The classical Bloch equations Martin Frimmer and Lukas Novotny ETH Zurich,€ Photonics Laboratory, 8093 Zurich,€ Switzerland (www.photonics.ethz.ch) (Received 13 October 2013; accepted 7 May 2014) Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes, and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled- harmonic-oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times (T1 and T2) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4878621] I. INTRODUCTION offers students an intuitive entry into the field and prepares them with the basic concepts of quantum coherent control. The harmonic oscillator is arguably the most fundamental building block at the core of our understanding of both clas- sical and quantum physics. Interestingly, a host of phenom- II. THE MECHANICAL ATOM ena originally encountered in quantum mechanics and A. Equations of motion initially thought to be of purely quantum-mechanical nature have been successfully modelled using coupled classical har- Throughout this paper, we consider two oscillators, as monic oscillators. Among these phenomena are electromag- illustrated in Fig. 1, with masses mA and mB, spring constants 1 2,3 netically induced transparency, rapid adiabatic passage, kA ¼ k – Dk(t) and kB ¼ k þ Dk(t) with a small detuning Dk(t) and Landau-Zener tunneling.4 A particularly rich subset of that can be time dependent, and coupled by a spring with experiments is enabled by the coherent manipulation of a spring constant j, which is weak compared to k. Oscillator A quantum-mechanical two-level system, providing access to can be externally driven by a force F(t), and both oscillators fascinating effects including Rabi oscillations, Ramsey are weakly damped at a rate c. Because of the coupling j, the fringes, and Hahn echoes.5 Remarkably, equipped with the dynamics of oscillator A couples over to oscillator B. Such models and ideas gained from studying quantum-mechanical two coupled harmonic oscillators are a generic model system systems, researchers have returned to construct classical ana- applicable to diverse fields of physics. For example, in molec- logues of two-level systems.6,7 Coherent control of such a ular physics oscillators A and B correspond to a pair of atoms. classical two-level system has been beautifully demonstrated Similarly, in cavity quantum electrodynamics, A is a two- for an “optical atom” consisting of two coupled modes of a level atom and B is a cavity field. In cavity optomechanics, cavity.8,9 Recently, coherent control of classical two-level oscillator A would represent a mechanical oscillator, such as systems has been achieved with coupled micromechanical a membrane or cantilever, and B an optical resonator. For the oscillators.10,11 With the analogy between a two-level system following, we assume that the masses of the oscillators are and a coupled pair of classical harmonic oscillators well equal (mA ¼ mB ¼ m). Then, in terms of the coordinates xA established, it is surprising that this analogy has not been and xB of the two oscillators, the equations of motion are used to familiarize students with the concepts of coherent control and to provide an accessible analogue to a variety of k þ j DkðtÞ j FðtÞ x€ þ cx_ þ À x À x ¼ ; quantum optical phenomena. Furthermore, exploring the lim- A A m m A m B m (1) its of any analogue typically illustrates very strikingly the k þ j DkðtÞ j genuine features of a physical theory that are not present in x€B þ cx_B þ þ xB À xA ¼ 0: the theory in which the analogy is phrased.12 m m m In this paper, we consider a pair of two parametrically driven coupled mechanical (harmonic) oscillators. From Newton’s second law, we derive a set of equations of motion for the eigenmode amplitudes that are formally equivalent to the time-dependent Schrodinger€ equation of a two-level atom. We then derive a set of coupled differential equations that are formally identical with the quantum Bloch equa- tions, with the exception that the longitudinal and transverse relaxation times are equal. We illustrate the concept of the Bloch sphere and provide an intuitive understanding of coherent control experiments by discussing Rabi oscillations, Fig. 1. Coupled mechanical oscillators with masses m , m and spring con- Ramsey fringes, and Hahn echoes. Finally, we point out the A B stants kA ¼ k – Dk, kB ¼ k þ Dk with a detuning Dk that can be time depend- distinct differences between our mechanical analogue and a ent. Oscillator A can be driven by an external force F(t), and the oscillators true quantum-mechanical two-level system. Our approach are coupled with a spring (with spring constant j). 947 Am. J. Phys. 82 (10), October 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 947 For ease of notation, we introduce the carrier frequency X0,the Here, Xþ denotes the frequency of the symmetric eigen- detuning frequency Xd, and the coupling frequency Xc as mode, which is lower than the frequency X– of the antisym- metric eigenmode. Thus, after transformation, Eq. (3) yields 2 k þ j two independent differential equations for the normal mode X0 ¼ ; m coordinates xe1 and xe2: 2 Dk (2) Xd ¼ ; d2 d m þ c þ X2 x ¼ U f ðtÞ j dt2 dt þ e1 11 X2 ¼ ; (7) c m d2 d þ c þ X2 x ¼ U f ðtÞ: dt2 dt À e2 21 and represent the coupled differential equations in Eq. (1) in matrix form as In Fig. 2(a), we plot the eigenfrequencies X6 as a function "#"#"#"# of the detuning Dk. In the absence of coupling, the two oscil- 2 2 2 d d 2 xA ÀXd ÀXc xA f ðtÞ lators are independent and their eigenfrequencies follow the 2 þ c þ X0 þ 2 2 ¼ ; dt dt xB ÀX X xB 0 straight lines that intersect at Dk ¼ 0. However, in the pres- c d ence of finite coupling, the two curves no longer intersect. (3) Instead, there is a characteristic anti-crossing of the eigenfre- quencies. The frequency splitting at resonance (Dk ¼ 0) is where f(t) ¼ F(t)/m. This system of equations describes the full dynamics of the coupled oscillator problem. 2 Xc DX ¼ XÀ À Xþ ; (8) B. Eigenmodes for constant detuning X0 We first consider the case of constant detuning where we made use of the fact that Xc X0. Thus, the split- (Dk ¼ constant) and solve for the eigenmodes of the system ting is proportional to the coupling strength j. If the separa- and their respective eigenfrequencies. To this end, we diago- tion of the frequency branches DX can be discriminated nalize the matrix in Eq. (3). The eigenmodes xe1 and xe2 of against their width, which scales with the damping constant the system can be derived from the coordinates of the two c, one considers the system to be in the so-called strong cou- 4 oscillators xA and xB as pling regime. Turning to the eigenmodes of the system, we find that on resonance the transformation matrix reads xA xe1 ¼ UÀ1 ; (4) x x 11 B e2 UðDk ¼ 0Þ¼ ; (9) 1 À1 where U is a transformation matrix, whose rows are the eigenvectors of the matrix in Eq. (3). We find and the eigenmodes become U11 U12 x ¼ xe1j ¼ xA þ xB U ¼ þ Dk¼0 (10) U21 U22 x ¼ x j ¼ x À x : 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 À e2 Dk¼0 A B 2 4 6 1 ðXd=XcÞ þ 1 þðXd=XcÞ 7 Thus, on resonance, the eigenmodes of the system are sym- ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; (5) 2 4 metric and antisymmetric superpositions of the two individ- 1 ðXd=XcÞ À 1 þðXd=XcÞ ual oscillators. For xþ the two masses swing in phase and for x out of phase, i.e., against each other.pffiffiffiffiffiffiffiffiffiffi The eigenfrequency and the eigenfrequencies turn out to be – of the symmetric mode is Xþ ¼ k = m, which is the fre- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 quency in the absence of coupling. This result is obvious since the coupling spring plays no role for the symmetric X X2 7 X4 X4 : (6) 6 ¼ 0 d þ c mode. The eigenfrequency of the antisymmetric mode is Fig. 2. (a) Eigenfrequencies Xþ and X– of the coupled oscillators as a function of the detuning Dk. The dashed lines show the eigenfrequencies in the absence of coupling. The frequency splitting DX at resonance (Dk ¼ 0) is proportional to the coupling strength j. (b) The energy of the system can be swapped between the eigenmodes by harmonically modulating the spring detuning Dk(t). 948 Am. J. Phys., Vol. 82, No. 10, October 2014 M. Frimmer and L. Novotny 948 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XÀ ¼ ðk þ 2jÞ = m. This frequency is higher than Xþ Note that, for the case of vanishing damping (c ¼ 0), Eq. (13) because each oscillator feels the coupling spring. resembles the time dependent Schrodinger€ equation We have thus far considered a static detuning Dk. ih @tjWi¼H^jWi for a state vector jWi¼aðtÞjgiþbðtÞjei; Intriguing effects happen when Dk becomes time dependent.