WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.1 Measuring Spin-Lattice Relaxation Time L.W. Lupinski, R. Paudel, and M.J. Madsen Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: December 16, 2009) NMR quantum computing has been a promising research line in recent years[3]. Before we implement any quantum information processing experiments, we need to investigate the possibility of coherence in the nuclear spins. In this project, we measure the spin-lattice relaxation time(t 1 ), which is an important property to investigate the decoherence time, of the 13 C labeled chloroform using an NMR spectrometer. We found the T 1 relaxation time to be 6.679 ±.064s (95%,CI). The T 1 relaxation time is an important feature when running quantum information experiments. Quantum information theory has become important recently due do to quantum computing [1]. Quantum computers would have some advantages over classical ones, for instance it has been shown both theoretically and experimentally that a quantum computer can factor large prime numbers much faster than classical computers [2, 3]. Systems where these quantum effects can be observed are ultra-cold neutral atoms, cold ions, nuclear spins(using NMR), optical lattices, photons, and quantum dots [4]. In this project we used an NMR spectrometer to measure the spin-lattice relaxation time of the proton in chloroform molecule, which is an important property for implementing quantum information processing[5]. A NMR (Nuclear Magnetic Resonance) spectrometer works by applying a strong magnetic field, causing the magnetic moment of the nuclei of the atoms within the spectrometer to align with this external field [6]. Because the spin angular momentum is quantized, the atoms are found in either spin up (lower energy) or spin down (higher energy) states. As the temperature of the atoms decreases, fewer atoms are found in the spin down state, until at absolute zero all of the atoms are in the spin up state. The distribution of atoms in these states are given by the Maxwell-Boltzmann distribution: N(T ) e γb 0 k B T (1) where γ is the gyromagnetic ratio of the nucleus, B 0 is the magnetic field, is the Planck s constant, k B is the Boltzmann s constant, and T is temperature. The precessional frequency
WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.2 of the nuclei, ω, is given by, ω = γb 0, (2) where ω is also known as the Larmor frequency. A π/2 radiofrequency pulse rotates the magnetization of the nuclear spins from z-direction to xy-plane. Since there is no magnetic field in the xy-plane the spins are free to precess on the plane with some frequency, ω. If this frequency matches the Larmor frequency, then we will have a resonating condition (the R of the NMR corresponds to this condition). The nuclei will be excited to the spin down state. Eventually, the nuclei relaxes back to the spin up state scattering photons. We can detect this by using a set of coils (experimentally we will use the same coil to generate an rf-pulse and receive the rf-pulse from the nuclei). The method of receiving this signal is called a Free Induction Decay (FID). By performing Fourier transformations on this data from the time domain to the frequency space, we measure the coupling of each proton or C-13 atoms in the molecule [7]. In NMR, we usually consider two energy-level states, the lower state with protons spin up, or aligned with the external field, and the higher state with protons spin down, or aligned against the external field [8]. The rf region of the electromagnetic spectrum causes protons to jump to higher energy levels. The nuclear spin returns to the state of equilibrium, that is the lower energy level by flipping its spin, and the excess energy is lost to the surroundings, also called the lattice in the form of heat [8]. The characteristic lifetime of a spin in the upper state is called the spin-lattice relaxation time T 1, also called the longitudinal relaxation time. T 1 is the average length of time that a proton remains in the same energy level[10]. When we apply a π pulse to nuclei aligned in a uniform B 0 ẑ field, the spin of the nuclei is rotated to the -z-direction. However, as the time progresses the nuclei will eventually come back to the initial state, that is aligned with the z-axis. This amount of time is characterized by the T 1 relaxation time[9]. We chose 13 C labeled CHCl 3 as our molecule to measure the spin-lattice relaxation time. 13 C labeled CHCl 3 is interesting since it has two spins one from the proton and one from the 13 labeled carbon atom. This will be useful in doing two-bit quantum information experiments in the future[5]. We prepared a 7% by weight solution of 13 C labeled CHCl 3 in deuterated acetone. The deuterium in the solvent is used to produce a lock signal which is picked up by the NMR probe (JEOL 400MHz) and is used in an active feedback loop to
WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.3 stabilize the axial magnetic field from drift [5]. The sample spins in almost homogeneous magnetic field B 0 = 9.4T in the NMR spectrometer. 7x10 7 6x10 7 B C Intensity (Arb. Units) 5x10 7 4x10 7 3x10 7 2x10 7 A 1x10 7 0 0 5 10 Frequency (ppm) FIG. 1: The figure shows the spectrum obtained from a 13 C-labeled chloroform molecules in d-6 acetone using a 400 MHz NMR spectrometer scanned through 15 ppm. The frequency is measured in ppm which is the ratio of the resonance frequency to the reference frequency (400 MHz) multiplied by the scan width. The peak B and C at 7.70 ppm and 8.24 ppm represent the signal obtained from the proton in the chloroform molecule. It is a doublet because of the spin-spin coupling between carbon and proton. Peaks A correspond to the impurities of the solvent. We ran a one-pulse with a pulse length t = 15.79 µs and θ = π/4, the parameters are included in the NMR-experiment parameter list. We got the spectrum shown in Figure 1. The two peaks at 7.70 ppm and 8.24 ppm is coming from the proton of the chloroform molecule. We get two peaks because of the spin-spin coupling between the proton and the 13 C. We see a third peak in between these two peaks because of the coupling. However, we expect it to be much larger compared to the other two peaks because it is the superposition
WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.4 Cl Cl 13 C Cl H FIG. 2: The figure shows the chemical structure of a chloroform molecule. Both the H and 13 C have spins of 1/2. of two up-down spin states. Since we did not use a pure d-6 acetone, we expect some H-6 acetone in the solution. The peak at 2.08 ppm is from the protons from acetone[14]. We need to do more analysis in the spectrum in the future. The next step in our project was to measure the T 1 time for the proton in the chloroform molecule. We use a standard two-pulse sequence, π τ π/2, to measure T1 of the sample [9]. The process is described above and is shown in Figure 3. The NMR probe then acquires π π/2 τ Time Delay FIG. 3: First the π pulse is applied, then after a time τ the π/2 degree pulse is applied. The intensity of the transverse magnetic field is then measured. the transverse magnetic field data. The field intensity can then be fit to the equation [9], f(t) = f( ) ( 1 H 0 e ) τ/t 1, (3) where f(t) is the field intensity at time t, f( ) is the field intensity right before the π-pulse is applied, H 0 is an iteratively determined scaling factor that helps to correct for an imperfect f( ), and T 1 is the relaxation time. Running this analysis, Figure 3, for the proton of the hydrogen atom we find a T 1 value of 6.679 ±.064s (95%,CI). Our value agrees with measurement done by Berkowitz [12] and Yannoni [13] using different NMR spectrometer
WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.5 frequency. 4x10 7 3x10 7 Intensity (Arb. Units) 2x10 7 1x10 7 0-1x10 7-2x10 7 0 2 4 6 8 10 12 14 16 Time (s) FIG. 4: The equation fitting this graph is of the form of equation 1, where f( ) = 3.91856 10 7, H 0 = 1.55199, and T 1 of 6.67926 s. We found that our molecule has a long T 1 time, which means the decoherence time is relatively long compared to the rf-pulses which means that we can do a significant number of quantum manipulations before the spins undergo decorence. Hence, 13 C-labeled chloroform is a promising sample to use in quantum information experiments such as a CNOT-gate, a Deutsch-Jozsa algorithm, or a Grover algorithm[5]. This line of research follows the current interest in quantum computing [3]. [1] DiVincenzo, D.P., Science. 270 255 (1995) [2] Shor, P., PRA 52 4 (1995) [3] Vandersypen, L.M.K., Chuang, I.L., et. al., Nature, 414, 883-887 (2001) [4] Michael A. Nielsen and Isaac Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000
WJP, PHY381 (2009) Wabash Journal of Physics v4.0, p.6 [5] Quantum Information Processing with NMR, MIT Department of Physics (2009). [6] Bloch, F., Phys. Rev., 70 7-8 (1946) [7] Klein, W., Am. J. Phys. 58 143 (1990) [8] www.colby.edu/chemistry/pchem/lab/spinlattice.pdf [9] ECA Experiment Note: T 1 Measurement by Inversion Recovery, Joel USA, Inc. (2007) [10] Donnally, B.L., et. al., Am. J. Phys. 31 779 (1963) [11] http://www.cem.msu.edu/ reusch/virtualtext/spectrpy/nmr/nmr1.htm [12] www.eberkowitz.com/work/qip2.pdf [13] Yannoni, C., et. al., Appl. Phy. Lett. 75, 3563 (1999) [14] Gottlieb, H.E., et. al, J. Org. Chem. 62 21 (1997)