The Classical Bloch Equations

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 classical 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 x 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 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 equations, 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 antisymmetric 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 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 Xd Xc xA f ðtÞ lators are independent and their eigenfrequencies follow the 2 þ c þ X0 þ 2 2 ¼ ; dt dt xB X 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 eigenfrequencies. 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 X 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 U1 ; (4) x x 11 B e2 UðDk ¼ 0Þ¼ ; (9) 1 1 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 x : 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 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 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 X 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. that is, a superposition of a ground state jgi and an excited For example, if oscillator A is excited and the detuning Dk is state jei separated in energy by hDX. The two states are swept through the anti-crossing region, then, depending on coupled by hejH^jgi¼hxd=2. Accordingly, our system of how fast the parameter Dk is varied, one can transfer the coupled harmonic oscillators can be considered as a energy to oscillator B or keep it in oscillator A. The former is “mechanical atom,” whose ground state (excited state) is referred to as an adiabatic transition and the latter as a dia- represented by the symmetric eigenmode xþ (antisymmetric batic transition. In a diabatic transition, the system jumps eigenmode x–). Importantly, the detuning Dk of the oscilla- from one branch in Fig. 2(a) to the other, a process referred tors leads to a coupling of the eigenmodes xþ and x–. to as a Landau-Zener transition.4,9 In this paper, instead of linearly sweeping Dk, we consider a detuning that varies har- D. Parametrically driven system monically in time. Before doing this, we introduce the slowly varying envelope approximation to establish the formal cor- We now investigate the dynamics of the “mechanical respondence between the mechanical oscillator system and a atom” for a time harmonic detuning quantum mechanical two-level system. DkðtÞ¼2X0mA cosðxdrivetÞ; (15) C. The slowly varying envelope approximation such that xd ¼A½expðixdrivetÞþexpðixdrivetÞ . The am- We are interested in the dynamics of the coupled oscilla- plitude A corresponds to the magnitude of the external mod- tors when they are tuned close to resonance (Dk ¼ 0). ulation, whereas xdrive is the frequency of the modulation Therefore, we transform Eq. (3) to the (xþ, x–) basis and (c.f. Fig. 1). To ease the notation, we apply the obtain transformation "#"#"# 2 2 2 a ¼ aðtÞeixdrivet=2; d d 2 xþ Xc Xd xþ f ðtÞ (16) þ c þ X þ ¼ ; þixdrivet=2 2 0 x 2 2 b ¼ bðtÞe : dt dt Xd Xc x f ðtÞ (11) Here, a and b are the slowly varying amplitudes of the symmetric and antisymmetric eigenmodes in a coordinate frame where we have used the transformation matrix U(Dk ¼ 0) rotating at the driving frequency. This transformation gener- given in Eq. (9). Note that the transformation to the (reso- ates terms exp½63ixdrivet=2 in Eq. (13) that are rapidly nant) eigenmodes has interchanged the roles of detuning and oscillating and that we neglect because they average out on coupling in the matrix in Eq. (11) as compared to Eq. (3).A the time scales of interest. This approximation is commonly driving force f(t) can be used to excite the system in any referred to as the rotating wave approximation (RWA).5 In eigenmode or superposition of eigenmodes. However, since the RWA and after transformation into the rotating coordi- we are interested in the dynamics of the system after its initi- nate frame Eq. (13) reads alization, we will from now on set f(t) ¼ 0. "# To understand the evolution of the eigenmodes we write a_ 1 d ic A a i ¼ ; (17) ÈÉ _ 2 A d ic b iX0t b xþ ¼ Re aðtÞe ÈÉ (12) iX0t x ¼ Re bðtÞe ; where we have defined the detuning d between the driving frequency and the level splitting where each mode is rapidly oscillating at the carrier frequency X0 and modulated by the slowly varying complex d ¼ DX xdrive: (18) amplitudes a(t) and b(t). Upon inserting Eq. (12) into the coupled equations of motion (11), we assume that the ampli- Note that the RWA and the transformation into a rotating tude functions a(t) and b(t) do not change appreciably during coordinate frame have turned the problem of two parametri- an oscillation period 2p/X0, which allows us to neglect terms cally driven modes into a simple problem of two modes with containing second time derivatives. This procedure is called a static coupling. With the initial conditions aðt ¼ 0Þ¼a0 the slowly varying envelope approximation (SVEA). and bðt ¼ 0Þ¼b0 the solutions of Eq. (17) are Furthermore, since we consider weak damping, we use 2iX 0 þ c 2iX . With these approximations, we arrive at the fol- 0 A XRt lowing equations of motion for the eigenmode amplitudes aðtÞ¼ i sin b X 2 0 R a_ 1 DX icxd a XRt d XRt ct=2 i ¼ : (13) þ cos i sin a0 e ; _ x DX c 2 X 2 b 2 d i b R A XRt To simplify notation, we have introduced the rescaled detun- bðtÞ¼ i sin a0 X 2 ing frequency R XRt d XRt ct=2 2 þ cos þ i sin b0 e ; Xd 2 XR 2 xd ¼ : (14) X0 (19)

949 Am. J. Phys., Vol. 82, No. 10, October 2014 M. Frimmer and L. Novotny 949 where we have introduced the generalized Rabi-frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 XR ¼ A þ d : (20)

Equations (19) together with Eqs. (10), (12), and (16) are the general solutions to the problem of two coupled harmonic oscillators under a time harmonic detuning. Note that our solutions rely on the validity of the RWA, which is reﬂected by the fact that Eq. (19) only retains dynamics on the time scale given by the generalized Rabi-frequency XR, and neglects any fast dynamics on time scales set by xdrive. Accordingly, our solutions are only valid for driving amplitudes A and detunings d small enough to ensure XR xdrive.

E. The Bloch sphere and the Bloch equations While we have found the general solutions to the equations of motion for the parametrically driven system, it is worth representing these solutions in a vectorial form that Fig. 3. The Bloch sphere. A pair of amplitudes (a; b) is represented by the was originally introduced by Felix Bloch in the context of T 2 2 13 Bloch vector s ¼ (sx, sy, sz) . Normalized amplitude pairs (jaj þjbj ¼ 1) nuclear magnetic resonance. We introduce the Bloch vec- lie on the surface of the Bloch sphere, which has unit radius. All amplitude T tor s ¼ (sx, sy, sz) with components pairs ða; bÞexpðiuÞ with arbitrary u are mapped onto the same point s. 2 3 2 32 3 sx ¼ ab þ a b ¼ 2Refab g¼2jajjbjcosð/Þ; s c d 0 s d x x 4 5 4 54 5 s ¼ iðab a bÞ¼2Imfab g¼2jajjbjsinð/Þ; sy ¼ d c A sy : (22) y dt 2 2 sz 0 A c sz sz ¼ aa bb ¼jaj jbj : (21) This system of equations can be represented in compact form as The Bloch vector s encodes in its three real-valued components the state of the coupled oscillators, which is repre- s_ ¼ R s cs; (23) sented by the amplitudes jaj; jbj and the relative phase /. Importantly, every state ða; bÞ of the oscillator system can be where we defined the rotation vector R ¼ (A,0,d)T. The multiplied by an arbitrary phase factor expðiuÞ without equation of motion s_ ¼ R s describes the precession of the changing the corresponding Bloch vector s. Discarding this Bloch vector s around the rotation vector R with the angular absolute phase of the complex amplitudes a; b reduces the frequency XR defined in Eq. (20), where XR equals the length degrees of freedom from four (two real amplitudes and two of R.5 Equation (22) defines the classical Bloch equations phases for a and b) to three, such that the state of the oscilla- which are the main result of this paper. The classical Bloch tor system can be represented in the three dimensional Bloch equations are formally identical to the quantum Bloch equa- vector space. tions with the exception of the damping terms.5 For an undamped system (c ¼ 0) and appropriately nor- Let us briefly recap here. We are dealing with a system of malized amplitudes (jaj2 þjbj2 ¼ 1), the tip of the Bloch two coupled harmonic oscillators. The state of this system is vector always lies on a unit sphere, called the Bloch sphere. entirely defined by the complex amplitudes of its eigenmo- For the sake of example, let us consider a few distinct points des, which correspond to distinct points on the Bloch sphere. on the Bloch sphere, sketched in Fig. 3. The north pole of According to Eq. (23), we can bring the system from any the Bloch sphere s ¼ (0, 0, 1)T corresponds to the state vector starting point to any other point on the Bloch sphere simply ða; bÞ¼ð1; 0Þ. In this state only the symmetric eigenmode by choosing the right rotation vector R and waiting for the of the system is excited—the mechanical atom is in its right time to achieve the desired amount of rotation. This ground state. Accordingly, when only the antisymmetric idea is at the core of the concept of coherent control. eigenmode is excited and the mechanical atom is in its Importantly, the rotation vector R enabling such coherent excited state, corresponding to ða; bÞ¼ð0; 1Þ, the tip of the control is entirely defined by the parametric driving applied Bloch vector is located at the south pole of the Bloch sphere to the spring constants of the system. Remember that A is s ¼ (0, 0, 1)T. All points on the equator of the Bloch sphere nothing else but the (rescaled) amplitude of the spring modu- correspond to equal superpositions of the two eigenmodes, lation Dk, while d is the difference between the frequency at but with varyingpffiffiffi relative phase /. For example, the state which we modulate the spring constant and the frequency ða; bÞ¼ð1; 1Þ= 2 lies at the intersection of the ex-axis and splitting of the mechanical atom DX. In Fig. 2(b), we have T the Bloch spherepffiffiffi [vector s ¼ (1, 0, 0) ], whereas the state visualized the coherent redistribution of amplitudes between ða; bÞ¼ð1; iÞ= 2 lies at the intersection of the Bloch sphere the eigenmodes by a parametric driving of Dk. T with the ey-axis [vector s ¼ (0, 1, 0) ]. Having pointed out the similarities between a mechanical It is instructive to express the dynamics of the coupled os- oscillator system and a quantum-mechanical two-level sys- cillator system in terms of the Bloch vector s. Using Eqs. tem we note an important difference. The damping c ulti- (17) and (21), we can show that the time evolution of the mately forces the system into the state ða; bÞ¼ð0; 0Þ, which Bloch vector is given by means that the Bloch vector disappears. In contrast, due to

950 Am. J. Phys., Vol. 82, No. 10, October 2014 M. Frimmer and L. Novotny 950 spontaneous emission, a quantum two-level system will always end up in its ground state after a long time. Clearly, while the concept of coherent control can be applied to entirely classical systems, spontaneous emission is a process that is genuinely quantum mechanical in nature and cannot be recovered in a system governed by classical mechanics. It must be emphasized that spontaneous emission requires a fully quantized theory and cannot be derived by semi- classical quantum mechanics. Even Bloch added the decay constants semi-phenomenologically in his treatment of nuclear spins.5 In quantum-mechanical systems, one commonly distinguishes two decay rates. The first—the longitudinal decay rate c1 ¼ 1/T1—describes the decay of the inversion sz, while the second—the transverse decay rate c2 ¼ 1/T2—governs the decay of the components sx and sy of the Bloch vector. For the mechanical oscillator system, we find c1 ¼ c2 ¼ c, which explains the recent experimental find- ing by Faust et al. in a micromechanical oscillator system.11 Finally, we stress that the analogy between a pair of coupled mechanical oscillators and a quantum two-level system relies on the SVEA and accordingly breaks down whenever the amplitudes a; b change on a time scale comparable to the carrier frequency X0.

III. OPERATIONS ON THE BLOCH SPHERE We have seen that by parametrically modulating the detuning of the coupled oscillators we can control the trajectory of the Bloch vector on the Bloch sphere at will. Three protocols for Bloch vector manipulation have proven particularly useful in the field of coherent control, leading to phenomena called Rabi oscillations, Ramsey fringes, and Hahn echo. We will now briefly discuss these phenomena. Fig. 4. Controlling the dynamics of coupled oscillators with pulses of different duration and amplitude. (a) Starting in the ground state ða; bÞ¼ð1; 0Þ,a A. Rabi oscillations p/2-pulse leaves both modes equally excited; (b) A p-pulse transfers the energy of xþ to x–; and (c) a 2p pulse brings the system back to where it In 1937, Rabi studied the dynamics of a spin in a static started. In all cases we assumed no damping (c ¼ 0) and no detuning (d ¼ 0). magnetic field that is modulated by a radio-frequency field and he found that the spin vector periodically oscillates with large amplitude A. For a continuous parametric driving, between parallel and anti-parallel directions with respect to starting at s ¼ (0, 0, 1)T, the system is oscillating between its the static magnetic field.14 These oscillations are referred to two eigenmodes at the resonant Rabi-frequency XR ¼ A.We as Rabi oscillations, or Rabi flopping. The mechanical atom can explicitly convince ourselves that the picture of the reproduces the basic physics of this Rabi flopping. Bloch sphere yields the correct result by considering the Let us neglect damping for the moment (c ¼ 0) and time evolution of the population of the eigenmodes, given by assume a resonant (d ¼ 0) parametric driving Dk / 2 2 jaj and jbj in Eq. (19). For ða0; b0Þ¼ð1; 0Þ, we obtain A cosðDX tÞ to our system, such that the Bloch vector, starting at the north pole s ¼ (0, 0, 1)T, rotates around the axis XRt R ¼Ae at a frequency X ¼ A according to Eq. (23). After jaðtÞj2 ¼ cos2 ect x R 2 a time tp ¼ p/A the Bloch vector will have rotated to the (24) T south pole s ¼ (0, 0, 1) . This means that the symmetric 2 XRt jbðtÞj ¼ sin2 ect: eigenmode now has zero amplitude while the antisymmetric 2 eigenmode has unit amplitude: ða; bÞ¼ð0; 1Þ. Obviously, parametric driving for a time tp (called a p-pulse) inverts our We plot the trajectory of the Bloch vector for a resonantly system. Accordingly, after parametrically driving the system driven system in Fig. 5(a) and the populations jaj2 and jbj2 in for a time t2p ¼ 2p/A it has returned to its initial state at the Fig. 5(b). Indeed, for zero damping (c ¼ 0) the energy oscil- north pole of the Bloch sphere. lates back and forth between the two eigenmodes of the sys- We have plotted the populations of the eigenmodes for tem at a frequency XR. However, a finite damping c makes three different pulse durations in Fig. 4. In general, for a the population of both eigenmodes die out with progressing H-pulse the parametric driving with amplitude A is turned time and the length of the Bloch vector is no longer con- on for a time tH ¼ H/A. Remember that the driving signal served. For finite detuning (d 6¼ 0), we find that even without oscillates at a frequency xdrive according to Eq. (15) and damping the population inversion is reduced and no longer therefore undergoes many oscillations during the pulse. reaches a value of one. The population of the antisymmetric 2 2 2 Importantly, within the validity of the RWA, we can make eigenmode reaches a maximum value of jbðtpÞj ¼ A =XR. every pulse arbitrarily short by applying a driving signal The fact that the Rabi oscillations do not lead to a complete

951 Am. J. Phys., Vol. 82, No. 10, October 2014 M. Frimmer and L. Novotny 951 Fig. 5. (a) Bloch sphere with trajectory of Bloch vector during resonant Rabi oscillations marked in gray (red online). Starting from the north pole the Bloch vector rotates around the ex-axis. To rotate the Bloch vector by the angle H the driving field with amplitude A has to be turned on for a time tH ¼ H/A. (b) Rabi 2 2 oscillations of the populations jaj and jbj for zero detuning (xdrive ¼ DX) and damping c ¼ XR/25. The energy flops back and forth between the two oscillation modes xþ and x–. The Rabi frequency XR defines the flopping rate and is given by the rescaled modulation amplitude A of the detuning Dk. inversion of the system for finite detuning can be understood the first is the frequency difference DX between the eigenm- by considering the rotation of the Bloch vector on the Bloch odes of the system, and the second is the coordinate transfor- sphere. For finite detuning, the rotation vector R has a com- mations we applied. In the transformation (12), we separated ponent in the ez-direction such that a rotation of the Bloch out all fast oscillations at the carrier frequency X0. vector s starting at the north pole of the Bloch sphere no lon- Accordingly, since the eigenmodes xþ, x– oscillate at fre- ger reaches the south pole. Another important observation is quencies X6, the amplitudes a, b still contain oscillations at that the non-resonant Rabi-oscillations always proceed at a 6DX/2. This frequency difference between the eigenmodes res frequency larger than in the resonant case (XR > XR for all means that as time passes they acquire a relative phase dif- d 6¼ 0) according to Eq. (20). ference. By transforming into a coordinate system rotating at the driving frequency in Eq. (16), we exactly compensate for B. Ramsey fringes the phase difference between the eigenmodes if the driving frequency xdrive exactly corresponds to the frequency split- We have seen that Rabi oscillations correspond to rotating DX between the levels. Thus, in this rotating frame, the tions around the ex-axis. What about rotations around other Bloch vector is constant. If, however, we choose a finite d, axes? Clearly, any detuning d leads to an ez-component of we are transforming into a reference system that is detuned the rotation vector R, that is, a rotation around the ez-axis. with respect to the transition frequency DX and the Bloch Assume we prepare the mechanical oscillator system in a vector will rotate even in the absence of driving. state s ¼ (0, 1, 0)T. As we know from the previous section, The detuning d can be made visible in an experimental we can reach this point by applying a p/2-pulse to a system scheme devised by Ramsey.15 The Ramsey method consists in the state s ¼ (0, 0, 1)T. If we allow for a finite detuning d, of three steps, as illustrated in the inset of Fig. 6(b). First, we according to Eq. (22) the system will evolve away from the bring the system from the north pole of the Bloch sphere to state s ¼ (0, 1, 0)T even if the driving is off (A ¼ 0). In fact, the state s ¼ (0, 1, 0)T with a short p/2-pulse with detuning d; for A ¼ 0 the Bloch vector rotates around the Bloch sphere’s second, we wait for a time T; and finally, we apply another equator at an angular frequency d, progressing by an angle p/2-pulse. In case of zero detuning, the Bloch vector has not U ¼ dt in the equatorial plane within a time t. moved during the waiting time and accordingly we end up It is puzzling at first sight that the Bloch vector is rotating on the south pole of the Bloch sphere. However, if the detun- even though no driving is applied to the system. To under- ing is finite the Bloch vector will precess around the ez-axis stand the precession of the Bloch vector for a finite detuning, by an angle U ¼ dT before being rotated by p/2 around the it is necessary to remind ourselves of two important facts: ex-axis by the second pulse. For example, after a time

Fig. 6. (a) Trajectory of Bloch vector in a Ramsey experiment with detuning d and waiting time T ¼ p/d during which the Bloch vector makes a half rotation on the equator. During a waiting time t ¼ U/d, the Bloch vector precesses by an angle U in the equatorial plane. (b) Ramsey fringes for ﬁnite damping c 6¼ 0. The populations jaj2 and jbj2 are inverted after T ¼ np/d, with n being an integer. The inset illustrates the applied pulse sequence that consists of two p/2 pulses separated by a waiting time T. The Ramsey fringes result from a phase difference acquired by the (mechanical) atom, which is due to the drive frequency xdrive (dashed in inset while driving is off) being different from the transition frequency DX (solid).

952 Am. J. Phys., Vol. 82, No. 10, October 2014 M. Frimmer and L. Novotny 952 Fig. 7. Hahn echo experiment. (a) The sequence of pulses and wait times represented as a trajectory on the Bloch sphere. The path starts at the north pole s ¼ (0, 0, 1)T and ends at the south pole s ¼ (0, 0, 1)T. (b) Measurement outcome of the Hahn experiment plotted as the populations jaj2 and jbj2. The population of the symmetric mode xþ is zero for all delay times T while the population of the antisymmetric mode x– falls off exponentially with the damping constant c. Inset: Illustration of the pulse sequence.

T ¼ p/d the Bloch vector will have rotated to the point the Bloch vector in the Hahn echo experiment occurs only in s ¼ (0, 1, 0)T such that the second p/2-pulse will bring the the xy-plane (the pulses can be made arbitrarily short by system back to the north pole s ¼ (0, 0, 1)T, as plotted in increasing their amplitude), the Hahn echo experiment is Fig. 6(a). If we measure the populations jaj2 and jbj2 after able to eliminate the contribution due to energy relaxation the second p/2-pulse we find the characteristic Ramsey and only render the damping due to phase decoherence.5 fringes, plotted in Fig. 6(b), where we also included a finite damping rate c that leads to an exponential decay of both populations. IV. CONCLUSIONS We have shown that a pair of coupled mechanical oscilla- C. Hahn-echo tors with parametrically driven detuning can serve as a classical analogue to a quantum-mechanical two-level system. Having established that a sequence of pulses and wait The correspondence between the classical and the quantum times can be represented by a trajectory on the Bloch sphere, system is established by using the slowly varying envelope we now consider the dynamics of the so-called Hahn echo approximation, which casts the Newtonian equations of experiment.16 The experiment involves a sequence of three motion of the coupled oscillators into a form resembling the short pulses [see inset of Fig. 7(b)]. As before, we start with Schrodinger€ equation for a two-level atom. The mechanical the system in the lower energy eigenmode corresponding oscillator analogue features only a single decay rate for both to the Bloch vector s ¼ (0, 0, 1)T. Similar to the Ramsey longitudinal and transverse damping in contrast to a experiment, we bring the system with a short p/2-pulse to quantum-mechanical two-level system that can show differ- T s ¼ (0, 1, 0) , which corresponds to a superposition of the xþ ent decay rates due to spontaneous emission and dephasing and x– modes. During a waiting time T/2 the Bloch vector processes. Also, the classical model does not conserve the rotates around the ez-axis according to the detuning d by an total population of the levels since both eigenmodes are angle U ¼ dT/2 until one applies a short p-pulse. This pulse equally damped. Our treatment provides an easy introduction flips the Bloch vector around the ex-axis as illustrated in to the field of coherent control in the context of both classical Fig. 7(a). Another wait time of T/2 rotates the Bloch vector and quantum systems. An extension of our discussion of the to s ¼ (0, 1, 0)T, and from there a 3p/2 pulse brings it to the mechanical atom lends itself to introduce students to Pauli south pole s ¼ (0, 0, 1)T. However, this is only true without matrices, density operators, and rotation operators. damping. The complex amplitudes ða; bÞ of the oscillation modes are both damped by a factor expðct=2Þ according to ACKNOWLEDGMENTS Eq. (19) and hence The authors thank Lo€ıc Rondin for fruitful discussions and jaðTÞj2 ¼ 0 are most grateful for financial support by ETH Zurich€ and (25) ERC-QMES (No. 338763). jbðTÞj2 ¼ ecT:

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