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A Theoretical Study of Quantum Memories in Ensemble-Based Media

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A Theoretical Study of Quantum Memories in Ensemble-Based Media 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 ¤-type level structure, as well as other control ¯elds that a®ect the transfer of the quantum signal ¯eld to a material storage state. Ensembles have the advantage that the coupling of the signal ¯eld 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 ¯elds, and propose a ¯gure of merit that measures how well such a memory preserves entanglement. We derive an analytical expression for the entanglement ¯delity in terms of fluctuations in the stochastic Hamiltonian parameters, and show how this ¯gure could be measured experimentally. We then examine the operation of an o®-resonant Raman quantum memory, in which a single photon and a control ¯eld propagate co-linearly and interact with an ensemble of ¤-atoms. We show that, in the Raman limit, a universal mode decom- position allows one to ¯nd the conditions, in particular the control pulseshape, under which storage of a given signal photon is optimal. Furthermore, the storage e±ciency is speci¯ed by a single parameter. Reading out the stored excitation is ine±cient unless the coupling for read-out is prohibitively high. We propose a scheme that solves this read-out problem, allowing e±cient 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 ¯eld from the control ¯elds. We go on to show that the above proposal for e±cient 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 di®erent components of the signal are spatially separated from each other, and from the control ¯elds. 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 ¯eld group velocity due the periodic refractive index structure induced. This leads to an enhanced coupling, and hence a higher memory e±ciency 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 ¤-type absorbers . 19 2.4 Figures of merit for a quantum memory . 22 2.4.1 Input-output ¯delity . 22 2.4.2 Entanglement ¯delity . 24 Chapter 3. Publication: Entanglement ¯delity of quantum memories 29 Chapter 4. Entanglement ¯delity of quantum memories - methods 43 4.1 Model . 43 4.2 Solution . 47 4.2.1 Ensemble-photon interaction . 48 4.2.2 Entanglement ¯delity . 51 4.3 Experimental setup . 53 4.4 Entanglement ¯delity 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 arti¯cially . 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: E±cient spatially-resolved multimode quantum memory 131 Contents v Chapter 8. E±cient 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 ¯elds . 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 signi¯cant 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 e®orts 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 ¯bre. 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 ¯bre 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 su±ciently 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 puri¯cation [9, 10]. The result is that Alice and Bob share the entangled state over a large distance, with in principle an arbitrarily high ¯delity (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.
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