WO2004106961A1

WO2004106961A1 - Method and apparatus for increasing the coherence time of an atomic spin system

Info

Publication number: WO2004106961A1
Application number: PCT/AU2004/000741
Authority: WO
Inventors: Elliot Fraval; Matthew Sellars; Jevon Longdell
Original assignee: The Australian National University
Priority date: 2003-06-03
Filing date: 2004-06-03
Publication date: 2004-12-09
Also published as: AU2003902762A0

Abstract

A method and apparatus for increasing the coherence time of a transition in a spin system with an angular momentum >1. Under the influence of a fluctuating local magnetic field in a solid state host. An external magnetic field is selectively applied to the spin system, to cause the Zeeman shift of the spin transition to be substantially at a critical point in all three spatial dimensions.

Description

METHOD AND APPARATUS FOR INCREASING THE COHERENCE TIME OF AN ATOMIC SPIN SYSTEM

Technical Field

This invention relates to quantum computing, and more particularly increasing the coherence time of transitions in solid-state hosts for quantum information processing. Although the invention will primarily be described in relation to increasing the coherence times of nuclear spin transitions it is also applicable to increasing coherence times of electron spin transitions and transitions involving spin states in different electronic states. Throughout this specification the term "coherence time" is the lifetime of a superposition state. Throughout this specification the term "local magnetic field" will refer to the magnetic field at the location of the spin transition of interest for coherence time extension. The specification also contains references to a "small magnetic field". In the context of this invention the magnitude of a magnetic field is considered small when applied in any direction the Zeeman shift has a smaller second order contribution than the first order contribution due to inhomogeneity of the Zeeman and Quadrupole interactions at FWHM of the distribution.

Background of the Invention

To date the most sophisticated demonstration of quantum logic operations have been performed using liquid state nuclear magnetic resonance techniques, with demonstrations involving up to seven qubits. It is unlikely though that liquid state NMR will provide a direct path to the development of a scalable quantum computer. Liquid state NMR employs an ensemble approach where the individual quantum systems are molecules in a solvent and the qubits are nuclear spins on the molecule. As it is required to operate at or near room temperature it necessary to distill a pseudo pure state from the thermal state of the ensemble. The size of this distilled ensemble drops drastically with the number of qubits. Interaction between qubits is achieved through the molecular bonds and as such is not tuneable and decreases in proportion to the distance between the qubits. This poses a significant problem for scalability since adding more qubits to approach the desired complexity for practical quantum computing slows the computation rate drastically. Ideally quantum computers would operate at cryogenic temperatures so that the quantum system relaxes toward a single state given the available thermal energy. This is clearly not possible for liquid phase NMR approaches.

A major limitation to using solid state systems is the requirement that the spin states have long coherence times. Long nuclear spin coherence times are challenging to achieve in a solid state system due to interactions with other spins within the environment. Spin free hosts can be utilized, however not all species of interest are chemically compatible with such hosts. Therefore solid state candidate quantum computing schemes can suffer from dephasing due to fluctuating local magnetic fields caused by mutual nuclear spin flips of the host nuclei. Eliminating this problem in a generalised manner would provide a significant step toward viable solid state quantum computing.

Disclosure of Invention

Accordingly, in one aspect this invention provides a method of increasing the coherence time of a transition in a spin system with an angular momentum >1, under the influence of a fluctuating local magnetic field in a solid state host including the step of selectively applying an external magnetic field to the spin system, to cause the Zeeman shift of the spin transition to be at a critical point in all three spatial dimensions.

As a result of the applied external magnetic field small fluctuations in local magnetic field result in a vanishingly small frequency shift of the spin transition.

In another aspect this invention provides an apparatus for increasing the coherence time of a transition in a spin system with an angular momentum >1, under the influence of a fluctuating local magnetic field in a solid state host by the application of an external magnetic field, said apparatus including a selectively operable magnetic field generator to apply an external magnetic field to the system and produces a critical point in the Zeeman shift of the spin transition in all three spatial dimensions.

The preferred critical point is defined when the external magnetic field is accurate in all dimensions to better than the sum of the host magnetic field fluctuations and better than the distribution of critical point magnetic fields due to inhomogeneous broadening of the ensemble.

Close to the critical point is defined as when the applied magnetic field produces Zeeman shifts comparable to the zero magnetic field hyperfine splittings and the magnetic field is within the basin of the critical point minimum (attractor). The basin of the critical point minimum (attractor) is defined as the region where any trajectory defined by negative of the gradient field (due to being a minimum) of the geometric mean of first order frequency shift due to magnetic field perturbations terminates at the critical point. The termination point is where the sign of the derivative must change to leave the point.

At the critical point the magnetic dipole moments of both spin states involved in the transition are equal for small magnetic field fluctuations. As a result small magnetic field fluctuations do not result in a significant change of transition frequency and therefore do not dephase the transition.

The present invention has particular application to solid state qubits in a host with nonzero spin. In such systems the fluctuations in local magnetic field due to spin flips within the host are often a major source of dephasing for the spin states of interest.

In one form the method can be applied to a spin system that forms part of a solid state nuclear magnetic resonance quantum computer (NMRQC). Preferably, the NMRQC is an optically detected NMRQC. In another form the method can be applied to a quantum system that forms part of a quantum memory, including but not limited to holographic based memory or Electromagnetically Induced Transparency (EIT) or "stopped light" memories. The transition in the quantum system can be between spin levels within an electronic state or between spin states in different electronic states. In one form of the invention the spin states are nuclear spin states. In another form of the invention the spin states are electron spin states. In another form of the invention the spin states involve both nuclear and electron spin states.

The external magnetic field is preferably supplied by three individually adjustable orthogonal electromagnets allowing application to the sample incorporating the spin system of the desired magnetic field in all three spatial dimensions.

It is further preferred that the external field adjusted firstly by reference to theoretical critical point simulations and secondly by an iterative adjustment of magnetic field strength and direction.

In a preferred form of the invention state manipulation and/or coherence excitation is accomplished using optical and/or RF pulses. In some forms of the invention state manipulation and/or coherence excitation can involve the use of a state other than the states associated with the transition at the critical point as an intermediary. For example, Raman process can be used to excite and manipulate transition of interest.

In one form of the invention the quantum system involves a hyperfine ground state spin transition of praseodymium ions in Pr³⁺:Y₂SiO₅.

In another form of the invention the quantum system involves a hyperfine ground state spin transition of europium ions in Eu³⁺:Y₂SiO₅.

In another form of the invention the quantum system involves an optical transition (m!=l 3/2<->l 5/2) of erbium ions in Er³⁺: Y₂SiO₅.

The present invention provides the advantage of decoupling the transition of interest from host spin flips, thereby substantially eliminating the dephasing effect of the host spin flip- flops. This significantly increases the coherence time so that in its application to quantum computing or quantum information processing the invention provides the following advantages:

significantly more gate operations can be performed before the system must be reinitialised

the fidelity of quantum gate operations is increased

the fidelity of quantum memory read or write operations is increased

- the storage time of quantum memory is increased the fidelity of quantum information transfer from optical or RF radiation to a spin state is increased.

Furthermore the technique can also be incorporated with other existing strategies for coherence time extension. Liquid phase NMR techniques including but not limited to Phase Cycling or magic angle line narrowing are compatible with this invention. Spin polarisation techniques and schemes that distribute a single spin state across an ensemble such as Shor's patent (5,768,297) are also able to be incorporated with this invention.

One embodiment of the invention will now be described with reference to the accompanying drawings. Although the invention will be explained in its application to increasing the coherence time of a ground state transition of praseodymium ions in Pr³⁺:Y₂SiO₅ it will be apparent that the invention is equally applicable to other spin systems.

Brief Description of the Drawings:

Figure 1 is a schematic drawing of an experimental arrangement used to demonstrate the invention.

Figure 2 schematically shows the timing and duration of two and three pulse echo systems relative to pumping laser operation used to measure coherence time and spectral in an embodiment of the invention;

Figure 3 is a graph showing the spectrum of hyperfine transition in ground states of praseodymium ions in Pr³⁺:Y₂SiO₅ showing experimentally measured points as external magnetic field is increased;

Figure 4 is a graph of two pulse echo sequences taken for three different transitions of praseodymium in Pr³⁺:Y₂SiO₅ and a reference 500μs decay line;

Best Mode for Carrying out the Invention

This invention will be further explained using the nuclear spin states of Pr³⁺:Y₂SiO₅ using Raman Heterodyne two and three pulse nuclear spin echos to investigate the effect on the coherence time, T₂.

There is currently great interest in using nuclear spin states in a solid state host to store and manipulate quantum information. Applications such as quantum computing and stopped light are being investigated. These applications require the spin states to have long coherence times. Long nuclear spin coherence times are challenging to achieve in a solid state system due to interactions with other spins within the environment. Spin free hosts can be utilized, however not all species of interest are chemically compatible with such hosts.

A limitation to Pr³⁺:Y₂SiO₅ being a viable quantum computing medium is the coherence time (T₂), stated in the literature as 500μs. Given that the hyperfine transition frequency is -10MHz, Rabi frequencies will need to be limited to <lMHz for the driving field to be transition specific. Therefore, only -1000 operations can be completed within T₂ which is considered to be insufficient for practical quantum computing.

The dephasing of the praseodymium hyperfine ground states in Y₂SiO₅ is dominated by their interactions with yttrium nuclear spins. The yttrium ions are undergoing mutual spin- flip interactions, causing the magnetic field at the praseodymium site to fluctuate. This fluctuating magnetic field randomly Zeeman shifts the praseodymium hyperfine levels.

To increase T₂ it is therefore necessary to decouple the praseodymium transition from the yttrium. Magic angle spinning is the standard solid state NMR technique to achieve this decoupling. Sample spinning is likely to be problematic for most quantum computing architectures when a specific, localised ensemble needs to be addressed. As such a static technique is more desirable. According to the present invention a magnetic field is applied such that the transition of interest is no longer susceptible to small magnetic field fluctuations.

The hyperfine ground state interactions for Praseodymium dopant ions in Y₂SiO₅ are described by the following Hamiltonian:

H = B »M »I + I » Q»I eq(l) where B is the magnetic field vector, I is the vector of nuclear spin operators, M is the effective Zeeman tensor combining nuclear and electronic Zeeman interactions and Q is the effective quadrupole tensor combining the quadrupole and second order magnetic hyperfine interaction, known as the pseudo-quadrupole. M and Q have been determined for Pr³⁺:Y₂SiO₅ to be:

where Ε=0.5624Mhz and D=4.4450Mhz, (g_x,g_y,g_z)=(2.86,3.05,11.56)kHz/G, and the Euler angles are (αβγ)=(-99.7,55.7,-40). These values are for the crystal aligned with the C₂ axis in the y direction, and the z axis is the direction of linear polarization of the praseodymium optical transitions. These tensors are highly anisotropic due to the low symmetry of the site.

Using these parameters a number of values for the magnetic field magnitude and direction were found with vanishingly small first order Zeeman shift. This is due to the magnetic dipoles of both states involved in the transition being identical for small field fluctuations. The situation investigated was at a magnetic field of

on the mι=+l/2<H>+3/2 transition as it had the smallest second order Zeeman shift. Around this magnetic field value the transition energy as a function of magnetic field has a turning point in the y and z axis while the x axis has a slow inflection point. For an ion with the magnetic field perfectly at the critical point the geometric mean second order sensitivity is 124.13Hz/G² giving an estimated line width of 11 Hz due to a fluctuating magnetic field produced by the yttrium of 0.3G.

A second line width contribution is due to the inhomogeneity of the pseudo-quadrupole and Zeeman tensors. The inhomogeneity results in the critical point field being different for each praseodymium ion, causing a non zero first order sensitivity to magnetic field fluctuations. Inhomogeneity of the pseudo-quadrupole and Zeeman tensors can be estimated from comparing the frequency of the transition to the inhomogeneous width of the transition (140kHz FWHM). This results in an ion at HWHM having a first order transition sensitivity of 6.5Hz/G which contributes ~2Hz to linewidth due to yttrium spin flips. Therefore the second order Zeeman shift is the dominant linewidth contribution for magnetic field fluctuations produced by yttrium spin flips.

Raman heterodyne two pulse spin echos were used to measure the T₂ of the system.

Figure 1 schematically shows the experimental arrangement used to demonstrate the invention. A cryostat 3 contains a sample 4 positioned at the centre of three pairs of superconducting magnetics 6a, 6b; 7a, 7b; 8a, 8b. The axis of the pair 6a, 6b is designated the x axis. The axis of the pair 7a, 7b is designated the y axis and the axis of the pair 8a, 8b is designated the z axis. The axes are arranged to be mutually orthogonal so that the magnetic field can be controlled in all three dimensions by control of the currents in respective pairs of magnets using standard controllers 6c, 7c, 8c. RF coils 5 are positioned each side of sample 4 to supply the R.F. pulse sequence shown in Figure 1. The pulse sequence is generated using a further generator 13 and RF switch 11 under the control of a tracking generator 14 contained in a signal processing and R.F. control module 10. The output of RF switch 12 is supplied to coils 5 via an amplifier 12.

A laser 1 provides an output that is directed through the sample 4 to a detector 9. The laser output is gated by an acoustic optical modulator (AOM) 2. AOM 2 is controlled by output of an R.F. switch 11 through and amplifier 12. The R.F. switch 11 is driven by an R.F. oscillator 18 and pulse generator 13.

The output of detector 9 is provided via a spectrum analyser 15 to a cathode ray osilliscope 16 and compactor 17.

The experiment was performed using a Coherent 699 frequency stabilized (IMHz FWHM) tunable dye laser tuned to the ³H -^^!D₂ transition at 605.977nm. The laser was gated using a lOOMHz AOM such that there was no laser radiation applied to the sample during the RF pulse sequence shown in Figure 2. The laser power incident on the crystal was 40mW, focused to ~100μm. The Pr³⁺:Y₂SiO₅ (0.05% concentration) crystal was held at -1.2K for the duration of the experiment. The laser prepared a population difference in the excited ions for 5 s before the pulse sequence and was scanned over 1.2GHz of the optical transition to avoid hole burning effects. The laser is off during the RF pulse sequence to minimize coherence loss from optical pumping.

The magnetic fields were supplied by two superconducting magnets along the x and z axis. The sample rod was then rotated by 13+0.5° to provide the correct ratio of fields along the x and y axes for the critical point in magnetic field space. Course adjustment of the field was done by comparing the Raman heterodyne spectrum to theoretical simulations as shown in Figure 3. Fine adjustment of the magnetic field utilised perturbing coils along each axis and associated lock-in amplifiers to perform field sensitivity measurements. The rotation of the sample rod and magnetic fields were iteratively adjusted to minimize the sensitivity of the desired transition. Final adjustments of the field values were made by optimizing the length of the echo decay. The magnetic field values were accurate to within ±2G of the critical point. In Y₂SiO₅ there are four possible sites for the praseodymium to occupy, grouped into crystallographically identical pairs termed site 1 and site 2. All of the spectra are taken using site 1 ions, however since the y axis field is created by rotating the sample rod one of the pair will experience the opposite y axis field to the other. The ions experiencing the critical point external field will be referred to as site la while those with the y axis field opposite will be referred to as site lb.

The two pulse echo data of Figure 4 shows three echo sequences at the critical point magnetic field configuration with the zero field echo sequence for reference. The transition 1=4-1/2 <→+3/2 at site la is shown by (+); the I=+l/2<→+3/2 at site lb is shown by (x); the I=+l/2 <→+3/2 transition is shown by (o). Since the field configuration is transition specific the mι=+l/2<→-+3/2 transition is at a critical point while the two other transitions are not. The normal exponential dependence of the echo decay is shown by the transitions not at a critical point. The transition at the critical point clearly has a nonexponential echo decay and continued for an order of magnitude longer than echo series taken on other transitions. When a spin echo series has a time dependence of the form given in equation, it is generally no longer limited by dephasing of the coherence but rather diffusion of the coherence. In this situation the phase memory, T_M, not the decoherence time is the defining characteristic of the system. The phase memory was calculated as 82.1ms using equation (3), corresponding to a linewidth of 12.2Hz.

The two pulse echo results taken on the transition at the critical point show that the inhomogeneity of the ensemble was not large enough to have a significant impact on the echo intensity. Large inhomogeneity would cause a portion of the ions involved in the echo to quickly decay, leaving a long lived tail of the ions at the critical point. To fit to an echo series with large inhomogeneity exponential terms with linear and quadratic time dependence would be required. The quality of the fit with only quadratic time dependence allows us to conclude that diffusion is the dominant loss of coherence in the system. Earlier estimates of second order linewidth contribution were conservative given that they assumed any configuration of yttrium spins was possible. Since the time scale of the echo is shorter than the reconfiguration time of the yttrium ions all configurations of the yttrium spins will not be present, thereby reducing the contribution to linewidth.

For echo sequences taken on other transitions with the same field configuration (shown in Figure 4) the mι=+3/2«-*-3/2 transition T₂ was 5.86ms while for the mι=+]J2<→+3/2 transition at site lb it was 9.98ms. The increase in the dephasing time for transitions not at the critical point compared to the zero magnetic field case is due the external field defining the quantisation axis such that the magnetic field change of a yttrium spin flip can be considered a small perturbation on the praseodymium.

The foregoing demonstrates that a phase memory time of 82.1ms can be achieved for Pr³⁺:Y₂SiO₅ by using an external magnetic field to minimize the transition sensitivity to magnetic field fluctuations. It has also been shown that the residual dephasing mechanism is due to an spectral diffusion mechanism that is based on electric field interactions, not magnetic field fluctuations.

The foregoing describes only one embodiment of the invention and modifications can be made without departing from the scope of the invention.

Claims

1. A method of increasing the coherence time of a transition in a spin system with an angular momentum >1, under the influence of a fluctuating local magnetic field in a solid state host including the step of selectively applying an external magnetic field to the spin system, to cause the Zeeman shift of the spin transition to be substantially at a critical point in all three spatial dimensions.

2. A method as claimed in claim 1 wherein said spin system includes solid state qubits in a host with non zero spin.

3. A method as claimed in claim 1 or 2 wherein the spin system forms part of a solid state nuclear magnetic resonance quantum computer.

4. A method as claimed in claim 3 wherein the computer is an optically detected nuclear magnetic resonance quantum computer.

5. A method as claimed in claim 1 wherein the spin system forms part of a quantum memory.

6. A method as claimed in claim 5 wherein said memory is holographic based memory.

7. A method as claimed in claim 5 wherein said memory is electromagnetically induced transparency memory.

8. A method as claimed in claim 5 wherein said memory is stopped light based memory.

9. A method as claimed in any one of claims 1 to 8 wherein the spin system includes spin levels within an electronic state.

10. A method as claimed in any one of claims 1 to 8 wherein the spin system includes spin levels between spin states in different electronic states.

11. A method as claimed in any one of claims 1 to 8 wherein the spin states are nuclear spin states.

12. A method as claimed in any one of claims 1 to 8 wherein the spin states are electron spin states.

13. A method as claimed in any one of claims 1 to 8 wherein the spin states include both nuclear and electron spin states.

14. A method as claimed in any one of claims 1 to 13 wherein the external magnetic field is applied by three individually adjustable orthogonally oriented electromagnets.

15. A method as claimed in claim 1 wherein the spin system involves a hyperfine ground state spin transition of praseodymium ions in Pr³⁺:Y₂SiO₅.

16. A method as claimed in claim 1 wherein the spin system involves a hyperfine ground state spin transition of eurobions in Eu³⁺:Y₂SiO₅.

17. A method as claimed in claim 1 wherein the spin system involves a hyperfine ground state spin transition of erbium ions in Er³⁺: Y₂SiO₅.

18. A method as claimed in claim 1 wherein the spin system involves an optical transition of erbium ions in Er : Y₂SιO₅.

19. An apparatus for increasing the coherence time of a transition in a spin system with an angular momentum >1, under the influence of a fluctuating local magnetic field in a solid state host by the application of an external magnetic field, said apparatus including a selectively operable magnetic field generator to apply an external magnetic field to the system and produces a critical point in the Zeeman shift of the spin transition substantially in all three spatial dimensions.

20. An apparatus as claimed in claim 19 wherein said spin system includes solid state qubits in a host with non zero spin.

21. An apparatus as claimed in claim 19 or 20 wherein the spin system forms part of a solid state nuclear magnetic resonance quantum computer.

22. An apparatus as claimed in claim 21 wherein the computer is an optically detected nuclear magnetic resonance quantum computer.

23. An apparatus as claimed in claim 19 wherein the spin system forms part of a quantum memory.

24. An apparatus as claimed in claim 23 wherein said memory is holographic based memory.

25. An apparatus as claimed in claim 23 wherein said memory is electromagnetically induced transparency memory.

26. An apparatus as claimed in claim 23 wherein said memory is stopped light based memory.

27. An apparatus as claimed in any one of claims 19 to 26 wherein the spin system includes spin levels within an electronic state.

28. An apparatus as claimed in any one of claims 19 to 26 wherein the spin system includes spin levels between spin states in different electronic states.

29. An apparatus as claimed in any one of claims 19 to 26 wherein the spin states are nuclear spin states.

30. An apparatus as claimed in any one of claims 19 to 26 wherein the spin states are electron spin states.

31. An apparatus as claimed in any one of claims 19 to 26 wherein the spin states include both nuclear and electron spin states.

32. An apparatus as claimed in any one of claims 19 to 31 wherein the external magnetic field is applied by three individually adjustable orthogonally oriented electromagnets.

33. An apparatus as claimed in claim 19 wherein the spin system involves a hyperfine ground state spin transition of praseodymium ions in Pr³⁺:Y₂SiO₅.

34. An apparatus as claimed in claim 19 wherein the spin system involves a hyperfine ground state spin transition of eurobions in Eu³⁺:Y₂SiO₅.

35. An apparatus as claimed in claim 19 wherein the spin system involves a hyperfine ground state spin transition of erbium ions in Er : Y₂SιO₅.

36. An apparatus as claimed in claim 19 wherein the spin system involves an optical transition of erbium ions in Er³⁺:Y₂SiO5.