mitigate for imperfect memories. One scheme used multiplexing of memory resources to make the repeater robust against memory loss [14]. Another examined the entanglement swapping process in a repeater, and showed that using a two-dimensional lattice, it is theoretically possible to create a Bell state with unit probability from a finite num- 186 Conclusions ber of imperfectly entangled qubits [15]. These kinds of ideas could greatly reduce the memory performance required for scalable quantum communication, and it is this direction that the theoretical side of the field seems to be taking.

So one way forward in this field is to design quantum communication and quantum information processing schemes where the required memory fidelity is lower, and where fewer storage units are required. However, as we saw in our optical lattice study, there is still scope for proposals to improve on the memory fidelity given a finite optical depth. Such advances could also result in an experiment where the memory fidelity is sufficiently high for use in quantum networks. In terms of realizing quantum memories for photons however, the main challenges now are experimental. Indeed, various dif- ferent groups have demonstrated working quantum memories [7–9], however as stated before a high efficiency has thus far been elusive. The goal of storing broadband pulses

(. 10−12s in duration) also remains. One of the other main areas of future research will be the continuing quest to implement a solid-state quantum memory. It will be interesting to see whether such a memory fulfils the promise that the initial proposals offered.

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