Multiple-period Floquet states and time-translation symmetry breaking in quantum oscillators
Yaxing Zhang J. Gosner S. M. Girvin J. Ankerhold M. I. Dykman




Department of Physics, Yale University, New Haven, CT 06511, USA

Institute for Complex Quantum Systems and IQST, Ulm University, 89069 Ulm, Germany

Department of Physics, Yale University, New Haven, CT 06511, USA

Institute for Complex Quantum Systems and IQST, Ulm University, 89069 Ulm, Germany

Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA


We study the breaking of the discrete time-translation symmetry in small periodically driven quantum systems. Such systems are intermediate between large closed systems and small dissipative systems, which both display the symmetry breaking, but have qualitatively different dynamics. As a nontrivial example we consider period tripling in a quantum nonlinear oscillator. We show that, for moderately strong driving, the period tripling is robust on an exponentially long time scale, which is further extended by an even weak decoherence. 




Multiple-period Floquet states and time-translation symmetry breaking
 in quantum oscillators
    M. Dykman
    December 30, 2023
=============================================================================================



The breaking of translation symmetry in time, first proposed by Wilczek <cit.>, has been attracting much attention recently. Such symmetry breaking can occur only away from thermal equilibrium <cit.>. It is of particular interest for periodically driven systems, which have a discrete time-translation symmetry imposed by the driving. Here, the time symmetry breaking is manifested in the onset of oscillations with a period that is a multiple of the driving period t_F. Oscillations with period 2t_F due to simultaneously initialized protected boundary states were studied in photonic quantum walks <cit.>; period-two oscillations can also be expected from the coexistence of Floquet Majorana fermions with quasienergies 0 and ħπ/t_F in a cold-atom system <cit.>. The onset of period-two phases was predicted and analyzed <cit.> in Floquet many-body localized systems, and the first observations of oscillations at multiples of the driving period  in disordered systems were reported <cit.>.  

In systems coupled to a thermal bath, on the other hand,  the effect of period doubling has been well-known. A textbook example is a classical oscillator modulated close to twice its eigenfrequency and displaying vibrations with period 2t_F <cit.>. The oscillator has two states of such vibrations; they have opposite phases, reminiscent of a ferromagnet with two orientations of the magnetization. Several aspects of the dynamics of a parametric oscillator in the quantum regime have been studied theoretically, cf. <cit.>, and in experiments, cf. <cit.>. For a sufficiently strong driving field, a quantum dissipative oscillator, like a classical oscillator, mostly performs vibrations with period 2t_F. The interplay of quantum fluctuations and dissipation leads to transitions between the period-two vibrational states, but the rate of these transitions is exponentially small <cit.>.

The goal of this paper is to study time symmetry breaking in isolated or almost isolated driven quantum systems with a few degrees of freedom. They are intermediate between large closed systems and dissipative systems, where the nature of the symmetry breaking is very different. To this end, we analyze a driven nonlinear quantum oscillator. Time symmetry breaking in this system should not be limited to period doubling. As 
an illustration of a behavior qualitatively different from period doubling, we consider period tripling and find the conditions where it occurs. We also address the role of decoherence and the connection between the time symmetry breaking in the coherent and incoherent regimes.

Floquet (quasienergy) states ψ_(t) are eigenstates of the operator  of time translation by t_F, ψ_(t) ≡ψ_ (t+t_F)=exp(-i t_F/ħ)ψ_(t). For a broken-symmetry state ψ_K,_K with K>1, time translation by t_F is not described by the factor exp(-i t_F/ħ). Instead, ψ_K,_K(t+Kt_F) = exp(-Ki_K t_F/ħ)ψ_K,_K(t). We call ψ_K,_K a period-K Floquet state. It is an eigenstate of T_Kt_F = ()^K, but not T_t_F. 

Multiple-period states naturally occur if the number of states of the system ℕ→∞, as in the case of an oscillator. For such systems the quasienergy spectrum is generally dense, cf. <cit.>. Then we can find states ψ_ and ψ_' with the difference of the quasienergies | - '| infinitesimally close to ħω_F/K with integer K>1 (or to ħω_F k/K with k<K); ω_F=2π/t_F is the driving frequency. A linear combination αψ_(t) +α'ψ_'(t) is a period-K state. The expectation value of dynamical variables in such a state oscillates with period Kt_F. However, the oscillation amplitude will be very small as, generally, the functions ψ_ and ψ_' will be of a very different form.

The situation is different for an oscillator driven close to an overtone of its eigenfrequency ω_0, i.e., for ω_F ≈ Kω_0. Such an oscillator has several sets of quasienergy states where the quasienergy differences within a set are very close to ħω_F/K in a broad parameter range, and are exactly equal to ħω_F/K for some interrelations between the parameters, whereas off-diagonal matrix elements of the dynamical variables are large, see Fig. <ref>. Such states result from tunnel splitting of the  states localized at the minima of the oscillator Hamiltonian in the rotating frame shown in Fig.<ref>(c). These localized states correspond to period-K vibrations in the laboratory frame, see below. 



In a way, for a parametric oscillator (K=2) the occurrence of a period-2 state could be inferred from the results <cit.>. However, this state was not identified there and the time symmetry breaking was not addressed. In different terms, sets of states separated by ≈ħω_F/K  were found numerically for K≫ 1 for a special model of an oscillator in the interesting paper <cit.>; the considered states did not break time symmetry.  

The period tripling (K=3) considered here for a driven oscillator is particularly interesting. It differs from the continuous Landau-type symmetry-breaking transition that occurs for period doubling, cf. <cit.>. In the presence of dissipation, the fully-symmetric (zero-amplitude) state does not loose stability. Also, in the quantum regime, there emerges a geometric phase between the broken-symmetry states localized at the minima of the effective Hamiltonian function in phase space, cf. Fig. <ref>(c). Thus, the period-tripling in an oscillator allows one to reveal, using a simple and physically relevant model, the generic conditions for the onset of strongly overlapping multiple-period states and to relate them to the underlying nontrivial symmetry. It also provides a platform for studying quantum tunneling  between localized states in phase space. This problem is considerably different from the classical problem  of tunneling in a symmetric double-well potential <cit.> (see also <cit.>). 

We study a most commonly used model of a nonlinear oscillator, the Duffing model, which describes parametric resonance and, as we will see, can describe period tripling; this model refers to a broad range of systems, including trapped relativistic electrons, cold atomic clouds, Josephson junction based systems, and  nanomechanical systems <cit.>. Its Hamiltonian reads


    H=H_0 + H_F,      H_0=1/2p^2 + 1/2ω_0^2 q^2 +1/4γ q^4,


where q and p are the oscillator coordinate and momentum. The term H_F≡ H_F(t) describes the driving. In the analysis of parametric resonance, one chooses H_F=-12 q^2 Fcosω_Ft with ω_F≈ 2ω_0. Here we consider H_F=-13q^3Fcosω_Ft with ω_F≈ 3ω_0; the results describe also a drive H_F'=-qF'cosω_Ft with F→ 3γ F'/8ω_0^2. 

If the driving is not too strong, so that for the states of interest the expectation values of H_F and the nonlinear term ∝ q^4 are small compared to the harmonic part of H_0, the resonant oscillator dynamics can be described in the rotating wave approximation (RWA) <cit.>. For an oscillator driven close to the Kth overtone of its eigenfrequency, one makes a canonical transformation U(t) =exp(- i a^† a  ω_F t/K), where a and a^† are the ladder operators. 








The RWA Hamiltonian H_ RWA is obtained by time-averaging the transformed Hamiltonian H_K(t) =U^† (t) H(t) U(t) - iħ U^†(t)U̇(t), 


    H_ RWA=(Kt_F)^-1∫_0^Kt_Fdt H_K(t).


Clearly, H_ RWA is  independent of time. 

We now establish the relation between the eigenvalues of H_ RWA and the quasienergies. If ϕ_(t) is an eigenfunction of H_ RWA, i.e., H_ RWAϕ_ = Eϕ_,  then the corresponding wave function in the lab frame is ψ(t)=U(t)ϕ_(t), and


    ψ(t) = e^-iEt_F/ħU(t+t_F)ϕ_(t)= e^-i Et_F/ħN_Kψ(t).


We call E the RWA energy. In Eq. (<ref>)


    N_K = exp(-2π i a^† a/K),     [N_K,H_ RWA]=0.


The above commutation relation follows from the relation H_K(t+t_F) = N_K^† H_K(t)N_K and Eq. (<ref>). Using the explicit form of H_ RWA, the commutation relation (<ref>) was found in Ref. Guo2013a for the same operator as N_K.

Operators N_K^k with k=0,1,...,K-1 form a cyclic group. Since eigenfunctions of H_ RWA are also eigenfunctions of N_K, one can label them by a superscript k, 


    N_Kϕ^(k) = exp(-2π i k_ /K)ϕ^(k),    0≤ k≤ K-1.


Note that H_ RWA has eigenfunctions with the same k, but different E. By comparing Eqs. (<ref>) and (<ref>) one finds that a wave function ϕ_^(k) with RWA energy E^(k) corresponds to a usual Floquet state with quasienergy 


    ^(k) = (E^(k)+ħω_F k_/K)  mod (ħω_F).


As we will see, for sufficiently strong drive the eigenstates of H_ RWA form multiplets with close eigenvalues E^(k) but different k. The quasienergies of different states in the multiplets differ by ≈ħω_F/K.

Equation (<ref>) allows one to write the functions ϕ^(k) in terms of the Fock states of the oscillator  |n⟩ defined by the condition a^† a|n⟩ = n|n⟩. Only one out of each K Fock states contributes to ϕ^(k),  ϕ^(k) = ∑_nC_n^(k)|Kn+k_⟩. This relation significantly simplifies numerical diagonalization of H_ RWA, as the coefficients C_n^(k) with different k are uncoupled. More importantly, it shows that the RWA energy levels of states with different k can cross when the parameters of the system vary. This crossing is seen in Fig. <ref>. In contrast, the RWA levels of states with the same k avoid crossing.

The motion in the rotating frame is  conveniently described by the coordinate Q  and momentum P, which are related to q and p as 


    U^† (t)[q+i(K/ω_F)p]U(t) = C(Q+iP)e^-iω_Ft/K.


The parameter C is the scaling factor that makes Q and P dimensionless, 


    [Q,P]=iλ,     λ=ħ K/ω_FC^2.


The dimensionless Planck constant λ and the parameter C in the case of a parametric oscillator, K=2, are given in <cit.>. For the case of period tripling, C=(8ω_Fδω/9γ)^1/2, where δω = 1/3ω_F - ω_0 is the frequency detuning from the resonance, |δω|≪ω_F. In this case H_ RWA = [8ω_F^2(δω)^2/27γ)]ĝ(Q,-iλ∂_Q) with


    g(Q,P) = 1/4(Q^2+P^2 -1)^2 -1/3 f(Q^3 - 3PQP),


where f=F/(8ω_F γδω)^1/2 is the scaled amplitude of the driving. Of interest is the region γδω >0, and we choose γ >0 and δω>0. 

The function g(Q,P) is the dimensionless Hamiltonian function in the rotating frame. It is plotted in Fig. <ref>. It has a three-fold rotational symmetry in the (Q,P)-plane. This symmetry follows from Eqs. (<ref>) and (<ref>), since N_K is an operator of rotation by angle 2π/K in phase plane; the K-fold symmetry of H_ RWA was also seen in <cit.>. 

For moderately strong fields, g(Q,P) has three well-separated minima positioned at the vertices of an equilateral triangle (Q_m,P_m); we count m=0,1,2 counterclockwise and
set m=0 for the vertex with P_0=0.  The eigenstates of the operator ĝ≡ g(Q,-iλ∂_Q) with the lowest RWA energies are localized near  (Q_m,P_m). In the absence of tunneling, ĝ has three degenerate eigenstates Ψ_m. Near their maxima, functions Ψ_m have the form of squeezed ground states of a harmonic oscillator centered at (Q_m,P_m) [see Supplemental Material for the details of the calculation]. 

The oscillator in a state Ψ_m has a broken time symmetry. The expectation values of dynamical variables oscillate at frequency ω_F/3. Indeed, from Eq. (<ref>) time translation by t_F transforms  Ψ_m→ N_3Ψ_m=Ψ_m-1≡Ψ_m+2. To come back to state Ψ_m, one has to increment time by 3t_F. The relation Ψ_m+1= N_3^†Ψ_m gives the phase shift between functions Ψ_m+1 and Ψ_m. Since N_3 is a rotation operator, this phase shift is geometric in nature [34]. 

Tunneling between the minima lifts the degeneracy of the ground state of the operator ĝ. In contrast to the problem of tunneling in a symmetric double-well potential <cit.>, g(Q,P) is not even in Q, it has three extrema, and two of them lie at nonzero momenta P. 

To find the tunnel splitting, we write the wave functions in the coordinate representation, Ψ_m≡Ψ_m(Q). The three normalized eigenstates ϕ^(k) of ĝ with the smallest eigenvalues g^(k) (k=0,1,2) have the form  


    ϕ^(k)(Q) = 1/√(3(1+δ^(k)))∑_m=0,1,2Ψ_m (Q)e^-2mkπ i/3.


where δ^(k)= 2 Re[⟨Ψ_0|Ψ_1⟩exp(-2π i k/3)]≪ 1. We choose Ψ_0(Q) to be real and normalized.  Since Ψ_m+1 = N_3^†Ψ_m, we have Ψ_2(Q)=Ψ_1^*(Q). Due to the symmetry,  the functions  ϕ^(k) can be shown to be orthogonal.

In the spirit of <cit.>, we calculate g^(k) using the relation


    ∫_∞^Q_* dQ[ϕ^(k)(Q)(ĝ- g_0)Ψ_0(Q) .
       . - Ψ_0(Q)(ĝ -g^(k)) ϕ^(k)(Q)]=0
 

with g_0 being the eigenvalue of ĝ in the state Ψ_0, g_0≈min g(Q,P) [34].
The difference g^(k)-g_0 is exponentially small for a small dimensionless Planck constant λ.

To choose the upper limit Q_* of the integral (<ref>), we note that the functions Ψ_m(Q) fall off exponentially away from the respective Q_m, with Ψ_0 and Ψ_1,2 falling off in the opposite directions in the interval (Q_1,Q_0). We choose Q_* within this interval and in such a way that Ψ_0,1,2(Q_*) are all of the same order of magnitude and thus can be kept in Eq. (<ref>) for ϕ^(k)(Q). The result of integration (<ref>) should be independent of Q_*.

The WKB wave functions Ψ_0,1(Q) in the classically forbidden region between Q_1 and Q_0 have the form


    Ψ_m(Q)=C_m(i∂_P g)^-1/2e^iS_m(Q)/λ  (m=0,1), 
       ∂_Q S_m = (-1)^m (Q),      g(Q,) = g_0,


where S_0,1(Q) is the  classical action and constants C_0,1 are found from the matching to the corresponding intrawell wave functions. 

It is critical for understanding the tunneling that, because the effective Hamiltonian function g(Q,P) is quartic in the momentum P,  (Q) has a branch point Q_B in the interval (Q_1,Q_0). For Q_1 < Q< Q_B, (Q) has both imaginary and real parts. So does the action S_m(Q). This leads to oscillations of the wave functions in the classically forbidden region. In S_m(Q) one should keep the root with the smallest | Im |. To describe Ψ_0, Eq. (<ref>)  has to be modified  by allowing for a complex conjugate term [34]. 

Calculating the integrals in Eq. (<ref>) by parts, we find


    g^(k) - g_0 = C_ tune^-S_ tun/λcos (λ^-1Φ_ tun-2π k/3),


where Φ_ tun + i S_ tun = ∫_Q_0^Q_1dQ'(Q') + P_1Q_1/2 + λ G with  given by equation g(Q,)=min g(Q,P), G being independent of λ and having a contribution from the geometric phase, and C_ tun∝λ^1/2 [34].

Equation (<ref>) shows that the splitting of the eigenvalues of H_RWA oscillates as the system parameters vary. Two eigenvalues cross each time λ^-1Φ_ tun = (n+n'/3)π with integer n,n'. Such crossings are seen in Fig. <ref>. Where the eigenvalues do not cross, they stay exponentially close to each other.

If the oscillator is in a superposition of two states ϕ^(k) and ϕ^(k'), the expectation values of its variables have period 3t_F provided the observation time is smaller than the exponentially long time |Ω_kk'|^-1, where  the frequency Ω_kk'=λ^-1[g^(k)-g^(k')]δω is determined by the tunnel splitting. The Fourier spectra of the expectation values generally have components at frequency ω_F/3 ±Ω_kk'; in particular, the coordinate and momentum have just one of these components. This behavior is characteristic also of the oscillator in intrawell states Ψ_m, which are superpositions of ϕ^(1,2,3). The oscillator fluorescence spectrum will display peaks at ω_F/3 ±Ω_kk' as well.

It is instructive to compare these results with the period-doubling associated with the topologically protected Floquet boundary states in extended systems <cit.>. To some extent, such states are analogous to the symmetry-protected states ϕ^(k). If tunneling between the Floquet boundary states can be disregarded, similar to disregarding oscillator tunneling, their combination becomes a  multiple-period state. However, their overlap is exponentially small, in contrast to the functions ϕ^(k).

The intrawell states Ψ_m are particularly important in the presence of dissipation. Even if the dissipation rate Γ is extremely small, but exceeds the exponentially small frequencies  Ω_kk', instead of coherent tunneling between the wells of g(Q,P), the oscillator performs incoherent interwell hopping with typical rate W<|Ω_kk'| [34]. This hopping corresponds to flips of the vibration phase. On times small compared to  W^-1 the oscillator stays in the multiple-period state inside a well. This is the exact analog of the classical behavior of a dissipative oscillator, including a parametric oscillator, where the multiple-period state is seen on times short compared to the reciprocal rate of interstate switching.

A promising type of oscillator for observing period tripling are modes of microwave cavities coupled to Josephson junctions. Recently there have been studied systems where inelastic Cooper pair tunneling leads to an effective driving of a cavity mode that nonlinearly depends on the mode coordinate and has a tunable frequency 2eV/ħ determined by the voltage V across the Josephson junction <cit.>. There are also other possibilities to resonantly excite multiple-period modes in microwave cavities [P. Delsing, D. Esteve, and  F. Portier, private communications].


In conclusion, we studied a quantum oscillator driven close to an overtone of its eigenfrequency and showed that a small quantum system can display coherent multiple-period dynamics. We explicitly described this dynamics for the previously unexplored nontrivial case of period tripling and established the relation to protected boundary Floquet states in extended systems and to multiple-period states in dissipative systems. 

We are grateful to G. Refael, M. Rudner, and S. Sondhi for the discussions and correspondence.  YZ and SMG were supported by the U.S. Army Research Office (W911NF1410011) and by the National Science Foundation (DMR-1609326).;  JG and JA were supported in part by the German Science Foundation through SFB/TRR 21 and the Center for Integrated Quantum Science and Technology (IQST);  MID was supported in part by the National Science Foundation (Grant No. DMR-1514591). 



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P̅



Supplemental Material




§ THE INTRAWELL WAVE FUNCTIONS OF THE RWA HAMILTONIAN


We consider the dynamics of the oscillator driven close to three times its eigenfrequency in the rotating wave approximation (RWA). The scaled RWA Hamiltonian function g(Q,P), which is given by Eq. (10) of the main text and is plotted there in Fig. 1, has three symmetrically located minima at  points (Q_m,P_m) with m=0,1,2,


    Q_0=1/2[f+(f^2+4)^1/2],      Q_1 = Q_2 = -Q_0/2,
       P_0=0,     P_1 = -P_2 = √(3) Q_0/2,


The minimal value g_min of g(Q,P) and the dimensionless frequency of classical vibrations about a minimum  ω_min = ([∂^2_x_i x_j g(x_1,x_2)])^1/2 (the derivatives are calculated at a minimum of g) are


    g_min= -1/12fQ_0(Q_0^2+3),  ω_min= [3fQ_0(Q_0^2 +1)]^1/2  .


The frequency ω_min is the same for all minima. So is also the lowest eigenvalue g_0 of the Hamiltonian ĝ(Q,-iλ∂_Q) in the neglect of tunneling. To the lowest order in the dimensionless Planck constant λ it corresponds to the lowest eigenvalue of a harmonic oscillator, 
    g_0=g_min+ 12λω_min.


The calculation of the tunnel splitting is done below by first finding the intrawell wave functions Ψ_m(Q) near their maxima inside the well, then finding the geometric phase shift between different Ψ_m, and then explicitly writing down the WKB tails of functions Ψ_m in the classically forbidden regions, which are given by Eq. (13) of the main text. Since Ψ_2(Q) = Ψ_1^*(Q), we only need to find Ψ_0(Q) and Ψ_1(Q).



 §.§ The wave function Ψ_0(Q)

Near the minimum (Q_0, P_0) we have g(Q,P)≈ g_min+1/2(Q_0^2+1)(Q-Q_0)^2 + 3/2fQ_0P^2.
The wave function Ψ_0(Q) is Gaussian for |Q-Q_0|≪ |Q_1-Q_0| and can be chosen to be real,


    Ψ_0(Q) = (√(π) l_q)^-1/2exp[-(Q-Q_0)^2/2l_q^2],


with l_q=[λω_min/(Q_0^2+1)]^1/2 being the localization length.

We are interested in the tail of Ψ_0 for Q between the minima of g(Q,P), i.e., for Q_1 < Q < Q_0-l_q. The WKB form of Ψ_0(Q) is given by Eq. (13) of the main text, which we here write explicitly,


    Ψ_0(Q)=C_0(i∂_P g)^-1/2exp[iS_0(Q)/λ], 
       S_0(Q)=∫_Q_0-l_q^Q dQ'(Q'),


where (Q) is given by equation g(Q,) = g_0 and ∂_Pg is calculated for P=(Q). For the branch of  that we are interested in


    (Q)^2= A(Q) + B^1/2(Q),      A(Q)= 1- Q^2 -2fQ, 
       B(Q) = A^2(Q) -4 [g(Q,0)-g_0],


with Im <0 for Q<Q_0; we keep the correction ∝λ to secure matching to Eq. (<ref>).

For Q close to Q_0 and Q<Q_0-l_q, we have A(Q)<0, B(Q)>0, and A(Q)+ B^1/2(Q) <0. Therefore (Q) is purely imaginary and the same is true for the function


    ∂_Pg = (Q)B^1/2(Q)


 with i∂_Pg >0. Accordingly, Ψ_0(Q) exponentially decays with increasing Q_0-Q. The prefactor C_0 is determined by matching Eqs. (<ref>) and (<ref>) for Q close to Q_0 but Q_0-Q≫ l_q,


    C_0=(ω_min/2√(π e))^1/2.


As Q decreases, first B(Q) becomes equal to zero at point Q_B. To the leading order in λ≪ 1

    Q_B ≈ Q_0-3/4f.


For still smaller Q,  A(Q) changes sign to positive. This happens for  Q_B> Q>Q_1≡ -Q_0/2. Importantly,


    A(Q_1)= P_1^2 >0,     B(Q_1) =2λω_min  .


In the explicit form, the imaginary part of the momentum in the classically forbidden region is


    Im(Q)=-[-A(Q)-B^1/2(Q)]^1/2  (Q_B<Q<Q_0)
        Im(Q)= -[(A^2+|B|)^1/2-A]^1/2/√(2)  (Q<Q_B).


As discussed in the main text, the level splitting crucially depends on the oscillations of the wave function under the barrier. These oscillations start with the decreasing Q at Q=Q_B. Near Q_B we have B(Q)≈∂_Q B(Q_B)(Q-Q_B), whereas A(Q_B)<0. Therefore  ≈ -i|A(Q_B)|^1/2 + (i/2)|∂_QB(Q_B)/A(Q_B)|^1/2(Q-Q_B)^1/2 for small Q-Q_B>0, i.e., Q_B is a branching point of (Q). We have to go around above and below this point in the complex plane to obtain the wave function for Q<Q_B, following the standard procedure <cit.>. As a result, we find for Q<Q_B


    Ψ_0(Q)≈ 2C_0|∂_Pg|^-1/2exp[- Im S_0(Q)/λ]cosΦ_0(Q),
       Φ_0(Q) = Φ_0'(Q)+Φ”_0(Q).


Here, the phase Φ'_0(Q) comes from the real part of the action,


    Φ_0'(Q) = λ^-1∫_Q_B^Q dQ'  Re(Q'),
        Re(Q) = -[(A^2+|B|)^1/2+A]^1/2/√(2),


whereas Φ_0”(Q) comes from the prefactor, with account taken of going around Q_B in the complex plane,


    Φ_0”(Q) = -1/2arcsin[ Re(Q)/|(Q)|] - π/4.


The choice of Re and Im in Eqs. (<ref>) and (<ref>) corresponds to writing B^1/2 = i|B|^1/2 in Eq. (<ref>) for ^2 in the region where B(Q)<0.

The WKB approximation (<ref>) breaks down near Q_1, as B(Q) becomes ∼λ and |∂_Pg| becomes small. However, we do not need to calculate the wave function Ψ_0(Q) in this region, as seen from Eq. (12) of the main text.



 §.§ The wave function Ψ_1(Q)


The minimum of g(Q,P) at (Q_1,P_1) corresponds to a nonzero momentum P_1>0. Therefore the wave function Ψ_1 centered at Q_1 is complex valued even near its maximum. Calculating Ψ_1 involves three steps: finding it inside the well of g(Q,P) near Q_1,P_1; finding the geometric phase, that relates Ψ_1 and Ψ_0 given that Ψ_0 is chosen in the form (<ref>), and then finding the tail of Ψ_1 in the classically forbidden range.



  §.§.§ The intra-well wave function and the geometric phase


Using the explicit form (<ref>) of Q_1,P_1, to the second order in δ Q = Q-Q_1, δ P=P-P_1 we write the Hamiltonian near (Q_1,P_1) as


    g(Q,P)≈ g_min   + 3/4(1+fQ_0)δ P^2 +1/4(1+5fQ_0)δ Q^2
        +(√(3)/4)(fQ_0-1)[δ Q δ P + h.c.].


The expression for Ψ_1 for |δ Q|≪ Q_0-Q_1 then reads


    Ψ_1(Q)=C_1, intraexp[(iP_1δ Q- 1/2βδ Q^2)/λ],
       β =[2ω_min  +i √(3)(fQ_0-1)]/3Q_0^2.


The Gaussian-width parameter β  is now complex-valued. So is also the prefactor C_1, intra, which has a phase factor exp(iθ_1).

The phase θ_1 has a geometric nature. It is determined by the fact that, as indicated in the main text, Ψ_1 and Ψ_0 are related by the transformation of rotation in phase plane, Ψ_1=N_3^† Ψ_0. Here, N_3 = exp(-2π ia^† a/3) with a = (2λ)^-1/2(Q+ iP)≡ (2λ)^-1/2(Q+ λ∂_Q). To calculate θ_1, we consider a coherent state in the coordinate representation


    |α⟩ = 1/(πλ)^1/4exp{-1/2(|α|^2-α^2) -
    [Q-(2λ)^1/2α]^2/2λ}


and set α =Q_0/√(2λ), so that the wave function |α⟩ is centered at Q_0 and thus  strongly overlaps with Ψ_0. Since the function Ψ_1 is obtained from Ψ_0 by applying to Ψ_0 the operator N_3^†,  we can write the overlap integral as ⟨α|Ψ_0⟩ =⟨α|N_3Ψ_1⟩= ⟨αexp(2π i/3)|Ψ_1⟩. The “rotated" state |αexp(2π i/3)⟩ strongly overlaps with Ψ_1. Therefore the above overlap integrals can be calculated using the explicit Gaussian form of Ψ_0 and  Ψ_1 near their maxima. With account taken of the normalization of Ψ_1, this gives


    C_1, intra=[ Reβ/πλ]^1/4exp(iθ_1),
       θ_1 = 1/2 (β +1)  +P_1Q_1/2λ.




  §.§.§ The wave function Ψ_1 in the classically forbidden region


It is clear from Eq. (12) of the main text that we need to find the tail  of the wave function Ψ_1 in the classically forbidden region only for Q > Q_1. It is given by Eq. (13) of the main text. In a more explicit form


    Ψ_1(Q)=C_1(i∂_P g)^-1/2exp[iS_1(Q)/λ], 
       S_1(Q)=-∫_Q_1 + l_q'^Q dQ' (Q'),


where (Q) is given by Eqs. (<ref>) and (<ref>), l_q' =[λ/ Re β]^1/2 . Equation (<ref>) corresponds to choosing B^1/2(Q) = i|B(Q)|^1/2 for B(Q) < 0 and to ∂_Pg calculated for P(Q)=(Q), i.e., ∂_Pg= B^1/2.  For Q_B-Q≫ Q-Q_1≫ l'_q we have -(Q) ≈ P_1+ iβ(Q-Q_1), as expected from Eq. (<ref>). By matching Eqs. (<ref>) and (<ref>),  we find


    C_1=(ω_min/2√(π e))^1/2exp(iθ_1'),
       θ_1' = θ_1 -λ^-1[ (l_q^' 2/2) Imβ-P_1l_q' ].


Because we count the action S_1 off from Q_1 + l_q', there emerges an extra phase factor in C_1 due to the oscillations of the wave function inside the “potential well" centered at (Q_1,P_1).



§ TUNNEL SPLITTING OF THE SCALED RWA ENERGY LEVELS


The scaled RWA energies g^(k) give the values of the quasienergies ^(k) of the driven oscillator, ^(k) = (Ξ g^(k) + ħω_F k/3)(ħω_F), where Ξ = |8ω_F^2(δω)^2/27γ|, see Eqs. (6) and the text above Eq. (9) of the main text. 
The explicit expressions for the wave functions (<ref>) and  (<ref>) allow us to calculate the scaled energies g^(k) using  Eq. (12) of the main text, which we reproduce here for convenience,


    ∫_∞^Q_* dQ[ϕ^(k)(Q)(ĝ- g_0)Ψ_0(Q) .
       . - Ψ_0(Q)(ĝ -g^(k)) ϕ^(k)(Q)]=0,


Functions ϕ^(k) are sums of functions Ψ_m(Q) weighted with factors exp(-2π i mk/3)/√(3). For Q_* well inside the interval (Q_1,Q_0) , we have ∫_∞^Q_*Ψ_0^2(Q)dQ =-1. Taking into account that overlapping of the functions Ψ_1,2(Q) with Ψ_0(Q) is exponentially small, we rewrite Eq. (<ref>) as


    g^(k)-g_0 ≈   [ ∫_∞ ^Q_*dQ  Ψ_1(Q)ĝΨ_0 - ∫_∞ ^Q_*dQ  Ψ_0ĝΨ_1(Q)] 
       ×exp(-2kπ i/3) +  c.c.


 It is important that the product Ψ_0(Q)Ψ_1(Q) has two terms. One of them is ∝exp{i[S_0(Q) + S_1(Q)]}. It smoothly depends on Q, because S_0(Q) + S_1(Q) = const for Q_1 < Q < Q_0. The other term is  ∝exp{-i[S_0^*(Q) - S_1(Q)]}, it is a fast oscillating function of Q. The contribution of this term to the integrals (<ref>)  is exponentially small and exponentially sensitive to the change of Q_* on the scale ∝λ. Therefore this term  should be disregarded.

Using the explicit form of the operator g(Q, -iλ∂_Q) and integrating by parts, from Eq. (<ref>) we obtain


    g^(k) - g_0 =- 2λ C_0|C_1|exp(-S_λ/λ)cos(Φ_λ/λ-2kπ/3),
       S_λ =  -∫_Q_1+l_q'^Q_0-l_qdQ   Im(Q),
       Φ^(k)_λ = -∫_Q_1 + l_q'^Q_BdQ   Re(Q)  +λθ_1'.




This expression is somewhat inconvenient, as  is calculated with account taken of the term ∝λ. It is easy to see that (Q) ≈(Q) +1/2λω_min/∂_Pg, where  is given by the value of  calculated for λ=0. This approximation breaks down near Q_0, Q_B and Q_1 where ∂_Pg goes to zero. Similar to Ref. Garg2000_1,  for Q_0>Q>Q_B one can write


    ∫_Q_0-l_q^QdQ'  (Q')≈∫_Q_0^Q dQ'[(Q') + λ Y(Q',Q_0) ]
        -iλ/2log|Q-Q_0|/l_q -iλ/4-iλ/2log 2,
        Y(Q,Q_m)= ω_min/2(Q)B_ cl^1/2(Q) -i/2|Q-Q_m|  .


Here, B_ cl(Q)= (16f/3)(Q-Q_1)^2(Q- Q_B) is the value of B(Q) calculated for λ=0. A similar transformation can be made for ∫_Q_1+ l_q'^Q dQ'(Q') in the region Q_1<Q<Q_B.


We now have to consider the vicinity of Q_B. Formally, the quantum correction to  diverges at Q_B. However, the divergence is integrable. Therefore Eq. (<ref>) applies all the way till Q=Q_B, and one can use the value of Q_B given by Eq. (<ref>).

The final result for the difference of the scaled RWA energies is Eq. (14) of the main text,


    g^(k) - g_0 = C_ tun  e^-S_ tun/λcos(λ^-1Φ_ tun -2π k/3)  ,


with real S_ tun and Φ_ tun,


    S_ tun = ∫_Q_0^Q_1dQ  Im(Q) + λ  Im  K_ tun
       Φ_ tun= ∫_ Q_B^Q_1dQ   Re(Q)  + λ Re  K_ tun +λθ_1,
       C_ tun=-3/2√(λ) ω_min[2(Q_0^2 +1)/3π^2 Q_0^2]^1/4[f(2Q_0-f)]^1/2.

Here,

    K_ tun = ∫_Q_0^ Q_BdQ  Y(Q,Q_0) + ∫_ Q_B^Q_1dQ  Y(Q,Q_1)


and θ_1 is given in (<ref>).



The explicit expression (<ref>) is in an extremely good agreement with the numerical calculations. This can  be seen from Fig. 1 in the main text. A more detailed comparison is shown in Fig. <ref>.  Equation (<ref>) simplifies in the limit of comparatively strong drive, f≫ 1. The leading order terms in S_ tun and in Φ_ tun are quadratic in f. Numerically, the asymptotic regime is reached for comparatively large f, where the tunneling amplitude becomes very small.



§ QUANTUM DIFFUSION OVER THE BROKEN-SYMMETRY STATES


The dynamics of the driven oscillator system can be strongly changed by an already very weak dissipation. Two types of dissipative processes can be conditionally separated. One of them causes transitions between the states that belong to the same multiplet  formed by the tunnel splitting of a quantized state of motion inside  a well of g(Q,P). In particular, in this paper we considered such multiplet ϕ^(k) formed by the tunnel splitting of the lowest quantized intrawell state. The other dissipative process leads to transitions between the intrawell states. 

In terms of the dissipation mechanisms, an important type of physical  dissipative processes are transitions between the Fock states of the oscillator with emission or absorption of excitations of the thermal reservoir to which the oscillator is coupled. Another mechanism is fluctuations of the oscillator eigenfrequency due to the coupling to a reservoir or due to an external noise. It leads to dephasing of the vibrations, but not to an appreciable energy exchange with the reservoir. There may be also dissipation channels that are induced by the driving field; however, for the considered comparatively weak resonant field they are not important. 

We note first that the dephasing does not mix the states within the tunnel-split multiplets. Indeed, as indicated in the main text, the wave functions ϕ^(k) can be written in terms of the Fock states of the oscillator |n⟩ as ϕ^(k) = ∑_nC_n^(k)|3n+k⟩. The coupling to a thermal bath, which leads to dephasing, has the form H^ (ph) = a^† a H_b^ (ph), where a,a^† are the oscillator ladder operators and H_b^ (ph) is an operator that depends on the dynamical variables of the bath only. Clearly, such coupling is diagonal in the ϕ^(k) basis.

The simplest coupling that leads to the oscillator energy relaxation is linear in a,a^†. To the lowest order of the perturbation theory, for the well-understood conditions, it is described by the term ρ̇_d in the equation for the oscillator density matrix ρ in slow time compared to ω_F^-1, 


    ρ̇_d = -Γ[(n̅ +1) (a^† aρ -2 a ρ a^† + ρ a^† a) .
       . +n̅ (a a^†ρ -2 a^†ρ a + ρ aa^†) ],


where 2Γ is the energy decay rate of the oscillator and n̅=[exp(ħω_0/k_BT) -1]^-1 is the oscillator Planck number. In what follows we assume that n̅=0; an extension to a nonzero Planck number is straightforward and does not affect the result.

The goal of this section is to show that, even where Γ is extremely small, but exceeds the tunnelling frequency Ω_kk' =λ^-1δω (g^(k) - g^(k')), the oscillator dynamics changes qualitatively compared to the coherent dynamics. Instead of coherent tunneling between the intrawell states Ψ_m, which have broken time translation symmetry, the oscillator performs random hopping between the wells.

We first discuss the effect of the dissipation (<ref>) by disregarding the dissipation-induced transitions between the intrawell states. In this approximation, one can describe the evolution of the oscillator in terms of the kinetic equation for the matrix elements ρ_mm'≡⟨Ψ_m|ρ|Ψ_m'⟩. The interwell tunneling can be mapped onto the tight-binding model with Hamiltonian 


    H_ tun =t_ tun∑_m=0,1,2 |Ψ_m⟩⟨Ψ_m+1| +  H.c.,


where we use the convention |Ψ_3⟩≡ |Ψ_0⟩. The hopping integral is t_ tun= (ħδω/2λ)C_ tunexp[(-S_ tun+iΦ_ tun)/λ] with C_ tun, S_ tun, and Φ_ tun given by Eq. (<ref>).

To the leading order in λ, we have ⟨Ψ_m|a|Ψ_m'⟩ = (2λ)^-1/2(Q_m+iP_m)δ_mm'. Therefore, from Eq. (<ref>), off-diagonal matrix elements ρ_mm' decay with rate ∝Γ/λ. If this rate exceeds |Ω_kk'| ∼ |t_ tun|/ħ, then over time ∼λ/Γ the off-diagonal matrix elements decay to their quasi-stationary values, which are determined by the diagonal matrix elements ρ_mm. The latter vary much slower, 


    ρ̇_mm = W∑_m'≠ mρ_m'm'- 2Wρ_mm, 
        W=λ |t_ tun|^2/ħ^2 Γ Q_0^2.


Parameter W is the rate of hopping between the wells of g(Q,P), it is much smaller than the tunneling frequency |Ω_kk'|. The hopping is a Poisson process in the slow time, it is incoherent and is a discrete analog of diffusion. The above analysis is in the spirit of the theory of quantum diffusion in solids <cit.> and its analog in systems with a small number of potential wells <cit.>.

The role of the dissipation-induced intrawell transitions is more subtle. Even for T=0, these transitions lead to an occupation of excited intrawell states, cf. <cit.>. On the time scale determined by 1/Γ, near the minimum of a well there is progressively formed a Boltzmann-type distribution over the states. The stationary ratio of the populations of the neighboring states can be shown to be (1+2fQ_0-ω_min)/(1+2fQ_0+ω_min). 

The  tunnel splitting increases for higher-lying intrawell states. However, near the minimum of g(Q,P) this increase is slow. The tunneling action S_ tun(n) varies with the intrawell level number n as |∂ S_ tun(n)/∂ n| =λω_minτ_n. Here, τ_n is the dimensionless imaginary time of interwell tunneling given by Im ∫ dQ/∂_Pg, where the classical momentum is calculated  for g(Q,P) = g_min +λω_min(n+1/2). This time is logarithmically large for small n. Therefore, for small Γ but still Γ≫ |t_ tun|/ħ, tunneling via excited intrawell states weakly renormalizes the rate W in Eq. (<ref>). 

Even if for highly excited intrawell states, with n exceeding some critical n_ cr∝ 1/λ, the hopping integral exceeds Γ/ħ, interwell switching via these states will be very slow, as the occupation of these states will be small. We note that, if ħΓ exceeds the hopping integral for almost all intrawell states, interwell switching may occur via dissipation-induced transitions over the interwell barrier of g(Q,P), i.e., over the saddle point of g(Q,P) seen in Fig. 1 of the main text. This is the dominating switching mechanism for a parametric oscillator <cit.>.

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