Time Operators and Time Crystals

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Nov 28, 2017 - ˆTR is obtained by taking the infinite radius limit of ˆTS1 . In this letter, we derive a ... tional Aharonov-Bohm time operator is obtained by tak-.
Time Operators and Time Crystals K. Nakatsugawa1,2, T. Fujii2,3 , A. Saxena4 , and S. Tanda1,2 1

arXiv:1711.10179v1 [quant-ph] 28 Nov 2017

2

Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo 060-8628, Japan 3 Department of Physics, Asahikawa Medical University, Asahikawa 078-8510, Japan 4 Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: November 29, 2017) We investigate time operators in the context of quantum time crystals in ring systems. We demonstrate that a self-adjoint time operator with a periodic time evolution can be derived for a free particle on a ring system: The conventional Aharonov-Bohm time operator is obtained by taking the infinite-radius limit. We also reveal the relationship between our time operator and a PT -symmetric time operator. We find that both time operators indeed describe the periodic time evolution of a quantum time crystal.

Introduction— In the framework of the standard quantum mechanics, time is not an observable but just a parameter. One of the reason for this is the difficulty to define a self-adjoint time operator Tˆ conjugate to a Hamilˆ which satisfies the canonical commutonian operator H tation relation [1–3] ˆ Tˆ] = i~. [H,

(1)

This difficulty lies in the difference between self-adjoint operators and symmetric operators even though these are both Hermitian operators [2]. The existence of orthogonal eigenstates and real eigenvalues is ensured for self-adjoint operators but not ensured for symmetric operators. Hence, although observables have to be represented by self-adjoint operators, most of the time operators which satisfy Eq. (1) are symmetric operators [3]. For example, the Aharonov-Bohm time operator [4] m xpˆ−1 + pˆ−1 x ˆ) TˆR = − (ˆ 2

(2)

which describes the arrival time of a free particle on a line (R) is a symmetric operator. How to define a self-adjoint time operators is still an open problem. We consider the above problem in the context of a quantum time crystal (QTC). A QTC is a quantum mechanical state which spontaneously breaks time translation symmetry [5–12]. The idea of a QTC ground state [5–7] was extended to our previous work of decoherenceinduced QTC [13]. A QTC promotes time from a parameter to a physical quantity. So, it is desirable to consider a time operator for a QTC. However, if we regard Eq. (1) as the equation of motion of the Heisenberg ˆ ˆ operator Tˆ(t) = eiHt/~ Tˆe−iHt/~ , then it is clear from i ˆ ˆ d ˆ dt T (t) = ~ [H, T (t)] = −1 that the time described by Eq. (1) is linear and not periodic. Therefore, one needs a self-adjoint time operator with the periodicity of a QTC in order to solve these problems in quantum physics.

FIG. 1. The Aharonov-Bohm time operator TˆR (Eq. (2)) is a symmetric operator which describes the arrival time of a free particle on a line R [4]. We derive the self-adjoint operator TˆS 1 (Eq. (8)) for a free particle on a ring system S 1 which reflects the periodic time evolution of a quantum time crystal. TˆR is obtained by taking the infinite radius limit of TˆS 1 .

In this letter, we derive a self-adjoint time operator for a free particle on a ring system (S 1 ) which satisfies a generalized form of Eq. (1) [14] and reflects the periodic time evolution of our model of QTC [13]: The conventional Aharonov-Bohm time operator is obtained by taking the infinite-radius limit (Fig. 1). This time operator is self-adjoint because bounded symmetric operators are self-adjoint operators [15, 16]. We also reveal the relationship between this time operator to a PT -symmetric time operator [17, 18] with orthogonal eigenstates and real eigenvalues which reduces to a non-Hermitian time operator in the infinite-radius limit [19]. We conclude that our time operators indeed describe the periodic time evolution of a QTC in a ring system. Quantum Mechanics on S 1 — Let H be the Hilbert space of square-integrable functions with the periodic boundary condition hθ|ψi = ψ(θ) = ψ(θ + 2π), |ψi ∈ H , θ ∈ S 1 . The time operator in Eq. (1) was defined such that it satisfies a commutation relation similar to the position-momentum commutation relation [ˆ x, pˆ] = i~. ˆπ However, the canonical commutation relation [θ, ˆθ ] = i~ with the angular position operator θˆ and the canonical

2 angular momentum operator π ˆθ does not hold because θˆ is a multivalued operator and its eigenvalues are illdefined [20]. In order to solve this problem, we use the ˆ the sine operator Sˆ [20] and the unicosine operator C, iθ = Cˆ + iS ˆ = ec ˆ [21] which satisfy the tary operator W commutation relations on H ˆ = −i~C, ˆ [ˆ πθ , S]

ˆ = i~S, ˆ [ˆ πθ , C]

ˆ ] = ~W ˆ, [ˆ πθ , W

(3)

and ˆ |ψl i = |ψl+1 i , (4) π ˆθ |ψl i = l~ |ψl i , W ˆ |θi = eiθ |θi , Cˆ |θi = cos θ |θi , Sˆ |θi = sin θ |θi , (5) W where ~ is the reduced Planck constant and l is an inte∂ ger. Through the position representation π ˆθ → −i~ ∂θ , it is clear that the commutation relations in Eq. (3) are the ∂ ∂ sin θ = cos θ, ∂θ cos θ = − sin θ, operator versions of ∂θ ∂ iθ iθ and ∂θ e = ie , respectively. A physical example of Cˆ is the charge density amplitude of an incommensurate charge density wave which is what we used to model a QTC [13]. The complete orthonormal set of momentum eigenstates {|ψl i}∞ such that any state l=−∞ spans H P∞ |ψi ∈ H can be written as |ψi = l=−∞ cl |ψl i. Moreover, let Fˆ (ˆ πθ ) be any function of π ˆθ , then it immediately follows from (4) that ˆ ] |ψl i = W ˆ δ Fˆ (ˆ [Fˆ (ˆ πθ ), W πθ ) |ψl i , ˆ ˆ δ F (ˆ πθ ) ≡ F (ˆ πθ + ~) − Fˆ (ˆ πθ ).

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Self-Adjoint Time Operator on S 1 — Let Tˆ be a symmetric or a self-adjoint operator on a Hilbert space H and ˆ be a Hamiltonian operator on H. Then, we say that H ˆ if (Tˆ, H, ˆ K(t)) ˆ Tˆ is a generalized time operator of H satisfies the generalized weak Weyl relation (GWWR) [14] Tˆe

ˆ −itH/~

|ψi = e

ˆ −itH/~

ˆ (Tˆ + K(t)) |ψi

(7)

ˆ with |ψi ∈ D(Tˆ) and D(K(t)) = H (where D(·) denotes operator domain). The bounded self-adjoint operator ˆ K(t) is called the commutation factor of the GWWR. ˆ Moreover, if K(t) is differentiable with respect to t, then ˙ ˆ ˆ ˆ (T , H, K(0)) (where the dot denotes time derivative) satisfies the generalized canonical commutation relation ˆ˙ ˆ Tˆ] |ψi = −i~K(0) |ψi [H, ˆ ∩ D(H ˆ Tˆ). In our case, we require for |ψi ∈ D(TˆH) ˙ˆ K(0) = −Cˆ for a reason that will become clear in a moˆ ment. The case K(t) = −t in the GWWR gives the weak ˆ ˆ ˆ Weyl relation T e−itH/~ |ψi = e−itH/~ (Tˆ − t) |ψi which is a stronger version of Eq. (1) [22]. In general, there is a hierarchy of time operators which satisfy stronger and

weaker versions of the canonical commutation relation [14, 23]. Now, let us consider a free particle on S 1 with a mo2 ˆ = πˆθ . Then, it ment of inertia I and a Hamiltonian H 2I immediately follows from Eq. (6) that ˆ W ˆ]=W ˆ δH ˆ ⇒ [H, ˆ W ˆ (δ H) ˆ −1 ] = W ˆ, [H, ˆ W ˆ † ] = −δ H ˆW ˆ † ⇒ [H, ˆ (δ H) ˆ −1 W ˆ † ] = −W ˆ† [H, holds in H . Consequently, we can define the time operator ˆ (δ H) ˆ −1 ] TˆS 1 = −Im[~W ~ ˆ −1 W ˆ†−W ˆ (δ H) ˆ −1 ], = [(δ H) 2i ˆ TˆS 1 ] = i~Cˆ [H,

(8) (9)

with D(TˆS 1 ) = H . The commutation relation Eq. (9) ˆ = −i~Cˆ in Eq. (3). Besides, has the same form as [ˆ πθ , S] it is a known fact from spectral analysis that the real and imaginary parts of a bounded operator are self-adjoint ˆ and (∆H) ˆ −1 are bounded operators. So, because W ˆ (δ H) ˆ −1 ] is a self-adjoint time operators, TˆS 1 = −Im[~W operator which satisfies Eq. (7). Infinite-Radius Limit— Next, we show that TˆS 1 reduces to TˆR in the infinite-radius limit R Rπ R ∞→ ∞. Using the identity operators −π dθ |θi hθ| = 1, −∞ dx |xi hx| = 1 R∞ P l∈Z |ψl i hψl | = 1 and −∞ dk |ki hk| = 1 with x = Rθ, k = l/R, p = ~k, hθ|ψl i = hx|ki, I = mR2 , µl = ˆ −1 |ψl i, eiθ ≈ 1+iθ and µl /R → m/p as R → ∞ hψl |~(δ H) we obtain " # XZ π TˆS 1 ≈ −Im dθ |θi hθ| (1 + iθ)µl |li hl| l∈Z

= −Re

" XZ

→ −Re

−π π

dθ |θi hθ| θµl |li hl|

l∈Z −π Z ∞

dk

−∞

Z



−∞

dx |xi hx|

#

mx |ki hk| ~k

m = − (ˆ xpˆ−1 + pˆ−1 xˆ). 2



(10)

Similarly, one can show that RSˆ → xˆ, Cˆ → 1 as R → ∞ and pˆ = π ˆθ /R, hence TˆS 1 → TˆR , ˆπ [S, ˆθ ] = i~Cˆ → [ˆ x, pˆ] = i~, ˆ TˆS 1 ] = i~Cˆ → [H, ˆ TˆR ] = i~, [H, as R → ∞. Therefore, we conclude that TˆS 1 is indeed a self-adjoint analogue of the Aharonov-Bohm time operator TˆR on S 1 .

3 Time Evolution of the Time Operator— For a free particle on a ring we haveP1/µl = (1 + 2l)/µ0 . So, using the identity operator 1 = l |ψl i hψl |, one can show that the ˆ ˆ Heisenberg operator TˆS 1 (t) = eiHt/~ TˆS 1 e−iHt/~ becomes " # X i(2l+1)t/µ0 ˆ TS 1 (t) = −Im |ψl+1 i hψl | , (11) µl e

TˆS 1 is defined in Eq. (8). The real part TˆSRe 1 is given by

ˆ hence TˆS 1 (t), K(t) = TˆS 1 (t) − TˆS 1 and their expectation values with a general state |ψi ∈ H are periodic with the period 2πµ0 because (2l + 1) is an integer. The periodic time evolution of TˆS 1 (t) is also present in the Heisenberg equation of motion

ˆ Tˆ Re ˆ [H, S 1 ] = −i~S.

l

i ˆ ˆ d ˆ ˆ TS 1 (t) = [H, TS 1 (t)] = −C(t) dt ~ " # X i(2l+1)t/µ0 = −Re |ψl+1 i hψl | . e

which satisfies the commutation relation ˆ ˆ Tˆ PT [H, S 1 ] |ψl i = −~W |ψl i .

ˆ (δ H) ˆ −1 + (δ H) ˆ −1 W ˆ †] ˆ −1 − ~ [W ˆPT TˆSRe 1 = Re(TS 1 ) = ~(δ H) 2 (15) and satisfies the commutation relation

l

Extension to PT -Symmetric Time Operators— The ˆ ] = ~W ˆ in Eq. (3) also recommutation relation [ˆ πθ , W duces to [ˆ x, pˆ] = i~ in the infinite-radius limit R → ∞. So, we also consider a time operator which satisfies a simlar commutation relation. In particular, we consider a time operator TˆSPT with PT (space-time inver1 sion) symmetry: Operators with PT -symmetry have real eigenvalues if their eigenstates are PT -symmetric as well [17, 18]. The parity operator P in S 1 satisfies the folˆ P −1 = W ˆ †, lowing properties [24]: P † P = P 2 = 1, P W −1 Pπ ˆθ P = −ˆ πθ , and P |ψl i = |ψ−l i. We show that the time reversal operator T shares the same properties. The time reversal operator for a bosonic particle is an antiunitary operator which satisfies T † T = T 2 = 1. Time reversal changes the direction of motion of a particle on S 1 , i.e. π ˆθ T |ψl i = −l~T |ψl i should be satisfied. Therefore, we have T |ψl i = al |ψ−l i with a coefficient al . Antiunitarity of T implies al = 1. Then, using Eq. (4) and Eq. (3) ˆ T −1 = W ˆ †. we readily obtain T π ˆθ T −1 = −ˆ πθ and T W ˆ Therefore, π ˆθ , W and |ψl i ∈ H are PT -symmetric: ˆ ] = 0, [PT , π [PT , W ˆθ ] = 0, PT |ψl i = |ψl i .

(12)

Now, we can define the PT -symmetric time operaˆ (δ H) ˆ −1 , where fˆ is a real PT tor by TˆSPT = fˆ − ~W 1 ˆ From a symmetric operator which commutes with H. calculation similar to Eq. (10) we see that TˆSPT diverges 1 −1 ˆ ˆ as R → ∞ unless we choose f = ~(δ H) . So, we obtain the PT -symmetric Aharonov-Bohm time operator on S 1 ˆ )(δ H) ˆ −1 = ~(1 − W TˆSPT 1 ˆ = Tˆ Re 1 + iTS 1 S

(13)

(16)

The eigenstates and eigenvalues of TˆSPT are calculated 1 using biorthogonal quantum mechanics [25]: Suppose that the time operator TˆSPT and its Hermitian conjugate 1 † (TˆSPT ) satisfy the eigenvalue equations 1 TˆSPT 1 |φl i = τl |φl i ,

We note that the period is proportional to the particle’s moment of inertia which diverges as R → ∞. Therefore, this periodicity, which was already discovered in our previous work on QTC [13], is intrinsic to ring systems.

(14)

† (TˆSPT 1 )

|χl i =

τl∗

|χl i .

(17) (18)

∂ Let us adopt the position representation π ˆθ → −i~ ∂θ  2 ˆ → ~ −i ∂ + 1 . Then, Eq. (17) which implies δ H I ∂θ 2 and Eq. (18) are equivalent to the following differential equations:   ∂ 1 ~ −i Φl (θ), + φl (θ) = I ∂θ 2   τl ~ ∂ 1 (1 − eiθ )Φl (θ) = −i Φl (θ), + I ∂θ 2   ∂ 1 τ ∗~ −i χl (θ), + (1 − e−iθ )χl (θ) = l I ∂θ 2

which have the orthonormal solutions  iθ 1 φl (θ) = φl0 1 − eiθ eiθνl e−(νl + 2 )e , ∗



1

−iθ

χl (θ) = χl0 eiθνl e(νl + 2 )e , Z π dθχ∗l (θ)φm (θ) = 2πχ∗l0 φm0 δl,m = δl,m −π

where νl = τIl ~ − 12 was introduced for brevity. The periodic boundary condition requires νl to be an integer. Therefore, we obtain the eigenvalues τl = τl∗ =

2I = µνl . (2νl + 1)~

(19)

I = mR2 implies that these eigenvalues are interpreted as the time required for a free particle with velocity v = ~ to move a distance R on the ring; i.e. the time (νl + 12 ) mR required to make a full rotation is 2πτl = 2πR/v (Fig. 2, Fig. 3). This period is expected to be proportional to the angular momentum l~, so we set νl ∼ l. Next, we calculate the large radius limit of TˆSPT 1 . The eigenvalues τl diverge as R → ∞ because it takes an infinite amount of time to move an infinite distance. Instead, if TˆSPT has maximally broken PT symmetry; that 1

4

FIG. 2. The eigenvalues τl of TˆSPT describes the periodicity 1 in time of a free particle moving on a ring with a constant velocity.

FIG. 4. TˆS 1 (Eq. (8)) is a self-adjoint time operators and TˆSPT 1 (Eq. (13)) is a PT -symmetric time operator. In the large radius limit (from S 1 to R), TˆS 1 reduces to the AharonovBohm time operator TˆR (Eq. (2)) and TˆSPT reduces to the 1 non-Hermitian operator TˆRNH (Eq.(20)) [26].

FIG. 3. The expectation values hφl |TˆS 1 (t)|φl i / hφl |φl i is shown for l = 0, 1, 5 and 100 for µ0 = 2 × 10−6 sec. The amplitude is proportional to µl and the period is P = 2πµ0 .

is, if all of its eigenvalues τl are pure-imaginary complexconjugate pairs, then −iTˆSPT 1 (which is not defined in H ) has real eigenvalues. In this case, a calculation similar to Eq. (10) gives m~ −2 pˆ . (20) → (TˆRNH )† = TˆR + i −iTˆSPT 1 2 TˆRNH and (TˆRNH )† satisfy the canonical commutation relation ˆ TˆRNH ] = [H, ˆ (TˆRNH )† ] = [H, ˆ TˆR ] = i~. [H,

(21)

The non-Hermitian time operator TˆRNH (which is not PT -symmetric) was studied in reference [19]. (Here, the fact that −iTˆSPT reduces to (TˆRNH )† is merely a problem 1 of definition. Since the eigenvalues of TˆSPT are real in 1 PT † ˆ H , we can say that (TS 1 ) is the actual PT -symmetric operator and TˆSPT is its complex conjugate. Then, 1 PT † ˆ −i(TS 1 ) reduces to TˆRNH in the large radius limit.) Discussion and Conclusion— First, we summarize our results. We defined a self-adjoint time operator TˆS 1 and a PT -symmetric time operator TˆSPT for a free particle 1 on a ring system. TˆS 1 is a generalized time operator which satisfies Eq. (7) and reflects the periodic time evolution of a QTC. The eigenstates and eigenvalues of TˆSPT 1 are calculated using biorthogonal quantum mechanics. A summary of the time operators is given in Fig. 4. Second, we discuss the significance of considering a ˆ ring system. K(t) of a ring system is periodic with a radius-dependent period. On the other hand, for a oneˆ dimensional system (R) we have K(t) = t which implies

ˆ Eq. (1) or K(t) = tCˆ with a bounded self-adjoint opˆ ˆ = iCˆ [14], hence time erator C which implies [Tˆ, H] for a one-dimensional system is not necessarily periodic. ˆ Therefore, K(t) may be interpreted as a function which gives the “temporal structure” of a quantum system. In other words, a QTC can be formed in ring systems. Third, Galapon et al. also defined a self-adjoint time operators for a confined system with and without periodic boundary conditions which satisfy Eq. (1) [15, 16]. However, our results imply that time operators should be defined based on the real-space topology of a quantum system. So, Eq. (1) may not be the only possibility to define self-adjoint time operators, especially if we want time operators for time crystals. Our work may also be generalized to relativistic particles, but quantization of constrained systems is still a subject with active research and the commutation relations Eq. (3) are not guaranteed to hold for relativistic systems as well. Fourth, an operator being PT -symmetric does not necessarily mean that it has real eigenvalues, but it can have real eigenvalues. Typically, in PT -symmetric systems with gain and loss (demoted by ±γ) all eigenvalues are real below a critical value γ < γc and PT -symmetry is spontaneously broken otherwise. Therefore, it would be an interesting problem to consider spontaneous breaking of PT -symmetry (of the Hamiltonian or the time operator) in periodically driven Floquet time crystals [8–12]. Moreover, one of the motivations to define time operators is to derive time-energy uncertainty relations. For the self-adjoint operator TˆS 1 the conventional Robertson ˆ TˆS 1 ≥ 1 | h[H, ˆ TˆS 1 ]i | = ~ | hCi ˆ | uncertainty relation ∆H∆ 2 2 PT ˆ is satisfied [27], and for TS 1 non-Hermitian analogues [24, 28] can be obtained. In all cases, the phase-angular momentum uncertainty relations and the time-energy uncertainty relations for the self-adjoint parts and for the PT -symmetric parts are completely equivalent. This fact is a direct consequence of the similarities between the

5 commutation relations. Finally, we discuss the results in this paper in the context of experiments. The order parameter of a charge density wave (CDW) or a superconductor is a complex scalar ∆ = |∆|eiθ . Suppose that we can quantize the iθ . Then the commutation relations ˆ = ec phase θ by W Eq. (9), Eq. (16), and Eq. (14) give the Heisenberg equad ˆ ˆ tion of motions of the time operators: dt TS 1 (t) = −C(t), d ˆ Re d ˆ PT ˆ ˆ dt TS 1 (t) = S(t), and dt TS 1 (t) = −iW (t), respectively. ˆ (t) was already Actually, the periodic oscillation of W used in our previous work of decoherence-induced QTC which consists of a ring-shaped incommensurate CDW coupled to a fluctuating magnetic flux [13]. Therefore, our results can be tested in various CDW or superconducting systems, in superfluid systems with spatial periodicity [7, 29] and in other systems which can be described by a single particle on S 1 . We expect that our results have many important applications and insights, such as advancement in the physics of constrained systems (ring systems, M¨ obius systems, etc.) [30, 31] or understanding the space-time structure of the early (topologically non-trivial) universe [32]. We thank Asao Arai, Izumi Tsutsui, Yoshimasa Hidaka, Akio Hosoya, Yuji Hasegawa, Toyoki Matsuyama, Kohkichi Konno, Kousuke Yakubo, and Koichi Ichimura for stimulating and valuable discussions.

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