Periodically modulated quantum nonlinear oscillators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Periodically modulated quantumnonlinear oscillators
M. I. Dykman
Vibrational systems have been attracting much attention in physics. Such systems arealways nonlinear, at least to some extent. For weak damping, even small nonlinearitycan become important. For example, classically, the nonlinearity-induced dependenceof the vibration frequency on amplitude can lead to bistability of forced resonant vi-brations (Landau and Lifshitz, 2004), see Fig. 7.1. Quantum mechanically, the nonlin-earity makes the frequencies of transitions between adjacent energy levels different andthus enables spectroscopic identification and selective excitation of these transitions.The interest in quantum effects in oscillators significantly increased recently in thecontext of nonlinear vibrations in Josephson junction based systems and applicationsof these systems in quantum information (Wallraff et al. , 2004; Siddiqi et al. , 2005;Lupa¸scu et al. , 2006; Steffen et al. , 2006; Metcalfe et al. , 2007; Schreier et al. , 2008; Watanabe et al. , 2009;Mallet et al. , 2009; Vijay et al. , 2009; Wilson et al. , 2010; Bishop et al. , 2010; Reed et al. , 2010).The long-sought (Blencowe, 2004; Schwab and Roukes, 2005) quantum regime has beenreached also in nanomechanical resonators (O’Connell et al. , 2010; Riviere et al. , 2011).This development makes it possible to study quantum effects in individual vibrationalsystems rather than ensembles.Besides being interesting on their own, nonlinear oscillators allow addressing somefairly general physics problems. One of them is classical and quantum fluctuationsfar from thermal equilibrium and whether they have features that have no analog insystems close to equilibrium. A resonantly modulated nonlinear oscillator could be thefirst well-characterized physical system with no detailed balance, which was used tostudy an important class of fluctuation phenomena, the fluctuation-induced switchingbetween coexisting stable states, both in the classical and quantum regimes, and toreveal some of such features . For the theory and experiment on switching of a resonantly modulated oscillator with no de-tailed balance see, in particular, Dykman and Krivoglaz 1979; Dmitriev and Dyakonov 1986 a ;Vogel and Risken 1990; Dykman et al. et al. et al. et al. a ; Almog et al. et al. et al. et al. a ; Wilson et al. Periodically modulated quantum nonlinear oscillators
An important source of quantum fluctuations in an oscillator is coupling to athermal bath. The coupling leads to oscillator relaxation via emission of excitationsin the bath (photons, phonons, etc) accompanied by transitions between the oscillatorenergy levels. If the coupling is weak, the transition rates are small compared to theenergy transferred in a transition, in frequency units. In the classical limit, the energylevels are not resolved and the transitions lead to friction.At the quantum level, one should take into account that the transitions happenat random. The randomness gives rise to a peculiar quantum noise. For a resonantlymodulated oscillator, it leads to diffusion over the quantum states, which are time-dependent because of the modulation. The result of this diffusion is quantum heatingof the oscillator. It is seen, in particular, in a nonzero width of the distribution over theoscillator states even where the temperature of the thermal reservoir is zero. Quantumheating is qualitatively different from the familiar Joule heating, which characterizesthe power absorbed from the modulating field. In contrast to the Joule heating, theresulting distribution over the states turns out to be independent of the oscillatorrelaxation rate, for weak damping. (a) Β B1 Β B2 Β A osc2 (b) - Μ B1 Μ B2 Μ A osc Fig. 7.1
Bistability of forced vibrations for additive (a) and parametric (b) modulation atfrequency ω F close to the oscillator eigenfrequency ω and 2 ω , respectively; A osc is the scaledvibration amplitude. In (a), β is the scaled squared modulation amplitude, and in (b) µ is thescaled frequency detuning ω F − ω . Parameters β, µ and the scaling factor of the vibrationamplitude C are defined in Table 7.1. Solid and dashed lines show the stable and unstablestationary vibrational states; in (b) there are two states with the same nonzero A osc and thephases that differ by π . The bifurcation parameter values β B , , µ B , indicate where thenumber of stable vibrational states changes. The scaled decay rate κ in (a) and (b) is 0.25and 0.3, respectively. A consequence of quantum heating is quantum activation (Dykman and Smelyansky, 1988;Marthaler and Dykman, 2006; Dykman, 2007; Katz et al. , 2007; Peano and Thorwart, 2010 b ).This is a mechanism of switching between coexisting stable states of forced vibrationsshown in Fig. 7.1. Similar to quantum heating, the switching is due to the quantumnoise that accompanies relaxation. It occurs via transitions over an effective barrierthat separates the vibrational states. The mechanism differs from quantum tunneling,and moreover, leads to an exponentially larger switching rate even for low tempera-tures. It differs also from thermal activation, which becomes important in the classicalregime of high temperatures.Quantum activation has no analog in systems close to thermal equilibrium. Theswitching rate has a characteristic dependence on the parameters and displays scaling esonant modulation: Quantum heating behavior with characteristic exponents. This has made it possible to identify quantumactivation in the experiment (Vijay et al. , 2009). The physics of quantum heating andquantum activation is explained in Secs. 7.2 and 7.3, respectively.Quantum heating is manifested also in the power spectra of resonantly modulatedoscillators and the spectra of their response to a weak additional field. Spectroscopyhas been recognized as a means of studying the dynamics of modulated oscillators and,more recently, of using oscillators for quantum measurements (Dykman and Krivoglaz, 1979;Drummond and Walls, 1980; Collett and Walls, 1985; Dykman et al. , 1994; Stambaugh and Chan, 2006 b ;Chan and Stambaugh, 2006; Nation et al. , 2008; Wilson et al. , 2010; Vierheilig and Grifoni, 2010;Boissonneault et al. , 2010; Laflamme and Clerk, 2011). Because of the interplay ofmany interstate transitions at close frequencies, the spectra of modulated quantum os-cillators have a characteristic shape. They can have a fine structure, in which case theydirectly provide the quantum-heating induced distribution over the oscillator states(Dykman et al. , 2011). The power spectra are also important for understanding the dy-namics of resonantly driven oscillators coupled to two-level systems (Picot et al. , 2008;Serban et al. , 2010); interesting spectral manifestations of quantum heating in suchcoupled systems have been recently found . The spectra of modulated oscillators arediscussed in Sec. 7.4.Nonresonant modulation can also have pronounced effect on the oscillator dy-namics. Recently such modulation attracted much attention in optomechanics, whereintracavity modes are coupled to mechanical vibrations, for example, to the vibrationsof a mirror in the cavity (Kippenberg and Vahala, 2008). Modulation can lead to cool-ing and heating of an oscillator, or excite self-sustained vibrations. In contrast to thequantum heating discussed above, where quantum fluctuations broaden the distribu-tion over the oscillator states in a strong resonant field, here the issue is the change ofthe distribution over the Fock states of the oscillator. An interesting feature of heatingand cooling is that, in the important case where the effective friction is linear (as forstandard viscous friction), the distribution over the Fock states remains of the Boltz-mann form, but the temperature differs from the bath temperature (Dykman, 1978;Clerk, 2004; Wilson-Rae et al. , 2007; Marquardt et al. , 2007). In Sec. 7.5 we extendthe previous analysis (Dykman, 1978) to take into consideration both the nonlinearityof the coupling of the oscillator to other degrees of freedom and the nonlinearity ofthe interaction with the modulating field. For weakly damped oscillators, vibration nonlinearity becomes important once thechange of the vibration frequency due to the nonlinearity ∆ ω becomes comparable tothe oscillator decay rate Γ, which characterizes frequency uncertainty. This happenswhere ∆ ω is still small compared to the oscillator eigenfrequency ω and the nonlinearpart of the vibration energy is small compared to the harmonic part. Respectively, thevibrations remain almost sinusoidal, which significantly simplifies the analysis. At thesame time, for weak damping an already moderately strong resonant modulation can F. R. Ong e t al., in preparation (experiment) and M. Boissonneault e t al., in preparation (theory);we are grateful to P. Bertet for informing us about this work Periodically modulated quantum nonlinear oscillators drive an oscillator into the amplitude range where ∆ ω & Γ. This makes underdampedoscillators advantageous for studying quantum phenomena far from equilibrium.
The most frequently used types of resonant modulation of an oscillator are modulationby a resonant additive force A cos ω F t with frequency ω F close to ω and parametricmodulation by force F cos ω F t with ω F close 2 ω . For moderately strong modulationit is often sufficient to take into account only the leading-order oscillator nonlinearitywhich leads to the amplitude dependence of the vibration frequency. It is sometimescalled Kerr nonlinearity, and the corresponding model of the oscillator is called theDuffing model. The Hamiltonian of the Duffing oscillator is H = 12 p + 12 ω q + 14 γq + H F ( t ) , (7.1)where q and p are the oscillator coordinate and momentum, the mass is set equal toone, and γ is the anharmonicity parameter.The modulation term H F in eq. (7.1) for additive ( H F = H add ) and parametric( H F = H par ) modulation has the form H add = − qA cos ω F t, H par = 12 q F cos ω F t. (7.2)The conditions that the modulation is resonant and not too strong are | δω | ≪ ω , δω = ω M − ω ; | γ |h q i ≪ ω . (7.3)Here, ω M is equal to ω F and ω F / γ, F >
0; for additive driving, theoscillator can be bistable for γδω >
0; we assume δω > U ( t ) = exp (cid:0) − ia † a ω M t (cid:1) , where a † and a are the raising and lowering oper-ators of the oscillator. We introduce slowly varying in time dimensionless coordinate Q and momentum P , using as a scaling factor the characteristic amplitude of forcedvibrations C , see Table 7.1, U † ( t ) qU ( t ) = C ( Q cos ϕ + P sin ϕ ) ,U † ( t ) pU ( t ) = − Cω M ( Q sin ϕ − P cos ϕ ) . (7.4)For additive and parametric modulation ϕ ≡ ϕ add = ω F t and ϕ ≡ ϕ par = ( ω F t + π ) / P and Q has the form[ P, Q ] = − iλ, λ = ~ / ( ω M C ) . (7.5)Parameter λ ∝ ~ plays the role of the Planck constant in the quantum dynamics in therotating frame. It is determined by the oscillator nonlinearity, λ ∝ γ , see Table 7.1.For characteristic | Q | , | P | .
1, where h q i . C , the last inequality in eqn (7.3) esonant modulation: Quantum heating Table 7.1
Parameters of a resonantly modulated oscillator
Additive driving Parametric drivingAmplitude scale C = [8 ω F ( ω F − ω ) / γ ] / C = | F/ γ | / Scaled Planck constant λ = 3 ~ γ/ ω F ( ω F − ω ) λ = 3 ~ ω − F | γ/F | Control parameter β = 3 γA / ω F ( ω F − ω ) µ = ω F ( ω F − ω ) / | F | Scaled decay rate ∗ κ = Ω − = Γ / | ω F − ω | κ = ζ − = 2Γ ω F / | F | ∗ Notations Ω and ζ were used in some of our previous papers, cf. Dykman (2007). coincides with the first inequality in this equation for additive modulation, whereasfor parametric modulation is gives condition F ≪ ω .In the range (7.3) the oscillator dynamics can be analyzed in the rotating waveapproximation (RWA). The Hamiltonian in the rotating frame is˜ H = U † H U − i ~ U † ˙ U ≈ (3 E sl / g, E sl = γC , (7.6)where E sl ∼ γ h q i is the characteristic energy of motion in the rotating frame. Thismotion is slow on the time scale ω − F . Operator ˆ g = g ( Q, P ) in eqn (7.6) is independentof time. For additive and parametric modulation, respectively, we have g add ( Q, P ) = 14 (cid:0) P + Q − (cid:1) − β / Q, (7.7)and g par ( Q, P ) = 14 (cid:0) P + Q (cid:1) + 12 (1 − µ ) P −
12 (1 + µ ) Q . (7.8)In Fig. 7.2 we show g add ( Q, P ) and g par ( Q, P ) as functions of classical coordinate
Fig. 7.2 (Color) The dimensionless Hamiltonian functions of the oscillator for additive (leftpanel) and parametric (right panel) modulation. The plots refer, respectively, to β = 0 . µ = − .
1. In the presence of weak dissipation, the minimum and the local maximumof g add and the minima of g par become classically stable states of forced vibrations in the labframe. and momentum. Each of these functions depends on one dimensionless parameter, β and µ , respectively, which characterizes the ratio of the modulation strength tothe frequency detuning. These parameters are given in Table 7.1. For 0 < β < / g add has the form of a tilted Mexican hat, with a local maximum and with Periodically modulated quantum nonlinear oscillators a minimum at the lowest point of the rim. In the presence of weak dissipation theseextrema correspond to classically stable states of forced vibrations with small andlarge amplitude, respectively, see Fig. 7.1(a).For − < µ <
1, function g par has two minima, which in the presence of weakdissipation correspond to stable vibrational states, see Fig. 7.1(b). Function g par hassymmetry g par ( Q, P ) = g par ( − Q, − P ). This is a consequence of the time-translationsymmetry H ( t ) = H ( t + 2 π/ω F ), as seen from eqn (7.2) and (7.4). Respectively, thevibration amplitudes in the stable states are the same, but the vibration phases differby π , characteristic of parametric resonance. Operator ˆ g plays the role of dimensionless Hamiltonian of the modulated oscillator inthe rotating frame. In the RWA, the Schr¨odinger equation in dimensionless slow time τ reads iλ ˙ ψ ≡ iλ∂ τ ψ = ˆ gψ, τ = tλγC / ~ ≡ ( λE sl / ~ ) t ; (7.9) τ = t δω and τ = tF/ ω F for additive and parametric modulation, respectively.Operator ˆ g has a discrete spectrum, ˆ g | n i = g n | n i . The eigenvalues g n have sim-ple physical meaning. A periodically modulated oscillator does not have stationarystates with conserved energy in the lab frame. It is rather described by the Floquet,or quasienergy states Ψ ε ( t ) = U ( t ) ψ ε ( τ ), which is a consequence of the periodic-ity of the Hamiltonian H ( t ) = H ( t + 2 π/ω F ). One can seek a solution of the fullSchr¨odinger equation i ~ ∂ t Ψ = H ( t )Ψ in the form Ψ ε ( t + t M ) = exp( − iεt M / ~ )Ψ ε ( t ),where t M = 2 π/ω M ( t M = 2 π/ω F and t M = 4 π/ω F for additive and parametricmodulation, respectively). This expression defines quasienergy ε . We note that, forparametric modulation, we use the doubled modulation period when defining ε ; thisis convenient for the description of period-two states of the oscillator.From eqns (7.2) and (7.9), in the RWA the oscillator quasienergies are simplyrelated to the eigenvalues of ˆ g , ε n = (3 E sl / g n , i.e., g n is a scaled quasienergy. Herewe are using an extended ε -axis rather than limiting ε to the analog of the first Brillouinzone 0 ≤ ε < ~ ω M . The scaled quasienergy spectrum for parametric modulation in theneglect of tunneling is sketched in the right panel of Fig. 7.3. In the region of bistability, − < µ <
1, the states with g n < g par ( Q, P = 0) has aform of a symmetric double-well potential, and the states with g n < g par in Fig. 7.2 by planes g par = g n .The structure of quasienergy states of an additively driven oscillator can be un-derstood in a similar way by thinking of the cross-sections of the surface g add ( Q, P )in Fig. 7.2 by planes g add = g n . The eigenstates localized near the local maximumof g add ( Q, P ) correspond to semiclassical orbits on the surface of the “inner dome”of g add ( Q, P ); these states become stronger localized as g n increases toward the localmaximum of g add ( Q, P ). This is in contrast with the conventional picture of a particlein a potential well, where the localization becomes stronger with decreasing energy.Many interesting and unusual features of the dynamics described by eqn (7.9)follow from the fact that functions g add , g par do not have the form of a sum of the esonant modulation: Quantum heating (cid:1)(cid:0)(cid:2)(cid:3)(cid:4) (cid:5)(cid:3)(cid:5) (cid:2)(cid:3)(cid:4)(cid:6)(cid:7)(cid:8)(cid:9)(cid:0)(cid:5)(cid:3)(cid:10)(cid:5)(cid:3)(cid:5)(cid:5)(cid:3)(cid:10) (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:14)(cid:16)(cid:17)(cid:18)(cid:19) (cid:20)(cid:12)(cid:14)(cid:16)(cid:17)(cid:19)(cid:21)(cid:22)(cid:23)(cid:14)(cid:19)(cid:16)(cid:23)(cid:24)(cid:25)(cid:26)(cid:25)(cid:27)(cid:25)(cid:28)(cid:25)(cid:29)(cid:25)(cid:29)(cid:30)(cid:27) (cid:31) !" Fig. 7.3
Panel (a): oscillator relaxation and excitation. Relaxation is due to transitionsbetween the Fock states with energies E N ≈ ~ ω ( N + 1 /
2) accompanied by emission (orabsorption, for nonzero temperature) of excitations in the bath, e.g., photons. The station-ary vibrational state is formed on balance between relaxation and excitation by periodicmodulation F ( t ). Panel (b): the effective RWA Hamiltonian of the parametrically modulatedoscillator g par ( Q, P = 0) for µ = − .
1, with sketched quasienergy levels. The dashed arrowsindicate transitions between quasienergy states, which are due to emission of excitations inthe bath. The solid arrows indicate the change of g in relaxation and quantum heating. kinetic and potential energies. These features are seen, in particular, in tunneling,which is significantly modified compared to the conventional picture, because fora given g ( Q, P ) the momentum P as function of coordinate Q has 4 rather than2 branches (Dmitriev and Dyakonov, 1986 b ; Serban and Wilhelm, 2007). One of theconsequences is that, for example, for a parametric oscillator, decay of the wave func-tion in the classically inaccessible region of Q can be accompanied by oscillations,leading to under-barrier interference (Marthaler and Dykman, 2007).As seen from eqn (7.9), the quasienergy level spacing is ∝ λE sl . From eqn (7.3), itis small compared to the distance between the oscillator energy levels in the absenceof modulation, | ε n − ε n +1 | ∼ λE sl ≪ ~ ω . Quantum heating is most easy to understand in the case where the oscillator decayrate Γ is small not only compared to ω , but also to the distance between quasienergylevels in frequency units, Γ ≪ λE sl / ~ . In this Section we consider relaxation due tocoupling to a bosonic thermal bath with the coupling Hamiltonian H i linear in theoscillator coordinate and momentum and thus in the oscillator ladder operators a, a † (Schwinger, 1961), H i = ah b + H . c ., Γ ≡ Γ( ω ) = ~ − Re Z ∞ dt h [ h † b ( t ) , h b (0)] i b e iω t , (7.10)where h b depends on the bath variables only and h . . . i b denotes thermal averagingover the bath states; in what follows we assume h h b i b = 0.For the interaction (7.10), in the absence of modulation relaxation is due to tran-sitions between adjacent oscillator Fock states | N i . For zero bath temperature, thesetransitions occur only downward in energy, with emission of excitations in the bath, seethe left panel of Fig. 7.3. For a smooth density of states of the bath, resonant modula-tion does not change the decay rate, Γ( ω M ) ≈ Γ( ω ). However, it excites the oscillator, Periodically modulated quantum nonlinear oscillators as sketched in Fig. 7.3. In the stationary vibrational state the energy provided by themodulation is balanced by relaxation.As outlined in Sec. 7.1, the randomness of the transitions, i.e., the quantum noisethat accompanies relaxation, leads to quantum heating. The heating can be understoodfrom Fig. 7.3 by noticing that the quasienergy states | n i (ˆ g | n i = g n | n i ) sketched inthe right panel are linear combinations of the Fock states | N i in the left panel, | n i = P N a nN | N i . Therefore transitions between the Fock states downward in oscillatorenergy correspond to transitions both downward and upward in quasienergy, withdifferent rates. If the state with minimal g n is the stable state, transitions downward aremore likely, but upward transitions still have nonzero rates. The outcome is diffusion over quasienergy states away from the minimum of g (or the extremum of g , foran additively modulated oscillator, see Fig. 7.2), that accompanies drift (relaxation)toward the minimum (extremum) of g .For a thermal equilibrium system with nondegenerate energy levels E α , the ratioof the rates of interstate transitions | α i → | β i and | β i → | α i due to weak coupling toa bath is W th αβ /W th βα = exp[( E α − E β ) /k B T ] and is fully determined by temperature.Similarly, the ratio W nm /W mn of the transition rates between quasienergy states | n i →| m i and | m i → | n i characterizes the effective temperature T e of the distribution overthese states. It is nonzero even where the bath temperature T = 0.A modulated oscillator does not have detailed balance for T >
0. The transitions | n i → | m i are not limited to m = n ± W nm /W mn cannot be written as exp[( ε n − ε m ) / T e ] withthe same effective temperature T e for all n and m . In other words, the stationarydistribution is generally not of the Boltzmann form, it can be described by an ε - or,equivalently, g -dependent temperature.A complete analysis of the distribution can be done using the master equation forthe oscillator density matrix ρ . In slow dimensionless time τ , for the coupling to athermal reservoir (7.10) this equation reads˙ ρ ≡ ∂ τ ρ = iλ − [ ρ, ˆ g ] − ˆ κρ, ˆ κρ = κ (¯ n + 1)( a † aρ − aρa † + ρa † a )+ κ ¯ n ( aa † ρ − a † ρa + ρaa † ) , κ = ~ Γ /λE sl . (7.11)Here, the term ∝ [ ρ, ˆ g ] describes dissipation-free motion, cf. eqn. (7.9). Operator ˆ κρ describes dissipation and has the same form as in the absence of oscillator modulation(Mandel and Wolf, 1995; Dykman and Krivoglaz, 1984), κ is the dimensionless decayrate, see Table 7.1, a is the lowering operator, and ¯ n is the oscillator Planck number, a = (2 λ ) − / ( Q + iP ) , ¯ n ≡ ¯ n ( ω ) = [exp( ~ ω /k B T ) − − (7.12)[in the lab frame, a has an extra factor exp( − iω M t )]; the oscillator eigenfrequency ω is defined so as to incorporate the renormalization due to the coupling (7.10). Of utmost interest is the parameter range where quantum fluctuations about the statesof the forced vibrations of the oscillator are small compared to the interstate distancein phase space and the oscillator motion in the rotating frame is semiclassical. In esonant modulation: Quantum heating terms of Fig. 7.3, it means that the number of states in the wells of g par ( Q, P ) is large.Similarly, a large number of states are localized near the extrema of g add ( Q, P ), foradditive modulation. As seen from eqns (7.7) - (7.9), this requires the dimensionlessPlanck constant to be small, λ ≪ . (7.13)In the range (7.13) the rate of switching between the stable states of forced vibrations W sw is exponentially small, − ln W tun ∝ λ − for T = 0, see Sec. 7.3. We will assumethat W sw ≪ Γ. Then over time ∼ Γ − the oscillator will reach the state of forcedvibrations in the vicinity of which it was prepared initially and will then fluctuateabout it; switching between the states takes an exponentially longer time, see Sec. 7.3. Near the extrema function g ( Q, P ) is parabolic. The oscillator motion in the rotat-ing frame is mostly harmonic vibrations about these extrema provided the quantumsmearing ∝ λ / is small compared to the scale in phase space where the nonparabol-icity of g ( Q, P ) becomes substantial (see also Sec. 7.4). If we disregard dissipation, themotion is described by the Heisenberg equations˙ Q ≡ dQ/dτ = − iλ − [ Q, ˆ g ] , ˙ P ≡ dP/dτ = − iλ − [ P, ˆ g ] (Γ → Q − Q , P . Here, Q , P = 0 is the position of the considered extremum of g .The dimensionless vibration frequency is ν = | g QQ g P P | / [the derivatives of g ( Q, P )are calculated for Q = Q , P = 0]. Its dependence on the oscillator parameters isshown in Fig. 7.9. In the ground vibrational state the average values of ( Q − Q ) and P are different, which indicates that forced vibrations in the lab frame are squeezed.It is convenient to change from Q − Q , P to the appropriate raising and loweringoperators b † and b using the standard squeezing transformation Q − Q + iP = (2 λ ) / ( b cosh ϕ ∗ − b † sinh ϕ ∗ ) , ˆ g ≈ g ( Q ,
0) + λν (cid:0) b † b + 1 / (cid:1) sgn g QQ , ν = | g QQ g P P | / , (7.14)which makes the mapping on small-amplitude vibrations explicit; in eqn (7.14), tanh ϕ ∗ =( | g QQ | / − | g P P | / ) / ( | g QQ | / + | g P P | / ).The distribution over quasienergy states near the extrema of g ( Q, P ) can be foundfrom eqn (7.11). For small damping, κ ≪ ν , off-diagonal matrix elements of ρ in thebasis of quasienergy wave functions are small, | ρ nm | ≪ ρ nn , ρ mm for m = n . To findthe diagonal matrix elements one should substitute into eqn (7.11) ˆ g from (7.14) and,using eqns. (7.12) and (7.14), express a, a † in ˆ κρ in terms of operators b, b † keepingonly bilinear terms that contain both b and b † while disregarding terms with b , ( b † ) as well as the terms linear in b, b † . Then operator ˆ κρ in terms of b, b † becomes of thesame form as in terms of a, a † , except that ¯ n is replaced with ¯ n e ,¯ n e = ¯ n + (2¯ n + 1) sinh ϕ ∗ , T e = λν / ln[(¯ n e + 1) / ¯ n e ] . (7.15)The stationary solution of the resulting equation is of the Boltzmann type, ρ (st) = (¯ n e +1) − exp( − λν b † b/ T e ) (Dykman et al. , 2011). Parameter T e given in eqn. (7.15) is the Periodically modulated quantum nonlinear oscillators n = = =
227 427 Β n e Additive driving, small - amplitude state n = = =
227 427 Β n e Additive driving, large - amplitude state n = = = - Μ n e Parametric modulation
Fig. 7.4
The effective Planck number ¯ n e for vibrations about the state of forced vibra-tions. The left, central, and right panels refer, respectively, to the small- and large-amplitudestates of an additively modulated oscillator and to a parametrically modulated oscillator.The damping is assumed to be small, κ ≪ ν in eqn (7.11). effective dimensionless temperature of vibrations about the stable state in the rotatingframe. For ¯ n = 0 the result coincides with that for a driven oscillator resonantly coupledto a two-level system (Peano and Thorwart 2010 a , 2010 b ). The distributions overquasienergy states for other systems and other relaxation mechanisms were discussedby Verso and Ankerhold (2010) and Ketzmerick and Wustmann (2010).From eqn (7.15), ¯ n e = sinh ϕ ∗ > k B T ≫ ~ ω , we have T e ∝ T , and ρ (st) looks like a Boltzmann distribution of an oscillator with frequency ω / (1 + 2 sinh ϕ ∗ ) < ω . The dependence of ¯ n e on the parameter that characterizesa resonantly modulated oscillator in the small damping limit is shown in Fig. 7.4.Expression (7.15) is simplified also near the bifurcation point (the bifurcation pa-rameter value) where the corresponding stable vibrational state disappears. If dissipa-tion is disregarded, at the bifurcation point the corresponding extremum (two extrema,for parametric oscillator) and the saddle point of g ( Q, P ) merge, whereas for the large-amplitude state for additive modulation, the values of g ( Q, P ) at the extremum andthe saddle point coincide. As the parameters β or µ approach their bifurcation values β B or µ B , see Fig. 7.1, g QQ → β → β B ), g P P →
0. Near a bifurcation point ν ≪
1. Then T e ≈ ( λ/ n + 1) | g P P | ; for the large-amplitude state of an additively modulatedoscillator g P P should be replaced with g QQ . As we see, T e displays a characteristictemperature dependence described by the factor 2¯ n + 1. To the leading order, it isindependent of the distance to the bifurcation point η = β − β B or η = µ − µ B . Incontrast, ¯ n e ∝ ν − sharply increases with decreasing | η | . This is because the systembecomes “soft” near a bifurcation point, and respectively, the distribution broadens.The scaling of ¯ n e with η is ¯ n e ∝ ν − ∝ | η | − ξ T with ξ T = 1 / ξ T = 1 / T e apply for not too small | η | . One constraint on | η | is classical, ν ≫ κ . The other is quantum: the dimensionless spacing of quasienergy levels λν ∝| η | ξ T should be small compared to the difference ∆ g between the values of g at theextremum and the saddle point of g ( Q, P ) that merge at the bifurcation point; for thecorresponding extremum, for parametric modulation ∆ g ∝ η , whereas for additivedriving ∆ g ∝ η / for β → β B and ∆ g ∝ η / for β → β B . uantum activation The diffusion over quasienergy states, that underlies quantum heating, also populatesstates near the top of the quasienergy barrier in Fig. 7.3(b). As a result, if the sys-tem was initially occupying the left well of g ( Q, P ), for example, it will diffuse to thebarrier top and switch to the right well. Such an overbarrier transition reminds theconventional switching via thermal activation in systems close to thermal equilibrium(Kramers, 1940), where the states near the barrier top are populated as a result ofthermal fluctuations. In the case of a modulated oscillator, for low temperature theeffect is due to quantum fluctuations. Respectively, the switching mechanism can becalled quantum activation. It applies to both parametrically and additively modu-lated oscillators: in the latter case switching occurs, with probability ∼ /
2, once theoscillator located initially near an extremum of g ( Q, P ) reaches the saddle, see Fig. 7.2.For small effective Planck constant λ , the switching rate is exponentially small, W sw ∝ exp( − R A /λ ) . (7.16)This estimate can be easily understood from Fig. 7.3(b). If the ratio of the typical ratesof transitions up and down in quasienergy is W ↑ /W ↓ < ∼ ( W ↑ /W ↓ ) M , where M is the number of intrawell states.Since the dimensionless level spacing g n +1 − g n ∼ λ and the well depth g max − g min ∼ M ∼ λ − , which immediately gives − ln W sw ∼ λ − for low temperatures.This estimate applies to both additively and parametrically modulated oscillators.Even though the effective activation energy for a given stable state R A is deter-mined by quantum fluctuations for low T , quantum activation is not tunneling, it is theresult of coupling of the oscillator to a thermal bath and the quantum noise that accom-panies relaxation. Therefore finding R A requires solving the master equation (7.11). Forsmall λ this can be done using the WKB approximation. The problem is qualitativelydifferent from that of switching in systems close to thermal equilibrium, which can beefficiently approached using the instanton technique (Langer, 1967; Coleman, 1977;Affleck, 1981; Caldeira and Leggett, 1983). A modulated oscillator does not have de-tailed balance, generally, and its distribution is not of the Boltzmann form and is notcharacterized by the partition function.For small relaxation rate, switching of a modulated oscillator can also occur viadynamical tunneling with constant quasienergy, for example via tunneling betweenequal-quasienergy states in the left and right wells of g ( Q, P ) in Fig. 7.3(b), seeLarsen and Bloembergen 1976; Sazonov and Finkelstein 1976; Dmitriev and Dyakonov 1986 b ;Vogel and Risken 1988; Wielinga and Milburn 1993; Peano and Thorwart 2006; Marthaler and Dykman 2007;Serban and Wilhelm 2007. As we explain below, switching via tunneling becomes sub-stantial only where the relaxation rate of the oscillator is exponentially small, otherwiseswitching occurs via quantum activation. The rate of switching from a given vibrational state can be found from the quasi-stationary solution of the master equation. It is formed on times Γ − ≪ t ≪ W − Periodically modulated quantum nonlinear oscillators in a broad range of phase space. As mentioned previously, in this time domain, theoscillator prepared at t = 0 near the considered stable state in phase space will havecome to local equilibrium, but most likely will not have switched to another state.The physical picture of switching can be understood from the classical phase por-trait for the range of bistability shown in Fig. 7.5. The positions of the stable stateson the oscillator phase plane in the rotating frame ( Q a , P a ) are given by the stablesolutions of classical equations of motion˙ Q = ∂ P g ( Q, P ) − κQ, ˙ P = − ∂ Q g ( Q, P ) − κP. (7.17)These equations immediately follow from master equation (7.11) if in the equations forthe average values h Q i = Tr Qρ, h P i = Tr P ρ one disregards fluctuations, h Q n P m i →h Q i n h P i m → Q n P m . S H a L a a - QP a H b L S a - - QP Fig. 7.5
The phase portraits of the classical additively (a) and parametrically (b) modulatedoscillators in the rotating frame. The attractors a , correspond to the stable states of forcedvibrations in the lab frame. Their positions ( Q a , P a ) give the scaled vibration amplitudes A osc = ( Q + P ) / in Fig. 7.1. The separatrix that goes through the saddle point S is theboundary of the basins of attraction to different attractors. The phase portrait in (b) hasinversion symmetry. The parameters in (a) are β = 1 / , κ = 0 .
15; in (b) µ = 0 . , κ = 0 . For t ≪ W − , the oscillator is mostly localized in phase space near the initiallyoccupied stable state, with localization length ∼ λ / for low temperatures. The tail ofits quasi-stationary distribution is formed by large rare fluctuations. In the semiclassi-cal picture, switching occurs as a result of a large fluctuation that brings the oscillatorto the basin of attraction of the initially empty stable state. From there, the oscillatorwill most likely move to this state staying within ∝ λ / from the classical trajectory(7.17). The rate W sw is determined by the maximal rate of an appropriate fluctuation.In this section we will be interested in finding the switching exponent, i.e., theleading-order term in ln W sw . This can be done (Dykman and Smelyansky, 1988) us-ing the density matrix in the coordinate representation ρ ( Q , Q ) ≡ h Q | ρ | Q i . Equa-tion (7.11) in this representation has the form˙ ρ ( Q , Q ) = − iλ − H ( Q , Q , − iλ∂ Q , − iλ∂ Q ) ρ ( Q , Q ) , where H ( Q , Q , P , P ) = g ( Q , P ) − g ( Q , P ) − κ ( P Q + P Q − iλ ) − iκ (2¯ n + 1) (cid:2) ( Q − Q ) + ( P + P ) (cid:3) . (7.18) uantum activation Along with ρ ( Q , Q ) it is convenient to consider the density matrix in the Wignerrepresentation ρ W ( Q, P ) = Z dξe − iξP/λ ρ ( Q + ξ/ , Q − ξ/ . (7.19)For the oscillator, function ρ W in the quasi-stationary regime has a Gaussian peak atthe initially occupied stable state ( Q a , P a ); this form of ρ W is generic for semiclassicalsystems (for κ ≪ ν it follows from the results of Sec. 7.2.5). It rapidly falls off awayfrom Q a , P a . The switching exponent is determined by the maximal value of ρ W ( Q, P )for
Q, P inside the basin of attraction of the initially empty stable state or on thebasin boundary, i.e., it is determined by the tail of ρ W ( Q, P ) and in turn, by the tailof ρ ( Q , Q ).In the spirit of the WKB approximation, one can seek ρ ( Q , Q ) on the tail in theeikonal form. To the leading order in λ in the quasi-stationary regime ( ˙ ρ = 0) we have ρ ( Q , Q ) = exp[ iS ( Q , Q ) /λ ] , H ( Q , Q , ∂ Q S, ∂ Q S ) = 0 . (7.20)Equation (7.20) has the form of the Hamilton-Jacobi equation for an auxiliary classical system with coordinates Q , Q and action S , with equations of motion˙ Q j = ∂ P j H ( Q , Q , P , P ) , ˙ P j = − ∂ Q j H ( Q , Q , P , P ) ( j = 1 , . (7.21)Equations (7.20) and (7.21) map the problem of finding the tail of the densitymatrix of a dissipative quantum oscillator onto the problem of finding classical Hamil-tonian trajectories. The initial conditions for these trajectories follow from the Gaus-sian form of ρ W near Q a , P a . From (7.19) and (7.20), for the trajectories comingfrom the stable state of the oscillator the initial conditions are Q = Q = Q a and P = − P = P a , where P , = ∂ Q , S . The analysis shows that the final point on theswitching trajectory (7.21) is Q = Q = Q S and P = − P = P S , where ( Q S , P S ) isthe saddle point of the oscillator.The effective activation energy for switching from a given stable state is determinedby action S calculated along the switching trajectory, R A = Im Z ∞−∞ dτ X j =1 , P j ˙ Q j . (7.22)We took into account that the initial and final points on the trajectory are stationarystates of the auxiliary system, therefore the time integral goes from −∞ to ∞ .The formulation (7.18) - (7.22) reminds the conventional instanton formulation.The major distinction is that the motion occurs in real rather than imaginary time. Atthe same time, the switching trajectory is complex , as are also H and S . It is instructiveto compare this formulation with the theory of noise-induced switching in classicalsystems. There, for Gaussian noise, the switching rate displays activation dependenceon the noise intensity ( k B T , for thermal noise). The effective activation energy can becalculated as action of an auxiliary Hamiltonian system (Freidlin and Wentzell, 1998;Dykman, 1990; Kamenev, 2011). The trajectory followed by the auxiliary system isreal and gives the most probable trajectory that the initial noise-driven system follows Periodically modulated quantum nonlinear oscillators in switching. Such trajectory has been seen in experiment, see the chapter by H. B.Chan and C. Stambaugh in this book and references therein.One can see that, due to the symmetry of the Hamiltonian H , eqn (7.21) hasa solution Q ( t ) = Q ∗ ( t ) , P ( t ) = − P ∗ ( t ), which satisfies the boundary conditions.This solution was used to find the activation energy R A for an additively modulatedoscillator as function of β for small damping (Dykman and Smelyansky, 1988). We now discuss an alternative approach, which immediately gives the switching rateand the distribution of the oscillator in the small damping limit. If the broadening ofthe quasienergy levels is small compared to the interlevel distance, one can disregardoff-diagonal matrix elements ρ nm ( n = m ) in the basis of quasienergy wave functions.Then eqn (7.11) is reduced to a balance equation for state populations˙ ρ nn = X m ( W mn ρ mm − W nm ρ nn ) , W mn = 2 κ (cid:2) (¯ n + 1) | a nm | + ¯ n | a mn | (cid:3) , (7.23)where a nm ≡ h n | a | m i (we remind that a is the oscillator lowering operator). Wedisregard tunneling when defining functions | n i ≡ ψ n ( Q ), i.e., we use the “intrawell”wave functions in Fig. 7.3(b); the effect of tunneling is exponentially small for λ ≪ a mn can be calculated in an explicit form using the WKB ap-proximation. It relates the problem to that of classical conservative motion ˙ Q = ∂ P g, ˙ P = − ∂ Q g . Such motion is periodic oscillations in time Q ( τ ; g ) , P ( τ ; g ) withgiven g ( Q, P ) = g and with dimensionless frequency ν ( g ) that depends on g . For nottoo large | m − n | , matrix element a mn is given by the ( m − n )th Fourier component ofthe periodic function a ( τ ; g n ) = (2 λ ) − / [ Q ( τ ; g ) + iP ( τ ; g )] calculated for the classi-cal orbit g ( Q, P ) = g n . Formally, we require that | m − n | ≪ n , but the results applyalso near the extrema of g ( Q, P ) where n ∼
1, since ψ n ( Q ) are close to the wavefunctions of a harmonic oscillator for such n .The evaluation of a mn simplifies if one notices that g ( Q, P ) is quartic in
Q, P andeven in P . Because of that, the orbits Q ( τ ; g ) , P ( τ ; g ) are described by the Jacobi el-liptic functions and are double periodic in τ . To calculate a mn one can then integrate a ( τ ; g n ) exp[ iτ ( n − m ) ν ( g n )] along an appropriately chosen closed contour on the com-plex τ -plane. The result is determined by the pole of a ( τ ; g n ) and has a simple form(Marthaler and Dykman, 2006). In particular, W mn exponentially decays with | m − n | for 1 ≪ | m − n | , with the exponent that depends on the sign of m − n .The quasi-stationary populations of neighboring states n and n ± ρ nn is a smooth function of n .Respectively, we seek the quasi-stationary distribution in the eikonal form, ρ nn =exp( − R n /λ ), R n ≡ R ( g n ). To the leading order in λ , eqn (7.23) then reads X k W n + k n { − exp [ − kν ( g n ) R ′ ( g n )] } = 0 , R ′ ( g ) ≡ dR/dg, (7.24)where we used g n + k ≈ g n + λkν ( g n ) , R n + k ≈ R n + λkν ( g n ) R ′ ( g n ) and W n n − k ≈ W n + k n for | k | ≪ n .From eqn (7.24), R ′ ( g ) is independent of λ and is given by a solution of a polynomialequation. For g close to its value g a = g ( Q a , P a ) at a stable state ( P a → κ → uantum activation the solution of eqn (7.24) is of the Boltzmann form, ρ nn ∝ exp[ − λν nR ′ ( g a )]; as onecan show, it coincides with the result of Sec. 7.2.5, with R ′ ( g a ) = λ/ T e . However, this isonly the asymptotic solution, generally the distribution is not described by an effectivetemperature, because R ′ ( g ) varies with g . We note that the corrections disregarded inderiving eqn (7.24) are ∝ λ , which justifies this equation for λ ≪ g S ≡ g ( Q S , P S ). Therefore, to the leading order in λR A = Z g S g a dgR ′ ( g ) . (7.25)In Fig. 7.6 we show the activation energy of switching between period-two states of aparametric oscillator obtained from eqns (7.24) and (7.25) (Marthaler and Dykman, 2006).For small damping, R A depends on two parameters, the scaled frequency detuning ofthe modulating field µ and the Planck number of the oscillator ¯ n . As seen in the rightpanel, the value of (2¯ n + 1) R A decreases with increasing temperature and alreadyfor ¯ n = 1 becomes very close to the result for the classical range ¯ n ≫
1. In thisrange, switching is thermally activated and R A ∝ /T (Dykman et al. , 1998); cf. theexperiments by Lapidus et al. (1999) and Chan and Stambaugh (2007). Both in theclassical limit and near bifurcation points | ν ( g ) R ′ ( g ) | ≪ R ′ ( g ). - 1 0 1024 tun R A ,n=0 R A ,n 0 - 1 0 1 n 0n=1 n=0.5classical R A ( + ) Fig. 7.6
The tunneling exponent S tun for tunneling between the extrema of g ( Q, P ) of aparametric oscillator and the switching activation energy R A in the limit of small dampingfor different values of the Planck number (Marthaler and Dykman, 2006). The cross indicatesthe value of S tun obtained by Wielinga and Milburn (1993). Near the bifurcation point µ B in Fig. 7.1 where the stable period two statesmerge together and disappear, R A ∝ (2¯ n + 1) − ( µ − µ B ) . For additive driving, forswitching from the large and small-amplitude vibrational states in Fig. 7.1 near thecorresponding bifurcation points R A ∝ (2¯ n +1) − ( β − β B ) and R A ∝ (2¯ n +1) − ( β B − β ) / , respectively (Dykman and Smelyansky, 1988). This scaling applies not too closeto bifurcation points, where still ν ≡ ν ( g a ) ≫ κ . Detailed balance for T = 0 . An important feature of the semiclassical matrix el-ements, which follows from the double-periodicity of a ( τ ; g ), is that | a mn /a nm | = Periodically modulated quantum nonlinear oscillators exp[( m − n ) c ( g n )] for | m − n | ≪ n , with c ( g n ) that smoothly depends on n . There-fore for T = 0 the system has detailed balance: the ratio of the transition rates ispath-independent, W nm W mk / ( W km W mn ) = W nk /W kn , and eqn (7.23) has a quasi-stationary solution ρ mm /ρ nn = W nm /W mn . This gives the quasi-stationary popula-tion of states near the saddle point ( Q S , P S ) relative to that near the initially occupiedstable state ( Q a , P a ), and thus the activation energy R A = − R g S g a dgν − ( g ) c ( g ). For aparametric oscillator R A for T = 0 is shown in Fig. 7.6.A remarkable property of the detailed balance solution seen from Fig. 7.6 is fragility:the value of R A for T = 0 differs from R A for T →
0. The fragility emerges for small λ in the limit where the relaxation rate Γ ∝ κ →
0. For nonzero κ , the transition from T = 0 to nonzero T solutions should be continuous (Dykman and Smelyansky, 1988);more work is required to study this transition. The detailed balance condition for T = 0applies for arbitrary κ . This was used to find the stationary probability distribution forboth additively and parametrically modulated oscillators (Drummond and Walls, 1980;Kryuchkyan and Kheruntsyan, 1996) and the switching rate in the overdamped limitfor parametric modulation (Drummond and Kinsler, 1989).We now compare switching via quantum activation and dynamical tunneling. Therate of switching via tunneling is ∝ exp( − S tun /λ ), where S tun is the tunneling ac-tion near the corresponding extremum of g ( Q, P ). The tunneling rate prefactor is ∝ ν λE sl / ~ . The prefactor in the rate of switching via quantum activation for smalldamping is ∝ Γ ∝ κ . From Fig. 7.6 and from similar results for additively modulatedoscillator, 2 S tun > R A (Dykman and Smelyansky, 1988; Marthaler and Dykman, 2006).Therefore unless κ is exponentially small, oscillator switches via quantum activationrather than tunneling. The analysis of switching near bifurcation parameter values is particularly impor-tant. In this range the switching rates display universal, model-independent features;also, the range is interesting for many applications, in particular, for the Josephsonbifurcation amplifiers (Vijay et al. , 2009). Since the frequency of vibrations about ametastable state ν rapidly decreases with the decreasing distance to a bifurcationpoint η , see Sec. 7.2.5, for small η the oscillator motion in the rotating frame is oftenoverdamped, ν ≪ κ , which we will assume in this subsection to be the case.Near a bifurcation point, the behavior of the system is controlled by a “soft mode”,a dynamical variable that slowly changes in time. Without fluctuations the occurrenceof such variable is well-known in classical dynamics (Guckenheimer and Holmes, 1997).It emerges because, for small η , the stable and unstable states of the system are closeto each other in phase space. In Fig. 7.5(a) the attractor corresponding to stableforced vibrations with large or small amplitude becomes close to the saddle point (thesaddle-node bifurcation), whereas in Fig. 7.5(b) for small µ − µ B the attractors becomeclose to each other and to the saddle point between them (the supercritical pitchforkbifurcation). In appropriately scaled variables, slow motion along the direction betweenthe close states is described by equation ˙ x = − U ′ ( x ). The potentials U ( x ) for thebifurcations of interest are sketched in Fig. 7.7. uantum activation &’( ) (’)*+)*))*+&’+ ) +’+)+, ,-./ -0/ Fig. 7.7
Effective potentials for overdamped motion near (a) the saddle-node and (b) the su-percritical pitchfork (onset of stable period-two vibrations) bifurcations. The equation of mo-tion in scaled variables is ˙ x = − U ′ ( x ) with U ( x ) = − x / ηx in (a) and U ( x ) = x / − ηx / η = 1. The minimum and maximum of U in (a) correspond to stableand unstable states of forced vibrations for additive modulation; the minima in (b) corre-spond to the stable vibrations with opposite phase, for parametric modulation. The specificform of U ( x ) is obtained by keeping the lowest order terms in x and η compatible with thecondition of merging of one or two stable and an unstable state of the system for η = 0. A theory of the switching rate based on the master equation in the Wigner repre-sentation was developed earlier (Dykman, 2007). Here we sketch a somewhat simplerderivation based on the quantum Langevin equation. Even without writing this equa-tion one can see that, since the motion is overdamped and is characterized by one slowdynamical variable with no conjugate variable, commutation relations for this slowvariable are irrelevant. Then its fluctuations are the same as in the case of a classicaloscillator (Dykman and Krivoglaz, 1980), the only difference being that the fluctua-tion intensity is determined by quantum rather than classical noise from the thermalbath. From Fig. 7.3, for nonzero bath temperature the noise intensity is proportionalto the overall rate of the bath-induced transitions up and down between the Fockstates of the oscillator, which in turn is proportional to 2¯ n + 1 (the Einstein relation).Respectively, using the quantum to classical correspondence for high temperature, inthe expression for the rate of classical thermally activated switching one should replace k B T with (2¯ n + 1) ~ ω /
2, which indeed gives the right answer.The quantum Langevin equation in the rotating frame is an extension of the Heisen-berg equation for
Q, P that includes the effect of coupling to a thermal bath. In thesame approximation that led to the Markov master equation (7.11), in slow time τ ˙ Q = − iλ − [ Q, ˆ g ] − κQ + ˆ f Q ( τ ) , ˙ P = − iλ − [ P, ˆ g ] − κP + ˆ f P ( τ ) . (7.26)Here, ˆ f Q,P are quantum noise operators. Equation (7.26) is well known (Ford et al. , 1965)for a harmonic oscillator linearly coupled to a bath of harmonic oscillators. It appliesalso for a more general form of the coupling to the bath H i , eqn (7.10). To the leadingorder in H i it can be obtained just by iterating the Heisenberg equations of motionfor the bath (Lax, 1966). The nonlinearity of our oscillator is relatively weak and doesnot affect the form of the dissipative and noise terms in eqn (7.26).From eqn (7.10), ˆ f Q , ˆ f P are linear combinations of operators h b ( t ) exp( − iω M t ), h † b ( t ) exp( iω M t ) calculated disregarding the coupling to the oscillator. For a smootharound ω M power spectrum of h b , the noise is δ -correlated in slow time, h ˆ f Q ( τ ) ˆ f Q ( τ ′ ) i b = h ˆ f P ( τ ) ˆ f P ( τ ′ ) i b = λκ (2¯ n + 1) δ ( τ − τ ′ ) , (7.27) Periodically modulated quantum nonlinear oscillators and h [ ˆ f Q ( τ ) , ˆ f P ( τ ′ )] i b = 2 iλκδ ( τ − τ ′ ). This commutation condition guarantees thatthe commutation relation [ Q, P ] = iλ does not change in time. The noise correlatorsare understood here in the Stratonovich sense (Van Kampen, 2007); in particular, h [ ˆ f Q ( τ ) , P ( τ )] i b = h [ Q ( τ ) , ˆ f P ( τ )] i b = iλκ .Near a bifurcation point we can simplify eqn (7.26) using essentially the sameapproach as for classical systems (Dykman and Krivoglaz, 1980). We can change tooperators Q − Q B and P − P B ; here Q B and P B are the classical values of Q and P at the bifurcation point given by the appropriate stationary solutions of the classicalnoise-free equations of motion (7.17). We can then further change from Q − Q B , P − P B to Q ′ , P ′ by rotating coordinates in the ( Q, P ) plane. We choose the angle of rotationin such a way that at the bifurcation point the equation for ˙ P ′ does not contain linearin Q ′ , P ′ terms. The equation of motion for Q ′ , on the other hand, has the form˙ Q ′ = − A QQ Q ′ + A QP P ′ + (nonlinear terms in Q ′ , P ′ ) + ˆ f Q ′ . Therefore, for small expectation values of Q ′ , P ′ , the operator P ′ is slowly varyingin time compared to Q ′ . We note that, for an additively modulated oscillator, therotation is not needed, P ′ = P − P B , and for a parametrically modulated oscillator Q B = P B = 0.We will study the slow dynamics of P ′ for a small deviation η of the controlparameter from its bifurcation value. For | η | ≪
1, in dimensionless time 1 /A QQ ∼ Q ′ reaches its adiabatic form ≈ ( A QP /A P P ) P ′ + O ( η ), while P ′ remainsunchanged. The expression for Q ′ can be then substituted into equation for ˙ P ′ . Theresulting equation for ˙ P ′ reads˙ P ′ ≈ B η + B ηP ′ + B P ′ + B P ′ + ˆ f P ′ ( τ ) . (7.28)Here, near the pitchfork bifurcation point for a parametrically modulated oscillator B = B = 0 by symmetry and B , ∼
1; near the saddle-node bifurcation pointfor an additively driven oscillator B , ∼ ∝ ηP ′ , P ′ can be dis-regarded, cf. Guckenheimer and Holmes (1997). The explicit form of B , , , followsfrom eqns (7.26).The correlator of ˆ f ′ P is the same as of ˆ f P in eqn (7.27), whereas ˆ f Q ′ drops out fromeqn (7.28), to the leading order in η ( ˆ f Q ′ enters the equation for ˙ P ′ with a coefficient ∝ P ′ ). Therefore the δ -correlated noise ˆ f P ′ behaves as classical, as it commutes withitself, and then P ′ behaves as a classical variable. However, the intensity of ˆ f P ′ is ∝ n + 1, so that the fluctuations still have quantum origin.Equation (7.28) can be written as ˙ P ′ = − ∂ P ′ U ( P ′ ) + ˆ f P ′ ( τ ). It maps the oscil-lator dynamics onto the dynamics of an overdamped classical Brownian particle in apotential U . The form of the potential depends on the nature of the bifurcation. Foran additively modulated oscillator, where the stable and unstable states of forced vi-brations merge for β = β B , this potential in rescaled variables is shown in Fig. 7.7(a),whereas for a parametrically modulated oscillator where the period-two states mergefor µ = µ B , it is shown in Fig. 7.7(b); the dynamics for small µ − µ B is described bya potential of the opposite sign. ower spectra of modulated quantum oscillators Table 7.2
Quantum-activated switching near bifurcation points, W sw = Ω sw exp( − R A /λ ) Additive driving Parametric drivingBifurcation points β B , = h κ ∓ (cid:0) − κ (cid:1) / i µ B , = ∓ (1 − κ ) / Squared amplitudeat bifurcation points ( A ) B , = (cid:2) ± (1 − κ ) / (cid:3) ( A ) B , = 0Distance to bifurcation η = β − β B η = µ − µ B Activation energy R A √ κ | b | / β / B | η | / / (2¯ n + 1) | µ B | η / (2¯ n + 1)Prefactor Ω sw | δω | ( bη/ / /πβ / B Γ | ηµ B | [1+Θ( µ B )]2 / πκ Auxiliary parameter b = β / B [3( A ) B − / κ Switching from a metastable vibrational state corresponds to a quantum-activatedescape from the corresponding minimum of the potential U . The rate of escape viaquantum tunneling is exponentially smaller (Dykman, 2007). The activation exponent R A is determined by the height of the potential barrier, and therefore it displays acharacteristic scaling dependence on the distance to the bifurcation point η = β − β B or η = µ − µ B . The results are summarized in Table 7.2.The scaling behavior of ln W sw with the distance to the bifurcation point for clas-sical oscillators has been seen for additive driving near the saddle-node bifurcationpoints (Siddiqi et al. , 2006; Stambaugh and Chan, 2006 a ) and near the critical point( β = 8 / , κ = 1 / √
3) where both stable vibrational states and the unstable statein Fig. 7.5(a) merge together (Aldridge and Cleland, 2005) , and for parametricallymodulated oscillators near the pitchfork bifurcations (Chan and Stambaugh 2007; seealso the chapter by Chan and Stambaugh in this book). Recently the scaling behav-ior near the saddle-node bifurcation and the characteristic temperature dependenceln W sw ∝ (2¯ n + 1) − were found also in the quantum regime (Vijay et al. et al. in this book). The results provide direct evidence insupport of the mechanism of quantum activation. Of significant interest are spectra of a resonantly modulated oscillator, including thepower spectrum and the spectrum of the response to an additional field (Dykman and Krivoglaz, 1979;Drummond and Walls, 1980; Collett and Walls, 1985). Among other characteristics,the power spectrum determines the emission spectrum of the oscillator and relaxationof a qubit coupled to the oscillator (Serban et al. , 2010). Spectral measurements havebeen reported both in the classical (Stambaugh and Chan, 2006 b ; Almog et al. , 2007) The results on the switching rates for an additively modulated classical oscillator near the criticalpoint (Dykman and Krivoglaz, 1980) extend to the quantum regime if one replaces k B T → ~ ω (¯ n +1 / Periodically modulated quantum nonlinear oscillators and quantum regimes (Wilson et al. , 2010); see also the chapters by Chan and Stam-baugh and by Wilson et al. .The spectra of interest are described by functions hh K, L ii ω = R ∞ dte iωt hh K ( t ) L (0) ii , hh K ( t ) L (0) ii = ω M π R π/ω M dt i h [ K ( t + t i ) − h K ( t + t i ) i ][ L ( t i ) − h L ( t i ) i ] i ; (7.29)recall that ω M is ω F for additive and ω F / ω M is close to ω . We will consider spectra near resonance, with | ω | ≈ ω ; the operators K and L willbe the ladder operators a or a † . In particular, the peak in the spectrum of spontaneousradiation emission by the oscillator is determined by Re hh a † , a ii ω with ω ≈ − ω , as inthe absence of periodic modulation, cf. Mandel and Wolf (1995). A physical exampleis radiation from a nonlinear cavity, with the oscillator being the cavity mode modu-lated by an incident electromagnetic field (Drummond and Walls, 1980) or excited bymodulating the boundary of the cavity (Wilson et al. , 2010).An additional weak resonant force A ′ exp( − iωt )+ c.c. causes the oscillator to vi-brate at frequencies ω and 2 ω M − ω . The vibrations at frequency ω are describedby the scaled susceptibility χ ( ω ), which determines the corresponding displacement h δq i = ( A ′ / ω M ) χ ( ω ) exp( − iωt )+ c.c., χ ( ω ) = i (cid:2) hh a, a † ii ω − hh a † , a ii ∗− ω (cid:3) . (7.30)In the absence of modulation eqn (7.30) goes over into the standard expression for theoscillator susceptibility. Function χ ( ω ) includes an extra factor 2 ω M compared to thenotation we used previously (Dykman et al. , 1994).Spectra of a modulated oscillator have two major contributions. One comes frommotion in the vicinity of stable vibrational states, where the oscillator spends much ofthe time, and the other comes from fluctuation-induced transitions between the states.We will consider them separately. The contribution to the oscillator spectra from motion near a stable vibrational stateshould be analyzed differently depending on the dimensionless oscillator relaxationrate κ . For κ ≫ λ (see below), one can linearize and then solve quantum equationsof motion (7.26) near the position ( Q a , P a ) of a given stable state, similar to theclassical case (Dykman and Krivoglaz, 1979; Dykman et al. , 1994) and to how it isdone in the coherent state representation (Drummond and Walls, 1980). Equivalently,one can linearize in ( Q − Q a , P − P a ) the drift term in the master equation for thedensity matrix in the Wigner representation (7.19). The resulting contribution to thepower spectrum is (Serban et al. , 2010; Dykman et al. , 2011)Re hh a, a † ii (a) ω ≈ κ Γ (¯ n + 1) h ( ν − c ) + κ i + ¯ nc ( ν − ν ) + 4 κ ν , ν = κ Γ ( ω − ω M ) . (7.31)The parameters in eqn (7.31) are expressed in terms of the scaled squared vibrationamplitude in the considered stable state r = Q + P , ower spectra of modulated quantum oscillators ( ν ) add = κ + 3 r − r + 1 , ( c ) add = 1 − r , ( c ) add = r , ( ν ) par = 4 r ( r − µ ) , ( c ) par = µ − r , ( c ) par = κ + µ , (7.32)where subscripts “add” and “par” refer to additive and parametric modulation, re-spectively; in eqn (7.31) we use the factor Γ /κ as the frequency scale, this factor isindependent of the decay rate Γ. In the limit of small damping, κ → ν a goes overinto the dimensionless frequency ν of vibrations about the attractor, eqn (7.14).The contribution of small-amplitude fluctuations to Re hh a † , a ii − ω for ω close to ω is also given by eqn (7.31) in which one should interchange ¯ n ⇌ ¯ n + 1. Thisrelation, together with eqn (7.30), give the imaginary part of the susceptibility χ ( ω ). Inevaluating these local contributions we replace in the definition of the power spectrum(7.29) h a ( t ) i → h a ( t ) i (a) = (2 λ ) − / ( Q a + iP a ) exp( − iω M t ).The susceptibility for not too small κ has the same form in the quantum and clas-sical case. It can be obtained by adding to the linearized in Q − Q a , P − P a eqns (7.26)a weak-driving term ∝ A ′ , which oscillates in the rotating frame at frequency ω − ω M .One then finds the solution that oscillates at this frequency. Since the equations arelinear, the noise term, which is the only term that has a different form in the classicaland quantum case, drops out on averaging, giving for κ ≫ λχ a ( ω ) ≈ i κ Γ κ − i ( ν − c ) ν − ν − iκν , ν = κ Γ ( ω − ω M ) . (7.33)For underdamped vibrations about the stable state in the rotating frame, where κ ≪ ν (but still κ ≫ λ ), the spectra (7.31) and Im χ a ( ω ) have peaks at frequencydetuning ω − ω M = ± (Γ /κ ) ν . There are generally two peaks on the opposite sides of ω M and they generally have different amplitudes. Their shape is close to Lorentzian,with halfwidth Γ. As the ratio ν /κ decrease the peaks start overlapping. - - - H Ω-Ω F L(cid:144) ∆ΩF a Additive driving, power spectrum - - - - H Ω-Ω F L(cid:144) ∆ΩΧŽ a '' Additive driving, absorption spectrum
12 3 - - - H Ω-Ω F L Ω F (cid:144) F F a Parametric driving, power spectrum
Fig. 7.8
The contributions to the scaled power spectra Φ a ( ω ) = Re ( κ/ Γ) hh a † , a ii (a) − ω andthe susceptibility, ˜ χ ′′ a ( ω ) = Im ( κ/ Γ) χ a ( ω ) from fluctuations about stable vibrational statesfor κ ≫ λ . The scaled decay rate is κ = 0 .
3. For the additively driven oscillator, curves 1and 2 refer to the small- and large-amplitude vibrational states; the parameters are β = 0 . n = 0 .
3. Curves 1 to 3 for the parametrically driven oscillator refer to µ = − . n = 0 , . , . The spectra are illustrated in Fig. 7.8. As seen from the figure and eqn (7.31), theemission spectrum hh a † , a ii (a) − ω is symmetric for ¯ n = 0; this holds for both additive(Drummond and Walls, 1980) and parametrically modulated oscillators. The onset of Periodically modulated quantum nonlinear oscillators emission for ¯ n = 0 is related to quantum heating. In terms of quantum optics, if themodulating field is electromagnetic radiation and one considers emission of photons bythe modulated oscillator, the emission can be thought of as resulting from multi-waveparametric process. Both the emission intensity and the positions of the spectral peaksdepend on the modulation strength in a complicated way.An interesting feature of the susceptibility seen from Fig. 7.8 is that Im χ ( ω ) canbecome negative (Dykman and Krivoglaz, 1979). In the corresponding frequency rangean additional weak field is amplified by the strong field, which can be also consideredas a parametric multi-wave process. The amplification occurs in spite the fact that theabsorption coefficient integrated over the whole spectrum is positive: from eqn (7.33) R dωχ ′′ a ( ω ) = π . This sum rule holds because, for any Fock state | N i (cf. Fig. 7.3), aninduced dipolar transition up in energy has a larger amplitude [ ∝ ( N + 1) / ] thandown in energy ( ∝ N / ). For a modulated nonlinear oscillator, the spectral regions ofabsorption and amplification are separated; weak field amplification generally occursfor both additive and parametric modulation. Interesting quantum effects emerge in the spectra for small κ , where not only κ ≪ ν ,but also κ . λ ≪ ν . In the whole range κ ≪ ν the stable states of the oscillatorare at the extrema of g ( Q, P ). However, the approach of the previous section doesnot apply for κ . λ . To consider the range κ . λ , one has to take into account thenonequidistance of the quasienergy levels g n , which results from the nonlinearity ofvibrations about the extrema. For a given extremum at Q = Q , P = 0, to the leadingorder in the nonlinearity g n ≈ g + (cid:2) λν ( n + 1 /
2) + λ V n ( n + 1) / (cid:3) sgn g QQ , (7.34)where g = g ( Q , V = ν ( dν/dg ) g is determined by the slope of thedimensionless frequency ν ( g ) of vibrations in the rotating frame with given g ; thequantum correction ∼ λ to the frequency ν = ν ( g ) is assumed to be incorporated.The parameters ν and V are plotted in Fig. 7.9. For small λ the vibration nonlinearityis small, λ | V | ≪ ν . The quasienergy spectrum (7.34) is sketched in Fig. 7.10(a). Ν - V H a L
227 427 Β Additive driving, small - amplitude state Ν - V H b L
227 427 Β Additive driving, large - amplitude state Ν - V H c L - Μ Parametric driving
Fig. 7.9
The scaled eigenfrequency of vibrations about the stable states in the rotating frame ν and the parameter of nonequidistance of quasienergy levels V , eqn (7.34), for additive andparametric modulation. The power spectrum of the oscillator in a given stable state is formed by transi-tions | n i → | n ± i between neighboring quasienergy states in Fig. 7.10(a). One might ower spectra of modulated quantum oscillators expect that the spectrum is then a superposition of partial spectra that correspond toindividual transitions, with width determined by the reciprocal lifetime of the respec-tive quasienergy states. However, to spectrally resolve the transitions one has to waitfor time & ( κ/ Γ) | λV | − . If this time becomes comparable or smaller than the lifetime,the transitions are not independent, the transition amplitudes interfere. The overalloscillator spectrum is then formed by many interfering transitions.The typical number of states that contribute to the spectrum is determined bythe effective Planck number ¯ n e , eqn (7.15). We assume that λ | V | ¯ n e ≪ ν , so thatall dimensionless transition frequencies are close to ± ν . Therefore the spectrum haswell-separated peaks at ω − ω M ≈ ± (Γ /κ ) ν . Near the peaks, eqn (7.29) can be sim-plified using the interrelation (7.14) between operators Q, P and operators b, b † thatdescribe vibrations about the stable state in the rotating frame. The operators b, b † are the ladder operators of an auxiliary oscillator in thermal equilibrium, with scaledeigenfrequency ν , energy spectrum (7.34), and temperature T e . The power spectrumof this oscillator isΦ bb † ( ν ) = Re Z ∞ dte iντ h b ( τ ) b † (0) i = e λν / T e Re Z ∞ dte − iντ h b † ( τ ) b (0) i . (7.35)Equation (7.35) is written for | ν − ν sgn g QQ | ≪ ν where Φ bb † ( ν ) has a narrow peak.The occurrence of such a peak can be understood by noticing that, if one disregardsnonlinearity and decay, b ( τ ) = exp( − iν τ sgn g QQ ) b (0), and then Φ bb † ( ν ) becomes a δ -function. The nonlinearity and decay lead to broadening of the δ -function.From eqns (7.14) and (7.29), near the peak of the power spectrum of the originalmodulated oscillator on the high-frequency (low-frequency, for sgn g QQ <
0) side ofthe forced-vibration frequency, ω ≈ ω M + (Γ /κ ) ν sgn g QQ , κ Γ Re hh a, a † ii (a) ω ≈ κ Γ e λν / T e Re hh a † , a ii (a) − ω ≈ cosh ϕ ∗ Φ bb † ( ν ) , (7.36)where, as in eqns (7.31) and (7.33), ν = ( κ/ Γ)( ω − ω M ).Similarly, near the peak in the power spectrum on the low-frequency (high-frequency,for sgn g QQ <
0) side of ω M , where ν = ( κ/ Γ)( ω − ω M ) is close to − ν sgn g QQ , κ Γ e λν / T e Re hh a, a † ii (a) ω ≈ κ Γ Re hh a † , a ii (a) − ω ≈ sinh ϕ ∗ Φ bb † ( − ν ) . (7.37)Equations (7.30), (7.36 ), and (7.37) also describe resonant peaks in the oscillatorabsorption spectrum Im χ a ( ω ). The shapes of the peaks are determined by functionΦ bb † ( ν ). The peak near frequency ω M + (Γ /κ ) ν sgn g QQ corresponds to absorption ofthe additional field, Im χ a ( ω ) >
0, whereas the one near ω M − (Γ /κ ) ν sgn g QQ , withIm χ a ( ω ) <
0, corresponds to amplification and has a smaller area.Equations (7.36) and (7.37) reduce the problem of the spectra of modulated oscil-lator for weak damping, κ ≪ ν , to calculating the power spectrum of an auxiliaryequilibrium oscillator Φ bb † ( ν ) (Dykman et al. , 2011). This problem was discussed pre-viously (Dykman and Krivoglaz, 1984). The result for the spectrum near its maximumcan be presented in the formΦ bb † ( ν ) = (¯ n e + 1)Re P ∞ n =1 φ ( n, ν ); (7.38) φ ( n, ν ) = 4 n (Λ − n − (Λ + 1) − ( n +1) [ κ (2 ℵ n − − i ( ν − ν sgn g QQ )] − , Periodically modulated quantum nonlinear oscillators
12 3 (b) H Ν-Ν L(cid:144) Κ F bb + (cid:144) H n - e + L Fig. 7.10 (a) A sketch of the quasienergy spectrum and dimensionless transition frequenciesof weakly nonlinear vibrations about the stable state in the rotating frame for g QQ > ν as given byeqn (7.38) for a comparatively large ratio of the level nonequidistance to the decay rate, ϑ = 10 and for g QQ >
0. The curves 1 to 3 refer to ¯ n e = 0 . , . where Λ = ℵ − [1 + iϑ (2¯ n e + 1)] and ℵ = (cid:2) iϑ (2¯ n e + 1) − ϑ (cid:3) / (Re ℵ > ϑ = ( λV / κ ) sgn g QQ is determined by the interrelation between thenonequidistance of the transition frequencies in Fig. 7.10(a) and the decay broadening κ of the quasienergy levels,.Equation (7.38) represents the spectrum as a sum of effective partial spectraRe φ ( n, ν ) that can be provisionally associated with transitions | n − i → | n i betweenthe quasienergy levels in Fig. 7.10(a). Functions φ ( n, ν ) depend on two parameters, ϑ and ¯ n e . The form of φ ( n, ν ) is particularly simple for a comparatively large levelnonequidistance or small damping, λ | V | ≫ κ , in which case φ ( n, ν ) ≈ n (¯ n e + 1) exp[ − λν ( n − / T e ] κ n − i [ ν − ν ( g n − )] , | ϑ | ≡ λ | V | κ ≫ . (7.39)Here, ν ( g n − ) = ( ν + λV n )sgn g QQ is the frequency of the | n − i → | n i transition inFig. 7.10(a) and κ n = κ [2 n (2¯ n e + 1) −
1] is the total halfwidth of levels | n − i and | n i . Therefore Re φ ( n, ν ) has a conventional form of a partial spectrum.For | ϑ | ≫ bb † ( ν ) has a fine structure. The intensities ofthe individual lines (7.39) immediately give the effective quantum temperature T e .However, the fine structure is pronounced only in a limited range of the effectivePlanck numbers ¯ n e . This is seen from Eq. (7.39). For ¯ n e ≪ φ (1 , ν ) has anappreciable intensity while Re κφ ( n, ν ) ≪ n >
1. On the other hand, for large¯ n e the linewidth κ n becomes large and spectral lines with different n overlap, startingwith large n . Not only are they overlapping, but their shape is also changed comparedto eqn (7.39) due to the interference of transitions. The evolution of the fine structurewith varying ¯ n e as given by eqn (7.38) is illustrated in Fig 7.10(b).As | ϑ | decreases all partial spectra start to overlap, and for | ϑ | . | ϑ | → φ ( n, ν ) ∝ δ n, , and the spectrum has the form of a single Lorentzian peakof dimensionless halfwidth κ , Φ bb † ( ν ) = (¯ n e + 1) κ (cid:2) κ + ( ν − ν sgn g QQ ) (cid:3) − . Thisexpression, with account taken of eqn (7.36), agrees with eqn (7.31) in the range λ | V | ≪ κ ≪ ν where both apply. Generally, because of the nonlinearity of the auxil-iary oscillator, the shape of the spectrum depends on ¯ n e even where there is no fine ower spectra of modulated quantum oscillators structure. The spectrum is non-Lorentzian and displays a characteristic asymmetryfor large ¯ n e , where | ϑ | ¯ n e >
1. This asymmetry is described by eqn (7.38) and providesa way of determining quantum temperature.Interference of transitions occurs also in various quantum oscillators and oscillator-type systems in the absence of modulation, from localized vibrations in solids to largespins in strong magnetic fields and Josephson-junction based systems, to mention buta few. In all these systems the spectra strongly depend on the interrelation between thelevel nonequidistance and the decay rate. Interference of transitions is important alsofor classical vibrational systems with fluctuating frequency, like nano- and microme-chanical resonators with a fluctuating number and/or positions of attached molecules(Vig and Kim, 1999; Yang et al. , 2011). The spectra of such systems can also be asym-metric and display a fine structure (Dykman et al. , 2010). On the formal side, theserecent results indicate that, for different systems and physical mechanisms, interfer-ence of transitions can be described by linear equations for coupled partial spectra.These equations are convenient for numerical solution.
Along with fluctuations about stable vibrational states, quantum noise leads to occa-sional interstate switching discussed in Sec. 7.3. Important manifestations of switchingare additional peaks in the oscillator power spectrum and susceptibility. The peaks arecentered at the forced-vibration frequency ω M and are supernarrow in the sense thattheir width is much smaller than the oscillator decay rate Γ.To describe the peak in the power spectrum we note that the populations w and w = 1 − w of the stable vibrational states 1 and 2 satisfy the balance equation dw /dt = − W ( w − ¯ w ) , W = W (12)sw + W (21)sw , ¯ w = W (21)sw /W, (7.40)where W ( ij )sw is the rate of switching from i th to j th state and ¯ w is the mean statepopulation. Equation (7.40) has the same form as in the classical case, except that theswitching rates are determined by quantum fluctuations.Fluctuations of the state populations lead to fluctuations of the expectation valuesof the operators a ( t ) , a † ( t ) averaged over time ∼ Γ − , which switch between theirstable-states values a i ( t ) , a † i ( t ), i = 1 ,
2; in the lab frame in the i th stable state a i ( t ) = a i exp( − iω M t ) with a i = (2 λ ) − / ( Q a i + iP a i ) (Dykman et al. , 2011). From eqn (7.40),similar to the case of a classical oscillator (Dykman et al. , 1994), for the contributionfrom interstate switching to the power spectrum we obtainRe hh a, a † ii (sw) ω ≈ Re hh a † , a ii (sw) ω ≈ | a − a | ¯ w ¯ w W/ [ W + ( ω − ω M ) ] . (7.41)The typical width of the spectral peak (7.41) is given by the total switching rate W ,it is exponentially smaller than the decay rate, cf. eqn (7.16). The intensity (area) ofthe peak is determined by the factor ¯ w ¯ w . For a parametrically modulated oscillatorfor µ < µ B the populations of the states are equal by symmetry and this factor isequal 1/4. A peak in the power spectrum related to interstate transitions was seen inthe radiation from a parametrically modulated microwave cavity (Wilson et al. , 2010).For additive driving, on the other hand, the populations ¯ w and ¯ w are exponen-tially different, and thus ¯ w ¯ w is exponentially small everywhere except for a narrow Periodically modulated quantum nonlinear oscillators parameter range where the switching rates W (12)sw and W (21)sw are almost equal. Thisrange corresponds to a smeared kinetic “phase transition” (Bonifacio and Lugiato, 1978;Dykman and Krivoglaz, 1979; Lugiato, 1984), with the stable states playing the roleof coexisting phases in a thermodynamic system. The onset of the supernarrow peak(7.41) is an indicator of the transition (Dykman et al. , 1994). For a classical oscillator,such peak was observed by Stambaugh and Chan (2006 b ).The susceptibility of the oscillator also displays a supernarrow peak. The analysis ofthis peak for additive driving is similar to that for a classical oscillator (Dykman and Krivoglaz, 1979;Dykman et al. , 1994). One notices that an extra force A ′ exp( − iωt )+ c.c. with fre-quency very close to the strong-force frequency, | ω − ω F | ≪ Γ, can be describedby making the parameter β in eqn (7.7) slowly dependent on time, β → β ( t ) = β + (2 β/A ) { A ′ exp[ − i ( ω − ω F ) t ] + c . c . } + . . . [we do not consider terms at the mirrorfrequency 2 ω F − ω in β ( t )]. One can then think of the switching rates becoming para-metrically dependent on time via β ( t ). From eqn (7.16), the major part of this timedependence comes from the modulation of the activation energies R A and R A forswitching from states 1 and 2, respectively. The state populations also become timedependent, which gives the switching-induced contribution to the susceptibility χ (sw) ( ω ) ≈ ¯ w ¯ w WW − i ( ω − ω F ) a − a ω F − ω (2 β/λ ) / ( ∂ β R A − ∂ β R A ) . (7.42)Function χ (sw) ( ω ) displays resonant structure in a frequency range determined bythe switching rate, | ω − ω F | . W . The amplitude of χ (sw) ( ω ) is proportional to a largefactor ∼ R A , /λ ≫
1. It also contains factor ¯ w ¯ w , which steeply depends on the dis-tance to the kinetic phase transition. For a classical oscillator, the switching-inducedpeak of the response has been seen in the experiment (Chan and Stambaugh, 2006;Almog et al. , 2007). The susceptibility of a parametrically modulated oscillator alsodisplays a supernarrow peak for ω = ω M with amplitude ∝ R A /λ ≫
1. It was consid-ered previously for a classical oscillator (Ryvkine and Dykman, 2006).The amplitude of the supernarrow peak of the susceptibility increases with de-creasing temperature, as the ratio R A /λ increases. The very onset of this peak forlow temperatures, as well as the supernarrow peaks in the power spectrum, is due toquantum fluctuation-induced interstate switching. We now briefly discuss oscillator dynamics in the presence of a moderately strongnonresonant modulation, where the modulation frequency ω F is significantly differentfrom the oscillator eigenfrequency ω . The dynamics of a modulated linear oscillatorwith linear in coordinate q and momentum p interaction with a thermal bath (7.10) wasstudied earlier (Schwinger, 1961; Zeldovich et al. , 1970). It can change dramatically ifthe interaction is nonlinear (Dykman, 1978) and/or the bath is also modulated.A major effect of nonresonant modulation is the creation of a new relaxation mecha-nism. The relaxation discussed earlier in this chapter, see Fig. 7.3, was due to oscillatortransitions between its neighboring energy levels. In a transition, the energy ≈ ~ ω istransferred to or taken from excitations in the bath. In the absence of modulation this onresonant modulation: oscillator heating and cooling imposes on the oscillator the thermal distribution of bath excitations with energy ~ ω ,leading to the Boltzmann distribution ρ NN ∝ exp( − N ~ ω /k B T ) over Fock states | N i .In the presence of nonresonant modulation, in a bath-induced interlevel transitionthe part ~ ω F of the energy can come from the modulation, see Fig. 7.11. The energyof the involved bath excitations is then ~ | ω ± ω F | . These excitations should imposeon the oscillator their thermal distribution. It corresponds to oscillator temperature T ∗ = T ω / ( ω ± ω F ). This temperature is positive if transitions | N i → | N − i correspond to emission of bath excitations and are thus more probable than transitions | N − i → | N i , see Fig. 7.11(a), (b). If, on the other hand, bath excitations are emittedin transitions | N − i → | N i , then T ∗ becomes negative. This case is sketched inFig. 7.11(c). It corresponds to relaxation processes in which the energy of a “photon”of the modulation goes into excitation of the oscillator and the bath. ?@A BC?DABEBFBE BFBC?GABE BFBC Fig. 7.11
A sketch of modulation-induced relaxation processes leading to cooling (a), heating(b), and population inversion (c) in the oscillator; ω , ω F , and ω b are the oscillator frequency,the modulation frequency, and the frequency of the bath excitation, respectively. A sufficientlystrong modulation imposes on the oscillator the probability distribution of the bath modesinvolved in the scattering in (a) and (b) and leads to population inversion of low-lying energylevels and self-sustained vibrations of the oscillator in (c). Interestingly, the distribution of a nonresonantly modulated quantum oscillatorover Fock states can remain of the Boltzmann form in a broad parameter range(Dykman, 1978). We will show this first for the case where the Hamiltonian of theinteraction with the bath along with the linear in a, a † term (7.10) has a quadraticterm H (2) i = q h (2)b . (7.43)The modulation term in the Hamiltonian of the isolated oscillator (7.1) is H F = H add = − qA cos ω F t ; ω F , | ω F − ω | ≫ Γ , | γ |h q i /ω . (7.44)We now go to the interaction representation using canonical transformation U nr ( t ) = T t exp {− i ~ − R t dt ′ [ H ( t ′ ) + H b ] } , where H ( t ) is given by eqns (7.1) and (7.44), H b is the Hamiltonian of the thermal bath, and T t is the chronological ordering operator.For nonresonant modulation U † nr ( t ) qU nr ( t ) ≈ q ( t ) + A osc cos ω F t, A osc = A/ ( ω − ω F ) . (7.45) The interaction (7.43) leads to nonlinear friction (Dykman and Krivoglaz, 1984), which can playan important role in oscillator dynamics, see the chapter by Moser et al. , and in particular determinethe characteristics of self-sustained vibrations; we do not discuss this dissipation mechanism here. Periodically modulated quantum nonlinear oscillators
Here, q ( t ) is operator q in the interaction representation in the absence of the modu-lating field. Similarly, operator a in the interaction representation is a sum of operator a ( t ) calculated for A = 0 and the terms oscillating as exp( ± iω F t ); the ac-Stark shiftof the eigenfrequency ∼ | γ | A osc2 /ω is assumed small compared to ω .From eqns (7.43) and (7.45), the nonlinear coupling to the bath H (2) i in the interac-tion representation contains a term 2 q ( t ) h (2)b ( t ) A osc cos ω F t , which has the same struc-ture as the linear coupling, eqn (7.10), except for the time-dependent factor ∝ A osc .Therefore the contribution of the field-induced scattering to the decay rate Γ is de-scribed by an expression similar to equation (7.10) except that one should replace ω → ω ± ω F . This gives the overall rate as Γ F = Γ + Γ + + Γ − − Γ i , withΓ ± = A ~ ω Re Z ∞ dt h [ h (2)b ( t ) , h (2)b (0)] i b e i ( ω ± ω F ) t ( ω ± ω F > − Γ i is formally given by eqn (7.46) for Γ − in which ω − ω F <
0; therefore Γ i > − = 0 for ω F − ω >
0, whereas Γ − > i = 0 for ω − ω F >
0. The termsΓ ± lead to the increase of the overall decay rate, whereas Γ i reduces the rate and, ifdominates, can make Γ F negative.The mapping on the effectively linear in a, a † coupling suggests that, in the presenceof nonresonant modulation, the master equation in the interaction representation hasthe conventional form for an oscillator with no modulation, cf. eqn (7.11), ∂ t ρ = − Γ F (¯ n F + 1)( a † a ρ − a ρa † + ρa † a ) − Γ F ¯ n F ( a a † ρ − a † ρa + ρa a † ) , ¯ n F = { Γ¯ n ( ω ) + Γ + ¯ n ( ω + ω F ) + Γ − ¯ n ( ω − ω F ) + Γ i [¯ n ( ω F − ω ) + 1] } / Γ F (7.47)where a ≡ a ( t ) , a † ≡ a † ( t ). Each relaxation mechanism in Fig. 7.11 gives an additivecontribution to eqn (7.47), with the Planck number at the energy transferred to thebath in the corresponding elementary process.The stationary solution of eqn (7.47) has the form of the Boltzmann distributionwith temperature T ∗ = ~ ω /k B ln[(¯ n F + 1) / ¯ n F ] . This expression goes into the discussed above limits if one of the scattering mechanismsdominates, which requires that parameters Γ ± , Γ i significantly differ from each other.The difference between parameters Γ ± , Γ i is determined by the difference betweenthe density of states of the thermal bath at the appropriate frequencies. It can be largeif the oscillator is coupled to the thermal bath via an underdamped vibrational mode,with h (2)b being proportional to the coordinate of this mode q m (Dykman, 1978). Themode decay rate Γ m should largely exceed Γ , | Γ F | , but still it can be small compared tothe mode frequency ω m and | ω m − ω | . One can then selectively tune ω F to resonanceand for example, for ω F ≈ ω m − ω ≫ ω achieve significant oscillator cooling.Similar effects occur and similar description applies if the coupling to the bath islinear in a, a † , eqn (7.10), but nonresonant modulation is parametric, with Hamiltonian H par , eqn(7.2), and ω F , | ω F − ω | ≫ Γ , | γ |h q i /ω . Parameters Γ ± , Γ i in this case arealso quadratic in the modulation amplitude F and are determined by the correlatorsof h b , h † b at frequencies ω ± ω F . onresonant modulation: oscillator heating and cooling The possibility of reducing the vibration temperature by modulation has beenattracting much attention recently, in particular in the context of optomechanics,(Braginsky and Vyatchanin, 2002; Kippenberg and Vahala, 2008). In an optomechan-ical system, the oscillator (a vibrating mirror) is coupled to a cavity mode, whichis driven by external radiation. A quantum theory of cooling of the mirror in thiscase was developed by Wilson-Rae et al. (2007) and Marquardt et al. (2007). If theincident radiation is classical, in the appropriately scaled variables the coupling andmodulation are described by Hamiltonians H ( m ) i and H ( m ) F , respectively, with H ( m ) i = c m qq m , H ( m ) F = − q m A cos ω F t, (7.48)where q and q m are the coordinates of the mirror and the mode. In cavity optome-chanics one usually writes H ( m ) i = c m qa † m a m ; the following discussion immediatelyextends to this form of the interaction.In the absence of coupling to the mirror the cavity mode is a linear system, hence q m ( t ) = q m ( t ) + [ χ m ( ω F ) exp( − iω F t ) + c . c . ] A/
2, where q m ( t ) is the mode coor-dinate in the absence of modulation and χ m ( ω ) is the susceptibility of the mode(Marquardt et al. , 2007). The coupling H ( m ) i in the interaction representation thenhas a cross-term ∝ q ( t ) q m ( t ) exp( ± iω F t ). Since the cavity mode serves as a thermalbath for the mirror, this term is fully analogous to the similar cross-term in H (2) i thatcomes from modulation of the oscillator, with c m q m ( t ) playing the role of h (2) b ( t ).One can then describe the dynamics of the mirror by eqns (7.46) and (7.47) in which h (2)b ( t ) is replaced with c m q m ( t ) and A is replaced with | Aχ m ( ω F ) | .Another potentially important contribution to nonresonant cooling, heating, orexcitation of the oscillator can come from the direct nonlinear interaction of the oscil-lator and the thermal bath with the modulation. For an electromagnetic modulationsuch interaction is due to nonlinear polarizability. For modulation A cos ω F t the cor-responding interaction Hamiltonian is H ( F ) i = − qh ( F )b A cos ω F t. (7.49)For example, here h ( F )b can be the coordinate of a comparatively quickly decayingmode of a nanomechanical resonator.The effect of interaction (7.49) is again described by eqns (7.46) and (7.47) in which h (2)b ( t ) is replaced with h ( F )b ( t ) and A is replaced with A /
4. If more than one of theabove mechanisms is relevant, in calculating the rates of modulation-induced decayone should take into account the interference terms, i.e., write the overall effectivecoupling in the interaction representation as a ( t ) exp( ± iω F t )˜ h b ( t ) + H . c . and thenexpress the rates Γ ± , Γ i in terms of the commutator of ˜ h b ( t ) similar to eqn (7.46).Nonresonant modulation modifies the power spectrum of the oscillator (Dykman, 1978;Dykman and Krivoglaz, 1984). Cooling of a nonlinear oscillator can lead to narrowingof its spectrum at frequency ω even though Γ F > Γ. Unless there are symmetry con-straints, the modulation can also lead to spontaneous emission of photons or phononsat frequencies | ω ± ω F | . It can also lead to amplification of a weak external field atfrequency ω F − ω for T ∗ > Periodically modulated quantum nonlinear oscillators
We described the dynamics of a modulated nonlinear oscillator for resonant and non-resonant modulation. Two types of resonant modulation were considered, an additiveforce with frequency ω F close to the oscillator eigenfrequency ω and parametric mod-ulation with frequency ω F close to 2 ω . For resonant modulation, quantum dynamicsin the rotating frame is characterized by three parameters, the scaled modulation in-tensity, decay rate, and Planck constant, which are given in Table 7.1. Of primaryinterest was the parameter range where the oscillator displays bistability of forcedvibrations. Fluctuations were assumed small on average, so that the smearing of theclassically stable states in phase space is small compared to the interstate distance.For T > T →
0. We foundthe distribution for weak damping, where the width of the quasienergy levels is smallcompared to the level spacing. Near its maximum the distribution is Boltzmann-like.The far tail is of non-Boltzmann form. It determines the exponent in the rate of switch-ing between the stable vibrational states. The switching occurs via transitions over theeffective barrier that separates the states in phase space. We called this quantum ac-tivation and studied the corresponding effective activation energy. Remarkably, evenfor T → eferences Affleck, I. (1981).
Phys. Rev. Lett. , , 388.Aldridge, J. S. and Cleland, A. N. (2005). Phys. Rev. Lett. , , 156403.Almog, R., Zaitsev, S., Shtempluck, O., and Buks, E. (2007). Appl. Phys. Lett. , ,013508.Bishop, Lev S., Ginossar, Eran, and Girvin, S. M. (2010). Phys. Rev. Lett. , ,100505.Blencowe, M. (2004). Phys. Rep. , , 159.Boissonneault, M., Gambetta, J. M., and Blais, A. (2010). Phys. Rev. Lett. , ,100504.Bonifacio, R. and Lugiato, L. A. (1978). Phys. Rev. Lett. , , 1023.Braginsky, V.B. and Vyatchanin, S.P. (2002). Phys. Lett. A , , 228.Caldeira, A. O. and Leggett, A. J. (1983). Ann. Phys. (N.Y.) , , 374.Chan, H. B. and Stambaugh, C. (2006). Phys. Rev. B , , 224301.Chan, H. B. and Stambaugh, C. (2007). Phys. Rev. Lett. , , 060601.Clerk, A. A. (2004). Phys. Rev. B , , 245306.Coleman, S. (1977). Phys. Rev. D , , 2929.Collett, M. J. and Walls, D. F. (1985). Phys. Rev. A , , 2887.Dmitriev, A. P. and Dyakonov, M. I. (1986 a ). Zh. Eksp. Teor. Fiz. , , 1430.Dmitriev, A. P. and Dyakonov, M. I. (1986 b ). JETP Lett. , , 84.Drummond, P. D. and Kinsler, P. (1989). Phys. Rev. A , , 4813.Drummond, P. D. and Walls, D. F. (1980). J. Phys. A , , 725.Dykman, M. I. (1978). Sov. Phys. Solid State , , 1306.Dykman, M. I. (1990). Phys. Rev. A , , 2020.Dykman, M. I. (2007). Phys. Rev. E , , 011101.Dykman, M. I., Khasin, M., Portman, J., and Shaw, S. W. (2010, December). Phys.Rev. Lett. , , 230601.Dykman, M. I. and Krivoglaz, M. A. (1979). Zh. Eksp. Teor. Fiz. , , 60.Dykman, M. I. and Krivoglaz, M. A. (1980). Physica A , , 480.Dykman, M. I. and Krivoglaz, M. A. (1984). In Sov. Phys. Reviews (ed. I. M. Kha-latnikov), Volume 5, pp. 265–441. Harwood Academic, New York.Dykman, M. I., Luchinsky, D. G., Mannella, R., McClintock, P. V. E., Stein, N. D.,and Stocks, N. G. (1994).
Phys. Rev. E , , 1198.Dykman, M. I., Maloney, C. M., Smelyanskiy, V. N., and Silverstein, M. (1998). Phys.Rev. E , , 5202.Dykman, M. I., Marthaler, M., and Peano, V. (2011). Phys. Rev. A , , 052115.Dykman, M. I. and Smelyansky, V. N. (1988). Zh. Eksp. Teor. Fiz. , , 61.Ford, G. W., Kac, M., and Mazur, P. (1965). J. Math. Phys. , , 504.Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical References
Systems (2nd edn). Springer-Verlag, New York.Guckenheimer, J. and Holmes, P. (1997).
Nonlinear Oscillators, Dynamical Systemsand Bifurcations of Vector Fields . Springer-Verlag, New York.Kamenev, A. (2011).
Field theory of non-equilibrium systems . Cambridge UniversityPress, Cambridge.Katz, I., Retzker, A., Straub, R., and Lifshitz, R. (2007).
Phys. Rev. Lett. , , 040404.Ketzmerick, R. and Wustmann, W. (2010). Phys. Rev. E , , 021114.Kim, K., Heo, M. S., Lee, K. H., Ha, H. J., Jang, K., Noh, H. R., and Jhe, W. (2005). Phys. Rev. A , , 053402.Kinsler, P. and Drummond, P. D. (1991). Phys. Rev. A , , 6194.Kippenberg, T. J. and Vahala, K. J. (2008). Science , , 1172.Kramers, H. (1940). Physica (Utrecht) , , 284.Kryuchkyan, G. Y. and Kheruntsyan, K. V. (1996). Opt. Commun. , , 230.Laflamme, C. and Clerk, A. A. (2011). Phys. Rev. A , , 033803.Landau, L. D. and Lifshitz, E. M. (2004). Mechanics (3rd edn). Elsevier, Amsterdam.Langer, J. S. (1967).
Ann. Phys. , , 108.Lapidus, L. J., Enzer, D., and Gabrielse, G. (1999). Phys. Rev. Lett. , , 899.Larsen, David M. and Bloembergen, N. (1976, June). Opt. Commun. , , 254.Lax, M. (1966). Phys. Rev. , , 110.Lugiato, L. A. (1984). Prog. Opt. , , 69.Lupa¸scu, A., Driessen, E. F. C., Roschier, L., Harmans, C. J. P. M., and Mooij, J. E.(2006). Phys. Rev. Lett. , , 127003.Mallet, F., Ong, F. R., Palacios-Laloy, A., Nguyen, F., Bertet, P., Vion, D., andEsteve, D. (2009). Nature Physics , , 791.Mandel, L. and Wolf, E. (Camridge, 1995). Optical Coherence and Quantum Optics .Cambirdge University Press.Marquardt, F., Chen, J. P., Clerk, A. A., and Girvin, S. M. (2007).
Phys. Rev.Lett. , , 093902.Marthaler, M. and Dykman, M. I. (2006). Phys. Rev. A , , 042108.Marthaler, M. and Dykman, M. I. (2007). Phys. Rev. A , , 010102R.Metcalfe, M., Boaknin, E., Manucharyan, V., Vijay, R., Siddiqi, I., Rigetti, C., Frun-zio, L., Schoelkopf, R. J., and Devoret, M. H. (2007). Phys. Rev. B , , 174516.Nation, P. D., Blencowe, M. P., and Buks, E. (2008). Phys. Rev. B , , 104516.O’Connell, A. D., Hofheinz, M., Ansmann, M., Bialczak, R. C., Lenander, M., Lucero,E., Neeley, M., Sank, D., Wang, H., Weides, M., Wenner, J., Martinis, J. M., andCleland, A. N. (2010). Nature , , 697.Peano, V. and Thorwart, M. (2006). New J. Phys. , , 021.Peano, V. and Thorwart, M. (2010 a ). EPL , , 17008.Peano, V. and Thorwart, M. (2010 b ). Phys. Rev. B , , 155129.Picot, T., Lupa¸scu, A., Saito, S., Harmans, C. J. P. M., and Mooij, J. E. (2008). Phys. Rev. B , , 132508.Reed, M. D., DiCarlo, L., Johnson, B. R., Sun, L., Schuster, D. I., Frunzio, L., andSchoelkopf, R. J. (2010). Phys. Rev. Lett. , , 173601.Riviere, R., Deleglise, S., Weis, S., Gavartin, E., Arcizet, O., Schliesser, A., andKippenberg, T. J. (2011). Phys. Rev. A , , 063835. eferences Ryvkine, D. and Dykman, M. I. (2006).
Phys. Rev. E , , 061118.Sazonov, V. N. and Finkelstein, V. I. (1976). Doklady Akad. Nauk SSSR , , 78.Schreier, J. A., Houck, A. A., Koch, J., Schuster, D. I., Johnson, B. R., Chow,J. M., Gambetta, J. M., Majer, J., Frunzio, L., Devoret, M. H., Girvin, S. M.,and Schoelkopf, R. J. (2008). Phys. Rev. B , , 180502.Schwab, K. C. and Roukes, M. L. (2005). Phys. Today , , 36.Schwinger, J. (1961). J. Math. Phys. , , 407.Serban, I., Dykman, M. I., and Wilhelm, F. K. (2010). Phys. Rev. A , , 022305.Serban, I. and Wilhelm, F. K. (2007). Phys. Rev. Lett. , , 137001.Siddiqi, I., Vijay, R., Pierre, F, Wilson, C. M., Frunzio, L, Metcalfe, M., Rigetti, C.,and Devoret, M. H. (2006). In Quantum Computation in Solid State Systems (ed.B. Ruggiero, P. Delsing, C. Granata, Y. Pashkin, and P. Silvertrini), pp. 28–37.Springer, NY.Siddiqi, I., Vijay, R., Pierre, F., Wilson, C. M., Frunzio, L., Metcalfe, M., Rigetti,C., Schoelkopf, R. J., Devoret, M. H., Vion, D., and Esteve, D. (2005).
Phys. Rev.Lett. , , 027005.Stambaugh, C. and Chan, H. B. (2006 a ). Phys. Rev. B , , 172302.Stambaugh, C. and Chan, H. B. (2006 b ). Phys. Rev. Lett. , , 110602.Steffen, M., Ansmann, M., Bialczak, R. C., Katz, N., Lucero, E., McDermott, R.,Neeley, M., Weig, E. M., Cleland, A. N., and Martinis, J. M. (2006). Science , ,1423.Van Kampen, N. G. (2007). Stochastic Processes in Physics and Chemistry (3rdedn). Elsevier, Amsterdam.Verso, Alvise and Ankerhold, Joachim (2010).
Phys. Rev. A , , 022110.Vierheilig, C. and Grifoni, M. (2010). Chem. Phys. , , 216.Vig, J. R. and Kim, Y. (1999). IEEE Trans Ultrason Ferroelectr Freq Control , ,1558.Vijay, R., Devoret, M. H., and Siddiqi, I. (2009). Rev. Sci. Instr. , (11), 111101.Vogel, K. and Risken, H. (1988). Phys. Rev. A , , 2409.Vogel, K. and Risken, H. (1990). Phys. Rev. A , , 627.Wallraff, A., Schuster, D. I., Blais, A., Frunzio, L., Huang, R. S., Majer, J., Kumar,S., Girvin, S. M., and Schoelkopf, R. J. (2004). Nature , , 162.Watanabe, M., Inomata, K., Yamamoto, T., and Tsai, J.-S. (2009). Phys. Rev. B , ,174502.Wielinga, B. and Milburn, G. J. (1993). Phys. Rev. A , , 2494.Wilson, C. M., Duty, T., Sandberg, M., Persson, F., Shumeiko, V., and Delsing, P.(2010). Phys. Rev. Lett. , , 233907.Wilson-Rae, I., Nooshi, N., Zwerger, W., and Kippenberg, T. J. (2007). Phys. Rev.Lett. , , 093901.Yang, Y. T., Callegari, C., Feng, X. L., and Roukes, M. L. (2011). Nano Lett. , ,1753.Zeldovich, B. Ya., Perelomov, A. M., and Popov, V. S. (1970). JETP ,30